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Discrete analysis method for random vibration of structures subjected to spatially correlated filtered white noises
Cornpurerr d Structures Vol. 43, No. 6, pp. 1051-1056. 1992 Printed in Great Britain.
0
OW5-?949/92 S5.00 + 0.00 1992 Pergamon Press Ltd
DISCRETE ANALYSIS METHOD FOR RANDOM VIBRATION OF STRUCTURES SUBJECTED TO SPATIALLY CORRELATED FILTERED WHITE NOISES D~NGYAO
TAN, QINGSHAN YANG and CHENZHAO
Harbin Architectural and Civil Engineering Institute, P.O. Box 340, Harbin 150006, P.R. China (Received 24 May 1991)
Abstract-Random vibration analysis of large-span space structures or high-rise structures which are subjected to spatially correlated filtered white noise excitations such as wind load and earthquake motion, has been a difficult problem in engineering computation. Based on the idea of the discreteanalysismethod of random vibration, this paper attempts to solve this problem. The formulae of calculating structural mean and mean square responses are given. As an example, the wind-induced vibration of a cable roof structure is analysed by using these formulae.
NOTATION load area of jth node of a cable roof structure structural damping matrix coefficient matrix in structural state equation pressure coefficient at node j variable in power spectrum function of gust wind velocity filtered white noises equivalent wind load perpendicularly adding on node j identity matrix ,. structural stiffness matrix coefficient matrix in recurrence Tormulae roughness coefficient of the ground structural mass matrix coefficient matrix in structural state equation number of time step observation vector of a filter coefficient matrix in structural state equation matrix which describes the position of the excitations spatial correlation matrix of white noises ith row and jth column element in Q, spatial correlation matrix of the excitations ith row and jth column element in R, correlation function matrix of white noises correlation matrix of responses represented by subscripts a and b excitation vector and its mean given power spectrum of a filtered white noise power spectrum of a generated filtered white noise power spectrum matrix of spatially correlated fIltered white noises coetBcient matrix in structural state equation characteristic matrix of a filter wind speed at node j mean wind speed at 10 meters high a white noise process 1051
x(r), x’(t), x”(r) structural displacement, velocity and acceleration vectors mean responses of structural displacement and velocity state vector of a filter and its derivative x,(0. x;(t) structural state vector and its derivative v(r), v’(r) mean response of structural state vector Y coefficient matrix in state equation of the z filter delta function a(r) density of the air P frequency Zr time step transpose of a matrix or a vector time instance at n Ar INTRODUCI’ION In the investigation of structural random vibration, there have been numerous research reports, state-ofthe-art articles and books which discussed or introduced various analysis methods. Although a great number of problems in structural random vibration have satisfactory solutions, the random vibration analysis for large-span space structures and high-rise structures which are subjected to wind load or multidimensional earthquake motions input from different sports is still a very difficult problem in practical engineering computation. This is because the random vibration analysis methods currently used in engineering computation are frequency-domain methods which can not efficiently carry out the random response analysis for a large-span space structure whose natural frequencies are densely distributed, or for a high-rise structure whose bend, shear and twist deformations are coupled together. The discrete analysis method of random vibration is a numerical method for calculating the stationary or non-stationary random vibration responses. This method depends on the recurrence formulae obtained by discretizing the vibration equation ‘in the time
1052
DONGYAOTAN et
domain, and uses the mean process and correlation characteristics of the excitations to calculate the mean and mean square random vibration responses directly [l]. With this idea, this paper discusses the random vibration analysis for structures which are excited by spatially correlated filtered white noises. The practical calculation formulae are derived. They can be used to solve the above problems efficiently. BASIC FORMULAE
OF CALCULATION
Structural vibration equation can be expressed as:
mx”(t) + cx’(t) + kx(t) = qx(t),
al. GENERATION OF SPATIALLY CORRELATED FILTERED WHITE NOISES
This section discusses how to generate the filtered white noises whose spatial correlativity is given from the white noises whose spatial correlativity is going to be determined. Only after such generation has been made, could the formulae of calculating structural mean and mean square responses given above be used. The power spectrum matrix of the spatially correlated filtered white noises which are going to be generated has the following expression
(1) S&J) = R&c-i),
where m, c, k are structural mass, damping and stiffness matrices, respectively. x(t), x’(r), x”(t) are structural displacement, velocity and acceleration vectors. s(t) is the excitation vector, q is a matrix which describes the position of the excitations. Transform eqn (1) into state equations as follows:
y’(t) = M-G(r)
+ M-‘Us(t),
(2)
where
where R, is a matrix which represents the spatial correlativity of the excitations. S(o) is the identical power spectrum for each excitation, such as the well-known Davenport’s wind velocity spectrum [3] in wind engineering and Tajimi’s earthquake acceleration spectrum in earthquake engineering. A filtered white noise process f;(t) having power spectrum S(o) can be generated from a white noise process w,(t) by using an appropriate filter. The properties of the filter are usually described by a linear differential equation x$(t) = u/x8(r) + zw,(r),
and y(t) is a state vector [xr(r), x”(r)lT, U is [Or, q7’. According to the results of [l] and [2], the unconditionally stable recurrence formulae which are able to calculate the accurate structural stationary mean and mean square responses are
(6)
where the spectral density of w,(r) is 1/2x, v, is a matrix which describes the characteristics of the filter, z is a constant vector, xfi is the state vector of the filter. The generated filtered white noise is an observation of the filter, which can be expressed as follows: f;(t) = p+(r),
jj(n) = [2(M - 0.5 Arc)-IM
(5)
(7)
- Ijy(n - 1)
+ Ar(M - 0.5 Arc)-’ R,,(n) = [2(M - 0.5 AK)-‘M x [2(M - 0.5 Arc)-‘M
Us(n - 0.5)
(3)
- I]R,,(n - 1)
Sdo) = ]p(ioZ - v,)-‘z (2,
- IjT
+Ar(M - 0.5 Arc)-’ x UQo U=(M - 0.5 AtC)-?
