Research Article Time Delay Effect on Regenerative …downloads.hindawi.com/journals/sv/2016/4025650.pdfResearch Article Time Delay Effect on Regenerative Chatter in Tandem Rolling

Research ArticleTime Delay Effect on Regenerative Chatter inTandem Rolling Mills

Xiaochan Liu Yong Zang Zhiying Gao and Lingqiang Zeng

School of Mechanical Engineering University of Science and Technology Beijing Beijing 100083 China

Correspondence should be addressed to Yong Zang yzangustbeducn

Received 1 September 2015 Revised 8 December 2015 Accepted 17 December 2015

Academic Editor Kumar V Singh

Copyright copy 2016 Xiaochan Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The interstand tension coupling effect and strip gauge variation passed on to next stand with time delay are the main causes forregenerative chatter in tandem rolling mills To study the effect of different factors on the stability of tandem rolling mills differentmodels considering different interstand factors were built Through stability analysis of these models by employing the Lyapunovindirectmethod and integral criterionmore detailed and quantitative explanation is put forward to regenerative chattermechanismin rolling To study the time delay effect as a single factor on the stability of tandem rollingmills stability charts of the chatter modelincluding the time delay effect and model neglecting the delay time were compared The results show that the time delay effectreduces the critical velocity of multistand mills slightly in the big picture But it alters the relationship between two adjacent standsby worsening the downstream stand stability To get preferable rolling process parameter configuration for the tandem rollingmillsthe time delay effect in rolling must be involved

1 Introduction

Chatter in high speed rolling mills results in unacceptablegauge variations affects the surface quality of rolled strip andmay damage the mill stand Many studies have been con-ducted to explore the chatter mechanism and put forwardmethods to suppress the vibration [1] The self-excited vibra-tion in rolling is caused by interaction between mill standstructure and rolling process Rolling chatter has three maindifferent types third octave fifth octave and torsional vibra-tion The third octave is the most detrimental [2 3] Thechatter mechanism in tandem rolling mills can be concludedas four reasons namely model matching effect negativedamping effect mode coupling effect and regenerative effectRegenerative effect is themost complexmechanism for insta-bility in tandem rolling mills [4 5]

Regenerative effect in rolling mainly considers the inter-action between adjacent stands in tandem rolling mills Hufirst defined the regenerative chatter in rolling process inanalogy to chatter inmetal cutting process According toHursquosdefinition the regenerative effect in strip rolling refers to thephenomenon that the vibration of a certain stand at a prior

time causes or aggravates vibration of the same stand at thecurrent time through interstand interactions [6] By buildinga dynamic rolling process model and coupling it with standstructure model a multistand regenerative chatter modelwas proposed Simulation and analysis were conducted toreveal the regenerative mechanism The results have shownthat due to the needed time for the strip to travel from onestand to another a tandem mill can become unstable even ifindividual mill stands are stable [5]

Considering the work hardening effect of the rolled pieceand work roll flattening during rolling process a new rollingprocess model was proposed by Zhao and EhmannThroughcoupling it withmill standmodel three differentmodels werebuilt to explore the regenerative mechanism The first one isa single stand chatter model The second one is a two-standchatter model which only considers the interstand tensioncoupling effect [7] The last one is a two-stand chatter modelwhich considers both the interstand tension coupling effectand the strip gauge variation from an upstream stand passedon to downstream stand with time delayThe Routh criterionwas employed to formulate the critical velocity of the first andsecond model Due to the time delay effect the third model

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 4025650 15 pageshttpdxdoiorg10115520164025650

2 Shock and Vibration

becomes a delay differential system and integral criterion wasemployed to find its critical velocity Comparative analysisof the three models presented a potential explanation to theregenerative mechanism [4]

Based on Zhao and Ehmannrsquos andHursquosmodel more com-plex regenerative chattermodels are proposed in recent yearsKimura et al constructed a five-stand regenerative chattermodel by coupling a five-degree freedom mill stand modeland the dynamic rollingmodel built in [7] togetherThe simu-lation results show that there exists an optimal friction of thefifth stand which relates to the maximum critical velocity ofthe whole tandem rolling mills [8] Instead of static analysisNiroomand et al used wave propagation theory in elasticsolids to formulate the dynamic tensile stress variationbetween two consequent stands and built a new two-standregenerative chatter model The simulation results show thatthe errors of critical velocity and chatter frequency are smallerwith the experimental results when the wave propagationtheory is applied [9] Considering the complex friction con-dition in the rolling gap Heidari et al built a regenerativechatter model using the unsteady lubrication model assum-ing that the rolling gap is in full film regime The effects ofrolling lubricant on the stability of tandem rolling mills werediscussed [10] Based on these regenerative chatter modelsoptimization of multistand rolling process parameters wereconducted to avoid chatter phenomenon using the combina-tion of neural networks and genetic algorithms [11 12]

All these regenerative chatter models have considered thetime delay effect which makes the tandem rolling mills adelay differential system Itmeans thatmore complex stabilitycriterion has to be employed to analyze the delay differen-tial multistand system which results in huge computationHowever the time delay effect on the stability of tandemrolling mills has not been analyzed as an independent factorMore research work has to be done to decide whether theeffect of time delay itself on the stability of tandem rollingmills is worth the great amount of computation Besidesthe relationship between consequent stands is not clearlydescribed though the interaction factors between stands havebeen pointed out The major objective pursed in this paper isto investigate the time delay effect as an independent factorin cold rolling tandem mills and put forward more detailedexplanation to the regenerative chatter mechanism and therelations between consecutive stands

2 Chatter Model

The dynamic rolling gap is shown as in Figure 1 when onlythe vertical roll vibration is considered Set the coordinates 119909and 119910 onto the center line of rolled piece and the work rollcenter line And the intersection point is set as the origin ofcoordinates 120590

119890and 120590

119889are the entry and exit tensile stress

V119890and V119889are the entry and exit strip velocity ℎ

119890and ℎ

119889are

the strip thickness at entry and exit V119903is the work roll linear

velocity ℎ119899is the strip thickness at neutral point 119909

119890 119909119899 and

119909119889are the position of entry point neutral point and exit

point

y

x

e

120590e he hn hd

d

R120601

120590d

xexn

xd

+

Figure 1 Dynamic rolling gap

The strip thickness at any 119909 location in the rolling gap isas follows

ℎ = ℎ119889+

1199092

119877

(1)

Due to the roll vertical vibration the continuity equationis modified as

Vℎ = V119890ℎ119890+ (119909 minus 119909

119890)ℎ119889 (2)

According to (1) and (2) the strip entry velocity V119890 strip

exit velocity V119889 strip entry position 119909

119890 and exit position 119909

119889

can be calculated as follows

V119890=

1

ℎ119890

(V119903ℎ119889+

V119903119909119899

2

119877

+ (119909119890minus 119909119899)ℎ119889)

V119889=

V119890ℎ119890+ (119909119889minus 119909119890)ℎ119889

ℎ119889+ 119909119889

2119877

119909119890= radic119877 (ℎ

119890minus ℎ119889)

119909119889=

119877ℎ119889

ℎ119889

2 (V119890ℎ119890minus 119909119890

ℎ119889)

(3)

Figure 2 illustrates the stresses acting on a vertical slabelement inside the rolling gap Employing the coulombfriction model 120591

119904= 120583119901 and the yield criterion 120590

119909= 119896119891minus 119901

the equilibrium equation for an elemental vertical section ofthe strip in the rolling gap in the 119909 direction is obtained

119889119901

119889119909

∓

2120583119901

ℎ

minus

1

ℎ

119889 (ℎ119896119891)

