DTI I COPY

NASA AVSCOMTechnical Memorandum 102349 Technical Memorandum 89-C-006

Dynamic Analysis of Geared00 Rotors by Finite Elements0

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Ahmet Kahraman, H. Nevzat Ozguven, and Donald R. HouserThe Ohio State UniversityColumbus, Ohio .1 Cand ~~E T

• , 6 EC 3 11990James J. ZakrajsekLewis Research CenterCleveland, Ohio

January 1990

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NSA AVIATION DSYSTEMS COMMAND

~ AVIATION R&T ACIVITY

DYNAMIC ANALYSIS OF GEARED ROTORS BY FINITE ELEMENTS

Ahmet Kahraman, H. Nevzat Ozguven, Donald R. HouserGear Dynamics and Gear Noise Research Laboratory

Department of Mechanical EngineeringThe Ohio State University

Columbus, Ohio

and

James ZakrajsekNational Aeronautics and Space Administration

Lewis Research CenterCleveland, Ohio

SUMMARY

A finite-element model of a geared rotor system on flexible bearings hasbeen developed. The model includes the rotary inertia of shaft elements, theaxial loading on shafts, flexibility and damping of bearings, material dampingof shafts and the stiffness and the damping of gear mesh. The couplingbetween the torsional and transverse vibrations of gears were considered inthe model. A constant mesh stiffness was assumed. The analysis procedure can

0 be used for forced vibration analysis of geared rotors by calculating thecritical speeds and determining the response of any point on the shaft to massunbalances, geometric eccentricities of gears and displacement transmissionerror excitation at the mesh point. The dynamic mesh forces due to theseexcitations can also be calculated. The model has been applied to severalsystems for the demonstration of its accuracy and for studying the effect ofbearing compliances on system dynamics.

INTRODUCTION

Even though there have been numerous studies on both rotor dynamics andgear dynamics, the studies on geared rotor dynamics have been rather recent.The study of the dynamic behavior of geared rotor systems usually requiresthat torsional and transverse vibration modes be coupled in the model , aproblem not present for studies of for rotors without gears.

For rotor dynamics studies, the finite-element method seems to be ahighly efficient modeling method. An early finite-element modeling method(Nelson and McVaugh, 1976) used a Rayleigh beam finite-element, which includedthe effects of translational and rotary inertia, gyroscopic moments, and axialload. Zorzi and Nelson (1977) generalized the Nelson and McVaugh study toinclude internal damping. Later, Nelson (1980) developed a Timoshenko beam by Eladding shear deformation to his earlier work. The Timoshenko model wasextended by Ozguven and Ozkan (1983) to include effects such as transverse and ...rotary inertia, gyroscopic moments, axial load, internal hysteretic, viscousdamping, and shear deformations in a single model. None of these models canhandle geared rotor systems, although they are capable of determining thedynamic behavior of rotors consisting of shafts supported at several points ,1esand carrying rigid disks at several locations.

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Gear dynamics studies, on the other hand, have usually neglected thelateral vibrations of the shafts and bearings and have typically representedthe system with a torsional model. Although neglecting lateral vibrationsmight provide a good approximation for systems having shafts with smallcompliances, the dynamic coupling between the transverse and torsionalvibrations due to the gear mesh affects the system behavior considerably whenthe shafts have high compliances (Mitchell and Mellen, 1975). This fact leadinvestigators to the include lateral vibrations of the shafts and bearings intheir models. Lund (1978) included influence coefficients at each gear meshby using the Holzer method for torsional vibrations and the Myklestad-Prohlmethod for lateral vibrations, thus, obtaining critical speeds and a forcedvibration response.

Early geared rotor dynamics models concentrated on the effects of massimbalance and eccentricity of the gear on the shaft, virtually neglecting theactual dynamics of the gear mesh. Hamad and Seireg (1980) studied thewhirling of geared rotor systems supported on hydrodynamic bearings.Torsional vibrations were not considered in this model and the shaft of thegear was assumed to be rigid. Iida, et al. (1980), who considered the sameproblem, by assuming one of the shafts to be rigid and neglecting thecompliance of the gear mesh obtained a three-degree-of-freedom model thatdetermined the first three vibration modes and the forced vibration responsedue to the unbalance and the geometric eccentricity of one of the gears. Theyalso showed that their theoretical results confirmed experimentalmeasurements. Later, Iida, et al. (1984, 1985, 1986) applied their model to alarger system consisting of three shafts coupled by two gear meshes.Hagiwara, Iida, and Kikuchi (1981) developed a simple model that included thetransverse flexibilities of the shafts by using discrete stiffness values thattook the damping and compliances of the journal bearings into account and thatassumed the mesh stiffness to be constant. With their model they studied theforced response of geared shafts due to unbalances and runout errors.

