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Higher order statistics of the response of MDOF linear systems excited by linearly parametric white noises and external excitations
Prob. EngngMech. Vol. 12, No. 3, pp. 179-188, 1997 ELSEVIER
PII:
S0266-8920(96)00041-0
© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0266-8920/97 $17.00 + 0.00
Higher order statistics of the response of MDOF linear systems excited by linearly parametric white noises and external excitations M. Di Paolat Dipartimento di Ingegneria Strutturale e Geotecnica, Viale delle Scienze, 1 90128 Palermo, Italy
&
G. Falsone Istituto di Scienza delle Costruzioni, Viale A. Doria n.6, 1 95125 Catania, Italy
The aim of this paper is the evaluation of higher order statistics of the response of linear systems subjected to external excitations and to linearly parametric white noise. The external excitations considered are deterministic or filtered white noise processes. The procedure implies the knowledge of the transition matrix connected to the linear system; this, however, has already been evaluated for obtaining the statistics at single times. The method, which avoids making further integrations for the evaluation of the higher order statistics, is very advantageous from a computational point of view. © 1997 Elsevier Science Ltd.
1 INTRODUCTION
delta-correlated inputs, it is possible to obtain the response correlations of any order once the cumulants of the corresponding order are known, by means of simple relationships where the fundamental matrix 4 of the linear system appears. This represents a great advantage from a computational point of view because it is surely simpler to find and to solve the differential equations governing the response moments or the cumulants rather than the differential equations governing the higher order statistics. Recently, analogous relationships have been found for a larger and very important class of systems: the linear systems subjected to linearly parametric normal white noise inputs. In particular, the relationship between A M T and moments of every order has been found for an SDOF system governed by a first order differential equation and excited by both a parametric white noise process and an external deterministic force. 5 In this relationship, the fundamental matrix of the system appears, in which the presence of the W o n g Zakai correction terms is taken into account. The importance of these systems has been underlined in the above cited paper; 5 here we want to repeat that any linear system excited by polynomials of filtered white noise can be exactly replaced by a linear system excited
Performing the stochastic analysis of linear or nonlinear systems subjected to randomly varying loads means finding the full probabilistic description of the stochastic response process once the probabilistic characterizaton of the load processes is known. It is well-known that any stochastic process is identified, from a probabilistic point of view, by means of the knowledge of its correlation functions, or, equally, of the averages at multiple times (AMT) 1'2 of order r (with r = 1 , 2 , . . . , oo), which are functions o f r variables (the instant times tl, t 2 , . . . , t r ) . These quantities represent the higher order statistics which depend on r time variables, in contrast with the cumulants and the moments which represent the corresponding first order statistics and depend on a single time variable only and giving a partial description of the process. It is well-known that only the first and the second order correlations are enough to characterize normal processes, while, for non-normal processes, the knowledge of higher order correlations is required. In a recent paper, 3 is was shown that for linear systems under either normal or non-normal external t Author to whom correspondence should be addressed. 179
M. Di Paola,G. Falsone
180
by linearly parametric white noise; moreover, any nonlinear system can be optimally approximated by a linear system excited by linearly parametric inputs. These concepts will be extensively treated in the companion paper. 6 Besides the above cited works, in the literature there are some papers devoted to the evaluation of the response correlations of linear systems subjected to linearly parametric white noise inputs, 7'8 but they are limited to the second order correlations and to the cases in which the Wong-Zakai correction terms do not arise; these restrictions are overcome here as well as in the previously cited paper. 5 The aim of the present paper is to extend this procedure in order to examine M D O F linear systems subjected to linearly parametric white noise inputs and external inputs that can be deterministic, white noise or filtered processes. This extension has been made possible thanks to the extensive use of the Kronecker algebra, 9 whose fundamentals are reported in Appendix A.
2 PRELIMINARY CONCEPTS
W(t)G(t)]Z(t) + f(t)
(1)
where the dot means derivative with respect to the time t, Z(t) is the response vector, A(t) and G(t) are two deterministic time-dependent matrices, W(t) is a parametric Gaussian white noise input with intensity q and f(t) is a vector of external excitations that can be deterministic or stochastic. By extending the results given in a recent work, 5 valid for SDOF first order differential equations, to the case of M D O F systems, it can easily be shown that the solution of eqn (1) can be written in the following form: Z(t) = ~(t, to)Zo ÷ It t~( t, T)f(T)dT, J to
Z 0 = Z(to)
(2) with to being the initial temporal instant, while ~(t, T) is the fundamental matrix related to eqn (l) and given by • (t,T) = exp [It A(p)dp +
JtG(p)dB(p)]
(3)
where B(t) is the Wiener process connected to the Gaussian white noise W(t) by means of the following formal relationship: ]°
W(t)
dB(t) --
dt
3 D E T E R M I N I S T I C EXTERNAL EXCITATIONS In this section, we suppose that the external excitations vector f(t) is deterministic; in this case, due to the presence of the Wong-Zakai correction terms, H the It6 differential equation corresponding to the eqn (1) is
dZ(t) = [A(t) + ~G2(t)q]Z(t)dt + G(t)Z(t)dB(t) + f(t)dt The first order moments of Z(t), that
(4)
Due to the fact that the fundamental matrix cI,(t, ~-) and, eventually, f(t) are stochastic quantities, even if the initial conditions vector Z0 is deterministic, the response vector Z(t) is stochastic. In the following sections, the
(5)
is its means, can be obtained by both applying the mean operator to the members of eqn (2), or solving the differential equation governing them; the latter can be written by applying the It6 differential rule 12 and has the following form: /~[Z(t)] = [A(t)+
Let us consider the following system of differential equations: Z(t) = [A(t) +
method for evaluating the higher order statistics of Z(t) will be seen in the various cases in which the external excitations are deterministic, normal white noise or filtered processes.
~G2(t)q]E[Z(t)] + f ( t )
(6)
where the symbol E[. ] indicates the mean operator. By applying both the above cited procedures, it can be shown that in the simpler case in which Z0 = 0, the first moments E[Z(t)] are given by E[Z(t)] =
;0 o (t, T)f(7)dT-
(7)
where the deterministic matrix O(t, ~-) is, at the same time, the mean of ~(t, 7) and the fundamental matrix related to the differential eqn (5), that is O(t, 7-)= E[,(t, ~-)] = e x p ( Ji [ A ( p ) +
~G2(p)qJdp} (8)
In order to find the second order statistics, it seems opportune to apply the method which is called alternative in the earlier cited paper. 5 For this purpose, in the following methodology, we will deal with the averages at multiple times (AMT) of the process Z(t) = Z(t) - E[Z(t)] in order to obtain a simpler formulation. It is easy to show that, by subtracting the mean ofeqn (5) from the same equation, Z(t) is characterized by the following It6 differential equation: dZ(t) = [ A ( t ) +
~G2(t)q]Z(t)dt+G(t)Z(t)dB(t) (9)
It is important to note that: (i) the A M T of ;L(t) coincide with the central averages at multiple times (CAMT) of Z(t); (ii) the second and third order C A M T coincides with the corresponding correlation functions, while for greater orders simple relationships between these quantities can be written.
