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Structural random vibration analysis by using recursive formulas
ProbabilisticEngineering Mechanics10 (1995) 11-21
~~
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ELSEVIER
0266-8920(94)00003-4
© 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0266-8920/95/$09.50
Structural random vibration analysis by using recursive formulas Dongyao Tan Beijing University of Aeronautics and Astronautics, School of Astronautics, P.O. Box 504, Beijing 100083, China
&
Qingshan Yang Harbin Architectural and Civil Engineering Institute, Harbin 150006, China
(Received October 1992; revised March 1994; accepted April 1994) This paper discusses the application of/3-recursive formula to structural random vibration analysis. This recursive formula is able to calculate the accurate stationary mean and mean square responses of a linear or equivalent linearized nonlinear structure subjected to stationary excitations. To enhance the computational efficiency, the basic formulae for calculating structural mean and mean square responses which are obtained from /3-recursive formula are transformed into other forms which directly use the band-shape and symmetric structural dynamic characteristic matrices in calculation. The detailed formulations are given according to different excitation environments, such as white noise excitations, time-domain correlated white noise excitations, independent filtered white noise excitations and spatially correlated filtered white noise excitations. Some numerical examples are given to show the applications of the recursive formula.
carry out the accurate stationary mean and mean square responses for any linear or equivalent linearized nonlinear dynamic system. 9'1° It also has second order accuracy in calculating non-stationary mean and mean square responses. 1° This paper discusses its application in structural random vibration analysis. Unlike structural deterministic dynamic analysis, the detailed calculation formulations in structural random vibration analysis depend largely on the types of excitations to which the structures are subjected. Here, the detailed and practical calculation formulations by using /3recursive formula to calculate structural mean and mean square responses are deduced with respect to four important excitation environments. The first is white noise excitation which roughly describes the engineering background of multi-dimensional earthquake motions exciting a structure simultaneously. The second is time-domain correlated white noise excitation which describes the situation of a propagating earthquake motion exciting a large span structure from its different supports. The third is independent filtered white noise excitation which describes in some detail the earthquake motion or any other coloured
INTRODUCTION Structural random vibration analysis by using numerical methods became a feature in the field of random vibration theory in the last decade. For examples, Di Paola et al.1 gave an unconditionally stable step-by-step method, based on the eigenvalue analysis, to calculate the mean square response of a multidegree of freedom linear or nonlinear system subjected to nonstationary filtered noises, C. W. S. To 5'6 proposed explicit and implicit direct integrators to calculate the random response of multidegree of freedom systems. Various numerical examples have been given to illustrate the efficiency of these methods. This paper uses/3-recursive formula which was proposed and studied by Dongyao Tan et al. TM to calculate the stationary and nonstationary mean and mean square responses for large linear or equivalent linearized nonlinear structures. /3-recursive formula was obtained by discretizing a general state equation in time domain with the assumption that the state vector changes linearly between two adjacent discrete time instants. 7 It has been proved to be unconditionally stable, and it can 11
12
D o n g y a o Tan, Qingshan Yang
noise which excites a structure. The fourth is spatially correlated filtered white noise excitation which describes the gust wind that excites large space structures or highrise structures.
BASIC F O R M U L A S OF CALCULATION Structural vibration equations can be expressed as: (1)
m x " ( t ) + cx'(t) + k x ( t ) = qe(t)s(t)
where m, c, k are structural mass, damping and stiffness matrices, respectively, x(t), x'(t), x " ( t ) are structural displacement, velocity and acceleration vectors, e(t) is a diagonal matrix which consists of the deterministic envelope functions that describe the non-stationary property of the excitation vector, s(t) is a stationary random vector, q is the influence matrix of the excitations. Since the /3-recursive formula was obtained for a dynamic system whose differential equation is expressed in the form of state equation, to use this formula, structural vibration eqn (1) should be transformed into a state equation as follows: (2)
M y ' ( t ) = Cy(t) + Ue(t)s(t)
where
m] C=
['1 -k
auto-correlation matrix of random vector s(t). Rys(n-1,n-0.5) is the cross-correlation matrix of structural responses at time instant ( n - 1)At with the random vector s(t) at time instant (n - 0.5)At. y(n) and Ryy(n) are structural mean and mean square responses at time instant n A t . Equations (3) and (4) are the basic formulas obtained from 3-recursive formula for calculating structural mean and mean square responses.
CALCULATION FOR STRUCTURAL MEAN VALUE RESPONSES It can be seen from eqn (3.1) that structural mean value responses are determined by the mean values of the excitations alone. So, this recursive formula can be used to calculate the mean value responses for structures subjected to any kind of excitation. However, it is disadvantageous to use the original form of eqn (3.1) to calculate structural mean value responses because it contains the inverse of matrix /-/3 which is nonsymmetric and order doubled comparing with structural dynamic characteristic matrices. In order to make the symmetric and banded-shape structural dynamic characteristic matrices in eqn (1) directly participate in the calculation so as to save computational time and storage, it is necessary to make a transformation for eqn (3.1). Using matrix Ha to multiply both of its sides of eqn (3. l) a new equation is obtained and its partitioned form can be expressed as:
-c
and y(t) is state vector [xr(t), x'r(t)] r, g is [Or, qr] r. According to the results of Tan et al., 9' 10 the unconditionally stable recursive formulas which are able to calculate the accurate structural stationary mean and mean square responses are y(n) = H l y ( n - 1) + A t H E U e ( n - 0.5)g
x Rsy(n-
[ ,
0.SAtk [2'(n-l)
J +At
-0.5Atk
e(n-0"5)~=
[b(n
G2
1) '
where a(n - 1) ----2(n - 1) + 0.5AtSt(n - 1)
+ A t H 2 U e ( n - 0.5)
1 (5)
(3.1)
Ryy(n) = H1Ryy(n- 1)H T -[-AtH1gys(n- 1, n - 0-5) x e(n - 0 . 5 ) U r H f
,
(6.1)
b(n - 1) = - 0 . 5 A t k 2 ( n - 1)
0 . 5 , n - 1)H~"
+ Gz2P(n - 1) + A t q e ( n - 0.5)~
(6.2)
+ A t 2 H 2 U e ( n - 0.5)Rss(n - 0-5, n - 0"5) × e(n - 0 - 5 ) U r H f ,
(3.2)
G l = m+O.5Atc
(7.1)
G2 = m - - O . 5 A t c .
(7.2)
where H1 = 2 H 2 M - I
(4.1)
H 2 = Hj -1
(4.2)
Ha = M - 0-5AtC
(4.3)
Here, ~ and Rss(n - 0'5, n - 0-5) are the mean value and
Here, a ( n - 1 ) and b ( n - 1 ) are known for time instant n A t . The solutions of the above matrix algebraic equation are Y,'(n) = K - l [ b ( n - 1) - 0 . 5 A t k a ( n - 1)]
(8)
2(n) = a(n - 1) + 0.5AtOP(n),
(9)
J
Structural random vibration analys& where
K = m + 0.5Arc + 0.25At2k.
(10)
Equations (6)-(10) are practical recursive expressions for calculating structural mean value responses.
13
structural displacement and velocity, respectively. Rxe is their cross-correlation matrix. For time instant nAt, rl(n -- 1), r2(n -- 1) and r3(n - 1) are known, they are
rl(n - 1) = Rxx(n - 1) +O.5AtRxx,(n - 1) + 0.5AtRrxx,(n - 1) + 0.25At2Rx, e(n - 1)
CALCULATION FOR MEAN SQUARE RESPONSES OF STRUCTURES SUBJECTED TO WHITE NOISE EXCITATIONS
White noise is an idealized stochastic process. Some excitations encountered in engineering can be taken as white noises roughly, such as the bedrock movement of an earthquake. Other excitations which cannot be taken as white noises, however, can usually be generated from them. Therefore, the random vibration analysis for structures subjected to white noise excitations is a fundamental problem in structural random vibration analysis. In addition, the mean and mean square responses of a linear oscillator subjected to stationary white noise excitation have closed form solutions, which have often been used to check the accuracy of the numerical methods. Substituting the coefficient matrices in eqn (2) into eqn (3.2), the recursive formula for calculating the mean square responses of a structure subjected to white noise excitations are obtained as follows:7'1° Ryy(n) = n l R y y ( n -
(13)
r2(n - 1) = - 0.5AtRxx(n - 1)k + Rxx,(n - 1)G2
- 0 . 2 5 A ? g ; e ( n - 1)k + 0"5AtRx, x,(n - 1)G2
(14)
r3(n - 1) = 0.25At2kRxx(n - 1)k - 0"5AtkRxx,(n - 1)G2 - 0.5AtG2Rrxx,(n - 1)k + GzRx, x,(n - 1)G2 + Atqe(n - 0.5)Qsse(n - 0.5)q r.
(15)
The solution of the matrix algebraic equation (12) is 1°
Rxx(n) = K -l [O'25At2r3(n- 1)+ 0 " 5 A t r T ( n - 1)G 1 + 0"5AtGlr2(n - 1) + Glrl(n - 1)G1]K -1 (16) 1 Rxe(n) = 0.5AtRxx(n)
1)a T
+ AtH2Ue(n - 0"5)Qse(n - 0-5) UrH2r,(11) where Qs is the coefficient matrix in the auto-correlation function matrix R~s(~')= Qsr('r) of the white noise excitations. Similarly, to make eqn (11) have higher computational efficiency, it is necessary to transform it into another form which is of benefit to practical calculation. Multiply both sides of eqn (11) with matrix Ha and its transpose from left and right, respectively, and the obtained equation in its partitioned form is
[I
-0.5Atll[Rxx(n
0"5Atk I
G1
)
[ Rx, x(n)
--0"5AtI] T= [
0"5Atk
G1
Rxx(n-1)
+
Rx, x,(n) 0"5AtI"
-0'5Atk
G2
Rxx,(n-1)][
r2(n-1)]
IrWin- 1)
r3(,-
1/J '
(17)
Rx, x , ( n ) -
1 0.5AtRxx,(n) 1
0.5At [r2(n - 1) - 0.5Atrl(n -- 1)k]K -1.