Some of the studies used the transfer matrix method to couple the gearmesh dynamic with system dynamics. Daws (1979) developed a three-dimensionalmodel that considered mesh stiffness as a time-varying, three-dimensionaltensor. He included the force coupling due the Interaction of gear deflectionand time varying stiffness, but he neglected the dynamic coupling. As acontinuation of the Daws study, Mitchell and David (1985) showed that dynamiccoupling terms dominate the dynamic behavior of the system. Another model inwhich the transfer matrix method was used is the model of Iwatsubo, Arii, andKawai (1984a) in which the forced response due to only mass unbalance wascalculated for a constant mesh stiffness. Later, they (1984b) included theeffects of periodic variation of mesh stiffness and profile errors of bschgears.

Other studies used lumped mass and finite-element methods to couple thelateral and torsional dynamics typical of geared rotor systems. Nerlya, Bhat,and Sankar (1984) extended the model of lida et al. (1980) by representing asingle gear by a two-mass, two-spring, two-damper system which used a constantmesh stiffness. The gear shafts were assumed to be massless, and equivalentvalies for the lateral and torsional stiffnesses of shafts were used to obtaina discrete model. As a continuation of this study, Neriya, et al. (1985) usedthe finite-element method to find the dynamic behavior of geared rotors. Theyalso found the forced vibration response of the system due to mass unbalances

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and runout errors of the gears by using modal summation. Bagci andRajavenkateswaran (1987) used a spatial finite line-element technique toperform mode shape and frequency analysis of coupled torsional, flexural, andlongitudinal vibratory systems with special application to multicylinderengines. They concluded that coupled torsional and flexural modal analysis isthe best procedure to find natural frequencies and corresponding mode shapes.

An extensive survey of mathematical models used in gear dynamics analysesis given in a recent paper by Ozguven and Houser (1988a).

The major goal of this study was to develop a finite-element model forthe dynamic analysis of geared rotor systems and to study the effect ofbearing flexibility, which is usually neglected in simpler gear dynamicsmidels, on the dynamics of the system. The formulation of rotor elements,except for gears, used the rotor dynamics program ROT-VIB, which was developedby Ozguven and Ozkan (1983) and Ozkan (1983). However, because of thecoupling between torsional and transverse vibration modes, a torsional degreeof freedom has been added to the formulation, and some special features ofROT-VIB have been omitted.

SYMBOL LIST

[C] damping matrix of the system

CxxCyy bearing damping coefficients in the x and y directions, respectively

cm mesh damping coefficient

cs modal damping value of sth mode

dl,d 2 diameters of driving and driven shafts, respectively

E, G modulus of elasticity and shear modulus, respectively

egep geometric eccentricities of driven and driving gears, respectively

et amplitude of the harmonic excitation

Fs average value of force transrmiitted (static load)

(Ft} total force vector of the system

Ig'lp mass moment of inertias of driven and driving gears, respectively

i imaginary number

Jd,Jm mass moment of inertias of load and motor, respectively

Ktcl torsional compliance of the flexible coupling

km mesh stiffness coefficient

kxx,kyy bearing stiffness values

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LI,L 2 lengths of driving and driven shafts, respectively

mg,mp masses of driven and driving gears, respectively

Np tooth number of driving gearPI

{q} total response of the system

rg,rp base circle radii of driven and driving gears, respectively

t time

UgUp mass unbalances of driven and driving gears, respectively

xg,xp coordinates perpendicular to the pressure line at the centers of the

driven and driving gears, respectively

yg'yp coordinates in the direction of the pressure line at the centers of

the driven and driving gears, respectively

01,E)2 total angular rotations of driving and driven gears, respectively

epeg fluctuating parts of e1 and e2, respectively

[4'] modal matrix

[S] sth normalized eigenvector

(p,Wg rotational speeds of driving and driven shafts, respectively

wr rth natural frequency

THEORY

A typical geared rotor system, as shown in figure 1, consists of a motorconnected to one of the shafts by a coupling, a load at the other end of theother shaft, and a gear pair which couples the shafts. Shafts are supportedat several locations by bearings. Hence, a geared rotor system consists ofthe following elements: (1) shafts, (2) rigid disks, (3) flexible bearings,and (4) gears. When two shafts are not coupled, each gear can be modeled as arigid disk. However, when they are in mesh, these rigid disks are connectedby a spring-damper element representing the mesh stiffness and damping.

For the formulation of the first three elements listed above, theexisting program ROT-VIB, (Ozguven and Ozkan, 1983) was used. ROT-VIB is ageneral-purpose rotor dynamics program that calculates whirl speeds,corresponding mode shapes, and the unbalance response of shaft, rigid-disk,bearing systems by including the effects of rotary and transverse inertia,shear deformations, internal hysteretic and viscous damping, axial load, andgyroscopic moments. In ROT-VIB the classical linearized model with eight

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spring and damping coefficients is used for modeling bearings, and finiteelements with four degrees of freedom at each node (excluding axial motion andtorsional rotation) are employed for the shaft elements.

