stiffness estimation from random response in multi-mass...

E L S E V I E R PII: S 0 2 6 6 - 8 9 2 0 ( 9 7 ) 0 0 0 2 1 - 0

Prob. Engng. Mech. Vol. 13, No. 4, pp. 255-268, 1998 © 1998 Elsevier Science Ltd

All rights reserved. Printed in Great Britain 0266-8920/98 $19.00+0.00

Stiffness estimation from random response in multi-mass rotor bearing systems

Rajiv Tiwari* & Nalinaksh S. Vyas Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India

This paper describes a procedure for extraction of the linear and nonlinear stiffness parameters in rotors with multiple discs, supported in rolling element bearings. The analysis puts forth a technique, which can be employed on-line, for processing the rotor vibrations picked up at the bearing caps, as it does not require an a priori knowledge of the excitation force. The problem is formulated for a multi-degree, nonlinear, balanced rotor system experiencing random excitations from the bearings, caused due to imperfections and deterioration of the rolling surfaces as well as from the other random sources, like inaccuracies in alignment, etc. The governing equations of motion are subjected to coordinate transformation and subsequently modeled into the Fokker-Planck equations through the Markov vector approach. The solution procedure for the Fokker-Planck equation and the assumptions involved are outlined. A curve fitting algorithm is proposed to process the solution to the Fokker-Planck equation for the inverse problem of parameter estimation from the measured response. The technique is illustrated on a laboratory rotor rig and the estimated parameters are compared with those obtained through analytical models available for isolated bearings. © 1998 Elsevier Science Ltd. All rights reserved

1 INTRODUCTION

Rolling element bearings are known to possess highly nonlinear stiffness characteristics. Estimation of linear stiffness parameters of isolated rolling element bearings involves establishing a relationship between the load carried by the bearing and its deformation through classical solution for the local stress and deformation of two elastic bodies apparently contacting at a single point, l The early studies 2'3 on bearings concern vibrations caused due to geometric imperfections of contact surfaces. Procedures 4'5 are available for estimation of bearing stiffness under static loading conditions. The problem of identifying the nonlinear stiffness characteristics was approached by Kononenko and Plakhtienko 6 through the Krylov-Bogoliubov-Mitropolsky procedure. Kraus et al. 7 presented a method for the extraction of rolling element bearing stiffness and damping under operating conditions. The method is based on experimental modal analysis combined with a mathematical model of the rotor-bearing-support system. The method is applied for investigation of the effect of speed, preload and free outer race bearings on stiffness and damping. Muszyanska 8 has developed a perturbation technique for estimation of these parameters. The technique involves a controlled input excitation to be given to the bearings. Goodwin 9 reviewed the

*Corresponding author.

255

experimental approaches to rotor support impedance measurement. Nonlinear stochastic contact vibrations and friction at a Hertzian contact have been studied by Hess et al. i t The experimentation involves excitation of the bearings either externally by a white Gaussian random normal load or within the contact region by a rough surface input and the analytical approach is based on the solution of the Fokker-Planck equation.

A technique for estimation of nonlinear stiffness in a rotor-bearing system, based on analysis of its random response has been developed by Tiwari and Vyas. 11 The governing nonlinear equation with a random excitation force, resulting due to random imperfections of the bearing surfaces and assembly is modeled through the Fokker- Planck equation. The solution of the Fokker-Planck equation is further processed by the linear and nonlinear bearing stiffness parameters. The technique has an advantage over other existing ones as it does not require an estimate of the excitation forces and works directly on the response signals from the bearing caps. The analysis involved a rotor with a rigid shaft carrying a single disc at its midspan. The problem becomes more involved for rotors with flexible shafts carrying more than one disc. The present study attempts the inverse problem of parameter estimation for such multi- mass flexible systems governed by multi-degree of freedom nonlinear differential equations. The excitation to the balanced rotor is taken to be random in nature, primarily

256 R. Tiwari, N. S. Vyas

/ Taking the shaft parameters to be linear, the shaft stiffness and proportional damping forces, respectively, are

~"n + 2 Fs~ = ~ = 1 k i j x j (3)

n+2 Fd, = Z 0/ijtJ i = 1,2 . . . . . ( n + 2 )

j=l

mn+~

j'

XI

Fig. 1. Multiple disc rotor on rolling element bearings.

arising out of bearing and assembly imperfections. The governing equations are subjected to a coordinate transformation, so as to enable the governing equations to be modeled as a Markov process through the Fokker-Planck equations. Certain engineering assumptions are made to obtain the solution to the Fokker-Planck equation. A curve fitting algorithm is developed, to process the statistical response of the system obtained by the solution of the Fokker-Planck equation, to extract the rotor-bearing stiffness parameters. The procedure is illustrated for a laboratory test-rig with two discs and the experimental results are compared with the analytical guidelines of Harris 5 and Ragulskis et al. 4 The algorithm developed is tested by Monte Carlo numerical simulation procedure.

where shaft stiffness parameter kij is defined as the ith force corresponding to a unit j deflection with all other deflec- tions held to zero and can be obtained from the Strength of Materials formulae. Proportional shaft damping parameter, 0/~j, is defined in a manner similar to k o.