(18)
I
0.5AtI"
-0.5Atk
G2
CALCULATION FOR MEAN SQUARE RESPONSES OF STRUCTURES SUBJECTED TO TIME-DOMAIN CORRELATED WHITE NOISE EXCITATIONS
0 l qe(n - 0"5)Q~ e(n - 0-5)q r
[rl(n-1)
(n - 1) + O.~-~tGlrl(n-1)]
Equations (13)-(18) are the recursive formulas for practically calculating the mean square responses of structures subjected to white noise excitations.
I
R~,x(n- 1) Rx, x , ( n - 1)
[0°
Rxx,(n) l
-K-l[rf
(12)
where Rxx and Rex, are auto-correlation matrices of
In the last section, the recursive formulas for calculating structural mean square responses are obtained under the assumption that any two white noise excitations at different time instants are un-correlated. In some engineering problems, however, excitations do not act on structures simultaneously but have certain timedelay relationships. These time-delay relationships will
Dongyao Tan, Qingshan Yang
14
cause the correlation between the excitations. A typical example is that the earthquake motion excites a large span structure from its multiple supports. This section discusses structural random vibration analysis in such an excitation environment. Assume that there are nonstationary white noise excitations Ulel(t)sl(t), U2e2(t)s2(t),..., ULeL(t)sL(t) which act on a structure sequentially. Their time-delay relationships are
Sp(t) = sl ( t - N e A t )
(P = 2, 3 , . . . , L),
= [U1, U 2 , . . .
lj, liAt<<,tli+lAt )
e(t)s(t) = [el(t)sT(t), e2(t)sT(t),..., ej(t)sf (t)] r, and the excitation vector in eqn (2) is J
U(t)e(t)s(t) = E
0.5, n - 0 . 5 )
-
- i - 0"5)e(n - i - 0.5)s(n - i - 0-5)
= E{
× [v(n - 0.5)e(n - 0.5)s(n - 0.5)] r} Utel(n - i - 0"5)st(n - i - 0.5) I.t=l
Ukek(n--O'5)Sk(n--0"5)] T } I.k= 1 Jl
J2
= Z
E
Ulel(n - i - 0.5)E[sl(n - i - 0-5)
l=1 k = l
X s k ( n -- 0 " 5 ) ] e k ( n Jl
Uiei(t)si(t).
(20)
i=1
By substituting the coefficient matrices in eqn (2) and the above excitation vector into eqn (3.2), structural mean square responses can be consequently calculated. To obtain the detailed recursive formulas, we first separate Ryy(n) into three parts, i.e.
Ryy(n) = Rl(n) + R2(n) + RT(n),
(21)
J2
= Z
E
Utet(n-i--0"5)Qstsk 6(Nk -- Nt - i )
/=1 k = l
Is = min(n - 1,max INk -- Ntl). Then, eqn (23) can be rewritten as RE(n) = AtE Z
Rl(n ) --- H 1 R y y ( n - 1)H T
i=1
+ At2H2Ree(n - 0"5, n - 0 . 5 ) H f
J2
E
Ute,(n- i-0"5)Qsm
l=lk=l
Using the discrete approximation for 6-function, i.e. A t r ( N k -- NI - i) = 6k(N k -- Nt - i), where 6k( ) is the Kronecker symbol, and following notations:
= AtH~Ry~(n - 2,n - 0.5)H f + At2H1H2Ree(n -- 1 -- 0-5, n - 0"5)H~"
Reeo(n - i - 0"5,n - 0"5) l~
= E
= AtH~Rye(-0"5 , n - 0"5)HT
12
Z
qlel(n -- i-- 0"5)Qs,~k
l=lk=l
n-l
H~H2Ree(n- i-0"5,n-
Jl
H~H2 Z
x 6(Nk -- Nt - i)ek(n -- 0 " 5 ) U T H T.
(22)
Rz(n) = AtH1Rye(n - 1,n - 0.5)H T
(24)
where Qsm is the coefficient matrix in the crosscorrelation matrix Rstsk('r) = Qstskr[(Nk -- N t ) A t - 7-] of st(t) and sk(t). U(n - i - 0.5)e(n - i - 0.5)s(n - i - 0.5) and U(n - 0.5)e(n - 0.5)s(n - 0"5) are total excitation vectors at time instants (n - i - 0.5)At and (n - 0-5)At, respectively. It is known from eqn (24) that i f / > INk -- Ntl, there is 6 ( N k - N t - i)=-O, which means there need not add new terms in the summation in eqn (23). So, the maximum value of i should be
Is
where
+ At2Z
0"5)U~"
-
× ek(n -- 0"5)U f ,
, Vj]
(U t = [ 0 r qir ]T ,l<<-J, • • l1 <12...<
Ree (n - i
(19)
where At is a time unit which can be chosen as the time step in numerical calculation. Np (P = 2, 3 , . . . L ) are different integers representing the time-delay of the excitations. ' = ' only represents the time-delay relationships between the excitations. Assume again that there are j ( j < L) excitation vectors which have acted on the structure at time instants llAt, 12At,..., lyAt before time instant t = n a t . The total excitations acting on the structure at time t can then be expressed as U(t)e(t)s(t), where
U(t)
obtain
0"5)HT.
i=1
× 6k(Nk -- Nl -- i)ek(n -- 0"5)qT, (/Jl ~ < n - - i - - 0 . 5 < /j~+l;/h~
(23) Assume the excitations are independent with the initial conditions. Therefore, the first term in Rz(n ) becomes zero. Then, use the characteristics of 6-function of the auto-correlation matrix function o f sl(t) to
matrix R2(n) can also be expressed as R2(n ) = At Z i=1 [
-AtK-lk
2K-ira - I ]
(25)
Structural random vibration analysis
x
K-1G1
0.5AtK -1 ]
_0.5AtK-lk
K -1
x0
Equations (16)-(18), (13) and (28)-(29) are recursive formulas for practically calculating the mean square responses of structures subjected to time-domain correlated white noise excitations.
]
0
15
]
0 Reeo(n- i - 0"5, n - 0"5) x
G1K-1 0.5AtK -l
-0"5AtkKq ] . K -1
(26)
For simplification, the following notations are used
C3i C4iJ
[-AtK-lk
2K-lm-IJ
Ree2i = K - l Reeo(n - i - 0"5, n - 0"5) Reeli = 0"5 A t Ree2i With these notations, R2(n) has the following expression
Is [ eli
C2i] [ :
R2(n ) = At Z
i= 1 C3i C4i ]
x [ G1K-I [0.5AtK -1
Reeli] Ree2i I
-0"5AtkK-a]. K -1
(27)
Equation (21) can be solved partitionally. To do so, multiply both of its sides with Ha and its transpose from left and fight. A matrix algebraic equation having the same form as that of eqn (12) can be obtained. Its solution is also expressed by eqns (16)-(18) in which r l ( n - 1) remains in the same form but r 2 ( n - 1) and ra(n - l) should be changed into:
rE(n - 1) = -- 0"5AtRxxk + Rxx,(n - 1)G2 - 0.25At2Rrxx,(n - 1)k + 0"5AtR,e,~(n - 1)G2 + Z At(CliReeli + C2iRee2i) i=1
-- 0"5At Z At(C3iReeli + C4iRee2i) (28) i=1 rs (n - 1) = 0.25At2kR~x (n - 1 ) k - 0.5AtkRxx, (n - 1) G2 - 0"5AtG2Rfx,(n - 1)k + G2R,e,e(n - 1)G2 + AtRee(n - 0"5, n - 0"5) Is + 0.5Atk E At(CliReeli + C2iRee2i) i=1 Is
+
GlZ
i=1
White noises are idealized random processes. Only a few excitations occurring in practical engineering problems can be roughly taken as white noises, most of them should be treated as coloured noises according to their power spectral characteristics. Usually, the coloured noises can be generated from white noises by using filters. So, the coloured noises are also called filtered white noises. To do the random vibration analysis for a structure subjected to filtered white noise excitations, a way that is easily taken is to replace the filtered white noise excitations with white noise excitations. In this way, the available recursive formulas obtained before can be used. Therefore, it is required to generate filtered white noises from white noises. Generation of stationary filtered white noise processes
Generating filtered white noises from white noises means to represent the former whose statistic characteristics are known from the latter whose statistic characteristics are determined. Generally, the statistic characteristics of the excitations occurring in engineering are obtained in frequency domain. For this reason, represent the power spectral matrices of J independent filtered white noise excitations sj(t) (j = 1,2,..., J) as Sj(w), ( j = 1 , 2 , . . . , J ) . A filtered white noise process sj(t) with power spectrum Sj(w) can be generated from a white noise process wj(t) with spectral density 1/Dr through an appropriate filtering process. The characteristics of the filter are often expressed by linear differential equations,
x'fj(t) = vfjxfj(t) + zjwj(t),
(30)
where vfj is the characteristic matrix of thejth filter, za. is a constant vector, x~ is the state vector of the filter. The generated filtered white noise process can be expressed as an observable value of the filter,
At(C3iReeli -t- C4iRee2i)
sat ) = pjx:j(t),
i, + 0"5At E At(CliReeli + C2iRee2i)Tk
(31)
where pj is an observable vector. The power spectrum of the generated filtered white noise process can be deduced from eqns (30) and (31) as
i=1
1, + E At(C3iReeli + C4iRee2i)TGl" i=1
CALCULATION FOR MEAN SQUARE RESPONSES OF STRUCTURES SUBJECTED TO INDEPENDENT FILTERED WHITE NOISE EXCITATIONS
(29)
S:j(w) = Ipj(iwI - vfj)-lzjl 2.