119889119909

= 0(4)

where the positive sign is for the exit side and the negativesign is for the entry side

Considering the strain hardening effect the deformationresistance of the rolled strip can be expressed as follows

119896119891= 1205900(119860 + ln(

ℎ0

ℎ

))

119899

(5)

Shock and Vibration 3

p p

120591120591

120590x 120590x

h h

120591 120591

p p

120590x +120597120590x120597x

dx 120590x +120597120590x120597x

dx

h +120597h

120597xh +

120597h

120597x

Figure 2 Slab analysis on a volume element of the entry region andexit region

1205900 119860 and 119899 are the material property parameter of the

rolled strip and can be decided by experiments ℎ0is the entry

strip thickness of multistand rolling millsApplying the boundary conditions

1199011003816100381610038161003816119909=119909119890

= 119896119891119890minus 120590119890

1199011003816100381610038161003816119909=119909119889

= 119896119891119889minus 120590119889

(6)

The rolling pressure at entry and exit side can be obtained

119901119890= (119896119891119890minus 120590119890)

ℎ119896119891

ℎ119890119896119891119890

exp(119906(2radic 119877ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

minus 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)))

119901119889= (119896119891119889minus 120590119889)

ℎ119896119891

ℎ119889119896119891119889

exp(119906(2radic 119877ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

minus 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)))

(7)

Letting 119901119890= 119901119889 the neutral point can be formulated as

follows

119909119899= radic119877ℎ

119889tan

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

minus

1

120583

ln(119896119891119889minus 120590119889

119896119891119890minus 120590119890

ℎ119890119896119891119890

ℎ119889119896119891119889

))

(8)

By integrating the rolling stresses the rolling force can bedetermined

119865 = int

119909119890

119909119899

119901119890119889119909 + int

119909119899

119909119889

119901119889119889119909 (9)

M

KC

1

2dhd

dF

Figure 3 Mill stand structure model

The dynamic rolling process model can be obtained byutilizing the first-order Taylor expansion

119889119865 = 1198651119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

(10)

119865119895119875119895119876119895 119895 = 1 2 3 4 5 are the partial derivatives of119865 V

119890 V119889

with respect to ℎ119889 ℎ119889 120590119890 120590119889 ℎ119890 respectively The calculation

of 119865119895 119875119895 119876119895 119895 = 1 2 3 4 5 can be found in the Appendix

Considering the work roll flattening effect in rolling theequivalent work roll radius is calculated by employing theHitchcock formula

119877 = 119877119908(1 +

16 (1 minus 1205822

)

120587119864119908

119865

119903

) (11)

where 119877119908is initial work roll radius 120582 is Poissonrsquos ratio for the

work roll material 119865 is rolling pressure 119903 is reduction and119864119908is elasticity modulus of work roll materialChatter in rolling is the result of the interaction between

rolling process and mill stand structure The rolling processmodel and stand structure model are coupled togetherthrough mechanical rolling parameters A one degree offreedom mill stand structure model is shown in Figure 3

By coupling the rolling processmodel and stand structuremodel together the single stand vibrationmodel is as follows

119872119889ℎ119889+ 119862119889

ℎ119889+ 119870119889ℎ

119889= 2119908119889119865

119889119890=

119864

119871119890

119889V119890

119889120590119889= minus

119864

119871119889

119889V119889

(12)

where 119871119890is the distance between the stand and the upstream

stand and 119871119889is the distance between the stand and down-

stream stand


Table 1 The initial rolling process parameters for the 1st and 2nd stand

ℎ1198901

(mm) ℎ1198891

ℎ1198902

(mm) ℎ1198892

(mm) 1205901198901

(Mpa) 1205901198891

(Mpa) 1205901198902

(Mpa)045 028 028 019 137 137 1371205901198892

(Mpa) 1199061

1199062

1198771199081

(mm) 1198771199082

(mm) 1205900(Mpa) 119860

98 0018 0015 276 291 8116 8116119899 119871

1198901(m) 119871

1198902(m) 119871

1198892(m) 119864

119908(Gpa) 119864 (Gpa) 119908 (m)

024 45 45 45 210 210 09

The regenerative chatter model for a two-stand rollingmill is as follows

119872119889ℎ1198891+ 119862119889

ℎ1198891+ 119870119889ℎ

1198891= 2119908119889119865

1

1198891198901=

119864

1198711198901

119889V1198901

1198891198891=

119864

1198711198902

(119889V1198902minus 119889V1198891)

119872119889ℎ1198892+ 119862119889

ℎ1198892+ 119870119889ℎ

1198892= 2119908119889119865

2

1198891198892= minus

119864

1198711198892

119889V1198892

(13)

where the second subscripts stand for the number of stands119864 is the elasticity modulus of roll piece material and 119908 is therolled strip width The rolled strip width spread is neglectedSo 119908 is constant

The relationship between the 2nd stand entry strip gaugevariation and the 1st stand exit strip gauge variation is asfollows

119889ℎ1198902(119905) = 119889ℎ

1198891(119905 minus 120591) (14)

where 120591 is the delay time decided by the interstand distanceand the entry velocity of 2nd stand

120591 =

1198711198902

V1198902

(15)

The initial rolling process parameters for the 1st and 2ndstand come from [7] and are shown in Table 1

To explore the effects of interstand coupling factors onregenerative chatter in detail four models were built in thispaperThe first one is a single stand vibrationmodelThe sec-ond one is a two-stand regenerative chattermodel which onlyconsiders the interstand tension coupling effectThe third oneis a two-stand regenerative chatter model which considersthe interstand tension coupling effect and the strip gaugevariation passed on to next stand but neglects the delay timeThe fourth one is a two-stand regenerative chatter modelwhich considers the interstand tension coupling effect and thestrip gauge variation passed on to next stand with time delayeffect

3 Stability Analysis

31 Stability Criterion To study the stabilities of the fourmodels proposed in Section 2 different stability criteria have

to be used to calculate the critical velocity for each modelThe first threemodels are nondelay differential systemsTheircritical velocities can be calculated by employing the Lya-punov indirect method According to the Lyapunov indirectmethod for a linear system = 119860119909 if all the real parts ofeigenvalues of matrix 119860 are negative the system is stable ifnot the system is unstable [13] Based on this method criticalvelocities of the first three models are obtained namely3482ms for the single 2nd stand model 3600ms for thesecond model and 3023ms for the third model Similar tothe single 2nd stand model the critical velocity of the single1st stand is 2478ms

The fourth model is a delay differential system Thestability criterion of time delay system is muchmore complexthan nondelay system The integral criterion is employed toanalyze the fourth model According to the integral criterionif the following inequation is true the delay differentialsystem is asymptotic stable [14]

int

infin

0

119871 (119908) gt

(119873 minus 1)

2

120587 (16)

where

119871 (119908) =

119877 (119908) 1198781015840

(119908) minus 119878 (119908) 1198771015840

(119908)

119877 (119908)2

+ 119878 (119908)2

(17)

119877(119908) and 119878(119908) are the real and imaginary parts of thedeterminant of the systemmatrix respectively119873 is the orderof the system For the fourth model 119873 is 7 The integrationis calculated with the help of MATLAB When the linearvelocity of the 2nd stand is 292ms

int

5000

0

119871 (119908) = 107 (18)

When the linear velocity of the 2nd stand is 293ms

int

5000

0

119871 (119908) = 45 (19)