The excitation to the system is taken to be random in nature. The bearing surface imperfections, caused by the random deviations from their standard theoretical design and progressive surface and subsurface deteriorations are large enough to cause measurable levels of vibration and can be the primary source of these excitations. In addition excitation can be contributed to by inaccuracies in the rotor- beating-housing assembly, etc. Taking Sl and s2 as the effective random displacements at the bearings, primarily due to surface imperfections and inaccuracies in the rotor- bearing-housing assembly, the bearing forces as masses m l and m2 are written as

Fo~ = {kLi(Xi -- s i) .A¢_ kNLia(x i _ Si)} i = 1,2 (4;

In the above kL and kuL are the unknown linear and nonlinear bearing stiffness parameters and G can be polynom- inal in x.

Eqn (4) can be rewritten, more generally, as

Fb~ = {kLiXi -Jv kNLiG(xi ) } - - F i ( t ) i = 1,2 (51

where Fi(t) is the random component of the bearin~ force.

Using eqns (3) and (5) the equations of motion (1 and 2) can be expressed as

[MI{X)(} + [AI {Jr} + {OV/OX} = {F} (6

2 EQUATIONS OF M O T I O N

A balanced rotor, with n discs mounted on a massless flexible shaft supported in nonlinear bearings at ends is shown in Fig. 1. The shaft is treated as a free-free body, carrying unknown effective bearing masses m l and m2 at its ends and the known disc masses m3,m4 .... mn+2. The bearings are incorporated through external "forces", Fb, acting on masses m~ and m> Taking the shaft stiffness and damping forces as F, and Fe, respectively, the equations of motion are written as

- - F d i - - Fs, - F b i - ~ mixi i = 1,2 (1)

-- Fd, -- F,, = miYc i i = 3, 4 . . . . . (n + 2) (2)

where

[M] =

[A]=

[i l° m2

0

I 0/11

0/21.

0/(n + 2) 1

0/12

0/22

0/(n + 2)2

0/1(n+2) 1 0/2( I + 2) /

0/(n + 2)(n + 2) -]

(i

Stiffness estimation from random response in multi-mass rotor bearing systems 257

OV(Xl, x2 . . . . . X(n + 2))/OXl }

{ O v } O V ( x t , x 2 X ( n + 2 ) ) / O X 2 ..... =

3V(x1, X2 ..... X(n + 2))]~X(n + 2) {x} {l} x2 F2

{g} = ;{F} = • ,

x(~ + 2) 0

1 V(Xl , x 2 . . . . . X(n+2)) = a-{X} T([K] + [KL]){X}

d.

+ { g(x) } r [KNL] { g(x) }

[K] =

[KL] =

[ kll k12 kl(n+2)

k21. k22 k2(n + 2)

t-k(n+2)l k(n + 2)2 k(n+2)(n+2)

[kL0 i1 i k/.:

1_ 0

[KNL ] :

{ g (x ) } =

0 lCNL:

L o

( o G(~)d~)l/2

G(~)d~) m ( o"

0

(8)

The Markov vector approach extended to nonlinear multi-degree-of-freedom systems 12 is adopted for the solution of eqn (6). Equations of motion, (6), with damping and force Fi set to zero, are solved for eigenvalues p2, i = I, 2 . . . . . (n + 2) and orthonormal modal matrix [U], such that

[U]r[K][U] = [p2] (9)

[uIT[MI[U] : [I1

where [p2] is the diagonal eigenvalue matrix, while [/] is an identity matrix.