Statistics of excited linear systems Moreover, for M D O F systems, it seems opportune to apply the Kronecker algebra, whose fundamental properties are given in Appendix A. Hence, in order to find the differential equations governing the second order CAMT, E[Z(tl) ® Z(t2)] (in which the symbol ® means Kronecker product), by using eqn (9) and by taking into account the properties of the Kronecker algebra, we can write E[Z(tl) @dZ(t2)] : ( I n ® [A(t2)+ ~G2(t2)q] } E[Z(fi) ® Z(t2)]dt2 + [In @ G(t2)] E[Z(fi) ® Z(t2)dB(t2) ]
(10)
OE[Z(tl)@Z(t2)]=Ot2 ( I n ® [A(t2)+ 1G2(t2)q] } (11)
whose solution, by considering t2 as variable and tl as its initial temporal instant of the variable t2, is E[Z(fi) ® Z(t2)] = [In ® O(t2,
fi)]E[Z[2](tl)],
t2 ~>tl
(12)
E[Z(t)][2]
(13)
in which the means E[Z(t)] are given by eqn (7) and the second order moments E[Z[2](t)] are given by the solution of the corresponding differential equation. The latter can be written by applying the It6 differential rule connected to the process Z(t); it has the following expression: /~[Z [2](t)] = [A2 (t) + G [21(t)q]E[Z [23(t)]
+ Q2{E[Z(t)] ® f(t)}
A2(t) = In @ A1 (t) + kl (t) @ I n
Al(t) = A(t) + ~G2(t)q
(15)
It has been shown 13 that the fundamental matrix O2(t, 7-) related to the matrix A2(t) given in eqn (15) is such that (16)
By extending the methodology used before, for the evaluation of the third order CAMT, we can write, for t~ < t 2 < 13 E[Z(q) ® Z(/2) ® dZ(/3)]
={I~]®[A(ta)+lG2(t3)q]} x E[Z(tl) ® Z(t2) ® Z(t3)]dt 3
(17)
where it has been considered that the mean E[Z(tl)® Z(t2) @ Z(ta)dB(t3)] is zero because the vector function Z(tl) ® Z(t2) @ Z(t3) is non-anticipating with respect to B(t3). The solution of eqn (17), by considering t 3 as variable and tl as its initial instant, has the following expression: E[Z(tl) @ Z(t2) @ Z(t3)] -- [I[n2] ® O(t3, tl)]
where O(t2, tl) is the fundamental matrix given into eqn (8) and the exponent in square brackets means Kronecker power; that is, Z[2] = Z ® Z. The relationship (12) gives the second order C A M T of a M D O F linear system subjected to a linearly parametric white noise input and to external deterministic inputs, once the second order central moments E[Z [21(t)] are known (note that the central moments are statistics of the first order). Equation (12) can be considered as the generalization of the relationship given in Ref. 3 in order to consider systems with parametric white noise input, and as the generalization of that given in Ref. 5 in order to consider M D O F systems. It is important to note that the second order central moments E[Z[Z](t)] can be easily obtained as E[Z [2](t)] = E[Z [2](t)] -
in Appendix A, while A2(t ) is given by
O2(t, r) = O[2](t, r)
I n being the identity matrix of order n × n (where n is the order of Z(t)). If we assume that tl < t2 and use the properties of the It6 calculus, noting that the last term of eqn (10) is zero (because, if tl < t2 the vector function Z(tl) @ Z(t2) is non-anticipating with respect to B(t2)), we obtain the following differential equation:
x E[Z(fi) ® Z(t2)]
181
(14)
where Q2 is a matrix of natural numbers which is given
x E[Z(q) ® Z(t2) @ Z(tl)]
(18)
which needs the knowledge of E[Z(tl) @ Z(tz) ® 7.(q)]. The latter can be evaluated by using the same procedure considered before, obtaining the following relationship: E[Z(/1) @ Z(t2) ® Z(tl)] = [In @ O(t2, tl) ® In] x E[Z I3](tl)]
(19)
If we substitute eqn (19) into eqn (18), we can express the third order C A M T E[Z(q) ® Z(t2) @ Z(t3)] in terms of the central moments E[Z [3](t)] as follows: E[Z(tl) ® Z(t2) ® Z(t3)]
= [I. ® O(t2, tl) ® O(t3, q)]E[Z[31(q)]
(20)
In this equation, the third order central moments can be evaluated in terms of moments up to the third order; these last ones are governed by a differential equation, obtained by applying the It6 differential rule and having the following expression: L'[Z[3I(t)] =
{ A3(t) + ~Q3[Q2 , ® I,][G[2l(t)q ® In] }
E[Z [3](t)] + Q3 { E[ Z[2] (/)] ® f(t) }
(21)
where Q3 is given in Appendix A, while A 3(t) is given by A3(t) = a2(t) ® I n -~ I [2] ® Al(t)
(22)
M. Di Paola, G. Falsone
182
In general, for the kth order CAMT, E[Z(tl) ® Z(t2) • " ® Z(tk)], with tl < t2 < "" < tt, by using the same approach considered before, it is possible to obtain the following relationship: E[z(t
E[Z(q)QdZ(t2)]= (In® [A(t2)+ ~G2(t2)q,] } × E[Z(q) @ Z(t2)]dt2 + E[Z(q)dBz(t2)] ® g(/2)
) ® z(t0) • • • ® z(tk)]
: [In ® O(t2, tl ) ® " - ® O(tk, tl)]E[Z [k](tl)]
relationship:
(23)
which needs the knowledge of the kth order central moments E[Z It] (t)]. These quantities are simply related to the moments up the kth order, which are governed by the corresponding differential equation, written by means of the It6 differential rule and having the following expression:
+ [In @ G(t2)]E[Z(tl) ® Z(t2)dBl (t2)]
By assuming tl < t2 and by using the properties of the It6 calculus related to the non-anticipating functions, the last equation leads to the following differential equation:
OE[Z(q)®Z(t2)]Ot2
Ot {e[zt - l (t)] ® fit) }
(24)
where At(t) is recursively given by At(t) = At_l(t) ® In + I[~k-ll ® Al(t)
(25)
4 WHITE NOISE EXTERNAL EXCITATIONS In this section, we consider that the external excitation f(t) is given by the product of a deterministic vector g(t) and a normal white noise process. The system is then governed by the following differential equation: Z(t) = [A(/) + W1 (t)G(t)]Z(t) + g(t) W2(t)
(26)
where Wl(t) and W2(t) are two white noises with intensity ql and q2, respectively. If they are not independent, their reciprocal intensity q12 is not zero, that is: E[W(fi) W(t2)] = q,26(t2 - tl). The corresponding It6 differential equation has the following expression: =
A(t) +
1 G2(t)ql]Z(t)dt
+ G(t)Z(t)dB 1(t) + g(t)dB2(t)
E[Z(h) ® Z(t2)] = [In @ O(t2, tl)]E[Z[2](tl)]
(27)
(28)
whose solution, for zero initial conditions (Z0 = 0), always gives E[Z(t)] = 0. As the response Z(t) has zero mean, its C A M T are coincident with the corresponding AMT. The second order A M T can be evaluated by means of the procedure given in the previous section, obtaining the following
(31)
This gives the relationship between the second order correlations and the corresponding moments directly. In this equation, O(t2, tl) has the same expression given in the previous section. The second order moments E[Z [2](t)] can be obtained by solving the corresponding differential equation that can be written by applying the It6 differential rule and has the following form: /~[Z [2](t)] = {A2(t ) + G [2](t)q I }E[Z [2)(t)] + ~Q2[G(t) ® g(t)qlz]E[Z(t)] + g[2](t)q2
(32)
where A2(t) has been introduced into eqn (15) and where it has to be remembered that E[Z(t)] = 0. Because of the result obtained for the second order A M T and because of what was stated in the previous section for C A M T of non-zero-mean response processes, it is not difficult to realize that the relationship between the kth order A M T and the corresponding moments has the following simple form (for tl < t2 < ... < tt): e[z(t
where B1 (t) and B2(t) are the Wiener processes related to Wl(t) and W2(t), respectively. The differential equation governing the behaviour of the first order moments E[Z(t)] is /~[Z(t)] = ( A ( t ) + 1G2(t)ql }E[Z(t)]
(30)
whose solution, by considering t2 as variable and tl initial instant for t2, is
I, Ak(t) + ~ Qk[Qk-1 ® In] [G[2](t)q ®
dZ(t)
= {In®[A(t2)+lGE(t2)ql]}
× E[Z(h) ® Z(t2)]
E[Zt~I (t)] =
+
(29)
) ® z(/:)
• • • ® z(tt)]
= [In ® O(t2, tl) ® " - ® O(tk,
tl)]E[ztt](tl)]
(33)
For the evaluation of the moments E[Z [k]] we note that they can be evaluated as the solution of the corresponding differential equation, which is written by means of the It6 differential rule and has the following expression: E[ZttJ (t)] = Ak(t) + ~ Qk [Qk-1 ® In] [G [2](t)ql ® I~ -2]] } E[Z [k](t)]
1
+ ~ Qk[Qk-1 ® In] [G(t) @ g(t)q12]E[Z tt-~l (/)]
1 + ~Qk[Qt-I Gin] {E[Z [k-2] (t)] ® g[21(t)q2}
(34)
Statistics of excited linear systems Hence, once the kth order moments are known by means of the solution of the differential equation (34), the corresponding A M T can be simply obtained by means of eqn (33). It is clear that, if in eqn (26), W2(t) = W 1(t), all the previous relationships hold in which ql = q e = q l e , while q12:0 if Wl(t ) and We(t) are independent.
5 FILTERED EXTERNAL EXCITATIONS
In this section, we suppose that the external excitations f(t) are filtered processes and, in particular, are given by f(t) = H(t)X(t)
(35)
where H(t) is an n × m deterministic matrix, while X(t) is the m-vector response of the following filter equation:
Y,(t) : Av(t)X(t) + gF(t) We(t)
(36)
where Av(t) and gF(t) are a matrix and a vector, respectively, and depend on the characteristics of the filter, while We(t) is normal white noise. Hence, the considered problem is governed by the two following systems of differential equations: Z(t) : [A(t)
+ Wt (t)G(t)]Z(t) + HX(t)
X(t) : AF(t)X(t) - gF(t) We(t)
(37a) (37b)
The simplest way to treat this case is to include the response vector Z(t) and X(t) in a single super-vector Y(t) of order n + m in such a way that Y+(t) = [Za'(t) XX(t)]. As a consequence, eqns (37a, b) can be replaced by the following equation: Y(t) = [~,(t)
+ W1 (t)ff,(t)]Y(t) + g(t)We(t )
(38)
where it is simple to show that .i,(t), I~,(t) and fg(t) are given by A(t):
g(t)=
(A(0t)
(o)
Av(t)H(t))
0)0 (39a-c)
gv(t)
183
matrix ,~(t) +
O(t, 7-) = It [A.(P) + ~G2(P)ql]dp
E[v(t,)
® v(te).