In the present analysis the formulation used in ROT-VIB for theseelements was modified. First, in order to avoid nonsymmetric system matriceswhich result in a complex eigenvalue problem, the gyroscopic moment effect wasignored and internal damping of the shaft was only included in the dampingmatrix. Second, the gear mesh causes coupling between the torsional andtransverse vibrations of the system, which makes It necessary to include thetorsional degree of freedom. Therefore, the mass and stiffness matrices ofthe system, which are taken from ROT-VIB, have been expanded in this study toinclude the torsional motion of the shafts. Hence, five degrees of freedomhave been defined at each node with only axial motion being excluded. Thismotion, which would be important for helical gears, could easily be Includedin later analyses.

Gear Mesh Formulation

A typical gear mesh can be represented by a pair of rigid disks connectedby a spring and a damper along the pressure line which is tangent to the basecircles of the gears (fig. 2). In this model, both mesh stiffness and dampingvalues are assumed to be constant, and tooth separation is not considered,since the gears are assumed to be constant, and heavily loaded. By choosingthe y axis on the pressure line and the x axis perpendicular to the pressureline, the transverse vibrations in the x direction are uncoupled from both thetorsional vibrations and the transverse vibrations in the y direction. Forthe system of figure 2, the mesh forces in the y direction can be written as

I = cm(y p + rpO 1 + epWpcos El - r - egWg cos E2 - etNp p cos(NpEl))

+ km(y + rp0 + e sin 01 - y- re - e sin2 - e sin<Np2 (1)m p p 1 p e1 g g2 g 9 2 t p 2

N2 = - N1 (2)

where Wl and N2 are mesh forces in the yy and yg directions at thedriving and driven gear locations, respectLvely cm and km are mesh dampingand mesh stiffness values; ep and e are geometric eccentricities ofdriving and driven gears; and rp an rg are base circle radii of thedriving and driven gears. The angles e1 and e2 are the total angularrotations of the driving and driven gears, respectively, and are equal to

el = ep + Wpt (3)

e2 = eg + Wgt (4)

where ep and e are the alternating parts of rotations and wp and gare the spin speeds of the driving and driven shafts, respectively. Thedisplacement, et, which may be considered to be a transmission errorexcitation, is applied at the mesh point. This displacement is usually takento be sinusoidal at the gear mesh frequency but could include higher harmonicsof this frequency. Ozguven and Houser (1988b) have shown that it is possible

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to simulate the variable mesh stiffness, approximately, by using a constantmesh stiffness with a displacement excitation representing loaded statictransmission error. Thus, by choosing et as the amplitude of the loadedstatic transmission error, the effect of variable mesh stiffness can beapproximately considered in the model.

Mesh forces also cause moments about dynamic centers of the gears which

are equal to

M1 = Wl(rp + ep cos e1) (5)

M2 = W2(rg + eg cos (2) (6)

Here, the initial angular positions of geometric eccentricities are taken tobe zero. The mesh stiffness and damping matrices and the force vector of thesystem due to gear errors and unbalances can be obtained by writing the forcetransmitted as the summation of the average transmitted force (static load),Fs, and a fluctuating component, and then neglecting high order termsfollowing the substitution of equations (1) and (2) into equations (5)and (6). By defining the degrees of freedom of the system at which thecoupling effect appears, as

(ql} = [yp ep yg eg]T (7)

the additional mesh stiffness matrix which causes the coupling effect andcorresponds to {ql} can be obtained from equations (1), (2), (5), and (6) tobe

km kmr -km -kmrg

kmrp kmr 2 -kmr -kmrpr

[Km] = km -kmrp km kmrg (8)

-kr -kmrpr kmr kmr2

m g mpg 9 m g m gJ

Similarly, the mesh damping matrix can be found to be

cm c mrp -c -Cmrg

cmrp cm r -cmrp -cmrprg

[C m = _cm _cmrp cm cmrg (9)

2-c r -crr c r c r2mrg mrpg inrg mrg

The other degrees of freedom defined at nodes p and g have not beenincluded in the vector (ql} since elements of [Km] and [Cm] corresponding tothese degrees of freedom are all zero. For the degrees of freedom expressed as

{q2} = FYp xp ep Yg Xg eg]T (10)

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the force vector due to runout, transmission errors and mass unbalances aregiven by

Upo Wsin opt + F

2UpW cos 0ptP P p

-Fe cos wpt + r F{F} p (11)

Ug W sin gt F1

2Ug03 cos Dgt

Fe cos t -r Fl

where

F1 = cm(egog cos ogt - ep&p cos Wpt + etNp cos(Ncat))

+ km(eg sin wgt - ep sin opt + et sin(N(opt)) (12)

Adding the mesh stiffness matrix given by equation (8) to the stiffnessmatrix of the uncoupled rotor system yields the total stiffness matrix of thesystem. The natural frequencies wr and the mode shapes fur} of the systemcan be determined by solving the eigenvalue problem by considering thehomogeneous part of the system equation. In the solution, the SequentialThreshold Jacobi method was used.