Application of coordinate transformation

{X} = [U]{~7} (10)

and premultiplication by [U] r, the equations of motion, (6), yield,

1 ~t "~ ~i~i "~- -~iiOV(~l, 172 ..... ~(n+ 2))/O~i :-- qi

i = 1 , 2 , . . . , n + 2 (11)

Mi is the modal mass• Modal damping matrix [/3] and the generalized force vector {q} are

[13] = [U]r[c~][U] {q} = [U]r{F} (12)

and the potential energy, in generalised coordinates, can be expressed as

1 V ( r / l , r] 2 . . . . . r/(n + 2)) : "~{rl}T[u]T([K] q'- [KLI)[U] {r/}

+ {g([g]{rl})}rIkNL]{g([U]{n})} (13)

3 R E S P O N S E

The approach to obtaining the response of the system is greatly simplified if the random excitation to the system is assumed to be such that the generalised forces, qi, in eqn (l 1) can be treated as ideal white noise• Many engineering appli- cations are based on this idealization and it turns out that the responses obtained through such models are quite accepta- ble if the time scale of the excitation is much smaller than the time scale of the response. 13 The time scale of the excitation is the correlation time, roughly defined as the length of time separation beyond which the excitation process is nearly uncorrelated. The time scale of the response is the measure of the memory duration of the system which is generally about one quarter of the natural period of a mode which contributes significantly to the response. Treat- ing the excitation of eqn (11) as uncorrelated Gaussian white random forces with the following properties

E[qi(t)] = 0; (14)

E[qi(t)qi(t2)] = 27r~bi~(t2 - tl);

where ~b i denotes the excitation intensity factor• The joint probability density function, P0/1,~2,'", ~¢n+2), 9~, 92,'",9¢n+2)), for the motion governed by eqn (11) can be described by the Fokker-Planck equation 14

i= 1 -- ~i ~i~i Mi Oni O~i "~ ~ [~iiiliP -+- 7rq~i~i~i

ap (15) Ot


For a stationary case eqn (15) reduces to

(16)

With the energy equi-partition assumption 14

[Jl 1 [1141 ~b I = ~221M2 ~2 . . . . . . . /~(n + 2)(n + 2)[M(n + 2)~(n + 2)

-----3"

eqn (16) can be solved to obtain the joint probability density of displacements and velocities in terms of the trans- formed coordinate system as 14

PO/1, "q2, " ' ", l'/(n + 2), ~ 1,172, " " ", ~(n + 2))

=cexp[-~{~{iT'r[M'{il}+V(~l,~lz,'",Tl(n+Z))}l

(17)

Performing the inverse orthonormal transformation and noting that the term

2{/lIr[M]{/l} + e(nl,n2, " " , n (n+2) )

in the bracket on the right hand side of eqn (17) represents the total energy of the system and that 3" is a constant, the joint probability density of displacements and velocities in the original set of coordinates is

p(xl, x2,'", X(n + 2), 21, X2, " " ", X(n + 2))

=cexp[-~(l{x}T[M]{~2}-t-V(Xl,X2,'",X(n+2))}]

(18)

The joint probability functions p(xl,X2, "",X(n+2)) and p(21,22, ...,2(,+2)) are obtained from eqn (18) as 15

)< p ( x l , x 2 , ' " , X(n + 2), 2 1 , 2 2 , " "X(n + 2))dXl dx2"" "dJc(n + 2)

~-clexp[- ~{V(Xl,X2,'",X(n+2))}] (19)

with

- 1 Cl -- L

× ex [- 7r

X dxldX2"--dx(n + 2 )

Also

.. L × p(xb X 2, "", X(n + 2), 21 ,22 , "" "2(n + 2))dxI dx2"" "dx(n + 2)

[ 1 / ~ (n+2) = V/mlm2" "'m(n +2)

× exp [ - ~{ I{X,r[M]{X, }] (20)

The joint probability function p(x l.X2) is obtained as

X p(Xl , X2, "", X(n + 2))dx3dx4' " .dx(n + 2)

~-~C 1 f ~ I~_ "'" I ~ exp[ - ~V(Xl,X2,'",X(n+2,)]

X dx3dx4" • .dx(n + 2/

-~-C2 Xexp[-- ~{~(MK22[MKl122)x~-~-~(MxII]MKl,22) x

(21

with

3' × ex [-

[r] = [K] -t- [KL]

and M. denoting the minors (Appendix) of the matrix [r] The probability density functions p(2 I) and put2) are

p(21)= f ~ ~_ "" f~ p(21,22, "",X(n+2))dx2d.~3

. ..d2(n + 2, __= [ 1V/~3`] exp [ 3 ̀I" 1 . ~ ~mlXl }] .2

(22

p(22)~- ~ I ~ " " f~ p(21,22,'",2(n+2))d21d23

3" 1 .2 ""d2(n+2)~-[l~exp[-;{-~m2x2}]

(25

The variances of velocity responses 21 and 22 are obtaine as

~, = _ 212p(21 )d21 = rr/ml3 , (2z

2 • • ° a~2 = p(x2)d22 = 7rim23̀ (2:


Combining eqn (24) eqn (25)