= [In+m ® ~)(t2,
••®
t2) ®"" ® O(tk, tl)]E[Y[k](tl)] (40)
where O(t,'r) is the fundamental matrix related to the
(41)
6 STATIONARY R E S P O N S E
In this section, the fundamental relationships between the C A M T or the A M T and the corresponding central moment or moments, introduced in the previous sections for the three different types of external excitation considered, will be considered for the case of stationary response. For this purpose, we assume that the deterministic matrices A and G of eqn (1) do not depend on time. Thus, the fundamental matrix defined in eqn (8) does not depend on the instants tl and t2 taken separately, but depends only on their difference tl - t2; that is, O(tl, t2) --- O(tl - t2). In the following, we analyze again, but in the stationary case, the three cases of external excitations considered before. 6.1 Deterministic external excitations
In this case, we consider that the external deterministic excitations f(t), after a transient temporal shift, assume a constant value Is- Then, the response moments of every order are characterized by stationary values that can be obtained starting from the differential equations governing them; for example, for the response means, starting from eqn (6), by putting zero as its first member, we obtain the stationary values E[Z] as
E [ Z ] = - A+
G2q
fs
(42)
By using an analogous procedure for the second order response moments, starting from eqn (14), we obtain the stationary value E[Z [2]] as follows: E[ zI2]] = -[A2 + G[2]q]-IQ2{E[Z] ® fs}
(43)
and, in general, for the kth order moments, starting from eqn (24), we obtain the stationary value E[Z [k]] as follows: E[Z [k]] = -
It is now clear that the system of differential equations given in eqn (38) has the same form as the one considered in Section 4; that is, a linear system excited by a parametric and an external normal white noise. Thus, the same considerations can be made in order to evaluate the A M T of Y(t) once its moments are known. That is, the following fundamental relationship holds:
1/2C,2(t)ql, that is;
{Ak + ~Qk[Qk-~ 1 ® ln][G[2lq ® I~ -2]] ),
x Qk{E[Z [/'-']] ® fs }
(44)
If the stationary means and the stationary second order moments are known, then it is possible, by means of eqn (13), to find the stationary second order central moments E[Z [2]] and, hence, by means of eqn (12), the second order stationary C A M T E[Z[2]]r ' (where ~-z : t2 - tt and the subscript indicates the variable on which the C A M T depends) as follows: E[Z[21]~, = [In ® O(T1)]E[Z [2J]
(45)
184
M. Di Paola, G. Falsone
We remember that the second order stationary C A M T coincides with the second order stationary correlations (and both depend on a single time variable "r1), while the second order central moments coincide with the second order cumulants (and both do not depend on time). Analogous particularization can be made for higher order statistics; for example, in the case of a zero-mean response process, the particularization of eqn (23) gives
E[z[k]]TI,7. 2..... Tk_l =
[I, ® O('7-1) ® ' ' " ®
@)("i-k_l)]E[ff~[k]] (46)
It is important to note that the kth A M T depends on k - 1 variables -~ = tj - q. 6.2 White noise external excitations
By taking into account eqn (26), if A, G and g are independent of t, as well as the white noise intensities ql, q2 and q~2, it is possible to evaluate the stationary statistics of the response. In the case of zero initial conditions, the response process has zero mean, while the second order stationary moments can be obtained by putting to zero the first member of eqn (32), that is E[Z [2]] = - [ A 2 + G[2]ql]-lgi2lq2
(47)
second order differential equation: X(t) + [2(w + aWl(t)]J((t) + [w2 +/3W2(t)]X(t ) = "7 (49) where ~ is the damping coefficient, w is the natural radian frequency, a, /3 and "7 are constants and Wl (t) and W2(t) are two independent white noises. The necessity to consider two parametric excitations, the first one acting on the velocity and the second one acting on the displacement, lies in the fact that, if only the first, Wl(t), is present, it can be shown that the stationary response is deterministic; while if only the second, W2(t), is present, no Wong-Zakai correction term arises. Hence, the stationary response is stochastic and it needs the presence of the Wong-Zakai correction term in the solution only if both the parametric white noises act on the system. By introducing the state variable vector z T ( t ) = [X(t)J((t)], eqn (49) can be rewritten as follows: Z(t) = [A + W1 (t)Gl + W2(t)G2]Z(t)+ where A
=
The moments of higher order can be obtained by putting zero to the first member of eqn (34), that is E[ Z[k]] = -
{1
},
Ak + ~Qk[Qk-I ® In][G[21ql ® I[~-211
-2{w
_032
'
G1 =
0
-~
The It6 differential equation corresponding to eqn (50) has the following form: dZ(t) =
(48)
I'
l
A+~G12ql Z(t)dt
+ [GIdB l + G2dB2]Z(t ) + fdt The stationary A M T of every order can be evaluated by particularizing eqn (33) which leads to an analogous expression for the A M T of that given in eqn (46) for the CAMT. 6.3 Filtered external excitations
Here we consider that the quantities H, Av and gv, appearing in Section 5, do not depend on t as well. Under this assumption, the response is stationary. Hence, once the matrices ii,, I] and g are introduced (as in Section 5), the analysis of the statistics of the stationary response Y can be made as seen in the previous subsection.
(52)
In fact, the Wong-Zakai correction term connected to the presence of W2(t) is zero because, as can be easily shown, G ~ - - 0 . Thus, the mean of the response is governed by the following differential equation: /~[Z(t)] =
[ 1 2 ] A + ~Glql E[Z(t)] + f
(53)
and it is characterized by the following stationary value: E[Z l = - IA' + ~ Gl2q, ]-1 f
(54)
The second order response moments are governed by the following differential equation:
E[Z [2](t)] 7 NUMERICAL EXAMPLES
= [A2 + Gl2]ql + G~Zlqz]E[Z [2](t)]
+ Q2{E[Z(t)I Q f}
(55)
where The first example considered is a linear oscillator excited by two parametric white noises and by an external constant excitation, and is governed by the following
(50)
(0 1) (0 0) :), f=(:)
× ~Qk[Qk-1 @In]{[G @gql2]E[z[k-l] l +E[Z[k-2]] ® g[Zlq2}
f
1
2
1
2
Statistics of excited linear systems 2E-4 -
-
C o r r e c t i v e terms included C o r r e c t i v e terms n e g l e c t e d
-
......
-
0.11
C o r r e c t i v e terms included C o r r e c t i v e terms neglected
-
......
X
A
0.0~
0
•~ 7E-21
185
'
i
VVv
,, v
o
O -IE-4
-2E-4
-0.0:
I 2
0
I 4
I 6
I 8
-0.10
I 10
I 0
I 4
2
T
,.¢
I 6
I 8
I 10
Fig. 1. Stationary correlation of X(t) of system (49); w = 5, ~= 0.1, ~' = 1, c~ =0.9,/3= 1, ql = q2 = 1, q|2 =0.
Fig. 3. Stationary correlation of X(t) of system (60); w = 5, ( = 0 . 1 , a = 0 - 9 , / 3 = 1, q| = 1.
The stationary value of E[ZP](t)] is given by
By introducing the vector Z(t) of the state variables, eqn (60) can be rewritten in the form of eqn (26), where
E [ Z [2]] =
-[A 2 +
Gl2]ql
+
G~Z]q2]-IQ2{E[Z] @f} (57)
In order to evaluate the stationary second order correlations of the response, we remember that they coincide with the corresponding CAMT; hence, we can use eqn (45), where the stationary second central moments are given by e [ z I21] = E[Z I21]- E[Z] E21
(58)
Finally, the fundamental matrix O('q) is given by O(rl) = exp{ [A + ~G~q,] ~-1}
(59)
In Figs 1 and 2, the exact displacement and velocity stationary correlations are reported for particular values of the coefficients that assure the stability of the system. In order to show the importance of the Wong-Zakai correction terms and the great errors that can be made if they are neglected, in the figures, the results are compared with those obtained by such neglect. The second example considered is an oscillator excited by parametric and external independent white noises and that is governed by the following differential equation: )((t) + [2~w + C~Wl(t)]8(t) + w2X(t) = flW2(t)
--~3 2
-2(w
'
G=
(:o)
,
g=
(;)
(61) In order to evaluate the stationary second correlations of the response, we remember that, as the system has zero-mean response, the second central moments are coincident with the second moments whose stationary value is given by eqn (47). Once the moments are known, the correlations can be obtained by means of the transition matrix O(~-) which has the same expression as that given in eqn (59) (in which G1 is replaced by G). In Figs 3 and 4, the exact displacement and velocity second order correlations are reported for particular values of the coefficients, compared again with those evaluated without taking into account the Wong-Zakai correction terms. Finally, the third example considered is represented by the following oscillator: J((t) + [2~w + a W1 (t)])((t) + J X ( t ) = 3F(t)
(62)
where F(t) is a filtered process which is the response of
C o r r e c t i v e terms i n c l u d e d C o r r e c t i v e terms n e g l e c t e d
- ......