(32)
Dongyao Tan, Qingshan Yang
16
By selecting the characteristic matrix vfj and the observable vector Pi of the filter properly, the power spectrum Sfi(a; ) of the generated filtered white noise process could approach that of the given filtered white noise process si(t ) with sufficient accuracy. The entire process of generating J independent filtered white noise excitations can then be expressed as
Sl(t) [Pl
40=
ix,,,,1
=
•
•
Ls/t)
pJ
.
.
.
D1 = 2D2 - I D 2 = ( I - 0"5AtV) -1
(I
- 0"5AtVfl) -1 (I - 0'5Atvfj) -1
= exs(t )
LxiJ(t) J
(33)
x}(o
where
The expression of R~(n) can be obtained by expanding eqn (36). Then, use/-/3 and its transpose to multiply both of its sides from left and right, respectively, and to obtain
H3Ryy(n)H T = n4Ryy(n -
.
L x~fJ(t) J
vfj L x~/t)
1)H4r
+ At{UePD2Rxly(n - 1)H~" + H4Ryxs[UePD2] r } + A t 2 UeeO2 [Rxsxr (n - 1) + 0 . 2 5 A t Z Z r][UeeO2] r,
+
•
zj
(37)
Lwj(t)j
where
= VXf(t) + Z W ( t ) .
(34)
With the spectral density of each independent white noise process wj being specified as 1/27r, the autocorrelation function matrix of white noise vector W(t) must be Rww(T) = Ir('r).
H a = M + 0.5ARC.
(38)
Similarly, the expression of Ryxf (n) can be obtained from eqn (36). Then, use Ha to multiply both of its sides from left to obtain
H3RyxI(n ) = H 4 R y x i ( n - 1)D T Recursive formulas for calculating mean square responses
+ AtUeeD2RxsxI(n - I)D~" According to eqn (32), only the stationary filtered white noise processes can be generated by the filters. Therefore, in the following discussion, the excitations are assumed to be stationary. Combining eqns (2), (33) and (34), the generalized state equation which couples structural state equation and the filtering process is
EM 1] I'v'(t) ] C U~.P]IY(t) (35)
The recursive formulas for calculating structural mean square responses are obtained by substituting eqn (35) into eqn (11),
+ 0"5At2U e P D 2 Z Z T D f .
Finally, the recursive formula for calculating Rxfxs is derived as:
Rxlxi(n) = D1Rxlxs(n - 1)D1r + A t D 2 Z Z T D f .
Rxlxi(n) j
X [Ryy ( n - l ) [R~y(n
Ryxf(n--l)] [01
(40)
Referring to the recursive formula for practically calculating structural mean value responses given before, the recursive formulas for practically calculating Ryxl (n) are derived as:
Rx, xi(n ) = K-l[rs(n - 1) - 0"5Atkr4(n - 1)] Rxx f = r4(n - 1) + 0.5AtR~xi(n),
A t H 2 UePD 2" Rxyy(n)
(39)
(41) (42)
where
D1 A tH 2 UePD 2 ]
r 4 ( n - 1 ) = [RxxI(n- 1)+ 0"5AtR~,x~(n- 1)]D1 (43)
]
rs(n - 1) = [ - 0 " 5 A t k R x x i ( n - 1)+G1R~xi(n - 1)]D1r
1) g x l x i ( n - 1)J
D1
+ AtqePD2Rxexf (n - 1)Dr1 + 0.5At2qePD2ZZrDf.
T ,
(36)
(44)
As eqn (37) and eqn (12) are identical in the form of expression, its solution can then be expressed by eqns (16)-(18) in which rl(n - 1) remains in the same form
17
Structural random vibration analysis
calculated as:
but r2(n - 1) and r3(n - 1) should be changed to r 2 ( n - 1) = - 0 " 5 A t R x x ( n -
1)k + R x z ( n -
R , f ( T ) = R~Rf(~-),
1)G2
- 0.25At2RxT"~,( n - 1 ) k + O . 5 A t R z z ( n -
1)G2
+ A t[Rxx I (n - 1) + 0"5AtR~xr (n - 1)] [qePD2] T
(45) r3(n - 1) -- 0 . 2 5 A t 2 k R x x ( n - 1)k - 0 . 5 A t k R x x , ( n - 1)G2
where RU(-C) is the Fourier transformation of S(w). Structural mean square responses can be calculated by using eqn (3.2). To obtain a concise expression for Ryy(n), we separate it into three parts as described by eqn (21), in which R1 (n) keeps the same expression but R2(n ) changes to n-1
R2(n) = At 2 Z
- 0.5AtG2Rrx~,(n - 1)k + G2Rx, x,(n - 1)G2
(48)
HIH2Ue(n - i - 0"5)RcRi(iAt )
i=l
+ A t q e P D 2 [Rf~xl (n - 1)G2 - 0.5AtRr~xl (n - 1)k]
× e(n - 0"5) UrH~".
+ A t [ G 2 R e x s (n - 1) - 0 . 5 A t k R x x s (n - 1)] [qePO2] T + At2qePD2 [Rxzx~ (n - 1) + 0 . 2 5 A t Z Z r] [qePD2] r.
(46) Equations (16)-(18), (13) and (40)-(46) construct the recursive formulas for practically calculating the mean square responses of structures subjected to independent filtered white noise excitations.
(49)
Substitute eqns (49) and (22) into eqn (21), and multiply the both of its sides of the obtained equation by H a and its transpose from left and right, respectively. A matrix algebraic equation is obtained which also has the same form as that of eqn (12), its solution can be expressed by eqns (16)-(18) and (27)-(29), in which Ree2i should be calculated by the following equation: Ree2i
=
A t K - l qe(n - i - 0 . 5 ) R c R f ( i A t ) e ( n - 0.5)q r.
(50) CALCULATION FOR MEAN SQUARE RESPONSES OF STRUCTURES SUBJECTED TO SPATIALLY CORRELATED FILTERED WHITE NOISE EXCITATIONS
The recursive formulas deduced in the last section are usually used for aseismic structures where the number of earthquake excitations are relatively fewer. If the number of excitations acting on the structures is larger, the order of the state equations of the entire filtering process would probably be near to or even greater than the order of structural dynamic differential equations. This is because each excitation needs a filter. For example, in random vibration analysis for space structures subjected to spatially correlated wind loads, the number of wind loads is equal to the number of forced nodes on the structure which is the same order as the degrees of freedom of the structure. In this case, those recursive formulas become powerless. Assume the random vector s(t) in eqn (1) is spatially correlated filtered white noise vector. Its power spectrum matrix has the following expression: Ssf(O3) = ReS(o3)
(47)
If R f ( n A t ) is small enough, R2(n) is approximately equal to R 2 ( n - 1 ) . Although both of the recursive formulas given in the last section and this section can be used to calculate the stationary mean square responses of structures subjected to filtered white noise excitations, the recursive formulas in this section have a distinct merit in that they are able to calculate the nonstationary mean square responses of structures subjected to non-stationary filtered white noise excitations. When the excitations are stationary, R2(n) can be calculated recursively. It is:
R2(n) = R 2 ( n - 1) + At2G(n - 1)H2UeRcRf(nAt)eUTH
T
where G(i) = H ] . If R f ( T ) is decomposible 1, i.e. P
R ~ ( (n - i - 0 " 5 ) A t ) R ~ ( (n - 0.5)At),
Rf(iAt) = Z l=l
R2(n ) can also be calculated recursively even if the excitations are non-stationary. It is: R2(n) = A t 2 ~ [ ~ H I H 2 U e ( n - i - 0 " 5 ) 1=1 [.i=1
where R~ is a matrix which represents the spatial correlativity of s(t). S ( ~ ) is the identical power spectrum for each random process in s(t). It can be the well-known Davenport's wind velocity spectrum in wind engineering and Tajimi's earthquake acceleration spectrum in earthquake engineering. The correlation function matrix of the spatially correlated filtered white noise vector can then be
x R c R ~ ( ( n - i - 0.5)At)] x R~2((n - 0 . 5 ) A t ) e ( n -- 0.5)UTH2r P
= At2 Z l=1
R~t(n)R~ 2((n - 0 . 5 ) A t ) e ( n -- 0.5)UTH z
Dongyao Tan, Qingshan Yang
18
Table 1. (~0 = 0"05,600 = 27r,g = 10)
Sg(w) = 1/27r
e(t) -- 1-0
I
II
III
IV
(t/3) 1
e(t)=
1 + 4~2(W/Wg)2 ~2 Sg@') = [1 -(fa~g)~+--~g(w/wg) i~~
Types of excitation environments
e 0'26(t-13)
(~g ---- 0 ' 6 S , w g
0 ~< t < 3 3~
where
=
18.85, S02= 1/27r)
square responses and (2) it is able to calculate the accurate structural stationary mean and mean square responses. Here, several examples are given to show its numerical accuracy and application in complicated structural random vibration problems. Example 1. The vibration equation of a linear oscillator subjected to a non-stationary excitation is
R~l(n) = HIR~_I(n- 1) + H1H2e(n- 1.5)RcR~71((n- 1.5)At).
NUMERICAL EXAMPLES
x"(t) + 2~oWoX'(t) + wZx(t) = e(t)s(t), The unconditional stability of/3-recursive formula, from which the recursive formulas for calculating structural mean and mean square responses are deduced, has been proved. 9:° Its numerical accuracy concluded as follows: l° (1) it has the second order accuracy in calculating structural non-stationary mean and mean 0.5 -
20.0 f Numerical solution (which coiaeides with the accurate solution in this Figure)
-
0.4
t(lIl)
0.3
0.2
"-.
,t'/J v " 1'/
0.1
"
"-.
/
~x~x.
~(III) r 10.0
0.0
t(I)
Fig. 1. Mean value response of the oscillator.