7r 1 (26)

the joint probability density function for the displacement responses xl and x2, from eqn (21) eqn (26), can be written as

[ I 1 = c2exp - z(M~2~/M.~ ~22)x~

+ l(M~lffM~1122)X 2 q'- (MxI2[M~1122)×ix2

..]_ kNL, g2 (X 1 ) -k- kNl a g2 (X 2) } ] (27)

with

C2

I × exp V(Xl,X2,'",x(.+2fl dx3dx4...dx("

Defining

~kXl =Xlq+ o --Xli'~ ~kX 2 =X2o+~) --X2j

for small ~x I and Ax 2, one can write

p(x~¢,+,7, x2~+ ,~) = csexp [ 1

try, ax~

{ 1 M 2-(MKll/M.l122)x2 X .~( r221Mrl122)x2i + 1

+(M~12]Mrl122)XllX2~-k-kNL, g2(Xli)+kNLsg2(x2~)}]

1 × exp

°X~ O'~t2 k / ~ m2

( (M~221M~ m)x~/Xx~ + (M~l ~/MK1122)xuax 2 ×

+ (M~21M~ m)(Xl~X~ + x~/Xx~) + kNC, G(Xl)AX~

+kNtaG(x2i)Ax2}] (30)

Combining eqn (30) eqn (28) gives

4 E X T R A C T I O N O F B E A R I N G P A R A M E T E R S

Based on the above analysis, the bearing parameters. namely, the linear stiffness parameters kL,. krs. the nonlinear stiffness parameters kuL,, kuc~ and the bearing masses m I and ms. are extracted from experimentally obtained random response. These parameters are obtained for both the vertical and horizontal directions. The problem formulation, in the horizontal direction, remains identical to that in the vertical direction.

The joint probability density function p(xl. x2) for a set of displacements (Xli,X2j)(Xl(i+,),X2q+,l) , (Xl~i+,~ > Xli and x2~+,~ > xs~) from eqn (27) are

[ o~, 0~2 ~ 1 {1 p(xl~, x2~) = c2exp - ~(M~22/MK1122)x~

1 + ~(M.l l/M.1122)xZj + (M~I2/M.1122)Xl~Xej + kul., g2(x b)

+kNLzg2(x2j)} 1 (28)

p(xl. +,1. XS~ +,~ ) = C2 exp [ 1

a~ I o~ 2 m v / - ~

1 × (M~22/M.II22)x2o+,~ + "~(M.II/M.1122)x~.~+,

+ (M~ls/M~1122)x~.+ .xs~+ ,7 + kN~, g2(x~.+ .)

+kNlag2(x2o+,,)}l (29)

p(XlII+ ,7' X2~+ ,)) =P(XlI' xzJ )exp

- 1 M X [ O - , O - x 2 ~ { ( x22[MKl122)Xll ~kXl

+ (Mrl l]Mr1122)x2j~.'~2 -~ (MK12[Mr1122) × (xl,Ax2 +xs/XXl) + kuL, G(xl,)AXl

+kutaG(xsj)~xs}] (31)

For N values of each displacement. Xl,.Xl: ........... XlN. and xs,.x22 ........... xsN. eqn (31) can be expressed as a set of (N - 1) 2 linear simultaneous algebraic equations, as,

Lp( + kL j

1,L,J l ~ l J ~ k ~ , J

- [(M~22/M~, ~22) (~-~2) + (M~,,/MK,,22) ( ~ )

+ (MKI2/M,d I22) ( ~ + ~2 ) ] { ~--~ }

\zXx~j \zx~2 ~ j /

i = 1.2 ..... ( N - 1) j = 1.2 ..... ( N - 1) (32)

eqn (32) can be solved for kL~. kL:. kuL,, kurd. and ~ . using the least squares fit technique.


Fig. 2. Two disc rotor-bearing set-up.

the vertical and horizontal directions by accelerometers mounted on both of the bearing housings. The nonlinear spring force provided by the rolling element bearings is taken to be cubic in nature 4 i.e. G(x)=x 3. The stiffness- matrix of the shaft model of the rig is taken as 16

[K] =

12El LI 10 Ol -01 _011 ] L3 ] - 1 0 2

0 - 1 - 1 2

E l = 1.03X 1 0 8 N - m m 2 ; L = 1 6 5 . 0 m m (35)

and the disc masses are m3 = 0.515 Kg; m4 = 0.755 Kg. (The procedure, however, does not require a knowledge

of the disc masses). Typical experimentally obtained displacement and velo-

city signals (Xx,X2,~t 1 and ±2), in the vertical direction, picked up by the accelerometer are given in Figs 4-7. The joint probability function, p(xl,x2) and variances, or2 and

XI 2 a~ 2 are computed from the measured responses. The joint

probability density function, p(xl, x2), of the displacements is shown in Fig. 8. The bearing parameters estimated from eqn (32) are given in Table 1.