where p is called the observation vector. The power spectrum of a generated filtered white noise process can be derived by using eqns (6) and (7) as follows:
(4)
Here, the random excitations are assumed to be the stationary white noises. f(r) and R,(r) = Q, d(7) are their mean processes and auto-correlation function matrix, respectively. j(n) and Ry,(n) are structural mean and mean square responses. If the random excitations are not white noises, but filtered white noises, they should be generated from white noise first, and eqns (3) and (4) can then be used.
(8)
where I is an identity matrix. Therefore, the elements in the characteristic matrix vf of the filter and in its observation vector p should be selected appropriately to make the power spectrum of the generated filtered white noise process S’(o) approach its given power spectrum S(o) as close as possible. According to eqns (6) and (7), the stationary correlation response Rh4 of any two generated filtered white noises A(r) and f;(t) satisfies the following equations (9)
Discrete analysis method for structural random vibration
where R,,, is the stationary correlation response of the filters i and i which satisfies the Lyapunov equation [4] Of%,
+ R,fl,$- = zQdi,i)z
‘,
(10)
where QD(i,j) is the cross-correlation intensity of two white noises w,(t) and w,(t). Since R,& is equal to the element in matrix R, at ith row and jth column, i.e. R/J, = Wj),
f(n) = a(n - 1) + 0.5 At?(n),
PRACITCAL CALCULATION FOR STRUCTURAL RANDOM RESPONSES
Since the calculation for the inverse of (M - 0.5 Arc) in eqns (3) and (4) is quite time-consuming for its widely banded, unsymmetric and order doubled disadvantages comparing with structural dynamic characteristic matrices, it is necessary to make an appropriate transformation for it so that the banded shape and symmetry properties of structural dynamic characteristic matrices can be used and the computation efficiency of discrete analysis method of structural random vibration is improved.
(12.2)
where a(n - 1) = n(n - 1) + 0.5 Atf’(n - 1)
(12.3)
b(n - 1) = -0.5 AtkZ(n - 1) + (m - 0.5 Arc) x n’(n - 1) + Atq.F(n - 0.5)
(11)
the spatial correlation matrix Q, of the white noises can then be calculated from the given spatial correlation matrix R, of the filtered white noises by using eqns (9)-(11).
1053
K = m + 0.5 Ate + 0.25 At2k,
(12.4) (12.5)
S(n - 0.5) is the mean values of the filtered white noises at time instance (n - 0.5) At. Practical formula for calculating structural mean square responses Combine eqns (2), (6) and (7), the coupled equation of structural dynamic equation with the state equation of the filters is
[
IL 1 v’(t)’
M
1
q(t),
= c
1 L
where P
Practical formula for calculating structural mean responses It can be seen from eqn (3) that structural mean responses only relate to the mean processes of the excitations. Hence, the calculation formulae for mean responses of the structures subjected to filtered white noises are identical with those used for the structures subjected to white noises. The practical formulae for calculating structural mean responses given in [S] can then be directly applied here. They are Z’(n) = K-‘[b(n
- 1) - 0.5 Atka(n - l)]
R,,(n) R,,,(n) -_ 2(M - 0.5 Arc)-‘M Rx,&)
R,,,,(n) I[
(12.1)
- I
P = [ 3
v= [
Apply eqn (4) to eqn (13), structural mean square responses can be calculated as follows:
At(M - 0.5 Arc)-‘UP(I
1
- 0.5 AtV)-’
