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Discrete analysis method for random vibration
Computers & Strucfures Vol. 49, No. 3. pp. 52M30. Printed in Great Britain.
DISCRETE
1993 0
ANALYSIS METHOD VIBRATION
0045.7949/93 $6.00 + 0.00 1993 Pergamon Press Ltd
FOR RANDOM
TAN DONGYAO School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100083, People’s Republic of China (Received 27 May 1992)
Abstract-A discrete analysis method which uses recurrent formulas to calculate the stationary and non-stationary random vibration responses for multi-degree of freedom (MDOF) dynamic systems is proposed in this paper. The unconditional stability of the recurrent formulas and their property of being able to calculate the accurate stationary mean and mean square responses are proved. An optimum time step is suggested with which the recurrent formulas can quickly carry out accurate stationary random responses. As an example, these recurrent formulas are applied to MDOF structures. Their practical calculation forms are derived which use the banded shape and symmetric properties of structural dynamic matrices to effectively calculate structural stationary or non-stationary mean and mean square responses. Some numerical examples are given to show the application of the method.
NOTATION
c
coefficient matrix in state equation structural damping matrix excitation vector and its mean value a(l) expectation of A -W 1 identity matrices I, I, , IO K coefficient matrix k structural stiffness matrix A4 coefficient matrix structural mass matrix T transformation matrix coefficient matrix QD influence matrix of excitations 4 correlation matrix auto-correlation matrix function of the LT, excitations coefficient matrices rl , r2, r3 block matrices in the Jordan form of C r,, rJr, r. u coefficient matrix real part of the eigenvalues of matrix C 11th complex part of the eigenvalue of matrix u&k C X(l), x’(r), x”(r) structural displacement, velocity and acceleration vectors Z(f), f’(f) mean values of x(t) and x’(r) state vector and its mean value Nh Y(f) parameter !, eigenvalue of the coefficient matrix in the recurrent formula At time step d-function 6(r)
b,
INTRODUCTION
With the increasing development of computer technology the analysis of continuum systems using discretization methods has gained a wide scope of application, The finite element method proposed in the early 195Os, which discretizes a continuum medium in space, established the foundations for quantitatively and routinely analysing the mechanical, thermal and flowing properties of multi-dimen-
sional complex solids or fluid structures. Later, the boundary element method was proposed as a counterpart of finite element method. In the 1960s the discretization idea was introduced into the deterministic dynamic analysis for structures and the direct integration method was proposed which discretized structural dynamic differential equations in the time
domain. The discrete analysis method for random vibration discussed in this paper is a further application of the discretization idea in random vibration theory. Developing such a method aims at finding a more powerful and general random vibration analysis method which may use the flexibility and routinely computational advantages of the discretization method. This method is applicable to both stationary and non-stationary linear or equivalent linearized non-linear random vibration problems. In the last decade, many workers focused their attention on investigating non-linear random vibration problems. It was a clear tendency in these investigations that most of the researchers abandoned the traditional way of solving FPK equations and adopted the equivalent linearization method. As many direct numerical simulations have proved that the equivalent linearization method could lead to very good results for a wide scope of non-linear problems, it gradually becomes a practical analysis method for non-linear random vibration problems. In this sense, analysing non-linear random vibration problems will also depend on the method used in analysing linear random vibration problems. There are many difficulties which should be overcome, not only for non-linear random vibration problems, but even for linear random vibration problems. Such problems which have not been well solved are the random vibration of soil-structure interaction systems with non-classical damping, 523
TAND~NGYAO
524
random response analysis for structures subjected to earthquake excitations input from multiple supports considering their time delay relationships and the structural wind induced random vibration considering the spatial correlativity of the wind loads. To solve these problems, it is necessary to establish a general analysis method for random vibration. The proposed discrete analysis method for random vibration is such an effort in this direction. The discrete analysis method for random vibration uses the mean value process and correlation function of the excitations directly to calculate the nonstationary and stationary mean and mean square responses of a dynamic system step by step by using recurrent formulas obtained by discretizing the dynamic differential equation of the system in the time domain. The recurrent formulas discussed in this paper have already been used to analyse the random responses of structures subjected to filtered white noises and white noises which correlate in the time domain [IA]. Here, we study their properties as the basis for a discrete analysis method for random vibration, i.e. their stability and accuracy in numerical calculation. Also, we study their practical application to structural dynamic systems. In the practical recurrent formulations the symmetric and banded shape properties of structural mass, damping and stiffness matrices are used and the computational time is greatly reduced.
where (n - 1)At < t 4 (n - 1 + With the above assumption, everywhere between (n - I)Ar Hence, it can be substituted into the system at time instant (n following equation is obtained y(n-l+B)--y(n-1)
+ D(n - 1 + /j).
y(n - 1 + 8) = (I - fiArC)-‘[y(n + pAtD(n
+ D(t),
(1)
+ D(n),
(2)
where y(n), y’(n) and D(n) represent the state vector, its derivative and excitation vector at time instant n At, respectively. Assume the state vector of the system varies linearly between two adjacent discrete time instants. Thus, the variation rate of the state vector in this time interval can be calculated below from the state vectors at these two discrete time instants r’(t)
=
Y(n--l+B)-y(n-1) BAt
- 1) -
1+ 8)1, (5)
where I is an identity matrix. It is known from eqn (3) that the variation rates of the state vectors at time instants nAt and (n - 1 + /?)At are identical so that y(n) -y(n
- 1)
At
y(n - I + S) -y(n
=
- 1)
(6)
flAt
Substituting eqn (5) into (6) to eliminate y(n - 1 + /I), the relationship between the state vectors at time instants nAt and (n - I)At is finally derived r, ., I 1 1+$(1-/JAtC)-’
(n - 1) P
where y(t) is the state vector of the system, C is the coefficient matrix which describes the dynamic characteristics of the system and D(t) is the excitation vector. The discrete time instants n At (n = 0, 1,2, . . ) can be obtained by using the time step At to equally discretize the time axis, and the corresponding state equation of the system at each discrete time instant is y’(n) = Q(n)
(4)
The state vector y(n - I + 8) can be derived from the above equation
FORMULA
The state equation is a general form for describing the dynamic behaviour of a system. For a linear time-invariant dynamic system the state equation can be written as y’(t) = Q(t)
=Cy(n-l+B)
BAr
y(n) = /I-RECtiRRENT
p)At and 0 < b < I. eqn (3) is satisfied and (n - 1 + /?)A[. the state equation of - 1 + /?)At, and the
’
(3)
+ (I - bArC)-’
ArD(n - 1 + ,0).