0.02434
0.02
0.02016
.//.~
where e(t) is an envelope function and s(t) is a stationary random process of which the mean value and spectral function of its centralized process are ~ and Sg(w), respectively. The parameters of the oscillator and different excitation environments are listed in Table 1. For comparison, the accurate transient displacement mean value response whose stationary value is 0.2532 and the centralized mean square response of the oscillator whose stationary value is 0.02016 are calculated according to the closed form solutions given in the Appendix. In addition, to use eqns (16)-(18), (27)-(29) and (50) to calculate the mean square response of the oscillator subjected to excitation environments II and IV, the correlation function of the excitation should be deduced. It is also given in the Appendix. The numerical results are shown in Figs 1-3. To compare with the existing examples, the variance function of Duffing oscillator given by Di Paola 1 is reanalysed by using the recursive formulas in this paper. The equivalent linearized damping and stiffness coefficients of the oscillator are 2~0w0 and w2 = a;2(1 + 0"9crZx), where ~0 = 1, w0 = 2. The excitation is nonstationary filtered white noise whose envelope function is e(t)= 0.906te -q3 and correlation function
.-.--
~;'~
I
/r "-6~" 0.01
! ,I . ~ t'"~[ / / ~
\
0.02
....
I (Numerical solution coincides with the accurate solution in this Figure) II(Eqs.(44)-(46)) n(Eqs.(50), (27)-(29))
// /
At =0.025 j 0.0 0.00
410
, 3.0
.... IV(Eqs.(44)-(46)) ---IV(Eq s.(50),(27)-(29)) A.t=0~25 , 5.0
15.0
N~X" ~.,.. 20.0
t
t~
Fig. 2. Centralized mean square responses of the oscillator in excitation environments I and II.
Fig. 3. Centralized mean square responses of the oscillator in excitation environments III and IV.
Structural random vibration analysis 2 Oxx 2.00
\ --
19
~1( x 0.01292) o2~2(x0.0050)
1.50
0.8 %
1.00
-
Linear
0.6
0.50 / 0.00 0.00
0.4
4.00
2.00
6.00
8.00
10.00
12.00
Fig. 4. Variance function of the Duffing oscillator. has the same form as that given in the Appendix, but the magnitude should be multiplied by 6"262. The parameters ~g and Wg are 0-6 and 57r, time step is 0.02. The results, which seem quite close to that given by Di Paola,1 are shown in Fig. 4. Example 2. A spring-body dynamic system with twodegree-of-freedom is shown in Fig. 5. Its structural parameters are Mass: m 1 = m2 -- 2.0 Stiffness: kl = k2 = 80.0 Damping: classical damping (a = 0.1,13 = 0.05) Load condition: two white noise excitations Sl(t) and S2(t ) = S l ( t - A T ) having identical power spectrum 1/27r act on the system, where A T is the time delay between the two excitations. Time step: At = 0"1 The stationary mean square responses of the structure with the variation of the time delay A T are calculated by using eqns (16)-(18), (25) and (27)-(29) and depicted in Fig. 6. Example 3. A bridge with four spans is excited by an SH earthquake wave (see Fig. 7) which propagates along the bridge. Only the lateral vibration of the bridge is considered. The structural characteristics of the bridge are:
00
01s
116
~T
Fig. 6. Mean square displacement responses of the dynamic system. Stiffness (x 109 N/m): k d = 3.6336, kc = 0.39375, kg : 2-200; Damping (× 106 N.s/m): Cd = 6"576, cc = 0"6551, Cg = 25"840. The earthquake wave is characterized by the following three functions: (1) correlation function
where er~ is the mean square ground displacement. B relates to the wave number of the earthquake wave and is chosen as 0.16 in this example. (2) decaying function fd(x)
= e
where 7 is the decay rate of the earthquake wave, which is 0"005 in this example, x refers to the distance away from the point of interest. (3) envelope function. It is the same as that given in the bottom row of Table 1. With the above three functions, the correlation function of the earthquake wave that propagates at any two supports of the bridge can be expressed as:
Rfij( ti, 7-) = e( ti) e-~la°l Rf ( 7- - dij / V )e( ti -t- 7- - dij / V ),
Mass (x 106 kg): me = 0"2975, md = 0"595, m3 = 0"818, my = 1.214;
////////
k~
Tx2
X Fig. 5. A spring-body dynamic system with two-degree-of-
freedom.
where d U is the distance from support i to support j, and the reference time is set at support i. V is the propagation velocity of the earthquake wave. The relative displacement mean-square-root responses of the beams and columns numbered from 1 to 6 are calculated according to eqns (16)-(19), (27)-(29) and (50). The time step is 0.02. Both stationary and nonstationary responses are carried out. The results are shown in Figs 8 and 9. In Fig. 8, the standard mean-square-root responses or, are calculated in the idealized case that V ~ oo, i.e. the earthquake wave simultaneously acts on the five supports of the bridge. The results are listed in Table 2. Figure 8 indicates that the first mode of the bridge, which is approximately symmetric half sinusoidal wave,
Dongyao Tan, Qingshan Yang
20 me
md 1
me
2
kd
3
.
v__ 40.0
40.0
*,
4'
40.0
L =
,1"
40.0
A"
Fig. 7. The model of the bridge. is not favourably excited out by the earthquake wave which propagates at velocity 1000(m/s). This means that in this case the phases of the first mode and the earthquake wave are opposite. So, we can estimate that the natural period of the bridge is about 0-16(s). When the bridge is excited by the earthquake wave which 1.01 a/*, \
--
1 3
--
'\ 0.8 ' ~ \ , ' , , \ ',,\, \.
4 5
0.4 0.00
CONCLUSIONS This paper applies the fundamental ¢3-recursive formula in discrete analysis method for random vibration to structural random vibration problems. The recursive formulas for practically calculating the mean and mean square responses of structures subjected to four typical excitation environments which are often encountered in engineering are derived. Some programming techniques which can enhance the computational efficiency are also introduced. In the recursive formulas for practical
/ ~-~-...
///
\\\
0:(M
0.08
propagates at a velocity between 400(m/s) and 1000(m/s), the responses ratio of columns 4 and 6 are approximately equal. This means that the second mode of the bridge, which is approximately asymmetric full sinusoidal wave, is excited out. So, we can estimate that the period of the second mode is about 0.14(s).
0.'12
0.'16
L/V
2. Standard relative displacement mean-square-roots
Table
Elements Fig. 8. Stationary relative displacement mean-square-root responses with variation of V.
as/trg
1
2
3
4
5
6
0'41
0"06
0"41
1.52
1'80
1"52
~.:(x 10-3) 5.0
\,',.,,, o.o
3.o
i
5.0
13.0
20.0 - t
0-2(x I0-l) 2.0 f.
\
!
I
- -
\
....
\
f
"x \ \ \
4 5
-----6
1.0
0.0
3.0
5.0
13.0
20.0 t
Fig. 9. Transient displacement mean square responses of the bridge.
21
Structural random vibration analysis
calculation, the symmetric and banded-shape structural dynamic characteristic matrices participate in the calculations directly so that the computational time and storage are saved and the computational efficiency is increased. This resursive formula can also be applied, without special difficulties, to the calculation for the auto-correlation function matrix of structural random responses. But this is out of the scope of this paper. It will be discussed in our following paper.
REFERENCES 1. M. D. Paola, M. Ioppolo and G. Muscolino, Stochastic seismic analysis of Multi-degree-of-freedom systems, Engineering Structures, 6, 113-118 (1984). 2. M. Hoshiya, K. Ishii and S. Nagata, Recursive covariance of structural responses, J. Engng Mech., Proc., ASCE 110(12), 1743-1755 (1984). 3. Y. K. Lin, Ruichong Zhang and Yan Yong, Multiply supported pipeline under seismic wave excitations, Journal of Engng Mechanics, 116(5), 1094-1108 (1990). 4. Federico Perotti, Structural response to non-stationary multiple-support random excitation, Earthquake Engng Structural Dynamics, 19, 513-527 (1990). 5. C. W. S. To, Direct integration operators and their stability for random response of multidegree of freedom systems, Computers and Structures, 30, 865-874 (1988). 6. C. W. S. To, An implicit direct integrator for random response of multidegree of freedom systems, Computers and Structures, 33, 73-77 (1989). 7. Dongyao Tan and Hua Guo, A procedure of discrete analysis for nonstationary random vibration, Earthquake Engng Engng Vibr., 6(3), 12-18 (1986). 8. Dongyao Tan, Qingshan Yang and Chen Zhao, Discrete analysis method for random vibration of structures subjected to spatially correlated filtered white noises, Computers and Structures, 43(6), 1051-1056 (1992). 9. Dongyao Tan, Qingshan Yang and Pei Lu, A proof of the stability of the discrete analysis method for random vibration, J. of Harbin Architectural and Civil Engineering Institute, 1, 12-23 (1988). 10. Dongyao Tan, Discrete analysis method of random vibration, ACTA MECHANICA SINICA, 24(4), 473479 (1992).
11. Dongyao Tan and Qingshan Yang, Discrete analysis method for random vibration of structures under excitations correlated in time domain, Earthquake Engng Engng Vibr, 10(2), 37-46 (1990).