Forced Response

The total force vector can be obtained by combining the force vector dueto the mass unbalances of the shafts and the other disks and the force vectordue to the mass unbalances of gears and gear errors as given in equation(11). This vector Is the sum of harmonic components with three differentfrequencies wp, wg, and (Npwp), and has the following general form:

{Ft} = (Fsp} sin wpt + (Fcp} cos cpt + {Fsg} sin ogt + {Fcg} cos ogt

+ (Fsm} sln(Nplpt) + (Fcm} cos(Npopt) (13)

The total response of the system to this excitation can be written as

{q} = [ap](Fsp} sin wpt + (ap](Fcp} cos wpt + [Cg](Fsg} sin ogt

+ [(ag](Fcg} cos wgt + [am]{Fsm} sln(Npwpt) + [cm]{Fcm} cos(Npwpt) (14)

where [a], [ag] and [am ] are the receptance matrices corresponding to theexciting frequencies, wp, wg, and (Npwp), respectively, and given by

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nT

[ap] = 2 c(15)2 P

s= Ws - W + 1WpC s

[ag] = 2 (16)

S= S -g +igCss=l s

n TI: {ts}{[(s}T (16)

[am] Z 2 2 1

s - N + iNWC s

s=lpp

Here, {s} represents the sth mass matrix normalized modal vector, n is thetotal number of the degrees of freedom of the system, I is the unit imaginarynumber, and cs is the sth modal damping value given by the sth diagonalelement of the transformed damping matrix CC] where

[C] = [c]T[C][4] (18)

and where E@] is the normalized modal matrix. In this approach it is assumedthat the damping matrix is the proportional type, which is usually not correctfor such systems. When the damping is not proportional, the transformeddamping atrix [C] will not be diagonal, in which case cs will still be thesth dlagoal element, and all nonzero, off-diagonal elements are simplyignored when using the classical, uncoupled-mode superposition method.Another approach for including damping in the dynamic analysis of such systemswould be to assume a modal damping, s, for each mode and then replace cs inequations (14) and (15) by 2(sw s. However, it is believed that using theactual values for damping, when they are known, and using an approximatesolution technique may give more realistic results than assuming a modaldamping value for each mode.

APPLICATIONS AND NUMERICAL RESULTS

Comparison With An Experimental Study

As the first application, the experimental setup of lida et al. (1980)was modeled (fig. 1). The gear system consists of two geared rotors: one isconnected to a motor with a mass moment of inertia of Jm, and the other isconnected to a load with a mass moment of inertia of Jd. Each shaft issupported by a pair of ball bearings. The parameters of the system are listedin table I. The gears with inertias 1P and I are both mounted on themiddle of the shafts of lengths Ll and L2 an diameters dl and d2,respectively. The driving and driven gears' respective base circle radii arer and rg and their masses are mp and mg. In their study, Ilda et al.(T980) did not specify the length of the second shaft, L2 , and the propertiesof bearings and couplings. Instead, they gave the total torsional stiffness

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values for driving and driven parts of the system and a total transversestiffness value for the second shaft. Therefore, we have estimated the lengthof the second shaft, L2, and the torsional stiffness of the first coupling,Ktcl, in our model so that the total values given by Iida et al. (1980) wereobtained. The forced vibration response due to a geometric eccentricity egand a mass unbalance U is shown in figure 3, along with the experimentalresults of Ilida et al. ?1980). Since no information is given about thedamping values of the system, a modal damping of 0.02 has been used at eachmode in the computations. As seen in figure 3, predictions from theanalytical model show good correlation with the experimental results.

Response Due to Geometric Eccentricities, Mass Unbalances, Static

Transmission Error and Mesh Stiffness Variation

As a second application, the system used by Neriya et al. (1985) wasstudied to investigate the effects of geometric eccentricities and massunbalances of the gears on the forced response of the system. The naturalfrequencies, mode shapes, and the responses at both gear locations due togeometric eccentricities and mass unbalances of gears obtained were almostidentical to those documented by Neriya. The results of this analysis havenot been included in this study since gear eccentricities and unbalancesexcite the system at the shaft rotational frequencies as was shown in thefirst example. The contribution of such low-frequency excitations on thegenerated gear noise is usually negligible when compared with that ofhigh-frequency excitations caused by transmission errors and mesh stiffnessvariations.

On the other hand, the system shown in figure 4 has been modeled toobtain the dynamic mesh force due to a harmonic displacement excitation ofamplitude et and frequency (Npwp) representing the mesh stiffnessvariation. Dimensions of the rotors shown given in figure 4 and other systemparameters are listed in table II. The bearings are assumed to be identical,and geometric eccentricities and mass unbalances for gears are assumed to bezero, so that the only excitation causing a forced response is the harmonicdisplacement excitation defined. Since the displacement input approximatesthe loaded static transmission error, the value of et was taken as theamplitude of the loaded static transmission error. Figure 5 shows thevariation with rotational speed of the ratio of dynamic to static mesh loadfor three different bearing compliances. The first two small peaks offigure 5 correspond to torsional modes of shafts, and the third peakcorresponds to the coupled lateral/torsional mode governed by the gear mesh.