(7)
This is a recurrent formula containing a parameter ,8, called the B-recurrent formula. It is a fundamental recurrent formula in the discrete analysis method for random vibration. NUMERICAL
STABILITY
OF P-RECURRENT
FORMULA
Firstly, the dynamic system is assumed to be physically stable. Then the numerical stability of the b-recurrent formula requires that the spectral radius of the coefficient matrix [(l - Be’)1 + p-‘(I - /3AtC)-‘1 (i.e. the maximum absolute value of its eigenvalues) in eqn (7) must not be greater than one [5]. With knowledge of linear algebra and theory of ordinary differential equations, we know that in the most general case the coefficient matrix C, in the state equation of the dynamic system, has repeated complex eigenvalues, and there always exists a nonsingular matrix P which can transform C into the Jordan form, i.e.
P_‘CP
=
r
r,
(8)
525
Discrete analysis method for random vibration where
P-1
=
ri I,
.. ri =
(
ra 10
,
i=1,2
,...,
P
I+;(&BAtP-VP)-’
1 -f
..
1
I+;(I-gAtC)-i
r.
> ...
. . .
=
. . . IO
1 -’ (
ril
B>
Ii+!-(li-/?Atri))’ B
(9)
Denote the block matrices on the diagonal position in above matrix as
In above matrix, I, is a second-order identity matrix, and
-I
Z,,- bAtrio -BAtI,, ..
.. --/IA&
rik=
%k
[ -v*
V* uik
1 ’
k = 1,2,. . . ,j
ai, = ui2= . . . = Uik= . . . = u,, v,, = viL=
’ = uik= . . . = Vii’
(10)
where uik and vik are real and complex parts of the eigenvalues of matrix C. For a physically stable dynamic system, the real parts of the eigenvalues of matrix C must not be positive, i.e. 44
GO, i=l,2,...
, r; k = 1,2,. . . ,j.
(1 - /!?:tu,f:iz:;
I
.. .
1, - /?Atr,
1
where Ii is an identity matrix of the same order as that of matrix r,. It is known from linear algebra that the eigenvalues of matrix [(l - p-i)1 + B-‘(1- /?AtC)-‘1 are equal to the eigenvalues of matrices e,(i = 1,2, . . . , r). It can be proved that e, has the partitioned low triangular form after the inverse matrix in above equation is carried out, so that the eigenvalues of the block matrices on the diagonal position of matrix ei are the eigenvalues of itself. These block matrices are
(11)
Atv&)’ + ’ - ’
(1 - ~At~~)$?Atr;,)’
j?Atv,
(1 - BAtUik)’+ (8AtVik)2
To simplify the notation, the identical block matrices on the diagonal of the matrix in eqn (9) are denoted as r,, thus rn = rik(k = 1,2,. . . , j). Multiply matrix ((1 - B -‘)I + /I -‘(I - /IArC)-‘] from the left and the right with P-’ and P, respectively, and use eqn (8) to derive
Their eigenvalues can be easily obtained as
1 ,+= j
1 - /.IAtu, +/I-1 (1 - /YAtuik)2+ (bAtvti)2
1
526
TAN D~NGYAO
Atv, ’ ’ (1 - Z?Atua)* + (/?A~v,~)*’
Since the /l-recurrent formula is unconditionally stable, it can then be expanded as
i=l,2
)...,
j.
Their modules can be derived below by taking complex operation
xy(n-l)+At
I+
[l - (fi - l)AtuJ* + [(b - l)Atv,]* o.5 IAl
=
(1 - fiAtuik)* + (BAru,)’ i=l,2
’ ,...)
r.
or
a > 0.5
(12)
(13)
the spectral radius of matrix [(l - z!~‘)Z + fl-‘(Z - fiAtC))‘] will not be greater than one for any time step At, i.e. the p-recurrent formula is unconditionally stable for p > 0.5. ACCURACY
OF THE b-RECURRENT
FORMULA
There are two meanings to the accuracy of the p-recurrent formula here: firstly, it refers to the accuracy when it is used to solve the state equation of a dynamic system. This accuracy can be evaluated by the accurate solution of the state equation. Secondly, it refers to the accuracy when it is used in analysing the random responses of the dynamic system. This accuracy can be evaluated by the accurate solutions of the stationary mean and mean square responses of the system.
= [I + AK - /I(AtC)*
+ fi*(AtC)’ -
of the state equation
.]
xy(n-l)+At[Z+fiAtC-@Arc)*+ x D(n - If
...I
p).
(16)
Comparing eqns (15) and (16), it can be concluded that the second-order accuracy of the p-recurrent formula for solving the state equation of the dynamic system can be achieved when fl = 0.5. Accuracy in calculating the stationary mean value responses
Taking mathematical expectation at both sides of eqn (7), the equation for the b-recurrent formula to calculate the mean value responses for the system can be obtained as j(n)=
I+!-(I-BAtC))’ + (I - /3AtC)-‘At&n
- 1 + ,(I). (17)
When the mean value responses enter their stationary states they are
Accuracy in solving the state equation
The accurate solution is (61
1
xD(n--l+B)
According to the above discussion, the eigenvalues obtained are identical to those of matrix [(l - j-‘)Z + b-‘(I - /IArC))‘]. So its spectral radius can also be expressed by eqn (12). Under the condition that the dynamic system is physically stable, i.e. if eqn (11) holds, it is found that if parameter fi satisfies /I > 1 -/
f (-l)“-~‘(~AtC)m II!=,
lim j(n) = lim j(n - 1) = j, n-cC n-a
(1)
limb(n-I+fl)=D n-e y(t) = ec’y(0) +
f0
’ e”‘-“‘D(s)
ds.
(14)
The relationship between any two state vectors at adjacent discrete time instants can be obtained below according to the above accurate solution “A, y(n) = e”“y(n - 1) + =
(I - /?AtC)g = (I - /IAtC + AtC)jj + AtB. That is
e C(“A’ _ “)D(s ) ds 5 ,n-
substitute them into eqn (17) and use (I - /?AtC) to multiply both. The following equation is derived
I)Ar
Ji = -C-ID.
[I + AtC - ;(AtC)* + $AtC)’ - . . b(n - 1) + At[Z + 0.5AtC - +(O.sAtC)* -+ . . .] x D(n - 0.5).
(15)
This is the accurate solution for the stationary mean value responses of the system [7]. Thus, it verifies that the /3-recurrent formula can carry out accurate stationary mean value responses.