1) 1) -1) R&J -1) 'Rx,& R,,(n R,.x,(n 1 2(1- 0.5 AtV)-’ -I
2(M - 0.5 Arc)-‘M
X
+At (M x(M [
0.5 Arc)-’
[
0.5 Arc)-’
-I
At(h4 - 0.5 Arc)-‘UP(I 2(1- 0.5 AtV)-’ -I
0.5 At(M - 0.5 Arc)-‘UP(Z (I - 0.5 AtV)-’ 0.5 At(M - 0.5 Arc)-‘UP(I (I-0.5AtV)-’
1 ILZQLI I.
- 0.5 AtV)-’
- 0.5 AtV)-’
- 0.5 AtV)-’
T
0
=
(14)
1054
Dot4OV~0
After R,,,(n) is obtained from the above equation explicitly, the following equation can be derived by multiplying both of its sides from left and right by (A4 - 0.5 AK) and its transpose, respectively (M - 0.5 AtC)R,,(n)(M
+At(M
r,(n - 1) = [ - 0.5 ArkR,(n
- 1)](2D - I)T
- 1)(2D - Z)r
+0.5 Ar2qPDZQ,[DZ]?
- l)(M + 0.5 ArC)r
+0.5 ArC)R,(n
- 1)
+(m - 0.5 Arc)R,,,,(n +ArqPDR,x,(n
(20.2)
According to the solution of mean square responses of structures which are subjected to white noises given in [5], the practical formulae for calculating eqn (15) are
- l)(M + 0.5 AtC)r
+Ar2UPD[Rx,x,(n
et al
- 0.5 AtC)r
=(A4 + 0.5 AtC)RJn +ArUPDR,,(n
TAN
- l)[UPDlr
- 1) R,,r(n) = K-‘[0.25 Ar2r,(n - 1)
+0.25 ArZQ,Zrl[UPD]‘,
(15) +0.5 Arrr(n - l)(m + 0.5 Arc)
where D is defined as +0.5 Ar(m + 0.5 Arc)r,(n - 1)
D = (I - 0.5 AtV)-’ -I
r(I - 0.5 Am,)-’ =
+ (m + 0.5 Arc)r, (n - 1)
‘..
L
x (m + 0.5 Arc)]K-’
(I - 0.5 Atv,)-’
Similarly, multiply from R&r) (M - 0.5 AK) after it has been obtained (14) explicitly, there is
left by from eqn
x
(M - 0.5 ArC)R,,(n) =(M + 0.5 AtC)R,,,(n +ArUPDR,,,,(n
(21)
rr(n - 1) + &
1
(m + 0.5 Arc)r,(n - 1)
- 1)(2D - I)r
- 1)(2D - f)’
+0.5 Ar2CJPDZQDZTDr. Finally, the equation for calculating R,, be derived from eqn (14)
(16) (n ) can also
-&
[r2(n - 1) - 0.5 Arr,(n - l)k]K-‘,
(23)
where R,,,,(n) = (20 - 0R,,,,(n
- l)(2D - 1)’ + ArDZQ,ZTDr.
(17)
Since equation (16) is similar to the equation used in calculating the mean responses for structures, the results in [5] can then be used and the practical formulae for calculating R,,x,(n) are
rl(n - 1) = R,,(n - 1) + 0.5 ArR,.,(n - 1) + 0.5 ArR$,(n - 1) -t 0.25 Ar’R,,,,(n - 1) (24.1) r2(n - 1) = -0.5 ArR,,(n - 1)k + R,,,(n - l)(m - 0.5 Arc)
R,,,(n)
= K-‘[r,(n - 1) - 0.5 Atkr,(n - l)]
(18)
R,,(n)
= r&r - 1) + 0.5 ArR.,(n),
(19)
- 0.25 Ar2R~,(n - 1)k +0.5 ArR,.,,(n - l)(m - 0.5 Arc)
where
+Ar(R,,(n
r&r - 1) = [R,,,(n - 1) +0.5 ArR,,,,(n - 1)](2D - I)
(20.1)
x WV7
- 1) + 0.5 ArR,,
(n - l)] (24.2)
1055
Discrete analysis method for structural random vibration
rj(n - 1) = 0.25 At%R,,(n -0.5 ArkR,,(n
- 1)k - l)(m - 0.5 Arc)
-0.5 At(m - 0.5 Atc)Rs,(n
- 1)k
+(m - 0.5 Atc)R,.,.(n - l)(m - 0.5 Ate) +AtqPD[Rs,(n -0.5 AtR&(n
- l)(m - 0.5 Arc) - l)k]
+At[(m - 0.5 Atc)R,,,(n -0.5
AtkR,,(n
- 1)
- l)][qPD]’
+ArZqPD[Rx,x,(n - 1) (24.3)
+0.25 ArZQ,Z7[qPD]‘.
It can be seen from eqns (12) and (18)-(24) that structural dynamic characteristic matrices participate in the calculation for structural mean and mean square responses directly, so their banded shape and symmetry properties are efficiently used and the requirement for practical calculation is satisfied. WIND-INDUCED RANDOM VIBRATION OF A CABLE ROOF STRUCTURE
The wind loads which act on a large-span space structure is a typical random excitations with spatial correlativity. In this section, a hypherbolic paraboliod cable roof structure subjected to wind blowing is taken as an example to demonstrate the application of the discrete analysis method of random vibration to such complicated engineering computation. The cable roof structure is shown in Fig. 1, and structural parameters are listed in Table 1. The power spectrum of gust wind velocity is selected as the one given by Davenport [3]. It is
Fig. I(c). Vertical view of the cable roof structure.