4E-3
(60)
(0 ,)
A=
3-
C o r r e c t i v e terms included C o r r e c t i v e terms n e g l e c t e d
- ......
2 "~
2E-3
.X
O
o
~ IE-19
-2E-3
-4E-3| 0
,., ? V v l
I
2
,
v--
I
I
4
6
I
8
I
10
T
Fig. 2. Stationary correlation of .~(t) of system (49).
-3/ 0
I 2
[ 4
I 6
[ 8
1 10
T
Fig. 4. Stationary correlation of ~'(t) of system (60).
186
M. Di Paola, G. Falsone
1.2E-3~
- ......
0.020 -
Corrective terms i n c l u d e d Corrective t e r m s n e g l e c t e d
0.015
8.0E-4
Corrective t e r m s i n c l u d e d Corrective terms neglected
......
•X 0.01o
X "~ 4.0E-4
," 0.005
e. o
O
2.7E-20 O
r,9 -4.0E-4 -8.0E-4
0
2
4
6
8
- - J
10
AA °it,,// V o
A
2
A
A
I 4
I 6
A~
A_~
I 8
I 10
T
Fig. 5. Stationary correlation of X(t) of system (62); w = 5, ~=0.1, a=0-9, f l = l , q l = q 2 = l , ql2=0, WF=2"5, ~F = 0"1, ~ :
1.
where 1- 2 A2 = [~k + ~G2ql] ® I4 + I4 ® [~k + ~G ql]
the following filter: F(t) + 2(FWFP(t) + w~F(t) =- "~W2(t)
(63)
These two equations can be unified into an equation having the same expression of that given into eqn (38), where
A:(: +")
Y=
Fig. 6. Stationary correlation of )((t) of system (62).
(0 1 0 0) /:ooo) o=(: :) /°>(!) -2{w
3
0
0
0
0
1
0
0
__2
0 0
0
0 0
0
0 0
E[~([2]]0" = [In ® ~)(0-)]E[Y[zl]
(67)
where the fundamental matrix ~)(0-) is given by (68)
For this example, the stationary second correlations of the displacement and of the velocity of the oscillator are reported in Figs 5 and 6. We see again that neglecting the Wong-Zakai correction terms may lead to very different results.
8 CONCLUSIONS
(64)
gF
Even in this case, for zero initial conditions, the mean of the response is zero; hence, the second moments coincide with the second central moments, while the second AMT coincides with the second correlations. In order to find the stationary response second CAMT, it is first necessary to find the stationary second central moments that, as stated previously, coincide with the second moments; these last ones are obtained by means of the following relationship: E[Y[21] = -[~,2 + Cj[2lql]-lg[2]q2
The CAMT are given by
O('r) = exp{ I-~ + ~ C,2ql]'r }
__2~F WF
-a
(66)
(65)
In this paper, a method has been introduced in order to evaluate the higher order statistic (correlations and CAMT) of the response of linear systems excited by external stochastic loads and linearly parametric white noise excitations. In this method, a very important role is played by the fundamental matrix related to the dynamical matrices of the systems, that, due to the presence of linearly parametric white noise excitations, must take into account the presence of Wong-Zakai correction terms. By applying this method, the response moments at a fixed time are first evaluated by means of an integration (for a non-stationary response) or a simple matrix inversion (in the stationary case), and then the higher order statistics are evaluated by means of simple relationships in which the fundamental matrices appear. The relationships evaluated here are an extension of those in the literature that hold in the case of linear systems and purely external excitations. The extension to the case in which linearly parametric white noise is also present in the motion equation is very important because, as can be seen in the companion paper 6, many problems can be included in this class of systems.
Statistics of excited linear systems
187
REFERENCES
The Kronecker product has the following properties:
1. Lin, Y. K., Probabilistic Theory of Structural Dynamics. McGraw-Hill, New York, 1967. 2. Stratonovich, R. L., Topics in the Theory of Random Noise. Gordon and Breach, New York, 1963. 3. Falsone, G., Cumulants and correlations for linear systems under non-stationary delta-correlated processes. Probabilistic Engineering Mechanics, 1994, 9, 157-165. 4. Muller, P. C. and Schielen, W. O., Linear Vibrations. Martinus Nijhoff, Dordrecht, 1985. 5. Di Paola, M. and Falsone, G., Response in terms of correlations of linear systems under linearly parametric white noises. Proceedings of the IUTAM Conference on Nonlinear Stochastic Mechanics, Trondheim, Norway, July 1995. 6. Di Paola, M. and Falsone, G., Multiple times response statistics of M D O F linear systems under polynomials of filtered normal white noise. Probabilistic Engineering Mechanics, 1997, 12, 189-196. 7. Benaroya, H. and Rehak, M., Response and stability of a random differential equation. Part I: moment equation method. Journal of Applied Mechanics ASME, 1989, 56, 192-195. 8. Hou, Z. K. and Iwan, W. D., Nonstationary response of linear systems under uncorrelated parametric and external excitations. Probabilistic Engineering Mechanics, 1991, 6, 74-81. 9. Graham, A., Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Chichester, UK, 1981. 10. Grigoriu, M., White noise processes. Journal of the Engineering and Mechanical Division ASCE, 1987, 119, 757-765. 11. Wong, E. and Zakai, M., On the relation between ordinary and stochastic differential calculus. International Journal of Engineering Science, 1965, 3, 213-229. 12. Itr, K., Stochastic differential equations. Memoirs of the American Mathematical Society, 1951, 4. 13. Di Paola, M., Moments of non-linear systems. In Proceedings of the 5th ASCE Conference on Probabilistic Methods in Civil Engineering, Blacksburg, 1988, pp. 285-288.
M ® ( N ® P ) -----( M ® N ) (M+N)
=M®P+M®Q+N®P+NQQ
(A3)
(M ® N)(P ® Q) = (MP) ® (NQ)
(A4)
(M ® N) T = M r ® N T
(A5)
(m ® N) -1 = M -1 @ N -1
(A6)
where P and Q are two given matrices. Equations (A4) and (A6) are valid, provided the various quantities exist. A fundamental role in the Kronecker algebra is played by the so-called permutation matrix El,j, which is a square matrix of order (ij x/j), consisting of (j × i) arrays of elementary submatrices E kl of order (i ×j); each of these matrices has the value one in the (k, l)th position and the value zero in all the other positions. Hence, Ei,j has the following expression:
E 12
E 22
...
.
.
".
E y2 .
E 2i
...
E j~
Ei,) =
\ E li
The Kronecker product of the matrices M and N, of order (p x q) and (s x t), respectively, is denoted by M ® N and gives the matrix of order (ps x qt) defined by the following relationship:
m ® N ----
ml2N
'
"
\mplN
"-" mlqN "..
mp2N ...
"
mpqN
with mij being the (i,j)th element of M.
(A7)
where, for example, the matrix E 12 is given by
0
0
•
.
0
0
-..
E 12 =
(A8) "..
-..
It is important to note that the matrix Ei,j exhibits a single 1 in each row and in each column, placed following the rule given in equation (A7). Its name is due to the fact that the following fundamental relationship holds: ® N)Et, q
(A9)
The matrices Qk, that are present in this paper, are strictly connected with the permutation matrices by means of the following relationship:
A.1 Some properties of Kronecker algebra
mxlN
(A2)
® ( V + Q)
N ® M = Ep, s(M APPENDIX A
®P
(A1)
k-1
Qk = Z
E,,-j,,j
(A10)
j=0
It is easy to show that these matrices exhibit only zeros and natural numbers in such a way that the sum of the elements of each row, or of each column, is equal to k. The matrices Qk are present in the expressions regarding the derivatives of vectors. In fact, let us consider an n-vector Z and the differential operation Vz, presenting the element O/OZj in the jth position;
188
M. Di Paola, G. Falsone
thus, the following relationships hold: V ~ ® Z ----In
(A11)
V~ ~) Z [k] ~ Qk(Z [k-1] ~ In)
I n is the identity matrix o f order n x n. Finally, if Qk premultiplies the kth K r o n e c k e r power o f a vector a, it
gives Qka [k] = ka
= Z (k-l] ® In + Z [k-2] ® In ® Z + ..- I n ® Z [k-l] (A12)
(A13)
PII:
S0266-8920(96)00041-0
© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0266-8920/97 $17.00 + 0.00
Higher order statistics of the response of MDOF linear systems excited by linearly parametric white noises and external excitations M. Di Paolat Dipartimento di Ingegneria Strutturale e Geotecnica, Viale delle Scienze, 1 90128 Palermo, Italy
&
G. Falsone Istituto di Scienza delle Costruzioni, Viale A. Doria n.6, 1 95125 Catania, Italy
The aim of this paper is the evaluation of higher order statistics of the response of linear systems subjected to external excitations and to linearly parametric white noise. The external excitations considered are deterministic or filtered white noise processes. The procedure implies the knowledge of the transition matrix connected to the linear system; this, however, has already been evaluated for obtaining the statistics at single times. The method, which avoids making further integrations for the evaluation of the higher order statistics, is very advantageous from a computational point of view. © 1997 Elsevier Science Ltd.
