Dynamics of Rotor Bearing Systems Using Finite Elements

7/29/2019 Dynamics of Rotor Bearing Systems Using Finite Elements

1/8

H. D. N@IsonProfessor of Engineering Science,

Arizona State University,Tempe, Ariz.MEM. ASf.lE

J , M. MeVaughAssistant Group Supervisor-System Dynamics,

Airesearch Manufacturing Company of Arizona,Phoenix, Ariz.

MEM. ASME

The Dynamics of Rotor-BearingSystems Using Finite ElementsA procedure is presented for dynamic modeling of rotor-bearing systems which consist ofrigid disks, distributed parameter finite rotor elements, and discrete bearings. The formulation is presented in both a fixed and rotating frame of reference. A finite elementmodel including the effects of rotatory inertia, gyroscopic moments, and axial load is developed using the consistent matrix approach. A reduction of coordinates procedure isutilized to model elements with variable cross-section properties. The bearings may benonlinear, however, only the linear stiffness and viscous damping case is considered. Thenatural w hirl speeds and unbalance response of a typical overhung system is presentedfor two sets of bearing parameters: (i) undamped isotropic, (ii) undamped orthotropic. Acomparison of results is made with an independent lumped mass analysis.

In t roduct ionFlexible rotor-bearing systems have been analyzed by many dif

fe ren t ma themat ica l me thods . An exce l len t comprehens ive rev iewof these methods with reference to specific contributors is conta ined in a recen t work by E sh leman [ l ] , 1 hence, need not be repeated here. Recently , the uti l ization of f inite e lement models ,such as introdu ced by Archer [2], in the area of rotor dynam ics h asyielded highly successful results . I t is the intent of this paper toprovide additional information on the use of f inite e lements for analyzing rotor-bearing systems.

Of primary interest re lative to this paper, is a contribution byRuhl [3], [4] who uti l ized a f inite e lement model of a turborotorsystem to s tudy stabil i ty and unbalance response. Ruhl 's f inite e lement includes only elastic bending energy and translational kineticenergy, while the effects of rotatory inertia , gyroscopic moments,axial load, axial torque, and shear deformation are neglected.These effects can be quite s ignificant for some configurations asindicated by several investigators referenced in [1]. Thi s parti cularpaper generalizes Ruhl 's e lement by including the effects of rotatory inertia , gyroscopic moments, and axial load. In addition the element and system equations are presented in both a f ixed and rotating reference frame. An example, which uti l izes a coordinate re-

1 Numbers in brackets designate References at end of paper.Contributed by the Vibration and Sound Committee of the Design Engineering Division and presented at the Winter Annual Meeting, Houston,Texas, November 30-December 5, 1975, of THE AMERICAN SOCIETYOF MECHANICAL ENGINEERS. Manuscript received at ASME Headquarters August 5,1975. Paper No. 75-WA/DE-19.

Journal of Engineering for Industry

duction procedure, is provided to indicate the accuracy of the elem e n t .

System Configurat ion and CoordinatesThe typical f lexible rotor-bearing system to be analyzed consistsof a rotor composed of discrete disks, rotor segments with distr ibuted mass and elastic ity , and discrete bearings. Such a system is i llustr ated in Fig. 1 along with the two reference frames tha t are util ized to describe the system motion. The XYZ: EF (xyz:Q{) tr iad is afixed (rotating) reference with the X a nd x axes being colinear andcoincident w ith the und eform ed rotor center l ine. 01 is defined relative to 'S by a s ingle rotation co t a b o u t X with


2/8

Fig. 2 Cross section rotation angles

relative to *5 by the t rans la t ions V (s, t) a n d W (s, t) in the Y an dZ directions respectively to locate the elastic centerl ine and smallang le ro ta t ions B (s, t) and r (s , t) a b o u t Y an d Z respectively toorient the plane of the cross-section. The cross-section also spinsnor ma l to its face relativ e to 9?. Th e abc: Q tr iad is a ttached to thecross-section with the "a" axis normal to the cross-section. S is defined by the three successive rotations, i l lustr ated in Fig. 2,1 r a b o u t Z defines a"b"c"

2 B a b o u t b" defines a'b'c'3 (j> a b o u t a' defines abc,

and the angular rate of re lative to ?f isr " - s i n B 1 0c o s B s i n 0 0 c o s 4>. c o s B c o s 0 - s i n .LB (1 )For small deformations the (B, T) ro ta t ions a re approx ima te ly

colinear with the (Y, Z) axes respectively. The spin angle fa for aconstant speed system and negligible torsional deformation, is f i tw h e r e 2 deno tes the ro to r sp in speed .