S(o)=~-,4k Go f 0 (1 +fY3 where f = 60O~/n Vi,, , o is frequency, k, is the roughness coefficient of the ground which is 0.189 for urban areas in this example. VI0 is the mean velocity at 10 m high, 25 m/set is used in the computation to simulated the velocity of X degree wind. The equivalent wind loads which act perpendicularly on the nodes of the cable roof structure can be expressed as R,(t) = PCjA,Vi,e/(t), where p = 1.25 kg/m* is the density of air, U,(t) is the velocity of gust wind at nodej, A, is the load area of nodej, C, is the pressure coefficient at node j which has been measured in wind tunnel test. Their distribution on the surface of the cable roof is shown in Fig. 2. The ith row and jth column element in the matrix which represents the spatial correlativity of wind loads is calculated by following equation R,(i, j) = exp _ [8(x, - x,)2 + 1601,- yj)* + 1qz, - Zj)70.-’ VI0
{ c
w
1OOm
where (xi, yi, z,) and (x,, JJ,, z,) are coordinates of two nodes on the cable roof structure.
Fig. l(a). Front view of the cable roof structure.
Table 1. Structural parameters
Sai3
-
1OOm Fig. I(b). Side view of the cable roof structure.
,
1
Height Span Cross-sectional area of cables Young’s modulus for cables Roof weight Area of static load Cable pretension Number of degrees of freedom
15m 30m 1OOm 3ooomm2 200 kN/mm* 1 kN/m* 100m*/joint 5000kN 123
1056
DONGYAO TAN er al. -0.5 -1.1
C
B
-1.5
D N
Fig. 2(a). Pressure coefficient distribution when wind blows in direction I.
A
Fig. 2(b). Pressure coefficient distribution when wind blows in direction II.
Table 2. Square roots of vertical node displacement on cables AC and BD Node number cmi (cm)
I II
Node number Qld (cm)
I II
I
13
19
25
31
31
43
49
55
4.67 1.84
6.64 3.26
9.66 5.00
10.87 6.86
11.90 8.04
10.87 8.34
9.66 7.33
6.64 5.69
4.67 3.78
11
16
21
26
31
36
41
46
51
2.30 1.58
3.09 2.52
4.89 4.33
8.39 6.68
11.90 8.04
10.09 6.28
6.60 4.07
3.98 2.51
2.35 1.63
In this example, a filter which is described by following second order linear differential equation is used to generate the required wind velocity processes whose power spectrum is given above
f(t) = [0, 390.161 ;;;; [
-G,(t) [ I[
1
0
G,(t)
=
-0.855
x lo-’
1 1
-0.268
where w(t) is a white noise process whose autocorrelation function is S(r). The relative error between the power spectrum of the generated wind velocity process and the Davenport’s power spectrum is less than 10e3. According to eqns (9) and (1 1), the spatial correlation matrix Q, of the white noise processes can be calculated with the spatial correlation matrix R,of the gust wind as follows: QD = 5.03 x lo-‘R,. -
u”*(r)
Aoc Fig. 3. The mean square responses in vertical direction of cables AC and ED.
The stationary mean square responses of the cable roof structure is obtained. Some of the results are listed in Table 2 and shown in Fig. 3. SUMMARY
The discrete analysis method of random vibration for structures which are subjected to spatially correlated filtered white noise is discussed in this paper. The practical formulae for calculating structural mean and mean square responses are given. The advantage of discrete analysis method over the frequency-domain methods which are presently used in engineering computation is that it can calculate the accurate stationary mean and mean square responses with relatively less computation time for large-span space structures whose natural frequencies are densely distributed or for high-rise structures whose bend, shear and twist deformations are coupled together. The purpose of giving the numerical example of analysing the wind induced random vibration of a cable roof structure in this paper is just to demonstrate this advantage. The obtained results are satisfactory and provide more information about the random vibration behaviours of the cable roof structure. REFERENCES
1. Tan Dongyao and Guo Hua, A procedure of discrete analysis for nonstationary random vibration. Eurrhquake Engng Engng Vibr. 3, 12-18 (1986). cI Tan Dontzvao. Discrete analysis method of random vibration.-.&& me& sin. (to be published). H. A. Buchholdt, An Introduction to Cable Roof Structures. Cambridge University Press (1985). N. C. Nigam, Introduction to Random Vibrations. MIT Press, Cambridge, MA (1983). Tan Dongyao and Yang Qingshan, The practical form of the discrete analysis method of random vibration. J. Harbin Architectural Civ. Engng Inst. 2, 21-37 (1989). 6. Tan Dongyao and Yang Qingshan, Discrete analysis method for random vibration of structures under excitations correlated in time domain. Earthquake Engng Engng Vibr. 2, 3746 (1990).
