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Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations
Probabilistic Engineering Mechanics 27 (2012) 75–81
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Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations Cho W. Solomon To Department of Mechanical Engineering, University of Nebraska, N104 Scott Engineering Center, Lincoln, NE 68588-0656, USA
article
info
Article history: Available online 8 May 2011 Keywords: Nonlinear Multi-degree-of-freedom systems Uncertain properties Non-Gaussian Nonstationary random excitations
abstract The investigation reported in this paper is concerned with the development of an approach for response analysis of multi-degree-of-freedom (mdof) nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The developed approach makes use of the stochastic central difference (SCD) method, time co-ordinate transformation (TCT), and adaptive time schemes (ATS). It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations is considered. Computed results obtained for the system with and without uncertain natural frequencies, and under Gaussian and nonGaussian nonstationary random excitations are presented. It is concluded that the approach is relatively simple, accurate and efficient to apply. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction For safety and economic reasons, many modern structural systems such as those that house nuclear reactors, tall buildings, naval and aerospace installations, and their components have to be designed to withstand various intensive and complicated loadings which can only be realistically modeled as random processes. Until very recently, these latter processes have generally been treated as Gaussian random processes. In practice, particularly in the designs and analysis of space shuttles and many other vehicles, there is a need to deal with the excitation processes as non-Gaussian ones. Specifically, the MIL-STD-810F DOD Test Method Standard [1], and Defence Standard 00-35 of the Ministry of Defence [2] require consideration of the non-Gaussian behavior in simulation and testing environments. In offshore structure design such as the tension-leg platform the response has been known to be nonGaussian. Wind-generated waves in finite water depth have long been recognized as non-Gaussian random processes. In addition, in the context of structural dynamics and particularly in the dynamic analysis of laminated composite plate and shell structures, uncertain system properties such as Young’s modulus of elasticity and damping have presented a formidable challenge to the designer. While the analysis of non-Gaussian random processes [3–5] and dynamic systems with uncertain properties [6–8] have generated
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considerable amount of interests in recent years in the field of random vibration and it is understood that there are approaches available in the literature for the analysis of nonlinear systems excited by non-Gaussian random disturbances it should be emphasized that these approaches are not applicable to the efficient analysis of geometrically complicated nonlinear systems represented by the FEM and under non-Gaussian nonstationary random excitations. Those entirely or partially based on the application of Monte Carlo simulation (MCS) are relatively inefficient for nonlinear systems with a large number of dof, if not computationally infeasible. Consequently, there is a definite need to provide an approach for the efficient, relatively general, and accurate analysis of nonlinear multi-degree-of-freedom (mdof) systems with uncertain properties or parameters of large variations and subjected to non-Gaussian random excitations. The investigation being reported here was therefore concerned with addressing such a need to provide a means of efficient and accurate analysis, and predicting responses of highly nonlinear mdof systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The emphasis of the investigation was on the formulation and application of mdof nonlinear systems idealized by the FEM. This is a natural extension of the work reported recently by the author [9]. Owing to their abilities to provide efficient and accurate responses of mdof nonlinear systems and discretized nonlinear shell structures under Gaussian random excitations, the stochastic central difference (SCD) method [6], the time co-ordinate transformation (TCT) [10,11], and the adaptive time schemes (ATS) [12] were combined to form a novel procedure that has been developed in the investigation.
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C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitation is considered. This same nonlinear tdof system without uncertain properties and subjected to a non-Gaussian nonstationary random excitation was studied and reported by the author recently [9]. In Section 2, the SCD method for the mdof nonlinear systems under nonstationary Gaussian random excitations is included for completeness. Section 3 is concerned with the SCD method for the mdof nonlinear systems with uncertain properties and under Gaussian nonstationary random excitations. The theoretical development of the approach for nonlinear mdof systems with uncertain properties and under non-Gaussian nonstationary random excitations is presented in Section 4. Computed results and comparisons between those obtained for systems with and without uncertain properties of large variations and under Gaussian and non-Gaussian nonstationary random excitations are included in Section 5. Conclusion is presented in the final section, Section 6. 2. Nonlinear systems under Gaussian excitations For completeness and in order to provide a foundation of the formulation and presentation of the systems with uncertain properties of large variations as well as subjected to non-Gaussian random excitations, to be presented in subsequent sections, analysis of mdof nonlinear systems under Gaussian random excitations is included in this section. The governing matrix equation of motion is given by M x¨ + C x˙ + Kx = P
(1)
where M , C and K are, respectively the nonlinear assembled mass, damping and stiffness matrices of the system; P is the external random excitation vector which, in general, includes Gaussian nonstationary random forces; x is the generalized displacement vector; the over-dot and double over-dot denote, respectively the first and second derivatives with respect to time t. A typical example for P is the product of modulating function vector and a zero-mean Gaussian white noise process. Discretizing Eq. (1) in the time domain, one has Ms x¨ s + Cs x˙ s + Ks xs = Ps ,
(2)
where the subscript s denotes the time step; for instance, xs is the value of x at time step ts such that the time step size 1t = ts+1 − ts and t0 = 0. While the recursive equations for mean squares of displacement vector have been obtained and presented in [6], for example, in order to provide a better understand in the derivation of recursive mean squares of displacements for nonlinear mdof systems subjected to nonstationary non-Gaussian random excitations and for completeness detailed steps are presented in the following. By Taylor’s theorem, the displacement vector at the next time step ts+1 is xs+1 = x(t + 1t ) = xs + (1t )˙xs +
1
xs−1 = x(t − 1t ) = xs − (1t )˙xs +
1
(1t )2 x¨ s + · · · .
