Dynamics of machine foundations with random parameters

Journal of Sound and Vibration (1987) 112(1), 23-30

DYNAMICS OF MACHINE F O U N D A T I O N S WITH R A N D O M PARAMETERS W. MIRONOWICZ AND P.

SNIADY

Institute of Civil Engineering, Technical University of Wroclaw, 50-370 Wroclaw Poland ( Receired 16 November 1985) Solution of the problem of free and forced vibration of a discrete system with random parameters, as a model of a machine foundation, is considered. Expressions for the expected values and correlation functions of the solution are formulated, in terms of the probabilistic characteristics of the structure, initial conditions and excitation, which are assumed to be known. The stochastic linearization method is used. The solution is illustrated by an example of a block foundation with random parameters. 1. INTRODUCTION Vibration problems o f structures in which random factors are involved have been extensively studied, with the randomness being most often assumed to be introduced by the Ioadings [1-4]. However, in a few papers the random character of structural parameters has been taken into account [5, 6]. The sensitivity o f the system dynamic response to changes in its parameters depends on the kind of load. It is less in the case of random wide band excitation and greater for deterministic harmonic excitation, as it is then possible for resonance to occur [7]. The parameters o f a machine foundation are often subject to random variations, of technological, operational and structural causes. Foundation dynamic analysis thus can be improved by taking them into account. In what follows the vibration of a discrete system with random parameters is considered as a model of the machine foundation. General solutions are given for the eigenvalue problem, and the free and forced vibration problems. Expressions are derived for the expected values and correlation functions, it being assumed that the appropriate probabilistic characteristics of the system as well as the initial conditions and excitation are known. The first order probabilistic method is used [8].

2. PROBLEM FORMULATION The problem to be considered is that o f the vibration of a discrete system of n degrees of freedom, governed by the equation

B?j(t) + C~t(t) + Kq(t) = f ( t ) ,

(2.1)

where q(t) is the vector of generalized co-ordinates, f ( t ) is the random excitation vector, and B, C and K are random matrices of inertia, stiffness and damping. It is assumed that C =/.tB + KK,

(2.2)

with ~ and ,r non-negative real constants. The probabilistic characteristics, such as the expected values and the covariance functions of the matrix constituents B, K, C and the 23 0022-460x/87/010023+05 s03.00/0 (~ 1987 Academic Press Inc. (London) Limited

24

w. M I R O N O W I C Z AND P. S N I A D Y

vectorf(t) are assumed to be known. The random elements of equation (2.1) are expressed as sums of their expected values and the random fluctuations: B=/~+B,

C = C'+ C',

f(t) =J~(t) + f ( t ) ,

K =/~+/~,

q(t) = 4(t) + @(t).

(2.3)

Here the expected values are denoted as follows: E[B] =/~,

E[C] = C,

E[K] = / ( ,

E[f(t)] =.f(t),

E[q(t)] = t~(t).

(2.4)

The random fluctuations of the constituent parts of the matrices B, C, K and the vector

q(t) are assumed to be small enough relative to their expected values for the first order probabilistic method [8] to be a good approximation of the solution. In this case the product of the random values X and Y has the form XY

= 9(~'+ X Y +

9~9.

(2.5)

Formula (2.5) can be considered as an expansion of the p r o d u c t X Y in a double Taylor's series, about the expected values 9(, 'r', in the fluctuations X, Y, with only the first two terms of the expansion being retained. This is similar to the immediate linearization method [4] applied to non-linear problems. After expression (2.3) have been substituted in equation (2.1) and the approximation (2.5) has been used one obtains

(2.6) For the eigenvalue problem equation (2.1) is transformed to the form ( K - t a 2 B ) q = 0.

(2.7)

Hence, after approximation (2.5) has been used, one obtains (/~ - A/~)c~+ (/~ - A/}- ~/~)4 + (/~ - A/~)t~ =0,

(2.8)

where to2 = Z = A + %,, E[~] = 0. Equations (2.6) and (2.8) can be solved to give expressions for the probabilistic characteristics of the displacements and frequencies of the system. 3. SYSTEM NATURAL FREQUENCIES, FREE VIBRATION Performing the expected value operation on equation (2.8) gives a deterministic relation, (/( - A,~)~ = 0,

(3.1)

from which eigenvalues Ai ( i = 1 , 2 , . . . , n) and the normal mode vector t~= ~,~ can be determined by using well known methods. These vectors can be used to construct a normal mode matrix. Taking into account expression (3.1) in equation (2.8) where the unknown factors ~, t~ occur, and after left-hand premultiplication by the vectors ~T (i = 1, 2 , . . . , n) in order to eliminate the vector q, one obtains the vector of random fluctuations of the eigenvalues, = {/~-'}( l~/T/~IV - I~VT/~IV{Ai}),

(3.2)

where {/;,} = l~Vr/~l~/. Symbols (')T and {. } denote the transposed vector or matrix and the diagonal matrix, respectively.