0
OW5-?949/92 S5.00 + 0.00 1992 Pergamon Press Ltd
DISCRETE ANALYSIS METHOD FOR RANDOM VIBRATION OF STRUCTURES SUBJECTED TO SPATIALLY CORRELATED FILTERED WHITE NOISES D~NGYAO
TAN, QINGSHAN YANG and CHENZHAO
Harbin Architectural and Civil Engineering Institute, P.O. Box 340, Harbin 150006, P.R. China (Received 24 May 1991)
Abstract-Random vibration analysis of large-span space structures or high-rise structures which are subjected to spatially correlated filtered white noise excitations such as wind load and earthquake motion, has been a difficult problem in engineering computation. Based on the idea of the discreteanalysismethod of random vibration, this paper attempts to solve this problem. The formulae of calculating structural mean and mean square responses are given. As an example, the wind-induced vibration of a cable roof structure is analysed by using these formulae.
NOTATION load area of jth node of a cable roof structure structural damping matrix coefficient matrix in structural state equation pressure coefficient at node j variable in power spectrum function of gust wind velocity filtered white noises equivalent wind load perpendicularly adding on node j identity matrix ,. structural stiffness matrix coefficient matrix in recurrence Tormulae roughness coefficient of the ground structural mass matrix coefficient matrix in structural state equation number of time step observation vector of a filter coefficient matrix in structural state equation matrix which describes the position of the excitations spatial correlation matrix of white noises ith row and jth column element in Q, spatial correlation matrix of the excitations ith row and jth column element in R, correlation function matrix of white noises correlation matrix of responses represented by subscripts a and b excitation vector and its mean given power spectrum of a filtered white noise power spectrum of a generated filtered white noise power spectrum matrix of spatially correlated fIltered white noises coetBcient matrix in structural state equation characteristic matrix of a filter wind speed at node j mean wind speed at 10 meters high a white noise process 1051
x(r), x’(t), x”(r) structural displacement, velocity and acceleration vectors mean responses of structural displacement and velocity state vector of a filter and its derivative x,(0. x;(t) structural state vector and its derivative v(r), v’(r) mean response of structural state vector Y coefficient matrix in state equation of the z filter delta function a(r) density of the air P frequency Zr time step transpose of a matrix or a vector time instance at n Ar INTRODUCI’ION In the investigation of structural random vibration, there have been numerous research reports, state-ofthe-art articles and books which discussed or introduced various analysis methods. Although a great number of problems in structural random vibration have satisfactory solutions, the random vibration analysis for large-span space structures and high-rise structures which are subjected to wind load or multidimensional earthquake motions input from different sports is still a very difficult problem in practical engineering computation. This is because the random vibration analysis methods currently used in engineering computation are frequency-domain methods which can not efficiently carry out the random response analysis for a large-span space structure whose natural frequencies are densely distributed, or for a high-rise structure whose bend, shear and twist deformations are coupled together. The discrete analysis method of random vibration is a numerical method for calculating the stationary or non-stationary random vibration responses. This method depends on the recurrence formulae obtained by discretizing the vibration equation ‘in the time
1052
DONGYAOTAN et
domain, and uses the mean process and correlation characteristics of the excitations to calculate the mean and mean square random vibration responses directly [l]. With this idea, this paper discusses the random vibration analysis for structures which are excited by spatially correlated filtered white noises. The practical calculation formulae are derived. They can be used to solve the above problems efficiently. BASIC FORMULAE
OF CALCULATION
Structural vibration equation can be expressed as:
mx”(t) + cx’(t) + kx(t) = qx(t),
al. GENERATION OF SPATIALLY CORRELATED FILTERED WHITE NOISES
This section discusses how to generate the filtered white noises whose spatial correlativity is given from the white noises whose spatial correlativity is going to be determined. Only after such generation has been made, could the formulae of calculating structural mean and mean square responses given above be used. The power spectrum matrix of the spatially correlated filtered white noises which are going to be generated has the following expression
(1) S&J) = R&c-i),
where m, c, k are structural mass, damping and stiffness matrices, respectively. x(t), x’(r), x”(t) are structural displacement, velocity and acceleration vectors. s(t) is the excitation vector, q is a matrix which describes the position of the excitations. Transform eqn (1) into state equations as follows:
y’(t) = M-G(r)
+ M-‘Us(t),
(2)
where
where R, is a matrix which represents the spatial correlativity of the excitations. S(o) is the identical power spectrum for each excitation, such as the well-known Davenport’s wind velocity spectrum [3] in wind engineering and Tajimi’s earthquake acceleration spectrum in earthquake engineering. A filtered white noise process f;(t) having power spectrum S(o) can be generated from a white noise process w,(t) by using an appropriate filter. The properties of the filter are usually described by a linear differential equation x$(t) = u/x8(r) + zw,(r),
and y(t) is a state vector [xr(r), x”(r)lT, U is [Or, q7’. According to the results of [l] and [2], the unconditionally stable recurrence formulae which are able to calculate the accurate structural stationary mean and mean square responses are
(6)
where the spectral density of w,(r) is 1/2x, v, is a matrix which describes the characteristics of the filter, z is a constant vector, xfi is the state vector of the filter. The generated filtered white noise is an observation of the filter, which can be expressed as follows: f;(t) = p+(r),
jj(n) = [2(M - 0.5 Arc)-IM
(5)
(7)
- Ijy(n - 1)
+ Ar(M - 0.5 Arc)-’ R,,(n) = [2(M - 0.5 AK)-‘M x [2(M - 0.5 Arc)-‘M
Us(n - 0.5)
(3)
- I]R,,(n - 1)
Sdo) = ]p(ioZ - v,)-‘z (2,
- IjT
+Ar(M - 0.5 Arc)-’ x UQo U=(M - 0.5 AtC)-?