As shown in figure 5, when the bearing stiffnesses are decreased, thedynamic force also decreases considerably because of a resulting decrease inthe relative angular rotations of the two gears. Although the displacementsin the y direction increase slightly, they do not appreciably affect thedynamic force. In this example, a mesh damping corresponding to a modaldamping of 0.1 in the mode of gear mesh has been used. This was the valueused by several investigators for the same problem.

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The Effect of Bearing Compliances on Gear Dynamics

A parametric study of the system shown in figure 6 was performed. Theeffects of bearing compliances on the natural frequencies and the forcedresponse of the system to the harmonic excitation representing the statictransmission error and the mesh stiffness variation were studied. The systemparameters are given in table III. The natural frequencies and the physicaldescriptions of the corresponding modes for a value of bearing stiffnesseskxx kvy = l.Ox1O9 N/m are presented in table IV. The forced response at thepinion ocation in both the transverse (pressure line) and rotationaldirections, and dynamic mesh forces are plotted in figures 7 to 9. Figures 7and 8 show that the system has peak responses only at two natural frequenciesin the range analyzed. Mode shapes corresponding to these two naturalfrequencies are presented in figure 10. When the free vibrationcharacteristics of these two modes is investigated in detail, it is seen thatthe dynamic coupling between the transverse and torsional vibrations at thesetwo modes are dominant. It is also seen that dynamic loads are high at onlythe second one of these two modes as shown in figure 9. The reason for thisis that the transverse and torsional vibrations for the second mode consideredapply at the same direction at the mesh point. This results in large relativedeflections at the mesh point which implies that this mode is governed by gearmesh. It is also seen from these figures that lowering the values of bearingstiffnesses causes a decrease in both the values of the natural frequenciesand the amplitudes of the peak responses and dynamic loads.

Figure 11 shows the variation of these natural frequencies with bearingstiffnesses for three shaft compllances: (1) long shafts (low stiffness) withdimensions given in figure 6, (2) moderately compliant shafts with half thelength of the long shafts, (3) very short (stiff) shafts. The shaft and thebearings supporting the gears can be thought of as two springs connected inseries. When one of these components is very stiff compared with the other,its effect on the overall dynamic behavior becomes negligible. When the modeshapes for these two modes are examined for the case of short shafts and stiffbearings, the first of these two modes becomes purely torsional, while thelateral vibrations become more important in the second mode. As shown infigure 11(a), since the mode considered becomes purely torsional in the caseof short shafts and stiff bearings, the value of this natural frequency doesnot change as bearing stiffnesses exceed a limiting value. For the other modeconsidered, the natoiral frequency becomes very high when a short bearing isused with a very stiff bearing, since the lateral vibrations are more dominantthan torsional vibrations in this mode (fig. 11(b)). Similarly, when theshafts are flexible enough, the effect of bearing stiffnesses on the naturalfrequency becomes negligible above a limiting value of bearing stiffness.

CONCLUSION

A finite-element model was developed to investigate the dynamic behaviorof geared rotor systems. In the analysis, transverse and torsional vibrationsof the shafts and the transverse vibrations of the bearings have beenconsidered. Effects such as transverse and rotary inertia an axial load, wereincluded in the model, and internal damping of the shafts was included only inthe damping matrix. The gear mesh was modeled by a pair of rigid disksconnected by a spring and a damper with constant values that represent average

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mesh values. Tooth separation was not considered. The model developed findsthe natural frequencies, corresponding mode shapes, and forced response of thesystem to mass unbalances and to the geometric eccentricities of gears andtransmission error excitations. Although a constant mesh stiffness wasassumed, the self-excitation effect of a real gear mesh was included in theanalysis by using a displacement excitation representing the statictransmission error.

Although it may be justified to solve nonlinear equations in simplermodels, for large models such as the ones used in this study, avoidingnonlinearitles and transient solutions saves ccnsiderable computation time.In the example problems only the first harmonic of the static transmissionerror was considered and good predictions were obtained.

Finally, it has been shown that the bearing compliances can greatlyaffect the dynamics of geared systems. Decreasing the stiffness values ofbearings beyond a certain value lowers the natural frequency governed by thegear mesh considerably. However, in the case of compliant shafts, when thebearing stiffnesses are above a certain value, the natural frequencycorresponding to the gear mesh does not change considerably by increasingbearing stiffnesses. On the other hand, it has been seen that the amplitudesof dynamic to static load ratio and the deflections at the torsional andtransverse directions are decreased by using bearings with higher compliances,which shows that the bearing compliance may also affect the dynamic toothload, depending upon the relative compliances of the other elements in thesystem.