2 Similarly, the displacement vector at the previous time step is
By substituting the last two equations into Eq. (2), and eliminating the velocity and acceleration terms, one has xs+1 = (1t )2 N1 Ps + N2 xs + N3 xs−1 , where the coefficient matrices are
[
1
, [ ] 1 N3 = N1 (1t )Cs − Ms . N1 = Ms +
2
(1t )Cs
] −1
N2 = N1 2Ms − (1t )2 Ks ,
2
The last recursive displacement vector is generally known as the central difference algorithm. Taking the transpose of the last recursive displacement vector, one obtains xTs+1 = (1t )2 PsT N1T + xTs N2T + xTs−1 N3T , where the superscript T designates ‘‘the transpose of’’. Performing the matrix operation xs+1 xTs+1 , taking the ensemble average or mathematical expectation of the resulting matrix, and re-arranging terms, one can show that the mean square matrix of generalized displacement vector becomes Rs+1 = N2 Rs N2T + N3 Rs−1 N3T + (1t )4 N1 Bs N1T + N2 Ds N3T
+ N3 DTs N2T + (1t )2 N2 ⟨xs PsT ⟩N1T + (1t )2 N1 ⟨Ps xTs ⟩N2T + (1t )2 N3 ⟨xs−1 PsT ⟩N1T + (1t )2 N1 ⟨Ps xTs−1 ⟩N3T ,
(3)
in which Rs = ⟨xs xTs ⟩, Ps PsT
Bs = ⟨
(4a)
⟩,
(4b)
xs xTs−1
(4c)
Ds = ⟨
⟩,
⟨xs+1 ⟩ = (1t )2 N1 ⟨Ps ⟩ + N2 ⟨xs ⟩ + N3 ⟨xs−1 ⟩,
(4d)
where the angular brackets denote the mathematical expectation or ensemble average. Eq. (3) contains the recursive covariance matrix for three consecutive time steps. The terms containing Ps xTs , Bs , Ds and their transposes on the right-hand side (RHS) of Eq. (3) require further algebraic manipulation and expansion. Without loss of generality, the external random excitation vector P in Eq. (1) may be defined by p = Θ (t )w(t ),
(5)
where Θ (t ) is the vector of deterministic modulating functions in time t, every element or entry of the deterministic modulating function vector can be written as
θi (t ) = Eri (e−α1i t − e−α2i t ) (6) in which α1i , i = 1, 2, 3, . . ., and α2i are positive constants satisfying the condition, α1i ≤ α2i , and Eri is a constant applied to normalize θi (t ) such that max{θi (t )} = 1.0; w(t ) is the zero-mean Gaussian white noise whose discrete variance,
⟨ws2 (0)⟩ = 2π S0 δ(0) (7) where δ(·) is the Kronecker delta function such that δ(0) = 1; S0 is
xs+1 − xs−1 = 2(1t )˙xs ,
the spectral density of the discrete Gaussian white noise (DGWN) process. It may be appropriate to note that while the excitation process is assumed to be Gaussian at every time step the nonlinear responses at every time step are not necessary Gaussian. For the case with non-Gaussian random excitations, it will be considered in Section 4. Of course, for stationary random excitations θi (t ) = 1 which are just special cases of Eq. (6). Applying Eqs. (5) through (7) to Eq. (3) and after some algebraic manipulation, one can show that
with a leading error of order (1t )2 .