where p is called the observation vector. The power spectrum of a generated filtered white noise process can be derived by using eqns (6) and (7) as follows:
(4)
Here, the random excitations are assumed to be the stationary white noises. f(r) and R,(r) = Q, d(7) are their mean processes and auto-correlation function matrix, respectively. j(n) and Ry,(n) are structural mean and mean square responses. If the random excitations are not white noises, but filtered white noises, they should be generated from white noise first, and eqns (3) and (4) can then be used.
(8)
where I is an identity matrix. Therefore, the elements in the characteristic matrix vf of the filter and in its observation vector p should be selected appropriately to make the power spectrum of the generated filtered white noise process S’(o) approach its given power spectrum S(o) as close as possible. According to eqns (6) and (7), the stationary correlation response Rh4 of any two generated filtered white noises A(r) and f;(t) satisfies the following equations (9)
Discrete analysis method for structural random vibration
where R,,, is the stationary correlation response of the filters i and i which satisfies the Lyapunov equation [4] Of%,
+ R,fl,$- = zQdi,i)z
‘,
(10)
where QD(i,j) is the cross-correlation intensity of two white noises w,(t) and w,(t). Since R,& is equal to the element in matrix R, at ith row and jth column, i.e. R/J, = Wj),
f(n) = a(n - 1) + 0.5 At?(n),
PRACITCAL CALCULATION FOR STRUCTURAL RANDOM RESPONSES
Since the calculation for the inverse of (M - 0.5 Arc) in eqns (3) and (4) is quite time-consuming for its widely banded, unsymmetric and order doubled disadvantages comparing with structural dynamic characteristic matrices, it is necessary to make an appropriate transformation for it so that the banded shape and symmetry properties of structural dynamic characteristic matrices can be used and the computation efficiency of discrete analysis method of structural random vibration is improved.
(12.2)
where a(n - 1) = n(n - 1) + 0.5 Atf’(n - 1)
(12.3)
b(n - 1) = -0.5 AtkZ(n - 1) + (m - 0.5 Arc) x n’(n - 1) + Atq.F(n - 0.5)
(11)
the spatial correlation matrix Q, of the white noises can then be calculated from the given spatial correlation matrix R, of the filtered white noises by using eqns (9)-(11).
1053
K = m + 0.5 Ate + 0.25 At2k,
(12.4) (12.5)
S(n - 0.5) is the mean values of the filtered white noises at time instance (n - 0.5) At. Practical formula for calculating structural mean square responses Combine eqns (2), (6) and (7), the coupled equation of structural dynamic equation with the state equation of the filters is
[
IL 1 v’(t)’
M
1
q(t),
= c
1 L
where P
Practical formula for calculating structural mean responses It can be seen from eqn (3) that structural mean responses only relate to the mean processes of the excitations. Hence, the calculation formulae for mean responses of the structures subjected to filtered white noises are identical with those used for the structures subjected to white noises. The practical formulae for calculating structural mean responses given in [S] can then be directly applied here. They are Z’(n) = K-‘[b(n
- 1) - 0.5 Atka(n - l)]
R,,(n) R,,,(n) -_ 2(M - 0.5 Arc)-‘M Rx,&)
R,,,,(n) I[
(12.1)
- I
P = [ 3
v= [
Apply eqn (4) to eqn (13), structural mean square responses can be calculated as follows:
At(M - 0.5 Arc)-‘UP(I
1
- 0.5 AtV)-’
1) 1) -1) R&J -1) 'Rx,& R,,(n R,.x,(n 1 2(1- 0.5 AtV)-’ -I
2(M - 0.5 Arc)-‘M
X
+At (M x(M [
0.5 Arc)-’
[
0.5 Arc)-’
-I
At(h4 - 0.5 Arc)-‘UP(I 2(1- 0.5 AtV)-’ -I
0.5 At(M - 0.5 Arc)-‘UP(Z (I - 0.5 AtV)-’ 0.5 At(M - 0.5 Arc)-‘UP(I (I-0.5AtV)-’
1 ILZQLI I.
- 0.5 AtV)-’
- 0.5 AtV)-’
- 0.5 AtV)-’
T
0
=
(14)
1054
Dot4OV~0
After R,,,(n) is obtained from the above equation explicitly, the following equation can be derived by multiplying both of its sides from left and right by (A4 - 0.5 AK) and its transpose, respectively (M - 0.5 AtC)R,,(n)(M
+At(M
r,(n - 1) = [ - 0.5 ArkR,(n
- 1)](2D - I)T
- 1)(2D - Z)r
+0.5 Ar2qPDZQ,[DZ]?