REFERENCES

I. Bagci, C.; and Rjavenkateswaran, S.K.: Critical Speed and Modal Analysesof Rotating Machinery Using Spatial Finite-Line Element Method.Proceedings of the Fifth International Modal Analysis Conference, D.J.Demichele, ed., Syracuse Univ. Press, Syracuse, New York, 1987,pp. 1708-1717.

2. Daws, J.W.: An Analytical Investigation of Three Dimensional Vibrationin Gear-Couple Rotor Systems. Ph.D. Thesis, Virginia PolytechnicInstitute and State University, VA, 1979.

3. Hagiwara, N.; Ida, M.; and Kikuchi, K.: Forced Vibration of aPinion-Gear System Supported on Journal Bearings. Proceedings ofInternational Symposium of Gearing and Power Transmissions, (Tokyo), N.Kikaigakkai, ed., Japan Society of Mechanical Engineers, Tokyo, 1981, pp.85-90.

4. Hamad, B.M.; and Seireg, A.: Simulation of Whirl Interaction InPinion-Gear Systems Supported on Oil Film Bearings. J. Eng. Power,vol. 102, no. 2, 1980, pp. 508-510.

5. Iida, H., et al.: Coupled Torsional-Flexural Vibration of a Shaft in aGeared System of Rotors (Ist Report). Bull. Japan. Soc. Mech. Eng.,vol. 23, no. 186, Dec. 1980, pp. 2111-2117.

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6. lida, H.; and Tamura, A.: Coupled Torsional-Flexural Vibration of aShaft in a Geared System. International Conference on Vibration inRotating Machinery, 3rd, Mechanical Engineering Publications, London,1984,pp. 67-72.

7. lida, H.; Tamura, A.; and Oonishi, M.: Coupled Dynamic Characteristicsof a Counter Shaft in a Gear Train System. Bull. Japan. Soc. Mech. Eng.,vol. 28. 1985, pp. 2694-2698.

8. lida, H.; lamura, A.; and Yamamoto, H.: Dynamic Characteristics of aGear Train System with Softly Supported Shafts. Bull. Japan. Soc. Mech.Eng., vol. 29, 1986, pp. 1811-1816.

9. Iwatsubo, T.; Arii, S.; and Kawai, R.: Coupled Lateral-TorsionalVibrations of Rotor Systems Trained by Gears (1. Analysis by TransferMatrix Method), Bull. Japan. Soc. Mech. Eng., vol. 27, 1984, pp. 271-277.

10. Iwatsubo, T.; Arii, S.; and Kawai, R.: Coupled Lateral TorsionalVibration of a Geared Rotor System. International Conference onVibrations in Rotating Machinery, 3rd, Mechanical EngineeringPublications, London, 1984, pp. 59-66.

11. Lund, J.W.: Critical Speeds, Stability and Response of a Geared Train ofRotors. J. Mech. Des., vol. 100, no. 3, 1978, pp. 535-538.

12. Mitchell, L.D.; and David, J.W.: Proposed Solution Methodology for theDynamically Coupled Nonlinear Geared Rotor Mechanics Equation. ASME,Paper 85-DET-90, Sept. 1983.

13. Mitchell, L.D.; and Mellen, D.M.: Torsional-Lateral Couple in a GearedHigh-Speed Rotor System. J. Mech. Eng., vol. 97, no. 12, 1975, p. 95.

14. Nelson, H.D.: A Finite Rotating Shaft Element Using Timoshenko BeamTheory. J. Mech. Des., vol. 102, no. 4, 1980, pp. 793-803.

15. Nelson, H.D.; and McVaugh, J.M.: The Dynamics of Rotor-Bearing SystemsUsing Finite Elements. J. Eng. Ind., vol. 98, no. 2, 1976, pp. 593-600.

16. Neriya, S.V.; Bhat, R.B.; and Sankar, T.S.: Effect of Coupled Torsional-Flexural Vibration of a Geared Shaft System on Dynamic Tooth Load. ShockVib. Bull., vol. 54, pt. 3, 1984, pp. 67-75.

17. Neriya, S.V.; Bhat, R.B.; and Sankar, T.S.: Coupled Torsional FlexuralVibration of a Geared Shaft System Using Finite Element Method. ShockVib. Bull., vol. 55, pt. 3, 1985, pp. 13-25.

18. Ozguven, H.N.; and Houser, D.R.: Mathematical Models Used in GearDynamics. J. Sound Vib., vol. 121, Mar. 1988, pp. 383-411.

19. Ozguven, H.N.; and Houser, D.R.: Dynamic Analysis of High Speed Gears byUsing Loaded Static Transmission Error. J. Sound Vib., vol. 125,Aug. 1988, pp. 71-83.

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20. Ozguven, H.N.; and Ozkan, Z.L.: Whirl Speeds and Unbalance Response ofMultibearing Rotors Using Finite Elements. ASME Paper 83-DET-89, Sept.1983.