Rs+1 = R(s1) + R(s2) + R(s3) ,
(1t )2 x¨ s − · · · .
2 By adding and subtracting the above two equations, one has, respectively xs+1 + xs−1 = 2xs + (1t )2 x¨s , and
(8)
C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
where R(s1) = N2 Rs N2T + N3 Rs−1 N3T ,
R(s2) = N2 Ds N3T + N3 DTs N2T ,
R(s3) = (1t )4 N1 Bs N1T . Eq. (8) is the recursive mean square of generalized displacement vector of the nonlinear system under discrete Gaussian nonstationary random excitations. It is noted that the parameter matrices N1 , N2 and N3 have to be updated at every time step since they are time-dependent for nonlinear systems. If the system is linear these parameter matrices are constant. The expression for the recursive covariance matrix of the generalized displacement vector can be obtained from Eq. (8) as Us+1 = Us(1) + Us(2) + Us(3) ,
(9)
where (1)
Us
=
77
where the angular brackets with a superscript x designate the ensemble average in the spatial domain while the inner angular brackets denote the ensemble average in the time domain. Eq. (12) is referred to as the extended average deterministic central difference (EADCD) scheme [7] and is used in the updating process for systems with finite deformations. Superficially, if Ps is Gaussian the updating process cannot be performed since ⟨Ps ⟩ = 0. However, starting strategies are available for this case and have been considered in [8]. By taking the ensemble average of Eq. (8) in the spatial domain, the recursive mean square matrix of the generalized displacement vector, or simply referred to as mean square matrix, becomes
⟨Rs+1 ⟩x = ⟨Rs+1 ⟩(1) + ⟨Rs+1 ⟩(n) , x
N2 Us N2T 4
N3 Us−1 N3T N1 Bs N1T xs xs T Am s
+
Us(3) = (1t )
,
(2)
Us
=
T N2 Am s N3
+
(
)
T T N3 Am N2 s
,
,
Us = Rs − ⟨ ⟩⟨ ⟩ ,
Consider the matrix equation of motion for a general nonlinear mdof system with uncertain properties and under Gaussian random excitations [7,8], (10)
where the symbols have already been defined in the last section except that the term r Φ is the associated stiffness with uncertain properties and whose definition will be presented later in this section. The temporally discretized form of Eq. (10) becomes (11)
in which the subscript s is an integer denoting the time step as defined in Section 2 above. For simple illustration, consider the uncertain property or spatially homogeneous stochastic modulus of elasticity which can be expressed as E˜ = E + r, where E is the deterministic component of Young’s modulus of elasticity and r is the spatially random component with zero ensemble average in the spatial domain. A practical example is the soil–structure interaction problem in which Young’s modulus of elasticity of soil varies stochastically in the spatial domain due to various rock formations. A further practical example can be found in laminated composite plate and shell structures where Young’s modulus of elasticity of every layer may vary spatially due to small air pockets created during the manufacturing process. This, in turn, causes the system natural frequencies to vary spatially. Then Φs in Eq. (11) is equal to Ks /E, in which Ks is the assembled tangent stiffness matrix for the system at ts without the spatial uncertainty. In the general case, the assembled tangent stiffness matrix at ts , Ks in the context of nonlinear finite element analysis (FEA) can readily be obtained in most, if not all, commercially available packages. It should be emphasized that many stochastic properties other than Young’s modulus of elasticity can be dealt with in a similar manner. Further, the uncertain properties considered above are of large variations. By taking the temporal and spatial ensemble averages of the central difference algorithm that was employed to the derivation of Eq. (3) in the foregoing, one obtains
⟨⟨xs+1 ⟩⟩x = (1t )2 N1 ⟨Ps ⟩ + N2 ⟨⟨xs ⟩⟩x + N3 ⟨⟨xs−1 ⟩⟩x ,
⟨Rs+1 ⟩
= =
x
N2 Rs N2T N2 Ds x N3T
⟨ ⟩
(12)
and
+ N3 ⟨Rs−1 ⟩x N3T ,
N3 DTs x N2T x xs Ps T N1T 1t 2 N1 x xs−1 Ps T N1T
⟨ ⟩
+
⟨
+ (1t ) N2 ⟨⟨ ( ) ⟩⟩ 2
3. Nonlinear systems with uncertain properties and under Gaussian excitations
Ms x¨ s + Cs x˙ s + (Ks + r Φs )xs = Ps
(1)
⟨Rs+1 ⟩
Before leaving this section it is noted that an efficient route of computing recursive covariances and mean squares of generalized displacement vector of the nonlinear system is to first apply Eq. (9) for the covariance matrix Us+1 and then evaluate the mean square matrix Rs and so on.