T he d isp lacemen ts (V, W, B, T) of a typical cross section relat ive to ff are transformed to corresponding displacements (u , w, /3,7) relative to (ft by the orthogo nal transfor matio n

w i t h

{q} = wBr -fc} =

{q} =fv\w

[y] , [ * ]

"coscrf -si n cot03

MM

sinarfcoswf00

00

CO S Ui fsinotf

00- sinatf

COSktf

(2)

(3)

and for la ter use the f irs t two time derivatives of equation (3) are{q} = a)[S]{ p} + [R]$\ (a)

= [R]{$\- o>2{p}}+ 2co[S ]{p} (6 )w i t h

[Sl=i[R} =- s i na r f - cosa r f 0 0

c o s a r t - s i n w i 0 00 0 - s ina r f - cosa t f0 0 cos cot - s inM?

(c) (4)

C o m p o n e n t E q u a t i o n s . The rotor-bearing system is considered to comprise a set of interconnecting components consisting ofrigid disks, rotor segments with distr ibuted mass and elastic ity ,and l inear bearings. In this section the r igid disk equation of motion is developed using a lagrangian formulation. The finite rotorelement equation of motion is developed in an analogous mannerby specifying spatia l shape functions and then treating the rotorelement as an integration of an infinite set of differentia l d isks.The bearing equations are not developed and only the l inear formof the equations as treated by Lund [5] are uti l ized in this paper.

Rigid Disfe . Th e kinetic energy of a typica l rigid disk with mass

. N o m e n c l a t u r e .' = differentia tion with respect to posit ion = differentia tion with respect to t imeSJ = fixed reference frame (XYZ)(R = ro tating reference frame (xyz)6 = cross-section reference frame (abc)3 = kinetic energy 20 = potential energy 23D , dp = element diametral and polar iner

t ia per unit length[M], [g], [x] = assembled mass, gyroscopic , and stiffness matrices 2a = eigenvalue( S , r ) , (/3,Y) = small angle rotations about(Y,Z), (y , z )axes4> = spin anglefi = spin speed = 4>

2 W he re a ppropr ia te the supe r sc r ip ts d, e, b, srefer to disk, e lement, bear ing, and system respe c t ive ly a nd the subsc r ip ts T, R, B, A refer totransla tional, rota tional, bonding, and axia l loadre spe c t ive ly .

X = whirl ratio = S2A["$?] = matr ix o f t rans la t ion d isp lacem en t

. functions; fa (s) , i = 1, 2, 3, 4[] = ma trix of rotat ion dis pla cem ent

functions; fa' (s), i = 1, 2, 3, 4u> = whirl speedo> a, ub , >c = angular rate components of 6relative to SFVd , id = location of disk mass center in b,c

d i rec t ionsn(s), f(s) = distr ibuted location of e lement

cross section mass center in b, c direct ions

riL, to = rj(O), f(0)VR , to = vV) W)M = element mass per unit length111. I P I = displacement vector relative to '5,

(R2(


3/8

Fig. 3 Typical finite rotor element and coordinates

center coincident with the elastic rotor centerl ine is given by theexpression

2 1 W ) ' m i 0 ". 0 mi_+{l\m D o o -o iD oLo o i,J (5a)T he use of equa t ions (1), with the re ten t ion of only second ordert e r m s , reduces equa t ion (5a) to

&d 2\wf L 0 md\ \WJ 2 \ r j- 4>TBIP

0 In(56)

T he lag rang ian equa t ion of motion of the r igid disk using equation(56) and the constant spin speed restr ic tion, = Q, is

([


4/8

By using equat ions (10), (13), equat ions (15) can be writ ten in thematr ix fo rms

d(?l = | E / { q W ' H * " ] M r f s (a)dS*A = - \ p { q e f [ * ' f [ * ' ] {q}ds (6 )

rf3 e = | M { q < T [ * ] r [ * ] { q e } d s

+ 1 ^ 4 , ^ + | & { q e } [ * ] r t * ] { q e } ^- n { 4 T [ * r ] r t * B ] { q c } ^ (c) (16)Th e energy of the com plete elemen t is obta ined by inte gratingequations (16) over the length of the element to obtain


5/8

ta ined from the prev ious deve lopmen t by either evaluating the integrals of equations (18), (21), using the var iab le p rope r t ie s or byt r ea t ing the e lemen t as an assemblage of un i fo rm sub-e lemen tequa t ions of motion as given in Append ix A. Th e assembled set ofsubelelements then possesses [4 X ( n u m b e r of sube lemen t s ta t ions)] coordinates which can be reduced by the following proced u r e .

