Research ArticleOn the Free Vibration Modeling of SpindleSystems A Calibrated Dynamic Stiffness Matrix

Omar Gaber and Seyed M Hashemi

Department of Aerospace Engineering Ryerson University 350 Victoria Street Toronto ON Canada M5B 2K3

Correspondence should be addressed to Seyed M Hashemi smhashemryersonca

Received 17 October 2013 Accepted 17 March 2014 Published 12 May 2014

Academic Editor Valder Steffen Jr

Copyright copy 2014 O Gaber and S M HashemiThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

The effect of bearings on the vibrational behavior of machine tool spindles is investigated This is done through the developmentof a calibrated dynamic stiffness matrix (CDSM) method where the bearings flexibility is represented by massless linear springelements with tuneable stiffness A dedicatedMATLAB code is written to develop and to assemble the element stiffnessmatrices forthe systemrsquosmultiple components and to apply the boundary conditionsThe developedmethod is applied to an illustrative exampleof spindle system When the spindle bearings are modeled as simply supported boundary conditions the DSM model results in afundamental frequencymuch higher than the systemrsquos nominal valueThe simply supported boundary conditions are then replacedby linear spring elements and the spring constants are adjusted such that the resulting calibrated CDSMmodel leads to the nominalfundamental frequency of the spindle system The spindle frequency results are also validated against the experimental data Theproposed method can be effectively applied to predict the vibration characteristics of spindle systems supported by bearings

1 Introduction

The booming aerospace industry and high levels of compe-tition have forced companies to constantly look for ways tooptimize their machining processes Cycle time has been amajor concern at various industries dealing with manufac-turing of airframe parts and subassemblies When trying tomachine a part as quickly as possible spindle speed or metalremoval rates are no longer the limiting factors but ratherthe chatter that occurs during the machining process It iswell established that chatter is directly linked to the naturalfrequency of the cutting system which includes the spindleshaft tool and hold combination

The first mention of chatter can be credited to Taylor in1907 [1] however the first comprehensive investigation ofthe problem was published by Arnold in 1946 [2] Arnoldrsquosexperiments were conducted on the turning process wherehe theorized that the machine could be modeled as a simpleoscillator and that the force on the tool decreased as the speedof the tool in relation to the work piece increased Gurneyand Tobias later theorized the nowwidely accepted belief thatchatter is caused by wave patterns traced onto the surfaceof the work piece by preceding tool passes [3] The phase

shift of the preceding wave to the wave currently being traceddetermineswhether there is any amplification in the tool headmovement If there exists a phase shift between the two toolpaths then the uncut chip cross-sectional area is varied overthe pass Thomas and Shabana [4] showed that the cuttingforce is dependent on the chips cross-sectional area andas a result a varying cutting force is produced This wasdemonstrated through an illustrative example of a grindingmachine modeled as a one-degree-of-freedom mass-springsystem as opposed to an oscillating systemThe spring-masssystem is also the widely used modeling theory for how avibrating tool should be characterized today

Prior to 1961 the machining procedures were regardedas steady state and discrete processes [5] This erroneousidea led to the creation of overly heavy and thick-walledmachines believed to provide high damping to the tooltipforces (assumed to be static) However one must note thatmachining is a continuous dynamic process with constantlyfluctuating tooltip forces Moreover while being identicalin construction the dynamics and response of similarmachines may be slightly different due to structural imper-fections imbalances and so forth Therefore the specificcharacteristics and dynamics of each machine must also be

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 787518 10 pageshttpdxdoiorg1011552014787518

2 Shock and Vibration

taken into consideration when performing chatter-relatedcalculations [6] The structural modes of the machine deter-mine the frequency and the direction that the tool is goingto vibrate at [7] Eisele [5] and Peng et al [8] stressed thatdesigners must investigate the mode-shapes weak pointsbearing clearances and self-inducing vibratory componentsof the machine to reduce chatter Butlin andWoodhouse alsoinvestigated the number of structural modes required to pro-duce accurate results [9]They showed that low ordermodelsincorporating only two modes are sufficiently accurate tomodel the machines

In 1981 Tlusty and Ismail published one of the first papersthat documented the nonlinearity of the vibratory systemoccurring during chatter [10] Kondo et al [11] added toTlusty and Ismailrsquos investigation the behaviour of the toolafter the onset of chatter An important study carried outby Lee et al in 1989 helped bring a more complete theoryof chatter to light [12] where they looked into the responseof the work piece to chatter They modeled the tool as atypical two-degree-of-freedom spring-mass system and thework piece as an Euler beamWhen the responses of both thetool and the work piece were incorporated into the typicalspring-mass equations the results obtained far surpassed theaccuracy of previously used equations They observed a largeincrease in surface finish quality while machining even withthe use of long end mills

While a comprehensive set of mathematical models havealready been derived to explain chatter using spring-mass andEuler beam systems which have led to dramatic increases insurface finish quality chatter is still an issue in the machiningindustry as the surface tolerances keep increasing Fromthe 1990s on this drove researchers to use more refinedmathematical models to characterize themachines [13]Withthe rise of computer aided calculations and simulations ithas become increasingly easier to perform more complexanalyses of the machining process [14] Tool wear is an often-overlooked factor that contributes to chatter With the aid ofmore powerful computers this variable can now be includedin simulations Clancy and Shin [15] investigated the wearof turning tools and how that affects the stability of thesystem It was found that as the tool becomes worn its limitsof stability increase As a result stability lobes have nowbecome a function of tool wear Li et al [16] determined thatthe coherence function of two crossed accelerations in thebending vibration of the tool shank approaches unity at theonset of chatter A threshold needs to be set [17] and thendetected using simple mechanism to alert the operator tochange the machining conditions and avoid increased toolwear

In most of the existing stability prediction methodsa frequency response function (FRF)mdashacquired using animpact testmdashis required to extract the system fundamentalfrequency used to perform the calculations In order toperform accurate impact tests the machine would need tobe shut off and allowed to cool down and the coolant dripoff This process could lead to several hours of down timeAn offline method of obtaining the FRF could greatly ben-efit manufacturing companies by eliminating the downtimeneeded for the impact tests While stability lobe graphs have

becomemore accurate the industry is trending towards com-puter simulations and numerical analyses With computingpower becoming relatively cheap and readily available thismethod has become feasible Providing the most accurateresults due to its constantly updated stability calculationsthis is the area of research that engineers will most likelyprogress towards Lee et al [18] suggested that the wholesystem comprising machine tool and work piece couldbe modeled using finite element method (FEM) analysisA computer simulation would be able to predict the FRFduring all phases of the machining process As the part ismachined and becomes thinner its response to vibrationchanges dramatically Most existing research papers assumethat the FRF remains constant throughout the whole processHowever a constantly updated FRF would allow for accuratereal time stability calculations

In an attempt to numerically model the entire spindlesystems for the vibration modeling and analysis and toincorporate the effects of bearings Cao and Altintas [19]modeled spindle bearing andmachine tool systems for virtualsimulation of milling operations where they presented ageneral integratedmodel of the spindle bearing andmachinetool system consisting of a rotating shaft tool holder angularcontact ball bearings housing and the machine tool mount-ingThe proposedmodel also verified experimentally allowsvirtual cutting of a work material with the numerical modelof the spindle during the design stage and predicts bearingstiffness mode shapes frequency response function (FRF)static and dynamic deflections along the cutter and spindleshaft and contact forces on the bearings with simulatedcutting forces before physically building and testing thespindles The study showed that the accuracy of predictingthe performance of the spindles requires integratedmodelingof all spindle elements and mounting on the machine toolJonesrsquo bearing model which considers the bearing balls andrings as elastic parts and the Hertzian contact theory wereused to calculate the contact force and displacement Patwariet al [20] presented a detailed systematic procedure forexperimental and analytical modal analysis techniques toevaluate structural dynamics of a vertical machining centreAbuthakeer et al [21] carried out the dynamic characteristicsanalysis of high speed motorized spindles More recentlyDelgado et al [22] used FEM to study thenonlinear dynamicbehavior of a high speed machine tool spindle

It is known that FEM predictions and the numericalresults are often different from test and experimental resultsHowever based on the model updating corrections can bemade on the FEM models by processing records of testresults Motthershead and Friswell [23] presented a surveyof model updating in structural dynamics Later Mares etal [24] investigated the model updating using robust esti-mation and more recently Mottershead et al [25] presenteda basic introduction to the most important procedures ofcomputational model updating including tutorial examplesand a large scale model updating example of a helicopterairframe and discussed the sensitivity method in FEMmodelupdating

To the authorsrsquo best knowledge except authorsrsquo previ-ous works [26 27] no studies on the calibratedupdated

Shock and Vibration 3

dynamic stiffness matrix (DSM) models of spindle systemshave appeared in the open literature The DSM method [28]provides an analytical solution to free vibration problemsachieved by combining the coupled governing differentialequations of motion of the system into a single higher-orderordinary differential equation Seeking the most generalsolution to the resulting higher-order ODE and enforcingthe end displacements and loads through extensive matrixmanipulations lead to the DSM of the structural elementUnlike limited nodal DOFs in conventional FEM [29] theDSM-based element is modeled as a continuous system withinfinite number of DOFs within the element The DSMformulation has been proven to provide exact frequency(within the limits of the theory) for various beam con-figurations and beam-structures with the use of a singlecontinuous element per segment (see eg [28 30ndash33])The objective of the present study is to determine thenatural frequenciesvibration characteristics of machine toolspindle systems by developing the corresponding dynamicstiffness matrix (DSM) [31 32] and the proper boundaryconditions These results would then be compared to theexisting results for a common cutting system to validatetunethe model developed The coupled bending-bending (B-B)vibration of a spinning beam is investigated A MATLABcode is developed to assemble the DSM element matrices formultiple components and to apply the boundary conditions(BC) The bearings are first modeled as simply supported (S-S) frictionless pins which are then replaced by linear springelements to incorporate the flexibility of bearings into themodel In comparison with the existing data on spindlersquosfundamental frequency the bearing stiffness coefficient 119870119878is then varied to achieve a calibrated (or updated) dynamicsstiffness matrix (CDSM) vibrational model The proposedformulation can also be extended to include torsional degree-of-freedom (DOF) for further modeling purposes

2 Problem Description andGoverning Equations

Computer numeric control (CNC) machines are becom-ing the norm in manufacturing plants worldwide Thesemachines are generally 3- 4- or 5-axis depending on thenumber of degree-of-freedoms the device has Having thetool translating in the 119909 119910 and 119911 directions accounts forthe first three degrees of freedom Rotation about the spindleaxes account for any further DOF A typical spindle containsthe motors that rotate the shaft connected to the tool holderthe bearings the tools and all the mechanisms that hold thetool in place (see Figure 1 for a sample spindle configuration)As mentioned earlier in this paper the fundamental naturalfrequency of this system dictates the limitation of the cuttingparameters If these limits are surpassed the system tends tovibrate which causes chatter Also as the bearings wear thesystem characteristics and spindlersquos natural frequency change

In what follows the derivation of the governing differen-tial equations for the coupled bending-bending vibration ofa spinning beam for a rotating shaft is first briefly discussedBearings restrain the spinning motion of the spindle system

Toolholder BearingsBearings Motor Spindle shaft

Spindle casing

Figure 1 Typical spindle configuration

Y

X

ZΩ

0

L

P

Figure 2 Spinning beam

in five degrees of freedom that is displacement in 119909 119910 and119911 directions and the rotation in 119909 and 119910 directionsThe beamtorsional rigidity 119866119869 is large enough so that the torsionalvibrations can be ignored [31 32] Figure 2 shows a cylindricalbeam in a right-handed rectangular Cartesian coordinatessystemThe beam length is denoted by119871 mass per unit lengthis 119898 = 120588119860 polar mass moment of inertia per unit length is119868120572 and the principal axes bending rigidities are EI for bothplanes At an arbitrary cross section along 119911-axis 119906 and V aredisplacement of a point 119875 in the 119909 and 119910 directionsThe crosssection is allowed to rotate or twist about the 119900119911 axis Theposition vector r of the point 119875 after deformation is given by(refer to Figures 2 and 3)

r = (119906 minus 120601119910) i + (V + 120601119909) j (1)

where i and j are unit vectors in the 119909 and 119910 directions Thevelocity of point 119875 is given by

v = r +Ω times r where Ω = Ωk (2)

The kinetic and potential energies of the beam (119879 and 119880) aregiven by

119879 =1

2int

119871

0

|v|2119898119889119911

=1

2119898int

119871

0

[2+ V2 + 2Ω (119906V minus V) + Ω

2(1199062+ V2)] 119889119911

119880 =1

2EI119909119909 int

119871

0

V101584010158402

119889119911 +1

2EI119910119910 int

119871

0

119906101584010158402

119889119911

(3)

4 Shock and Vibration

U1

V1

U2

V2

Θx1

Θx2

Θy1

Θy2

Φ1

Φ2

Figure 3 Degrees of freedom of the system

Using the Hamilton principle in the usual notation state

120575int

1199052

1199051

(119879 minus 119880) 119889119905 = 0 (4)

where 1199051and 119905

2are the time intervals in the dynamic

trajectory and 120575 is the variational operator Substituting thekinetic and potential energies of the beam in the Hamiltonprinciple collecting like terms and integrating by parts thefollowing differential equations are obtained

EI1199101199101199061015840101584010158401015840

minus 119898Ω2119906 + 119898 minus 2119898ΩV = 0

minus2119898Ω minus EI119909119909V1015840101584010158401015840

minus 119898V + 119898Ω2V = 0

(5)

which govern the bending-bending (B-B) vibration of abeam coupled by the spinning speedThe resulting loads alsoforce boundary conditions at free ends are then found to bein the following forms written for shear forces as

119878119909= EI119909119909119906101584010158401015840 119878

119910= EI119910119910V101584010158401015840 (6)

and for bending moments as

119872119909= EI119909119909V10158401015840 119872

119910= minusEI

11991011991011990610158401015840 (7)

Assuming simple harmonic motion

119906 (119911 119905) = 119880 (119911) cos120596119905 V (119911 119905) = 119880 (119911) cos120596119905 (8)

and substituting (8) into (5) they can be rewritten as

(EI1199101199101198801015840101584010158401015840

minus 119898(Ω2+ 1205962)119880) + 2119898Ω120596119881 = 0

(EI1199091199091198811015840101584010158401015840

minus 119898(Ω2+ 1205962)119881) + 2119898Ω120596119880 = 0

(9)

where 120596 is frequency of oscillation and 119880 119881 and Φ are theamplitudes of 119906 V and 120601 Introducing 120585 = 119911119871 and 119863 =

119889119889120585 which are nondimensional length and the differentialoperator and back substitution into (9) lead to

[1198634minus

119898(Ω2+ 1205962) 1198714

EI119910119910

]119880 +2119898Ω120596119871

4

EI119910119910

119881 = 0

[1198634minus

119898(Ω2+ 1205962) 1198714

EI119909119909]119881 +

2119898Ω1205961198714

EI119909119909119880 = 0

(10)

3 Solution of the Governing Equations

31 Dynamic Stiffness Matrix (DSM) Solution As suggestedby Banerjee and Su [31 32] the above equation (10) can becombined into one 8th-order differential equation written as

[1198638minus (1205822

119909+ 1205822

119910) (1 + 120578

2)1198634+ 1205822

1199091205822

119910(1 minus 120578

2)2

]119882 = 0

(11)

where ldquo119882rdquo represents both lateral displacements119880 and119881 and

1205822

119909=

11989812059621198714

EI119909119909 120582

2

119910=

11989812059621198714

EI119910119910 120578

2=

Ω2

1205962 (12)

The general solution of the differential equation is sought inthe form

119882 = 119890119903120585 (13)

Substituting (13) into (11) leads to

1199038minus (1205822

119909+ 1205822

119910) (1 + 120578

2) 1199034+ 1205822

1199091205822

119910(1 minus 120578

2)2

(14)

where 11990313 = plusmnradic120572 11990324 = plusmnradic120573 11990357 = plusmn119894radic120572 11990368 = plusmn119894radic120573 and

1205722=

1

2 (1205822

119909+ 1205822

119910) (1 + 120578

2)

+radic(1205822119909minus 1205822119910)2

(1 + 1205782)2+ 161205822

11990912058221199101205782

1205732=

1

2 (1205822

119909+ 1205822

119910) (1 + 120578

2)

minusradic(1205822119909minus 1205822119910)2

(1 + 1205782)2+ 161205822

11990912058221199101205782

(15)

Based on Euler-Bernoulli beam bending theory thecorresponding bending rotation that is slope about 119909-axisand 119910-axis Θ

119909and Θ

119910 respectively is given by (see the

appendix for further details)

Θ119909=

119889119881

119889119911= minus

1

119871

119889119881

119889120585 Θ

119910=

119889119880

119889119911=

1

119871

119889119880

119889120585 (16)

By doing similar substitutions in load (3) and (4) one finds

119878119909= (

EI119910119910

1198713)(

1198893119880

1198891205853) 119878

119910= (

EI119909119909

1198713)(

1198893119881

1198891205853)

119872119909 = (

EI119910119910

1198712)(

1198892119880

1198891205852) 119872119910 = (

EI1199091199091198712

)(1198892119881

1198891205852)

(17)

Substituting the boundary conditions into the governingequations enforced at 119911 = 0 (120585 = 0) and 119911 = 119871 (120585 = 1)leads to

120575 = BR (18)

Shock and Vibration 5

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

(19)

Substituting similarly for the force equation one finds

F = AR (20)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

(21)

The frequency-dependent dynamic stiffness matrix(DSM) of the spinning beam K(120596) can then be derivedby eliminating R We also know that the force amplitude isrelated to the displacement vector by

F = K (120596) 120575 where K (120596) = ABminus1 (22)

Explicitmatrix forms ofAB andR are given in the appendix

32 Classical Finite Element Method (FEM) For validationand comparison purposes a conventional finite elementformulation of the problem and aMATLAB-based FEM codewere developed to carry out the free vibration analysis of thespindle system [33] The Galerkin method of weighted resid-uals [29 33] was employed to develop the integral form ofthe governing equation (10) Performing integration by partson the resulting integral equations leads to the weak integralform of equations which also satisfy the principle of virtualwork The system is then discretized using beam elementswith four DOF per node (ie two lateral displacements 119880and119881 and two slopes1198801015840 and119881

1015840 eight degrees of freedom intotal) Introducing cubic Hermite-type polynomial approx-imations [29 33] for bending displacements in the elementweak integral form the classical finite element formulationis developed This formulation results in constant (or static)element stiffness mass and coupling matrices which arethen assembled to find the global matrices Finally enforcingthe boundary conditions leads to a linear eigenvalue problemwhich is solved to extract the system natural frequencies andmodes There has also been an attempt to develop a dynamicfinite element (DFE) [34] formulation for the problem inhand using frequency-dependent interpolation functionsand the preliminary results were found to be promising [35]

4 Application of the Theory andNumerical Results

TheDSM formulation was first validated against various ana-lytical and numerical frequency data available in the literature[31 32] for various uniform simple beam configurations TheDSM frequency data were also comparedwith those obtainedfrom commercial FEM-based software ABAQUS and theeffect of various element types on the convergence was alsoexamined In addition the conventional FEM formulation[29 33] was used for further comparisons however as the

in-house FEM-based code and commercial software both ledto similar results the FEM results have been omitted here forthe brevity

A typical spindle system model (see Figure 4) was thenconstructed using multiple segments As the nominal fun-damental frequency was available for nonspinning spindletherefore the effect of spinning speed was taken out by simplysetting Ω to zero in FEM models However one should notethat setting the spinning speed exactly to zero would causethe DSM formulation to collapse This is due to the factthat the DSM formulation pivots around combining the twoexpressions (10) into a single on (11) through the couplingterms involving Ω [31 32] However an extremely smallnumber (Ω cong 0) can be used to model the stationary spindle

The entire system was assumed to be made from thesame material that is tooling steel Initially it was alsoassumed that the system is simply supported at the bearinglocations that is the bearings modeled as frictionless pinsrigidly attached to the spindle casing with zero flexibilityallowing only for rotations in all directions This system wasmodeled using the DSM method described above as wellas finite element analysis (FEA) in ABAQUS commercialsoftware to evaluate the first six natural frequencies of thespindle system Several different element types andmesh sizeswere examined to ensure the convergence (Table 1) beamelement B33 uses a 2-node cubic beam (similar to the in-house FEM code) C3D20R is a general-purpose quadraticbrick element and C3D10M is a general-purpose tetrahedralelement (see Table 1 for results) Unlike the conventionalFEM formulation the DSM method results in exact solutionfor all natural frequencies of a spinning uniform beam [3132] Therefore as the spindle system in hand is made of12 uniform sections only a 12-element piecewise-uniform(stepped) DSM model was required As expected the resultsusing beam element type B33 yielded the best agreement withthe DSM results as the 3D and shear effects are neglectedin both However the fundamental frequency values werefound to be different from the nominal value that is 1393Hzcompared to 1000 plusmn 50Hz as provided by the manufacturer

In order to address this issue the DSM and FEM modelswere calibrated to take into account the flexibility of thebearing systems To this end the simply supported boundaryconditions were modified and replaced by simple linearspring elements attached to the spindle shaft in 119909 and119910 directions representing general flexibility of the bearingsystems (see Figure 4) The bearings were all assumed to beidentical with the same spring coefficient 119870119904 The springstiffness value 119870

119904 was then varied and a calibrated model

equippedwith spring elements of constant119870119904= 21times10

8Nmwas found to result in a fundamental bending frequencyof 1004Hz closely equivalent to the systemrsquos nominal value(Figure 5) The effect of spinning speed Ω on the systemrsquosfundamental frequency was also investigated In the absenceof spinning speed Ω = 0 the system exhibits uncoupledbehavior (see (7) and (8)) undergoing two separate bendingdisplacements in 119909 and 119910 directions As a result for a circularshaft with 119868119909119909 = 119868119910119910 the modal analysis results in repeatedfrequency values appearing in pairs (Table 1) By introducingthe spinning speedΩ = 0 the bending natural frequencies in

6 Shock and Vibration

Table 1 Spindle natural frequencies DSM versus FEA for a nonspinning (Ω cong 955RPM) with simply supported boundary conditions

Mode DSM FEA B33 element (Hz) FEA C3D20R (Hz) FEA C3D10M (Hz)12 sections 600 elements 1200 elements 16965 elements 20122 elements 34465 elements 45645 elements

1 139292 139300 139300 130800 130800 130100 1301002 139324 139300 139300 130800 130800 130100 1301003 407102 407100 407100 320800 321200 333500 3322004 407134 407100 407100 321000 321200 333500 3324005 566814 566900 566900 433900 431100 432200 4291006 566846 566900 566900 434100 431600 432300 429200

Table 2 Spindle natural frequencies versus various spinning speeds for flexible boundary conditions 119870119904= 21 times 10

8Nm

Mode955

(RPM)(Ω cong 0)

47842(RPM)

95589(RPM)

14334(RPM)

19108(RPM)

23883(RPM)

28657(RPM)

33432(RPM)