M x¨ + C x˙ + (K + r Φ )x = P
⟨Rs+1 ⟩x = ⟨⟨xs+1 xTs+1 ⟩⟩ , (n)
= Ds − ⟨xs ⟩⟨xs−1 ⟩T .
(13)
in which the expression on the left-hand side (LHS) is
+ (1t )2 N3 ⟨⟨
(14a)
⟩
+(
)
x
⟨⟨(Ps )xTs ⟩⟩ N2T
( ) ⟩⟩ x
+ (1t ) N1 ⟨⟨(Ps )xTs−1 ⟩⟩ N3T + (1t )4 N1 ⟨Bs ⟩x N1T . 2
(14b)
Eq. (13) is similar to (3) except that application of the ensemble averaging in the spatial domain has been made. The RHS of Eq. (13) consists of two terms. Mathematically, the first term ⟨Rs+1 ⟩(1) is not affected by the non-Gaussian property of the excitations while the second term includes those depend on non-Gaussian characteristics of the excitation processes. For the case with nonGaussian excitations Eq. (14b) has to be further evaluated. This latter case will be considered in Section 4. Returning to the present case, if Ps is a zero-mean Gaussian random excitation vector Eqs. (12) and (13) can further be simplified. In particular, after some lengthy algebraic manipulation, Eq. (13) reduces to
⟨Rs+1 ⟩x = N2 ⟨Rs ⟩x N2T + N3 ⟨Rs−1 ⟩x N3T + (1t )4 N1 ⟨Bs ⟩x N1T + N2 ⟨Ds ⟩x N3T + N3 ⟨DTs ⟩x N2T ,
(15)
where
⟨Bs ⟩x = Brs + Bx2 Φs ⟨Rs ⟩x ΦsT ,
(16a)
x
⟨Ds ⟩x = ⟨⟨xs xTs−1 ⟩⟩ = N2 ⟨Rs−1 ⟩x + N3 ⟨DTs−1 ⟩x ,
(16b)
Brs
= ⟨Ps (Ps ) ⟩,
(16c)
BX2
= ⟨r ⟩ .
(16d)
T
2 x
Eq. (15) is the recursive mean square of generalized displacements of the spatially and temporally stochastic nonlinear systems under temporally stochastic excitations. Note that Eq. (15) can be easily extended to cases with temporally and spatially stochastic excitations but it is not pursued here for brevity. The expression for the recursive covariance matrix associated with the recursive mean square matrix of Eq. (15) can be simply obtained, after some algebraic manipulation, as [7,9]
⟨Us+1 ⟩x = ⟨Us+1 ⟩(1) + ⟨Us+1 ⟩(2) + ⟨Us+1 ⟩(3) ,
(17)
where
⟨Us+1 ⟩(1) = (1t )4 N1 [Brs + Bx2 Φs (⟨Us ⟩x + ⟨⟨xs ⟩⟩x ⟨⟨xTs ⟩⟩ )ΦsT ]N1T , x
⟨Us+1 ⟩(2) = N2 ⟨Us ⟩x N2T + N3 ⟨Us−1 ⟩x N3T , x T m T x T ⟨Us+1 ⟩3 = N2 ⟨Am s ⟩ N3 + N3 ⟨(As ) ⟩ N2 , x x m T x ⟨Am s ⟩ = N2 ⟨Us−1 ⟩ + N3 ⟨(As−1 ) ⟩ x x T x = ⟨Ds ⟩ − ⟨⟨xs ⟩⟩ ⟨⟨xs−1 ⟩⟩ , x
⟨Us ⟩x = ⟨Rs ⟩x − ⟨⟨xs ⟩⟩x ⟨⟨xTs ⟩⟩ .
(18a) (18b)
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C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
Before leaving this section it should be noted that the foregoing procedure has been generalized to include deterministic components in the excitation vector and applied to the nonlinear random response analysis of shell structures represented by the FEM [7,13]. Further, an efficient route of computing recursive mean squares and covariances of generalized displacement vector is to first apply Eq. (17) for ⟨Us+1 ⟩x and then determine the mean square matrix ⟨Rs ⟩x by Eq. (18b). 4. Nonlinear systems with uncertain properties and under nonGaussian random excitations The SCD method for nonlinear mdof systems with uncertain properties of large variations and subjected to temporally Gaussian random excitations presented in Section 3 is extended to include the case with non-Gaussian random excitations in Section 4.1. The non-Gaussian random excitation model, TCT technique, and ATS are briefly introduced in Sections 4.2 through 4.4, respectively.
system without uncertain property presented in [9], the following specific non-Gaussian nonstationary random excitation process is employed. Consider the discretized excitation vector
Ps = P1s
P2s
P3s
···
Pns
T
,
(21)
where Pis is the ith element or entry of the non-Gaussian random excitation vector at time step ts . In the present investigation,
w 2 (ts ) w2 Pis = Pis (ts ) = [θi (ts )]2 √ = (θis )2 √ s . 6π S0 6π S0
(22)
The corresponding ensemble average is defined by
⟨Pis ⟩ =
2π S0 /3θis2 .