521
Discrete analysis method for random vibration Accuracy in calculating the stationary mean square responses If a dynamic system is subjected to stationary white noise excitations its accurate mean square responses will satisfy Lyapunov equation [A CR,+Z?,C*+Q,=O,
(18)
where R,, represents the auto-correlation matrix of the responses of the system. Its diagonal values are the mean squares of the responses; QD is the coefficient matrix in the auto-correlation matrix function R,(z) = Q,s(r) of the white noise excitations. If the dynamic system is subjected to filtered white noise excitations, the latter should be generated from white noise excitations. In this case, the coefficient matrix C and excitation vector D should be modified, but eqn (18) remains valid. Multiply both sides of the B-recurrent formula from the left-hand side with its own transposes and then take mathematical expectations. The mean square responses of the system, calculated by the discrete analysis method, can be derived as R,(n)=
1
Z+$(Z-,!lAtC)-i
R,,(n -1) T
z+$(z-gAtC)-'
X
1 + At(Z - /lAtC)-‘QD(Z - /!?AtC)-?
(19)
In deriving the above equation, the property that the responses of the system at time instant (n - l)At is uncorrelated with the white noise excitations at time instant (n - 0.5)At is used, i.e. E[y(n - l)DT(n - 0.5)] = 0 and following approximation
has been applied
AtR,(O) = Q,&O)At E Q,. Multiplying both sides of eqn (19) from the left and the right with (I - BAtC) and its transpose, the following equation is obtained (I - /3AtC)R,(n)(I =
Substituting
them back into eqn (20), there is
(I - fiAC)R,(Z - fiAtC)‘=
[AtC + (I - /T?AtC)]
x RJAtC + (I - BAtC)]r+ AtQD. That is CRY+R,Cr+Q,+At(l
-2j?)CRYCT=0.
(21)
This is the equation that the stationary mean square responses of the system will satisfy using the /?-recurrent formula. Obviously, this equation will be identical to the Lyapunov equation which the accurate mean square responses of the system satisfy when parameter ,!l = 0.5. This means that the B-recurrent formula can also perform accurate stationary mean square responses. OPTIMUMTIME STEP FOR QUICKLYCALCULATING STATIONARY RANDOMRESPONSES The speed of using the b-recurrent formula to calculate the stationary random responses depends on the value of spectral radius of the coefficient matrix [(l - Zl-‘)I + fi-‘(I - fiAtC)-‘1 in the formula. The smaller the spectral radius, the quicker the calculation. The spectral radius has a relation with the dynamic characteristics of the system, the time step used in obtaining the recurrent formula and the parameter ~!l.According to the discussion given in the above two sections, in order to make the p-recurrent formula unconditionally stable and accurate, parameter /I should equal 0.5. So in a real calculation, only the time step can be used to control the calculation speed of the recurrent formula. It is known from eqn (12) that the spectral radius is a continuous function of the time step which take its maximum value one when At values are zero and infinite. Thus, there must exist a time step At which should make the spectral radius smallest. Substitute /I = 0.5 into eqn (12) to obtain
1’
(1 + 0.5Atuik)’ + (0.5Atu,)’ o.s (1 - 0.5Atuik)2 + (0.5Atvik)2
- BAtC)’
(I-/IAtC)+?
(I-pAtC)+;Z
1
i=l,2
R,(n - 1)
1
r+AtQ,.
(20)
When the mean square responses of the system enter their stationary states, there exists limR,,(n)=limR,,(n-l)=R,. r-m n-cc
Its derivative corresponding
44
a
,...,
r.
to At is
uJ1 - (0.5At)2(uf, + u:)]
=
IW’ [(1 - 0.5Atu,)2 + (0.5Atvik)2]2’ i=1,2
,...,
r.
By setting the above derivative to be zero, a time step can be obtained below by which the stationary
TAND~NGYAO
528
random responses of the ith mode of the dynamic system can be carried out in the fastest time
R,(n) = [2(M - O.SAtC)-‘M - Z]R,(n - 1) x [2(M - O.SAtC)-rM - I]’
2 At=(U;+V;)P,5’
i=l,2
1...,
(22)
r.
+ At(M - 0.5AtC)-’ x UQ,tJT(M - 0.5A.tC)-T.
In this case (L,] has following expression )Lil=
J0 I+
2
‘+y*
i=1,2
,...,
r.
Since the denominator in eqn (22) is the frequency of the ith mode for structural dynamic systems, and the first mode usually plays a dominant role in structural random responses, the optimum time step for calculating the stationary responses for structures is then suggested as To/n, where T,, is the natural period of the structure. APPLICATION
TO STRUCTURAL ANALYSIS
RANDOM
VIBRATION
The state equation is a general form for describing the behaviour of a dynamic system. The second-order differential equation which governs the structural vibration can be expressed in the form of state equation. In this sense the p-recurrent formula, which is based on the state equation of the dynamic system, has a certain generality and can be used in a discrete analysis for structural random vibration. The vibration equation of a structural dynamic system can be expressed as
mx’Yt)+ cx’(t) + kx(t) = q&t),
Here, the excitations are assumed to be stationary white noise processes. Their auto-correlation matrix function is R,(r) = Q,6(7). It should be noticed that in eqns (25) and (26) the calculation for the inverse of (M - 0.5AtC) is quite time consuming because it is widely banded, unsymmetric and order-doubled compared with the structural dynamic characteristic matrices. This shortcoming of the original form of the b-recurrent formula prevents it being widely applied. Thus, it is necessary to make an appropriate transformation of the formula and then the banded shape and symmetric properties of the structural dynamic characteristic matrices can be used and the computational efficiency of the b-recurrent formula could be improved. Practical recurrent formulas for calculating structural mean value responses The partitioned matrix equation obtained by using (M - 0.5AtC) to multiply the both sides of eqn (25) from left-hand side is I 0.5Atk
(23)
where m, c and k are structural mass, damping and stiffness matrices, respectively. x, x’ and x” are structural displacement, velocity and acceleration vectors. D(t) is the excitation vector and q is the influence matrix of the excitations. The state equation form of structural vibration equation is
y’(t) = M-‘Cy(t) + W’UD(t),
(24)
(26)
=
-0.5AtI m + 0.5Atc
-O.:Atk
Z(n) I 1 Z’(n)
m ?)!%c][~~
1 ii]
+Ar[~~(n-0.5)=[~I::I:1],
(27)
where a(n - 1) = .?(n - 1) + 0.5AtzZ’(n - 1) b(n - 1) = -0.5Atkf(n
where
(28)
- I) + (m - 0.5Atc)
x T(n - 1) + AtqD(n - 0.5).
(29)
At time instant nAt, a(n - 1) and b(n - I) are known. The solution of eqn (27) is and y is a state vector [x’, x”]’ and U is [OT,qTT. According to eqns (17) and (19) the mean and mean square responses of the structural dynamic system can be calculated by using the b-recurrent formula. The detailed formulas are obtained below when the parameter fi = 0.5.
n’(n) = K-‘[b(n - 1~)- O.SAtka(n - l)]
(30)
z?(n) = a(n - 1) + 0.5Atx’(n),
(31)
K = m + 0.5Atc + 0.25At2k.