RANDOM

PARAMETER

MACIIINE

FOUNDATIONS

25

By making use of equation (3.2) the correlation matrix of the eigenvalues A, ( i = 1 , 2 , . . . , n) is obtained in the form K ~ = E[AA"r] = {/~-'} I~'TE[(/(~V--/~IV{A,})( I~T/~T__ {~j} i~,'r/~T)] l~{/~-t}.

(3.3)

In the particular case when/~ = a/~,/( = ),/(, where the parameters a and Y are random variables, one obtains, from formula (3.3), o2

Kaa = {A,}E[(a - 7)2].

(3.4)

In this case the matrix Kxx is diagonal, from which it follows that correlation between different eigenvalues does not exist. The expected value and the variance of the natural frequencies are approximately governed by the formulae [8] o~, ~ ff-~i,

2 o',o = (114,~,)o'].

(3.5)

The variance cry, is an element on the diagonal in the ith line of the matrix Kax. Given the fluctuations of the eigenvalues (3.2), the corresponding fluctuations of the normal mode vectors can be calculated from formula (2.8). Next the problem of free vibrations of the system can be considered, with damping taken into account. It is assumed that for t = 0 the initial random conditions are

q(O)=d=t*l+d,

(3.6)

4(0) = e = ~+,~.

Assuming](,) = f ( , ) = 0 and performing the expected value operation one obtains, from equation (2.6), /~c~'(t) + C'c~(t) +/(c~(t) = 0.

(3.7)

This is a deterministic, homogeneous set of differential equations satisfying the deterministic initial conditions 4(0) = d,

d(0) = ~.

(3.8)

Random fluctuations of the vector of generalized co-ordinates q(t) satisfy the equation /~'(t) + C'~(t) +/(t~(t) = - Bt~'(t) - C'q(t) - / ( 4 (t),

(3.9)

and the random initial conditions ,~(0) = d,

,~(0) = ~.

(3.10)

It is convenient to solve equations (3.7) and (3.9) with the initial conditions (3.8) and (3.10) being transferred to the expected values of the normal co-ordinates:

~(t)= I~')*(t),

t~(t)= l~,'f(t),

d= l,~'y(0)= lh,,

e= i~,')(0)= ;~'v. (3.11)

One thus obtains f t3, clio

2(t) = - ~'T~li'

I2

/t~,(t- ~')};"(T) d~ - ~'Tdl~'

fo

f 3,

Io

&~,l

{t~,(t -- ~')}~(~') d~

(3.12)

26

W. M I R O N O W I C Z A N D P. .SNIADY

where f,~(t) = e -a,' sin .f'/,t,

.0,~=,~-a, ~,

f~(t) = e -'~,' cos .0,t,

a,=~,/2~,,

{h,(t)} = {(1/8,.O,) e -a,' sin

:,~=:~,1:,,, .0:}.

(3.13)

As can be seen from equation (3.12) the random fluctuations of the normal co-ordinates are the sums of the fluctuations due to the randomness of the system parameters and the randomness of the initial conditions. Making use of equation (3.12) one can determine the correlation function of the normal co-ordinates from the formula

K~r(tl,

t2) = E[37(t1))~T(t2)],

(3.14)

and then the vector of expected values and the correlation matrix of the normal coordinates, ,i(t)

~ = W~,(t),

Kqq(t,, t2) =

E[q(tl)#T(t2)]

=

l:VKyy(t,,

t2) fiT.

(3.15)

4. SYSTEM FORCED VIBRATION For forced vibration the system to be considered is that described by equation (2.1) with the initial conditions q(0) = 0,

8(0) = 0.

(4.I)

From equation (2.6) one obtains expressions for the expected value and the random fluctuations of the vector q(t) in the forms

~:i(t)+ C:1(t)+ :c:1(t)=](t), /~c~(t)+ (~c~(t)+/~c~(t)=:(t) -/~'- 6"c~(t)-/~c](t). Applying the normal mode approach (3.11)to equations (4.2)gives :(t) = .~(0 =

(4.2)

Io

{1;,(t-~-)}~:.:0) tiT,

Io{~,(t--r)}I~'T](,)dr Io{h',(t--~')I(~'TN~'}(T)+ -

I~TcI'v~(7)+ I,VT/~I,V~(,))dr.