21. Ozkan, Z.L.: Whirl Speeds and Unbalance Responses of Rotating ShaftsUsing Finite Elements. M.S. Thesis, The Middle East TechnicalUniversity, Ankara, Turkey, 1983.

22. Zorzi, E.S.; and Nelson, H.D.: Finite Element Simulation ofRotor-Bearing Systems with Internal Damping. J. Eng. Power, vol. 99, no.1., 1977,pp. 71-76.

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TABLE 1. - PARAMETERS OF THE GEAR SYSTEM OF FIGURE 1

Moments of inertia:

Of motor, Jm' kg-m2 . . . . . . . . . . . . . . . . . . . . . . .. . 0.459

Of load, Jd, k9'M2 . . . . . . . . . . . . . . . . . . . . . . . . 0.549

Of driven gears, I., kg-m2 . . . . . . . . . . . . . . . . . 6.28.xlo-3

Of driven gears, I,, kg-m2 . . . . . . . . . . . . . . . . . . . . 0.030

Mass:

Of driven gears, ing, kg .. .... ...... ..... ...... 5.65

Of driving gears, inp, kg .. ...... ...... ........16.96

Basic circle radius:

Of driven gears, rg, m. .... ...... ...... ..... 0.1015

Of driving gears rp In.......................0.0564

Length:

Of driven shaft, L2 i.........................0.40

Of driving shaft, L, in........................0.78

Diameter:

Of driven shaft, d2, in........................0.02

Of driving shaft, dl, m .. .... ...... ..... ...... 0.03

Geometric eccentricity of driven gears, eg, i............1.2x10- 5

Mass unbalance of driven gear, UgV k9*m . .. ...... ..... 2.8x,0-4

Torsional compliance, Ktcl, N-in/rad. .. ...... ...... .. 115.0

Mesh stiffness coefficient, kin, N/in. .. ...... ........2.0x108

14

TABLE 11. -PARAMETERS OF THE GEAR SYSTEM OF FIGURE 4

(Variable hearing stiffness of values.]

Moment of inertia:

Of motor, im' kg-i 2 . . . . . . . . . . . . . . . . . . . . . . . 1.15x13-2

Of load, Jd, kgM2 . . . . . . . . . . . . . . . . . . . . . . . 5.75x]0-3

Of driven gears, 1g' km 2 . . . . . . . . . . . . . . . . .. . 5x10-3

Of driven gears I,1 kgm 2 . . . . . . . . . . . . . . . . . .. . 5x10-3

Mass:

Of motor, mm, kg.......................9.2

Of load, md, kg .. .... ...... ...... ..... ..... 4.6

Of driven gears, mg, kg....................0.92

Of driven gears, mp kg....................0.92

Basic circle radius:

Of driven gears, rg. m. ..... ..... ...... ...... 0.047

Of driving gears, rp, m .. .... ...... ...... ..... 0.047

Mesh stiffness coefficient, kin, N/M .. .... ...... ..... 2.0x108

Average values of force transmitted, Fs, N. ..... ...... .. 2500

15

TABLE III. - PARAMETERS OF THE GEAR SYSTEM OF FIGURE 6

(Variable bearing stiffness values.]

Moment of inertia:

Of driven gears, Ig, k9 m2 . . . . . . . . . .. . .. . .. .. . . . . . . 0.0018

Of driving gears, Ip, kg.m2. . . . . . . . . . . .. . .. . .. . . . . . . .. 0.0018

Mass:

Of driven gears, mg, kg .... ....................... .... 1.84

Of driving gears, mp, kg .................... 1.84

Base circle radius:

Of driven gears, rg, m ...... .................... ... 0.0445

Of driving gears, rp, m ..... ................... .... 0.0445

Amplitude of the harmonic excitation, et, m ..... .......... 9.3x10-6

Mesh stiffness coefficient, ki, N/r ............... 1.0×10 8

Tooth numbers of driving gear, N. .................. 28

TABLE IV. - FIRST 14 NAIURAL FREQUENCIES OF THE SYSTEM OF

FIGURE 6 FOR THE CASE OF kxx/km = 10

Natural frequency, Corresponding modeHz

0 Torsional rigid body

581 Transverse, torsional

687 Transverse, x dir., driving shaft

689 Transverse, y dir.

691 Transverse, x dir., driven shaft

2524 Transverse, torsional

3387 Transverse, y dir.

3387 Transverse, x dir., driving shaft

3421 Transverse, x dir., driven shaft

3421 Transverse, y dir.

6447 Torsional, driving shaft

6539 Torsional, driven shaft

6831 Transverse, x dir., driving shaft

6840 Transverse, y dir.

16

Yg GEAR

rYg r,2 e sn0

eg '02

Xg:

MOTOR PINION ,cm

n PINION

Yp +rpo +ep sini 0Yp 1

01

BERNGI SHAFT 2 LOAD

GEAR

Figure 1. - A typical gear-rotor system. Figure 2. -Modeling of a gear mesh.

.28

w - lda eta].24 PRESENT MODEL

0 20 /16 /

X .12

Z 080

LOA- .04

U.