(32)
where
j(n) = [2(M - O.SAtC)-‘M - ZF(n - 1) + At(M - 0.5AtC)-‘U&n
- 0.5)
(25)
It can be seen from eqns (28)-(32) that the structural dynamic characteristic matrices directly partici-
Discrete analysis method for random vibration
pate in the operation, so their banded shape and symmetric properties are efficiently used and the requirement for practical calculation is satisfied.
529
The solution of eqn (33) is R,,(n) = K-‘[0.25At2r,(n
- 1) + O.SAtr:(n - 1)
Practical recurrent formulas for calculating structural mean square responses
x (m + 0.5Atc) + 0.5At(m + 0.5Atc)
The partitioned matrix equation, obtained by multiplying both sides of eqn (26) with (M - OSAtC) and its transpose is
x r2(n - 1) + (m + O.SAtc)r,(n - 1)
I OSAtk
-0.5AtI m + OSAtc I
R,,(n) I[ 1 1 1 1 Rxxs (n1
R,.,(n)
-0SAtI
I
m +OSAtc
-0.5Atk
OSAtl m - OSAtc
x
X
&An) =
&R,(n)- K-‘[rT(n+&
-0.5Atk
rl(n - 1) = r,‘(n - 1)
0.5AtI T m - O.SAtc
1
r,(n - 1) = R,,(n - 1) + 0.5AtR,,(n
- &
- 1) + 0.25At2R,.,(n
- 1)
-
(38)
1)
(39)
It can be seen from eqns (34)-(39) that the structural dynamic characteristic matrices directly participate in the calculation of the structural mean square responses, so their banded shape and symmetric properties are efficiently used and the requirement for practical calculation is also satisfied. NUMERICAL
- 1)
[r2@
- 0.5Atr, (n - l)k]K-I.
r2(n - 1) rg(n - 1) ’
where R, and R,.,. are the auto-correlation matrices of structural displacement and velocity, respectively, R,,. is their cross-correlation matrix, and r,(n - 1), r2(n - 1) and r,(n - 1) are known for time instant n At. They are
+ O.SAtR:,(n
1)
(m + O.SAtc)r,(n - I)]
Rx.,. (n ) = &R,,(n)
Rx,@- 1) RA - 1) [ R,,,(n - 1) R,,,,(n - 1)
I
(37)
=
x [ OSAtk =
Rx.,.(n)1
x (m + O.SAtc)]K-’
EXAMPLES
First, the mean square response of a linear oscillator excited by zero mean stationary white noise is calculated, and the result is used to examine the validity of the recurrent formula. The frequency of the oscillator is w = 1.0, its damping ratio is r = 0.05. The spectral density of the excitation is S, = l/211. The accurate solution of the displacement mean square response is [7]
(34)
cr’;= -
nS0 = 5.0.
2503
r2(n - 1) = -O.SAtR,,(n
- 1)k + R,,,(n - 1) The numerical results are shown in Fig. 1, in which the transient mean square responses calculated by discrete analysis method with time step At = 0.2
x (m - 0.5Atc) - 0.25At2
x R$,(n - 1)k + O.SAtR,.,.(n - 1) x (m - 0.5Atc)
r,(n - 1) = 0.25At2kR,,(n
(35) - 1)k - 0.5Atk
_:;,:I
x R,,,(n - l)(m - 0.5Atc) - 0.5At(m - O.SAtc)R:,(n + (m - 0.5Atc)R,.,.(n
- 1)k
- 1)
x (m - 0.5Atc) + AtqQDqT.
j 1 I
(36)
50
loo
154
200
254
”
Fig. 1. Mean square response of the oscillator.
530
TANDONGYAO A.
CONCLUSION 0.1353 _-_-.
IL*,
0.4434 -I
I
m,
////////
-+--
XI
I
4
0.2057
l---k
X”.
Fig. 2. Structural sectional displacement mean square responses. covers the accurate transient mean square responses [7]. To compare the calculation speeds in
deriving the stationary mean square responses with different time steps, the horizontal axis in the figure is the number of recurrent calculations. It is obvious that there exists an optimum time step with which the recurrent formulas can most quickly carry out the accurate stationary solution. The second example is to calculate the stationary mean square responses of a three-storey plane shear frame excited by a horizontal earthquake motion. In this frame there is a sub-structure on the second floor. The structural parameters are: structural mass parameters: m,=m,=m,= 1.0, m,=0.08 structural damping parameters: c, = 8.0, c2 = 6.0, c, = 4.0, c4 = 0.25 structural stiffness parameters: k, = 200.0, k2 = 180.0, k, = 150.0, k., = 5.0.
The earthquake motion is modelled as a zero mean white noise process whose spectral density So is 3.183. This is a typical structural interaction problem in which the structure is non-classically damped. The calculated structural sectional displacement mean square responses are shown in Fig. 2 which shows that the vibration of the sub-structure has whip-tail effect.
A discrete analysis method for random vibration which is based on the b-recurrent formula is proposed in this paper. The unconditional stability of this recurrent formula and the property that it can carry out the accurate stationary mean and mean square responses for a linear time-invariant dynamic system are proved. In its application for structural dynamic systems, the practical calculation form of the p-recurrent formula is derived which can efficiently use the properties that the structural dynamic characteristic matrices are symmetric and banded, and computational time is saved. Although the calculation formulas in this paper are given under the assumption that the excitations are stationary random processes, they remain valid even when the excitations are nonstationary random processes. For the non-linear dynamic systems, if the equivalent linearized method is used, these formulas are also valid and an explicit recurrent procedure can be developed to calculate the accurate stationary mean and mean square responses for their equivalent linearized dynamic systems. REFERENCES
1. Tan Dongyao and Guo Hua, A procedure of discrete analysis for nonstationary random vibration. Earthquake Engng Engng Vibr. No. 3, 12-18 (in Chinese) (1986). 2. Tan Dongyao and Guo Hua, Discrete analysis for nonstationary vibration of nonlinear structures. Proc. ICCEM, Beijing, pp. 661467 (1987). 3. Tan Dongyao and Yang Qingshan, Discrete random vibration analysis for structures under excitations of filtered white noises. J. Harbin Architectural and Civil Engineering Inst. No. 1, 23-36 (in Chinese) (1989). 4. Tan Dongyao and Yang Qingshan, Discrete analysis method for random vibration of structures under excitations correlated in dime domain. Earthquake Engng Enanp Vibr. No. 2, 3746 (in Chinese) (1990). 5. K.-J: Bathe, Numerical Methods in Finite Element Analvsis. Prentice-Hall. Enelewood Cliffs. NJ (1976). 6. Yie ‘Yianlian, Introdktioi to Ordinary DifjPren;ial Equation, 2nd Edn. People’s Education Press (in Chinese) (1982). I. N. C. Nigam, Introduction to Random Vibration. MIT Press, Cambridge, MA (1983).