APPENDIX 1. The closed-form solution of the transient displacement mean value response of the oscillator $ ( t ) = ~oo 1-e-~°~°t c°sv/1-~°2w°t
+ /1~°_ ~o2 sin ~/~ ----~°2w°t)1 " 2. The closed-form solution of the transient centralized mean square responses of the oscillator in excitation environment I. 1 a2xx(t)= ~
{ 1
e -2~°w°t [ Joo--~o2) wo2(1-~o2)
+ 2(~oWosin ~/1 - ~2oWt)2 + wo2~oV/1 - ,o2sin 2V/1 ~o2----Wot] }. 3. The correlation function of the excitation acting on the oscillator Rf(r)=
[1 +4~ 2 cos V/1-~gwgr 4 e _~g~g,[----~g
1 - 4(~ sin ~ 1 -AI-~/1__~2 -~203gT] "
~~
; .:'All
ELSEVIER
0266-8920(94)00003-4
© 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0266-8920/95/$09.50
Structural random vibration analysis by using recursive formulas Dongyao Tan Beijing University of Aeronautics and Astronautics, School of Astronautics, P.O. Box 504, Beijing 100083, China
&
Qingshan Yang Harbin Architectural and Civil Engineering Institute, Harbin 150006, China
(Received October 1992; revised March 1994; accepted April 1994) This paper discusses the application of/3-recursive formula to structural random vibration analysis. This recursive formula is able to calculate the accurate stationary mean and mean square responses of a linear or equivalent linearized nonlinear structure subjected to stationary excitations. To enhance the computational efficiency, the basic formulae for calculating structural mean and mean square responses which are obtained from /3-recursive formula are transformed into other forms which directly use the band-shape and symmetric structural dynamic characteristic matrices in calculation. The detailed formulations are given according to different excitation environments, such as white noise excitations, time-domain correlated white noise excitations, independent filtered white noise excitations and spatially correlated filtered white noise excitations. Some numerical examples are given to show the applications of the recursive formula.
carry out the accurate stationary mean and mean square responses for any linear or equivalent linearized nonlinear dynamic system. 9'1° It also has second order accuracy in calculating non-stationary mean and mean square responses. 1° This paper discusses its application in structural random vibration analysis. Unlike structural deterministic dynamic analysis, the detailed calculation formulations in structural random vibration analysis depend largely on the types of excitations to which the structures are subjected. Here, the detailed and practical calculation formulations by using /3recursive formula to calculate structural mean and mean square responses are deduced with respect to four important excitation environments. The first is white noise excitation which roughly describes the engineering background of multi-dimensional earthquake motions exciting a structure simultaneously. The second is time-domain correlated white noise excitation which describes the situation of a propagating earthquake motion exciting a large span structure from its different supports. The third is independent filtered white noise excitation which describes in some detail the earthquake motion or any other coloured
INTRODUCTION Structural random vibration analysis by using numerical methods became a feature in the field of random vibration theory in the last decade. For examples, Di Paola et al.1 gave an unconditionally stable step-by-step method, based on the eigenvalue analysis, to calculate the mean square response of a multidegree of freedom linear or nonlinear system subjected to nonstationary filtered noises, C. W. S. To 5'6 proposed explicit and implicit direct integrators to calculate the random response of multidegree of freedom systems. Various numerical examples have been given to illustrate the efficiency of these methods. This paper uses/3-recursive formula which was proposed and studied by Dongyao Tan et al. TM to calculate the stationary and nonstationary mean and mean square responses for large linear or equivalent linearized nonlinear structures. /3-recursive formula was obtained by discretizing a general state equation in time domain with the assumption that the state vector changes linearly between two adjacent discrete time instants. 7 It has been proved to be unconditionally stable, and it can 11
12
D o n g y a o Tan, Qingshan Yang
noise which excites a structure. The fourth is spatially correlated filtered white noise excitation which describes the gust wind that excites large space structures or highrise structures.
BASIC F O R M U L A S OF CALCULATION Structural vibration equations can be expressed as: (1)
m x " ( t ) + cx'(t) + k x ( t ) = qe(t)s(t)
where m, c, k are structural mass, damping and stiffness matrices, respectively, x(t), x'(t), x " ( t ) are structural displacement, velocity and acceleration vectors, e(t) is a diagonal matrix which consists of the deterministic envelope functions that describe the non-stationary property of the excitation vector, s(t) is a stationary random vector, q is the influence matrix of the excitations. Since the /3-recursive formula was obtained for a dynamic system whose differential equation is expressed in the form of state equation, to use this formula, structural vibration eqn (1) should be transformed into a state equation as follows: (2)
M y ' ( t ) = Cy(t) + Ue(t)s(t)
where
m] C=
['1 -k
auto-correlation matrix of random vector s(t). Rys(n-1,n-0.5) is the cross-correlation matrix of structural responses at time instant ( n - 1)At with the random vector s(t) at time instant (n - 0.5)At. y(n) and Ryy(n) are structural mean and mean square responses at time instant n A t . Equations (3) and (4) are the basic formulas obtained from 3-recursive formula for calculating structural mean and mean square responses.
CALCULATION FOR STRUCTURAL MEAN VALUE RESPONSES It can be seen from eqn (3.1) that structural mean value responses are determined by the mean values of the excitations alone. So, this recursive formula can be used to calculate the mean value responses for structures subjected to any kind of excitation. However, it is disadvantageous to use the original form of eqn (3.1) to calculate structural mean value responses because it contains the inverse of matrix /-/3 which is nonsymmetric and order doubled comparing with structural dynamic characteristic matrices. In order to make the symmetric and banded-shape structural dynamic characteristic matrices in eqn (1) directly participate in the calculation so as to save computational time and storage, it is necessary to make a transformation for eqn (3.1). Using matrix Ha to multiply both of its sides of eqn (3. l) a new equation is obtained and its partitioned form can be expressed as:
-c
and y(t) is state vector [xr(t), x'r(t)] r, g is [Or, qr] r. According to the results of Tan et al., 9' 10 the unconditionally stable recursive formulas which are able to calculate the accurate structural stationary mean and mean square responses are y(n) = H l y ( n - 1) + A t H E U e ( n - 0.5)g
x Rsy(n-
[ ,
0.SAtk [2'(n-l)
J +At
-0.5Atk
e(n-0"5)~=
[b(n
G2
1) '
where a(n - 1) ----2(n - 1) + 0.5AtSt(n - 1)
+ A t H 2 U e ( n - 0.5)
1 (5)
(3.1)
Ryy(n) = H1Ryy(n- 1)H T -[-AtH1gys(n- 1, n - 0-5) x e(n - 0 . 5 ) U r H f
,
(6.1)
b(n - 1) = - 0 . 5 A t k 2 ( n - 1)
0 . 5 , n - 1)H~"
+ Gz2P(n - 1) + A t q e ( n - 0.5)~
(6.2)
+ A t 2 H 2 U e ( n - 0.5)Rss(n - 0-5, n - 0"5) × e(n - 0 - 5 ) U r H f ,
(3.2)
G l = m+O.5Atc
(7.1)
G2 = m - - O . 5 A t c .
(7.2)
where H1 = 2 H 2 M - I
(4.1)
H 2 = Hj -1
(4.2)
Ha = M - 0-5AtC
(4.3)
Here, ~ and Rss(n - 0'5, n - 0-5) are the mean value and
Here, a ( n - 1 ) and b ( n - 1 ) are known for time instant n A t . The solutions of the above matrix algebraic equation are Y,'(n) = K - l [ b ( n - 1) - 0 . 5 A t k a ( n - 1)]
(8)
2(n) = a(n - 1) + 0.5AtOP(n),
(9)
J
Structural random vibration analys& where
K = m + 0.5Arc + 0.25At2k.
(10)
Equations (6)-(10) are practical recursive expressions for calculating structural mean value responses.
13
structural displacement and velocity, respectively. Rxe is their cross-correlation matrix. For time instant nAt, rl(n -- 1), r2(n -- 1) and r3(n - 1) are known, they are
rl(n - 1) = Rxx(n - 1) +O.5AtRxx,(n - 1) + 0.5AtRrxx,(n - 1) + 0.25At2Rx, e(n - 1)
CALCULATION FOR MEAN SQUARE RESPONSES OF STRUCTURES SUBJECTED TO WHITE NOISE EXCITATIONS
White noise is an idealized stochastic process. Some excitations encountered in engineering can be taken as white noises roughly, such as the bedrock movement of an earthquake. Other excitations which cannot be taken as white noises, however, can usually be generated from them. Therefore, the random vibration analysis for structures subjected to white noise excitations is a fundamental problem in structural random vibration analysis. In addition, the mean and mean square responses of a linear oscillator subjected to stationary white noise excitation have closed form solutions, which have often been used to check the accuracy of the numerical methods. Substituting the coefficient matrices in eqn (2) into eqn (3.2), the recursive formula for calculating the mean square responses of a structure subjected to white noise excitations are obtained as follows:7'1° Ryy(n) = n l R y y ( n -
(13)
r2(n - 1) = - 0.5AtRxx(n - 1)k + Rxx,(n - 1)G2
- 0 . 2 5 A ? g ; e ( n - 1)k + 0"5AtRx, x,(n - 1)G2
(14)
r3(n - 1) = 0.25At2kRxx(n - 1)k - 0"5AtkRxx,(n - 1)G2 - 0.5AtG2Rrxx,(n - 1)k + GzRx, x,(n - 1)G2 + Atqe(n - 0.5)Qsse(n - 0.5)q r.
(15)
The solution of the matrix algebraic equation (12) is 1°
Rxx(n) = K -l [O'25At2r3(n- 1)+ 0 " 5 A t r T ( n - 1)G 1 + 0"5AtGlr2(n - 1) + Glrl(n - 1)G1]K -1 (16) 1 Rxe(n) = 0.5AtRxx(n)
1)a T
+ AtH2Ue(n - 0"5)Qse(n - 0-5) UrH2r,(11) where Qs is the coefficient matrix in the auto-correlation function matrix R~s(~')= Qsr('r) of the white noise excitations. Similarly, to make eqn (11) have higher computational efficiency, it is necessary to transform it into another form which is of benefit to practical calculation. Multiply both sides of eqn (11) with matrix Ha and its transpose from left and right, respectively, and the obtained equation in its partitioned form is
[I
-0.5Atll[Rxx(n
0"5Atk I
G1
)
[ Rx, x(n)
--0"5AtI] T= [
0"5Atk
G1
Rxx(n-1)
+
Rx, x,(n) 0"5AtI"
-0'5Atk
G2
Rxx,(n-1)][
r2(n-1)]
IrWin- 1)
r3(,-
1/J '
(17)
Rx, x , ( n ) -
1 0.5AtRxx,(n) 1
0.5At [r2(n - 1) - 0.5Atrl(n -- 1)k]K -1.