10 12 14 16 18 20 22 24 26 28 30 IKN 2.ROTATIONAL SPEED OF SHAFT 2, Hz MOTOR

Figure 3. - Comparison of the theoretical values of the dynamic Figure 4. - The system of the second example. (Dimensions aredeflection of the driven gear In the pressure line direction with In millimeters.)the experimental results given by lda et a]. (1980).

17

BEARINGSTIFFNESSVALUES

kxx AND kyy,N'm

2.2o lx10 15 (ALMOST RIGID)

Ix 010-I X109

127 127W PINIONM37 F

P: 1.8

0 1.6

Z 1.4 --.-

BEARING

91.2

1.0 I0 50 100 150 200 250 300 37 o.d.

ROTATIONAL SPEED, Hz GEAR Id.

Figure 5. -Variation of dynamic to static load ratio with frequency Figure 6. - The system analyzed as a third example.for three different bearing compliances. (Dimensions are In millImeters.)

7x10-4

--. 0035 i

BEARING BEARINGK COMPLIANCE, COMPLIANCE,

..6 - kxx/km 1 ' .0030 kxx/kr

0.1 0 0.15 -. o0025 .5

10 cc 1.000, 6 1.0

. 1 10.0 I , 0, 10.0

~.oo i i.A1; -j .0010

Q, I IIr000

2000 2500 3000 0 500 1000 1500 2000 2500 3000

EXCITATION FREQUENCY. Hz EXCITATION FREQUENCY, Hz

Fidgure 7, - Forced response of the system shown In figure 610o Figure 8.- Forced response of system shown in figure 610o thethe displacement excitation at the direction of pressure line displacement exdtatlon at the torsIonal direction (at pinion(at pinion locaton) for tour different bearIng compliances. location) for four different bearing co Ifplcancs.

Z W8

2.0

BEARING (a) Nafliral frequency, 581 Hz.

0 COMPLIANCE. i

cn= 1.8 kAOW 0.1W' .5ujQi 1.6 10.0 il

1.4

00 1.21

1,014

0 500 1000 150 2000 2500 3000EXCITATION FREQUENCY, Hz (b) Natural frequency. 2524 Hz.

Figure 9. - Dynamic to static load ratios for system shown in figure 6 Figure 10. - Mode shapes corresponding to rwiturall frequenciesduo to the dfispacement excitation for four different bearing at which the system shown In figure 6 has piak responses.compiances,

19

A

C'SHORT SHAFTS (L = .05 M)2500 - MODERATE SHAFTS(L=0.127M)

A LONG SHAFTS (L=0.254 m) ~

z

4, 500

LL

1 000

1500 I

(b) Second natural frequency

02

[ Aris- Report Documentation PageNatu~al Aetonaut~ca and

Space Adrsnatralon

1. Report No. NASA TM-102349 2. Government Accession No. 3. Recipient's Catalog No.

AVSCOM TM 89-C-0064. Title and Subtitle 5. Report Date

Dynamic Analysis of Geared Rotors by Finite Elements January 1990

6. Performing Organization Code

7. Author(s) 8. Performing Organization Report No.

Ahmet Kahraman, H. Nevzat Ozguven, Donald R. Houser, and E-5058James J. Zakrajsek

10. Work Unit No.9. Performing Organization Name and Address 505-63-51

NASA Lewis Research Center I L162209A47ACleveland, Ohio 44135-3191and 11 Contract or Grant NoPropulsion DirectorateU.S. Army Aviation Research and Technology Activity-AVSCOMCleveland, Ohio 44135-3127 13. Type of Report and Period Covered

12. Sponsoring Agency Name and Address Technical MemorandumNational Aeronautics and Space AdministrationWashington, D.C. 20546-0001 14. Sponsoring Agency Code

andU.S. Army Aviation Systems CommandSt. Louis, Mo. 63120-1798

115. Supplementary Notes -- -

116. Abstract

A finite-element model of a geared rotor system on flexible bearings has been developed. The model includes therotary inertia of shaft elements, the axial loading on shafts, flexibility and damping of bearings, material dampingof shafts and the stiffness and the damping of gear mesh. The coupling between the torsional and transversevibrations of gears were considered in the model. A constant mesh stiffness was assumed. The analysis procedurecan be used for forced vibration analysis of geared rotors by calculating the critical speeds and determining theresponse of any point on the shaft to mass unbalances, geometric eccentricities of gears and displacementtransmission error excitation at the mesh point. The dynamic mesh forces due to these excitations can also becalculated. The model has been applied to several systems for the demonstration of its accuracy and for studyingthe effect of bearing compliances on system dynamics.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Dynamic Unclassified - UnlimitedGears Subject Category 37RotorsFinite element

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of pages 22. Price*

Unclassified Unclassified 22 A03NASA FORM 1626 OCT 86 *For sale by the National Technical Information Service, Springfield. Virginia 22161