1 INTRODUCTION
delta-correlated inputs, it is possible to obtain the response correlations of any order once the cumulants of the corresponding order are known, by means of simple relationships where the fundamental matrix 4 of the linear system appears. This represents a great advantage from a computational point of view because it is surely simpler to find and to solve the differential equations governing the response moments or the cumulants rather than the differential equations governing the higher order statistics. Recently, analogous relationships have been found for a larger and very important class of systems: the linear systems subjected to linearly parametric normal white noise inputs. In particular, the relationship between A M T and moments of every order has been found for an SDOF system governed by a first order differential equation and excited by both a parametric white noise process and an external deterministic force. 5 In this relationship, the fundamental matrix of the system appears, in which the presence of the W o n g Zakai correction terms is taken into account. The importance of these systems has been underlined in the above cited paper; 5 here we want to repeat that any linear system excited by polynomials of filtered white noise can be exactly replaced by a linear system excited
Performing the stochastic analysis of linear or nonlinear systems subjected to randomly varying loads means finding the full probabilistic description of the stochastic response process once the probabilistic characterizaton of the load processes is known. It is well-known that any stochastic process is identified, from a probabilistic point of view, by means of the knowledge of its correlation functions, or, equally, of the averages at multiple times (AMT) 1'2 of order r (with r = 1 , 2 , . . . , oo), which are functions o f r variables (the instant times tl, t 2 , . . . , t r ) . These quantities represent the higher order statistics which depend on r time variables, in contrast with the cumulants and the moments which represent the corresponding first order statistics and depend on a single time variable only and giving a partial description of the process. It is well-known that only the first and the second order correlations are enough to characterize normal processes, while, for non-normal processes, the knowledge of higher order correlations is required. In a recent paper, 3 is was shown that for linear systems under either normal or non-normal external t Author to whom correspondence should be addressed. 179
M. Di Paola,G. Falsone
180
by linearly parametric white noise; moreover, any nonlinear system can be optimally approximated by a linear system excited by linearly parametric inputs. These concepts will be extensively treated in the companion paper. 6 Besides the above cited works, in the literature there are some papers devoted to the evaluation of the response correlations of linear systems subjected to linearly parametric white noise inputs, 7'8 but they are limited to the second order correlations and to the cases in which the Wong-Zakai correction terms do not arise; these restrictions are overcome here as well as in the previously cited paper. 5 The aim of the present paper is to extend this procedure in order to examine M D O F linear systems subjected to linearly parametric white noise inputs and external inputs that can be deterministic, white noise or filtered processes. This extension has been made possible thanks to the extensive use of the Kronecker algebra, 9 whose fundamentals are reported in Appendix A.
2 PRELIMINARY CONCEPTS
W(t)G(t)]Z(t) + f(t)
(1)
where the dot means derivative with respect to the time t, Z(t) is the response vector, A(t) and G(t) are two deterministic time-dependent matrices, W(t) is a parametric Gaussian white noise input with intensity q and f(t) is a vector of external excitations that can be deterministic or stochastic. By extending the results given in a recent work, 5 valid for SDOF first order differential equations, to the case of M D O F systems, it can easily be shown that the solution of eqn (1) can be written in the following form: Z(t) = ~(t, to)Zo ÷ It t~( t, T)f(T)dT, J to
Z 0 = Z(to)
(2) with to being the initial temporal instant, while ~(t, T) is the fundamental matrix related to eqn (l) and given by • (t,T) = exp [It A(p)dp +
JtG(p)dB(p)]
(3)
where B(t) is the Wiener process connected to the Gaussian white noise W(t) by means of the following formal relationship: ]°
W(t)
dB(t) --
dt
3 D E T E R M I N I S T I C EXTERNAL EXCITATIONS In this section, we suppose that the external excitations vector f(t) is deterministic; in this case, due to the presence of the Wong-Zakai correction terms, H the It6 differential equation corresponding to the eqn (1) is
dZ(t) = [A(t) + ~G2(t)q]Z(t)dt + G(t)Z(t)dB(t) + f(t)dt The first order moments of Z(t), that
(4)
Due to the fact that the fundamental matrix cI,(t, ~-) and, eventually, f(t) are stochastic quantities, even if the initial conditions vector Z0 is deterministic, the response vector Z(t) is stochastic. In the following sections, the
(5)
is its means, can be obtained by both applying the mean operator to the members of eqn (2), or solving the differential equation governing them; the latter can be written by applying the It6 differential rule 12 and has the following form: /~[Z(t)] = [A(t)+
Let us consider the following system of differential equations: Z(t) = [A(t) +
method for evaluating the higher order statistics of Z(t) will be seen in the various cases in which the external excitations are deterministic, normal white noise or filtered processes.
~G2(t)q]E[Z(t)] + f ( t )
(6)
where the symbol E[. ] indicates the mean operator. By applying both the above cited procedures, it can be shown that in the simpler case in which Z0 = 0, the first moments E[Z(t)] are given by E[Z(t)] =
;0 o (t, T)f(7)dT-
(7)
where the deterministic matrix O(t, ~-) is, at the same time, the mean of ~(t, 7) and the fundamental matrix related to the differential eqn (5), that is O(t, 7-)= E[,(t, ~-)] = e x p ( Ji [ A ( p ) +
~G2(p)qJdp} (8)
In order to find the second order statistics, it seems opportune to apply the method which is called alternative in the earlier cited paper. 5 For this purpose, in the following methodology, we will deal with the averages at multiple times (AMT) of the process Z(t) = Z(t) - E[Z(t)] in order to obtain a simpler formulation. It is easy to show that, by subtracting the mean ofeqn (5) from the same equation, Z(t) is characterized by the following It6 differential equation: dZ(t) = [ A ( t ) +
~G2(t)q]Z(t)dt+G(t)Z(t)dB(t) (9)
It is important to note that: (i) the A M T of ;L(t) coincide with the central averages at multiple times (CAMT) of Z(t); (ii) the second and third order C A M T coincides with the corresponding correlation functions, while for greater orders simple relationships between these quantities can be written.