1 100427 92469 84511 76554 68596 60638 52680 447232 100459 108416 114321 106363 98406 90448 82490 745323 130237 122279 1163741 113414 105456 97498 89541 815834 130268 129329 1213716 124332 132290 140247 146805 1388475 137287 138226 1461838 154142 154142 154762 148205 1561636 137319 145277 1532344 161192 162720 170057 178015 185973

Toolholder Motor Spindle shaftSpring Spring

Spindle casing

Figure 4 Modified boundary conditions

119909 and 119910 directions become coupled changing the repeatednatural frequencies to two distinct ones (see Table 2) withone decreasing and approaching zero (Figure 6) and the otherincreasing from their uncoupled values Further increasein the spinning speed would make one of the bendingfrequencies go to zero that is whirling instability alsoreported and discussed in [31 32] The first critical spindlespeed in the case of calibrated model was found to be Ωcr =6 times 10

5 RPM which is well above the operating speed of thespindle that is 35 times 104 RPM (see Figure 7)

In order to further validate the proposed CDSM modelthe tap testing method was used to experimentally collectthe frequency response function (FRF) of the spindle sys-tem in hand while being installed on its correspondingmachine tool A 1-inch diameter blank tool with a 2-inchprotrusion in a typical shrink fit tool holder was usedThe tool and holder were inserted into the spindle Thespindle was set to be horizontal (ie neutral) position andan accelerometer (352A21 08 g) was then attached to theedge of the tool holder The spindle was struck 10 timesin the 119909 direction using a hammer (086C04 5000N) and

0

200

400

600

800

1000

1200

1400

Syste

m n

atur

al fr

eque

ncy

(Hz)

Bearing equivalent spring constant K (Nm)

DSMSimply supportedANSYS

Spindle manufacturerrsquos designed natural frequency (1000Hz)

100E+00

100E+01

100E+02

100E+03

100E+04

100E+05

100E+06

100E+07

100E+08

100E+09

100E+10

Tool Simplesupports

Simplesupports

Motor Spindleshaftholder

MotorSpring SpringSpindle casing

ToolSpindle

shaftholder

Figure 5 Varying spring constant versus system natural frequencyusing DSM and FEA

the average FRF graph was generated (SIM3 Module Photon+ Data Acquisition) Similarly the spindle was also testedalong the 119910 direction The nonspinning spindlersquos naturalfrequencies extracted from the FRF graphs are presented inTable 3 alongside the CDSM results As can be seen themaximum difference between the first three CDSM flexural

Shock and Vibration 7

Table 3 Spindle natural frequencies experimental versus CDSM for a nonspinning spindle system with flexible boundary conditions119870119904=

21 times 108 Nm

Nonspinning bending modes (Hz) CDSM (Ω cong 955RPM) Experimental (x-axisy-axis) Difference ()(x-axisy-axis)

1205961 100427100459 9298192651 80841205962 130237130268 130151131067 0070061205963 137287137319 160144158386 143133

0 5000 10000 15000 20000 25000 30000 35000 400000

200

400

600

800

1000

1200

1400

1600

1800

2000

Nat

ural

freq

uenc

y (H

z)

Spindle speed (RPM)

Spindle speed versus natural frequency

1st mode2nd mode3rd mode

4th mode5th mode6th mode

Figure 6 Spindle speed versus natural frequency

0

200

400

600

800

1000

1200

1400

0 10000 20000 30000 40000 50000 60000 70000 80000

Nat

ural

freq

uenc

y (H

z)

Spindle RPM

Spindle RPM versus natural frequency

Spring BCSimply supported

Spindle RPM operational limit

Figure 7 Spindle natural frequency versus spindle RPM simplysupported and flexible bearing models

natural frequencies and the experimentally evaluated dataare respectively 8 007 and 143 in 119909 direction and84 006 and 133 in 119910 directionThe average differenceis found to be around 74 which is within the acceptablerange

5 Discussion and Conclusion

It has been established that the service life changes the vibra-tional behavior of spindle systems that is reduced naturalfrequency over the time associated with the bearings wearTherefore in order to generate an updated spindlemodel it isessential to include the effect of the bearings in the systemAnanalytical model of a multisegment spinning spindle basedon the dynamic stiffnessmatrix (DSM) formulation and exactwithin the limits of the Euler-Bernoulli beam bending theorywas developed The beam exhibits coupled bending-bending(B-B) vibration and as expected its fundamental coupledfrequency was found to decrease with increasing spinningspeed The bearings were included in the model using twodifferent models rigid simply supported frictionless pinsand flexible linear spring elements The simply supportedbearing model was found to yield an excessively rigid systemwith amuch higher natural frequency than the nominal valueof the system Replacing the simply supported boundarycondition with linear spring elements the spring stiffness119870119904 was then calibrated so that the fundamental frequency

of the system matched the nominal value that is a morerealistic and accurate representation of the spindle systemThe accuracy of the proposed calibrated dynamic stiffnessmatrix (CDSM) method was confirmed through compari-son with both numerical and experimental results Whencompared to the conventional finite element method (FEM)the CDSM frequencies were found to best match with thoseobtained from a cubic beamfinite element (B33 inABAQUS)Moreover in comparison with experimental data the CDSMresults were found to be within acceptable accuracy the aver-age error for the first three bending natural frequencies is lessthan 8

It is worth noting that the proposed CDSM can alsobe fine-tuned to match spindlersquos experimental fundamentalfrequency instead of the nominal value Further researchis underway to improve the CDSM by using more accu-rate material properties for different segments of thespindle The developed calibrated model could then beexploited in an attempt to take into account the effects

8 Shock and Vibration

of bearing wear in terms of spindlersquos service time numer-ically evaluate the equivalent cutting force and calcu-late updated stability lobes used to achieve chatter-freecuts

Appendix

From the solutions of 119880 and 119881 (see expressions (13)) thecorresponding bending rotation about 119909-axis and 119910-axis Θ

119909

and Θ119910 are respectively given by

Θ119909 =

119889119881

119889119911= minus

1

119871

119889119881

119889120585

= minus1

119871(radic120572119861

1cosradic120572120585 minus radic120572119861

2sinradic120572120585

+ radic1205721198613coshradic120572120585 + radic120572119861

1sinradic120572120585

+ radic1205731198615cosradic120573120585 minus radic120573119861

6sinradic120573120585

+radic1205731198617coshradic120573120585 + radic120573119861

8sinhradic120573120585)

Θ119910=

119889119880

119889119911=

1

119871

119889119880

119889120585

= minus1

119871(radic120572119860

1cosradic120572120585 minus radic120572119860

2sinradic120572120585

+ radic1205721198603 coshradic120572120585 + radic1205721198601 sinradic120572120585

+ radic1205731198605 cosradic120573120585 minus radic1205731198606 sinradic120573120585

+radic1205731198607coshradic120573120585 + radic120573119860

8sinhradic120573120585)

(A1)

By doing similar substitutions we find (see expressions (3)(4) and (14))

119878119909 = (

EI119910119910

1198713)(

minus120572radic1205721198601cosradic120572120585 + 120572radic120572119860

2sinradic120572120585 + 120572radic120572119860

3coshradic120572120585 + 120572radic120572119860

4sinhradic120572120585

minus120573radic1205731198605cosradic120573120585 + 120573radic120573119860

6sinradic120573120585 + 120573radic120573119860

7coshradic120573120585 + 120573radic120573119860

8sinhradic120573120585

)

119878119910 = (

EI1199091199091198713

)(minus120572radic120572119861

1cosradic120572120585 + 120572radic120572119861

2sinradic120572120585 + 120572radic120572119861

3coshradic120572120585 + 120572radic120572119861

4sinhradic120572120585

minus120573radic1205731198615cosradic120573120585 + 120573radic120573119861

6sinradic120573120585 + 120573radic120573119861

7coshradic120573120585 + 120573radic120573119861

8sinhradic120573120585

)

119872119909= (

EI119910119910

1198712)(

minus1205721198611 sinradic120572120585 minus 1205721198612 cosradic120572120585 + 1205721198613 sinhradic120572120585 + 1205721198614 coshradic120572120585

minus1205731198615sinradic120573120585 + 120573119861

6cosradic120573120585 + 120573119861

7sinhradic120573120585 + 120573119861

8coshradic120573120585

)

119872119910= (

EI119909119909

1198712)(

minus1205721198601 sinradic120572120585 minus 120572119860

2 cosradic120572120585 + 1205721198603 sinhradic120572120585 + 1205721198604 coshradic120572120585

minus1205731198605 sinradic120573120585 + 1205731198606 cosradic120573120585 + 1205731198607 sinhradic120573120585 + 1205731198608 coshradic120573120585)

(A2)

where1198601= 1198961205721198611 119860

2= 1198961205721198612

1198603= 1198961205721198613 119860

4= 1198961205721198614

1198605 = 1198961205731198615 1198606 = 1198961205731198616

1198607 = 1198961205731198617 1198608 = 1198961205731198618

119896120572=

21205822

119910120578

1205722 minus 1205822119910(1 + 1205782)

119896120573=

21205822

119910120578

1205732 minus 1205822119910(1 + 1205782)

(A3)

To obtain the dynamic stiffness matrix we introduce theboundary conditions into the governing equations For dis-placements

at 119911 = 0

119880 = 1198801 119881 = 119881

1 Θ

119909= Θ1199091 Θ

119910= Θ1199101

(A4)

at 119911 = 119871

119880 = 1198802 119881 = 119881

2 Θ

119909= Θ1199092 Θ

119910= Θ1199102

(A5)

And for forces we have the following

at 119911 = 0

119878119909= 1198781199091 119878

119910= 1198781199101 119872

119909= 1198721199091 119872

119910= 1198721199101

(A6)

at 119911 = 119871

119878119909= 1198781199092 119878

119910= 1198781199102 119872

119909= 1198721199092 119872

119910= 1198721199102

(A7)

Substituting the boundary conditions into the governingequations we find

120575 = BR (A8)

Shock and Vibration 9

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

B =

[[[[[[[[[[[[

[

0 119896120572

0 119896120572

0 119896120573

0 119896120573

0 1 0 1 0 1 0 1

minus120591120572

0 minus120591120572

0 minus120591120573

0 minus120591120573

0

120594120572

0 120594120572

0 120594120573

0 120594120573

0

119896120572119878120572

119896120572119862120572

119896120572119878ℎ120572

119896120572119862ℎ120572

119896120573119878120573

119896120573119862120573

119896120573119878ℎ120573

119896120573119862ℎ120573

119878120572 119862120572 119878ℎ120572

119862ℎ120572

119878120573 119862120573 119878ℎ120573

119862ℎ120573

minus120591120572119862120572

120591120572119878120572

minus120591120572119862ℎ120572

minus120591120572119878ℎ120572

minus120591120573119862120573

120591120573119878120573

minus120591120573119862ℎ120573

minus120591120573119878ℎ120573

120594120572119862120572

minus120594120572119878120572

120594120572119862ℎ120572

120594120572119878ℎ120572

120594120573119862120573

minus120594120573119878120573

120594120573119862ℎ120573

120594120573119878ℎ120573

]]]]]]]]]]]]

]

120591120572=

radic120572

119871 120591

120573= minus

radic120573

119871 120594

120572= 119896120572120591120572 120594

120573= 119896120573120591120573

(A9)

Also

119878120572= sinradic120572 119862

120572= cosradic120572

119878ℎ120572

= sinhradic120572 119862ℎ120572

= coshradic120572

119878120573= sinradic120573 119862

120573= cosradic120573

119878ℎ120573

= sinhradic120573 119862ℎ120573

= coshradic120573

(A10)

Substituting similarly for the force equation

F = AR (A11)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

A =

[[[[[[[[[[[[

[

minus120577120572 0 120577120572 0 minus120577120573 0 120577120573 0

minus120576120572 0 120576120572 0 minus120576120573 0 120576120573 0

0 minus120574120572

0 120574120572

0 minus120574120573

0 120574120573

0 120582120572

0 minus120582120572

0 120582120573

0 minus120582120573

120577120572119862120572

minus120577120572119878120572

minus120577120572119862ℎ120572

minus120577120572119878ℎ120572

120577120573119862120573

minus120577120573119878120573

minus120577120573119862ℎ120573

minus120577120573119878ℎ120573

120576120572119862120572

minus120576120572119878120572

minus120576120572119862ℎ120572

minus120576120572119878ℎ120572

120576120573119862120573

minus120576120573119878120573

minus120576120573119862ℎ120573

minus120576120573119878ℎ120573

120574120572119878120572

120574120572119862120572

minus120574120572119878ℎ120572

minus120574120572119862ℎ120572

120574120573119878120573

120574120573119862120573

minus120574120573119878ℎ120573

minus120574120573119862ℎ120573

minus120582120572119878120572 minus120582120572119862120572 120582120572119878ℎ120572

120582120572119862ℎ120572

minus120582120573119878120573 minus120582120573119862120573 120582120573119878ℎ120573

120582120573119862ℎ120573

]]]]]]]]]]]]

]

120577120572 = 119896120572120572radic120572

EI119910119910

1198713 120576120572 = 120572radic120572

EI1199091199091198713

120574120572= 120572

EI119909119909

1198712 120582

120572= 119896120572120572

EI119910119910

1198712

120577120573= 119896120573120573radic120573

EI119910119910

1198713 120576

120573= 120573radic120573

EI119909119909

1198713

120574120573 = 120573

EI1199091199091198712

120582120573 = 119896120573120573

EI119910119910

1198712

(A12)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the supports providedby Natural Sciences and Engineering Research Council ofCanada (NSERC) and Ryerson University

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

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Journal of

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International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

taken into consideration when performing chatter-relatedcalculations [6] The structural modes of the machine deter-mine the frequency and the direction that the tool is goingto vibrate at [7] Eisele [5] and Peng et al [8] stressed thatdesigners must investigate the mode-shapes weak pointsbearing clearances and self-inducing vibratory componentsof the machine to reduce chatter Butlin andWoodhouse alsoinvestigated the number of structural modes required to pro-duce accurate results [9]They showed that low ordermodelsincorporating only two modes are sufficiently accurate tomodel the machines

In 1981 Tlusty and Ismail published one of the first papersthat documented the nonlinearity of the vibratory systemoccurring during chatter [10] Kondo et al [11] added toTlusty and Ismailrsquos investigation the behaviour of the toolafter the onset of chatter An important study carried outby Lee et al in 1989 helped bring a more complete theoryof chatter to light [12] where they looked into the responseof the work piece to chatter They modeled the tool as atypical two-degree-of-freedom spring-mass system and thework piece as an Euler beamWhen the responses of both thetool and the work piece were incorporated into the typicalspring-mass equations the results obtained far surpassed theaccuracy of previously used equations They observed a largeincrease in surface finish quality while machining even withthe use of long end mills

While a comprehensive set of mathematical models havealready been derived to explain chatter using spring-mass andEuler beam systems which have led to dramatic increases insurface finish quality chatter is still an issue in the machiningindustry as the surface tolerances keep increasing Fromthe 1990s on this drove researchers to use more refinedmathematical models to characterize themachines [13]Withthe rise of computer aided calculations and simulations ithas become increasingly easier to perform more complexanalyses of the machining process [14] Tool wear is an often-overlooked factor that contributes to chatter With the aid ofmore powerful computers this variable can now be includedin simulations Clancy and Shin [15] investigated the wearof turning tools and how that affects the stability of thesystem It was found that as the tool becomes worn its limitsof stability increase As a result stability lobes have nowbecome a function of tool wear Li et al [16] determined thatthe coherence function of two crossed accelerations in thebending vibration of the tool shank approaches unity at theonset of chatter A threshold needs to be set [17] and thendetected using simple mechanism to alert the operator tochange the machining conditions and avoid increased toolwear

In most of the existing stability prediction methodsa frequency response function (FRF)mdashacquired using animpact testmdashis required to extract the system fundamentalfrequency used to perform the calculations In order toperform accurate impact tests the machine would need tobe shut off and allowed to cool down and the coolant dripoff This process could lead to several hours of down timeAn offline method of obtaining the FRF could greatly ben-efit manufacturing companies by eliminating the downtimeneeded for the impact tests While stability lobe graphs have

becomemore accurate the industry is trending towards com-puter simulations and numerical analyses With computingpower becoming relatively cheap and readily available thismethod has become feasible Providing the most accurateresults due to its constantly updated stability calculationsthis is the area of research that engineers will most likelyprogress towards Lee et al [18] suggested that the wholesystem comprising machine tool and work piece couldbe modeled using finite element method (FEM) analysisA computer simulation would be able to predict the FRFduring all phases of the machining process As the part ismachined and becomes thinner its response to vibrationchanges dramatically Most existing research papers assumethat the FRF remains constant throughout the whole processHowever a constantly updated FRF would allow for accuratereal time stability calculations

In an attempt to numerically model the entire spindlesystems for the vibration modeling and analysis and toincorporate the effects of bearings Cao and Altintas [19]modeled spindle bearing andmachine tool systems for virtualsimulation of milling operations where they presented ageneral integratedmodel of the spindle bearing andmachinetool system consisting of a rotating shaft tool holder angularcontact ball bearings housing and the machine tool mount-ingThe proposedmodel also verified experimentally allowsvirtual cutting of a work material with the numerical modelof the spindle during the design stage and predicts bearingstiffness mode shapes frequency response function (FRF)static and dynamic deflections along the cutter and spindleshaft and contact forces on the bearings with simulatedcutting forces before physically building and testing thespindles The study showed that the accuracy of predictingthe performance of the spindles requires integratedmodelingof all spindle elements and mounting on the machine toolJonesrsquo bearing model which considers the bearing balls andrings as elastic parts and the Hertzian contact theory wereused to calculate the contact force and displacement Patwariet al [20] presented a detailed systematic procedure forexperimental and analytical modal analysis techniques toevaluate structural dynamics of a vertical machining centreAbuthakeer et al [21] carried out the dynamic characteristicsanalysis of high speed motorized spindles More recentlyDelgado et al [22] used FEM to study thenonlinear dynamicbehavior of a high speed machine tool spindle

It is known that FEM predictions and the numericalresults are often different from test and experimental resultsHowever based on the model updating corrections can bemade on the FEM models by processing records of testresults Motthershead and Friswell [23] presented a surveyof model updating in structural dynamics Later Mares etal [24] investigated the model updating using robust esti-mation and more recently Mottershead et al [25] presenteda basic introduction to the most important procedures ofcomputational model updating including tutorial examplesand a large scale model updating example of a helicopterairframe and discussed the sensitivity method in FEMmodelupdating

To the authorsrsquo best knowledge except authorsrsquo previ-ous works [26 27] no studies on the calibratedupdated

Shock and Vibration 3

dynamic stiffness matrix (DSM) models of spindle systemshave appeared in the open literature The DSM method [28]provides an analytical solution to free vibration problemsachieved by combining the coupled governing differentialequations of motion of the system into a single higher-orderordinary differential equation Seeking the most generalsolution to the resulting higher-order ODE and enforcingthe end displacements and loads through extensive matrixmanipulations lead to the DSM of the structural elementUnlike limited nodal DOFs in conventional FEM [29] theDSM-based element is modeled as a continuous system withinfinite number of DOFs within the element The DSMformulation has been proven to provide exact frequency(within the limits of the theory) for various beam con-figurations and beam-structures with the use of a singlecontinuous element per segment (see eg [28 30ndash33])The objective of the present study is to determine thenatural frequenciesvibration characteristics of machine toolspindle systems by developing the corresponding dynamicstiffness matrix (DSM) [31 32] and the proper boundaryconditions These results would then be compared to theexisting results for a common cutting system to validatetunethe model developed The coupled bending-bending (B-B)vibration of a spinning beam is investigated A MATLABcode is developed to assemble the DSM element matrices formultiple components and to apply the boundary conditions(BC) The bearings are first modeled as simply supported (S-S) frictionless pins which are then replaced by linear springelements to incorporate the flexibility of bearings into themodel In comparison with the existing data on spindlersquosfundamental frequency the bearing stiffness coefficient 119870119878is then varied to achieve a calibrated (or updated) dynamicsstiffness matrix (CDSM) vibrational model The proposedformulation can also be extended to include torsional degree-of-freedom (DOF) for further modeling purposes

2 Problem Description andGoverning Equations

Computer numeric control (CNC) machines are becom-ing the norm in manufacturing plants worldwide Thesemachines are generally 3- 4- or 5-axis depending on thenumber of degree-of-freedoms the device has Having thetool translating in the 119909 119910 and 119911 directions accounts forthe first three degrees of freedom Rotation about the spindleaxes account for any further DOF A typical spindle containsthe motors that rotate the shaft connected to the tool holderthe bearings the tools and all the mechanisms that hold thetool in place (see Figure 1 for a sample spindle configuration)As mentioned earlier in this paper the fundamental naturalfrequency of this system dictates the limitation of the cuttingparameters If these limits are surpassed the system tends tovibrate which causes chatter Also as the bearings wear thesystem characteristics and spindlersquos natural frequency change

In what follows the derivation of the governing differen-tial equations for the coupled bending-bending vibration ofa spinning beam for a rotating shaft is first briefly discussedBearings restrain the spinning motion of the spindle system

Toolholder BearingsBearings Motor Spindle shaft

Spindle casing

Figure 1 Typical spindle configuration

Y

X

ZΩ

0

L

P

Figure 2 Spinning beam

in five degrees of freedom that is displacement in 119909 119910 and119911 directions and the rotation in 119909 and 119910 directionsThe beamtorsional rigidity 119866119869 is large enough so that the torsionalvibrations can be ignored [31 32] Figure 2 shows a cylindricalbeam in a right-handed rectangular Cartesian coordinatessystemThe beam length is denoted by119871 mass per unit lengthis 119898 = 120588119860 polar mass moment of inertia per unit length is119868120572 and the principal axes bending rigidities are EI for bothplanes At an arbitrary cross section along 119911-axis 119906 and V aredisplacement of a point 119875 in the 119909 and 119910 directionsThe crosssection is allowed to rotate or twist about the 119900119911 axis Theposition vector r of the point 119875 after deformation is given by(refer to Figures 2 and 3)

r = (119906 minus 120601119910) i + (V + 120601119909) j (1)

where i and j are unit vectors in the 119909 and 119910 directions Thevelocity of point 119875 is given by

v = r +Ω times r where Ω = Ωk (2)

The kinetic and potential energies of the beam (119879 and 119880) aregiven by

119879 =1

2int

119871

0

|v|2119898119889119911

=1

2119898int

119871

0

[2+ V2 + 2Ω (119906V minus V) + Ω

2(1199062+ V2)] 119889119911

119880 =1

2EI119909119909 int

119871

0

V101584010158402

119889119911 +1

2EI119910119910 int

119871

0

119906101584010158402

119889119911

(3)

4 Shock and Vibration

U1

V1

U2

V2

Θx1

Θx2

Θy1

Θy2

Φ1

Φ2

Figure 3 Degrees of freedom of the system

Using the Hamilton principle in the usual notation state

120575int

1199052

1199051

(119879 minus 119880) 119889119905 = 0 (4)

where 1199051and 119905

2are the time intervals in the dynamic

trajectory and 120575 is the variational operator Substituting thekinetic and potential energies of the beam in the Hamiltonprinciple collecting like terms and integrating by parts thefollowing differential equations are obtained