So the critical velocity of fourth model is about 2925msComparing the critical velocities of the four models it

can be concluded that the interstand tension has just a littleeffect on the critical velocity of the chatter model The stripgauge variation passed on between stands reduces the criticalvelocity dramatically However the time delay effect as anindependent factor on critical velocity is very limited


Frequency (Hz)0 200 400

0

1000

2000

Am

plitu

de (P

a)(e)

246

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

200

400

(f)

246

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

246

times10minus9

0 005 01

0

1

(a)

minus1

times10minus9dhd2

(m)

t (s)

0 005 01

0

5000

(b)

minus5000

d120590e2

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d2

(Pa)

t (s)

Figure 4 Time and frequency domain of the single 2nd stand when V1199032= 3482ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)

and (f)) exit tension

32 Simulation and Analysis To study the regenerative chat-ter mechanism and relationship between stands in detailtime and frequency domain simulations of the four modelswere conducted As shown in Figure 4 the single 2nd standvibrates periodically when V

1199032is 3482ms and the frequency

is 246Hz Similarly the single 1st stand vibrates periodicallywhen V

1199031is 2478ms and the frequency is 209Hz as shown

in Figure 5The second model only considers the interstand tension

coupling effect between stands As can been seen in Figure 6the 1st stand takes on periodic oscillation state and thefrequency is 209Hz It is the same with the single 1st standmodel The beat phenomena appear in the 2nd stand and theinterstand tension The frequencies are 209Hz and 251Hz209Hz is the frequency of the 1st stand and 251Hz isapproximate to the frequency of the 2nd stand It is obviousthat the beat phenomena are caused by the interstand tensioncoupling effect between the 1st stand and 2nd stand Besidesthe amplitude of 2nd stand is far less than the 1st stand

It suggests that the vibration in 2nd stand raised by theinterstand tension variation is not violent

The third model considers the interstand tension cou-pling effect and the strip gauge variation passed on to the nextstand but neglects the time delay effect As shown in Figure 7the beat phenomena in the second model disappear Thefrequency of the 1st stand the 2nd stand and the interstandtension are all 212HzThe amplitudes of the 1st and 2nd standare in the same magnitude It indicates that the strip gaugevariation passed on to the next stand makes the 2nd standvibrate more violently

The fourth model considers the interstand tension cou-pling effect between stands the strip gauge variation passedon to the next stand and the time delay effect As shown inFigure 8 the time domain of 1st stand 2nd stand and theinterstand tension are all gourd-shaped which consist withthe test results in [1] The time domain of the 2nd stand lagsbehind the 1st stand by the delay time Due to the time delayeffect the frequency domain of the fourth model is more



0

1000

2000

Am

plitu

de (P

a)

(e)

209

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

100

200

(f)

209

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

0 005 01

0

5000

(b)

minus5000

d120590e1

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d1

(Pa)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

Figure 5 Time and frequency domain of the single 1st stand when V1199031= 2478ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)


complex than the thirdmodelThemain frequency is 209Hza little smaller than the third model but still in the range ofthe third octave mode chatter

Comparing the above four models a more detailedexplanation for regenerative chattermechanism in rolling canbe presented Consequent stands in tandem rolling mills arecoupled together through rolled piece One of these standsbecomes unstable and oscillates first for example the 119894thstand The 119894th stand vibration gives rise to the interstandtension vibration The interstand vibration makes the 119894 + 1thstand vibrate gently at onceThe amplitude of the 119894+1th standis far smaller than that of the 119894th stand by this time But thestrip gauge variation generated by the 119894th stand is passed onto the 119894+1th stand after delay time It aggravates the vibrationof the 119894 + 1th stand and makes the 119894th and 119894 + 1th standvibrate in the same amplitude The oscillation of the 119894 + 1thstand intensifies vibration of the interstand tension andfinallymakes the 119894th stand vibrate again

It can be concluded from the above analysis that timedelay effect as an independent factor has very limited effects

on both critical velocity and chatter frequency while makingthe stability analysis more complex But it does not mean thattime delay effect can be neglected when modeling the rollingsystem Enough works have been done to compare the firstsecond and the fourth model in [4] To study the time delayeffect as a single factor on multistand rolling system stabilityanalysis comparison of the third model and fourth modelwas done in the next section To simplify the writing modethe third model is called the nondelay system and the fourthmodel is called the delay system

4 Effects of Rolling Process Parameters

41 Effects of Friction Stability of a single stand is very sen-sitive to friction The critical velocity of a single stand growsrapidly with the increase of friction coefficient [2 7] Butthings are different for multistand systems as shown inFigure 9

When 1199061is small enough stability of the 1st stand gets

worse and it becomes more unstable than the 2nd stand


Frequency (Hz)

Am

plitu

de (P

a)

0 200 400

(f)

209

251100

200

0

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

2

4

(e)

209251

times10minus11

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 005 01

0

5

(b)

minus5

times10minus11

dhd2

(m)

t (s)

0 005 01

0

500

(c)

minus500

d120590d1

(Pa)

t (s)

Figure 6 The time and frequency domain of the second model V1199032= 3600ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand

work roll ((c) and (f)) interstand tension

For example as shown in Table 2 when 1199061is 0014 the critical

velocity of single 1st stand is 229ms According to theprinciple of mass conservation the corresponding velocityof 2nd stand is 331ms which is smaller than the single2nd stand critical velocity 348ms The 1st stand becomesunstable before the 2nd standThe critical velocity of the two-stand system is determined by the 1st stand As 119906

1increases

the 1st stand becomes more stable and critical velocity of thetwo-stand system increases sharply until 119906

1reaches a certain

point

Table 2The critical velocities for different stands when 1199061changes

1199061

0014 0018Critical velocity of single 1st stand V

1198881

(ms) 229 248

Corresponding critical velocity ofsingle 2nd stand V

1198882(ms) 331 363

Critical velocity of single 2nd stand V1198882

(ms) 348 348

More unstable stand 1st stand 2nd stand


Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

1000

2000

(f)

212

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

212

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

05

1

(e)

212

times10minus9

0 005 01

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9

dhd1

(m)

t (s)

0 005 01

0

5000

(c)

minus5000

d120590d1

(Pa)

t (s)

Figure 7The time and frequency domain of the third model when V1199032= 3023ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand


When 1199061is big enough 2nd stand becomesmore unstable

than 1st stand For example as shown in Table 2 when 1199061

is 0018 the critical velocity of the single 1st stand goes upto 248ms and the corresponding velocity of the single 2ndstand rises to 363ms which is bigger than critical velocityof the 2nd stand The 2nd stand becomes unstable first Thecritical velocity of the two-stand system is determined by the2nd stand As 119906

1increases the 1st stand becomesmore stable

but it has little effect on the 2nd stand The stability of 2ndstand is much worse than the 1st stand so the critical velocityof the two-stand system goes down as shown in Figure 9

When 1199061goes up the overall trend of critical velocities

are the same for the delay system and nondelay systemThere exists an optimal 119906

1 which relates to the maximum

critical velocity given that other rolling parameters remainunchanged But the optimal 119906

1values are different for the

delay and nondelay system The optimal 1199061for the delay

system is about 0016 while the optimal 1199061for the nondelay

system is about 002 Critical velocities for the delay systemare smaller than the nondelay system especially when 119906

1is

larger It can be explained by the idea that time delay effect hasa far greater impact on 2nd stand than 1st stand as the strip


0 200 4000

50

100

Frequency (Hz)

Am

plitu

de (P

a)

(f)

209

0 200 4000

5

Frequency (Hz)A

mpl

itude

(m)

(e)

209

times10minus11

0 200 4000

5

Am

plitu

de (m

)

(d)

Frequency (Hz)