- l)(M + 0.5 ArC)r
+0.5 ArC)R,(n
- 1)
+(m - 0.5 Arc)R,,,,(n +ArqPDR,x,(n
(20.2)
According to the solution of mean square responses of structures which are subjected to white noises given in [5], the practical formulae for calculating eqn (15) are
- l)(M + 0.5 AtC)r
+Ar2UPD[Rx,x,(n
et al
- 0.5 AtC)r
=(A4 + 0.5 AtC)RJn +ArUPDR,,(n
TAN
- l)[UPDlr
- 1) R,,r(n) = K-‘[0.25 Ar2r,(n - 1)
+0.25 ArZQ,Zrl[UPD]‘,
(15) +0.5 Arrr(n - l)(m + 0.5 Arc)
where D is defined as +0.5 Ar(m + 0.5 Arc)r,(n - 1)
D = (I - 0.5 AtV)-’ -I
r(I - 0.5 Am,)-’ =
+ (m + 0.5 Arc)r, (n - 1)
‘..
L
x (m + 0.5 Arc)]K-’
(I - 0.5 Atv,)-’
Similarly, multiply from R&r) (M - 0.5 AK) after it has been obtained (14) explicitly, there is
left by from eqn
x
(M - 0.5 ArC)R,,(n) =(M + 0.5 AtC)R,,,(n +ArUPDR,,,,(n
(21)
rr(n - 1) + &
1
(m + 0.5 Arc)r,(n - 1)
- 1)(2D - I)r
- 1)(2D - f)’
+0.5 Ar2CJPDZQDZTDr. Finally, the equation for calculating R,, be derived from eqn (14)
(16) (n ) can also
-&
[r2(n - 1) - 0.5 Arr,(n - l)k]K-‘,
(23)
where R,,,,(n) = (20 - 0R,,,,(n
- l)(2D - 1)’ + ArDZQ,ZTDr.
(17)
Since equation (16) is similar to the equation used in calculating the mean responses for structures, the results in [5] can then be used and the practical formulae for calculating R,,x,(n) are
rl(n - 1) = R,,(n - 1) + 0.5 ArR,.,(n - 1) + 0.5 ArR$,(n - 1) -t 0.25 Ar’R,,,,(n - 1) (24.1) r2(n - 1) = -0.5 ArR,,(n - 1)k + R,,,(n - l)(m - 0.5 Arc)
R,,,(n)
= K-‘[r,(n - 1) - 0.5 Atkr,(n - l)]
(18)
R,,(n)
= r&r - 1) + 0.5 ArR.,(n),
(19)
- 0.25 Ar2R~,(n - 1)k +0.5 ArR,.,,(n - l)(m - 0.5 Arc)
where
+Ar(R,,(n
r&r - 1) = [R,,,(n - 1) +0.5 ArR,,,,(n - 1)](2D - I)
(20.1)
x WV7
- 1) + 0.5 ArR,,
(n - l)] (24.2)
1055
Discrete analysis method for structural random vibration
rj(n - 1) = 0.25 At%R,,(n -0.5 ArkR,,(n
- 1)k - l)(m - 0.5 Arc)
-0.5 At(m - 0.5 Atc)Rs,(n
- 1)k
+(m - 0.5 Atc)R,.,.(n - l)(m - 0.5 Ate) +AtqPD[Rs,(n -0.5 AtR&(n
- l)(m - 0.5 Arc) - l)k]
+At[(m - 0.5 Atc)R,,,(n -0.5
AtkR,,(n
- 1)
- l)][qPD]’
+ArZqPD[Rx,x,(n - 1) (24.3)
+0.25 ArZQ,Z7[qPD]‘.
It can be seen from eqns (12) and (18)-(24) that structural dynamic characteristic matrices participate in the calculation for structural mean and mean square responses directly, so their banded shape and symmetry properties are efficiently used and the requirement for practical calculation is satisfied. WIND-INDUCED RANDOM VIBRATION OF A CABLE ROOF STRUCTURE
The wind loads which act on a large-span space structure is a typical random excitations with spatial correlativity. In this section, a hypherbolic paraboliod cable roof structure subjected to wind blowing is taken as an example to demonstrate the application of the discrete analysis method of random vibration to such complicated engineering computation. The cable roof structure is shown in Fig. 1, and structural parameters are listed in Table 1. The power spectrum of gust wind velocity is selected as the one given by Davenport [3]. It is
Fig. I(c). Vertical view of the cable roof structure.