EI1199101199101199061015840101584010158401015840

minus 119898Ω2119906 + 119898 minus 2119898ΩV = 0

minus2119898Ω minus EI119909119909V1015840101584010158401015840

minus 119898V + 119898Ω2V = 0

(5)

which govern the bending-bending (B-B) vibration of abeam coupled by the spinning speedThe resulting loads alsoforce boundary conditions at free ends are then found to bein the following forms written for shear forces as

119878119909= EI119909119909119906101584010158401015840 119878

119910= EI119910119910V101584010158401015840 (6)

and for bending moments as

119872119909= EI119909119909V10158401015840 119872

119910= minusEI

11991011991011990610158401015840 (7)

Assuming simple harmonic motion

119906 (119911 119905) = 119880 (119911) cos120596119905 V (119911 119905) = 119880 (119911) cos120596119905 (8)

and substituting (8) into (5) they can be rewritten as

(EI1199101199101198801015840101584010158401015840

minus 119898(Ω2+ 1205962)119880) + 2119898Ω120596119881 = 0

(EI1199091199091198811015840101584010158401015840

minus 119898(Ω2+ 1205962)119881) + 2119898Ω120596119880 = 0

(9)

where 120596 is frequency of oscillation and 119880 119881 and Φ are theamplitudes of 119906 V and 120601 Introducing 120585 = 119911119871 and 119863 =

119889119889120585 which are nondimensional length and the differentialoperator and back substitution into (9) lead to

[1198634minus

119898(Ω2+ 1205962) 1198714

EI119910119910

]119880 +2119898Ω120596119871

4

EI119910119910

119881 = 0

[1198634minus

119898(Ω2+ 1205962) 1198714

EI119909119909]119881 +

2119898Ω1205961198714

EI119909119909119880 = 0

(10)

3 Solution of the Governing Equations

31 Dynamic Stiffness Matrix (DSM) Solution As suggestedby Banerjee and Su [31 32] the above equation (10) can becombined into one 8th-order differential equation written as

[1198638minus (1205822

119909+ 1205822

119910) (1 + 120578

2)1198634+ 1205822

1199091205822

119910(1 minus 120578

2)2

]119882 = 0

(11)

where ldquo119882rdquo represents both lateral displacements119880 and119881 and

1205822

119909=

11989812059621198714

EI119909119909 120582

2

119910=

11989812059621198714

EI119910119910 120578

2=

Ω2

1205962 (12)

The general solution of the differential equation is sought inthe form

119882 = 119890119903120585 (13)

Substituting (13) into (11) leads to

1199038minus (1205822

119909+ 1205822

119910) (1 + 120578

2) 1199034+ 1205822

1199091205822

119910(1 minus 120578

2)2

(14)

where 11990313 = plusmnradic120572 11990324 = plusmnradic120573 11990357 = plusmn119894radic120572 11990368 = plusmn119894radic120573 and

1205722=

1

2 (1205822

119909+ 1205822

119910) (1 + 120578

2)

+radic(1205822119909minus 1205822119910)2

(1 + 1205782)2+ 161205822

11990912058221199101205782

1205732=

1

2 (1205822

119909+ 1205822

119910) (1 + 120578

2)

minusradic(1205822119909minus 1205822119910)2

(1 + 1205782)2+ 161205822

11990912058221199101205782

(15)

Based on Euler-Bernoulli beam bending theory thecorresponding bending rotation that is slope about 119909-axisand 119910-axis Θ

119909and Θ

119910 respectively is given by (see the

appendix for further details)

Θ119909=

119889119881

119889119911= minus

1

119871

119889119881

119889120585 Θ

119910=

119889119880

119889119911=

1

119871

119889119880

119889120585 (16)

By doing similar substitutions in load (3) and (4) one finds

119878119909= (

EI119910119910

1198713)(

1198893119880

1198891205853) 119878

119910= (

EI119909119909

1198713)(

1198893119881

1198891205853)

119872119909 = (

EI119910119910

1198712)(

1198892119880

1198891205852) 119872119910 = (

EI1199091199091198712

)(1198892119881

1198891205852)

(17)

Substituting the boundary conditions into the governingequations enforced at 119911 = 0 (120585 = 0) and 119911 = 119871 (120585 = 1)leads to

120575 = BR (18)

Shock and Vibration 5

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

(19)

Substituting similarly for the force equation one finds

F = AR (20)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

(21)

The frequency-dependent dynamic stiffness matrix(DSM) of the spinning beam K(120596) can then be derivedby eliminating R We also know that the force amplitude isrelated to the displacement vector by

F = K (120596) 120575 where K (120596) = ABminus1 (22)

Explicitmatrix forms ofAB andR are given in the appendix

32 Classical Finite Element Method (FEM) For validationand comparison purposes a conventional finite elementformulation of the problem and aMATLAB-based FEM codewere developed to carry out the free vibration analysis of thespindle system [33] The Galerkin method of weighted resid-uals [29 33] was employed to develop the integral form ofthe governing equation (10) Performing integration by partson the resulting integral equations leads to the weak integralform of equations which also satisfy the principle of virtualwork The system is then discretized using beam elementswith four DOF per node (ie two lateral displacements 119880and119881 and two slopes1198801015840 and119881

1015840 eight degrees of freedom intotal) Introducing cubic Hermite-type polynomial approx-imations [29 33] for bending displacements in the elementweak integral form the classical finite element formulationis developed This formulation results in constant (or static)element stiffness mass and coupling matrices which arethen assembled to find the global matrices Finally enforcingthe boundary conditions leads to a linear eigenvalue problemwhich is solved to extract the system natural frequencies andmodes There has also been an attempt to develop a dynamicfinite element (DFE) [34] formulation for the problem inhand using frequency-dependent interpolation functionsand the preliminary results were found to be promising [35]

4 Application of the Theory andNumerical Results

TheDSM formulation was first validated against various ana-lytical and numerical frequency data available in the literature[31 32] for various uniform simple beam configurations TheDSM frequency data were also comparedwith those obtainedfrom commercial FEM-based software ABAQUS and theeffect of various element types on the convergence was alsoexamined In addition the conventional FEM formulation[29 33] was used for further comparisons however as the

in-house FEM-based code and commercial software both ledto similar results the FEM results have been omitted here forthe brevity

A typical spindle system model (see Figure 4) was thenconstructed using multiple segments As the nominal fun-damental frequency was available for nonspinning spindletherefore the effect of spinning speed was taken out by simplysetting Ω to zero in FEM models However one should notethat setting the spinning speed exactly to zero would causethe DSM formulation to collapse This is due to the factthat the DSM formulation pivots around combining the twoexpressions (10) into a single on (11) through the couplingterms involving Ω [31 32] However an extremely smallnumber (Ω cong 0) can be used to model the stationary spindle

The entire system was assumed to be made from thesame material that is tooling steel Initially it was alsoassumed that the system is simply supported at the bearinglocations that is the bearings modeled as frictionless pinsrigidly attached to the spindle casing with zero flexibilityallowing only for rotations in all directions This system wasmodeled using the DSM method described above as wellas finite element analysis (FEA) in ABAQUS commercialsoftware to evaluate the first six natural frequencies of thespindle system Several different element types andmesh sizeswere examined to ensure the convergence (Table 1) beamelement B33 uses a 2-node cubic beam (similar to the in-house FEM code) C3D20R is a general-purpose quadraticbrick element and C3D10M is a general-purpose tetrahedralelement (see Table 1 for results) Unlike the conventionalFEM formulation the DSM method results in exact solutionfor all natural frequencies of a spinning uniform beam [3132] Therefore as the spindle system in hand is made of12 uniform sections only a 12-element piecewise-uniform(stepped) DSM model was required As expected the resultsusing beam element type B33 yielded the best agreement withthe DSM results as the 3D and shear effects are neglectedin both However the fundamental frequency values werefound to be different from the nominal value that is 1393Hzcompared to 1000 plusmn 50Hz as provided by the manufacturer

In order to address this issue the DSM and FEM modelswere calibrated to take into account the flexibility of thebearing systems To this end the simply supported boundaryconditions were modified and replaced by simple linearspring elements attached to the spindle shaft in 119909 and119910 directions representing general flexibility of the bearingsystems (see Figure 4) The bearings were all assumed to beidentical with the same spring coefficient 119870119904 The springstiffness value 119870

119904 was then varied and a calibrated model

equippedwith spring elements of constant119870119904= 21times10

8Nmwas found to result in a fundamental bending frequencyof 1004Hz closely equivalent to the systemrsquos nominal value(Figure 5) The effect of spinning speed Ω on the systemrsquosfundamental frequency was also investigated In the absenceof spinning speed Ω = 0 the system exhibits uncoupledbehavior (see (7) and (8)) undergoing two separate bendingdisplacements in 119909 and 119910 directions As a result for a circularshaft with 119868119909119909 = 119868119910119910 the modal analysis results in repeatedfrequency values appearing in pairs (Table 1) By introducingthe spinning speedΩ = 0 the bending natural frequencies in

6 Shock and Vibration

Table 1 Spindle natural frequencies DSM versus FEA for a nonspinning (Ω cong 955RPM) with simply supported boundary conditions

Mode DSM FEA B33 element (Hz) FEA C3D20R (Hz) FEA C3D10M (Hz)12 sections 600 elements 1200 elements 16965 elements 20122 elements 34465 elements 45645 elements

1 139292 139300 139300 130800 130800 130100 1301002 139324 139300 139300 130800 130800 130100 1301003 407102 407100 407100 320800 321200 333500 3322004 407134 407100 407100 321000 321200 333500 3324005 566814 566900 566900 433900 431100 432200 4291006 566846 566900 566900 434100 431600 432300 429200

Table 2 Spindle natural frequencies versus various spinning speeds for flexible boundary conditions 119870119904= 21 times 10

8Nm

Mode955

(RPM)(Ω cong 0)

47842(RPM)

95589(RPM)

14334(RPM)

19108(RPM)

23883(RPM)

28657(RPM)

33432(RPM)

1 100427 92469 84511 76554 68596 60638 52680 447232 100459 108416 114321 106363 98406 90448 82490 745323 130237 122279 1163741 113414 105456 97498 89541 815834 130268 129329 1213716 124332 132290 140247 146805 1388475 137287 138226 1461838 154142 154142 154762 148205 1561636 137319 145277 1532344 161192 162720 170057 178015 185973

Toolholder Motor Spindle shaftSpring Spring

Spindle casing

Figure 4 Modified boundary conditions

119909 and 119910 directions become coupled changing the repeatednatural frequencies to two distinct ones (see Table 2) withone decreasing and approaching zero (Figure 6) and the otherincreasing from their uncoupled values Further increasein the spinning speed would make one of the bendingfrequencies go to zero that is whirling instability alsoreported and discussed in [31 32] The first critical spindlespeed in the case of calibrated model was found to be Ωcr =6 times 10

5 RPM which is well above the operating speed of thespindle that is 35 times 104 RPM (see Figure 7)

In order to further validate the proposed CDSM modelthe tap testing method was used to experimentally collectthe frequency response function (FRF) of the spindle sys-tem in hand while being installed on its correspondingmachine tool A 1-inch diameter blank tool with a 2-inchprotrusion in a typical shrink fit tool holder was usedThe tool and holder were inserted into the spindle Thespindle was set to be horizontal (ie neutral) position andan accelerometer (352A21 08 g) was then attached to theedge of the tool holder The spindle was struck 10 timesin the 119909 direction using a hammer (086C04 5000N) and

0

200

400

600

800

1000

1200

1400

Syste

m n

atur

al fr

eque

ncy

(Hz)

Bearing equivalent spring constant K (Nm)

DSMSimply supportedANSYS

Spindle manufacturerrsquos designed natural frequency (1000Hz)

100E+00

100E+01

100E+02

100E+03

100E+04

100E+05

100E+06

100E+07

100E+08

100E+09

100E+10

Tool Simplesupports

Simplesupports

Motor Spindleshaftholder

MotorSpring SpringSpindle casing

ToolSpindle

shaftholder

Figure 5 Varying spring constant versus system natural frequencyusing DSM and FEA

the average FRF graph was generated (SIM3 Module Photon+ Data Acquisition) Similarly the spindle was also testedalong the 119910 direction The nonspinning spindlersquos naturalfrequencies extracted from the FRF graphs are presented inTable 3 alongside the CDSM results As can be seen themaximum difference between the first three CDSM flexural

Shock and Vibration 7

Table 3 Spindle natural frequencies experimental versus CDSM for a nonspinning spindle system with flexible boundary conditions119870119904=

21 times 108 Nm

Nonspinning bending modes (Hz) CDSM (Ω cong 955RPM) Experimental (x-axisy-axis) Difference ()(x-axisy-axis)

1205961 100427100459 9298192651 80841205962 130237130268 130151131067 0070061205963 137287137319 160144158386 143133

0 5000 10000 15000 20000 25000 30000 35000 400000

200

400

600

800

1000

1200

1400

1600

1800

2000

Nat

ural

freq

uenc

y (H

z)

Spindle speed (RPM)

Spindle speed versus natural frequency

1st mode2nd mode3rd mode

4th mode5th mode6th mode

Figure 6 Spindle speed versus natural frequency

0

200

400

600

800

1000

1200

1400

0 10000 20000 30000 40000 50000 60000 70000 80000

Nat

ural

freq

uenc

y (H

z)

Spindle RPM

Spindle RPM versus natural frequency

Spring BCSimply supported

Spindle RPM operational limit

Figure 7 Spindle natural frequency versus spindle RPM simplysupported and flexible bearing models

natural frequencies and the experimentally evaluated dataare respectively 8 007 and 143 in 119909 direction and84 006 and 133 in 119910 directionThe average differenceis found to be around 74 which is within the acceptablerange

5 Discussion and Conclusion

It has been established that the service life changes the vibra-tional behavior of spindle systems that is reduced naturalfrequency over the time associated with the bearings wearTherefore in order to generate an updated spindlemodel it isessential to include the effect of the bearings in the systemAnanalytical model of a multisegment spinning spindle basedon the dynamic stiffnessmatrix (DSM) formulation and exactwithin the limits of the Euler-Bernoulli beam bending theorywas developed The beam exhibits coupled bending-bending(B-B) vibration and as expected its fundamental coupledfrequency was found to decrease with increasing spinningspeed The bearings were included in the model using twodifferent models rigid simply supported frictionless pinsand flexible linear spring elements The simply supportedbearing model was found to yield an excessively rigid systemwith amuch higher natural frequency than the nominal valueof the system Replacing the simply supported boundarycondition with linear spring elements the spring stiffness119870119904 was then calibrated so that the fundamental frequency

of the system matched the nominal value that is a morerealistic and accurate representation of the spindle systemThe accuracy of the proposed calibrated dynamic stiffnessmatrix (CDSM) method was confirmed through compari-son with both numerical and experimental results Whencompared to the conventional finite element method (FEM)the CDSM frequencies were found to best match with thoseobtained from a cubic beamfinite element (B33 inABAQUS)Moreover in comparison with experimental data the CDSMresults were found to be within acceptable accuracy the aver-age error for the first three bending natural frequencies is lessthan 8

It is worth noting that the proposed CDSM can alsobe fine-tuned to match spindlersquos experimental fundamentalfrequency instead of the nominal value Further researchis underway to improve the CDSM by using more accu-rate material properties for different segments of thespindle The developed calibrated model could then beexploited in an attempt to take into account the effects

8 Shock and Vibration

of bearing wear in terms of spindlersquos service time numer-ically evaluate the equivalent cutting force and calcu-late updated stability lobes used to achieve chatter-freecuts

Appendix

From the solutions of 119880 and 119881 (see expressions (13)) thecorresponding bending rotation about 119909-axis and 119910-axis Θ

119909

and Θ119910 are respectively given by

Θ119909 =

119889119881

119889119911= minus

1

119871

119889119881

119889120585

= minus1

119871(radic120572119861

1cosradic120572120585 minus radic120572119861

2sinradic120572120585

+ radic1205721198613coshradic120572120585 + radic120572119861

1sinradic120572120585

+ radic1205731198615cosradic120573120585 minus radic120573119861

6sinradic120573120585

+radic1205731198617coshradic120573120585 + radic120573119861

8sinhradic120573120585)

Θ119910=

119889119880

119889119911=

1

119871

119889119880

119889120585

= minus1

119871(radic120572119860

1cosradic120572120585 minus radic120572119860

2sinradic120572120585

+ radic1205721198603 coshradic120572120585 + radic1205721198601 sinradic120572120585

+ radic1205731198605 cosradic120573120585 minus radic1205731198606 sinradic120573120585

+radic1205731198607coshradic120573120585 + radic120573119860

8sinhradic120573120585)

(A1)

By doing similar substitutions we find (see expressions (3)(4) and (14))

119878119909 = (

EI119910119910

1198713)(

minus120572radic1205721198601cosradic120572120585 + 120572radic120572119860

2sinradic120572120585 + 120572radic120572119860

3coshradic120572120585 + 120572radic120572119860

4sinhradic120572120585

minus120573radic1205731198605cosradic120573120585 + 120573radic120573119860

6sinradic120573120585 + 120573radic120573119860

7coshradic120573120585 + 120573radic120573119860

8sinhradic120573120585

)

119878119910 = (

EI1199091199091198713

)(minus120572radic120572119861

1cosradic120572120585 + 120572radic120572119861

2sinradic120572120585 + 120572radic120572119861

3coshradic120572120585 + 120572radic120572119861

4sinhradic120572120585

minus120573radic1205731198615cosradic120573120585 + 120573radic120573119861

6sinradic120573120585 + 120573radic120573119861

7coshradic120573120585 + 120573radic120573119861

8sinhradic120573120585

)

119872119909= (

EI119910119910

1198712)(

minus1205721198611 sinradic120572120585 minus 1205721198612 cosradic120572120585 + 1205721198613 sinhradic120572120585 + 1205721198614 coshradic120572120585

minus1205731198615sinradic120573120585 + 120573119861

6cosradic120573120585 + 120573119861

7sinhradic120573120585 + 120573119861

8coshradic120573120585

)

119872119910= (

EI119909119909

1198712)(

minus1205721198601 sinradic120572120585 minus 120572119860

2 cosradic120572120585 + 1205721198603 sinhradic120572120585 + 1205721198604 coshradic120572120585

minus1205731198605 sinradic120573120585 + 1205731198606 cosradic120573120585 + 1205731198607 sinhradic120573120585 + 1205731198608 coshradic120573120585)

(A2)

where1198601= 1198961205721198611 119860

2= 1198961205721198612

1198603= 1198961205721198613 119860

4= 1198961205721198614

1198605 = 1198961205731198615 1198606 = 1198961205731198616

1198607 = 1198961205731198617 1198608 = 1198961205731198618

119896120572=

21205822

119910120578

1205722 minus 1205822119910(1 + 1205782)

119896120573=

21205822

119910120578

1205732 minus 1205822119910(1 + 1205782)

(A3)

To obtain the dynamic stiffness matrix we introduce theboundary conditions into the governing equations For dis-placements

at 119911 = 0

119880 = 1198801 119881 = 119881

1 Θ

119909= Θ1199091 Θ

119910= Θ1199101

(A4)

at 119911 = 119871

119880 = 1198802 119881 = 119881

2 Θ

119909= Θ1199092 Θ

119910= Θ1199102

(A5)

And for forces we have the following

at 119911 = 0

119878119909= 1198781199091 119878

119910= 1198781199101 119872

119909= 1198721199091 119872

119910= 1198721199101

(A6)

at 119911 = 119871

119878119909= 1198781199092 119878

119910= 1198781199102 119872

119909= 1198721199092 119872

119910= 1198721199102

(A7)

Substituting the boundary conditions into the governingequations we find

120575 = BR (A8)

Shock and Vibration 9

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

B =

[[[[[[[[[[[[

[

0 119896120572

0 119896120572

0 119896120573

0 119896120573

0 1 0 1 0 1 0 1

minus120591120572

0 minus120591120572

0 minus120591120573

0 minus120591120573

0

120594120572

0 120594120572

0 120594120573

0 120594120573

0

119896120572119878120572

119896120572119862120572

119896120572119878ℎ120572

119896120572119862ℎ120572

119896120573119878120573

119896120573119862120573

119896120573119878ℎ120573

119896120573119862ℎ120573

119878120572 119862120572 119878ℎ120572

119862ℎ120572

119878120573 119862120573 119878ℎ120573

119862ℎ120573

minus120591120572119862120572

120591120572119878120572

minus120591120572119862ℎ120572

minus120591120572119878ℎ120572

minus120591120573119862120573

120591120573119878120573

minus120591120573119862ℎ120573

minus120591120573119878ℎ120573

120594120572119862120572

minus120594120572119878120572

120594120572119862ℎ120572

120594120572119878ℎ120572

120594120573119862120573

minus120594120573119878120573

120594120573119862ℎ120573

120594120573119878ℎ120573

]]]]]]]]]]]]

]

120591120572=

radic120572

119871 120591

120573= minus

radic120573

119871 120594

120572= 119896120572120591120572 120594

120573= 119896120573120591120573

(A9)

Also

119878120572= sinradic120572 119862

120572= cosradic120572

119878ℎ120572

= sinhradic120572 119862ℎ120572

= coshradic120572

119878120573= sinradic120573 119862

120573= cosradic120573

119878ℎ120573

= sinhradic120573 119862ℎ120573

= coshradic120573

(A10)

Substituting similarly for the force equation

F = AR (A11)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

A =

[[[[[[[[[[[[

[

minus120577120572 0 120577120572 0 minus120577120573 0 120577120573 0

minus120576120572 0 120576120572 0 minus120576120573 0 120576120573 0

0 minus120574120572

0 120574120572

0 minus120574120573

0 120574120573

0 120582120572

0 minus120582120572

0 120582120573

0 minus120582120573

120577120572119862120572

minus120577120572119878120572

minus120577120572119862ℎ120572

minus120577120572119878ℎ120572

120577120573119862120573

minus120577120573119878120573

minus120577120573119862ℎ120573

minus120577120573119878ℎ120573

120576120572119862120572

minus120576120572119878120572

minus120576120572119862ℎ120572

minus120576120572119878ℎ120572

120576120573119862120573

minus120576120573119878120573

minus120576120573119862ℎ120573

minus120576120573119878ℎ120573

120574120572119878120572

120574120572119862120572

minus120574120572119878ℎ120572

minus120574120572119862ℎ120572

120574120573119878120573

120574120573119862120573

minus120574120573119878ℎ120573

minus120574120573119862ℎ120573

minus120582120572119878120572 minus120582120572119862120572 120582120572119878ℎ120572

120582120572119862ℎ120572

minus120582120573119878120573 minus120582120573119862120573 120582120573119878ℎ120573

120582120573119862ℎ120573

]]]]]]]]]]]]

]

120577120572 = 119896120572120572radic120572

EI119910119910

1198713 120576120572 = 120572radic120572

EI1199091199091198713

120574120572= 120572

EI119909119909

1198712 120582

120572= 119896120572120572

EI119910119910

1198712

120577120573= 119896120573120573radic120573

EI119910119910

1198713 120576

120573= 120573radic120573

EI119909119909

1198713

120574120573 = 120573

EI1199091199091198712

120582120573 = 119896120573120573

EI119910119910

1198712

(A12)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the supports providedby Natural Sciences and Engineering Research Council ofCanada (NSERC) and Ryerson University