209

times10minus11

0 05 1

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 05 1

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 05 1

0

2000

(c) t (s)

minus2000

d120590d1

(Pa)

Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd

stand work roll ((c) and (f)) interstand tension

gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction

Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906

2is small enough the 2nd stand is

more unstable and the critical velocity increases as 1199062goes

up When 1199062is big enough the stability of the whole system

becomes better as 1199062goes up The critical velocities for the

delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay

system is slightly smaller than that of the nondelay systemThe smaller the 119906

2is the more unstable and sensitive to 119906

2

the time delay system is

42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry

tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891

is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590

1198891 Therefore the critical

velocities of the delay and nondelay system both fall offas 1205901198891

increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590

1198891 as 120590

1198891is the entry

tension for the 2nd stand As can been seen from Table 3when 120590

1198891is rather small the 1st stand becomes unstable first

and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590

1198891is the exit

tension of the 1st stand But when 1205901198891

is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590

1198891is the entry tension of the 2nd stand


001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)

Delay systemNondelay system

u1

Figure 9 Comparison of stability charts for 1199061

001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2


When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590

1198891

43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871

1198902is the

distance between the 1st and 2nd stand If 1198711198902

is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871

1198902 the interstand tension variation decreases

and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising

50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)


Table 3 The critical velocities for different stands when 1205901198891

changes

1205901198891

(Mpa) 60 250Critical velocity of single 1st stand V

1198881(ms) 227 225

Corresponding critical velocity of single 2ndstand V

1198882(ms) 326 335


(ms) 367 331


At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871

1198902 the critical velocities of the 1st

and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871

1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)


the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902

is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871

1198902affects the exit

tension variation of 1st stand It can been seen in Table 4 thatwhen 119871

1198902is 35m the 2nd stand goes to unstable first but

when 1198711198902

is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871

1198902is quite bigger

than nondelay system for the delay time effect makes the 2ndstand more unstable

44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with

022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)

Figure 15 Comparison of stability charts for ℎ1198902


changes

1198711198902

(ms) 35 55Critical velocity of single 1st stand V

1198881

(ms) 2463 2487

Corresponding critical velocity of single2nd stand V

1198882(ms) 3605 3640


(ms) 3138 3789

More unstable stand 2nd stand 1st stand

the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width

45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ

1198902 is changed

If ℎ1198902

is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902

rises the 1st stand reduction gets smaller while the 2nd


Table 5The critical velocities for different standswhen ℎ1198902changes

ℎ1198902

(mm) 026 029Critical velocity of single 1st stand V

1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341


stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ

1198902

is 026mm the 1st stand is more unstable However whenℎ1198902

increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ

1198902for the

multistand system which is related to the maximum criticalvelocity as shown in Figure 11

The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ

1198902are different for the two systems

because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ

1198902

for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ

1198902for the delay system is

more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills

The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ

1198902is bigger than a certain point the critical

velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ

1198902value for the two-

stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex

5 Conclusions

In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to

the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved

Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)

The intermedia variables are given as follows

(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)

Conflict of Interests

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

Acknowledgments

The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367

References

[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998

[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998

[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015

[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013

[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001

[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998

[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013

[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003

[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015

[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014

[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013


[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014

[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999

[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986

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DistributedSensor Networks



becomes a delay differential system and integral criterion wasemployed to find its critical velocity Comparative analysisof the three models presented a potential explanation to theregenerative mechanism [4]

Based on Zhao and Ehmannrsquos andHursquosmodel more com-plex regenerative chattermodels are proposed in recent yearsKimura et al constructed a five-stand regenerative chattermodel by coupling a five-degree freedom mill stand modeland the dynamic rollingmodel built in [7] togetherThe simu-lation results show that there exists an optimal friction of thefifth stand which relates to the maximum critical velocity ofthe whole tandem rolling mills [8] Instead of static analysisNiroomand et al used wave propagation theory in elasticsolids to formulate the dynamic tensile stress variationbetween two consequent stands and built a new two-standregenerative chatter model The simulation results show thatthe errors of critical velocity and chatter frequency are smallerwith the experimental results when the wave propagationtheory is applied [9] Considering the complex friction con-dition in the rolling gap Heidari et al built a regenerativechatter model using the unsteady lubrication model assum-ing that the rolling gap is in full film regime The effects ofrolling lubricant on the stability of tandem rolling mills werediscussed [10] Based on these regenerative chatter modelsoptimization of multistand rolling process parameters wereconducted to avoid chatter phenomenon using the combina-tion of neural networks and genetic algorithms [11 12]

All these regenerative chatter models have considered thetime delay effect which makes the tandem rolling mills adelay differential system Itmeans thatmore complex stabilitycriterion has to be employed to analyze the delay differen-tial multistand system which results in huge computationHowever the time delay effect on the stability of tandemrolling mills has not been analyzed as an independent factorMore research work has to be done to decide whether theeffect of time delay itself on the stability of tandem rollingmills is worth the great amount of computation Besidesthe relationship between consequent stands is not clearlydescribed though the interaction factors between stands havebeen pointed out The major objective pursed in this paper isto investigate the time delay effect as an independent factorin cold rolling tandem mills and put forward more detailedexplanation to the regenerative chatter mechanism and therelations between consecutive stands

2 Chatter Model

The dynamic rolling gap is shown as in Figure 1 when onlythe vertical roll vibration is considered Set the coordinates 119909and 119910 onto the center line of rolled piece and the work rollcenter line And the intersection point is set as the origin ofcoordinates 120590

119890and 120590

119889are the entry and exit tensile stress

V119890and V119889are the entry and exit strip velocity ℎ

119890and ℎ

119889are

the strip thickness at entry and exit V119903is the work roll linear

velocity ℎ119899is the strip thickness at neutral point 119909

119890 119909119899 and

119909119889are the position of entry point neutral point and exit

point

y

x

e

120590e he hn hd

d

R120601

120590d

xexn

xd

+

Figure 1 Dynamic rolling gap

The strip thickness at any 119909 location in the rolling gap isas follows

ℎ = ℎ119889+

1199092

119877

(1)

Due to the roll vertical vibration the continuity equationis modified as

Vℎ = V119890ℎ119890+ (119909 minus 119909

119890)ℎ119889 (2)

According to (1) and (2) the strip entry velocity V119890 strip

exit velocity V119889 strip entry position 119909

119890 and exit position 119909

119889

can be calculated as follows

V119890=

1

ℎ119890

(V119903ℎ119889+

V119903119909119899

2

119877

+ (119909119890minus 119909119899)ℎ119889)

V119889=

V119890ℎ119890+ (119909119889minus 119909119890)ℎ119889

ℎ119889+ 119909119889

2119877

119909119890= radic119877 (ℎ

119890minus ℎ119889)

119909119889=

119877ℎ119889

ℎ119889

2 (V119890ℎ119890minus 119909119890

ℎ119889)

(3)

Figure 2 illustrates the stresses acting on a vertical slabelement inside the rolling gap Employing the coulombfriction model 120591

119904= 120583119901 and the yield criterion 120590

119909= 119896119891minus 119901

the equilibrium equation for an elemental vertical section ofthe strip in the rolling gap in the 119909 direction is obtained

119889119901

119889119909

∓

2120583119901

ℎ

minus

1

ℎ

119889 (ℎ119896119891)

119889119909

= 0(4)

where the positive sign is for the exit side and the negativesign is for the entry side

Considering the strain hardening effect the deformationresistance of the rolled strip can be expressed as follows

119896119891= 1205900(119860 + ln(

ℎ0

ℎ

))