DISCRETE
1993 0
ANALYSIS METHOD VIBRATION
0045.7949/93 $6.00 + 0.00 1993 Pergamon Press Ltd
FOR RANDOM
TAN DONGYAO School of Astronautics, Beijing University of Aeronautics and Astronautics, Beijing 100083, People’s Republic of China (Received 27 May 1992)
Abstract-A discrete analysis method which uses recurrent formulas to calculate the stationary and non-stationary random vibration responses for multi-degree of freedom (MDOF) dynamic systems is proposed in this paper. The unconditional stability of the recurrent formulas and their property of being able to calculate the accurate stationary mean and mean square responses are proved. An optimum time step is suggested with which the recurrent formulas can quickly carry out accurate stationary random responses. As an example, these recurrent formulas are applied to MDOF structures. Their practical calculation forms are derived which use the banded shape and symmetric properties of structural dynamic matrices to effectively calculate structural stationary or non-stationary mean and mean square responses. Some numerical examples are given to show the application of the method.
NOTATION
c
coefficient matrix in state equation structural damping matrix excitation vector and its mean value a(l) expectation of A -W 1 identity matrices I, I, , IO K coefficient matrix k structural stiffness matrix A4 coefficient matrix structural mass matrix T transformation matrix coefficient matrix QD influence matrix of excitations 4 correlation matrix auto-correlation matrix function of the LT, excitations coefficient matrices rl , r2, r3 block matrices in the Jordan form of C r,, rJr, r. u coefficient matrix real part of the eigenvalues of matrix C 11th complex part of the eigenvalue of matrix u&k C X(l), x’(r), x”(r) structural displacement, velocity and acceleration vectors Z(f), f’(f) mean values of x(t) and x’(r) state vector and its mean value Nh Y(f) parameter !, eigenvalue of the coefficient matrix in the recurrent formula At time step d-function 6(r)
b,
INTRODUCTION
With the increasing development of computer technology the analysis of continuum systems using discretization methods has gained a wide scope of application, The finite element method proposed in the early 195Os, which discretizes a continuum medium in space, established the foundations for quantitatively and routinely analysing the mechanical, thermal and flowing properties of multi-dimen-
sional complex solids or fluid structures. Later, the boundary element method was proposed as a counterpart of finite element method. In the 1960s the discretization idea was introduced into the deterministic dynamic analysis for structures and the direct integration method was proposed which discretized structural dynamic differential equations in the time
domain. The discrete analysis method for random vibration discussed in this paper is a further application of the discretization idea in random vibration theory. Developing such a method aims at finding a more powerful and general random vibration analysis method which may use the flexibility and routinely computational advantages of the discretization method. This method is applicable to both stationary and non-stationary linear or equivalent linearized non-linear random vibration problems. In the last decade, many workers focused their attention on investigating non-linear random vibration problems. It was a clear tendency in these investigations that most of the researchers abandoned the traditional way of solving FPK equations and adopted the equivalent linearization method. As many direct numerical simulations have proved that the equivalent linearization method could lead to very good results for a wide scope of non-linear problems, it gradually becomes a practical analysis method for non-linear random vibration problems. In this sense, analysing non-linear random vibration problems will also depend on the method used in analysing linear random vibration problems. There are many difficulties which should be overcome, not only for non-linear random vibration problems, but even for linear random vibration problems. Such problems which have not been well solved are the random vibration of soil-structure interaction systems with non-classical damping, 523
TAND~NGYAO
524
random response analysis for structures subjected to earthquake excitations input from multiple supports considering their time delay relationships and the structural wind induced random vibration considering the spatial correlativity of the wind loads. To solve these problems, it is necessary to establish a general analysis method for random vibration. The proposed discrete analysis method for random vibration is such an effort in this direction. The discrete analysis method for random vibration uses the mean value process and correlation function of the excitations directly to calculate the nonstationary and stationary mean and mean square responses of a dynamic system step by step by using recurrent formulas obtained by discretizing the dynamic differential equation of the system in the time domain. The recurrent formulas discussed in this paper have already been used to analyse the random responses of structures subjected to filtered white noises and white noises which correlate in the time domain [IA]. Here, we study their properties as the basis for a discrete analysis method for random vibration, i.e. their stability and accuracy in numerical calculation. Also, we study their practical application to structural dynamic systems. In the practical recurrent formulations the symmetric and banded shape properties of structural mass, damping and stiffness matrices are used and the computational time is greatly reduced.
where (n - 1)At < t 4 (n - 1 + With the above assumption, everywhere between (n - I)Ar Hence, it can be substituted into the system at time instant (n following equation is obtained y(n-l+B)--y(n-1)
+ D(n - 1 + /j).
y(n - 1 + 8) = (I - fiArC)-‘[y(n + pAtD(n
+ D(t),
(1)
+ D(n),
(2)
where y(n), y’(n) and D(n) represent the state vector, its derivative and excitation vector at time instant n At, respectively. Assume the state vector of the system varies linearly between two adjacent discrete time instants. Thus, the variation rate of the state vector in this time interval can be calculated below from the state vectors at these two discrete time instants r’(t)
=
Y(n--l+B)-y(n-1) BAt
- 1) -
1+ 8)1, (5)
where I is an identity matrix. It is known from eqn (3) that the variation rates of the state vectors at time instants nAt and (n - 1 + /?)At are identical so that y(n) -y(n
- 1)
At
y(n - I + S) -y(n
=
- 1)
(6)
flAt
Substituting eqn (5) into (6) to eliminate y(n - 1 + /I), the relationship between the state vectors at time instants nAt and (n - I)At is finally derived r, ., I 1 1+$(1-/JAtC)-’
(n - 1) P
where y(t) is the state vector of the system, C is the coefficient matrix which describes the dynamic characteristics of the system and D(t) is the excitation vector. The discrete time instants n At (n = 0, 1,2, . . ) can be obtained by using the time step At to equally discretize the time axis, and the corresponding state equation of the system at each discrete time instant is y’(n) = Q(n)
(4)
The state vector y(n - I + 8) can be derived from the above equation
FORMULA
The state equation is a general form for describing the dynamic behaviour of a system. For a linear time-invariant dynamic system the state equation can be written as y’(t) = Q(t)
=Cy(n-l+B)
BAr
y(n) = /I-RECtiRRENT
p)At and 0 < b < I. eqn (3) is satisfied and (n - 1 + /?)A[. the state equation of - 1 + /?)At, and the
’
(3)
+ (I - bArC)-’
ArD(n - 1 + ,0).