(4.3)

By using the second of equations (4.3) in formula (3.14) one can determine the correlation function. With the excitation v e c t o r f ( t ) assumed to be independent of the system random characteristics B, C, K one obtains

K,.r(t,, t2) =

io'io'o{h,(q-

r,)} l'~'TK/r(r,, z2) 14:{i*l)(tz- rz)} dr, dr2

- fo" fo2{h~(t,-r,)}fV-~E[R(~',)R'r(~'2)]fVIt~(t2-~',)} d~',dr2, where R(t) =/~g'.~'(t) +

CI:V}(t)+ ~I:Vf~(t).

(4.4)

RANI)OM

PARAMETER

IqACilINE

27

FOUNDATIONS

Similarly, for the generalized co-ordinates one obtains, respectively,

4(t) = q(t) =

l~':(t) =

fo

Io

~(t-~-)/(,)

/4(t- r)f(r) dr-

d~-,

Io

I?t(t- r)R(r) dr,

(4.5)

and Kqq(ll ' 12) = l~Ksy(ll, t2) ~T

=Io"Io'*l?f(ti-ri)Klr(ti,t2)fflX(t,-h)dridh -Io"~2l~l(t,-'r,)E[R(r,)[((h)llZl(t2-h)dr,

dh,

(4.6)

where H(t) = IV{/~i(t)} IV r is the matrix of impulse response functions. For the case of constant frequency due to harmonic excitation, which is important in practical applications, the excitation vector can be taken to be of the form

f(t) =f(t)+f(t) = d sin pt + d sin pt.

(4.7)

Assuming the solution in the form

~(t)=~ssinpt+~ccospt,

~(t)=~,sinpt+~ccospt,

(4.8)

and substituting this into equation (4.2) gives a set of algebraic equations for the expected values, ( Ko - / ~ ) q ,o

*" = ~, - pCqr

o

*" + ( l< - p : B)q< pCq, * =0,

(4.9)

and random fluctuations,

pCtl, + (/< -p2B)tT< =

-pCti, - (I< - p2/~) ~e.

(4.10)

After the equation sets (4.9) and (4.10) have been solved one obtains expected values of the generalized co-ordinates and their random fluctuations from which the correlation matrix can be determined. The presented theoretical solutions are going to be illustrated by the exemplary analysis of the block foundation of random parameters. 5. BLOCK FOUNDATION OF RANDOM PARAMETERS An example of the analysis can be provided by the vibration of a block foundation (see Figure 1) in the plane xz, for which

~=tg'~l--L-~,= L= i,

i ~

~ ~k~A~ ~ A ~ k~J~ ~

(5.1)

Here rh is the mass of the block, -~x.- and S~.z are the products of inertia of the block mass, iy is the moment of inertia of the block mass, ,4 is the area of the block base, Jr is the moment of inertia of the block base, and/~,,/~x a n d / ~ are the base stitinesses.

28

W. M I R O N O W I C Z

AND

P. S N I A D Y

Z 2

Jr(t)

t

2

2p

I

4

I

O,y

4

t

2

Figure 1. B l o c k f o u n d a t i o n .

The system random fluctuations are assumed to be given in the form of matrices B, K of the same structure as those of equations (5.1). With the notation

0 = ~,~,T,

0 ' = ~'(a,} ~,T,

0" = ~'{a,}{a,} IP,

the correlation matrix (3.3) becomes

The elements r, s of the constituent matrices (5.2) are given by the formulae 3

E[(/(C//(T),] = /~,,E[/~,,/~,,],

E[(/(CP/~T),,] = E O{,E[/~,,/~I,], I=1

3

3

E[(/3OI/(T),] = E 01,E[/~,f,i],

3

E[(/~/~/0/~T),~]= Y~ 2

I-I

0~,E[b,,bh,],

(5.3)

I=1 h = l

where the/~'s and /~'s are elements of the matrices /~,/~. As an example, free vibration at the block foundation, as shown in Figure 1, of the base random stiffness

g =

Tx,i,

(5.4)

has been considered. The correlation between the r, s elements of the matrix (5.4) is assumed to be of the form

K,~ = E[ k,.k~,] = IA I ~Z,, exp (-~,x2- x,D cos , ( x 2 - x,) dxt dx2 ,

(5.5)

2 where or,, ~', and ,f are constants. Figures 2(a) and (b) show the effect of the variations in the parameters ~', ~r on the correlation function (5.5) and consequently on the correlation matrix (5.2). In machine foundation design the ratio oJ'p is important because of the transitory resonance problem. It is convenient to introduce the resonance index given by

/3 = (c3 -p)/,,/crL + cry,

(5.6)

which is the analog of the reliability index for structural reliability problems. Here o~ and o'~ are the natural frequency and its variance, and p and o-p2 are the frequency of excitation and its variance, respectively. This index can be treated as the randomness measure at the first-order level of the transitory resonance problem. Figure 3 presents the variation of the index /3 =/3(~', ~:) for the foundation under consideration; here o-~,=/~x =0, o'~, = o']3 and lack of correlation between/~ and/~y have been assumed.