5 SIMULATION

Fig. 3. Accelerometer locations.

The procedure is illustrated for a laboratory rotor, with the shaft supported in bearings at the ends and carrying two equidistant discs, as shown in Figs 2 and 3. The shaft is driven through a flexible coupling by a motor and the vibration signals are picked up (after balancing the rotor) in both

Monte Carlo simulation is employed to check the accuracies involved in making the engineering assumptions in the algorithm. The experimentally obtained values of kL,, k&, kNL,, ksL2 and mx/Cm~ are used in eqn (6) to simulate the displacement and velocity responses, x l, x2,~tl and 22, through the fourth order Runge-Kutta numerical technique, for broad band excitation forces, Fl(t) and F2(t) with zero mean and Gaussian probability distribution as described in Figs 9-12. The vertical displacement and velocity responses resulting at the two bearings due to the simulated forces ot Figs 9-12 are given in Figs 13-16. The joint probability distribution of the simulated vertical displacement is shown

7.00

4.20

1.40

- - -t.40

-4.20 ..~

-7.00 0.00 0.25 0.50 0.75 1.00 1.25 1-~0 1__75 2.00 2.25 250

Time t (sec X 10 j)

Fig. 4. Displacement signal in the vertical direction at bearing 1.


. - - ,

• 11, i t

-7.00 i , I ,. ~ I = ,, l . . . . I

0.00 0.25 0.50 0.75 1.OO 1.25 1.50 1.75 2.0'3 2.25 250

Time t (sec X lO t )

Fig. 5. Velocity signal in the vertical direction at bearing 1.

><

pI

e~ .m

700

4 .20

1.40

-1.40

-4,20

-7.00 0.00 0.25 0.5O 0.75 1.00 1.25 1.50 L75 2.00 2.25 2.50

Time t (sec X 10')

Fig, 6. Displacement signal in the vertical direction at bearing 2.

7.00

>d 4.20 g

~ •

~ , 0.00 0.25 0,50 0.7'= 1.00 1.25 1.50 L75 2.r'JC 2.25 2.5(3

Time t (sec X 10 j)

Fig. 7. Velocity signal in the vertical direction at bearing 2.

in Fig. 17. The simulated response is now fed into eqn (32) to obtain the values of kL,, kt~,kuL,, kNt~ and ~ . A similar exercise is carded out to obtain the parameters in the horizontal direction. These values are listed in Table 2.

The fairly good agreement between the values of the bearing stiffness parameters, kL,,kG,kNL,,kNt~ and v / ~ m 2 , obtained by processing the experimental data and those from the Monte Carlo simulation, indicate the


~- 2.0

1

~ o.o

(,an, X t0') ~.o -s. ¢.

Fig. 8. Joint probability density distribution of vertical displacements at bearing 1 and bearing 2.

T a b l e 1. E x p e r i m e n t a l b e a r i n g s t i f f n e s s a n d m a s s p a r a m e t e r s

Parameters Vertical Horizontal

k/., (N/mm) 1.04 × 104 0.87 >( 104 kNL ' (N /mm 3) -- 5.10 × 10 l° - 3.50 X 101° kG (N/mm) 1.04 × 104 0.86 X 104 k N ~ m m 3) -- 3.62 × 101° -- 2.19 × 10 I° x/mlrn2 (Kg) 0.21 0.20

correctness of the experimental and algebraic exercises. It should be noted that the simulated values of the bearing stiffness parameters are obtained for an ideal white noise excitation, while the experimental ones are obtained by processing the actual response of the system, where the unknown excitation was idealized as white noise. It also needs to be pointed out that the values of the damping parameters aij, are not required for the estimation procedure (eqn (32)). Any convenient set of values of aij, can be employed in eqn (6) for the purpose of simulation.

6 V A L I D A T I O N

The analytical formulations of Harris 5 and Ragulskis et al. 4 which are based on Hertzian contact theory, are employed for comparison of the bearing stiffness parameters kL and kNL, obtained by the procedure developed.

The total elastic force at the points of contact of the ith ball with the inner and outer races is expressed as 4

Fi = Kn(g -Jr xcos~/i q- ysinrli) 3/2 (34)

and its projection along the line of action of the applied force is

F i = Kn(g ÷ xcosT//+ ysimli)3/2cosTli (35)

where g is the radial preload or pre-clearance between the ball and the races and x and y are the displacements of the moving ring in the direction of the radial load and perpendicular to the direction of the radial load respectively. ~7i is the angle between the lines of action of the radial load (direction of displacement of the moving ring) and the radius passing through the center of the ith ball. Kn is coefficient of proportionality depending on the geometric and material properties of the bearing. The specifications ol the test bearing are given below.