(7)
This is a recurrent formula containing a parameter ,8, called the B-recurrent formula. It is a fundamental recurrent formula in the discrete analysis method for random vibration. NUMERICAL
STABILITY
OF P-RECURRENT
FORMULA
Firstly, the dynamic system is assumed to be physically stable. Then the numerical stability of the b-recurrent formula requires that the spectral radius of the coefficient matrix [(l - Be’)1 + p-‘(I - /3AtC)-‘1 (i.e. the maximum absolute value of its eigenvalues) in eqn (7) must not be greater than one [5]. With knowledge of linear algebra and theory of ordinary differential equations, we know that in the most general case the coefficient matrix C, in the state equation of the dynamic system, has repeated complex eigenvalues, and there always exists a nonsingular matrix P which can transform C into the Jordan form, i.e.
P_‘CP
=
r
r,
(8)
525
Discrete analysis method for random vibration where
P-1
=
ri I,
.. ri =
(
ra 10
,
i=1,2
,...,
P
I+;(&BAtP-VP)-’
1 -f
..
1
I+;(I-gAtC)-i
r.
> ...
. . .
=
. . . IO
1 -’ (
ril
B>
Ii+!-(li-/?Atri))’ B
(9)
Denote the block matrices on the diagonal position in above matrix as
In above matrix, I, is a second-order identity matrix, and
-I
Z,,- bAtrio -BAtI,, ..
.. --/IA&
rik=
%k
[ -v*
V* uik
1 ’
k = 1,2,. . . ,j
ai, = ui2= . . . = Uik= . . . = u,, v,, = viL=
’ = uik= . . . = Vii’
(10)
where uik and vik are real and complex parts of the eigenvalues of matrix C. For a physically stable dynamic system, the real parts of the eigenvalues of matrix C must not be positive, i.e. 44
GO, i=l,2,...
, r; k = 1,2,. . . ,j.
(1 - /!?:tu,f:iz:;
I
.. .
1, - /?Atr,
1
where Ii is an identity matrix of the same order as that of matrix r,. It is known from linear algebra that the eigenvalues of matrix [(l - p-i)1 + B-‘(1- /?AtC)-‘1 are equal to the eigenvalues of matrices e,(i = 1,2, . . . , r). It can be proved that e, has the partitioned low triangular form after the inverse matrix in above equation is carried out, so that the eigenvalues of the block matrices on the diagonal position of matrix ei are the eigenvalues of itself. These block matrices are
(11)
Atv&)’ + ’ - ’
(1 - ~At~~)$?Atr;,)’
j?Atv,
(1 - BAtUik)’+ (8AtVik)2
To simplify the notation, the identical block matrices on the diagonal of the matrix in eqn (9) are denoted as r,, thus rn = rik(k = 1,2,. . . , j). Multiply matrix ((1 - B -‘)I + /I -‘(I - /IArC)-‘] from the left and the right with P-’ and P, respectively, and use eqn (8) to derive
Their eigenvalues can be easily obtained as
1 ,+= j
1 - /.IAtu, +/I-1 (1 - /YAtuik)2+ (bAtvti)2
1
526
TAN D~NGYAO
Atv, ’ ’ (1 - Z?Atua)* + (/?A~v,~)*’
Since the /l-recurrent formula is unconditionally stable, it can then be expanded as
i=l,2
)...,
j.
Their modules can be derived below by taking complex operation
xy(n-l)+At
I+
[l - (fi - l)AtuJ* + [(b - l)Atv,]* o.5 IAl
=
(1 - fiAtuik)* + (BAru,)’ i=l,2
’ ,...)
r.
or
a > 0.5
(12)
(13)
the spectral radius of matrix [(l - z!~‘)Z + fl-‘(Z - fiAtC))‘] will not be greater than one for any time step At, i.e. the p-recurrent formula is unconditionally stable for p > 0.5. ACCURACY
OF THE b-RECURRENT
FORMULA
There are two meanings to the accuracy of the p-recurrent formula here: firstly, it refers to the accuracy when it is used to solve the state equation of a dynamic system. This accuracy can be evaluated by the accurate solution of the state equation. Secondly, it refers to the accuracy when it is used in analysing the random responses of the dynamic system. This accuracy can be evaluated by the accurate solutions of the stationary mean and mean square responses of the system.
= [I + AK - /I(AtC)*
+ fi*(AtC)’ -
of the state equation
.]
xy(n-l)+At[Z+fiAtC-@Arc)*+ x D(n - If
...I
p).
(16)
Comparing eqns (15) and (16), it can be concluded that the second-order accuracy of the p-recurrent formula for solving the state equation of the dynamic system can be achieved when fl = 0.5. Accuracy in calculating the stationary mean value responses
Taking mathematical expectation at both sides of eqn (7), the equation for the b-recurrent formula to calculate the mean value responses for the system can be obtained as j(n)=
I+!-(I-BAtC))’ + (I - /3AtC)-‘At&n
- 1 + ,(I). (17)
When the mean value responses enter their stationary states they are
Accuracy in solving the state equation
The accurate solution is (61
1
xD(n--l+B)
According to the above discussion, the eigenvalues obtained are identical to those of matrix [(l - j-‘)Z + b-‘(I - /IArC))‘]. So its spectral radius can also be expressed by eqn (12). Under the condition that the dynamic system is physically stable, i.e. if eqn (11) holds, it is found that if parameter fi satisfies /I > 1 -/
f (-l)“-~‘(~AtC)m II!=,
lim j(n) = lim j(n - 1) = j, n-cC n-a
(1)
limb(n-I+fl)=D n-e y(t) = ec’y(0) +
f0
’ e”‘-“‘D(s)
ds.
(14)
The relationship between any two state vectors at adjacent discrete time instants can be obtained below according to the above accurate solution “A, y(n) = e”“y(n - 1) + =
(I - /?AtC)g = (I - /IAtC + AtC)jj + AtB. That is
e C(“A’ _ “)D(s ) ds 5 ,n-
substitute them into eqn (17) and use (I - /?AtC) to multiply both. The following equation is derived
I)Ar
Ji = -C-ID.
[I + AtC - ;(AtC)* + $AtC)’ - . . b(n - 1) + At[Z + 0.5AtC - +(O.sAtC)* -+ . . .] x D(n - 0.5).
(15)
This is the accurate solution for the stationary mean value responses of the system [7]. Thus, it verifies that the /3-recurrent formula can carry out accurate stationary mean value responses.
521
Discrete analysis method for random vibration Accuracy in calculating the stationary mean square responses If a dynamic system is subjected to stationary white noise excitations its accurate mean square responses will satisfy Lyapunov equation [A CR,+Z?,C*+Q,=O,
(18)
where R,, represents the auto-correlation matrix of the responses of the system. Its diagonal values are the mean squares of the responses; QD is the coefficient matrix in the auto-correlation matrix function R,(z) = Q,s(r) of the white noise excitations. If the dynamic system is subjected to filtered white noise excitations, the latter should be generated from white noise excitations. In this case, the coefficient matrix C and excitation vector D should be modified, but eqn (18) remains valid. Multiply both sides of the B-recurrent formula from the left-hand side with its own transposes and then take mathematical expectations. The mean square responses of the system, calculated by the discrete analysis method, can be derived as R,(n)=
1
Z+$(Z-,!lAtC)-i
R,,(n -1) T
z+$(z-gAtC)-'
X
1 + At(Z - /lAtC)-‘QD(Z - /!?AtC)-?