(18)
I
0.5AtI"
-0.5Atk
G2
CALCULATION FOR MEAN SQUARE RESPONSES OF STRUCTURES SUBJECTED TO TIME-DOMAIN CORRELATED WHITE NOISE EXCITATIONS
0 l qe(n - 0"5)Q~ e(n - 0-5)q r
[rl(n-1)
(n - 1) + O.~-~tGlrl(n-1)]
Equations (13)-(18) are the recursive formulas for practically calculating the mean square responses of structures subjected to white noise excitations.
I
R~,x(n- 1) Rx, x , ( n - 1)
[0°
Rxx,(n) l
-K-l[rf
(12)
where Rxx and Rex, are auto-correlation matrices of
In the last section, the recursive formulas for calculating structural mean square responses are obtained under the assumption that any two white noise excitations at different time instants are un-correlated. In some engineering problems, however, excitations do not act on structures simultaneously but have certain timedelay relationships. These time-delay relationships will
Dongyao Tan, Qingshan Yang
14
cause the correlation between the excitations. A typical example is that the earthquake motion excites a large span structure from its multiple supports. This section discusses structural random vibration analysis in such an excitation environment. Assume that there are nonstationary white noise excitations Ulel(t)sl(t), U2e2(t)s2(t),..., ULeL(t)sL(t) which act on a structure sequentially. Their time-delay relationships are
Sp(t) = sl ( t - N e A t )
(P = 2, 3 , . . . , L),
= [U1, U 2 , . . .
lj, liAt<<,tli+lAt )
e(t)s(t) = [el(t)sT(t), e2(t)sT(t),..., ej(t)sf (t)] r, and the excitation vector in eqn (2) is J
U(t)e(t)s(t) = E
0.5, n - 0 . 5 )
-
- i - 0"5)e(n - i - 0.5)s(n - i - 0-5)
= E{
× [v(n - 0.5)e(n - 0.5)s(n - 0.5)] r} Utel(n - i - 0"5)st(n - i - 0.5) I.t=l
Ukek(n--O'5)Sk(n--0"5)] T } I.k= 1 Jl
J2
= Z
E
Ulel(n - i - 0.5)E[sl(n - i - 0-5)
l=1 k = l
X s k ( n -- 0 " 5 ) ] e k ( n Jl
Uiei(t)si(t).
(20)
i=1
By substituting the coefficient matrices in eqn (2) and the above excitation vector into eqn (3.2), structural mean square responses can be consequently calculated. To obtain the detailed recursive formulas, we first separate Ryy(n) into three parts, i.e.
Ryy(n) = Rl(n) + R2(n) + RT(n),
(21)
J2
= Z
E
Utet(n-i--0"5)Qstsk 6(Nk -- Nt - i )
/=1 k = l
Is = min(n - 1,max INk -- Ntl). Then, eqn (23) can be rewritten as RE(n) = AtE Z
Rl(n ) --- H 1 R y y ( n - 1)H T
i=1
+ At2H2Ree(n - 0"5, n - 0 . 5 ) H f
J2
E
Ute,(n- i-0"5)Qsm
l=lk=l
Using the discrete approximation for 6-function, i.e. A t r ( N k -- NI - i) = 6k(N k -- Nt - i), where 6k( ) is the Kronecker symbol, and following notations:
= AtH~Ry~(n - 2,n - 0.5)H f + At2H1H2Ree(n -- 1 -- 0-5, n - 0"5)H~"
Reeo(n - i - 0"5,n - 0"5) l~
= E
= AtH~Rye(-0"5 , n - 0"5)HT
12
Z
qlel(n -- i-- 0"5)Qs,~k
l=lk=l
n-l
H~H2Ree(n- i-0"5,n-
Jl
H~H2 Z
x 6(Nk -- Nt - i)ek(n -- 0 " 5 ) U T H T.
(22)
Rz(n) = AtH1Rye(n - 1,n - 0.5)H T
(24)
where Qsm is the coefficient matrix in the crosscorrelation matrix Rstsk('r) = Qstskr[(Nk -- N t ) A t - 7-] of st(t) and sk(t). U(n - i - 0.5)e(n - i - 0.5)s(n - i - 0.5) and U(n - 0.5)e(n - 0.5)s(n - 0"5) are total excitation vectors at time instants (n - i - 0.5)At and (n - 0-5)At, respectively. It is known from eqn (24) that i f / > INk -- Ntl, there is 6 ( N k - N t - i)=-O, which means there need not add new terms in the summation in eqn (23). So, the maximum value of i should be
Is
where
+ At2Z
0"5)U~"
-
× ek(n -- 0"5)U f ,
, Vj]
(U t = [ 0 r qir ]T ,l<<-J, • • l1 <12...<
Ree (n - i
(19)
where At is a time unit which can be chosen as the time step in numerical calculation. Np (P = 2, 3 , . . . L ) are different integers representing the time-delay of the excitations. ' = ' only represents the time-delay relationships between the excitations. Assume again that there are j ( j < L) excitation vectors which have acted on the structure at time instants llAt, 12At,..., lyAt before time instant t = n a t . The total excitations acting on the structure at time t can then be expressed as U(t)e(t)s(t), where
U(t)
obtain
0"5)HT.
i=1
× 6k(Nk -- Nl -- i)ek(n -- 0"5)qT, (/Jl ~ < n - - i - - 0 . 5 < /j~+l;/h~
(23) Assume the excitations are independent with the initial conditions. Therefore, the first term in Rz(n ) becomes zero. Then, use the characteristics of 6-function of the auto-correlation matrix function o f sl(t) to
matrix R2(n) can also be expressed as R2(n ) = At Z i=1 [
-AtK-lk
2K-ira - I ]
(25)
Structural random vibration analysis
x
K-1G1
0.5AtK -1 ]
_0.5AtK-lk
K -1
x0
Equations (16)-(18), (13) and (28)-(29) are recursive formulas for practically calculating the mean square responses of structures subjected to time-domain correlated white noise excitations.
]
0
15
]
0 Reeo(n- i - 0"5, n - 0"5) x
G1K-1 0.5AtK -l
-0"5AtkKq ] . K -1
(26)
For simplification, the following notations are used
C3i C4iJ
[-AtK-lk
2K-lm-IJ
Ree2i = K - l Reeo(n - i - 0"5, n - 0"5) Reeli = 0"5 A t Ree2i With these notations, R2(n) has the following expression
Is [ eli
C2i] [ :
R2(n ) = At Z
i= 1 C3i C4i ]
x [ G1K-I [0.5AtK -1
Reeli] Ree2i I
-0"5AtkK-a]. K -1
(27)
Equation (21) can be solved partitionally. To do so, multiply both of its sides with Ha and its transpose from left and fight. A matrix algebraic equation having the same form as that of eqn (12) can be obtained. Its solution is also expressed by eqns (16)-(18) in which r l ( n - 1) remains in the same form but r 2 ( n - 1) and ra(n - l) should be changed into:
rE(n - 1) = -- 0"5AtRxxk + Rxx,(n - 1)G2 - 0.25At2Rrxx,(n - 1)k + 0"5AtR,e,~(n - 1)G2 + Z At(CliReeli + C2iRee2i) i=1
-- 0"5At Z At(C3iReeli + C4iRee2i) (28) i=1 rs (n - 1) = 0.25At2kR~x (n - 1 ) k - 0.5AtkRxx, (n - 1) G2 - 0"5AtG2Rfx,(n - 1)k + G2R,e,e(n - 1)G2 + AtRee(n - 0"5, n - 0"5) Is + 0.5Atk E At(CliReeli + C2iRee2i) i=1 Is
+
GlZ
i=1
White noises are idealized random processes. Only a few excitations occurring in practical engineering problems can be roughly taken as white noises, most of them should be treated as coloured noises according to their power spectral characteristics. Usually, the coloured noises can be generated from white noises by using filters. So, the coloured noises are also called filtered white noises. To do the random vibration analysis for a structure subjected to filtered white noise excitations, a way that is easily taken is to replace the filtered white noise excitations with white noise excitations. In this way, the available recursive formulas obtained before can be used. Therefore, it is required to generate filtered white noises from white noises. Generation of stationary filtered white noise processes
Generating filtered white noises from white noises means to represent the former whose statistic characteristics are known from the latter whose statistic characteristics are determined. Generally, the statistic characteristics of the excitations occurring in engineering are obtained in frequency domain. For this reason, represent the power spectral matrices of J independent filtered white noise excitations sj(t) (j = 1,2,..., J) as Sj(w), ( j = 1 , 2 , . . . , J ) . A filtered white noise process sj(t) with power spectrum Sj(w) can be generated from a white noise process wj(t) with spectral density 1/Dr through an appropriate filtering process. The characteristics of the filter are often expressed by linear differential equations,
x'fj(t) = vfjxfj(t) + zjwj(t),
(30)
where vfj is the characteristic matrix of thejth filter, za. is a constant vector, x~ is the state vector of the filter. The generated filtered white noise process can be expressed as an observable value of the filter,
At(C3iReeli -t- C4iRee2i)
sat ) = pjx:j(t),
i, + 0"5At E At(CliReeli + C2iRee2i)Tk
(31)
where pj is an observable vector. The power spectrum of the generated filtered white noise process can be deduced from eqns (30) and (31) as
i=1
1, + E At(C3iReeli + C4iRee2i)TGl" i=1
CALCULATION FOR MEAN SQUARE RESPONSES OF STRUCTURES SUBJECTED TO INDEPENDENT FILTERED WHITE NOISE EXCITATIONS
(29)
S:j(w) = Ipj(iwI - vfj)-lzjl 2.