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

International Journal of

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

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Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

dynamic stiffness matrix (DSM) models of spindle systemshave appeared in the open literature The DSM method [28]provides an analytical solution to free vibration problemsachieved by combining the coupled governing differentialequations of motion of the system into a single higher-orderordinary differential equation Seeking the most generalsolution to the resulting higher-order ODE and enforcingthe end displacements and loads through extensive matrixmanipulations lead to the DSM of the structural elementUnlike limited nodal DOFs in conventional FEM [29] theDSM-based element is modeled as a continuous system withinfinite number of DOFs within the element The DSMformulation has been proven to provide exact frequency(within the limits of the theory) for various beam con-figurations and beam-structures with the use of a singlecontinuous element per segment (see eg [28 30ndash33])The objective of the present study is to determine thenatural frequenciesvibration characteristics of machine toolspindle systems by developing the corresponding dynamicstiffness matrix (DSM) [31 32] and the proper boundaryconditions These results would then be compared to theexisting results for a common cutting system to validatetunethe model developed The coupled bending-bending (B-B)vibration of a spinning beam is investigated A MATLABcode is developed to assemble the DSM element matrices formultiple components and to apply the boundary conditions(BC) The bearings are first modeled as simply supported (S-S) frictionless pins which are then replaced by linear springelements to incorporate the flexibility of bearings into themodel In comparison with the existing data on spindlersquosfundamental frequency the bearing stiffness coefficient 119870119878is then varied to achieve a calibrated (or updated) dynamicsstiffness matrix (CDSM) vibrational model The proposedformulation can also be extended to include torsional degree-of-freedom (DOF) for further modeling purposes

2 Problem Description andGoverning Equations

Computer numeric control (CNC) machines are becom-ing the norm in manufacturing plants worldwide Thesemachines are generally 3- 4- or 5-axis depending on thenumber of degree-of-freedoms the device has Having thetool translating in the 119909 119910 and 119911 directions accounts forthe first three degrees of freedom Rotation about the spindleaxes account for any further DOF A typical spindle containsthe motors that rotate the shaft connected to the tool holderthe bearings the tools and all the mechanisms that hold thetool in place (see Figure 1 for a sample spindle configuration)As mentioned earlier in this paper the fundamental naturalfrequency of this system dictates the limitation of the cuttingparameters If these limits are surpassed the system tends tovibrate which causes chatter Also as the bearings wear thesystem characteristics and spindlersquos natural frequency change

In what follows the derivation of the governing differen-tial equations for the coupled bending-bending vibration ofa spinning beam for a rotating shaft is first briefly discussedBearings restrain the spinning motion of the spindle system

Toolholder BearingsBearings Motor Spindle shaft

Spindle casing

Figure 1 Typical spindle configuration

Y

X

ZΩ

0

L

P

Figure 2 Spinning beam

in five degrees of freedom that is displacement in 119909 119910 and119911 directions and the rotation in 119909 and 119910 directionsThe beamtorsional rigidity 119866119869 is large enough so that the torsionalvibrations can be ignored [31 32] Figure 2 shows a cylindricalbeam in a right-handed rectangular Cartesian coordinatessystemThe beam length is denoted by119871 mass per unit lengthis 119898 = 120588119860 polar mass moment of inertia per unit length is119868120572 and the principal axes bending rigidities are EI for bothplanes At an arbitrary cross section along 119911-axis 119906 and V aredisplacement of a point 119875 in the 119909 and 119910 directionsThe crosssection is allowed to rotate or twist about the 119900119911 axis Theposition vector r of the point 119875 after deformation is given by(refer to Figures 2 and 3)

r = (119906 minus 120601119910) i + (V + 120601119909) j (1)

where i and j are unit vectors in the 119909 and 119910 directions Thevelocity of point 119875 is given by

v = r +Ω times r where Ω = Ωk (2)

The kinetic and potential energies of the beam (119879 and 119880) aregiven by

119879 =1

2int

119871

0

|v|2119898119889119911

=1

2119898int

119871

0

[2+ V2 + 2Ω (119906V minus V) + Ω

2(1199062+ V2)] 119889119911

119880 =1

2EI119909119909 int

119871

0

V101584010158402

119889119911 +1

2EI119910119910 int

119871

0

119906101584010158402

119889119911

(3)

4 Shock and Vibration

U1

V1

U2

V2

Θx1

Θx2

Θy1

Θy2

Φ1

Φ2

Figure 3 Degrees of freedom of the system

Using the Hamilton principle in the usual notation state

120575int

1199052

1199051

(119879 minus 119880) 119889119905 = 0 (4)

where 1199051and 119905

2are the time intervals in the dynamic

trajectory and 120575 is the variational operator Substituting thekinetic and potential energies of the beam in the Hamiltonprinciple collecting like terms and integrating by parts thefollowing differential equations are obtained

EI1199101199101199061015840101584010158401015840

minus 119898Ω2119906 + 119898 minus 2119898ΩV = 0

minus2119898Ω minus EI119909119909V1015840101584010158401015840

minus 119898V + 119898Ω2V = 0

(5)

which govern the bending-bending (B-B) vibration of abeam coupled by the spinning speedThe resulting loads alsoforce boundary conditions at free ends are then found to bein the following forms written for shear forces as

119878119909= EI119909119909119906101584010158401015840 119878

119910= EI119910119910V101584010158401015840 (6)

and for bending moments as

119872119909= EI119909119909V10158401015840 119872

119910= minusEI

11991011991011990610158401015840 (7)

Assuming simple harmonic motion

119906 (119911 119905) = 119880 (119911) cos120596119905 V (119911 119905) = 119880 (119911) cos120596119905 (8)

and substituting (8) into (5) they can be rewritten as

(EI1199101199101198801015840101584010158401015840

minus 119898(Ω2+ 1205962)119880) + 2119898Ω120596119881 = 0

(EI1199091199091198811015840101584010158401015840

minus 119898(Ω2+ 1205962)119881) + 2119898Ω120596119880 = 0

(9)

where 120596 is frequency of oscillation and 119880 119881 and Φ are theamplitudes of 119906 V and 120601 Introducing 120585 = 119911119871 and 119863 =

119889119889120585 which are nondimensional length and the differentialoperator and back substitution into (9) lead to

[1198634minus

119898(Ω2+ 1205962) 1198714

EI119910119910

]119880 +2119898Ω120596119871

4

EI119910119910

119881 = 0

[1198634minus

119898(Ω2+ 1205962) 1198714

EI119909119909]119881 +

2119898Ω1205961198714

EI119909119909119880 = 0

(10)

3 Solution of the Governing Equations

31 Dynamic Stiffness Matrix (DSM) Solution As suggestedby Banerjee and Su [31 32] the above equation (10) can becombined into one 8th-order differential equation written as

[1198638minus (1205822

119909+ 1205822

119910) (1 + 120578

2)1198634+ 1205822

1199091205822

119910(1 minus 120578

2)2

]119882 = 0

(11)

where ldquo119882rdquo represents both lateral displacements119880 and119881 and

1205822

119909=

11989812059621198714

EI119909119909 120582

2

119910=

11989812059621198714

EI119910119910 120578

2=

Ω2

1205962 (12)

The general solution of the differential equation is sought inthe form

119882 = 119890119903120585 (13)

Substituting (13) into (11) leads to

1199038minus (1205822

119909+ 1205822

119910) (1 + 120578

2) 1199034+ 1205822

1199091205822

119910(1 minus 120578

2)2

(14)

where 11990313 = plusmnradic120572 11990324 = plusmnradic120573 11990357 = plusmn119894radic120572 11990368 = plusmn119894radic120573 and

1205722=

1

2 (1205822

119909+ 1205822

119910) (1 + 120578

2)

+radic(1205822119909minus 1205822119910)2

(1 + 1205782)2+ 161205822

11990912058221199101205782

1205732=

1

2 (1205822

119909+ 1205822

119910) (1 + 120578

2)

minusradic(1205822119909minus 1205822119910)2

(1 + 1205782)2+ 161205822

11990912058221199101205782

(15)

Based on Euler-Bernoulli beam bending theory thecorresponding bending rotation that is slope about 119909-axisand 119910-axis Θ

119909and Θ

119910 respectively is given by (see the

appendix for further details)

Θ119909=

119889119881

119889119911= minus

1

119871

119889119881

119889120585 Θ

119910=

119889119880

119889119911=

1

119871

119889119880

119889120585 (16)

By doing similar substitutions in load (3) and (4) one finds

119878119909= (

EI119910119910

1198713)(

1198893119880

1198891205853) 119878

119910= (

EI119909119909

1198713)(

1198893119881

1198891205853)

119872119909 = (

EI119910119910

1198712)(

1198892119880

1198891205852) 119872119910 = (

EI1199091199091198712

)(1198892119881

1198891205852)

(17)

Substituting the boundary conditions into the governingequations enforced at 119911 = 0 (120585 = 0) and 119911 = 119871 (120585 = 1)leads to

120575 = BR (18)

Shock and Vibration 5

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

(19)

Substituting similarly for the force equation one finds

F = AR (20)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

(21)

The frequency-dependent dynamic stiffness matrix(DSM) of the spinning beam K(120596) can then be derivedby eliminating R We also know that the force amplitude isrelated to the displacement vector by

F = K (120596) 120575 where K (120596) = ABminus1 (22)

Explicitmatrix forms ofAB andR are given in the appendix

32 Classical Finite Element Method (FEM) For validationand comparison purposes a conventional finite elementformulation of the problem and aMATLAB-based FEM codewere developed to carry out the free vibration analysis of thespindle system [33] The Galerkin method of weighted resid-uals [29 33] was employed to develop the integral form ofthe governing equation (10) Performing integration by partson the resulting integral equations leads to the weak integralform of equations which also satisfy the principle of virtualwork The system is then discretized using beam elementswith four DOF per node (ie two lateral displacements 119880and119881 and two slopes1198801015840 and119881

1015840 eight degrees of freedom intotal) Introducing cubic Hermite-type polynomial approx-imations [29 33] for bending displacements in the elementweak integral form the classical finite element formulationis developed This formulation results in constant (or static)element stiffness mass and coupling matrices which arethen assembled to find the global matrices Finally enforcingthe boundary conditions leads to a linear eigenvalue problemwhich is solved to extract the system natural frequencies andmodes There has also been an attempt to develop a dynamicfinite element (DFE) [34] formulation for the problem inhand using frequency-dependent interpolation functionsand the preliminary results were found to be promising [35]

4 Application of the Theory andNumerical Results

TheDSM formulation was first validated against various ana-lytical and numerical frequency data available in the literature[31 32] for various uniform simple beam configurations TheDSM frequency data were also comparedwith those obtainedfrom commercial FEM-based software ABAQUS and theeffect of various element types on the convergence was alsoexamined In addition the conventional FEM formulation[29 33] was used for further comparisons however as the

in-house FEM-based code and commercial software both ledto similar results the FEM results have been omitted here forthe brevity

A typical spindle system model (see Figure 4) was thenconstructed using multiple segments As the nominal fun-damental frequency was available for nonspinning spindletherefore the effect of spinning speed was taken out by simplysetting Ω to zero in FEM models However one should notethat setting the spinning speed exactly to zero would causethe DSM formulation to collapse This is due to the factthat the DSM formulation pivots around combining the twoexpressions (10) into a single on (11) through the couplingterms involving Ω [31 32] However an extremely smallnumber (Ω cong 0) can be used to model the stationary spindle

The entire system was assumed to be made from thesame material that is tooling steel Initially it was alsoassumed that the system is simply supported at the bearinglocations that is the bearings modeled as frictionless pinsrigidly attached to the spindle casing with zero flexibilityallowing only for rotations in all directions This system wasmodeled using the DSM method described above as wellas finite element analysis (FEA) in ABAQUS commercialsoftware to evaluate the first six natural frequencies of thespindle system Several different element types andmesh sizeswere examined to ensure the convergence (Table 1) beamelement B33 uses a 2-node cubic beam (similar to the in-house FEM code) C3D20R is a general-purpose quadraticbrick element and C3D10M is a general-purpose tetrahedralelement (see Table 1 for results) Unlike the conventionalFEM formulation the DSM method results in exact solutionfor all natural frequencies of a spinning uniform beam [3132] Therefore as the spindle system in hand is made of12 uniform sections only a 12-element piecewise-uniform(stepped) DSM model was required As expected the resultsusing beam element type B33 yielded the best agreement withthe DSM results as the 3D and shear effects are neglectedin both However the fundamental frequency values werefound to be different from the nominal value that is 1393Hzcompared to 1000 plusmn 50Hz as provided by the manufacturer

In order to address this issue the DSM and FEM modelswere calibrated to take into account the flexibility of thebearing systems To this end the simply supported boundaryconditions were modified and replaced by simple linearspring elements attached to the spindle shaft in 119909 and119910 directions representing general flexibility of the bearingsystems (see Figure 4) The bearings were all assumed to beidentical with the same spring coefficient 119870119904 The springstiffness value 119870

119904 was then varied and a calibrated model

equippedwith spring elements of constant119870119904= 21times10

8Nmwas found to result in a fundamental bending frequencyof 1004Hz closely equivalent to the systemrsquos nominal value(Figure 5) The effect of spinning speed Ω on the systemrsquosfundamental frequency was also investigated In the absenceof spinning speed Ω = 0 the system exhibits uncoupledbehavior (see (7) and (8)) undergoing two separate bendingdisplacements in 119909 and 119910 directions As a result for a circularshaft with 119868119909119909 = 119868119910119910 the modal analysis results in repeatedfrequency values appearing in pairs (Table 1) By introducingthe spinning speedΩ = 0 the bending natural frequencies in

6 Shock and Vibration

Table 1 Spindle natural frequencies DSM versus FEA for a nonspinning (Ω cong 955RPM) with simply supported boundary conditions

Mode DSM FEA B33 element (Hz) FEA C3D20R (Hz) FEA C3D10M (Hz)12 sections 600 elements 1200 elements 16965 elements 20122 elements 34465 elements 45645 elements

1 139292 139300 139300 130800 130800 130100 1301002 139324 139300 139300 130800 130800 130100 1301003 407102 407100 407100 320800 321200 333500 3322004 407134 407100 407100 321000 321200 333500 3324005 566814 566900 566900 433900 431100 432200 4291006 566846 566900 566900 434100 431600 432300 429200

Table 2 Spindle natural frequencies versus various spinning speeds for flexible boundary conditions 119870119904= 21 times 10

8Nm

Mode955

(RPM)(Ω cong 0)

47842(RPM)

95589(RPM)

14334(RPM)

19108(RPM)

23883(RPM)

28657(RPM)

33432(RPM)

1 100427 92469 84511 76554 68596 60638 52680 447232 100459 108416 114321 106363 98406 90448 82490 745323 130237 122279 1163741 113414 105456 97498 89541 815834 130268 129329 1213716 124332 132290 140247 146805 1388475 137287 138226 1461838 154142 154142 154762 148205 1561636 137319 145277 1532344 161192 162720 170057 178015 185973

Toolholder Motor Spindle shaftSpring Spring

Spindle casing

Figure 4 Modified boundary conditions

119909 and 119910 directions become coupled changing the repeatednatural frequencies to two distinct ones (see Table 2) withone decreasing and approaching zero (Figure 6) and the otherincreasing from their uncoupled values Further increasein the spinning speed would make one of the bendingfrequencies go to zero that is whirling instability alsoreported and discussed in [31 32] The first critical spindlespeed in the case of calibrated model was found to be Ωcr =6 times 10

5 RPM which is well above the operating speed of thespindle that is 35 times 104 RPM (see Figure 7)

In order to further validate the proposed CDSM modelthe tap testing method was used to experimentally collectthe frequency response function (FRF) of the spindle sys-tem in hand while being installed on its correspondingmachine tool A 1-inch diameter blank tool with a 2-inchprotrusion in a typical shrink fit tool holder was usedThe tool and holder were inserted into the spindle Thespindle was set to be horizontal (ie neutral) position andan accelerometer (352A21 08 g) was then attached to theedge of the tool holder The spindle was struck 10 timesin the 119909 direction using a hammer (086C04 5000N) and

0

200

400

600

800

1000

1200

1400

Syste

m n

atur

al fr

eque

ncy

(Hz)

Bearing equivalent spring constant K (Nm)

DSMSimply supportedANSYS

Spindle manufacturerrsquos designed natural frequency (1000Hz)

100E+00

100E+01

100E+02

100E+03

100E+04

100E+05

100E+06

100E+07

100E+08

100E+09

100E+10

Tool Simplesupports

Simplesupports

Motor Spindleshaftholder

MotorSpring SpringSpindle casing

ToolSpindle

shaftholder

Figure 5 Varying spring constant versus system natural frequencyusing DSM and FEA

the average FRF graph was generated (SIM3 Module Photon+ Data Acquisition) Similarly the spindle was also testedalong the 119910 direction The nonspinning spindlersquos naturalfrequencies extracted from the FRF graphs are presented inTable 3 alongside the CDSM results As can be seen themaximum difference between the first three CDSM flexural

Shock and Vibration 7

Table 3 Spindle natural frequencies experimental versus CDSM for a nonspinning spindle system with flexible boundary conditions119870119904=

21 times 108 Nm

Nonspinning bending modes (Hz) CDSM (Ω cong 955RPM) Experimental (x-axisy-axis) Difference ()(x-axisy-axis)

1205961 100427100459 9298192651 80841205962 130237130268 130151131067 0070061205963 137287137319 160144158386 143133

0 5000 10000 15000 20000 25000 30000 35000 400000

200

400

600

800

1000

1200

1400

1600

1800

2000

Nat

ural

freq

uenc

y (H

z)

Spindle speed (RPM)

Spindle speed versus natural frequency

1st mode2nd mode3rd mode

4th mode5th mode6th mode

Figure 6 Spindle speed versus natural frequency

0

200

400

600

800

1000

1200

1400

0 10000 20000 30000 40000 50000 60000 70000 80000

Nat

ural

freq

uenc

y (H

z)

Spindle RPM

Spindle RPM versus natural frequency

Spring BCSimply supported

Spindle RPM operational limit

Figure 7 Spindle natural frequency versus spindle RPM simplysupported and flexible bearing models

natural frequencies and the experimentally evaluated dataare respectively 8 007 and 143 in 119909 direction and84 006 and 133 in 119910 directionThe average differenceis found to be around 74 which is within the acceptablerange

5 Discussion and Conclusion

It has been established that the service life changes the vibra-tional behavior of spindle systems that is reduced naturalfrequency over the time associated with the bearings wearTherefore in order to generate an updated spindlemodel it isessential to include the effect of the bearings in the systemAnanalytical model of a multisegment spinning spindle basedon the dynamic stiffnessmatrix (DSM) formulation and exactwithin the limits of the Euler-Bernoulli beam bending theorywas developed The beam exhibits coupled bending-bending(B-B) vibration and as expected its fundamental coupledfrequency was found to decrease with increasing spinningspeed The bearings were included in the model using twodifferent models rigid simply supported frictionless pinsand flexible linear spring elements The simply supportedbearing model was found to yield an excessively rigid systemwith amuch higher natural frequency than the nominal valueof the system Replacing the simply supported boundarycondition with linear spring elements the spring stiffness119870119904 was then calibrated so that the fundamental frequency

of the system matched the nominal value that is a morerealistic and accurate representation of the spindle systemThe accuracy of the proposed calibrated dynamic stiffnessmatrix (CDSM) method was confirmed through compari-son with both numerical and experimental results Whencompared to the conventional finite element method (FEM)the CDSM frequencies were found to best match with thoseobtained from a cubic beamfinite element (B33 inABAQUS)Moreover in comparison with experimental data the CDSMresults were found to be within acceptable accuracy the aver-age error for the first three bending natural frequencies is lessthan 8

It is worth noting that the proposed CDSM can alsobe fine-tuned to match spindlersquos experimental fundamentalfrequency instead of the nominal value Further researchis underway to improve the CDSM by using more accu-rate material properties for different segments of thespindle The developed calibrated model could then beexploited in an attempt to take into account the effects

8 Shock and Vibration

of bearing wear in terms of spindlersquos service time numer-ically evaluate the equivalent cutting force and calcu-late updated stability lobes used to achieve chatter-freecuts

Appendix

From the solutions of 119880 and 119881 (see expressions (13)) thecorresponding bending rotation about 119909-axis and 119910-axis Θ

119909

and Θ119910 are respectively given by

Θ119909 =

119889119881

119889119911= minus

1

119871

119889119881

119889120585

= minus1

119871(radic120572119861

1cosradic120572120585 minus radic120572119861

2sinradic120572120585

+ radic1205721198613coshradic120572120585 + radic120572119861

1sinradic120572120585

+ radic1205731198615cosradic120573120585 minus radic120573119861

6sinradic120573120585

+radic1205731198617coshradic120573120585 + radic120573119861

8sinhradic120573120585)

Θ119910=

119889119880

119889119911=

1

119871

119889119880

119889120585

= minus1

119871(radic120572119860

1cosradic120572120585 minus radic120572119860

2sinradic120572120585

+ radic1205721198603 coshradic120572120585 + radic1205721198601 sinradic120572120585

+ radic1205731198605 cosradic120573120585 minus radic1205731198606 sinradic120573120585

+radic1205731198607coshradic120573120585 + radic120573119860

8sinhradic120573120585)

(A1)

By doing similar substitutions we find (see expressions (3)(4) and (14))

119878119909 = (

EI119910119910

1198713)(

minus120572radic1205721198601cosradic120572120585 + 120572radic120572119860

2sinradic120572120585 + 120572radic120572119860

3coshradic120572120585 + 120572radic120572119860

4sinhradic120572120585

minus120573radic1205731198605cosradic120573120585 + 120573radic120573119860

6sinradic120573120585 + 120573radic120573119860

7coshradic120573120585 + 120573radic120573119860

8sinhradic120573120585

)

119878119910 = (

EI1199091199091198713

)(minus120572radic120572119861

1cosradic120572120585 + 120572radic120572119861

2sinradic120572120585 + 120572radic120572119861

3coshradic120572120585 + 120572radic120572119861

4sinhradic120572120585

minus120573radic1205731198615cosradic120573120585 + 120573radic120573119861

6sinradic120573120585 + 120573radic120573119861

7coshradic120573120585 + 120573radic120573119861

8sinhradic120573120585

)

119872119909= (

EI119910119910

1198712)(

minus1205721198611 sinradic120572120585 minus 1205721198612 cosradic120572120585 + 1205721198613 sinhradic120572120585 + 1205721198614 coshradic120572120585

minus1205731198615sinradic120573120585 + 120573119861

6cosradic120573120585 + 120573119861

7sinhradic120573120585 + 120573119861

8coshradic120573120585

)

119872119910= (

EI119909119909

1198712)(

minus1205721198601 sinradic120572120585 minus 120572119860

2 cosradic120572120585 + 1205721198603 sinhradic120572120585 + 1205721198604 coshradic120572120585

minus1205731198605 sinradic120573120585 + 1205731198606 cosradic120573120585 + 1205731198607 sinhradic120573120585 + 1205731198608 coshradic120573120585)

