shel element

7/30/2019 Shel Element

1/5

PergamonCompurers & Srru~mrrs Vol. 62, No. 4. pp. 715-719, 1997

Copyright 0 1996 Published by Elsevier Science LfdPII: SO0457949(96)00214-3 Printed m Great Britain. All rights reserved0045.7949/97 $17.00 + 0.00

DISC FLEXIBILITY EFFECTS IN ROTOR BEARINGSYSTEMSD. Satish Kumar, C. Sujatha and N. Ganesan

Machine Dynamics Laboratory, Department of Applied Mechanics, Indian Institute of Technology,Madras 600 036, India(Received 10 May 1995)

Abstract-The importance of the inclusion of disk flexibility on the natural frequencies in a rotor dynamicanalysis has attracted the attention of many researchers in the recent past. Such an analysis helps to predictthe dynamic response of rotors more accurately from the natural frequencies and mode shapes calculatedhence. In the present analysis a semi-analytical conical shell finite element is used for the modelling ofthe rotor with the inclusion of the support bearing flexibility. Such an element could include the flexibilityof the disks, shaft and the bearings. A parametric study was also performed for looking into the frequencycharacteristics of the system. The study brings out the additional disk modes a well as the behaviour ofthe system at higher circumferential modes. Copyright 0 1996 Elsevier Science Ltd.

NOTATlON

coordinate directionsdisplacements in the coordinate directionsmiddle surface displacements in the coordinatedirectionsangles of rotation in the meridional andcircumferential directions of the normal to themiddle surface of the elementradius of the middle surface of the conical shellat the small endmeridional numbershape functionss/l, isoparametric coordinatelength of the shaftinner radius of the discouter rad,us of the discthickness of the discinner radius of the shaftouter radius of the shaftstrain energy contribution from the shaft and thediscstrain energy contribution from the bearings

1. INTRODUCTION

The advent of digital computers has acceleratedresearch in many a frontier over the last threedecades. The finite element method is no exception.With a growing demand for accuracy, a widespectrum of finite elements have been developed foruse in the fields of structural dynamics, rotordynamics, etc.

Typically in a rotor dynamic analysis, the shaftis modelled by a series of beam finite elements, theother components of the system being modelled aslumped masses. However, there is a recent body ofliterature which visualizes the importance of theinclusion of the disk flexibility in rotor dynamic

analysis. Beam elements are constrained by the factthat they may not be able to take into accountaccurately the large changes in diameter that occurwith the inclusion of the disc as a flexible member.Also they may not be able to provide an insightinto the higher circumferential modes of the disc.These modes play a significant role whenever thereis an axial excitation, such as the one that occursin helical gears.

Solid finite elements stand at a disadvantagebecause of the astronomical computational costinvolved in their use. A semi-analytical finiteelement procedure is envisaged. An importantaspect of the procedure to be described here is theline shell element which saves a lot of compu-tational effort compared to its two-dimensionalcounterpart. To the authors knowledge there is nopublished literature on the inclusion of bearingflexibility in a semi-analytical treatment. The aim ofthe present work is to study the effect of theinclusion of disk flexibility on the natural frequen-cies of the system with the inclusion of the supportbearing flexibility.

2. BACKGROUND

The amount of literature available on rotordynamic analysis is immense. The purpose of thiswork is not to review, but to provide a briefdescription of the literature available. Eshelman andEubanks [l] modelled the shaft as having a continu-ously distributed mass and elasticity and the disk asa lumped mass. Dopkin and Shoup [2] did somepioneering work in the inclusion of disc flexibility. Atransfer matrix analysis was used by them to obtain

715


2/5

the resonant frequencies. They had neglected theshear deformation and rotary inertia effects in theanalysis of the disc. Also, the higher circumferen-tial modes of the disc were not calculated. Thisfact precludes any attempt on the study of discs,having significantly high radial flexibility. Chivensand Nelson [3] investigated the influence of discflexibility on the transverse bending natural fre-quencies and concluded that the natural frequen-cies are significantly affected by the inclusion ofthe flexibility of the disc. The shaft in their modelwas a Euler-Bernoulli beam and the disc a thinflexible plate. Nevzat Ozguven and LeventOzkan [4] calculated the whirl speeds and unbal-ance response using finite elements, with sheardeformation, rotary inertia, gyroscopic moment,internal viscous and hysteretic damping effectsincluded. Hayashi and Iwatsuki [5] solved for thedisc modes to establish a method for accuratelyestimating the noise power radiated from gears andgear mechanisms.