(19)
In deriving the above equation, the property that the responses of the system at time instant (n - l)At is uncorrelated with the white noise excitations at time instant (n - 0.5)At is used, i.e. E[y(n - l)DT(n - 0.5)] = 0 and following approximation
has been applied
AtR,(O) = Q,&O)At E Q,. Multiplying both sides of eqn (19) from the left and the right with (I - BAtC) and its transpose, the following equation is obtained (I - /3AtC)R,(n)(I =
Substituting
them back into eqn (20), there is
(I - fiAC)R,(Z - fiAtC)‘=
[AtC + (I - /T?AtC)]
x RJAtC + (I - BAtC)]r+ AtQD. That is CRY+R,Cr+Q,+At(l
-2j?)CRYCT=0.
(21)
This is the equation that the stationary mean square responses of the system will satisfy using the /?-recurrent formula. Obviously, this equation will be identical to the Lyapunov equation which the accurate mean square responses of the system satisfy when parameter ,!l = 0.5. This means that the B-recurrent formula can also perform accurate stationary mean square responses. OPTIMUMTIME STEP FOR QUICKLYCALCULATING STATIONARY RANDOMRESPONSES The speed of using the b-recurrent formula to calculate the stationary random responses depends on the value of spectral radius of the coefficient matrix [(l - Zl-‘)I + fi-‘(I - fiAtC)-‘1 in the formula. The smaller the spectral radius, the quicker the calculation. The spectral radius has a relation with the dynamic characteristics of the system, the time step used in obtaining the recurrent formula and the parameter ~!l.According to the discussion given in the above two sections, in order to make the p-recurrent formula unconditionally stable and accurate, parameter /I should equal 0.5. So in a real calculation, only the time step can be used to control the calculation speed of the recurrent formula. It is known from eqn (12) that the spectral radius is a continuous function of the time step which take its maximum value one when At values are zero and infinite. Thus, there must exist a time step At which should make the spectral radius smallest. Substitute /I = 0.5 into eqn (12) to obtain
1’
(1 + 0.5Atuik)’ + (0.5Atu,)’ o.s (1 - 0.5Atuik)2 + (0.5Atvik)2
- BAtC)’
(I-/IAtC)+?
(I-pAtC)+;Z
1
i=l,2
R,(n - 1)
1
r+AtQ,.
(20)
When the mean square responses of the system enter their stationary states, there exists limR,,(n)=limR,,(n-l)=R,. r-m n-cc
Its derivative corresponding
44
a
,...,
r.
to At is
uJ1 - (0.5At)2(uf, + u:)]
=
IW’ [(1 - 0.5Atu,)2 + (0.5Atvik)2]2’ i=1,2
,...,
r.
By setting the above derivative to be zero, a time step can be obtained below by which the stationary
TAND~NGYAO
528
random responses of the ith mode of the dynamic system can be carried out in the fastest time
R,(n) = [2(M - O.SAtC)-‘M - Z]R,(n - 1) x [2(M - O.SAtC)-rM - I]’
2 At=(U;+V;)P,5’
i=l,2
1...,
(22)
r.
+ At(M - 0.5AtC)-’ x UQ,tJT(M - 0.5A.tC)-T.
In this case (L,] has following expression )Lil=
J0 I+
2
‘+y*
i=1,2
,...,
r.
Since the denominator in eqn (22) is the frequency of the ith mode for structural dynamic systems, and the first mode usually plays a dominant role in structural random responses, the optimum time step for calculating the stationary responses for structures is then suggested as To/n, where T,, is the natural period of the structure. APPLICATION
TO STRUCTURAL ANALYSIS
RANDOM
VIBRATION
The state equation is a general form for describing the behaviour of a dynamic system. The second-order differential equation which governs the structural vibration can be expressed in the form of state equation. In this sense the p-recurrent formula, which is based on the state equation of the dynamic system, has a certain generality and can be used in a discrete analysis for structural random vibration. The vibration equation of a structural dynamic system can be expressed as
mx’Yt)+ cx’(t) + kx(t) = q&t),
Here, the excitations are assumed to be stationary white noise processes. Their auto-correlation matrix function is R,(r) = Q,6(7). It should be noticed that in eqns (25) and (26) the calculation for the inverse of (M - 0.5AtC) is quite time consuming because it is widely banded, unsymmetric and order-doubled compared with the structural dynamic characteristic matrices. This shortcoming of the original form of the b-recurrent formula prevents it being widely applied. Thus, it is necessary to make an appropriate transformation of the formula and then the banded shape and symmetric properties of the structural dynamic characteristic matrices can be used and the computational efficiency of the b-recurrent formula could be improved. Practical recurrent formulas for calculating structural mean value responses The partitioned matrix equation obtained by using (M - 0.5AtC) to multiply the both sides of eqn (25) from left-hand side is I 0.5Atk
(23)
where m, c and k are structural mass, damping and stiffness matrices, respectively. x, x’ and x” are structural displacement, velocity and acceleration vectors. D(t) is the excitation vector and q is the influence matrix of the excitations. The state equation form of structural vibration equation is
y’(t) = M-‘Cy(t) + W’UD(t),
(24)
(26)
=
-0.5AtI m + 0.5Atc
-O.:Atk
Z(n) I 1 Z’(n)
m ?)!%c][~~
1 ii]
+Ar[~~(n-0.5)=[~I::I:1],
(27)
where a(n - 1) = .?(n - 1) + 0.5AtzZ’(n - 1) b(n - 1) = -0.5Atkf(n
where
(28)
- I) + (m - 0.5Atc)
x T(n - 1) + AtqD(n - 0.5).
(29)
At time instant nAt, a(n - 1) and b(n - I) are known. The solution of eqn (27) is and y is a state vector [x’, x”]’ and U is [OT,qTT. According to eqns (17) and (19) the mean and mean square responses of the structural dynamic system can be calculated by using the b-recurrent formula. The detailed formulas are obtained below when the parameter fi = 0.5.
n’(n) = K-‘[b(n - 1~)- O.SAtka(n - l)]
(30)
z?(n) = a(n - 1) + 0.5Atx’(n),
(31)
K = m + 0.5Atc + 0.25At2k.