S(o)=~-,4k Go f 0 (1 +fY3 where f = 60O~/n Vi,, , o is frequency, k, is the roughness coefficient of the ground which is 0.189 for urban areas in this example. VI0 is the mean velocity at 10 m high, 25 m/set is used in the computation to simulated the velocity of X degree wind. The equivalent wind loads which act perpendicularly on the nodes of the cable roof structure can be expressed as R,(t) = PCjA,Vi,e/(t), where p = 1.25 kg/m* is the density of air, U,(t) is the velocity of gust wind at nodej, A, is the load area of nodej, C, is the pressure coefficient at node j which has been measured in wind tunnel test. Their distribution on the surface of the cable roof is shown in Fig. 2. The ith row and jth column element in the matrix which represents the spatial correlativity of wind loads is calculated by following equation R,(i, j) = exp _ [8(x, - x,)2 + 1601,- yj)* + 1qz, - Zj)70.-’ VI0
{ c
w
1OOm
where (xi, yi, z,) and (x,, JJ,, z,) are coordinates of two nodes on the cable roof structure.
Fig. l(a). Front view of the cable roof structure.
Table 1. Structural parameters
Sai3
-
1OOm Fig. I(b). Side view of the cable roof structure.
,
1
Height Span Cross-sectional area of cables Young’s modulus for cables Roof weight Area of static load Cable pretension Number of degrees of freedom
15m 30m 1OOm 3ooomm2 200 kN/mm* 1 kN/m* 100m*/joint 5000kN 123
1056
DONGYAO TAN er al. -0.5 -1.1
C
B
-1.5
D N
Fig. 2(a). Pressure coefficient distribution when wind blows in direction I.
A
Fig. 2(b). Pressure coefficient distribution when wind blows in direction II.
Table 2. Square roots of vertical node displacement on cables AC and BD Node number cmi (cm)
I II
Node number Qld (cm)
I II
I
13
19
25
31
31
43
49
55
4.67 1.84
6.64 3.26
9.66 5.00
10.87 6.86
11.90 8.04
10.87 8.34
9.66 7.33
6.64 5.69
4.67 3.78
11
16
21
26
31
36
41
46
51
2.30 1.58
3.09 2.52
4.89 4.33
8.39 6.68
11.90 8.04
10.09 6.28
6.60 4.07
3.98 2.51
2.35 1.63
In this example, a filter which is described by following second order linear differential equation is used to generate the required wind velocity processes whose power spectrum is given above
f(t) = [0, 390.161 ;;;; [
-G,(t) [ I[
1
0
G,(t)
=
-0.855
x lo-’
1 1
-0.268
where w(t) is a white noise process whose autocorrelation function is S(r). The relative error between the power spectrum of the generated wind velocity process and the Davenport’s power spectrum is less than 10e3. According to eqns (9) and (1 1), the spatial correlation matrix Q, of the white noise processes can be calculated with the spatial correlation matrix R,of the gust wind as follows: QD = 5.03 x lo-‘R,. -
u”*(r)
Aoc Fig. 3. The mean square responses in vertical direction of cables AC and ED.
The stationary mean square responses of the cable roof structure is obtained. Some of the results are listed in Table 2 and shown in Fig. 3. SUMMARY
The discrete analysis method of random vibration for structures which are subjected to spatially correlated filtered white noise is discussed in this paper. The practical formulae for calculating structural mean and mean square responses are given. The advantage of discrete analysis method over the frequency-domain methods which are presently used in engineering computation is that it can calculate the accurate stationary mean and mean square responses with relatively less computation time for large-span space structures whose natural frequencies are densely distributed or for high-rise structures whose bend, shear and twist deformations are coupled together. The purpose of giving the numerical example of analysing the wind induced random vibration of a cable roof structure in this paper is just to demonstrate this advantage. The obtained results are satisfactory and provide more information about the random vibration behaviours of the cable roof structure. REFERENCES
1. Tan Dongyao and Guo Hua, A procedure of discrete analysis for nonstationary random vibration. Eurrhquake Engng Engng Vibr. 3, 12-18 (1986). cI Tan Dontzvao. Discrete analysis method of random vibration.-.&& me& sin. (to be published). H. A. Buchholdt, An Introduction to Cable Roof Structures. Cambridge University Press (1985). N. C. Nigam, Introduction to Random Vibrations. MIT Press, Cambridge, MA (1983). Tan Dongyao and Yang Qingshan, The practical form of the discrete analysis method of random vibration. J. Harbin Architectural Civ. Engng Inst. 2, 21-37 (1989). 6. Tan Dongyao and Yang Qingshan, Discrete analysis method for random vibration of structures under excitations correlated in time domain. Earthquake Engng Engng Vibr. 2, 3746 (1990).