(A2)

where1198601= 1198961205721198611 119860

2= 1198961205721198612

1198603= 1198961205721198613 119860

4= 1198961205721198614

1198605 = 1198961205731198615 1198606 = 1198961205731198616

1198607 = 1198961205731198617 1198608 = 1198961205731198618

119896120572=

21205822

119910120578

1205722 minus 1205822119910(1 + 1205782)

119896120573=

21205822

119910120578

1205732 minus 1205822119910(1 + 1205782)

(A3)

To obtain the dynamic stiffness matrix we introduce theboundary conditions into the governing equations For dis-placements

at 119911 = 0

119880 = 1198801 119881 = 119881

1 Θ

119909= Θ1199091 Θ

119910= Θ1199101

(A4)

at 119911 = 119871

119880 = 1198802 119881 = 119881

2 Θ

119909= Θ1199092 Θ

119910= Θ1199102

(A5)

And for forces we have the following

at 119911 = 0

119878119909= 1198781199091 119878

119910= 1198781199101 119872

119909= 1198721199091 119872

119910= 1198721199101

(A6)

at 119911 = 119871

119878119909= 1198781199092 119878

119910= 1198781199102 119872

119909= 1198721199092 119872

119910= 1198721199102

(A7)

Substituting the boundary conditions into the governingequations we find

120575 = BR (A8)

Shock and Vibration 9

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

B =

[[[[[[[[[[[[

[

0 119896120572

0 119896120572

0 119896120573

0 119896120573

0 1 0 1 0 1 0 1

minus120591120572

0 minus120591120572

0 minus120591120573

0 minus120591120573

0

120594120572

0 120594120572

0 120594120573

0 120594120573

0

119896120572119878120572

119896120572119862120572

119896120572119878ℎ120572

119896120572119862ℎ120572

119896120573119878120573

119896120573119862120573

119896120573119878ℎ120573

119896120573119862ℎ120573

119878120572 119862120572 119878ℎ120572

119862ℎ120572

119878120573 119862120573 119878ℎ120573

119862ℎ120573

minus120591120572119862120572

120591120572119878120572

minus120591120572119862ℎ120572

minus120591120572119878ℎ120572

minus120591120573119862120573

120591120573119878120573

minus120591120573119862ℎ120573

minus120591120573119878ℎ120573

120594120572119862120572

minus120594120572119878120572

120594120572119862ℎ120572

120594120572119878ℎ120572

120594120573119862120573

minus120594120573119878120573

120594120573119862ℎ120573

120594120573119878ℎ120573

]]]]]]]]]]]]

]

120591120572=

radic120572

119871 120591

120573= minus

radic120573

119871 120594

120572= 119896120572120591120572 120594

120573= 119896120573120591120573

(A9)

Also

119878120572= sinradic120572 119862

120572= cosradic120572

119878ℎ120572

= sinhradic120572 119862ℎ120572

= coshradic120572

119878120573= sinradic120573 119862

120573= cosradic120573

119878ℎ120573

= sinhradic120573 119862ℎ120573

= coshradic120573

(A10)

Substituting similarly for the force equation

F = AR (A11)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

A =

[[[[[[[[[[[[

[

minus120577120572 0 120577120572 0 minus120577120573 0 120577120573 0

minus120576120572 0 120576120572 0 minus120576120573 0 120576120573 0

0 minus120574120572

0 120574120572

0 minus120574120573

0 120574120573

0 120582120572

0 minus120582120572

0 120582120573

0 minus120582120573

120577120572119862120572

minus120577120572119878120572

minus120577120572119862ℎ120572

minus120577120572119878ℎ120572

120577120573119862120573

minus120577120573119878120573

minus120577120573119862ℎ120573

minus120577120573119878ℎ120573

120576120572119862120572

minus120576120572119878120572

minus120576120572119862ℎ120572

minus120576120572119878ℎ120572

120576120573119862120573

minus120576120573119878120573

minus120576120573119862ℎ120573

minus120576120573119878ℎ120573

120574120572119878120572

120574120572119862120572

minus120574120572119878ℎ120572

minus120574120572119862ℎ120572

120574120573119878120573

120574120573119862120573

minus120574120573119878ℎ120573

minus120574120573119862ℎ120573

minus120582120572119878120572 minus120582120572119862120572 120582120572119878ℎ120572

120582120572119862ℎ120572

minus120582120573119878120573 minus120582120573119862120573 120582120573119878ℎ120573

120582120573119862ℎ120573

]]]]]]]]]]]]

]

120577120572 = 119896120572120572radic120572

EI119910119910

1198713 120576120572 = 120572radic120572

EI1199091199091198713

120574120572= 120572

EI119909119909

1198712 120582

120572= 119896120572120572

EI119910119910

1198712

120577120573= 119896120573120573radic120573

EI119910119910

1198713 120576

120573= 120573radic120573

EI119909119909

1198713

120574120573 = 120573

EI1199091199091198712

120582120573 = 119896120573120573

EI119910119910

1198712

(A12)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the supports providedby Natural Sciences and Engineering Research Council ofCanada (NSERC) and Ryerson University

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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International Journal of

4 Shock and Vibration

U1

V1

U2

V2

Θx1

Θx2

Θy1

Θy2

Φ1

Φ2

Figure 3 Degrees of freedom of the system

Using the Hamilton principle in the usual notation state

120575int

1199052

1199051

(119879 minus 119880) 119889119905 = 0 (4)

where 1199051and 119905

2are the time intervals in the dynamic

trajectory and 120575 is the variational operator Substituting thekinetic and potential energies of the beam in the Hamiltonprinciple collecting like terms and integrating by parts thefollowing differential equations are obtained

EI1199101199101199061015840101584010158401015840

minus 119898Ω2119906 + 119898 minus 2119898ΩV = 0

minus2119898Ω minus EI119909119909V1015840101584010158401015840

minus 119898V + 119898Ω2V = 0

(5)

which govern the bending-bending (B-B) vibration of abeam coupled by the spinning speedThe resulting loads alsoforce boundary conditions at free ends are then found to bein the following forms written for shear forces as

119878119909= EI119909119909119906101584010158401015840 119878

119910= EI119910119910V101584010158401015840 (6)

and for bending moments as

119872119909= EI119909119909V10158401015840 119872

119910= minusEI

11991011991011990610158401015840 (7)

Assuming simple harmonic motion

119906 (119911 119905) = 119880 (119911) cos120596119905 V (119911 119905) = 119880 (119911) cos120596119905 (8)

and substituting (8) into (5) they can be rewritten as

(EI1199101199101198801015840101584010158401015840

minus 119898(Ω2+ 1205962)119880) + 2119898Ω120596119881 = 0

(EI1199091199091198811015840101584010158401015840

minus 119898(Ω2+ 1205962)119881) + 2119898Ω120596119880 = 0

(9)

where 120596 is frequency of oscillation and 119880 119881 and Φ are theamplitudes of 119906 V and 120601 Introducing 120585 = 119911119871 and 119863 =

119889119889120585 which are nondimensional length and the differentialoperator and back substitution into (9) lead to

[1198634minus

119898(Ω2+ 1205962) 1198714

EI119910119910

]119880 +2119898Ω120596119871

4

EI119910119910

119881 = 0

[1198634minus

119898(Ω2+ 1205962) 1198714

EI119909119909]119881 +

2119898Ω1205961198714

EI119909119909119880 = 0

(10)

3 Solution of the Governing Equations

31 Dynamic Stiffness Matrix (DSM) Solution As suggestedby Banerjee and Su [31 32] the above equation (10) can becombined into one 8th-order differential equation written as

[1198638minus (1205822

119909+ 1205822

119910) (1 + 120578

2)1198634+ 1205822

1199091205822

119910(1 minus 120578

2)2

]119882 = 0

(11)

where ldquo119882rdquo represents both lateral displacements119880 and119881 and

1205822

119909=

11989812059621198714

EI119909119909 120582

2

119910=

11989812059621198714

EI119910119910 120578

2=

Ω2

1205962 (12)

The general solution of the differential equation is sought inthe form

119882 = 119890119903120585 (13)

Substituting (13) into (11) leads to

1199038minus (1205822

119909+ 1205822

119910) (1 + 120578

2) 1199034+ 1205822

1199091205822

119910(1 minus 120578

2)2

(14)

where 11990313 = plusmnradic120572 11990324 = plusmnradic120573 11990357 = plusmn119894radic120572 11990368 = plusmn119894radic120573 and

1205722=

1

2 (1205822

119909+ 1205822

119910) (1 + 120578

2)

+radic(1205822119909minus 1205822119910)2

(1 + 1205782)2+ 161205822

11990912058221199101205782

1205732=

1

2 (1205822

119909+ 1205822

119910) (1 + 120578

2)

minusradic(1205822119909minus 1205822119910)2

(1 + 1205782)2+ 161205822

11990912058221199101205782

(15)

Based on Euler-Bernoulli beam bending theory thecorresponding bending rotation that is slope about 119909-axisand 119910-axis Θ

119909and Θ

119910 respectively is given by (see the

appendix for further details)

Θ119909=

119889119881

119889119911= minus

1

119871

119889119881

119889120585 Θ

119910=

119889119880

119889119911=

1

119871

119889119880

119889120585 (16)

By doing similar substitutions in load (3) and (4) one finds

119878119909= (

EI119910119910

1198713)(

1198893119880

1198891205853) 119878

119910= (

EI119909119909

1198713)(

1198893119881

1198891205853)

119872119909 = (

EI119910119910

1198712)(

1198892119880

1198891205852) 119872119910 = (

EI1199091199091198712

)(1198892119881

1198891205852)

(17)

Substituting the boundary conditions into the governingequations enforced at 119911 = 0 (120585 = 0) and 119911 = 119871 (120585 = 1)leads to

120575 = BR (18)

Shock and Vibration 5

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

(19)

Substituting similarly for the force equation one finds

F = AR (20)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

(21)

The frequency-dependent dynamic stiffness matrix(DSM) of the spinning beam K(120596) can then be derivedby eliminating R We also know that the force amplitude isrelated to the displacement vector by

F = K (120596) 120575 where K (120596) = ABminus1 (22)

Explicitmatrix forms ofAB andR are given in the appendix

32 Classical Finite Element Method (FEM) For validationand comparison purposes a conventional finite elementformulation of the problem and aMATLAB-based FEM codewere developed to carry out the free vibration analysis of thespindle system [33] The Galerkin method of weighted resid-uals [29 33] was employed to develop the integral form ofthe governing equation (10) Performing integration by partson the resulting integral equations leads to the weak integralform of equations which also satisfy the principle of virtualwork The system is then discretized using beam elementswith four DOF per node (ie two lateral displacements 119880and119881 and two slopes1198801015840 and119881

1015840 eight degrees of freedom intotal) Introducing cubic Hermite-type polynomial approx-imations [29 33] for bending displacements in the elementweak integral form the classical finite element formulationis developed This formulation results in constant (or static)element stiffness mass and coupling matrices which arethen assembled to find the global matrices Finally enforcingthe boundary conditions leads to a linear eigenvalue problemwhich is solved to extract the system natural frequencies andmodes There has also been an attempt to develop a dynamicfinite element (DFE) [34] formulation for the problem inhand using frequency-dependent interpolation functionsand the preliminary results were found to be promising [35]

4 Application of the Theory andNumerical Results

TheDSM formulation was first validated against various ana-lytical and numerical frequency data available in the literature[31 32] for various uniform simple beam configurations TheDSM frequency data were also comparedwith those obtainedfrom commercial FEM-based software ABAQUS and theeffect of various element types on the convergence was alsoexamined In addition the conventional FEM formulation[29 33] was used for further comparisons however as the

in-house FEM-based code and commercial software both ledto similar results the FEM results have been omitted here forthe brevity

A typical spindle system model (see Figure 4) was thenconstructed using multiple segments As the nominal fun-damental frequency was available for nonspinning spindletherefore the effect of spinning speed was taken out by simplysetting Ω to zero in FEM models However one should notethat setting the spinning speed exactly to zero would causethe DSM formulation to collapse This is due to the factthat the DSM formulation pivots around combining the twoexpressions (10) into a single on (11) through the couplingterms involving Ω [31 32] However an extremely smallnumber (Ω cong 0) can be used to model the stationary spindle

The entire system was assumed to be made from thesame material that is tooling steel Initially it was alsoassumed that the system is simply supported at the bearinglocations that is the bearings modeled as frictionless pinsrigidly attached to the spindle casing with zero flexibilityallowing only for rotations in all directions This system wasmodeled using the DSM method described above as wellas finite element analysis (FEA) in ABAQUS commercialsoftware to evaluate the first six natural frequencies of thespindle system Several different element types andmesh sizeswere examined to ensure the convergence (Table 1) beamelement B33 uses a 2-node cubic beam (similar to the in-house FEM code) C3D20R is a general-purpose quadraticbrick element and C3D10M is a general-purpose tetrahedralelement (see Table 1 for results) Unlike the conventionalFEM formulation the DSM method results in exact solutionfor all natural frequencies of a spinning uniform beam [3132] Therefore as the spindle system in hand is made of12 uniform sections only a 12-element piecewise-uniform(stepped) DSM model was required As expected the resultsusing beam element type B33 yielded the best agreement withthe DSM results as the 3D and shear effects are neglectedin both However the fundamental frequency values werefound to be different from the nominal value that is 1393Hzcompared to 1000 plusmn 50Hz as provided by the manufacturer

In order to address this issue the DSM and FEM modelswere calibrated to take into account the flexibility of thebearing systems To this end the simply supported boundaryconditions were modified and replaced by simple linearspring elements attached to the spindle shaft in 119909 and119910 directions representing general flexibility of the bearingsystems (see Figure 4) The bearings were all assumed to beidentical with the same spring coefficient 119870119904 The springstiffness value 119870

119904 was then varied and a calibrated model

equippedwith spring elements of constant119870119904= 21times10

8Nmwas found to result in a fundamental bending frequencyof 1004Hz closely equivalent to the systemrsquos nominal value(Figure 5) The effect of spinning speed Ω on the systemrsquosfundamental frequency was also investigated In the absenceof spinning speed Ω = 0 the system exhibits uncoupledbehavior (see (7) and (8)) undergoing two separate bendingdisplacements in 119909 and 119910 directions As a result for a circularshaft with 119868119909119909 = 119868119910119910 the modal analysis results in repeatedfrequency values appearing in pairs (Table 1) By introducingthe spinning speedΩ = 0 the bending natural frequencies in

6 Shock and Vibration

Table 1 Spindle natural frequencies DSM versus FEA for a nonspinning (Ω cong 955RPM) with simply supported boundary conditions

Mode DSM FEA B33 element (Hz) FEA C3D20R (Hz) FEA C3D10M (Hz)12 sections 600 elements 1200 elements 16965 elements 20122 elements 34465 elements 45645 elements

1 139292 139300 139300 130800 130800 130100 1301002 139324 139300 139300 130800 130800 130100 1301003 407102 407100 407100 320800 321200 333500 3322004 407134 407100 407100 321000 321200 333500 3324005 566814 566900 566900 433900 431100 432200 4291006 566846 566900 566900 434100 431600 432300 429200

Table 2 Spindle natural frequencies versus various spinning speeds for flexible boundary conditions 119870119904= 21 times 10

8Nm

Mode955

(RPM)(Ω cong 0)

47842(RPM)

95589(RPM)

14334(RPM)

19108(RPM)

23883(RPM)

28657(RPM)

33432(RPM)

1 100427 92469 84511 76554 68596 60638 52680 447232 100459 108416 114321 106363 98406 90448 82490 745323 130237 122279 1163741 113414 105456 97498 89541 815834 130268 129329 1213716 124332 132290 140247 146805 1388475 137287 138226 1461838 154142 154142 154762 148205 1561636 137319 145277 1532344 161192 162720 170057 178015 185973

Toolholder Motor Spindle shaftSpring Spring

Spindle casing

Figure 4 Modified boundary conditions

119909 and 119910 directions become coupled changing the repeatednatural frequencies to two distinct ones (see Table 2) withone decreasing and approaching zero (Figure 6) and the otherincreasing from their uncoupled values Further increasein the spinning speed would make one of the bendingfrequencies go to zero that is whirling instability alsoreported and discussed in [31 32] The first critical spindlespeed in the case of calibrated model was found to be Ωcr =6 times 10

5 RPM which is well above the operating speed of thespindle that is 35 times 104 RPM (see Figure 7)

In order to further validate the proposed CDSM modelthe tap testing method was used to experimentally collectthe frequency response function (FRF) of the spindle sys-tem in hand while being installed on its correspondingmachine tool A 1-inch diameter blank tool with a 2-inchprotrusion in a typical shrink fit tool holder was usedThe tool and holder were inserted into the spindle Thespindle was set to be horizontal (ie neutral) position andan accelerometer (352A21 08 g) was then attached to theedge of the tool holder The spindle was struck 10 timesin the 119909 direction using a hammer (086C04 5000N) and

0

200

400

600

800

1000

1200

1400

Syste

m n

atur

al fr

eque

ncy

(Hz)

Bearing equivalent spring constant K (Nm)

DSMSimply supportedANSYS

Spindle manufacturerrsquos designed natural frequency (1000Hz)

100E+00

100E+01

100E+02

100E+03

100E+04

100E+05

100E+06

100E+07

100E+08

100E+09

100E+10

Tool Simplesupports

Simplesupports

Motor Spindleshaftholder

MotorSpring SpringSpindle casing

ToolSpindle

shaftholder

Figure 5 Varying spring constant versus system natural frequencyusing DSM and FEA

the average FRF graph was generated (SIM3 Module Photon+ Data Acquisition) Similarly the spindle was also testedalong the 119910 direction The nonspinning spindlersquos naturalfrequencies extracted from the FRF graphs are presented inTable 3 alongside the CDSM results As can be seen themaximum difference between the first three CDSM flexural

Shock and Vibration 7

Table 3 Spindle natural frequencies experimental versus CDSM for a nonspinning spindle system with flexible boundary conditions119870119904=

21 times 108 Nm

Nonspinning bending modes (Hz) CDSM (Ω cong 955RPM) Experimental (x-axisy-axis) Difference ()(x-axisy-axis)

1205961 100427100459 9298192651 80841205962 130237130268 130151131067 0070061205963 137287137319 160144158386 143133

0 5000 10000 15000 20000 25000 30000 35000 400000

200

400

600

800

1000

1200

1400

1600

1800

2000

Nat

ural

freq

uenc

y (H

z)

Spindle speed (RPM)

Spindle speed versus natural frequency

1st mode2nd mode3rd mode

4th mode5th mode6th mode

Figure 6 Spindle speed versus natural frequency

0

200

400

600

800

1000

1200

1400

0 10000 20000 30000 40000 50000 60000 70000 80000

Nat

ural

freq

uenc

y (H

z)

Spindle RPM

Spindle RPM versus natural frequency

Spring BCSimply supported

Spindle RPM operational limit

Figure 7 Spindle natural frequency versus spindle RPM simplysupported and flexible bearing models

natural frequencies and the experimentally evaluated dataare respectively 8 007 and 143 in 119909 direction and84 006 and 133 in 119910 directionThe average differenceis found to be around 74 which is within the acceptablerange

5 Discussion and Conclusion

It has been established that the service life changes the vibra-tional behavior of spindle systems that is reduced naturalfrequency over the time associated with the bearings wearTherefore in order to generate an updated spindlemodel it isessential to include the effect of the bearings in the systemAnanalytical model of a multisegment spinning spindle basedon the dynamic stiffnessmatrix (DSM) formulation and exactwithin the limits of the Euler-Bernoulli beam bending theorywas developed The beam exhibits coupled bending-bending(B-B) vibration and as expected its fundamental coupledfrequency was found to decrease with increasing spinningspeed The bearings were included in the model using twodifferent models rigid simply supported frictionless pinsand flexible linear spring elements The simply supportedbearing model was found to yield an excessively rigid systemwith amuch higher natural frequency than the nominal valueof the system Replacing the simply supported boundarycondition with linear spring elements the spring stiffness119870119904 was then calibrated so that the fundamental frequency

of the system matched the nominal value that is a morerealistic and accurate representation of the spindle systemThe accuracy of the proposed calibrated dynamic stiffnessmatrix (CDSM) method was confirmed through compari-son with both numerical and experimental results Whencompared to the conventional finite element method (FEM)the CDSM frequencies were found to best match with thoseobtained from a cubic beamfinite element (B33 inABAQUS)Moreover in comparison with experimental data the CDSMresults were found to be within acceptable accuracy the aver-age error for the first three bending natural frequencies is lessthan 8

It is worth noting that the proposed CDSM can alsobe fine-tuned to match spindlersquos experimental fundamentalfrequency instead of the nominal value Further researchis underway to improve the CDSM by using more accu-rate material properties for different segments of thespindle The developed calibrated model could then beexploited in an attempt to take into account the effects

8 Shock and Vibration

of bearing wear in terms of spindlersquos service time numer-ically evaluate the equivalent cutting force and calcu-late updated stability lobes used to achieve chatter-freecuts

Appendix

From the solutions of 119880 and 119881 (see expressions (13)) thecorresponding bending rotation about 119909-axis and 119910-axis Θ

119909

and Θ119910 are respectively given by

Θ119909 =

119889119881

119889119911= minus

1

119871

119889119881

119889120585

= minus1

119871(radic120572119861

1cosradic120572120585 minus radic120572119861

2sinradic120572120585

+ radic1205721198613coshradic120572120585 + radic120572119861

1sinradic120572120585

+ radic1205731198615cosradic120573120585 minus radic120573119861

6sinradic120573120585

+radic1205731198617coshradic120573120585 + radic120573119861

8sinhradic120573120585)

Θ119910=

119889119880

119889119911=

1

119871

119889119880

119889120585

= minus1

119871(radic120572119860

1cosradic120572120585 minus radic120572119860

2sinradic120572120585

+ radic1205721198603 coshradic120572120585 + radic1205721198601 sinradic120572120585

+ radic1205731198605 cosradic120573120585 minus radic1205731198606 sinradic120573120585

+radic1205731198607coshradic120573120585 + radic120573119860

8sinhradic120573120585)

(A1)

By doing similar substitutions we find (see expressions (3)(4) and (14))

119878119909 = (

EI119910119910

1198713)(

minus120572radic1205721198601cosradic120572120585 + 120572radic120572119860

2sinradic120572120585 + 120572radic120572119860

3coshradic120572120585 + 120572radic120572119860

4sinhradic120572120585

minus120573radic1205731198605cosradic120573120585 + 120573radic120573119860

6sinradic120573120585 + 120573radic120573119860

7coshradic120573120585 + 120573radic120573119860

8sinhradic120573120585

)

119878119910 = (

EI1199091199091198713

)(minus120572radic120572119861

1cosradic120572120585 + 120572radic120572119861

2sinradic120572120585 + 120572radic120572119861

3coshradic120572120585 + 120572radic120572119861

4sinhradic120572120585

minus120573radic1205731198615cosradic120573120585 + 120573radic120573119861

6sinradic120573120585 + 120573radic120573119861

7coshradic120573120585 + 120573radic120573119861

8sinhradic120573120585

)