T he assembled set of sube lemen t equa t ions , where a ssembly isaccomplished by util iz ing the appropr ia te geometr ic d isp lacemen tcompa t ib i l i ty cons t ra in ts (see [7], p. 38) is of the form, in fixedframe coordinates,([MH + [MH]) (&}a

= {Qe} (29)

(4 x n u m b e r of s u b s t a t i o n s ) x 1A d isp lacemen t dependency be tween the in te rna l d i sp lacemen ts ,|qB)6, and the e lemen t endpo in t d i sp lacemen ts , jq e) a and |q e |C) canbe imposed by considering the static , homogeneous case of equation (29).

' [ * " ] [KU [K e}ac~[K]ba [K]bb [K]bc[K%a [K e]cb [K%c {0} (30)T he in te rna l d i sp lacem en t vec to r from the second row of equa t ion(30) is

M 6 = -[*&[*!.{}. - lK'l[K'\.{


6/8

00

bU

-=

A =

i/~'~\~"~^ - - -

. V 4

. - -A -

/P/IT""

/ !

/ 7^ 1 * - -

\//^

r- - - '

"~"~'/

= L ~ I

/"_=

*

\A i i

"/

5 1

r^

rPr

/

- - -

X -

- -\^^~~

^

i! (RPM x 10 )

- WHIRL FRAME ANALYSIS, CASE {a)FIXED FRAME ANALYSIS, CASE (a)

- FIXED FRAME ANALYSIS, CASE (b>

Fig. 6 System w hirl speeds

Table 1 Rotor configuration dataElemen tNo .

1

2

3

4

5

6

Sube lemen tNo . Ax i a l D i s t a n c eto Sube lemen t(cm)

0 . 0 01.27

5 .087 .62

8 .8910 .1610-6711 .4312 .7013 .46

16 .5119 .05

2 2 . 8 62 6 . 6 7

28 .7030 .4831 .5034 .54

I . D .(cm)

1.521.78

1.52

O.D.(cm)

0 . 5 11 .02

0 .762 . 0 3

2 .033 . 3 03 .302 . 5 12 . 5 41.27

1.271 .52

1.521 .27

1 .273 .812 . 0 32 . 0 3

Table 2 Whirl speeds Case (a)

Wh i r lRat io

2

1

1/2

1/4

0

Wh i r l

P o s i t i v e18,14851,430111,45517,15949,98396,45716,70049,20485,55216,48148,80080,649

Speed

16,26748,38476,382

(RPM)Negat ive

14,75844,69558,42415,47046,61264,75215,85847,52069,64016,06047,95772,737

SpinSpeed

70,000

Whirl SpeedsForward

19,83850,55591,320

Backward

128154559963990

e = eg eccent r i c i ty ofstation 3Fig. 7 System unbalance response

For whirl frame coordinates with isotropic bearings, the u n b a l ance response is obtained from equation (36) with to = Q (i.e., A =1) . In th is case the unbalance force in equa t ion (36) is a cons tan trelative to (R, henc e, the unba lance re sponse is also a cons tan t re la t ive to (R. From e i the r of the two p lanes of equa t ion (36) the undamped unba lance re sponse is

M = ([Ks] - Q2([M S] + [ G 5 ] ) ) - 1 ^ 8 } (41)2w X 1N u m e r i c a l E x a m p l e s . To d e m o n s t r a t e the app l ica t ion and

accuracy of the f inite e lement model, a typical rotor bearing syst e m as i l lu s t ra ted in Fig. 5 is analyzed to d e t e r m i n e its whir lspeeds and unba lance re sponse . A dens i ty of 7806. kg/m 3 and elast ic modulus 2 .078 X I 0 1 1 N / m 2 were used for the d is t r ibu ted ro to ra n d a concen t ra ted d isk wi th a mass of 1.401 kg, polar inertia0.0020 kg/m 3 and diametral inertia 0 .0136 kg-m 2 was located ats ta t ion 3. The d is t r ibu ted ro to r was mode led as a six e lemen tmember with each element consisting of seve ra l sub e lemen ts . Thegeometr ic da ta of these e lemen ts and sube lemen ts is l is ted inT a b l e 1.Two identical bearings, idealized as u n d a m p e d and l inear,were located at stations four and six. The following two cases ofbearing stiffness were analyzed:

(") kbyy(b) k"vv

= k" = 4.378 x 1 0 7 N / m , / 4 , = kbwr = 0.= kbww = 3.503 x 1 0 7 N / m , ^ w = kbm