Statistics of excited linear systems Moreover, for M D O F systems, it seems opportune to apply the Kronecker algebra, whose fundamental properties are given in Appendix A. Hence, in order to find the differential equations governing the second order CAMT, E[Z(tl) ® Z(t2)] (in which the symbol ® means Kronecker product), by using eqn (9) and by taking into account the properties of the Kronecker algebra, we can write E[Z(tl) @dZ(t2)] : ( I n ® [A(t2)+ ~G2(t2)q] } E[Z(fi) ® Z(t2)]dt2 + [In @ G(t2)] E[Z(fi) ® Z(t2)dB(t2) ]
(10)
OE[Z(tl)@Z(t2)]=Ot2 ( I n ® [A(t2)+ 1G2(t2)q] } (11)
whose solution, by considering t2 as variable and tl as its initial temporal instant of the variable t2, is E[Z(fi) ® Z(t2)] = [In ® O(t2,
fi)]E[Z[2](tl)],
t2 ~>tl
(12)
E[Z(t)][2]
(13)
in which the means E[Z(t)] are given by eqn (7) and the second order moments E[Z[2](t)] are given by the solution of the corresponding differential equation. The latter can be written by applying the It6 differential rule connected to the process Z(t); it has the following expression: /~[Z [2](t)] = [A2 (t) + G [21(t)q]E[Z [23(t)]
+ Q2{E[Z(t)] ® f(t)}
A2(t) = In @ A1 (t) + kl (t) @ I n
Al(t) = A(t) + ~G2(t)q
(15)
It has been shown 13 that the fundamental matrix O2(t, 7-) related to the matrix A2(t) given in eqn (15) is such that (16)
By extending the methodology used before, for the evaluation of the third order CAMT, we can write, for t~ < t 2 < 13 E[Z(q) ® Z(/2) ® dZ(/3)]
={I~]®[A(ta)+lG2(t3)q]} x E[Z(tl) ® Z(t2) ® Z(t3)]dt 3
(17)
where it has been considered that the mean E[Z(tl)® Z(t2) @ Z(ta)dB(t3)] is zero because the vector function Z(tl) ® Z(t2) @ Z(t3) is non-anticipating with respect to B(t3). The solution of eqn (17), by considering t 3 as variable and tl as its initial instant, has the following expression: E[Z(tl) @ Z(t2) @ Z(t3)] -- [I[n2] ® O(t3, tl)]
where O(t2, tl) is the fundamental matrix given into eqn (8) and the exponent in square brackets means Kronecker power; that is, Z[2] = Z ® Z. The relationship (12) gives the second order C A M T of a M D O F linear system subjected to a linearly parametric white noise input and to external deterministic inputs, once the second order central moments E[Z [21(t)] are known (note that the central moments are statistics of the first order). Equation (12) can be considered as the generalization of the relationship given in Ref. 3 in order to consider systems with parametric white noise input, and as the generalization of that given in Ref. 5 in order to consider M D O F systems. It is important to note that the second order central moments E[Z[Z](t)] can be easily obtained as E[Z [2](t)] = E[Z [2](t)] -
in Appendix A, while A2(t ) is given by
O2(t, r) = O[2](t, r)
I n being the identity matrix of order n × n (where n is the order of Z(t)). If we assume that tl < t2 and use the properties of the It6 calculus, noting that the last term of eqn (10) is zero (because, if tl < t2 the vector function Z(tl) @ Z(t2) is non-anticipating with respect to B(t2)), we obtain the following differential equation:
x E[Z(fi) ® Z(t2)]
181
(14)
where Q2 is a matrix of natural numbers which is given
x E[Z(q) ® Z(t2) @ Z(tl)]
(18)
which needs the knowledge of E[Z(tl) @ Z(tz) ® 7.(q)]. The latter can be evaluated by using the same procedure considered before, obtaining the following relationship: E[Z(/1) @ Z(t2) ® Z(tl)] = [In @ O(t2, tl) ® In] x E[Z I3](tl)]
(19)
If we substitute eqn (19) into eqn (18), we can express the third order C A M T E[Z(q) ® Z(t2) @ Z(t3)] in terms of the central moments E[Z [3](t)] as follows: E[Z(tl) ® Z(t2) ® Z(t3)]
= [I. ® O(t2, tl) ® O(t3, q)]E[Z[31(q)]
(20)
In this equation, the third order central moments can be evaluated in terms of moments up to the third order; these last ones are governed by a differential equation, obtained by applying the It6 differential rule and having the following expression: L'[Z[3I(t)] =
{ A3(t) + ~Q3[Q2 , ® I,][G[2l(t)q ® In] }
E[Z [3](t)] + Q3 { E[ Z[2] (/)] ® f(t) }
(21)
where Q3 is given in Appendix A, while A 3(t) is given by A3(t) = a2(t) ® I n -~ I [2] ® Al(t)
(22)
M. Di Paola, G. Falsone
182
In general, for the kth order CAMT, E[Z(tl) ® Z(t2) • " ® Z(tk)], with tl < t2 < "" < tt, by using the same approach considered before, it is possible to obtain the following relationship: E[z(t
E[Z(q)QdZ(t2)]= (In® [A(t2)+ ~G2(t2)q,] } × E[Z(q) @ Z(t2)]dt2 + E[Z(q)dBz(t2)] ® g(/2)
) ® z(t0) • • • ® z(tk)]
: [In ® O(t2, tl ) ® " - ® O(tk, tl)]E[Z [k](tl)]
relationship:
(23)
which needs the knowledge of the kth order central moments E[Z It] (t)]. These quantities are simply related to the moments up the kth order, which are governed by the corresponding differential equation, written by means of the It6 differential rule and having the following expression:
+ [In @ G(t2)]E[Z(tl) ® Z(t2)dBl (t2)]
By assuming tl < t2 and by using the properties of the It6 calculus related to the non-anticipating functions, the last equation leads to the following differential equation:
OE[Z(q)®Z(t2)]Ot2
Ot {e[zt - l (t)] ® fit) }
(24)
where At(t) is recursively given by At(t) = At_l(t) ® In + I[~k-ll ® Al(t)
(25)
4 WHITE NOISE EXTERNAL EXCITATIONS In this section, we consider that the external excitation f(t) is given by the product of a deterministic vector g(t) and a normal white noise process. The system is then governed by the following differential equation: Z(t) = [A(/) + W1 (t)G(t)]Z(t) + g(t) W2(t)
(26)
where Wl(t) and W2(t) are two white noises with intensity ql and q2, respectively. If they are not independent, their reciprocal intensity q12 is not zero, that is: E[W(fi) W(t2)] = q,26(t2 - tl). The corresponding It6 differential equation has the following expression: =
A(t) +
1 G2(t)ql]Z(t)dt
+ G(t)Z(t)dB 1(t) + g(t)dB2(t)
E[Z(h) ® Z(t2)] = [In @ O(t2, tl)]E[Z[2](tl)]
(27)
(28)
whose solution, for zero initial conditions (Z0 = 0), always gives E[Z(t)] = 0. As the response Z(t) has zero mean, its C A M T are coincident with the corresponding AMT. The second order A M T can be evaluated by means of the procedure given in the previous section, obtaining the following
(31)
This gives the relationship between the second order correlations and the corresponding moments directly. In this equation, O(t2, tl) has the same expression given in the previous section. The second order moments E[Z [2](t)] can be obtained by solving the corresponding differential equation that can be written by applying the It6 differential rule and has the following form: /~[Z [2](t)] = {A2(t ) + G [2](t)q I }E[Z [2)(t)] + ~Q2[G(t) ® g(t)qlz]E[Z(t)] + g[2](t)q2
(32)
where A2(t) has been introduced into eqn (15) and where it has to be remembered that E[Z(t)] = 0. Because of the result obtained for the second order A M T and because of what was stated in the previous section for C A M T of non-zero-mean response processes, it is not difficult to realize that the relationship between the kth order A M T and the corresponding moments has the following simple form (for tl < t2 < ... < tt): e[z(t
where B1 (t) and B2(t) are the Wiener processes related to Wl(t) and W2(t), respectively. The differential equation governing the behaviour of the first order moments E[Z(t)] is /~[Z(t)] = ( A ( t ) + 1G2(t)ql }E[Z(t)]
(30)
whose solution, by considering t2 as variable and tl initial instant for t2, is
I, Ak(t) + ~ Qk[Qk-1 ® In] [G[2](t)q ®
dZ(t)
= {In®[A(t2)+lGE(t2)ql]}
× E[Z(h) ® Z(t2)]
E[Zt~I (t)] =
+
(29)
) ® z(/:)
• • • ® z(tt)]
= [In ® O(t2, tl) ® " - ® O(tk,
tl)]E[ztt](tl)]
(33)
For the evaluation of the moments E[Z [k]] we note that they can be evaluated as the solution of the corresponding differential equation, which is written by means of the It6 differential rule and has the following expression: E[ZttJ (t)] = Ak(t) + ~ Qk [Qk-1 ® In] [G [2](t)ql ® I~ -2]] } E[Z [k](t)]
1
+ ~ Qk[Qk-1 ® In] [G(t) @ g(t)q12]E[Z tt-~l (/)]
1 + ~Qk[Qt-I Gin] {E[Z [k-2] (t)] ® g[21(t)q2}
(34)
Statistics of excited linear systems Hence, once the kth order moments are known by means of the solution of the differential equation (34), the corresponding A M T can be simply obtained by means of eqn (33). It is clear that, if in eqn (26), W2(t) = W 1(t), all the previous relationships hold in which ql = q e = q l e , while q12:0 if Wl(t ) and We(t) are independent.
5 FILTERED EXTERNAL EXCITATIONS
In this section, we suppose that the external excitations f(t) are filtered processes and, in particular, are given by f(t) = H(t)X(t)
(35)
where H(t) is an n × m deterministic matrix, while X(t) is the m-vector response of the following filter equation:
Y,(t) : Av(t)X(t) + gF(t) We(t)
(36)
where Av(t) and gF(t) are a matrix and a vector, respectively, and depend on the characteristics of the filter, while We(t) is normal white noise. Hence, the considered problem is governed by the two following systems of differential equations: Z(t) : [A(t)
+ Wt (t)G(t)]Z(t) + HX(t)
X(t) : AF(t)X(t) - gF(t) We(t)
(37a) (37b)
The simplest way to treat this case is to include the response vector Z(t) and X(t) in a single super-vector Y(t) of order n + m in such a way that Y+(t) = [Za'(t) XX(t)]. As a consequence, eqns (37a, b) can be replaced by the following equation: Y(t) = [~,(t)
+ W1 (t)ff,(t)]Y(t) + g(t)We(t )
(38)
where it is simple to show that .i,(t), I~,(t) and fg(t) are given by A(t):
g(t)=
(A(0t)
(o)
Av(t)H(t))
0)0 (39a-c)
gv(t)
183
matrix ,~(t) +
O(t, 7-) = It [A.(P) + ~G2(P)ql]dp
E[v(t,)
® v(te).