(32)
Dongyao Tan, Qingshan Yang
16
By selecting the characteristic matrix vfj and the observable vector Pi of the filter properly, the power spectrum Sfi(a; ) of the generated filtered white noise process could approach that of the given filtered white noise process si(t ) with sufficient accuracy. The entire process of generating J independent filtered white noise excitations can then be expressed as
Sl(t) [Pl
40=
ix,,,,1
=
•
•
Ls/t)
pJ
.
.
.
D1 = 2D2 - I D 2 = ( I - 0"5AtV) -1
(I
- 0"5AtVfl) -1 (I - 0'5Atvfj) -1
= exs(t )
LxiJ(t) J
(33)
x}(o
where
The expression of R~(n) can be obtained by expanding eqn (36). Then, use/-/3 and its transpose to multiply both of its sides from left and right, respectively, and to obtain
H3Ryy(n)H T = n4Ryy(n -
.
L x~fJ(t) J
vfj L x~/t)
1)H4r
+ At{UePD2Rxly(n - 1)H~" + H4Ryxs[UePD2] r } + A t 2 UeeO2 [Rxsxr (n - 1) + 0 . 2 5 A t Z Z r][UeeO2] r,
+
•
zj
(37)
Lwj(t)j
where
= VXf(t) + Z W ( t ) .
(34)
With the spectral density of each independent white noise process wj being specified as 1/27r, the autocorrelation function matrix of white noise vector W(t) must be Rww(T) = Ir('r).
H a = M + 0.5ARC.
(38)
Similarly, the expression of Ryxf (n) can be obtained from eqn (36). Then, use Ha to multiply both of its sides from left to obtain
H3RyxI(n ) = H 4 R y x i ( n - 1)D T Recursive formulas for calculating mean square responses
+ AtUeeD2RxsxI(n - I)D~" According to eqn (32), only the stationary filtered white noise processes can be generated by the filters. Therefore, in the following discussion, the excitations are assumed to be stationary. Combining eqns (2), (33) and (34), the generalized state equation which couples structural state equation and the filtering process is
EM 1] I'v'(t) ] C U~.P]IY(t) (35)
The recursive formulas for calculating structural mean square responses are obtained by substituting eqn (35) into eqn (11),
+ 0"5At2U e P D 2 Z Z T D f .
Finally, the recursive formula for calculating Rxfxs is derived as:
Rxlxi(n) = D1Rxlxs(n - 1)D1r + A t D 2 Z Z T D f .
Rxlxi(n) j
X [Ryy ( n - l ) [R~y(n
Ryxf(n--l)] [01
(40)
Referring to the recursive formula for practically calculating structural mean value responses given before, the recursive formulas for practically calculating Ryxl (n) are derived as:
Rx, xi(n ) = K-l[rs(n - 1) - 0"5Atkr4(n - 1)] Rxx f = r4(n - 1) + 0.5AtR~xi(n),
A t H 2 UePD 2" Rxyy(n)
(39)
(41) (42)
where
D1 A tH 2 UePD 2 ]
r 4 ( n - 1 ) = [RxxI(n- 1)+ 0"5AtR~,x~(n- 1)]D1 (43)
]
rs(n - 1) = [ - 0 " 5 A t k R x x i ( n - 1)+G1R~xi(n - 1)]D1r
1) g x l x i ( n - 1)J
D1
+ AtqePD2Rxexf (n - 1)Dr1 + 0.5At2qePD2ZZrDf.
T ,
(36)
(44)
As eqn (37) and eqn (12) are identical in the form of expression, its solution can then be expressed by eqns (16)-(18) in which rl(n - 1) remains in the same form
17
Structural random vibration analysis
calculated as:
but r2(n - 1) and r3(n - 1) should be changed to r 2 ( n - 1) = - 0 " 5 A t R x x ( n -
1)k + R x z ( n -
R , f ( T ) = R~Rf(~-),
1)G2
- 0.25At2RxT"~,( n - 1 ) k + O . 5 A t R z z ( n -
1)G2
+ A t[Rxx I (n - 1) + 0"5AtR~xr (n - 1)] [qePD2] T
(45) r3(n - 1) -- 0 . 2 5 A t 2 k R x x ( n - 1)k - 0 . 5 A t k R x x , ( n - 1)G2
where RU(-C) is the Fourier transformation of S(w). Structural mean square responses can be calculated by using eqn (3.2). To obtain a concise expression for Ryy(n), we separate it into three parts as described by eqn (21), in which R1 (n) keeps the same expression but R2(n ) changes to n-1
R2(n) = At 2 Z
- 0.5AtG2Rrx~,(n - 1)k + G2Rx, x,(n - 1)G2
(48)
HIH2Ue(n - i - 0"5)RcRi(iAt )
i=l
+ A t q e P D 2 [Rf~xl (n - 1)G2 - 0.5AtRr~xl (n - 1)k]
× e(n - 0"5) UrH~".
+ A t [ G 2 R e x s (n - 1) - 0 . 5 A t k R x x s (n - 1)] [qePO2] T + At2qePD2 [Rxzx~ (n - 1) + 0 . 2 5 A t Z Z r] [qePD2] r.
(46) Equations (16)-(18), (13) and (40)-(46) construct the recursive formulas for practically calculating the mean square responses of structures subjected to independent filtered white noise excitations.
(49)
Substitute eqns (49) and (22) into eqn (21), and multiply the both of its sides of the obtained equation by H a and its transpose from left and right, respectively. A matrix algebraic equation is obtained which also has the same form as that of eqn (12), its solution can be expressed by eqns (16)-(18) and (27)-(29), in which Ree2i should be calculated by the following equation: Ree2i
=
A t K - l qe(n - i - 0 . 5 ) R c R f ( i A t ) e ( n - 0.5)q r.
(50) CALCULATION FOR MEAN SQUARE RESPONSES OF STRUCTURES SUBJECTED TO SPATIALLY CORRELATED FILTERED WHITE NOISE EXCITATIONS
The recursive formulas deduced in the last section are usually used for aseismic structures where the number of earthquake excitations are relatively fewer. If the number of excitations acting on the structures is larger, the order of the state equations of the entire filtering process would probably be near to or even greater than the order of structural dynamic differential equations. This is because each excitation needs a filter. For example, in random vibration analysis for space structures subjected to spatially correlated wind loads, the number of wind loads is equal to the number of forced nodes on the structure which is the same order as the degrees of freedom of the structure. In this case, those recursive formulas become powerless. Assume the random vector s(t) in eqn (1) is spatially correlated filtered white noise vector. Its power spectrum matrix has the following expression: Ssf(O3) = ReS(o3)
(47)
If R f ( n A t ) is small enough, R2(n) is approximately equal to R 2 ( n - 1 ) . Although both of the recursive formulas given in the last section and this section can be used to calculate the stationary mean square responses of structures subjected to filtered white noise excitations, the recursive formulas in this section have a distinct merit in that they are able to calculate the nonstationary mean square responses of structures subjected to non-stationary filtered white noise excitations. When the excitations are stationary, R2(n) can be calculated recursively. It is:
R2(n) = R 2 ( n - 1) + At2G(n - 1)H2UeRcRf(nAt)eUTH
T
where G(i) = H ] . If R f ( T ) is decomposible 1, i.e. P
R ~ ( (n - i - 0 " 5 ) A t ) R ~ ( (n - 0.5)At),
Rf(iAt) = Z l=l
R2(n ) can also be calculated recursively even if the excitations are non-stationary. It is: R2(n) = A t 2 ~ [ ~ H I H 2 U e ( n - i - 0 " 5 ) 1=1 [.i=1
where R~ is a matrix which represents the spatial correlativity of s(t). S ( ~ ) is the identical power spectrum for each random process in s(t). It can be the well-known Davenport's wind velocity spectrum in wind engineering and Tajimi's earthquake acceleration spectrum in earthquake engineering. The correlation function matrix of the spatially correlated filtered white noise vector can then be
x R c R ~ ( ( n - i - 0.5)At)] x R~2((n - 0 . 5 ) A t ) e ( n -- 0.5)UTH2r P
= At2 Z l=1
R~t(n)R~ 2((n - 0 . 5 ) A t ) e ( n -- 0.5)UTH z
Dongyao Tan, Qingshan Yang
18
Table 1. (~0 = 0"05,600 = 27r,g = 10)
Sg(w) = 1/27r
e(t) -- 1-0
I
II
III
IV
(t/3) 1
e(t)=
1 + 4~2(W/Wg)2 ~2 Sg@') = [1 -(fa~g)~+--~g(w/wg) i~~
Types of excitation environments
e 0'26(t-13)
(~g ---- 0 ' 6 S , w g
0 ~< t < 3 3~
where
=
18.85, S02= 1/27r)
square responses and (2) it is able to calculate the accurate structural stationary mean and mean square responses. Here, several examples are given to show its numerical accuracy and application in complicated structural random vibration problems. Example 1. The vibration equation of a linear oscillator subjected to a non-stationary excitation is
R~l(n) = HIR~_I(n- 1) + H1H2e(n- 1.5)RcR~71((n- 1.5)At).
NUMERICAL EXAMPLES
x"(t) + 2~oWoX'(t) + wZx(t) = e(t)s(t), The unconditional stability of/3-recursive formula, from which the recursive formulas for calculating structural mean and mean square responses are deduced, has been proved. 9:° Its numerical accuracy concluded as follows: l° (1) it has the second order accuracy in calculating structural non-stationary mean and mean 0.5 -
20.0 f Numerical solution (which coiaeides with the accurate solution in this Figure)
-
0.4
t(lIl)
0.3
0.2
"-.
,t'/J v " 1'/
0.1
"
"-.
/
~x~x.
~(III) r 10.0
0.0
t(I)
Fig. 1. Mean value response of the oscillator.