119899

(5)


p p

120591120591

120590x 120590x

h h

120591 120591

p p

120590x +120597120590x120597x

dx 120590x +120597120590x120597x

dx

h +120597h

120597xh +

120597h

120597x





1199011003816100381610038161003816119909=119909119890

= 119896119891119890minus 120590119890

1199011003816100381610038161003816119909=119909119889

= 119896119891119889minus 120590119889

(6)


119901119890= (119896119891119890minus 120590119890)

ℎ119896119891

ℎ119890119896119891119890

exp(119906(2radic 119877ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

minus 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)))

119901119889= (119896119891119889minus 120590119889)

ℎ119896119891

ℎ119889119896119891119889

exp(119906(2radic 119877ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

minus 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)))

(7)


follows

119909119899= radic119877ℎ

119889tan

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

minus

1

120583

ln(119896119891119889minus 120590119889

119896119891119890minus 120590119890

ℎ119890119896119891119890

ℎ119889119896119891119889

))

(8)


119865 = int

119909119890

119909119899

119901119890119889119909 + int

119909119899

119909119889

119901119889119889119909 (9)

M

KC

1

2dhd

dF



119889119865 = 1198651119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

(10)


119890 V119889




119877 = 119877119908(1 +

16 (1 minus 1205822

)

120587119864119908

119865

119903

) (11)





119872119889ℎ119889+ 119862119889

ℎ119889+ 119870119889ℎ

119889= 2119908119889119865

119889119890=

119864

119871119890

119889V119890

119889120590119889= minus

119864

119871119889

119889V119889

(12)



stream stand



ℎ1198901

(mm) ℎ1198891

ℎ1198902

(mm) ℎ1198892

(mm) 1205901198901

(Mpa) 1205901198891

(Mpa) 1205901198902

(Mpa)045 028 028 019 137 137 1371205901198892

(Mpa) 1199061

1199062

1198771199081

(mm) 1198771199082

(mm) 1205900(Mpa) 119860

98 0018 0015 276 291 8116 8116119899 119871

1198901(m) 119871

1198902(m) 119871

1198892(m) 119864

119908(Gpa) 119864 (Gpa) 119908 (m)

024 45 45 45 210 210 09


119872119889ℎ1198891+ 119862119889

ℎ1198891+ 119870119889ℎ

1198891= 2119908119889119865

1

1198891198901=

119864

1198711198901

119889V1198901

1198891198891=

119864

1198711198902

(119889V1198902minus 119889V1198891)

119872119889ℎ1198892+ 119862119889

ℎ1198892+ 119870119889ℎ

1198892= 2119908119889119865

2

1198891198892= minus

119864

1198711198892

119889V1198892

(13)



119889ℎ1198902(119905) = 119889ℎ

1198891(119905 minus 120591) (14)


120591 =

1198711198902

V1198902

(15)







int

infin

0

119871 (119908) gt

(119873 minus 1)

2

120587 (16)

where

119871 (119908) =

119877 (119908) 1198781015840

(119908) minus 119878 (119908) 1198771015840

(119908)

119877 (119908)2

+ 119878 (119908)2

(17)


int

5000

0

119871 (119908) = 107 (18)


int

5000

0

119871 (119908) = 45 (19)





0

1000

2000

Am

plitu

de (P

a)(e)

246

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

200

400

(f)

246

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

246

times10minus9

0 005 01

0

1

(a)

minus1

times10minus9dhd2

(m)

t (s)

0 005 01

0

5000

(b)

minus5000

d120590e2

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d2

(Pa)

t (s)














0

1000

2000

Am

plitu

de (P

a)

(e)

209

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

100

200

(f)

209

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

0 005 01

0

5000

(b)

minus5000

d120590e1

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d1

(Pa)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)












Frequency (Hz)

Am

plitu

de (P

a)

0 200 400

(f)

209

251100

200

0

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

2

4

(e)

209251

times10minus11

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 005 01

0

5

(b)

minus5

times10minus11

dhd2

(m)

t (s)

0 005 01

0

500

(c)

minus500

d120590d1

(Pa)

t (s)





1increases


1reaches a certain

point


1199061


1198881

(ms) 229 248


1198882(ms) 331 363


(ms) 348 348



Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

1000

2000

(f)

212

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

212

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

05

1

(e)

212

times10minus9

0 005 01

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9

dhd1

(m)

t (s)

0 005 01

0

5000

(c)

minus5000

d120590d1

(Pa)

t (s)
















1is



0 200 4000

50

100

Frequency (Hz)

Am

plitu

de (P

a)

(f)

209

0 200 4000

5

Frequency (Hz)A

mpl

itude

(m)

(e)

209

times10minus11

0 200 4000

5

Am

plitu

de (m

)

(d)

Frequency (Hz)

209

times10minus11

0 05 1

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 05 1

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 05 1

0

2000

(c) t (s)

minus2000

d120590d1

(Pa)












2








1198891 as 120590

1198891is the entry




1198891is the exit





001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)


u1


001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2



1198891


1198902is the





50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)



changes

1205901198891


1198881(ms) 227 225


1198882(ms) 326 335


(ms) 367 331





1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










p p

120591120591

120590x 120590x

h h

120591 120591

p p

120590x +120597120590x120597x

dx 120590x +120597120590x120597x

dx

h +120597h

120597xh +

120597h

120597x





1199011003816100381610038161003816119909=119909119890

= 119896119891119890minus 120590119890

1199011003816100381610038161003816119909=119909119889

= 119896119891119889minus 120590119889

(6)


119901119890= (119896119891119890minus 120590119890)

ℎ119896119891

ℎ119890119896119891119890

exp(119906(2radic 119877ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

minus 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)))

119901119889= (119896119891119889minus 120590119889)

ℎ119896119891

ℎ119889119896119891119889

exp(119906(2radic 119877ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

minus 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)))

(7)


follows

119909119899= radic119877ℎ

119889tan

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

minus

1

120583

ln(119896119891119889minus 120590119889

119896119891119890minus 120590119890

ℎ119890119896119891119890

ℎ119889119896119891119889

))

(8)


119865 = int

119909119890

119909119899

119901119890119889119909 + int

119909119899

119909119889

119901119889119889119909 (9)

M

KC

1

2dhd

dF



119889119865 = 1198651119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

(10)


119890 V119889




119877 = 119877119908(1 +

16 (1 minus 1205822

)

120587119864119908

119865

119903

) (11)





119872119889ℎ119889+ 119862119889

ℎ119889+ 119870119889ℎ

119889= 2119908119889119865

119889119890=

119864

119871119890

119889V119890

119889120590119889= minus

119864

119871119889

119889V119889

(12)



stream stand



ℎ1198901

(mm) ℎ1198891

ℎ1198902

(mm) ℎ1198892

(mm) 1205901198901

(Mpa) 1205901198891

(Mpa) 1205901198902

(Mpa)045 028 028 019 137 137 1371205901198892

(Mpa) 1199061

1199062

1198771199081

(mm) 1198771199082

(mm) 1205900(Mpa) 119860

98 0018 0015 276 291 8116 8116119899 119871

1198901(m) 119871

1198902(m) 119871

1198892(m) 119864

119908(Gpa) 119864 (Gpa) 119908 (m)

024 45 45 45 210 210 09


119872119889ℎ1198891+ 119862119889

ℎ1198891+ 119870119889ℎ

1198891= 2119908119889119865

1

1198891198901=

119864

1198711198901

119889V1198901

1198891198891=

119864

1198711198902

(119889V1198902minus 119889V1198891)