= [In+m ® ~)(t2,
••®
t2) ®"" ® O(tk, tl)]E[Y[k](tl)] (40)
where O(t,'r) is the fundamental matrix related to the
(41)
6 STATIONARY R E S P O N S E
In this section, the fundamental relationships between the C A M T or the A M T and the corresponding central moment or moments, introduced in the previous sections for the three different types of external excitation considered, will be considered for the case of stationary response. For this purpose, we assume that the deterministic matrices A and G of eqn (1) do not depend on time. Thus, the fundamental matrix defined in eqn (8) does not depend on the instants tl and t2 taken separately, but depends only on their difference tl - t2; that is, O(tl, t2) --- O(tl - t2). In the following, we analyze again, but in the stationary case, the three cases of external excitations considered before. 6.1 Deterministic external excitations
In this case, we consider that the external deterministic excitations f(t), after a transient temporal shift, assume a constant value Is- Then, the response moments of every order are characterized by stationary values that can be obtained starting from the differential equations governing them; for example, for the response means, starting from eqn (6), by putting zero as its first member, we obtain the stationary values E[Z] as
E [ Z ] = - A+
G2q
fs
(42)
By using an analogous procedure for the second order response moments, starting from eqn (14), we obtain the stationary value E[Z [2]] as follows: E[ zI2]] = -[A2 + G[2]q]-IQ2{E[Z] ® fs}
(43)
and, in general, for the kth order moments, starting from eqn (24), we obtain the stationary value E[Z [k]] as follows: E[Z [k]] = -
It is now clear that the system of differential equations given in eqn (38) has the same form as the one considered in Section 4; that is, a linear system excited by a parametric and an external normal white noise. Thus, the same considerations can be made in order to evaluate the A M T of Y(t) once its moments are known. That is, the following fundamental relationship holds:
1/2C,2(t)ql, that is;
{Ak + ~Qk[Qk-~ 1 ® ln][G[2lq ® I~ -2]] ),
x Qk{E[Z [/'-']] ® fs }
(44)
If the stationary means and the stationary second order moments are known, then it is possible, by means of eqn (13), to find the stationary second order central moments E[Z [2]] and, hence, by means of eqn (12), the second order stationary C A M T E[Z[2]]r ' (where ~-z : t2 - tt and the subscript indicates the variable on which the C A M T depends) as follows: E[Z[21]~, = [In ® O(T1)]E[Z [2J]
(45)
184
M. Di Paola, G. Falsone
We remember that the second order stationary C A M T coincides with the second order stationary correlations (and both depend on a single time variable "r1), while the second order central moments coincide with the second order cumulants (and both do not depend on time). Analogous particularization can be made for higher order statistics; for example, in the case of a zero-mean response process, the particularization of eqn (23) gives
E[z[k]]TI,7. 2..... Tk_l =
[I, ® O('7-1) ® ' ' " ®
@)("i-k_l)]E[ff~[k]] (46)
It is important to note that the kth A M T depends on k - 1 variables -~ = tj - q. 6.2 White noise external excitations
By taking into account eqn (26), if A, G and g are independent of t, as well as the white noise intensities ql, q2 and q~2, it is possible to evaluate the stationary statistics of the response. In the case of zero initial conditions, the response process has zero mean, while the second order stationary moments can be obtained by putting to zero the first member of eqn (32), that is E[Z [2]] = - [ A 2 + G[2]ql]-lgi2lq2
(47)
second order differential equation: X(t) + [2(w + aWl(t)]J((t) + [w2 +/3W2(t)]X(t ) = "7 (49) where ~ is the damping coefficient, w is the natural radian frequency, a, /3 and "7 are constants and Wl (t) and W2(t) are two independent white noises. The necessity to consider two parametric excitations, the first one acting on the velocity and the second one acting on the displacement, lies in the fact that, if only the first, Wl(t), is present, it can be shown that the stationary response is deterministic; while if only the second, W2(t), is present, no Wong-Zakai correction term arises. Hence, the stationary response is stochastic and it needs the presence of the Wong-Zakai correction term in the solution only if both the parametric white noises act on the system. By introducing the state variable vector z T ( t ) = [X(t)J((t)], eqn (49) can be rewritten as follows: Z(t) = [A + W1 (t)Gl + W2(t)G2]Z(t)+ where A
=
The moments of higher order can be obtained by putting zero to the first member of eqn (34), that is E[ Z[k]] = -
{1
},
Ak + ~Qk[Qk-I ® In][G[21ql ® I[~-211
-2{w
_032
'
G1 =
0
-~
The It6 differential equation corresponding to eqn (50) has the following form: dZ(t) =
(48)
I'
l
A+~G12ql Z(t)dt
+ [GIdB l + G2dB2]Z(t ) + fdt The stationary A M T of every order can be evaluated by particularizing eqn (33) which leads to an analogous expression for the A M T of that given in eqn (46) for the CAMT. 6.3 Filtered external excitations
Here we consider that the quantities H, Av and gv, appearing in Section 5, do not depend on t as well. Under this assumption, the response is stationary. Hence, once the matrices ii,, I] and g are introduced (as in Section 5), the analysis of the statistics of the stationary response Y can be made as seen in the previous subsection.
(52)
In fact, the Wong-Zakai correction term connected to the presence of W2(t) is zero because, as can be easily shown, G ~ - - 0 . Thus, the mean of the response is governed by the following differential equation: /~[Z(t)] =
[ 1 2 ] A + ~Glql E[Z(t)] + f
(53)
and it is characterized by the following stationary value: E[Z l = - IA' + ~ Gl2q, ]-1 f
(54)
The second order response moments are governed by the following differential equation:
E[Z [2](t)] 7 NUMERICAL EXAMPLES
= [A2 + Gl2]ql + G~Zlqz]E[Z [2](t)]
+ Q2{E[Z(t)I Q f}
(55)
where The first example considered is a linear oscillator excited by two parametric white noises and by an external constant excitation, and is governed by the following
(50)
(0 1) (0 0) :), f=(:)
× ~Qk[Qk-1 @In]{[G @gql2]E[z[k-l] l +E[Z[k-2]] ® g[Zlq2}
f
1
2
1
2
Statistics of excited linear systems 2E-4 -
-
C o r r e c t i v e terms included C o r r e c t i v e terms n e g l e c t e d
-
......
-
0.11
C o r r e c t i v e terms included C o r r e c t i v e terms neglected
-
......
X
A
0.0~
0
•~ 7E-21
185
'
i
VVv
,, v
o
O -IE-4
-2E-4
-0.0:
I 2
0
I 4
I 6
I 8
-0.10
I 10
I 0
I 4
2
T
,.¢
I 6
I 8
I 10
Fig. 1. Stationary correlation of X(t) of system (49); w = 5, ~= 0.1, ~' = 1, c~ =0.9,/3= 1, ql = q2 = 1, q|2 =0.
Fig. 3. Stationary correlation of X(t) of system (60); w = 5, ( = 0 . 1 , a = 0 - 9 , / 3 = 1, q| = 1.
The stationary value of E[ZP](t)] is given by
By introducing the vector Z(t) of the state variables, eqn (60) can be rewritten in the form of eqn (26), where
E [ Z [2]] =
-[A 2 +
Gl2]ql
+
G~Z]q2]-IQ2{E[Z] @f} (57)
In order to evaluate the stationary second order correlations of the response, we remember that they coincide with the corresponding CAMT; hence, we can use eqn (45), where the stationary second central moments are given by e [ z I21] = E[Z I21]- E[Z] E21
(58)
Finally, the fundamental matrix O('q) is given by O(rl) = exp{ [A + ~G~q,] ~-1}
(59)
In Figs 1 and 2, the exact displacement and velocity stationary correlations are reported for particular values of the coefficients that assure the stability of the system. In order to show the importance of the Wong-Zakai correction terms and the great errors that can be made if they are neglected, in the figures, the results are compared with those obtained by such neglect. The second example considered is an oscillator excited by parametric and external independent white noises and that is governed by the following differential equation: )((t) + [2~w + C~Wl(t)]8(t) + w2X(t) = flW2(t)
--~3 2
-2(w
'
G=
(:o)
,
g=
(;)
(61) In order to evaluate the stationary second correlations of the response, we remember that, as the system has zero-mean response, the second central moments are coincident with the second moments whose stationary value is given by eqn (47). Once the moments are known, the correlations can be obtained by means of the transition matrix O(~-) which has the same expression as that given in eqn (59) (in which G1 is replaced by G). In Figs 3 and 4, the exact displacement and velocity second order correlations are reported for particular values of the coefficients, compared again with those evaluated without taking into account the Wong-Zakai correction terms. Finally, the third example considered is represented by the following oscillator: J((t) + [2~w + a W1 (t)])((t) + J X ( t ) = 3F(t)
(62)
where F(t) is a filtered process which is the response of
C o r r e c t i v e terms i n c l u d e d C o r r e c t i v e terms n e g l e c t e d
- ......