Ball bearing type SKF 6200

Number of balls 6 Ball diameter 6 mm Bore diameter 10 mm Outer diameter 30 mm Pitch diameter 20 mm Inner groove radius 3.09 mm Outer groove radius 3.09 mm Allowable pre-load 0-2 microns

The value of Kn, for the test bearing with the abow specifications, is estimated by the method suggested b,. Harris 5 as 2.82 × 105 N/mm 15.

6 . 0 0 . , ., , , i , ° , •

~, 1.,~0

HIt t I 'lrll I rwll! r'l -G.O0

0,00 0.20 0.40 0.60 0.80 1.00 120 1.40 1.60 t.80

T i m e t ( m - s e c )

Fig. 9. Simulated random force at bearing 1.

!.00

r.C r~

6~0

2.50

2.00

1.50

1.00

0.50

0.00 -BOO -6.40 -4.80 - 3 2 0 -t .60 0.00 1.60 3.20 4.80

£


Force F l (Nx10 "I)

Fig. 10. Probability density distribution of simulated force at bearing 1.

1.00

3.60

t .2o .I L

~ -360

-6.00 ' 0.00 020 0.40 0.60 0.80 1.00 120 1.40 1.60 1.80

Time t (m-see)

Fig. 11. Simulated random force at bearing 2.

!.00

2.00 | w " i , l i , i

'~ 1.60

120

0.80

'~0 0.4O

-8.00 --6.40 -4.80 -320 -1.~0 0.00 1.50 3.20 4.80 6~0 8,00

Force F~ (N x t0 "l)

Fig. 12. Probability density distribution of simulated force at bearing 2.


F~

.=_

7.00

4.20

1.40

-1.40

-4.20

-7.00 0.00 0.20 0.40 0.60 0.80 1.00 120 1.40 1.60 1.80 200

Time t (see X 101 )

Fig. 13. Simulated displacement signal at bearing l.

0

~4

M

0

7 ° ~ '

4.20 t

1.40

-1.40 I w

-4.20 F

-7.00 ~" 0.00

A A A A.A A A A A A A '- ,,-,~,~v"-,, vv V V V V V V v y V ~ /

I I I I ] I I T L

020 0.40 0.(50 0.80 1.00 120 1.40 1.15~, Leo 2.~

Time t (sec X lO t )

Fig. 14. Simulated velocity signal at bearing 1.

7.00 ~ ~

420

1.40

~ -1.40

-4.2.0

-7.00 0.00 020 0.40 0.60 0.80 1.00 120 1.40 1.60 1~0

Time t (sec X 10 t)

Fig. 15. Simulated displacement signal at bearing 2.

.00


N

E

°~

o i

b.

7.00

4.20

1.40

-140 J V ' I -4.20 [

-7.00 ' 0.00

vvV j

I

020

I ~ ~ t '

.,,,,,,-,.,. ? ? V v v,., .,,v v I; v,,,l!v

0.40 0.~0 0 8 0 1.120 1 2 0 1 . 4 0 1.60 I.eC

Time t (see X 10 x)

Fig. 16. Simulated velocity signal at beating 2.

2C~

3.0

1.5

d

o.o

0.0 L¢~O

Fig. 17. Joint probability density distribution of simulated vertical displacements at beating 1 and bearing 2.

The total elastic force in the direction of the applied force is

F = ~ Fi (36) i = 1

where n is the total number of balls in the bearing. Using the condition of zero elastic force in the

direction perpendicular to the elastic load, the deforma-

tion y, perpendicular to the radial force line is expressed as

y = ~ . [g + xcos(Bi)]3/2sin(~li)/~. [g + XCOS(~i)] 1/2

i = 1 i = 1

X sin207i) (37)

Eqns (35) and (37) are used in eqn (36) and the bearing stiffness is determined as a function of the deformation x as

k(x) = OF/Ox (38)

It can be seen that the bearing stiffness is critically depen- dent on the preloading, g, of the balls. While the manufac- turer, may, at times, provide the preload range, the exact value of the preloading of the bearing balls in the shaft- casing assembly, especially during operations which have involved wear and tear, would be difficult to determine. The stiffness of the test bearing is plotted in Figs 18 and 19 as a function of the radial deformation, x, for various allowable preload values, g. The bearing stiffness obtained experimentally using the procedure developed (shown in Figs 18 and 19) shows good accuracy with theoretically possible values. It is also to be noted that the theoretical stiffness calculations are based on formulations which analyse the bearing in isolation of the shaft. The comparison between the experimental and theoretically possible stiffness is also shown in Table 3. The expressions for the theoretical stiffness