Recently Flowers and Fang Sheng Wu [6] mod-

WJ v,eKS

TRa---- - -Fig. I. Coordinate system for a conical shell.

3. FORMULATION

u = x i, ,u,cos m6-sim = 0With the coordinate system shown in Fig. 1 for a

conical shell finite element, the displacement assump-tions [lo] are given by

elled the disc as a collection of four equally spacedmass elements connected to a central hub by linearsprings. Flowers and Ryan [7] in one of their latestworks developed a set of equations in terms of thegeneralized coordinates based on the mode shapes

I 3v= cT, NJ, sin me,8 0of the system for incorporating the disc flexibilityeffects and concluded that at rotor speeds signifi-cantly higher than the nominal operating speeds,the critical speeds are significantly affected by theflexibility of the disks. They also concluded that m = 0for super harmonic vibration which could beexcited by non-linear effects such as rubbing or x 3non-linear bearing compliance, the disc flexibility w= 1 N,w, cos me,plays a significant role. ,=I

Shahab and Thomas [S] studied the coupling cl=0effects of disc flexibility on the dynamic behaviourof multi-disc shaft systems with simply-supportedboundary conditions. Their three-dimensional finite I)$= 2element increases the computational cost whenlarge problems are solved even with modern c$,J++,,,in me,I!, = 0computing facilities. Stephenson ef al. [9] usedquadrilateral axisymmetric elements in the mod- where N, are the shape functions given byelling of the shaft disc system. However, the linetype of harmonic elements based on a shell theoryproposed here may still be more computationally N, = (52 N2 = 1 -


3/5

Disc flexibility effects 717Table 1. Comparison of theoretical and experimental [8] frequencies (Hz) for a hollow shaft corresponding to variouscircumferential modes

m =z 0 m= m=2-_____Theoretical Experimental Present Theoretical Experimental Present Theoretical Experimental Present

650 645 772 288 280 294 903 900 837907 928 670 660 576 5880 5830 53964852 4815 4129 1598 1590 1650 12123 12100 131645572 5580 561I 2621 2610 2693 13675 13767

5 = s\l is the isoparametric axial coordinate, I is theslant length of the conical shell and m is thecircumferential mode number.

The displacement vector is given byu,=(u v w Efi.5XP}2.3.

With the strain displacement as given by Ref. [IO]and the stressstrain relationships based on three-dimensional elasticity, the strain energy contributionfrom the element is given by

which upon substitution in the well-known Lagrangeequation results in the element stiffness matrix

[Kel = s [BIVWI du.tThe energy contributions from the bearings in the

v- and w-direction I& are given byf[K,,I[wfl and f[f&][uf],

where K,,.,,. nd K,.,. ire the stiffness coefficients in thew- and v-directions as shown in Fig. 2 and i is thenodal location corresponding to the bearings. Thesubstitution of these energy terms in the Lagrangeequation gives the forces transmitted to the bearingfor a shaft in lateral bending in the w- andv-directions. These forces are given by K,c,,.w os BandKrrusin 8. Now at 0 = 0, the force transmitted in thev-direction is zero, *asa result the bearing stiffness inthe w-direction alone is lumped for the bendingmodes. This results in a situation wherein the bendingmode in the z-direction alone is obtained. To obtain

Table 2. Comparison of theoreti-cal [4] bending frequencies (rad s-)for a solid shaftBeam Present521.79 519.501095.34 1091.802206.00 2232.004411.00 4952.30

the other mode in the v-direction, the antisymmetricassumptions in the Fourier series transformation areto be made.