4E-3
(60)
(0 ,)
A=
3-
C o r r e c t i v e terms included C o r r e c t i v e terms n e g l e c t e d
- ......
2 "~
2E-3
.X
O
o
~ IE-19
-2E-3
-4E-3| 0
,., ? V v l
I
2
,
v--
I
I
4
6
I
8
I
10
T
Fig. 2. Stationary correlation of .~(t) of system (49).
-3/ 0
I 2
[ 4
I 6
[ 8
1 10
T
Fig. 4. Stationary correlation of ~'(t) of system (60).
186
M. Di Paola, G. Falsone
1.2E-3~
- ......
0.020 -
Corrective terms i n c l u d e d Corrective t e r m s n e g l e c t e d
0.015
8.0E-4
Corrective t e r m s i n c l u d e d Corrective terms neglected
......
•X 0.01o
X "~ 4.0E-4
," 0.005
e. o
O
2.7E-20 O
r,9 -4.0E-4 -8.0E-4
0
2
4
6
8
- - J
10
AA °it,,// V o
A
2
A
A
I 4
I 6
A~
A_~
I 8
I 10
T
Fig. 5. Stationary correlation of X(t) of system (62); w = 5, ~=0.1, a=0-9, f l = l , q l = q 2 = l , ql2=0, WF=2"5, ~F = 0"1, ~ :
1.
where 1- 2 A2 = [~k + ~G2ql] ® I4 + I4 ® [~k + ~G ql]
the following filter: F(t) + 2(FWFP(t) + w~F(t) =- "~W2(t)
(63)
These two equations can be unified into an equation having the same expression of that given into eqn (38), where
A:(: +")
Y=
Fig. 6. Stationary correlation of )((t) of system (62).
(0 1 0 0) /:ooo) o=(: :) /°>(!) -2{w
3
0
0
0
0
1
0
0
__2
0 0
0
0 0
0
0 0
E[~([2]]0" = [In ® ~)(0-)]E[Y[zl]
(67)
where the fundamental matrix ~)(0-) is given by (68)
For this example, the stationary second correlations of the displacement and of the velocity of the oscillator are reported in Figs 5 and 6. We see again that neglecting the Wong-Zakai correction terms may lead to very different results.
8 CONCLUSIONS
(64)
gF
Even in this case, for zero initial conditions, the mean of the response is zero; hence, the second moments coincide with the second central moments, while the second AMT coincides with the second correlations. In order to find the stationary response second CAMT, it is first necessary to find the stationary second central moments that, as stated previously, coincide with the second moments; these last ones are obtained by means of the following relationship: E[Y[21] = -[~,2 + Cj[2lql]-lg[2]q2
The CAMT are given by
O('r) = exp{ I-~ + ~ C,2ql]'r }
__2~F WF
-a
(66)
(65)
In this paper, a method has been introduced in order to evaluate the higher order statistic (correlations and CAMT) of the response of linear systems excited by external stochastic loads and linearly parametric white noise excitations. In this method, a very important role is played by the fundamental matrix related to the dynamical matrices of the systems, that, due to the presence of linearly parametric white noise excitations, must take into account the presence of Wong-Zakai correction terms. By applying this method, the response moments at a fixed time are first evaluated by means of an integration (for a non-stationary response) or a simple matrix inversion (in the stationary case), and then the higher order statistics are evaluated by means of simple relationships in which the fundamental matrices appear. The relationships evaluated here are an extension of those in the literature that hold in the case of linear systems and purely external excitations. The extension to the case in which linearly parametric white noise is also present in the motion equation is very important because, as can be seen in the companion paper 6, many problems can be included in this class of systems.
Statistics of excited linear systems
187
REFERENCES
The Kronecker product has the following properties:
1. Lin, Y. K., Probabilistic Theory of Structural Dynamics. McGraw-Hill, New York, 1967. 2. Stratonovich, R. L., Topics in the Theory of Random Noise. Gordon and Breach, New York, 1963. 3. Falsone, G., Cumulants and correlations for linear systems under non-stationary delta-correlated processes. Probabilistic Engineering Mechanics, 1994, 9, 157-165. 4. Muller, P. C. and Schielen, W. O., Linear Vibrations. Martinus Nijhoff, Dordrecht, 1985. 5. Di Paola, M. and Falsone, G., Response in terms of correlations of linear systems under linearly parametric white noises. Proceedings of the IUTAM Conference on Nonlinear Stochastic Mechanics, Trondheim, Norway, July 1995. 6. Di Paola, M. and Falsone, G., Multiple times response statistics of M D O F linear systems under polynomials of filtered normal white noise. Probabilistic Engineering Mechanics, 1997, 12, 189-196. 7. Benaroya, H. and Rehak, M., Response and stability of a random differential equation. Part I: moment equation method. Journal of Applied Mechanics ASME, 1989, 56, 192-195. 8. Hou, Z. K. and Iwan, W. D., Nonstationary response of linear systems under uncorrelated parametric and external excitations. Probabilistic Engineering Mechanics, 1991, 6, 74-81. 9. Graham, A., Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Chichester, UK, 1981. 10. Grigoriu, M., White noise processes. Journal of the Engineering and Mechanical Division ASCE, 1987, 119, 757-765. 11. Wong, E. and Zakai, M., On the relation between ordinary and stochastic differential calculus. International Journal of Engineering Science, 1965, 3, 213-229. 12. Itr, K., Stochastic differential equations. Memoirs of the American Mathematical Society, 1951, 4. 13. Di Paola, M., Moments of non-linear systems. In Proceedings of the 5th ASCE Conference on Probabilistic Methods in Civil Engineering, Blacksburg, 1988, pp. 285-288.
M ® ( N ® P ) -----( M ® N ) (M+N)
=M®P+M®Q+N®P+NQQ
(A3)
(M ® N)(P ® Q) = (MP) ® (NQ)
(A4)
(M ® N) T = M r ® N T
(A5)
(m ® N) -1 = M -1 @ N -1
(A6)
where P and Q are two given matrices. Equations (A4) and (A6) are valid, provided the various quantities exist. A fundamental role in the Kronecker algebra is played by the so-called permutation matrix El,j, which is a square matrix of order (ij x/j), consisting of (j × i) arrays of elementary submatrices E kl of order (i ×j); each of these matrices has the value one in the (k, l)th position and the value zero in all the other positions. Hence, Ei,j has the following expression:
E 12
E 22
...
.
.
".
E y2 .
E 2i
...
E j~
Ei,) =
\ E li
The Kronecker product of the matrices M and N, of order (p x q) and (s x t), respectively, is denoted by M ® N and gives the matrix of order (ps x qt) defined by the following relationship:
m ® N ----
ml2N
'
"
\mplN
"-" mlqN "..
mp2N ...
"
mpqN
with mij being the (i,j)th element of M.
(A7)
where, for example, the matrix E 12 is given by
0
0
•
.
0
0
-..
E 12 =
(A8) "..
-..
It is important to note that the matrix Ei,j exhibits a single 1 in each row and in each column, placed following the rule given in equation (A7). Its name is due to the fact that the following fundamental relationship holds: ® N)Et, q
(A9)
The matrices Qk, that are present in this paper, are strictly connected with the permutation matrices by means of the following relationship:
A.1 Some properties of Kronecker algebra
mxlN
(A2)
® ( V + Q)
N ® M = Ep, s(M APPENDIX A
®P
(A1)
k-1
Qk = Z
E,,-j,,j
(A10)
j=0
It is easy to show that these matrices exhibit only zeros and natural numbers in such a way that the sum of the elements of each row, or of each column, is equal to k. The matrices Qk are present in the expressions regarding the derivatives of vectors. In fact, let us consider an n-vector Z and the differential operation Vz, presenting the element O/OZj in the jth position;
188
M. Di Paola, G. Falsone
thus, the following relationships hold: V ~ ® Z ----In
(A11)
V~ ~) Z [k] ~ Qk(Z [k-1] ~ In)
I n is the identity matrix o f order n x n. Finally, if Qk premultiplies the kth K r o n e c k e r power o f a vector a, it
gives Qka [k] = ka
= Z (k-l] ® In + Z [k-2] ® In ® Z + ..- I n ® Z [k-l] (A12)
(A13)