119872119909= (

EI119910119910

1198712)(

minus1205721198611 sinradic120572120585 minus 1205721198612 cosradic120572120585 + 1205721198613 sinhradic120572120585 + 1205721198614 coshradic120572120585

minus1205731198615sinradic120573120585 + 120573119861

6cosradic120573120585 + 120573119861

7sinhradic120573120585 + 120573119861

8coshradic120573120585

)

119872119910= (

EI119909119909

1198712)(

minus1205721198601 sinradic120572120585 minus 120572119860

2 cosradic120572120585 + 1205721198603 sinhradic120572120585 + 1205721198604 coshradic120572120585

minus1205731198605 sinradic120573120585 + 1205731198606 cosradic120573120585 + 1205731198607 sinhradic120573120585 + 1205731198608 coshradic120573120585)

(A2)

where1198601= 1198961205721198611 119860

2= 1198961205721198612

1198603= 1198961205721198613 119860

4= 1198961205721198614

1198605 = 1198961205731198615 1198606 = 1198961205731198616

1198607 = 1198961205731198617 1198608 = 1198961205731198618

119896120572=

21205822

119910120578

1205722 minus 1205822119910(1 + 1205782)

119896120573=

21205822

119910120578

1205732 minus 1205822119910(1 + 1205782)

(A3)

To obtain the dynamic stiffness matrix we introduce theboundary conditions into the governing equations For dis-placements

at 119911 = 0

119880 = 1198801 119881 = 119881

1 Θ

119909= Θ1199091 Θ

119910= Θ1199101

(A4)

at 119911 = 119871

119880 = 1198802 119881 = 119881

2 Θ

119909= Θ1199092 Θ

119910= Θ1199102

(A5)

And for forces we have the following

at 119911 = 0

119878119909= 1198781199091 119878

119910= 1198781199101 119872

119909= 1198721199091 119872

119910= 1198721199101

(A6)

at 119911 = 119871

119878119909= 1198781199092 119878

119910= 1198781199102 119872

119909= 1198721199092 119872

119910= 1198721199102

(A7)

Substituting the boundary conditions into the governingequations we find

120575 = BR (A8)

Shock and Vibration 9

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

B =

[[[[[[[[[[[[

[

0 119896120572

0 119896120572

0 119896120573

0 119896120573

0 1 0 1 0 1 0 1

minus120591120572

0 minus120591120572

0 minus120591120573

0 minus120591120573

0

120594120572

0 120594120572

0 120594120573

0 120594120573

0

119896120572119878120572

119896120572119862120572

119896120572119878ℎ120572

119896120572119862ℎ120572

119896120573119878120573

119896120573119862120573

119896120573119878ℎ120573

119896120573119862ℎ120573

119878120572 119862120572 119878ℎ120572

119862ℎ120572

119878120573 119862120573 119878ℎ120573

119862ℎ120573

minus120591120572119862120572

120591120572119878120572

minus120591120572119862ℎ120572

minus120591120572119878ℎ120572

minus120591120573119862120573

120591120573119878120573

minus120591120573119862ℎ120573

minus120591120573119878ℎ120573

120594120572119862120572

minus120594120572119878120572

120594120572119862ℎ120572

120594120572119878ℎ120572

120594120573119862120573

minus120594120573119878120573

120594120573119862ℎ120573

120594120573119878ℎ120573

]]]]]]]]]]]]

]

120591120572=

radic120572

119871 120591

120573= minus

radic120573

119871 120594

120572= 119896120572120591120572 120594

120573= 119896120573120591120573

(A9)

Also

119878120572= sinradic120572 119862

120572= cosradic120572

119878ℎ120572

= sinhradic120572 119862ℎ120572

= coshradic120572

119878120573= sinradic120573 119862

120573= cosradic120573

119878ℎ120573

= sinhradic120573 119862ℎ120573

= coshradic120573

(A10)

Substituting similarly for the force equation

F = AR (A11)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

A =

[[[[[[[[[[[[

[

minus120577120572 0 120577120572 0 minus120577120573 0 120577120573 0

minus120576120572 0 120576120572 0 minus120576120573 0 120576120573 0

0 minus120574120572

0 120574120572

0 minus120574120573

0 120574120573

0 120582120572

0 minus120582120572

0 120582120573

0 minus120582120573

120577120572119862120572

minus120577120572119878120572

minus120577120572119862ℎ120572

minus120577120572119878ℎ120572

120577120573119862120573

minus120577120573119878120573

minus120577120573119862ℎ120573

minus120577120573119878ℎ120573

120576120572119862120572

minus120576120572119878120572

minus120576120572119862ℎ120572

minus120576120572119878ℎ120572

120576120573119862120573

minus120576120573119878120573

minus120576120573119862ℎ120573

minus120576120573119878ℎ120573

120574120572119878120572

120574120572119862120572

minus120574120572119878ℎ120572

minus120574120572119862ℎ120572

120574120573119878120573

120574120573119862120573

minus120574120573119878ℎ120573

minus120574120573119862ℎ120573

minus120582120572119878120572 minus120582120572119862120572 120582120572119878ℎ120572

120582120572119862ℎ120572

minus120582120573119878120573 minus120582120573119862120573 120582120573119878ℎ120573

120582120573119862ℎ120573

]]]]]]]]]]]]

]

120577120572 = 119896120572120572radic120572

EI119910119910

1198713 120576120572 = 120572radic120572

EI1199091199091198713

120574120572= 120572

EI119909119909

1198712 120582

120572= 119896120572120572

EI119910119910

1198712

120577120573= 119896120573120573radic120573

EI119910119910

1198713 120576

120573= 120573radic120573

EI119909119909

1198713

120574120573 = 120573

EI1199091199091198712

120582120573 = 119896120573120573

EI119910119910

1198712

(A12)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the supports providedby Natural Sciences and Engineering Research Council ofCanada (NSERC) and Ryerson University

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

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Shock and Vibration

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International Journal of

Shock and Vibration 5

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

(19)

Substituting similarly for the force equation one finds

F = AR (20)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

(21)

The frequency-dependent dynamic stiffness matrix(DSM) of the spinning beam K(120596) can then be derivedby eliminating R We also know that the force amplitude isrelated to the displacement vector by

F = K (120596) 120575 where K (120596) = ABminus1 (22)

Explicitmatrix forms ofAB andR are given in the appendix

32 Classical Finite Element Method (FEM) For validationand comparison purposes a conventional finite elementformulation of the problem and aMATLAB-based FEM codewere developed to carry out the free vibration analysis of thespindle system [33] The Galerkin method of weighted resid-uals [29 33] was employed to develop the integral form ofthe governing equation (10) Performing integration by partson the resulting integral equations leads to the weak integralform of equations which also satisfy the principle of virtualwork The system is then discretized using beam elementswith four DOF per node (ie two lateral displacements 119880and119881 and two slopes1198801015840 and119881

1015840 eight degrees of freedom intotal) Introducing cubic Hermite-type polynomial approx-imations [29 33] for bending displacements in the elementweak integral form the classical finite element formulationis developed This formulation results in constant (or static)element stiffness mass and coupling matrices which arethen assembled to find the global matrices Finally enforcingthe boundary conditions leads to a linear eigenvalue problemwhich is solved to extract the system natural frequencies andmodes There has also been an attempt to develop a dynamicfinite element (DFE) [34] formulation for the problem inhand using frequency-dependent interpolation functionsand the preliminary results were found to be promising [35]

4 Application of the Theory andNumerical Results

TheDSM formulation was first validated against various ana-lytical and numerical frequency data available in the literature[31 32] for various uniform simple beam configurations TheDSM frequency data were also comparedwith those obtainedfrom commercial FEM-based software ABAQUS and theeffect of various element types on the convergence was alsoexamined In addition the conventional FEM formulation[29 33] was used for further comparisons however as the

in-house FEM-based code and commercial software both ledto similar results the FEM results have been omitted here forthe brevity

A typical spindle system model (see Figure 4) was thenconstructed using multiple segments As the nominal fun-damental frequency was available for nonspinning spindletherefore the effect of spinning speed was taken out by simplysetting Ω to zero in FEM models However one should notethat setting the spinning speed exactly to zero would causethe DSM formulation to collapse This is due to the factthat the DSM formulation pivots around combining the twoexpressions (10) into a single on (11) through the couplingterms involving Ω [31 32] However an extremely smallnumber (Ω cong 0) can be used to model the stationary spindle

The entire system was assumed to be made from thesame material that is tooling steel Initially it was alsoassumed that the system is simply supported at the bearinglocations that is the bearings modeled as frictionless pinsrigidly attached to the spindle casing with zero flexibilityallowing only for rotations in all directions This system wasmodeled using the DSM method described above as wellas finite element analysis (FEA) in ABAQUS commercialsoftware to evaluate the first six natural frequencies of thespindle system Several different element types andmesh sizeswere examined to ensure the convergence (Table 1) beamelement B33 uses a 2-node cubic beam (similar to the in-house FEM code) C3D20R is a general-purpose quadraticbrick element and C3D10M is a general-purpose tetrahedralelement (see Table 1 for results) Unlike the conventionalFEM formulation the DSM method results in exact solutionfor all natural frequencies of a spinning uniform beam [3132] Therefore as the spindle system in hand is made of12 uniform sections only a 12-element piecewise-uniform(stepped) DSM model was required As expected the resultsusing beam element type B33 yielded the best agreement withthe DSM results as the 3D and shear effects are neglectedin both However the fundamental frequency values werefound to be different from the nominal value that is 1393Hzcompared to 1000 plusmn 50Hz as provided by the manufacturer

In order to address this issue the DSM and FEM modelswere calibrated to take into account the flexibility of thebearing systems To this end the simply supported boundaryconditions were modified and replaced by simple linearspring elements attached to the spindle shaft in 119909 and119910 directions representing general flexibility of the bearingsystems (see Figure 4) The bearings were all assumed to beidentical with the same spring coefficient 119870119904 The springstiffness value 119870

119904 was then varied and a calibrated model

equippedwith spring elements of constant119870119904= 21times10

8Nmwas found to result in a fundamental bending frequencyof 1004Hz closely equivalent to the systemrsquos nominal value(Figure 5) The effect of spinning speed Ω on the systemrsquosfundamental frequency was also investigated In the absenceof spinning speed Ω = 0 the system exhibits uncoupledbehavior (see (7) and (8)) undergoing two separate bendingdisplacements in 119909 and 119910 directions As a result for a circularshaft with 119868119909119909 = 119868119910119910 the modal analysis results in repeatedfrequency values appearing in pairs (Table 1) By introducingthe spinning speedΩ = 0 the bending natural frequencies in

6 Shock and Vibration

Table 1 Spindle natural frequencies DSM versus FEA for a nonspinning (Ω cong 955RPM) with simply supported boundary conditions

Mode DSM FEA B33 element (Hz) FEA C3D20R (Hz) FEA C3D10M (Hz)12 sections 600 elements 1200 elements 16965 elements 20122 elements 34465 elements 45645 elements

1 139292 139300 139300 130800 130800 130100 1301002 139324 139300 139300 130800 130800 130100 1301003 407102 407100 407100 320800 321200 333500 3322004 407134 407100 407100 321000 321200 333500 3324005 566814 566900 566900 433900 431100 432200 4291006 566846 566900 566900 434100 431600 432300 429200

Table 2 Spindle natural frequencies versus various spinning speeds for flexible boundary conditions 119870119904= 21 times 10

8Nm

Mode955

(RPM)(Ω cong 0)

47842(RPM)

95589(RPM)

14334(RPM)

19108(RPM)

23883(RPM)

28657(RPM)

33432(RPM)

1 100427 92469 84511 76554 68596 60638 52680 447232 100459 108416 114321 106363 98406 90448 82490 745323 130237 122279 1163741 113414 105456 97498 89541 815834 130268 129329 1213716 124332 132290 140247 146805 1388475 137287 138226 1461838 154142 154142 154762 148205 1561636 137319 145277 1532344 161192 162720 170057 178015 185973

Toolholder Motor Spindle shaftSpring Spring

Spindle casing

Figure 4 Modified boundary conditions

119909 and 119910 directions become coupled changing the repeatednatural frequencies to two distinct ones (see Table 2) withone decreasing and approaching zero (Figure 6) and the otherincreasing from their uncoupled values Further increasein the spinning speed would make one of the bendingfrequencies go to zero that is whirling instability alsoreported and discussed in [31 32] The first critical spindlespeed in the case of calibrated model was found to be Ωcr =6 times 10

5 RPM which is well above the operating speed of thespindle that is 35 times 104 RPM (see Figure 7)

In order to further validate the proposed CDSM modelthe tap testing method was used to experimentally collectthe frequency response function (FRF) of the spindle sys-tem in hand while being installed on its correspondingmachine tool A 1-inch diameter blank tool with a 2-inchprotrusion in a typical shrink fit tool holder was usedThe tool and holder were inserted into the spindle Thespindle was set to be horizontal (ie neutral) position andan accelerometer (352A21 08 g) was then attached to theedge of the tool holder The spindle was struck 10 timesin the 119909 direction using a hammer (086C04 5000N) and

0

200

400

600

800

1000

1200

1400

Syste

m n

atur

al fr

eque

ncy

(Hz)

Bearing equivalent spring constant K (Nm)

DSMSimply supportedANSYS

Spindle manufacturerrsquos designed natural frequency (1000Hz)

100E+00

100E+01

100E+02

100E+03

100E+04

100E+05

100E+06

100E+07

100E+08

100E+09

100E+10

Tool Simplesupports

Simplesupports

Motor Spindleshaftholder

MotorSpring SpringSpindle casing

ToolSpindle

shaftholder

Figure 5 Varying spring constant versus system natural frequencyusing DSM and FEA

the average FRF graph was generated (SIM3 Module Photon+ Data Acquisition) Similarly the spindle was also testedalong the 119910 direction The nonspinning spindlersquos naturalfrequencies extracted from the FRF graphs are presented inTable 3 alongside the CDSM results As can be seen themaximum difference between the first three CDSM flexural

Shock and Vibration 7

Table 3 Spindle natural frequencies experimental versus CDSM for a nonspinning spindle system with flexible boundary conditions119870119904=

21 times 108 Nm

Nonspinning bending modes (Hz) CDSM (Ω cong 955RPM) Experimental (x-axisy-axis) Difference ()(x-axisy-axis)

1205961 100427100459 9298192651 80841205962 130237130268 130151131067 0070061205963 137287137319 160144158386 143133

0 5000 10000 15000 20000 25000 30000 35000 400000

200

400

600

800

1000

1200

1400

1600

1800

2000

Nat

ural

freq

uenc

y (H

z)

Spindle speed (RPM)

Spindle speed versus natural frequency

1st mode2nd mode3rd mode

4th mode5th mode6th mode

Figure 6 Spindle speed versus natural frequency

0

200

400

600

800

1000

1200

1400

0 10000 20000 30000 40000 50000 60000 70000 80000

Nat

ural

freq

uenc

y (H

z)

Spindle RPM

Spindle RPM versus natural frequency

Spring BCSimply supported

Spindle RPM operational limit

Figure 7 Spindle natural frequency versus spindle RPM simplysupported and flexible bearing models

natural frequencies and the experimentally evaluated dataare respectively 8 007 and 143 in 119909 direction and84 006 and 133 in 119910 directionThe average differenceis found to be around 74 which is within the acceptablerange

5 Discussion and Conclusion

It has been established that the service life changes the vibra-tional behavior of spindle systems that is reduced naturalfrequency over the time associated with the bearings wearTherefore in order to generate an updated spindlemodel it isessential to include the effect of the bearings in the systemAnanalytical model of a multisegment spinning spindle basedon the dynamic stiffnessmatrix (DSM) formulation and exactwithin the limits of the Euler-Bernoulli beam bending theorywas developed The beam exhibits coupled bending-bending(B-B) vibration and as expected its fundamental coupledfrequency was found to decrease with increasing spinningspeed The bearings were included in the model using twodifferent models rigid simply supported frictionless pinsand flexible linear spring elements The simply supportedbearing model was found to yield an excessively rigid systemwith amuch higher natural frequency than the nominal valueof the system Replacing the simply supported boundarycondition with linear spring elements the spring stiffness119870119904 was then calibrated so that the fundamental frequency

of the system matched the nominal value that is a morerealistic and accurate representation of the spindle systemThe accuracy of the proposed calibrated dynamic stiffnessmatrix (CDSM) method was confirmed through compari-son with both numerical and experimental results Whencompared to the conventional finite element method (FEM)the CDSM frequencies were found to best match with thoseobtained from a cubic beamfinite element (B33 inABAQUS)Moreover in comparison with experimental data the CDSMresults were found to be within acceptable accuracy the aver-age error for the first three bending natural frequencies is lessthan 8

It is worth noting that the proposed CDSM can alsobe fine-tuned to match spindlersquos experimental fundamentalfrequency instead of the nominal value Further researchis underway to improve the CDSM by using more accu-rate material properties for different segments of thespindle The developed calibrated model could then beexploited in an attempt to take into account the effects

8 Shock and Vibration

of bearing wear in terms of spindlersquos service time numer-ically evaluate the equivalent cutting force and calcu-late updated stability lobes used to achieve chatter-freecuts

Appendix

From the solutions of 119880 and 119881 (see expressions (13)) thecorresponding bending rotation about 119909-axis and 119910-axis Θ

119909

and Θ119910 are respectively given by

Θ119909 =

119889119881

119889119911= minus

1

119871

119889119881

119889120585

= minus1

119871(radic120572119861

1cosradic120572120585 minus radic120572119861

2sinradic120572120585

+ radic1205721198613coshradic120572120585 + radic120572119861

1sinradic120572120585

+ radic1205731198615cosradic120573120585 minus radic120573119861

6sinradic120573120585

+radic1205731198617coshradic120573120585 + radic120573119861

8sinhradic120573120585)

Θ119910=

119889119880

119889119911=

1

119871

119889119880

119889120585

= minus1

119871(radic120572119860

1cosradic120572120585 minus radic120572119860

2sinradic120572120585

+ radic1205721198603 coshradic120572120585 + radic1205721198601 sinradic120572120585

+ radic1205731198605 cosradic120573120585 minus radic1205731198606 sinradic120573120585

+radic1205731198607coshradic120573120585 + radic120573119860

8sinhradic120573120585)

(A1)

By doing similar substitutions we find (see expressions (3)(4) and (14))

119878119909 = (

EI119910119910

1198713)(

minus120572radic1205721198601cosradic120572120585 + 120572radic120572119860

2sinradic120572120585 + 120572radic120572119860

3coshradic120572120585 + 120572radic120572119860

4sinhradic120572120585

minus120573radic1205731198605cosradic120573120585 + 120573radic120573119860

6sinradic120573120585 + 120573radic120573119860

7coshradic120573120585 + 120573radic120573119860

8sinhradic120573120585

)

119878119910 = (

EI1199091199091198713

)(minus120572radic120572119861

1cosradic120572120585 + 120572radic120572119861

2sinradic120572120585 + 120572radic120572119861

3coshradic120572120585 + 120572radic120572119861

4sinhradic120572120585

minus120573radic1205731198615cosradic120573120585 + 120573radic120573119861

6sinradic120573120585 + 120573radic120573119861

7coshradic120573120585 + 120573radic120573119861

8sinhradic120573120585

)

119872119909= (

EI119910119910

1198712)(

minus1205721198611 sinradic120572120585 minus 1205721198612 cosradic120572120585 + 1205721198613 sinhradic120572120585 + 1205721198614 coshradic120572120585

minus1205731198615sinradic120573120585 + 120573119861

6cosradic120573120585 + 120573119861

7sinhradic120573120585 + 120573119861

8coshradic120573120585

)

119872119910= (

EI119909119909

1198712)(

minus1205721198601 sinradic120572120585 minus 120572119860

2 cosradic120572120585 + 1205721198603 sinhradic120572120585 + 1205721198604 coshradic120572120585

minus1205731198605 sinradic120573120585 + 1205731198606 cosradic120573120585 + 1205731198607 sinhradic120573120585 + 1205731198608 coshradic120573120585)

(A2)

where1198601= 1198961205721198611 119860

2= 1198961205721198612

1198603= 1198961205721198613 119860

4= 1198961205721198614

1198605 = 1198961205731198615 1198606 = 1198961205731198616

1198607 = 1198961205731198617 1198608 = 1198961205731198618

119896120572=

21205822

119910120578

1205722 minus 1205822119910(1 + 1205782)

119896120573=

21205822

119910120578

1205732 minus 1205822119910(1 + 1205782)

(A3)

To obtain the dynamic stiffness matrix we introduce theboundary conditions into the governing equations For dis-placements

at 119911 = 0

119880 = 1198801 119881 = 119881

1 Θ

119909= Θ1199091 Θ

119910= Θ1199101

(A4)

at 119911 = 119871

119880 = 1198802 119881 = 119881

2 Θ

119909= Θ1199092 Θ

119910= Θ1199102

(A5)

And for forces we have the following

at 119911 = 0

119878119909= 1198781199091 119878

119910= 1198781199101 119872

119909= 1198721199091 119872

119910= 1198721199101

(A6)

at 119911 = 119871

119878119909= 1198781199092 119878

119910= 1198781199102 119872

119909= 1198721199092 119872

119910= 1198721199102

(A7)

Substituting the boundary conditions into the governingequations we find

120575 = BR (A8)

Shock and Vibration 9

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

B =

[[[[[[[[[[[[

[

0 119896120572

0 119896120572

0 119896120573

0 119896120573

0 1 0 1 0 1 0 1

minus120591120572

0 minus120591120572

0 minus120591120573

0 minus120591120573

0

120594120572

0 120594120572

0 120594120573

0 120594120573

0

119896120572119878120572

119896120572119862120572

119896120572119878ℎ120572

119896120572119862ℎ120572

119896120573119878120573

119896120573119862120573

119896120573119878ℎ120573

119896120573119862ℎ120573

119878120572 119862120572 119878ℎ120572

119862ℎ120572

119878120573 119862120573 119878ℎ120573

119862ℎ120573

minus120591120572119862120572

120591120572119878120572

minus120591120572119862ℎ120572

minus120591120572119878ℎ120572

minus120591120573119862120573

120591120573119878120573

minus120591120573119862ℎ120573

minus120591120573119878ℎ120573

120594120572119862120572

minus120594120572119878120572

120594120572119862ℎ120572

120594120572119878ℎ120572

120594120573119862120573

minus120594120573119878120573

120594120573119862ℎ120573

120594120573119878ℎ120573

]]]]]]]]]]]]

]

120591120572=

radic120572

119871 120591

120573= minus

radic120573

119871 120594

120572= 119896120572120591120572 120594

120573= 119896120573120591120573

(A9)

Also

119878120572= sinradic120572 119862

120572= cosradic120572

119878ℎ120572

= sinhradic120572 119862ℎ120572

= coshradic120572

119878120573= sinradic120573 119862

120573= cosradic120573

119878ℎ120573

= sinhradic120573 119862ℎ120573

= coshradic120573

(A10)