4. VALIDATION

The validation study was initially performed on ahollow shaft disk system of Ref. [8] with simply-supported boundary conditions. The mechanicalproperties are given by the youngs modulus2.07 x IO N m-2, shear modulus 80 x lo9 N m-,density 7800 kg m-). The physical dimensions withthe disc symmetrically located at the middle of theshaft are given by L = 457 mm, b/a = 0.2, h/a = 0.05, Ri = 12.7 mm and R, = 17.7 mm. Theresults obtained are compared in Table I. Thetheoretical and experimental frequencies given inTable 1 are those from Ref. [8] and the third columnindicates the results obtained through the presentapproach. Another validation study was performedfor a solid shaft with clamped boundary con-ditions [S] whose physical dimensions areL = 180 mm, R, = 50 mm, plate thickness = 50 mm.Table 3 gives the comparison for this study. Theresults obtained are in good agreement with those inRef. [S].

In order to validate the inclusion of the bearingcompliance effects, a numerical example [4] wasused. The system consists of a 10.16 cm diameterand 127 cm long steel shaft without any discs,supported by identical bearings of stiffnessK,,,, = Krr = 1.7513 x 1ON m-, with the mechanicalproperties given by the Youngs modulus2.07 x 10 N m-2, density 7833 kg mm3. The frequen-cies given in Ref. [4] are the lateral bendingfrequencies of the shaft calculated using a sheardeformable beam element. However, with the presentelement the first circumferential mode or the m= 1mode is the lateral bending mode of the shaft. Theresults obtained were compared and are shown inTable 2. From these tables it is obvious that theagreement between the results is reasonably good. InTable 3. The natural frequencies (Hz) of flexural vibrationof a circular plate with a solid shaft+clamped endsMode ofvibration 0,o 0,I l,O l,l 2,OFEM 2825 10471 2106 12475 3461Present 2610 10273 2287 12517 3990


4/5

718 D. Satish Kumar et al.Table 4. Comparison of frequencies (rad s-l) by the two Table 6. Frequencies (rad s-l) corresponding to the variousapproaches (h/a = 0.1) circumferential modes for various L/r ratios

Present Mode Beam398.9 Shaft bending

1127.1 Coupled399.11052.0-

1959.0 Shaft bending 1957.02630.2 Couuled --5260.0 Coupled 3839.0-7573.0

general the shell used in the present study is supposedto give accurate results up to r/t = 5, where r is theradius and t is the thickness of the shell. Hence it ishighly applicable for hollow thick shafts. Addition-ally, the two case studies show that the element trulyrepresents the bending behaviour of solid shaft discsystem even for a solid shafts (r/t = 1).

5. PARAMETER VARIATION STUDIES

Initially for the sake of comparison with thesemi-analytical approach (m = 1 mode) a set ofresults obtained by using a shear deformable beamelement [4] with disks of various thickness consideredas lumped masses were obtained with spring-sup-ported boundary conditions. A typical case withh/a = 0.1; L/r = 25; K,,,, = 1.7513 x 1ON m- wastaken, the first five natural frequencies (rad SC)obtained by the two approaches is shown in Table 4.Table 5 shows another case of a highly flexible discwith h/a = 0.005. The variation in the naturalfrequencies obtained by the proposed approach andthe beam approach in Tables 4 and 5 gives us aninsight into the effect of inclusion of flexibility of thedisc on the system.

From Tables 4 and 5 it is apparent that the systemfrequencies are affected by the inclusion of theflexibility of the disc. The frequencies other thanthose obtained by the beam approach are the discmodes or the coupled modes. These modes as seenfrom the tables are not predicted by the beamapproach. As mentioned earlier the disc frequenciescome into relevance whenever there is an axialexcitation, as that in helical gears.