Table 2. Experimental and simulated bearing stiffness and mass parameters

Parameters Vertical Horizontal

Experimental Simulated Experimental Simulated

kL, (N/ram) 1.04 X l04 1.49 X 104 0.87 X 104 0.99 X 104 ku& (N/mm) - 5.10 x 10 ~° - 8.48 X 101° - 3.50 X 10 l° -4 .81 X 10 ~° kL~ (N/mm) 1.04 X 104 1.48 X l04 0.86 X 104 0.98 X 104

~ mm 3) - 3.62 x 10 ~° - 7.83 X 10 t° -- 2.19 X 10 j° - 3.23 x 10 l° (Kg) 0.21 0.26 0.20 0.24

Q


3 . 0 0

2 , 7 0

2. , ,01

.g- ~.4 2.10

1 . 8 0

1.50

1 . 2 0

0.90

0 . 6 0

0 . 3 0

' ' ' ~ I ' ' " - ' '

--7

~ 5

~ 4

~ - - ' - - ~ ~ ~ 3

0 . 0 0 , , i , I J n t i

o.oo o~o a.oo ~ . ~ zoo zso :too 3.so ~.oo ,.so

Displacement x t (mm X l0 ~)

~.00

Fig. 18. Comparison of rolling element bearing stiffness at bearing 1. (1,2)--Present study (in vertical and horizontal directions, respectively). (3-7)--Harris 5 and Ragulskis e t al. 4 with preload 0.0002, 0.0003, 0.0004, 0.0005 and 0.0006 mm, respectively (in any radial direction).

in Table 3 have been obtained by curve fitting the stiffness values obtained from eqn (32) through a quadratic in x.

In an earlier analysis by the authors, 11 where the shaft flexibility was not accounted for and was treated as a rigid body and the analysis was restricted to a single degree of freedom idealisation, the bearing stiffness for the same experimental set-up was found to be 1.32 X 104-5.08 x 101°x 2 N/mm and 2.23 X 104-8.50 X 101°x2 N/mm) in the horizontal and vertical directions, respectively. A comparison with the stiffness values of Table 3 reveals the influence of shaft flexibility.

While a good agreement on the bearings stiffness parameters is observed between the values generated following the method of Harris 5 and Ragulskis et al. 4 and those obtained experimentally through the present procedure, the values of the effective masses at the bearing ends obtained as by-products of the present procedure also appear reasonable. The experimentally obtained values of the effective masses ~ are 0.21 and 0.20 (Table 1) in the vertical and horizontal directions, respectively. If the two beatings were taken to be identical for each of the bearing ends, the values turn out to be 0.21 Kg for the vertical direction and 0.20 Kg for the horizontal direction. These values look reasonable in view of the fact that along with some contribution from the bearings themselves,

a division of the mass of the shaft, which in this case is 0.306 Kg, is seen at the two beating ends.

7 CONCLUSION

A procedure for estimation of linear and nonlinear stiffness parameters of the rolling element bearings supporting a generalised multi-disc flexible rotor, from the random response of the system, is presented. The procedure involves certain engineering approximations, including idealisation of the excitations from the bearing surface, assembly imperfections as white noise sources and the assumption of energy equi-partition (expression between eqn (16) and eqn (17)). The procedure, however, has a distinct advantage in that it does not require a knowledge of the excitation forces and works directly on the random response signals, which can be conveniently picked up at the rotor bearing caps.

A C K N O W L E D G E M E N T S

The authors wish to acknowledge the support provided by the Structures Panel of the Aeronautical Research and Development Board of India in carrying out this work.


3.00

2.70

-r 2.40 0

N 2.10

. ~ 1.50

1.20

0.60

0.30

. . . . . J I i i w v j w v i

~ 4

- - - . - _ _ _ . ~ ' ~ ~ 1

7

6

0.00 ...... J I I I I ~ v n 0.00 0.50 t.O0 1.50 2,.00 2 .50 3.00 3.50 " .00 4 .50 5.00

Displacement x~ (mm X l0 4)

Fig. 19. Comparison of rolling element bearing stiffness at bearing 2. (l,2)--Present study (in vertical and horizontal directions, respectively). (3-7)--Harris 5 and Ragulskis et al. 4 with preload 0.0002, 0.0003, 0.0004, 0.0005 and 0.0006 mm, respectively (in any

radial direction).