0.02434
0.02
0.02016
.//.~
where e(t) is an envelope function and s(t) is a stationary random process of which the mean value and spectral function of its centralized process are ~ and Sg(w), respectively. The parameters of the oscillator and different excitation environments are listed in Table 1. For comparison, the accurate transient displacement mean value response whose stationary value is 0.2532 and the centralized mean square response of the oscillator whose stationary value is 0.02016 are calculated according to the closed form solutions given in the Appendix. In addition, to use eqns (16)-(18), (27)-(29) and (50) to calculate the mean square response of the oscillator subjected to excitation environments II and IV, the correlation function of the excitation should be deduced. It is also given in the Appendix. The numerical results are shown in Figs 1-3. To compare with the existing examples, the variance function of Duffing oscillator given by Di Paola 1 is reanalysed by using the recursive formulas in this paper. The equivalent linearized damping and stiffness coefficients of the oscillator are 2~0w0 and w2 = a;2(1 + 0"9crZx), where ~0 = 1, w0 = 2. The excitation is nonstationary filtered white noise whose envelope function is e(t)= 0.906te -q3 and correlation function
.-.--
~;'~
I
/r "-6~" 0.01
! ,I . ~ t'"~[ / / ~
\
0.02
....
I (Numerical solution coincides with the accurate solution in this Figure) II(Eqs.(44)-(46)) n(Eqs.(50), (27)-(29))
// /
At =0.025 j 0.0 0.00
410
, 3.0
.... IV(Eqs.(44)-(46)) ---IV(Eq s.(50),(27)-(29)) A.t=0~25 , 5.0
15.0
N~X" ~.,.. 20.0
t
t~
Fig. 2. Centralized mean square responses of the oscillator in excitation environments I and II.
Fig. 3. Centralized mean square responses of the oscillator in excitation environments III and IV.
Structural random vibration analysis 2 Oxx 2.00
\ --
19
~1( x 0.01292) o2~2(x0.0050)
1.50
0.8 %
1.00
-
Linear
0.6
0.50 / 0.00 0.00
0.4
4.00
2.00
6.00
8.00
10.00
12.00
Fig. 4. Variance function of the Duffing oscillator. has the same form as that given in the Appendix, but the magnitude should be multiplied by 6"262. The parameters ~g and Wg are 0-6 and 57r, time step is 0.02. The results, which seem quite close to that given by Di Paola,1 are shown in Fig. 4. Example 2. A spring-body dynamic system with twodegree-of-freedom is shown in Fig. 5. Its structural parameters are Mass: m 1 = m2 -- 2.0 Stiffness: kl = k2 = 80.0 Damping: classical damping (a = 0.1,13 = 0.05) Load condition: two white noise excitations Sl(t) and S2(t ) = S l ( t - A T ) having identical power spectrum 1/27r act on the system, where A T is the time delay between the two excitations. Time step: At = 0"1 The stationary mean square responses of the structure with the variation of the time delay A T are calculated by using eqns (16)-(18), (25) and (27)-(29) and depicted in Fig. 6. Example 3. A bridge with four spans is excited by an SH earthquake wave (see Fig. 7) which propagates along the bridge. Only the lateral vibration of the bridge is considered. The structural characteristics of the bridge are:
00
01s
116
~T
Fig. 6. Mean square displacement responses of the dynamic system. Stiffness (x 109 N/m): k d = 3.6336, kc = 0.39375, kg : 2-200; Damping (× 106 N.s/m): Cd = 6"576, cc = 0"6551, Cg = 25"840. The earthquake wave is characterized by the following three functions: (1) correlation function
where er~ is the mean square ground displacement. B relates to the wave number of the earthquake wave and is chosen as 0.16 in this example. (2) decaying function fd(x)
= e
where 7 is the decay rate of the earthquake wave, which is 0"005 in this example, x refers to the distance away from the point of interest. (3) envelope function. It is the same as that given in the bottom row of Table 1. With the above three functions, the correlation function of the earthquake wave that propagates at any two supports of the bridge can be expressed as:
Rfij( ti, 7-) = e( ti) e-~la°l Rf ( 7- - dij / V )e( ti -t- 7- - dij / V ),
Mass (x 106 kg): me = 0"2975, md = 0"595, m3 = 0"818, my = 1.214;
////////
k~
Tx2
X Fig. 5. A spring-body dynamic system with two-degree-of-
freedom.
where d U is the distance from support i to support j, and the reference time is set at support i. V is the propagation velocity of the earthquake wave. The relative displacement mean-square-root responses of the beams and columns numbered from 1 to 6 are calculated according to eqns (16)-(19), (27)-(29) and (50). The time step is 0.02. Both stationary and nonstationary responses are carried out. The results are shown in Figs 8 and 9. In Fig. 8, the standard mean-square-root responses or, are calculated in the idealized case that V ~ oo, i.e. the earthquake wave simultaneously acts on the five supports of the bridge. The results are listed in Table 2. Figure 8 indicates that the first mode of the bridge, which is approximately symmetric half sinusoidal wave,
Dongyao Tan, Qingshan Yang
20 me
md 1
me
2
kd
3
.
v__ 40.0
40.0
*,
4'
40.0
L =
,1"
40.0
A"
Fig. 7. The model of the bridge. is not favourably excited out by the earthquake wave which propagates at velocity 1000(m/s). This means that in this case the phases of the first mode and the earthquake wave are opposite. So, we can estimate that the natural period of the bridge is about 0-16(s). When the bridge is excited by the earthquake wave which 1.01 a/*, \
--
1 3
--
'\ 0.8 ' ~ \ , ' , , \ ',,\, \.
4 5
0.4 0.00
CONCLUSIONS This paper applies the fundamental ¢3-recursive formula in discrete analysis method for random vibration to structural random vibration problems. The recursive formulas for practically calculating the mean and mean square responses of structures subjected to four typical excitation environments which are often encountered in engineering are derived. Some programming techniques which can enhance the computational efficiency are also introduced. In the recursive formulas for practical
/ ~-~-...
///
\\\
0:(M
0.08
propagates at a velocity between 400(m/s) and 1000(m/s), the responses ratio of columns 4 and 6 are approximately equal. This means that the second mode of the bridge, which is approximately asymmetric full sinusoidal wave, is excited out. So, we can estimate that the period of the second mode is about 0.14(s).
0.'12
0.'16
L/V
2. Standard relative displacement mean-square-roots
Table
Elements Fig. 8. Stationary relative displacement mean-square-root responses with variation of V.
as/trg
1
2
3
4
5
6
0'41
0"06
0"41
1.52
1'80
1"52
~.:(x 10-3) 5.0
\,',.,,, o.o
3.o
i
5.0
13.0
20.0 - t
0-2(x I0-l) 2.0 f.
\
!
I
- -
\
....
\
f
"x \ \ \
4 5
-----6
1.0
0.0
3.0
5.0
13.0
20.0 t
Fig. 9. Transient displacement mean square responses of the bridge.
21
Structural random vibration analysis
calculation, the symmetric and banded-shape structural dynamic characteristic matrices participate in the calculations directly so that the computational time and storage are saved and the computational efficiency is increased. This resursive formula can also be applied, without special difficulties, to the calculation for the auto-correlation function matrix of structural random responses. But this is out of the scope of this paper. It will be discussed in our following paper.
REFERENCES 1. M. D. Paola, M. Ioppolo and G. Muscolino, Stochastic seismic analysis of Multi-degree-of-freedom systems, Engineering Structures, 6, 113-118 (1984). 2. M. Hoshiya, K. Ishii and S. Nagata, Recursive covariance of structural responses, J. Engng Mech., Proc., ASCE 110(12), 1743-1755 (1984). 3. Y. K. Lin, Ruichong Zhang and Yan Yong, Multiply supported pipeline under seismic wave excitations, Journal of Engng Mechanics, 116(5), 1094-1108 (1990). 4. Federico Perotti, Structural response to non-stationary multiple-support random excitation, Earthquake Engng Structural Dynamics, 19, 513-527 (1990). 5. C. W. S. To, Direct integration operators and their stability for random response of multidegree of freedom systems, Computers and Structures, 30, 865-874 (1988). 6. C. W. S. To, An implicit direct integrator for random response of multidegree of freedom systems, Computers and Structures, 33, 73-77 (1989). 7. Dongyao Tan and Hua Guo, A procedure of discrete analysis for nonstationary random vibration, Earthquake Engng Engng Vibr., 6(3), 12-18 (1986). 8. Dongyao Tan, Qingshan Yang and Chen Zhao, Discrete analysis method for random vibration of structures subjected to spatially correlated filtered white noises, Computers and Structures, 43(6), 1051-1056 (1992). 9. Dongyao Tan, Qingshan Yang and Pei Lu, A proof of the stability of the discrete analysis method for random vibration, J. of Harbin Architectural and Civil Engineering Institute, 1, 12-23 (1988). 10. Dongyao Tan, Discrete analysis method of random vibration, ACTA MECHANICA SINICA, 24(4), 473479 (1992).
11. Dongyao Tan and Qingshan Yang, Discrete analysis method for random vibration of structures under excitations correlated in time domain, Earthquake Engng Engng Vibr, 10(2), 37-46 (1990).
APPENDIX 1. The closed-form solution of the transient displacement mean value response of the oscillator $ ( t ) = ~oo 1-e-~°~°t c°sv/1-~°2w°t
+ /1~°_ ~o2 sin ~/~ ----~°2w°t)1 " 2. The closed-form solution of the transient centralized mean square responses of the oscillator in excitation environment I. 1 a2xx(t)= ~
{ 1
e -2~°w°t [ Joo--~o2) wo2(1-~o2)
+ 2(~oWosin ~/1 - ~2oWt)2 + wo2~oV/1 - ,o2sin 2V/1 ~o2----Wot] }. 3. The correlation function of the excitation acting on the oscillator Rf(r)=
[1 +4~ 2 cos V/1-~gwgr 4 e _~g~g,[----~g
1 - 4(~ sin ~ 1 -AI-~/1__~2 -~203gT] "