= - 8 . 7 5 6 x 1 0 eN /mT his symmetr ic bea r ing is equ iva len t to an or tho t rop ic bea r ingwith principal axes oriented at (45, 135) relative to the X axis .Case (a) Isotropic Bearings. The undam ped wh ir l speeds werecomputed from the 14th order e igenvalue problem of equa t ion (37)for whirl ra tios of 0, \ %, 1 , 2. The f irs t three whirl speedsfor each whirl ra tio are l is ted in T a b l e 2 and p lo t ted in Fig. 6. Theunba lance re sponse for a disk mass center eccentric ity of 0.025 in.a t s ta t ion th ree was de te rmined f rom equa t ion (41) for the speedrange 0-30,000 rpm and is p lo t ted in Fig. 7. Also by using the 56 thorder e igenvalue problem of equation (35) the na tu ra l wh i r l speedsassociated with a spin speed of 70,000 rpm were computed . T hesespeeds are also l is ted in T a b l e 2 and p lo t ted in Fig. 6.

Case (6) Orthotropic Bearings. The undamped wh ir l speedswere computed from the 56th order e igenvalue problem of equat ion (35) for several spin speeds. The f irs t three whirl speeds are

598 / MAY 1976 Transactions of the ASftHE


7/8

T a b l e 3 W h i rl s p e e d s C a s e ( 6 )

Sp inSpeed(RPM)

1000

20000

40000

60000

80000

100000

Wh irl Speeds (RPM)

Forward

152584838476387

166064835278177

173104829381917

181024815786177

188974792390457

196554754994510

Backward

140994003366747

137623999765264

130633988962414

122803970459548

114973943657057

107473907755079

l is ted in Table 3 and are plotted in Fig. 6 . The unbalance responsewith the same unbalance as for case (a) was computed for thespeed range 0-30,000 rpm. The response orbits are ell ip tical withprincip al axes oriented a t (45 , 135) relative to the Y axis . Th esemima jor a nd sem imino r axes of the orbits for s ta tion 3, are plotted in Pig. 7 versus spin speed for the above sp eed range. For thiscase (nonisotropic) the backward whirl modes are excited as wellas the forward whirl modes by the unbalance force. Hence, i t is interesting to note that the system precession changes from forwardto backward to forward as the spin speed passes through the f irs tbackward cri t ical speed.

Summ ary and Conclus ionsA finite e lement model including the effects of rotatory inertia ,

gyroscopic moments, and axial load has been developed for a rotating shaft e lement. The equation of motion of the element is presented in both a f ixed and rotating frame of reference. The rotating frame equation is particularly useful for isotropic systems sincethe two planes of motion can be treated separately , while the f ixedframe equations provide tha generali ty of handling problems withnonsymmetr ic bea r ing s t i f fness and damping . T he ma jo r d isad vantag e of the f ixed frame finite e leme nt formulation is th at t heorder of the system matrices is larger thereby requiring a largec o m p u t e r .

The finite rotor e lement was applied to a typical rotor bearingsystem to i l lustrate the procedure and accuracy of the model. Natu ra l wh i r l speeds and modes were ca lcu la ted by us ing bo th thefixed and rotatin g frame formulatio ns. A sepa rate 26 statio nlumped mass analysis of the case (a) example was run to obtain acomparison with the f inite e lement results . The whirl speeds obtaine d using the 6 s ta ti on f inite e lem ent analysis were all h igherthan the 26 station lumped mass analysis . For the whirl speedslis ted in Table 2 , the largest difference be tween th e two analyseswas less than 4 percent for the third forward mode at a whirl ra tioof + 2 .

The finite e lement model can easily be uti l ized to model rotor-bearing systems for purposes of determining cri t ical speeds, s tabili ty , unba lance re sponse , t rans ien t re sponse , e tc . T he re su l t s ob

tained by the authors for the included examples and other rotorsystems indicates that the model is re liably accurate . The rotor e lement can be generalized to include the 'effects of shear deformation, axial torque, and various forms of internal damping. I t appea rs tha t the f in i te e lemen t app roach can p rov ide a va luab le newdimension to the analytical options available for s tudies of rotor-bea r ing sys tems .

A c k n o w l e d g m e n tThe authors wish to acknowledge the valuable assis tance ren

de red by Dr . E dwa rd Z orz i , Dynamics Spec ia l i s t , AiResea rch Man ufacturing C o. , durin g the pre para tion of this pap er.