Table 3. Experimental and theoretical 4~ bearing stiffness parameters

Preload (mm) Theoretical stiffness Experimental stiffness Experimental stiffness (Radial) (N/ram) (N/mm) (at hearing 1) (N/mm) (at bearing 2)

0.0002

0.0003

0.0004

0.0005

0.0006

1.20 X 104-4.01 X 101°x 2

1.47 × 104-2.18 X 101°x 2

1.69 X 104-1.42 X lOl°x 2

1.89 X 104-1.02 X lOI°x 2

2.08 X 104-6.09 X 109x 2

0.87 x 104-3.50 x 101°x 2 (horizontal)

1.04 X 104-5.10 X 101°x 2 (vertical)

0.86 X 104-2.19 x lO~°x 2 (horizontal)

1.04 × 104-3.62 X 10J°x 2 (horizontal)

R E F E R E N C E S

1. Hertz, H., On the contact of rigid elastic solids and on hardness. In Miscellaneous Papers. Macmillan, London, 1896, pp. 163-183.

2. Lohman, G., Untersuchung des Laufgerausches von Walzlgem. Konstruktion, 1953, 5.

3. Gustavsson, O. and Tillian, T., Detection of damage in assembled rolling element bearings. Trans. of the American Society of Locomotive Engineers, 1962, 5.

4. Ragulskis, K. M. Jurkauskas, A. Y., Atastupenas, V. V., Vitkute, A. Y. and Kulvec, A. P., Vibration of Bear- ings. Mintis Publishers, Vilnyus, 1974.

5. Harris, T. A., Rolling Bearing Analysis. Wiley, New York, 1984.

6. Kononenko, V. O. and Plakhtienko, N. P. Determination of nonlinear vibration system characteristics from analysis of vibrations. Prikladnaya Mekhanika, 1970, 6, 9.

7. Kraus, J., Blech, J. J. and Braun, S. G. In situ determination of rolling bearing stiffness and damping by


model analysis. Trans. of American Society of Mechan- ical Engineers, J. of Vibration, Acoustics, Stress, and Reliability in Design, 1987, 109, 235-240.

8. Muszynska, A. and Bentley, D. E. Frequency-swept rotating input perturbation techniques and identifica- tion of the fluid force models in rotor/bearing/seal systems and fluid handling machines• J. Sound and Vibration, 1990, 143, 103-124.

9. Goodwin, M. J. Experimental techniques for bearing impedance measurement• Trans. of the American Society of Mechanical Engineers, J. of Engineering for Industry, 1991, 113, 335-342•

10. Hess, D. P., Soom, A. and Kim, C. H. Normal vibrations and friction at a Hertzian contact under random excitation: theory and experiments. J. Sound and Vibration, 1991, 153(3), 491-508•

11. Tiwari, R. and Vyas, N. A. Estimation of nonlinear stiffness parameters of rolling element bearings from random response of rotor bearing systems• J. Sound and Vibration, 1995, 187(2), 229-239•

12. Nigam, N. C., Introduction to Random Vibrations. The MIT Press, Cambridge, Massachusetts, 1983.

13. Lin, Y. K., Stochastic aspects of dynamic systems. Stochastic Problems in Mechanics, Study No. 10, Solid Mechanics Division, University of Waterloo, 1973.

14. Caughey, T. K. Derivation and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white random excitation• J. Acous- tical Society of America, 1963, 35(11), 1683-1692.

15. Roberts, J. B. and Spanos, P. D., Random Vibration and Statistical Linearization. John Wiley, New York, 1990.

16. Childs, D., Turbomachinery Rotordynamics: Phenom- ena, Modeling and Analysis• A Wiley-Interscience Publication, John Wiley, New York, 1993.

APPENDIX A MIRRORS OF THE MATRIX [AK]

[K ] =

(k l l +kn,) k12 k13 kl(n + e)

kzl (ke2 + km 2) k23 k2(n + 2)

k31 k32 k33 k3(n + 2)

k(n+2)l k (n+2)2 k(n+2)3 k(n+2)(n+2)

Mrl I =

I (k22 + kn2) k23 k2(n + 2) 1

k32 k33 k3(n+2) /

[ L k(n + 2)2 k(n + 2)3 k(n + 2)(n + 2) .1

Mr22 =

(k]l +kn~) k13 kl(n+e) ]

k31 k33 k3(n+2) /

J L k(n+2)l k(n+2)3 k(n+2)(n+2) _1

" k33 k34 k3(n + 2)

k43 k44 k4(n + 2) MK1122 =

• k(n + 2)3 k(n + 2)4 k(n + 2)(n + 2)

stiffness estimation from random response in multi-mass...

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