119872119889ℎ1198892+ 119862119889

ℎ1198892+ 119870119889ℎ

1198892= 2119908119889119865

2

1198891198892= minus

119864

1198711198892

119889V1198892

(13)



119889ℎ1198902(119905) = 119889ℎ

1198891(119905 minus 120591) (14)


120591 =

1198711198902

V1198902

(15)







int

infin

0

119871 (119908) gt

(119873 minus 1)

2

120587 (16)

where

119871 (119908) =

119877 (119908) 1198781015840

(119908) minus 119878 (119908) 1198771015840

(119908)

119877 (119908)2

+ 119878 (119908)2

(17)


int

5000

0

119871 (119908) = 107 (18)


int

5000

0

119871 (119908) = 45 (19)





0

1000

2000

Am

plitu

de (P

a)(e)

246

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

200

400

(f)

246

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

246

times10minus9

0 005 01

0

1

(a)

minus1

times10minus9dhd2

(m)

t (s)

0 005 01

0

5000

(b)

minus5000

d120590e2

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d2

(Pa)

t (s)














0

1000

2000

Am

plitu

de (P

a)

(e)

209

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

100

200

(f)

209

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

0 005 01

0

5000

(b)

minus5000

d120590e1

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d1

(Pa)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)












Frequency (Hz)

Am

plitu

de (P

a)

0 200 400

(f)

209

251100

200

0

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

2

4

(e)

209251

times10minus11

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 005 01

0

5

(b)

minus5

times10minus11

dhd2

(m)

t (s)

0 005 01

0

500

(c)

minus500

d120590d1

(Pa)

t (s)





1increases


1reaches a certain

point


1199061


1198881

(ms) 229 248


1198882(ms) 331 363


(ms) 348 348



Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

1000

2000

(f)

212

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

212

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

05

1

(e)

212

times10minus9

0 005 01

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9

dhd1

(m)

t (s)

0 005 01

0

5000

(c)

minus5000

d120590d1

(Pa)

t (s)
















1is



0 200 4000

50

100

Frequency (Hz)

Am

plitu

de (P

a)

(f)

209

0 200 4000

5

Frequency (Hz)A

mpl

itude

(m)

(e)

209

times10minus11

0 200 4000

5

Am

plitu

de (m

)

(d)

Frequency (Hz)

209

times10minus11

0 05 1

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 05 1

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 05 1

0

2000

(c) t (s)

minus2000

d120590d1

(Pa)












2








1198891 as 120590

1198891is the entry




1198891is the exit





001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)


u1


001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2



1198891


1198902is the





50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)



changes

1205901198891


1198881(ms) 227 225


1198882(ms) 326 335


(ms) 367 331





1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











ℎ1198901

(mm) ℎ1198891

ℎ1198902

(mm) ℎ1198892

(mm) 1205901198901

(Mpa) 1205901198891

(Mpa) 1205901198902

(Mpa)045 028 028 019 137 137 1371205901198892

(Mpa) 1199061

1199062

1198771199081

(mm) 1198771199082

(mm) 1205900(Mpa) 119860

98 0018 0015 276 291 8116 8116119899 119871

1198901(m) 119871

1198902(m) 119871

1198892(m) 119864

119908(Gpa) 119864 (Gpa) 119908 (m)

024 45 45 45 210 210 09


119872119889ℎ1198891+ 119862119889

ℎ1198891+ 119870119889ℎ

1198891= 2119908119889119865

1

1198891198901=

119864

1198711198901

119889V1198901

1198891198891=

119864

1198711198902

(119889V1198902minus 119889V1198891)

119872119889ℎ1198892+ 119862119889

ℎ1198892+ 119870119889ℎ

1198892= 2119908119889119865

2

1198891198892= minus

119864

1198711198892

119889V1198892

(13)



119889ℎ1198902(119905) = 119889ℎ

1198891(119905 minus 120591) (14)


120591 =

1198711198902

V1198902

(15)







int

infin

0

119871 (119908) gt

(119873 minus 1)

2

120587 (16)

where

119871 (119908) =

119877 (119908) 1198781015840

(119908) minus 119878 (119908) 1198771015840

(119908)

119877 (119908)2

+ 119878 (119908)2

(17)


int

5000

0

119871 (119908) = 107 (18)


int

5000

0

119871 (119908) = 45 (19)





0

1000

2000

Am

plitu

de (P

a)(e)

246

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

200

400

(f)

246

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

246

times10minus9

0 005 01

0

1

(a)

minus1

times10minus9dhd2

(m)

t (s)

0 005 01

0

5000

(b)

minus5000

d120590e2

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d2

(Pa)

t (s)














0

1000

2000

Am

plitu

de (P

a)

(e)

209

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

100

200

(f)

209

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

0 005 01

0

5000

(b)

minus5000

d120590e1

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d1

(Pa)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)












Frequency (Hz)

Am

plitu

de (P

a)

0 200 400

(f)

209

251100

200

0

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

2

4

(e)

209251

times10minus11

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 005 01

0

5

(b)

minus5

times10minus11

dhd2

(m)

t (s)

0 005 01

0

500

(c)

minus500

d120590d1

(Pa)

t (s)





1increases


1reaches a certain

point


1199061


1198881

(ms) 229 248


1198882(ms) 331 363


(ms) 348 348



Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

1000

2000

(f)

212

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

212

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

05

1

(e)

212

times10minus9

0 005 01

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9

dhd1

(m)

t (s)

0 005 01

0

5000

(c)

minus5000

d120590d1

(Pa)

t (s)
















1is



0 200 4000

50

100

Frequency (Hz)

Am

plitu

de (P

a)

(f)

209

0 200 4000

5

Frequency (Hz)A

mpl

itude

(m)

(e)

209

times10minus11

0 200 4000

5

Am

plitu

de (m

)

(d)

Frequency (Hz)

209

times10minus11

0 05 1

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 05 1

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 05 1

0

2000

(c) t (s)

minus2000

d120590d1

(Pa)












2








1198891 as 120590

1198891is the entry




1198891is the exit





001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)


u1


001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2



1198891


1198902is the





50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)



changes

1205901198891


1198881(ms) 227 225


1198882(ms) 326 335


(ms) 367 331





1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

1000

2000

Am

plitu

de (P

a)(e)

246

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

200

400

(f)

246

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

246

times10minus9

0 005 01

0

1

(a)

minus1

times10minus9dhd2

(m)

t (s)

0 005 01

0

5000

(b)

minus5000

d120590e2

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d2

(Pa)

t (s)














0

1000

2000

Am

plitu

de (P

a)

(e)

209

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

100

200

(f)

209

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

0 005 01

0

5000

(b)

minus5000

d120590e1

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d1

(Pa)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)












Frequency (Hz)

Am

plitu

de (P

a)

0 200 400

(f)

209

251100

200

0

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

2

4

(e)

209251

times10minus11

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 005 01

0

5

(b)

minus5

times10minus11

dhd2

(m)

t (s)

0 005 01

0

500

(c)

minus500

d120590d1

(Pa)

t (s)





1increases


1reaches a certain

point


1199061


1198881

(ms) 229 248


1198882(ms) 331 363


(ms) 348 348



Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

1000

2000

(f)

212

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

212

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

05

1

(e)

212

times10minus9

0 005 01

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9

dhd1

(m)

t (s)

0 005 01

0

5000

(c)

minus5000

d120590d1

(Pa)

t (s)
















1is



0 200 4000

50

100

Frequency (Hz)

Am

plitu

de (P

a)

(f)

209

0 200 4000

5

Frequency (Hz)A

mpl

itude

(m)