R e f e r e n c e s1 Eshlem an, R. L., "Critica l Speeds and Response of Flexible Rotor Systems," Flexible Rotor-Bearing System D ynamics, ASME, Vol. 1,1972.2 . Archer, J. S., "Consistent Mass Matrix for Distributed Mass Systems," Journal of the Structural D ivision, Proceedings of the ASCE, Vol.89, ST4,1963, p. 161.3 Ruhl, R. L., "Dynamics of Distributed Parameter Turborotor Systems:Transfer M atrix and Finite Element Techniques," PhD Thesis, Cornell University, Ithaca, N. Y. Jan. 1970.4 Ruhl, R. L., and Booker, J. F., "A Finite Element Model for Distributed Parameter Turborotor Systems," JOURNAL OF ENGINEERING FORINDUSTRY , TRAN S. ASME, Feb. 1972, p. 126.5 Lund, J. W., Rotor-Bearing D ynamics D esign Technology, Parts III,IV, AFAPL-TR-65-45, May 1965.6 Green, R. B., "Gyroscopic Effects on the C ritical Speeds of FlexibleRotors," TRA NS. ASM E, Vol. 70,1958, pp. 369-376.7 Hurty, W. C , and Rubenstein, M. F., D ynamics of Structures, Prentice-Ha ll, Inc., 1964.8 Meirovitch, L., Analytical Methods in Vibrations, MacMillan BookCo., 1967.9 Ziegler, H., Principles of Structural Stability, Blaisdell PublishingCo., 1968.

A P P E N D I X ARigid Disk Matrices

[M*] =

[Mj] =

[G*] =

[ < ] =

IK1 =

"m 0 0 00 % 0 00 0 0 00 0 0 0ro o o o "10 0 0 00 0 ID 0_0 0 0 ID _ro o o o i0 0 0 00 0 0 / p0 0 Ip 0

r 0 ~m d 0 0m i 0 0 00 0 0 0_0 0 0 0"0 0 0 0 "0 0 0 00 0 0 -ID0 0 ID 0

~

F o r IP = 21 B, [MdR] = %[G

[& ] =

F o r Ip = 5

"0 0 0 0 '0 0 0 00 0 - / , 0_0 0 0 -Ip

ID , [G*] = - 2 M

Journal of Engineering for Industry MAY 1976 / 599


8/8

Fin i te Shaf t Element Matr ices

[Mfi =4/20

0- 1 3 /-3 /20

[Mg\ = 120/

[G] =

[Ki] = ~

15 60 1560 -22/id 221 042 0 54 00 540 13/L-1 3 Z 03600_ ju"r' 3/- 3 6003/036-3 1

2[ir2 0120/ 0- 3 6 0-3 1 0L 0 -3Z120 120 -6 / 4/26 / 0 0 4/2- 1 2 0 0 -6 /0 -12 6/ 00 -6/ 2/2 0

4/213 / 1560 0

03/ 2

0- 2 2 /

s y m

15622/ 4Z20 0 4/2.

\_mT\ /on

36-3 Z 4Z20 0 4Z20 0 -3Z 36- 3 6 3Z 0 0-3 1 -f0 0 0 0Z2

sy m

363Z 4/2

- 3 / 0 0 4/2sk ew sym00 0-3 1 4/236 - 3 / 0

0 - 3/ 360-f 3 / 00 3Z 4/2 0_

sy m

M = ih0 /

L 6/36003 /

- 3 60

0 0 2/2

120 120 6/ 4/2-61 0 0 4/2 Js y m

\x l42 0

015 6- 2 2 /005413 /0

00- 2 2 /- 5 4 0

0- 1 3 /

04/213/00-3 /2

00- 1 3 /

3/ 20

sk ew sym

015 6 022 / 0 00 -22/ '4/2 0

363 1 4/20 00 036 +3 /0 -3/ - /2L 31 0 0

4/2- 3 / 360 0 360 0 3/ 4/2-Z 2 - 3 / 0 0 4/2_

jiti1

{Q1} = ^

l o ^ + i o ^20 L1 + 20 tRl20 QL 30 QRf o nJ + l o v*fh^ l + h**120 QL 20 QRL~ t /2 + t 7230 QL 20 CR"30 ^ ~ 20 ^

' 7 3~ 2 0 ^ ~ 20 R1

| o vj + f o v JT o ^ - f o VR12

~ t /2 - t f20 QL 36 QR20 fei 20 QRfo V + | o "l o ^ 2 + f o v J 230 ^ + 20 * ? .

600 / MAY 1976 Transaction:-- of the ASME

Documents

Dynamics of Rotor Bearing Systems Using Finite Elements