A parameter variation study was performedfor different L/r ratios viz. 5, 15,25 with

Table 5. Comparison of frequencies (rad s-l) by the twoapproaches (h/a = 0.005)Present

150.7510.81092.6-1105.9-2211.0-

ModeDISCShaft bendingShaft bendingCoupledShaft bending

Beam-

511.91093.02211.0-4915.0-9025.0

L/r m5 015 025 05 115 125 15 215 225 25 315 325 35 415 425 4

Frequencies correspondingto circumferentialmodes3239 4497 19702 265453210 3724 11701 149133147 3475 7876 10129

793 1675 6496 19469578 1476 2893 4094398 1127 1959 26303731 22851 31442 533953734 22860 31514 533793737 22872 31639 536877449 30225 45607 608227449 30231 45644 608307450 30237 45682 60852

12733 39733 58990 7174712733 39775 58996 7175912733 39777 59001 71769

K ,,,, = 1.7513 x 10N m- and h/a = 0.1. As seenfrom Table 6, which shows the natural frequencies(rad s-) for different L/r ratios, only the lateralbending mode and the mode with no nodal diametersof the system becomes affected; an appreciabledifference in the other circumferential modes is notseen.

An obvious extension to the above irrelevance ofthe L/r ratio to higher modes of the system is that thebearings are passive at higher circumferential modes.This is due to the fact that an increase in the L/r ratioincreases the flexibility of the system similar to theaddition of bearing stiffness.

6. CONCLUSIONS

(1) The present proposal predicts that, in additionto the beam bending frequencies, the naturalfrequencies where the disk modes alone are prevalentare also obtained.

(2) The inclusion of the flexibility of the disk helpspredict the free-free modes of the system moreaccurately for accurate prediction of the response ofthe system.

(3) The variation of the length of the shaft doesnot have any remarkable influence on the naturalfrequencies corresponding to the modes other thanthe lateral bending mode and those with no nodaldiameters.

Acknowledgement-The authors wash to thank Prof. V.Ramamurti, Machine Dynamics Laboratory Department ofApplied Mechanics, Indian Institute of Technology, Madrasfor enlightening them on the subject.

REFERENCES

I. R. L. Eshelman and R. A. Eubanks. On the criticalspeeds of a continuous shaft disc system. Trans. ASMEJ . Engng Ind. 89, 645-652 (1967).


5/5

Disc flexibility effects 7192. J. A. Dopkin and T. E. Shoup, Rotor resonant speed disk rotor dynamics. Trans. ASME J. Engng Gas

reduction caused by flexibility of disk. Trans. ASME J. Turbines Power 115, 279-286 (1993).Engng Ind. 96, 1328-1333 (1974). 7. G. T. Flowers and S. G. Ryan, Development of a

3. D. R. Chivens and H. D. Nelson, The natural set of equations for incorporating the discfrequencies and critical speeds of a rotating flexible flexibility effects in rotor dynamic analyses. Trans.shaft disc system. Trans. ASME J. Engng Ind. 97, ASME J. Engng Gas Turbines Power 115, 227-233881-886 (1975). (1993).

4. H. Nevzat Ozguven and Z. Levent Ozkan, Whirl speedsand unbalance response of multibearing rotors usingfinite elements. Trans. ASME J. Vibr. Acoust. StressReliab. Des. 106, 72-79 (1984).

5. I. Hayashi and N. Iwatsuki, The theoretical modalanalysis of a circular plate with a solid shaft. J. SoundVibr. 173, 633-655 (1994).

8. A. A. S. Shahab and J. Thomas, Coupling effects of discflexibility on the dynamic behaviour of multi-disc shaftsystem. J. Sound Vibr. 114, 435452 (1987).

9. R. W. Stephenson, K. E. Rouch and R. Arora,Modelling of rotors with axisymmetric harmonicelements. J. Sound Vibr. 131, 431443 (1989).

IO. T. C. Ramesh and N. Ganesan, Finite element analysis6. G. T. Flowers and Fang Sheng Wu, A study of the

influence of bearing clearance on lateral coupled shaftof conical shells with a constrained vicoelastic layer. J.Sound Vibr. 171, 577-601 (1994).

shel element

Documents