(e)

209

times10minus11

0 200 4000

5

Am

plitu

de (m

)

(d)

Frequency (Hz)

209

times10minus11

0 05 1

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 05 1

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 05 1

0

2000

(c) t (s)

minus2000

d120590d1

(Pa)












2








1198891 as 120590

1198891is the entry




1198891is the exit





001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)


u1


001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2



1198891


1198902is the





50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)



changes

1205901198891


1198881(ms) 227 225


1198882(ms) 326 335


(ms) 367 331





1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

1000

2000

Am

plitu

de (P

a)

(e)

209

Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

100

200

(f)

209

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

0 005 01

0

5000

(b)

minus5000

d120590e1

(Pa)

t (s)

0 005 01

0

1000

(c)

minus1000

d120590d1

(Pa)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)












Frequency (Hz)

Am

plitu

de (P

a)

0 200 400

(f)

209

251100

200

0

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

2

4

(e)

209251

times10minus11

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 005 01

0

5

(b)

minus5

times10minus11

dhd2

(m)

t (s)

0 005 01

0

500

(c)

minus500

d120590d1

(Pa)

t (s)





1increases


1reaches a certain

point


1199061


1198881

(ms) 229 248


1198882(ms) 331 363


(ms) 348 348



Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

1000

2000

(f)

212

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

212

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

05

1

(e)

212

times10minus9

0 005 01

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9

dhd1

(m)

t (s)

0 005 01

0

5000

(c)

minus5000

d120590d1

(Pa)

t (s)
















1is



0 200 4000

50

100

Frequency (Hz)

Am

plitu

de (P

a)

(f)

209

0 200 4000

5

Frequency (Hz)A

mpl

itude

(m)

(e)

209

times10minus11

0 200 4000

5

Am

plitu

de (m

)

(d)

Frequency (Hz)

209

times10minus11

0 05 1

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 05 1

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 05 1

0

2000

(c) t (s)

minus2000

d120590d1

(Pa)












2








1198891 as 120590

1198891is the entry




1198891is the exit





001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)


u1


001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2



1198891


1198902is the





50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)



changes

1205901198891


1198881(ms) 227 225


1198882(ms) 326 335


(ms) 367 331





1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Frequency (Hz)

Am

plitu

de (P

a)

0 200 400

(f)

209

251100

200

0

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

209

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

2

4

(e)

209251

times10minus11

0 005 01

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 005 01

0

5

(b)

minus5

times10minus11

dhd2

(m)

t (s)

0 005 01

0

500

(c)

minus500

d120590d1

(Pa)

t (s)





1increases


1reaches a certain

point


1199061


1198881

(ms) 229 248


1198882(ms) 331 363


(ms) 348 348



Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

1000

2000

(f)

212

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

212

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

05

1

(e)

212

times10minus9

0 005 01

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9

dhd1

(m)

t (s)

0 005 01

0

5000

(c)

minus5000

d120590d1

(Pa)

t (s)
















1is



0 200 4000

50

100

Frequency (Hz)

Am

plitu

de (P

a)

(f)

209

0 200 4000

5

Frequency (Hz)A

mpl

itude

(m)

(e)

209

times10minus11

0 200 4000

5

Am

plitu

de (m

)

(d)

Frequency (Hz)

209

times10minus11

0 05 1

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 05 1

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 05 1

0

2000

(c) t (s)

minus2000

d120590d1

(Pa)












2








1198891 as 120590

1198891is the entry




1198891is the exit





001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)


u1


001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2



1198891


1198902is the





50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)



changes

1205901198891


1198881(ms) 227 225


1198882(ms) 326 335


(ms) 367 331





1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Frequency (Hz)

Am

plitu

de (P

a)

0 200 4000

1000

2000

(f)

212

0 200 4000

05

1

Frequency (Hz)

Am

plitu

de (m

)

(d)

212

times10minus9

Frequency (Hz)

Am

plitu

de (m

)

0 200 4000

05

1

(e)

212

times10minus9

0 005 01

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 005 01

0

1

(a)

minus1

times10minus9

dhd1

(m)

t (s)

0 005 01

0

5000

(c)

minus5000

d120590d1

(Pa)

t (s)
















1is



0 200 4000

50

100

Frequency (Hz)

Am

plitu

de (P

a)

(f)

209

0 200 4000

5

Frequency (Hz)A

mpl

itude

(m)

(e)

209

times10minus11

0 200 4000

5

Am

plitu

de (m

)

(d)

Frequency (Hz)

209

times10minus11

0 05 1

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 05 1

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 05 1

0

2000

(c) t (s)

minus2000

d120590d1

(Pa)












2








1198891 as 120590

1198891is the entry




1198891is the exit





001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)


u1


001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2



1198891


1198902is the





50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)



changes

1205901198891


1198881(ms) 227 225


1198882(ms) 326 335


(ms) 367 331





1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 200 4000

50

100

Frequency (Hz)

Am

plitu

de (P

a)

(f)

209

0 200 4000

5

Frequency (Hz)A

mpl

itude

(m)

(e)

209

times10minus11

0 200 4000

5

Am

plitu

de (m

)

(d)

Frequency (Hz)

209

times10minus11

0 05 1

0

1

(a)

minus1

times10minus9dhd1

(m)

t (s)

0 05 1

0

2

(b)

minus2

times10minus9

dhd2

(m)

t (s)

0 05 1

0

2000

(c) t (s)

minus2000

d120590d1

(Pa)












2








1198891 as 120590

1198891is the entry




1198891is the exit





001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)


u1


001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2



1198891


1198902is the





50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)



changes

1205901198891


1198881(ms) 227 225


1198882(ms) 326 335


(ms) 367 331





1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










001 0015 002 0025 00323

24

25

26

27

28

29

Criti

cal s

peed

(ms

)


u1


001 0015 002 0025 00322

24

26

28

30

32

34

Criti

cal s

peed

(ms

)


u2



1198891


1198902is the





50 100 150 200 250 30025

26

27

28

29

30

31

32

Criti

cal s

peed

(ms

)


120590d1 (MPa)


3 35 4 45 5 55 623

24

25

26

27

28

29

30Cr

itica

l spe

ed (m

s)


Le2 (m)



changes

1205901198891


1198881(ms) 227 225


1198882(ms) 326 335


(ms) 367 331





1198902affects


1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 15 2 25 3 35 420

25

30

35

40

45

Criti

cal s

peed

(ms

)


E (GPa)


05 1 1526

27

28

29

30

31

32

33

34

Criti

cal s

peed

(ms

)


w (m)







when 1198711198902





022 024 026 028 03 032 034 03624

26

28

30

32

34

36

38

40

Criti

cal s

peed

(ms

)


he2 (mm)



changes

1198711198902


1198881

(ms) 2463 2487


1198882(ms) 3605 3640


(ms) 3138 3789




1198902 is changed

If ℎ1198902





ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











ℎ1198902


1198881

(ms) 242 252


1198882(ms) 331 380


(ms) 368 341



1198902



1198902for the





1198902









5 Conclusions



Appendix

Consider119889119865 = 119865

1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890

1198651= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 + int

119909119890

119909119899

(

120597119901119890

120597ℎ119889

)

119904

119889119909

+ (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119889

)

119904

1198652= (

120597119865

120597ℎ119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597ℎ119889

)

119904

119889119909 minus (119896119891119889minus 120590119889)(

120597119909119889

120597ℎ119889

)

119904

1198653= (

120597119865

120597120590119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597120590119890

)

119904

119889119909

1198654= (

120597119865

120597120590119889

)