Substituting similarly for the force equation

F = AR (A11)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

A =

[[[[[[[[[[[[

[

minus120577120572 0 120577120572 0 minus120577120573 0 120577120573 0

minus120576120572 0 120576120572 0 minus120576120573 0 120576120573 0

0 minus120574120572

0 120574120572

0 minus120574120573

0 120574120573

0 120582120572

0 minus120582120572

0 120582120573

0 minus120582120573

120577120572119862120572

minus120577120572119878120572

minus120577120572119862ℎ120572

minus120577120572119878ℎ120572

120577120573119862120573

minus120577120573119878120573

minus120577120573119862ℎ120573

minus120577120573119878ℎ120573

120576120572119862120572

minus120576120572119878120572

minus120576120572119862ℎ120572

minus120576120572119878ℎ120572

120576120573119862120573

minus120576120573119878120573

minus120576120573119862ℎ120573

minus120576120573119878ℎ120573

120574120572119878120572

120574120572119862120572

minus120574120572119878ℎ120572

minus120574120572119862ℎ120572

120574120573119878120573

120574120573119862120573

minus120574120573119878ℎ120573

minus120574120573119862ℎ120573

minus120582120572119878120572 minus120582120572119862120572 120582120572119878ℎ120572

120582120572119862ℎ120572

minus120582120573119878120573 minus120582120573119862120573 120582120573119878ℎ120573

120582120573119862ℎ120573

]]]]]]]]]]]]

]

120577120572 = 119896120572120572radic120572

EI119910119910

1198713 120576120572 = 120572radic120572

EI1199091199091198713

120574120572= 120572

EI119909119909

1198712 120582

120572= 119896120572120572

EI119910119910

1198712

120577120573= 119896120573120573radic120573

EI119910119910

1198713 120576

120573= 120573radic120573

EI119909119909

1198713

120574120573 = 120573

EI1199091199091198712

120582120573 = 119896120573120573

EI119910119910

1198712

(A12)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the supports providedby Natural Sciences and Engineering Research Council ofCanada (NSERC) and Ryerson University

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

International Journal of

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

6 Shock and Vibration

Table 1 Spindle natural frequencies DSM versus FEA for a nonspinning (Ω cong 955RPM) with simply supported boundary conditions

Mode DSM FEA B33 element (Hz) FEA C3D20R (Hz) FEA C3D10M (Hz)12 sections 600 elements 1200 elements 16965 elements 20122 elements 34465 elements 45645 elements

1 139292 139300 139300 130800 130800 130100 1301002 139324 139300 139300 130800 130800 130100 1301003 407102 407100 407100 320800 321200 333500 3322004 407134 407100 407100 321000 321200 333500 3324005 566814 566900 566900 433900 431100 432200 4291006 566846 566900 566900 434100 431600 432300 429200

Table 2 Spindle natural frequencies versus various spinning speeds for flexible boundary conditions 119870119904= 21 times 10

8Nm

Mode955

(RPM)(Ω cong 0)

47842(RPM)

95589(RPM)

14334(RPM)

19108(RPM)

23883(RPM)

28657(RPM)

33432(RPM)

1 100427 92469 84511 76554 68596 60638 52680 447232 100459 108416 114321 106363 98406 90448 82490 745323 130237 122279 1163741 113414 105456 97498 89541 815834 130268 129329 1213716 124332 132290 140247 146805 1388475 137287 138226 1461838 154142 154142 154762 148205 1561636 137319 145277 1532344 161192 162720 170057 178015 185973

Toolholder Motor Spindle shaftSpring Spring

Spindle casing

Figure 4 Modified boundary conditions

119909 and 119910 directions become coupled changing the repeatednatural frequencies to two distinct ones (see Table 2) withone decreasing and approaching zero (Figure 6) and the otherincreasing from their uncoupled values Further increasein the spinning speed would make one of the bendingfrequencies go to zero that is whirling instability alsoreported and discussed in [31 32] The first critical spindlespeed in the case of calibrated model was found to be Ωcr =6 times 10

5 RPM which is well above the operating speed of thespindle that is 35 times 104 RPM (see Figure 7)

In order to further validate the proposed CDSM modelthe tap testing method was used to experimentally collectthe frequency response function (FRF) of the spindle sys-tem in hand while being installed on its correspondingmachine tool A 1-inch diameter blank tool with a 2-inchprotrusion in a typical shrink fit tool holder was usedThe tool and holder were inserted into the spindle Thespindle was set to be horizontal (ie neutral) position andan accelerometer (352A21 08 g) was then attached to theedge of the tool holder The spindle was struck 10 timesin the 119909 direction using a hammer (086C04 5000N) and

0

200

400

600

800

1000

1200

1400

Syste

m n

atur

al fr

eque

ncy

(Hz)

Bearing equivalent spring constant K (Nm)

DSMSimply supportedANSYS

Spindle manufacturerrsquos designed natural frequency (1000Hz)

100E+00

100E+01

100E+02

100E+03

100E+04

100E+05

100E+06

100E+07

100E+08

100E+09

100E+10

Tool Simplesupports

Simplesupports

Motor Spindleshaftholder

MotorSpring SpringSpindle casing

ToolSpindle

shaftholder

Figure 5 Varying spring constant versus system natural frequencyusing DSM and FEA

the average FRF graph was generated (SIM3 Module Photon+ Data Acquisition) Similarly the spindle was also testedalong the 119910 direction The nonspinning spindlersquos naturalfrequencies extracted from the FRF graphs are presented inTable 3 alongside the CDSM results As can be seen themaximum difference between the first three CDSM flexural

Shock and Vibration 7

Table 3 Spindle natural frequencies experimental versus CDSM for a nonspinning spindle system with flexible boundary conditions119870119904=

21 times 108 Nm

Nonspinning bending modes (Hz) CDSM (Ω cong 955RPM) Experimental (x-axisy-axis) Difference ()(x-axisy-axis)

1205961 100427100459 9298192651 80841205962 130237130268 130151131067 0070061205963 137287137319 160144158386 143133

0 5000 10000 15000 20000 25000 30000 35000 400000

200

400

600

800

1000

1200

1400

1600

1800

2000

Nat

ural

freq

uenc

y (H

z)

Spindle speed (RPM)

Spindle speed versus natural frequency

1st mode2nd mode3rd mode

4th mode5th mode6th mode

Figure 6 Spindle speed versus natural frequency

0

200

400

600

800

1000

1200

1400

0 10000 20000 30000 40000 50000 60000 70000 80000

Nat

ural

freq

uenc

y (H

z)

Spindle RPM

Spindle RPM versus natural frequency

Spring BCSimply supported

Spindle RPM operational limit

Figure 7 Spindle natural frequency versus spindle RPM simplysupported and flexible bearing models

natural frequencies and the experimentally evaluated dataare respectively 8 007 and 143 in 119909 direction and84 006 and 133 in 119910 directionThe average differenceis found to be around 74 which is within the acceptablerange

5 Discussion and Conclusion

It has been established that the service life changes the vibra-tional behavior of spindle systems that is reduced naturalfrequency over the time associated with the bearings wearTherefore in order to generate an updated spindlemodel it isessential to include the effect of the bearings in the systemAnanalytical model of a multisegment spinning spindle basedon the dynamic stiffnessmatrix (DSM) formulation and exactwithin the limits of the Euler-Bernoulli beam bending theorywas developed The beam exhibits coupled bending-bending(B-B) vibration and as expected its fundamental coupledfrequency was found to decrease with increasing spinningspeed The bearings were included in the model using twodifferent models rigid simply supported frictionless pinsand flexible linear spring elements The simply supportedbearing model was found to yield an excessively rigid systemwith amuch higher natural frequency than the nominal valueof the system Replacing the simply supported boundarycondition with linear spring elements the spring stiffness119870119904 was then calibrated so that the fundamental frequency

of the system matched the nominal value that is a morerealistic and accurate representation of the spindle systemThe accuracy of the proposed calibrated dynamic stiffnessmatrix (CDSM) method was confirmed through compari-son with both numerical and experimental results Whencompared to the conventional finite element method (FEM)the CDSM frequencies were found to best match with thoseobtained from a cubic beamfinite element (B33 inABAQUS)Moreover in comparison with experimental data the CDSMresults were found to be within acceptable accuracy the aver-age error for the first three bending natural frequencies is lessthan 8

It is worth noting that the proposed CDSM can alsobe fine-tuned to match spindlersquos experimental fundamentalfrequency instead of the nominal value Further researchis underway to improve the CDSM by using more accu-rate material properties for different segments of thespindle The developed calibrated model could then beexploited in an attempt to take into account the effects

8 Shock and Vibration

of bearing wear in terms of spindlersquos service time numer-ically evaluate the equivalent cutting force and calcu-late updated stability lobes used to achieve chatter-freecuts

Appendix

From the solutions of 119880 and 119881 (see expressions (13)) thecorresponding bending rotation about 119909-axis and 119910-axis Θ

119909

and Θ119910 are respectively given by

Θ119909 =

119889119881

119889119911= minus

1

119871

119889119881

119889120585

= minus1

119871(radic120572119861

1cosradic120572120585 minus radic120572119861

2sinradic120572120585

+ radic1205721198613coshradic120572120585 + radic120572119861

1sinradic120572120585

+ radic1205731198615cosradic120573120585 minus radic120573119861

6sinradic120573120585

+radic1205731198617coshradic120573120585 + radic120573119861

8sinhradic120573120585)

Θ119910=

119889119880

119889119911=

1

119871

119889119880

119889120585

= minus1

119871(radic120572119860

1cosradic120572120585 minus radic120572119860

2sinradic120572120585

+ radic1205721198603 coshradic120572120585 + radic1205721198601 sinradic120572120585

+ radic1205731198605 cosradic120573120585 minus radic1205731198606 sinradic120573120585

+radic1205731198607coshradic120573120585 + radic120573119860

8sinhradic120573120585)

(A1)

By doing similar substitutions we find (see expressions (3)(4) and (14))

119878119909 = (

EI119910119910

1198713)(

minus120572radic1205721198601cosradic120572120585 + 120572radic120572119860

2sinradic120572120585 + 120572radic120572119860

3coshradic120572120585 + 120572radic120572119860

4sinhradic120572120585

minus120573radic1205731198605cosradic120573120585 + 120573radic120573119860

6sinradic120573120585 + 120573radic120573119860

7coshradic120573120585 + 120573radic120573119860

8sinhradic120573120585

)

119878119910 = (

EI1199091199091198713

)(minus120572radic120572119861

1cosradic120572120585 + 120572radic120572119861

2sinradic120572120585 + 120572radic120572119861

3coshradic120572120585 + 120572radic120572119861

4sinhradic120572120585

minus120573radic1205731198615cosradic120573120585 + 120573radic120573119861

6sinradic120573120585 + 120573radic120573119861

7coshradic120573120585 + 120573radic120573119861

8sinhradic120573120585

)

119872119909= (

EI119910119910

1198712)(

minus1205721198611 sinradic120572120585 minus 1205721198612 cosradic120572120585 + 1205721198613 sinhradic120572120585 + 1205721198614 coshradic120572120585

minus1205731198615sinradic120573120585 + 120573119861

6cosradic120573120585 + 120573119861

7sinhradic120573120585 + 120573119861

8coshradic120573120585

)

119872119910= (

EI119909119909

1198712)(

minus1205721198601 sinradic120572120585 minus 120572119860

2 cosradic120572120585 + 1205721198603 sinhradic120572120585 + 1205721198604 coshradic120572120585

minus1205731198605 sinradic120573120585 + 1205731198606 cosradic120573120585 + 1205731198607 sinhradic120573120585 + 1205731198608 coshradic120573120585)

(A2)

where1198601= 1198961205721198611 119860

2= 1198961205721198612

1198603= 1198961205721198613 119860

4= 1198961205721198614

1198605 = 1198961205731198615 1198606 = 1198961205731198616

1198607 = 1198961205731198617 1198608 = 1198961205731198618

119896120572=

21205822

119910120578

1205722 minus 1205822119910(1 + 1205782)

119896120573=

21205822

119910120578

1205732 minus 1205822119910(1 + 1205782)

(A3)

To obtain the dynamic stiffness matrix we introduce theboundary conditions into the governing equations For dis-placements

at 119911 = 0

119880 = 1198801 119881 = 119881

1 Θ

119909= Θ1199091 Θ

119910= Θ1199101

(A4)

at 119911 = 119871

119880 = 1198802 119881 = 119881

2 Θ

119909= Θ1199092 Θ

119910= Θ1199102

(A5)

And for forces we have the following

at 119911 = 0

119878119909= 1198781199091 119878

119910= 1198781199101 119872

119909= 1198721199091 119872

119910= 1198721199101

(A6)

at 119911 = 119871

119878119909= 1198781199092 119878

119910= 1198781199102 119872

119909= 1198721199092 119872

119910= 1198721199102

(A7)

Substituting the boundary conditions into the governingequations we find

120575 = BR (A8)

Shock and Vibration 9

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

B =

[[[[[[[[[[[[

[

0 119896120572

0 119896120572

0 119896120573

0 119896120573

0 1 0 1 0 1 0 1

minus120591120572

0 minus120591120572

0 minus120591120573

0 minus120591120573

0

120594120572

0 120594120572

0 120594120573

0 120594120573

0

119896120572119878120572

119896120572119862120572

119896120572119878ℎ120572

119896120572119862ℎ120572

119896120573119878120573

119896120573119862120573

119896120573119878ℎ120573

119896120573119862ℎ120573

119878120572 119862120572 119878ℎ120572

119862ℎ120572

119878120573 119862120573 119878ℎ120573

119862ℎ120573

minus120591120572119862120572

120591120572119878120572

minus120591120572119862ℎ120572

minus120591120572119878ℎ120572

minus120591120573119862120573

120591120573119878120573

minus120591120573119862ℎ120573

minus120591120573119878ℎ120573

120594120572119862120572

minus120594120572119878120572

120594120572119862ℎ120572

120594120572119878ℎ120572

120594120573119862120573

minus120594120573119878120573

120594120573119862ℎ120573

120594120573119878ℎ120573

]]]]]]]]]]]]

]

120591120572=

radic120572

119871 120591

120573= minus

radic120573

119871 120594

120572= 119896120572120591120572 120594

120573= 119896120573120591120573

(A9)

Also

119878120572= sinradic120572 119862

120572= cosradic120572

119878ℎ120572

= sinhradic120572 119862ℎ120572

= coshradic120572

119878120573= sinradic120573 119862

120573= cosradic120573

119878ℎ120573

= sinhradic120573 119862ℎ120573

= coshradic120573

(A10)

Substituting similarly for the force equation

F = AR (A11)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

A =

[[[[[[[[[[[[

[

minus120577120572 0 120577120572 0 minus120577120573 0 120577120573 0

minus120576120572 0 120576120572 0 minus120576120573 0 120576120573 0

0 minus120574120572

0 120574120572

0 minus120574120573

0 120574120573

0 120582120572

0 minus120582120572

0 120582120573

0 minus120582120573

120577120572119862120572

minus120577120572119878120572

minus120577120572119862ℎ120572

minus120577120572119878ℎ120572

120577120573119862120573

minus120577120573119878120573

minus120577120573119862ℎ120573

minus120577120573119878ℎ120573

120576120572119862120572

minus120576120572119878120572

minus120576120572119862ℎ120572

minus120576120572119878ℎ120572

120576120573119862120573

minus120576120573119878120573

minus120576120573119862ℎ120573

minus120576120573119878ℎ120573

120574120572119878120572

120574120572119862120572

minus120574120572119878ℎ120572

minus120574120572119862ℎ120572

120574120573119878120573

120574120573119862120573

minus120574120573119878ℎ120573

minus120574120573119862ℎ120573

minus120582120572119878120572 minus120582120572119862120572 120582120572119878ℎ120572

120582120572119862ℎ120572

minus120582120573119878120573 minus120582120573119862120573 120582120573119878ℎ120573

120582120573119862ℎ120573

]]]]]]]]]]]]

]

120577120572 = 119896120572120572radic120572

EI119910119910

1198713 120576120572 = 120572radic120572

EI1199091199091198713

120574120572= 120572

EI119909119909

1198712 120582

120572= 119896120572120572

EI119910119910

1198712

120577120573= 119896120573120573radic120573

EI119910119910

1198713 120576

120573= 120573radic120573

EI119909119909

1198713

120574120573 = 120573

EI1199091199091198712

120582120573 = 119896120573120573

EI119910119910

1198712

(A12)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the supports providedby Natural Sciences and Engineering Research Council ofCanada (NSERC) and Ryerson University

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

Table 3 Spindle natural frequencies experimental versus CDSM for a nonspinning spindle system with flexible boundary conditions119870119904=

21 times 108 Nm

Nonspinning bending modes (Hz) CDSM (Ω cong 955RPM) Experimental (x-axisy-axis) Difference ()(x-axisy-axis)

1205961 100427100459 9298192651 80841205962 130237130268 130151131067 0070061205963 137287137319 160144158386 143133

0 5000 10000 15000 20000 25000 30000 35000 400000

200

400

600

800

1000

1200

1400

1600

1800

2000

Nat

ural

freq

uenc

y (H

z)

Spindle speed (RPM)

Spindle speed versus natural frequency

1st mode2nd mode3rd mode

4th mode5th mode6th mode

Figure 6 Spindle speed versus natural frequency

0

200

400

600

800

1000

1200

1400

0 10000 20000 30000 40000 50000 60000 70000 80000

Nat

ural

freq

uenc

y (H

z)

Spindle RPM

Spindle RPM versus natural frequency

Spring BCSimply supported

Spindle RPM operational limit

Figure 7 Spindle natural frequency versus spindle RPM simplysupported and flexible bearing models

natural frequencies and the experimentally evaluated dataare respectively 8 007 and 143 in 119909 direction and84 006 and 133 in 119910 directionThe average differenceis found to be around 74 which is within the acceptablerange

5 Discussion and Conclusion

It has been established that the service life changes the vibra-tional behavior of spindle systems that is reduced naturalfrequency over the time associated with the bearings wearTherefore in order to generate an updated spindlemodel it isessential to include the effect of the bearings in the systemAnanalytical model of a multisegment spinning spindle basedon the dynamic stiffnessmatrix (DSM) formulation and exactwithin the limits of the Euler-Bernoulli beam bending theorywas developed The beam exhibits coupled bending-bending(B-B) vibration and as expected its fundamental coupledfrequency was found to decrease with increasing spinningspeed The bearings were included in the model using twodifferent models rigid simply supported frictionless pinsand flexible linear spring elements The simply supportedbearing model was found to yield an excessively rigid systemwith amuch higher natural frequency than the nominal valueof the system Replacing the simply supported boundarycondition with linear spring elements the spring stiffness119870119904 was then calibrated so that the fundamental frequency

of the system matched the nominal value that is a morerealistic and accurate representation of the spindle systemThe accuracy of the proposed calibrated dynamic stiffnessmatrix (CDSM) method was confirmed through compari-son with both numerical and experimental results Whencompared to the conventional finite element method (FEM)the CDSM frequencies were found to best match with thoseobtained from a cubic beamfinite element (B33 inABAQUS)Moreover in comparison with experimental data the CDSMresults were found to be within acceptable accuracy the aver-age error for the first three bending natural frequencies is lessthan 8

It is worth noting that the proposed CDSM can alsobe fine-tuned to match spindlersquos experimental fundamentalfrequency instead of the nominal value Further researchis underway to improve the CDSM by using more accu-rate material properties for different segments of thespindle The developed calibrated model could then beexploited in an attempt to take into account the effects

8 Shock and Vibration

of bearing wear in terms of spindlersquos service time numer-ically evaluate the equivalent cutting force and calcu-late updated stability lobes used to achieve chatter-freecuts

Appendix

From the solutions of 119880 and 119881 (see expressions (13)) thecorresponding bending rotation about 119909-axis and 119910-axis Θ

119909

and Θ119910 are respectively given by

Θ119909 =

119889119881

119889119911= minus

1

119871

119889119881

119889120585

= minus1

119871(radic120572119861

1cosradic120572120585 minus radic120572119861

2sinradic120572120585

+ radic1205721198613coshradic120572120585 + radic120572119861

1sinradic120572120585

+ radic1205731198615cosradic120573120585 minus radic120573119861

6sinradic120573120585

+radic1205731198617coshradic120573120585 + radic120573119861

8sinhradic120573120585)

Θ119910=

119889119880

119889119911=

1

119871

119889119880

119889120585

= minus1

119871(radic120572119860

1cosradic120572120585 minus radic120572119860

2sinradic120572120585

+ radic1205721198603 coshradic120572120585 + radic1205721198601 sinradic120572120585

+ radic1205731198605 cosradic120573120585 minus radic1205731198606 sinradic120573120585

+radic1205731198607coshradic120573120585 + radic120573119860

8sinhradic120573120585)

(A1)

By doing similar substitutions we find (see expressions (3)(4) and (14))

119878119909 = (

EI119910119910

1198713)(

minus120572radic1205721198601cosradic120572120585 + 120572radic120572119860

2sinradic120572120585 + 120572radic120572119860

3coshradic120572120585 + 120572radic120572119860

4sinhradic120572120585

minus120573radic1205731198605cosradic120573120585 + 120573radic120573119860

6sinradic120573120585 + 120573radic120573119860

7coshradic120573120585 + 120573radic120573119860

8sinhradic120573120585

)

119878119910 = (

EI1199091199091198713

)(minus120572radic120572119861

1cosradic120572120585 + 120572radic120572119861

2sinradic120572120585 + 120572radic120572119861

3coshradic120572120585 + 120572radic120572119861

4sinhradic120572120585

minus120573radic1205731198615cosradic120573120585 + 120573radic120573119861

6sinradic120573120585 + 120573radic120573119861

7coshradic120573120585 + 120573radic120573119861

8sinhradic120573120585

)

119872119909= (

EI119910119910

1198712)(

minus1205721198611 sinradic120572120585 minus 1205721198612 cosradic120572120585 + 1205721198613 sinhradic120572120585 + 1205721198614 coshradic120572120585

minus1205731198615sinradic120573120585 + 120573119861

6cosradic120573120585 + 120573119861

7sinhradic120573120585 + 120573119861

8coshradic120573120585

)

119872119910= (

EI119909119909

1198712)(

minus1205721198601 sinradic120572120585 minus 120572119860

2 cosradic120572120585 + 1205721198603 sinhradic120572120585 + 1205721198604 coshradic120572120585

minus1205731198605 sinradic120573120585 + 1205731198606 cosradic120573120585 + 1205731198607 sinhradic120573120585 + 1205731198608 coshradic120573120585)

(A2)

where1198601= 1198961205721198611 119860

2= 1198961205721198612

1198603= 1198961205721198613 119860

4= 1198961205721198614

1198605 = 1198961205731198615 1198606 = 1198961205731198616

1198607 = 1198961205731198617 1198608 = 1198961205731198618

119896120572=

21205822

119910120578

1205722 minus 1205822119910(1 + 1205782)

119896120573=

21205822

119910120578

1205732 minus 1205822119910(1 + 1205782)

(A3)