RANDOM

PARAMETER MACHINE

29

FOUNDATIONS

(a) IO0 8O (=0

40

I 05

I

T

1

I

I

1.5

2

2'5

06

08

1.0

,oo~..~o

oo l-

-\

20

" 0.2

04

Figure 2. Correlation function (a) K,, = K,,(~) and (b) K,, = K,,(,~).

250 , ~ 200 *3

150

6- l o o 50 015

10

1 5

2 0

2 5

310

Figure 3. Resonance i n d e x / 3 = fl(~').

The forced vibration of the foundation in question has also been considered. For C = 0, p = 2500 kg m -3,/~ = 3 k , / ~ =2k,/~. = 7k, k = 2 x 107 Nm -3, p = 6.32 s -I, ~"= 0.5, ~: = 0.25, 2 -or,, - o" = constants, ao = [ 1,0, 2]Va, and the standard deviation of random fluctuation (4.7) equal to 0. I a, the results are ~, = [559"5

1.29

[3,30.4 72 32,.2] E[qqT] =

7-2 321.2

0.02 0.7

0.7[+ 32.9_1

57"3]Tx 10-t2a, "2473"9 10"9

10"9 .3"0

108.0

0.5

o

qc = 0,

108"01 \ 0.5[0"2 / x 10-24a 2. 1

4.9J

#

]

30

w. M I R O N O W I C Z A N D P. S N I A D Y

6. CONCLUSIONS The dynamic response o f a structure having parameters with small random fluctuations has been considered. A probabilistic analysis of such structures has been presented, providing estimates o f the response and the structural reliability. The general problem, in view of the randomness of the system parameters, is of a non-linear character. The solution presented here is based on the linearization method (probabilistic method of the first order) and hence is approximate. It is the more precise the smaller the fluctuations are in relation to their expected values. This solution can also be treated as a first iteration step. In the second step formula (2.5) should be taken in the form X Y = ,~"~" + ) ~ f ' + ,~"Y"+ .~Y.

(6.1)

Performing the expected value operation then gives E[X (2) Y(')] = ..Y~"+ E[.~ (') ~'(')],

(6.2)

where indices ('), (2) denote the first and second iteration steps, respectively. As an example a block foundation problem has been considered. Analysis of the eigenvalue problem has been carried out, with the stiffness matrix assumed to have small random fluctuations with a correlation function of the form (5.5). As can be seen from Figures 2(a) and (b) the structural correlation function is considerably affected for small values of ~', ~, but the disturbance to it stabilizes and diminishes as these parameters increase in value. Figure 3 shows the effect of the randomness on the resonance index fl =fl(~') (5.6). Significant effects of the parameters, especially when ~" has small values, are evident. An example of forced vibration analysis of the foundation under investigation has been given as well.

REFERENCES I. V. V. BOLOTIN 1965 Statistical Methods in Structural Mechanics. Moscow: Stroiizdat. 2. J. D. ROBSON 1964 An Introduction to Random Vibration. Edinburgh: Edinburgh University Press, Amsterdam: Elsevier. 3. Y. K. LIN 1967 Probabilistic Theory of Structural Dynamics. New York: McGraw-Hill. 4. B. SKALNIERSKI and A. TYLIKOWSKI 1982 Stochastic Processes in Dynamics. Warsaw: PWN-Polish Scientific Publishers. 5. J. O. COLLINS and W. T. THOMSON 1969 American Institute of Aeronautics and Astronautics Journal 7, 642-648. The eigenvalue problem for structural systems with statistical properties. 6. M. SHINOZUKA and C. J. ASTILL 1972 American Institute of Aeronautics and Astronautics Journal 10, 456-462. Random eigenvalue problems in structural analysis. 7. H. GRUNDMANN 1977 Munich ICOSSAR'77, 81-96. On the reliability of structures under periodic loading. 8. C. A. CORNELL 1970 Structural Reliability and Codified Design, SM, Study No. 3, University of Waterloo. A first order reliability theory of structural design.