119904

= int

119909119899

119909119889

(

120597119901119889

120597120590119889

)

119904

119889119909

1198655= (

120597119865

120597ℎ119890

)

119904

= int

119909119890

119909119899

(

120597119901119890

120597ℎ119890

)

119904

119889119909 + (119896119891119890minus 120590119890) (

120597119909119890

120597ℎ119890

)

119904

119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890

1198751= (

120597V119890

120597ℎ119889

)

119904

=

V119903

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198752= (

120597V119890

120597ℎ119889

)

119904

=

119909119890minus 119909119899

ℎ119890

+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119889

)

119904

1198753= (

120597V119890

120597120590119890

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119890

)

119904

1198754= (

120597V119890

120597120590119889

)

119904

=

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597120590119889

)

119904

1198755= (

120597V119890

120597ℎ119890

)

119904

= minus

V119903ℎ119899

ℎ119890

2+

2V119903119909119899

ℎ119890119877

(

120597119909119899

120597ℎ119890

)

119904


119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890

1198761= (

120597V119889

120597ℎ119889

)

119904

= minus

V119890ℎ119890

ℎ119889

2+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198762= (

120597V119889

120597ℎ119889

)

119904

=

minus119909119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119889

)

119904

1198763= (

120597V119889

120597120590119890

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119890

)

119904

1198764= (

120597V119889

120597120590119889

)

119904

=

ℎ119890

ℎ119889

(

120597V119890

120597120590119889

)

119904

1198765= (

120597V119889

120597ℎ119890

)

119904

=

V119890

ℎ119889

+

ℎ119890

ℎ119889

(

120597V119890

120597ℎ119890

)

119904

(A1)


(

120597119909119899

120597ℎ119889

)

119904

= (minus

1198921

8119906

+

1

4119906

minus

119909119890

4ℎ119890

) sec (119892)2 + radic 119877

4ℎ119889

sdot tan (119892) + (120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

+ (

120597119909119899

120597119896119891119889

)

119904

sdot (

120597119896119891119889

120597ℎ119889

)

119904

1198921 = log(ℎ119890119896119891119890

ℎ119889119896119891119889

119896119891119889minus 120590119889

119896119891119890minus 120590119890

)

119892 =

1

4

radicℎ119889

119877

(2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

+ 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

) minus

1198921

119906

)

(

120597119909119899

120597119909119890

)

119904

=

ℎ119889sec (119892)2

2ℎ119890

(

120597119909119899

120597119896119891119889

)

119904

=

ℎ119889

4119906

(

1

119896119891119889

minus

1

119896119891119889minus 120590119889

) sec (119892)2

(

120597119896119891119889

120597ℎ119889

) = minus

1198991205900

ℎ119889

(119860 + log(ℎ0

ℎ119889

))

119899minus1

(

120597119909119899

120597ℎ119889

)

119904

=

1

2

sec (119892)2 (120597119909119889

120597ℎ119889

)

119904

(

120597119909119899

120597120590119890

)

119904

= minus

ℎ119889sec (119892)2

4119906 (119896119891119890minus 120590119890)

(

120597119909119899

120597120590119889

)

119904

=

ℎ119889sec (119892)2

4119906 (119896119891119889minus 120590119889)

(

120597119909119899

120597ℎ119890

)

119904

= minus

ℎ119889sec (119892)2

4119906ℎ119890

+ (

120597119909119899

120597119909119890

)

119904

(

120597119909119890

120597ℎ119890

)

119904

+ (

120597119909119899

120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

(

120597119909119899

120597119896119891119890

)

119904

=

ℎ119889

4119906

(

1

119896119891119890minus 120590119890

minus

1

119896119891119890

) sec (119892)2

(

120597119896119891119890

120597ℎ119890

)

119904

= minus

1198991205900

ℎ119890

(119860 + log(ℎ0

ℎ119890

))

119899minus1

(

120597119909119890

120597ℎ119889

)

119904

= minusradic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119890

120597ℎ119890

)

119904

= radic

119877

4 (ℎ119890minus ℎ119889)

(

120597119909119889

120597ℎ119889

)

119904

=

119877ℎ119889

2V119903ℎ119899

(

120597119901119890

120597ℎ119889

)

119904

= (

120597119901119890

120597ℎ

)

119904

+ (

120597119901119890

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867119890

)

119904

sdot (

120597119867119890

120597ℎ119889

)

119904

+ (

120597119901119890

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

(

120597119901119890

120597ℎ

)

119904

=

(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119901119890

120597119896119891

)

119904

=

(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119896119891

120597ℎ119889

)

119904

= minus

1205900119899

ℎ

(119860 + ln(ℎ0

ℎ

))

119899minus1

(

120597119901119890

120597119867119890

)

119904

=

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867119890

120597ℎ119889

)

119904

= minus

119877arctan (119909119890radic119877ℎ

119889)

ℎ119889

2

radic119877ℎ119889

minus

119909119890

ℎ119889ℎ119890

+ (

120597119867119890

120597119909119890

)

119904

(

120597119909119890

120597ℎ119889

)

119904

(

120597119901119890

120597119867

)

119904

= minus

(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)

ℎ119890119896119891119890

(

120597119867

120597ℎ119889

)

119904

= minus

119877arctan (119909radic119877ℎ119889)

ℎ119889

2

radic119877ℎ119889

minus

119909

ℎ119889ℎ

(

120597119901119890

120597120590119890

)

119904

= minus

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890


(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(

120597119901e120597ℎe)

119904

= (

120597119901e120597119896119891119890

)

119904

(

120597119896119891119890

120597ℎ119890

)

119904

+ (

120597119901e120597119867119890

)

119904

(

120597119867119890

120597119909119890

)

119904

sdot (

120597119909119890

120597ℎ119890

)

119904

minus

(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890

2

119896119891119890

(

120597119901119890

120597119896119891119890

)

119904

=

ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

minus

(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)

ℎ119890119896119891119890

2

(

120597119867119890

120597119909119890

)

119904

=

2

ℎ119890

(

120597119901d120597ℎ119889

)

119904

= (

120597119901119889

120597119896119891119889

)

119904

(

120597119896119891119889

120597ℎ119889

)

119904

+ (

120597119901119889

120597ℎ

)

119904

+ (

120597119901119889

120597119896119891

)

119904

(

120597119896119891

120597ℎ119889

)

119904

+ (

120597119901119889

120597119867

)

119904

(

120597119867

120597ℎ119889

)

119904

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889

2

119896119891119889

(

120597119901119889

120597119896119891119889

)

119904

=

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

minus

(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

2

(

120597119901119889

120597ℎ

)

119904

=

(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119896119891

)

119904

=

(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867119889

)

119904

= minus

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597119867

)

119904

=

(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867

119889)

ℎ119889119896119891119889

(

120597119901119889

120597ℎ119889

)

119904

= (

120597119901119889

120597119867119889

)

119904

(

120597119867119889

120597119909119889

)

119904

(

120597119909119889

120597ℎ119889

)

119904

(

120597119867119889

120597119909119889

)

119904

=

2

ℎ119889

(

120597119901119889

120597120590119889

)

119904

= minus

ℎ119896119891119890119906(119867minus119867

119889)

ℎ119889119896119891119889

119867119890= 2radic

119877

ℎ119889

tanminus1 (119909119890

radic119877ℎ119889

)

119867119889= 2radic

119877

ℎ119889

tanminus1 (119909119889

radic119877ℎ119889

)

119867 = 2radic

119877

ℎ119889

tanminus1 ( 119909

radic119877ℎ119889

)

(A2)



Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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