To obtain the dynamic stiffness matrix we introduce theboundary conditions into the governing equations For dis-placements

at 119911 = 0

119880 = 1198801 119881 = 119881

1 Θ

119909= Θ1199091 Θ

119910= Θ1199101

(A4)

at 119911 = 119871

119880 = 1198802 119881 = 119881

2 Θ

119909= Θ1199092 Θ

119910= Θ1199102

(A5)

And for forces we have the following

at 119911 = 0

119878119909= 1198781199091 119878

119910= 1198781199101 119872

119909= 1198721199091 119872

119910= 1198721199101

(A6)

at 119911 = 119871

119878119909= 1198781199092 119878

119910= 1198781199102 119872

119909= 1198721199092 119872

119910= 1198721199102

(A7)

Substituting the boundary conditions into the governingequations we find

120575 = BR (A8)

Shock and Vibration 9

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

B =

[[[[[[[[[[[[

[

0 119896120572

0 119896120572

0 119896120573

0 119896120573

0 1 0 1 0 1 0 1

minus120591120572

0 minus120591120572

0 minus120591120573

0 minus120591120573

0

120594120572

0 120594120572

0 120594120573

0 120594120573

0

119896120572119878120572

119896120572119862120572

119896120572119878ℎ120572

119896120572119862ℎ120572

119896120573119878120573

119896120573119862120573

119896120573119878ℎ120573

119896120573119862ℎ120573

119878120572 119862120572 119878ℎ120572

119862ℎ120572

119878120573 119862120573 119878ℎ120573

119862ℎ120573

minus120591120572119862120572

120591120572119878120572

minus120591120572119862ℎ120572

minus120591120572119878ℎ120572

minus120591120573119862120573

120591120573119878120573

minus120591120573119862ℎ120573

minus120591120573119878ℎ120573

120594120572119862120572

minus120594120572119878120572

120594120572119862ℎ120572

120594120572119878ℎ120572

120594120573119862120573

minus120594120573119878120573

120594120573119862ℎ120573

120594120573119878ℎ120573

]]]]]]]]]]]]

]

120591120572=

radic120572

119871 120591

120573= minus

radic120573

119871 120594

120572= 119896120572120591120572 120594

120573= 119896120573120591120573

(A9)

Also

119878120572= sinradic120572 119862

120572= cosradic120572

119878ℎ120572

= sinhradic120572 119862ℎ120572

= coshradic120572

119878120573= sinradic120573 119862

120573= cosradic120573

119878ℎ120573

= sinhradic120573 119862ℎ120573

= coshradic120573

(A10)

Substituting similarly for the force equation

F = AR (A11)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

A =

[[[[[[[[[[[[

[

minus120577120572 0 120577120572 0 minus120577120573 0 120577120573 0

minus120576120572 0 120576120572 0 minus120576120573 0 120576120573 0

0 minus120574120572

0 120574120572

0 minus120574120573

0 120574120573

0 120582120572

0 minus120582120572

0 120582120573

0 minus120582120573

120577120572119862120572

minus120577120572119878120572

minus120577120572119862ℎ120572

minus120577120572119878ℎ120572

120577120573119862120573

minus120577120573119878120573

minus120577120573119862ℎ120573

minus120577120573119878ℎ120573

120576120572119862120572

minus120576120572119878120572

minus120576120572119862ℎ120572

minus120576120572119878ℎ120572

120576120573119862120573

minus120576120573119878120573

minus120576120573119862ℎ120573

minus120576120573119878ℎ120573

120574120572119878120572

120574120572119862120572

minus120574120572119878ℎ120572

minus120574120572119862ℎ120572

120574120573119878120573

120574120573119862120573

minus120574120573119878ℎ120573

minus120574120573119862ℎ120573

minus120582120572119878120572 minus120582120572119862120572 120582120572119878ℎ120572

120582120572119862ℎ120572

minus120582120573119878120573 minus120582120573119862120573 120582120573119878ℎ120573

120582120573119862ℎ120573

]]]]]]]]]]]]

]

120577120572 = 119896120572120572radic120572

EI119910119910

1198713 120576120572 = 120572radic120572

EI1199091199091198713

120574120572= 120572

EI119909119909

1198712 120582

120572= 119896120572120572

EI119910119910

1198712

120577120573= 119896120573120573radic120573

EI119910119910

1198713 120576

120573= 120573radic120573

EI119909119909

1198713

120574120573 = 120573

EI1199091199091198712

120582120573 = 119896120573120573

EI119910119910

1198712

(A12)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the supports providedby Natural Sciences and Engineering Research Council ofCanada (NSERC) and Ryerson University

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

of bearing wear in terms of spindlersquos service time numer-ically evaluate the equivalent cutting force and calcu-late updated stability lobes used to achieve chatter-freecuts

Appendix

From the solutions of 119880 and 119881 (see expressions (13)) thecorresponding bending rotation about 119909-axis and 119910-axis Θ

119909

and Θ119910 are respectively given by

Θ119909 =

119889119881

119889119911= minus

1

119871

119889119881

119889120585

= minus1

119871(radic120572119861

1cosradic120572120585 minus radic120572119861

2sinradic120572120585

+ radic1205721198613coshradic120572120585 + radic120572119861

1sinradic120572120585

+ radic1205731198615cosradic120573120585 minus radic120573119861

6sinradic120573120585

+radic1205731198617coshradic120573120585 + radic120573119861

8sinhradic120573120585)

Θ119910=

119889119880

119889119911=

1

119871

119889119880

119889120585

= minus1

119871(radic120572119860

1cosradic120572120585 minus radic120572119860

2sinradic120572120585

+ radic1205721198603 coshradic120572120585 + radic1205721198601 sinradic120572120585

+ radic1205731198605 cosradic120573120585 minus radic1205731198606 sinradic120573120585

+radic1205731198607coshradic120573120585 + radic120573119860

8sinhradic120573120585)

(A1)

By doing similar substitutions we find (see expressions (3)(4) and (14))

119878119909 = (

EI119910119910

1198713)(

minus120572radic1205721198601cosradic120572120585 + 120572radic120572119860

2sinradic120572120585 + 120572radic120572119860

3coshradic120572120585 + 120572radic120572119860

4sinhradic120572120585

minus120573radic1205731198605cosradic120573120585 + 120573radic120573119860

6sinradic120573120585 + 120573radic120573119860

7coshradic120573120585 + 120573radic120573119860

8sinhradic120573120585

)

119878119910 = (

EI1199091199091198713

)(minus120572radic120572119861

1cosradic120572120585 + 120572radic120572119861

2sinradic120572120585 + 120572radic120572119861

3coshradic120572120585 + 120572radic120572119861

4sinhradic120572120585

minus120573radic1205731198615cosradic120573120585 + 120573radic120573119861

6sinradic120573120585 + 120573radic120573119861

7coshradic120573120585 + 120573radic120573119861

8sinhradic120573120585

)

119872119909= (

EI119910119910

1198712)(

minus1205721198611 sinradic120572120585 minus 1205721198612 cosradic120572120585 + 1205721198613 sinhradic120572120585 + 1205721198614 coshradic120572120585

minus1205731198615sinradic120573120585 + 120573119861

6cosradic120573120585 + 120573119861

7sinhradic120573120585 + 120573119861

8coshradic120573120585

)

119872119910= (

EI119909119909

1198712)(

minus1205721198601 sinradic120572120585 minus 120572119860

2 cosradic120572120585 + 1205721198603 sinhradic120572120585 + 1205721198604 coshradic120572120585

minus1205731198605 sinradic120573120585 + 1205731198606 cosradic120573120585 + 1205731198607 sinhradic120573120585 + 1205731198608 coshradic120573120585)

(A2)

where1198601= 1198961205721198611 119860

2= 1198961205721198612

1198603= 1198961205721198613 119860

4= 1198961205721198614

1198605 = 1198961205731198615 1198606 = 1198961205731198616

1198607 = 1198961205731198617 1198608 = 1198961205731198618

119896120572=

21205822

119910120578

1205722 minus 1205822119910(1 + 1205782)

119896120573=

21205822

119910120578

1205732 minus 1205822119910(1 + 1205782)

(A3)

To obtain the dynamic stiffness matrix we introduce theboundary conditions into the governing equations For dis-placements

at 119911 = 0

119880 = 1198801 119881 = 119881

1 Θ

119909= Θ1199091 Θ

119910= Θ1199101

(A4)

at 119911 = 119871

119880 = 1198802 119881 = 119881

2 Θ

119909= Θ1199092 Θ

119910= Θ1199102

(A5)

And for forces we have the following

at 119911 = 0

119878119909= 1198781199091 119878

119910= 1198781199101 119872

119909= 1198721199091 119872

119910= 1198721199101

(A6)

at 119911 = 119871

119878119909= 1198781199092 119878

119910= 1198781199102 119872

119909= 1198721199092 119872

119910= 1198721199102

(A7)

Substituting the boundary conditions into the governingequations we find

120575 = BR (A8)

Shock and Vibration 9

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

B =

[[[[[[[[[[[[

[

0 119896120572

0 119896120572

0 119896120573

0 119896120573

0 1 0 1 0 1 0 1

minus120591120572

0 minus120591120572

0 minus120591120573

0 minus120591120573

0

120594120572

0 120594120572

0 120594120573

0 120594120573

0

119896120572119878120572

119896120572119862120572

119896120572119878ℎ120572

119896120572119862ℎ120572

119896120573119878120573

119896120573119862120573

119896120573119878ℎ120573

119896120573119862ℎ120573

119878120572 119862120572 119878ℎ120572

119862ℎ120572

119878120573 119862120573 119878ℎ120573

119862ℎ120573

minus120591120572119862120572

120591120572119878120572

minus120591120572119862ℎ120572

minus120591120572119878ℎ120572

minus120591120573119862120573

120591120573119878120573

minus120591120573119862ℎ120573

minus120591120573119878ℎ120573

120594120572119862120572

minus120594120572119878120572

120594120572119862ℎ120572

120594120572119878ℎ120572

120594120573119862120573

minus120594120573119878120573

120594120573119862ℎ120573

120594120573119878ℎ120573

]]]]]]]]]]]]

]

120591120572=

radic120572

119871 120591

120573= minus

radic120573

119871 120594

120572= 119896120572120591120572 120594

120573= 119896120573120591120573

(A9)

Also

119878120572= sinradic120572 119862

120572= cosradic120572

119878ℎ120572

= sinhradic120572 119862ℎ120572

= coshradic120572

119878120573= sinradic120573 119862

120573= cosradic120573

119878ℎ120573

= sinhradic120573 119862ℎ120573

= coshradic120573

(A10)

Substituting similarly for the force equation

F = AR (A11)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

A =

[[[[[[[[[[[[

[

minus120577120572 0 120577120572 0 minus120577120573 0 120577120573 0

minus120576120572 0 120576120572 0 minus120576120573 0 120576120573 0

0 minus120574120572

0 120574120572

0 minus120574120573

0 120574120573

0 120582120572

0 minus120582120572

0 120582120573

0 minus120582120573

120577120572119862120572

minus120577120572119878120572

minus120577120572119862ℎ120572

minus120577120572119878ℎ120572

120577120573119862120573

minus120577120573119878120573

minus120577120573119862ℎ120573

minus120577120573119878ℎ120573

120576120572119862120572

minus120576120572119878120572

minus120576120572119862ℎ120572

minus120576120572119878ℎ120572

120576120573119862120573

minus120576120573119878120573

minus120576120573119862ℎ120573

minus120576120573119878ℎ120573

120574120572119878120572

120574120572119862120572

minus120574120572119878ℎ120572

minus120574120572119862ℎ120572

120574120573119878120573

120574120573119862120573

minus120574120573119878ℎ120573

minus120574120573119862ℎ120573

minus120582120572119878120572 minus120582120572119862120572 120582120572119878ℎ120572

120582120572119862ℎ120572

minus120582120573119878120573 minus120582120573119862120573 120582120573119878ℎ120573

120582120573119862ℎ120573

]]]]]]]]]]]]

]

120577120572 = 119896120572120572radic120572

EI119910119910

1198713 120576120572 = 120572radic120572

EI1199091199091198713

120574120572= 120572

EI119909119909

1198712 120582

120572= 119896120572120572

EI119910119910

1198712

120577120573= 119896120573120573radic120573

EI119910119910

1198713 120576

120573= 120573radic120573

EI119909119909

1198713

120574120573 = 120573

EI1199091199091198712

120582120573 = 119896120573120573

EI119910119910

1198712

(A12)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the supports providedby Natural Sciences and Engineering Research Council ofCanada (NSERC) and Ryerson University

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

where

120575 = [1198801 1198811

Θ1199091

Θ1199101

1198802

1198812

Θ1199092

Θ1199102]119879

R = [1198771 1198772

1198773

1198774

1198775

1198776

1198777

1198778]119879

B =

[[[[[[[[[[[[

[

0 119896120572

0 119896120572

0 119896120573

0 119896120573

0 1 0 1 0 1 0 1

minus120591120572

0 minus120591120572

0 minus120591120573

0 minus120591120573

0

120594120572

0 120594120572

0 120594120573

0 120594120573

0

119896120572119878120572

119896120572119862120572

119896120572119878ℎ120572

119896120572119862ℎ120572

119896120573119878120573

119896120573119862120573

119896120573119878ℎ120573

119896120573119862ℎ120573

119878120572 119862120572 119878ℎ120572

119862ℎ120572

119878120573 119862120573 119878ℎ120573

119862ℎ120573

minus120591120572119862120572

120591120572119878120572

minus120591120572119862ℎ120572

minus120591120572119878ℎ120572

minus120591120573119862120573

120591120573119878120573

minus120591120573119862ℎ120573

minus120591120573119878ℎ120573

120594120572119862120572

minus120594120572119878120572

120594120572119862ℎ120572

120594120572119878ℎ120572

120594120573119862120573

minus120594120573119878120573

120594120573119862ℎ120573

120594120573119878ℎ120573

]]]]]]]]]]]]

]

120591120572=

radic120572

119871 120591

120573= minus

radic120573

119871 120594

120572= 119896120572120591120572 120594

120573= 119896120573120591120573

(A9)

Also

119878120572= sinradic120572 119862

120572= cosradic120572

119878ℎ120572

= sinhradic120572 119862ℎ120572

= coshradic120572

119878120573= sinradic120573 119862

120573= cosradic120573

119878ℎ120573

= sinhradic120573 119862ℎ120573

= coshradic120573

(A10)

Substituting similarly for the force equation

F = AR (A11)

where

F = [1198781199091 1198781199101

1198721199091

1198721199101

1198781199092

1198781199102

1198721199092

1198721199102]119879

A =

[[[[[[[[[[[[

[

minus120577120572 0 120577120572 0 minus120577120573 0 120577120573 0

minus120576120572 0 120576120572 0 minus120576120573 0 120576120573 0

0 minus120574120572

0 120574120572

0 minus120574120573

0 120574120573

0 120582120572

0 minus120582120572

0 120582120573

0 minus120582120573

120577120572119862120572

minus120577120572119878120572

minus120577120572119862ℎ120572

minus120577120572119878ℎ120572

120577120573119862120573

minus120577120573119878120573

minus120577120573119862ℎ120573

minus120577120573119878ℎ120573

120576120572119862120572

minus120576120572119878120572

minus120576120572119862ℎ120572

minus120576120572119878ℎ120572

120576120573119862120573

minus120576120573119878120573

minus120576120573119862ℎ120573

minus120576120573119878ℎ120573

120574120572119878120572

120574120572119862120572

minus120574120572119878ℎ120572

minus120574120572119862ℎ120572

120574120573119878120573

120574120573119862120573

minus120574120573119878ℎ120573

minus120574120573119862ℎ120573

minus120582120572119878120572 minus120582120572119862120572 120582120572119878ℎ120572

120582120572119862ℎ120572

minus120582120573119878120573 minus120582120573119862120573 120582120573119878ℎ120573

120582120573119862ℎ120573

]]]]]]]]]]]]

]

120577120572 = 119896120572120572radic120572

EI119910119910

1198713 120576120572 = 120572radic120572

EI1199091199091198713

120574120572= 120572

EI119909119909

1198712 120582

120572= 119896120572120572

EI119910119910

1198712

120577120573= 119896120573120573radic120573

EI119910119910

1198713 120576

120573= 120573radic120573

EI119909119909

1198713

120574120573 = 120573

EI1199091199091198712

120582120573 = 119896120573120573

EI119910119910

1198712

(A12)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to acknowledge the supports providedby Natural Sciences and Engineering Research Council ofCanada (NSERC) and Ryerson University

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

References

[1] F W Taylor ldquoOn the art of cutting metalrdquo Transactions of theAmerican Society of Mechanical Engineers vol 28 pp 31ndash3501907

[2] RNArnold ldquoDiscussion on themechanismof tool vibration inthe cutting of steelrdquo Proceedings of the Institution of MechanicalEngineers vol 154 pp 429ndash432 1946

[3] J P Gurney and S A Tobias ldquoA graphical method for thedetermination of the dynamic stability of machine toolsrdquoInternational Journal of Machine Tool Design and Research vol1 no 1-2 pp 148ndash156 1961

[4] B Thomas and A Shabana ldquoChatter vibration of flexiblemultibodymachine toolmechanismsrdquoMechanism andMachineTheory vol 22 no 4 pp 359ndash369 1987

[5] F Eisele ldquoMachine tool research at the Technological Universityof Munichrdquo International Journal of Machine Tool Design andResearch vol 1 no 4 pp 249ndash274 1961

[6] C Andrew and S A Tobias ldquoVibration in horizontal millingrdquoInternational Journal of Machine Tool Design and Research vol2 no 4 pp 369ndash378 1962

[7] T Insperger G Stepan P V Bayly and B P Mann ldquoMultiplechatter frequencies in milling processesrdquo Journal of Sound andVibration vol 262 no 2 pp 333ndash345 2003

[8] Z K PengM R Jackson L Z Guo RM Parkin andGMengldquoEffects of bearing clearance on the chatter stability of millingprocessrdquoNonlinear Analysis RealWorld Applications vol 11 no5 pp 3577ndash3589 2010

[9] T Butlin and J Woodhouse ldquoFriction-induced vibrationshould low-order models be believedrdquo Journal of Sound andVibration vol 328 no 1-2 pp 92ndash108 2009

[10] J Tlusty and F Ismail ldquoBasic non-linearity in machiningchatterrdquo CIRP AnnalsmdashManufacturing Technology vol 30 no1 pp 299ndash304 1981

[11] Y Kondo O Kawano and H Sato ldquoBehaviour of self excitedchatter due to multiple regenerative effectrdquo Journal of Engineer-ing for Industry vol 103 no 3 pp 324ndash329 1981

[12] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[13] I E Minis E B Magrab and I O Pandelidis ldquoImprovedmethods for the prediction of chatter in turning part 3 Ageneralized linear theoryrdquo Journal of Engineering for Industryvol 112 no 1 pp 28ndash35 1990

[14] H Li and X Li ldquoModelling and simulation of chatter in millingusing a predictive forcemodelrdquo International Journal ofMachineTools and Manufacture vol 40 no 14 pp 2047ndash2071 2000

[15] B E Clancy andYC Shin ldquoA comprehensive chatter predictionmodel for face turning operation including tool wear effectrdquoInternational Journal ofMachine Tools andManufacture vol 42no 9 pp 1035ndash1044 2002

[16] X Q Li Y S Wong and A Y C Nee ldquoTool wear and chatterdetection using the coherence function of two crossed accelera-tionsrdquo International Journal of Machine Tools and Manufacturevol 37 no 4 pp 425ndash435 1996

[17] M Rahman and Y Ito ldquoDetection of the onset of chattervibrationrdquo Journal of Sound and Vibration vol 109 no 2 pp193ndash205 1986

[18] A-C Lee C-S Liu and S-T Chiang ldquoAnalysis of chattervibration in a cutter-workpiece systemrdquo International Journal ofMachine Tools andManufacture vol 31 no 2 pp 221ndash234 1991

[19] Y Cao and Y Altintas ldquoModeling of spindle-bearing andmachine tool systems for virtual simulation of milling opera-tionsrdquo International Journal of Machine Tools and Manufacturevol 47 no 9 pp 1342ndash1350 2007

[20] A U Patwari W F Faris A K M Nurul Amin and S KLoh ldquoDynamic modal analysis of vertical machining centrecomponentsrdquo Advances in Acoustics and Vibration vol 2009Article ID 508076 10 pages 2009

[21] S S Abuthakeer P V Mohanram and G Mohan KumarldquoDynamic characteristics analysis of high speed motorizedspindlerdquo Annals of Faculty Engineering HunedorarmdashInternational Journal of Engineering 2011 Tome IX Fascicule2

[22] A S Delgado E Ozturk and N Sims ldquoAnalysis of non-linear machine tool synamic behaviourrdquo in Proceedings of the5thManufacturing Engineering Society International ConferenceZaragoza Spain June 2013

[23] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[24] CMaresM I Friswell and J EMottershead ldquoModel updatingusing robust estimationrdquo Mechanical Systems and Signal Pro-cessing vol 16 no 1 pp 169ndash183 2002

[25] J E Mottershead M Link and M I Friswell ldquoThe sensitivitymethod in finite element model updating a tutorialrdquoMechani-cal Systems and Signal Processing vol 25 no 7 pp 2275ndash22962011

[26] S M Hashemi and O Gaber ldquoFree vibration analysis of spin-ning spindles a calibrated dynamic stiffness matrix methodrdquo inAdvances in Vibration Engineering and Structural Dynamics FBeltran-Carbajal Ed chapter 4 p 369 InTech 2012

[27] O Gaber and S M Hashemi ldquoVibration modeling of machinetool spindles a calibrated dynamic stiffness matrix methodrdquoAdvanced Materials Research vol 651 pp 710ndash716 2013

[28] J R Banerjee ldquoDynamic stiffness formulation for structuralelements a general approachrdquo Computers amp Structures vol 63no 1 pp 101ndash103 1997

[29] K-J Bathe Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[30] J R Banerjee and F W Williams ldquoExact Bernoulli-Eulerdynamic stiffness matrix for a range of tapered beamsrdquo Inter-national Journal for Numerical Methods in Engineering vol 21no 12 pp 2289ndash2302 1985

[31] J R Banerjee and H Su ldquoDynamic stiffness formulation andfree vibration analysis of spinning beamsrdquo in Proceedings ofthe 6th International Conference on Computational StructuresTechnology (ICCST rsquo02) pp 77ndash78 Civil-Comp Press PragueCzech Republic September 2002

[32] J R Banerjee and H Su ldquoDevelopment of a dynamic stiffnessmatrix for free vibration analysis of spinning beamsrdquoComputersand Structures vol 82 no 23ndash26 pp 2189ndash2197 2004

[33] M Marciniak Modal analysis of a tool holder system for a fiveaxis milling machine [BEng thesis] Department of AerospaceEng Ryerson University Toronto Canada 2012

[34] S M Hashemi Free vibrational analysis of rotating beam-like structures a dynamic finite element approach [PhD the-sis] Department of Mechanical Engineering Laval UniversityQuebec Canada 1998

[35] S Samiezadeh ldquoFree vibratial analysis of spinning compositebeams using Dynamic Finite Element (DFE) methodrdquo InternalReport Department of Aerospace Eng Ryerson UniversityToronto Canada 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of