(32)
where
j(n) = [2(M - O.SAtC)-‘M - ZF(n - 1) + At(M - 0.5AtC)-‘U&n
- 0.5)
(25)
It can be seen from eqns (28)-(32) that the structural dynamic characteristic matrices directly partici-
Discrete analysis method for random vibration
pate in the operation, so their banded shape and symmetric properties are efficiently used and the requirement for practical calculation is satisfied.
529
The solution of eqn (33) is R,,(n) = K-‘[0.25At2r,(n
- 1) + O.SAtr:(n - 1)
Practical recurrent formulas for calculating structural mean square responses
x (m + 0.5Atc) + 0.5At(m + 0.5Atc)
The partitioned matrix equation, obtained by multiplying both sides of eqn (26) with (M - OSAtC) and its transpose is
x r2(n - 1) + (m + O.SAtc)r,(n - 1)
I OSAtk
-0.5AtI m + OSAtc I
R,,(n) I[ 1 1 1 1 Rxxs (n1
R,.,(n)
-0SAtI
I
m +OSAtc
-0.5Atk
OSAtl m - OSAtc
x
X
&An) =
&R,(n)- K-‘[rT(n+&
-0.5Atk
rl(n - 1) = r,‘(n - 1)
0.5AtI T m - O.SAtc
1
r,(n - 1) = R,,(n - 1) + 0.5AtR,,(n
- &
- 1) + 0.25At2R,.,(n
- 1)
-
(38)
1)
(39)
It can be seen from eqns (34)-(39) that the structural dynamic characteristic matrices directly participate in the calculation of the structural mean square responses, so their banded shape and symmetric properties are efficiently used and the requirement for practical calculation is also satisfied. NUMERICAL
- 1)
[r2@
- 0.5Atr, (n - l)k]K-I.
r2(n - 1) rg(n - 1) ’
where R, and R,.,. are the auto-correlation matrices of structural displacement and velocity, respectively, R,,. is their cross-correlation matrix, and r,(n - 1), r2(n - 1) and r,(n - 1) are known for time instant n At. They are
+ O.SAtR:,(n
1)
(m + O.SAtc)r,(n - I)]
Rx.,. (n ) = &R,,(n)
Rx,@- 1) RA - 1) [ R,,,(n - 1) R,,,,(n - 1)
I
(37)
=
x [ OSAtk =
Rx.,.(n)1
x (m + O.SAtc)]K-’
EXAMPLES
First, the mean square response of a linear oscillator excited by zero mean stationary white noise is calculated, and the result is used to examine the validity of the recurrent formula. The frequency of the oscillator is w = 1.0, its damping ratio is r = 0.05. The spectral density of the excitation is S, = l/211. The accurate solution of the displacement mean square response is [7]
(34)
cr’;= -
nS0 = 5.0.
2503
r2(n - 1) = -O.SAtR,,(n
- 1)k + R,,,(n - 1) The numerical results are shown in Fig. 1, in which the transient mean square responses calculated by discrete analysis method with time step At = 0.2
x (m - 0.5Atc) - 0.25At2
x R$,(n - 1)k + O.SAtR,.,.(n - 1) x (m - 0.5Atc)
r,(n - 1) = 0.25At2kR,,(n
(35) - 1)k - 0.5Atk
_:;,:I
x R,,,(n - l)(m - 0.5Atc) - 0.5At(m - O.SAtc)R:,(n + (m - 0.5Atc)R,.,.(n
- 1)k
- 1)
x (m - 0.5Atc) + AtqQDqT.
j 1 I
(36)
50
loo
154
200
254
”
Fig. 1. Mean square response of the oscillator.
530
TANDONGYAO A.
CONCLUSION 0.1353 _-_-.
IL*,
0.4434 -I
I
m,
////////
-+--
XI
I
4
0.2057
l---k
X”.
Fig. 2. Structural sectional displacement mean square responses. covers the accurate transient mean square responses [7]. To compare the calculation speeds in
deriving the stationary mean square responses with different time steps, the horizontal axis in the figure is the number of recurrent calculations. It is obvious that there exists an optimum time step with which the recurrent formulas can most quickly carry out the accurate stationary solution. The second example is to calculate the stationary mean square responses of a three-storey plane shear frame excited by a horizontal earthquake motion. In this frame there is a sub-structure on the second floor. The structural parameters are: structural mass parameters: m,=m,=m,= 1.0, m,=0.08 structural damping parameters: c, = 8.0, c2 = 6.0, c, = 4.0, c4 = 0.25 structural stiffness parameters: k, = 200.0, k2 = 180.0, k, = 150.0, k., = 5.0.
The earthquake motion is modelled as a zero mean white noise process whose spectral density So is 3.183. This is a typical structural interaction problem in which the structure is non-classically damped. The calculated structural sectional displacement mean square responses are shown in Fig. 2 which shows that the vibration of the sub-structure has whip-tail effect.
A discrete analysis method for random vibration which is based on the b-recurrent formula is proposed in this paper. The unconditional stability of this recurrent formula and the property that it can carry out the accurate stationary mean and mean square responses for a linear time-invariant dynamic system are proved. In its application for structural dynamic systems, the practical calculation form of the p-recurrent formula is derived which can efficiently use the properties that the structural dynamic characteristic matrices are symmetric and banded, and computational time is saved. Although the calculation formulas in this paper are given under the assumption that the excitations are stationary random processes, they remain valid even when the excitations are nonstationary random processes. For the non-linear dynamic systems, if the equivalent linearized method is used, these formulas are also valid and an explicit recurrent procedure can be developed to calculate the accurate stationary mean and mean square responses for their equivalent linearized dynamic systems. REFERENCES
1. Tan Dongyao and Guo Hua, A procedure of discrete analysis for nonstationary random vibration. Earthquake Engng Engng Vibr. No. 3, 12-18 (in Chinese) (1986). 2. Tan Dongyao and Guo Hua, Discrete analysis for nonstationary vibration of nonlinear structures. Proc. ICCEM, Beijing, pp. 661467 (1987). 3. Tan Dongyao and Yang Qingshan, Discrete random vibration analysis for structures under excitations of filtered white noises. J. Harbin Architectural and Civil Engineering Inst. No. 1, 23-36 (in Chinese) (1989). 4. Tan Dongyao and Yang Qingshan, Discrete analysis method for random vibration of structures under excitations correlated in dime domain. Earthquake Engng Enanp Vibr. No. 2, 3746 (in Chinese) (1990). 5. K.-J: Bathe, Numerical Methods in Finite Element Analvsis. Prentice-Hall. Enelewood Cliffs. NJ (1976). 6. Yie ‘Yianlian, Introdktioi to Ordinary DifjPren;ial Equation, 2nd Edn. People’s Education Press (in Chinese) (1982). I. N. C. Nigam, Introduction to Random Vibration. MIT Press, Cambridge, MA (1983).