(23)
The corresponding mean square becomes
⟨Pis2 ⟩ = 2π S0 θis4 .
(24)
Finally, the corresponding variance, however, is obtained as 4.1. Uncertain systems under non-Gaussian random excitations
⟨σis2 ⟩ = (4π S0 /3)θis4 .
In this case, the recursive expressions for the mean square matrix of generalized displacements are identical to those defined in Eq. (13) except that the second term on its RHS, that is Eq. (14b), has to be evaluated differently. As, in general, ⟨Ps ⟩ and ⟨xs ⟩ are jointly non-Gaussian such that the terms on the RHS of Eq. (14b) have to be dealt with as in the following. Firstly, let one writes
With the above results, Eqs. (13) and (20) can now be evaluated. It should be noted that the stochastic loading vector P, whose discrete form is defined by Eq. (21), represents the random excitations applied at the discrete masses in the case of lumped parameter models. In the case of distributed parameter models considered in the FEA, for example, P is the assembled applied nodal stochastic excitation vector. While the development of the present approach can easily be extended to cases involving spatially correlated excitations it is beyond the scope of the present investigation and will not be pursued in this report.
⟨Rs ⟩(n1) = (1t )2 N2 ⟨⟨xs ⟩⟩x ⟨PsT ⟩N1T , (n2)
⟨Rs ⟩
(n3)
⟨Rs ⟩
(n4)
⟨Rs ⟩
= (1t ) N3 ⟨⟨xs−1 ⟩⟩ ⟨ ⟩ 2
x
x
= N2 ⟨Ds ⟩
N3T
x
in which ⟨Ds ⟩ = ⟨⟨ N3 ⟨DTs−1 ⟩x , and
N1T
(19a)
,
(19b)
,
= (1t ) N1 ⟨Bs ⟩ 4
PsT
x
N1T
xs xTs−1
, x
⟩⟩ = (1t ) N1 ⟨Ps ⟩ ⟨⟨ 2
xTs−1
x
(19c)
4.3. Time co-ordinate transformation
(19d)
For stiff mdof nonlinear systems or systems with large numbers of dof, typically encountered in the FEA, a computational strategy, known as the TCT technique has been proposed by the author [10] to deal with computational instability [7] and efficiency, and its use in the FEA has been demonstrated by the author and his associate [11,13], for example. Another advantage of applying the TCT technique in the FEA is the fact that the finite element size or mesh dimensions can be relatively much coarser for accurate response computations since the dimensionless highest natural frequency is always equal to unity. Why this is so will be explained later in this subsection. For completeness, the TCT is briefly introduced in the following. In the TCT it is assumed that the linear counter part of the stiff nonlinear system governed by Eq. (2) or (11) has its highest natural frequency Ωs or simply written as Ω . Within every time step the system can be considered as linear. Therefore, one can apply the following matrix equation
x
⟩⟩ + N2 ⟨Rs−1 ⟩ +
⟨Bs ⟩x = Brs + Bx2 Φs ⟨Rs ⟩x ΦsT , where Brs and Bx2 have already been defined in Eqs. (16c) and (16d), respectively. Secondly, with the definitions in Eq. (19) the term on the LHS of Eq. (14b) becomes
⟨Rs+1 ⟩(n) = ⟨Rs ⟩(n) + (⟨Rs ⟩(n) )T + ⟨Rs ⟩(n4) ,
(25)
(20)
where
⟨Rs ⟩(n) = ⟨Rs ⟩(n1) + ⟨Rs ⟩(n2) + ⟨Rs ⟩(n3) . With Eqs. (20) and (13), and the particular definition of nonGaussian random excitations Ps the recursive mean square matrix for a mdof nonlinear system with uncertain properties of large variations and under non-Gaussian random excitations can be determined. Similar to the derivation of Eq. (17), the recursive covariance matrix associated with the mean square matrix in Eq. (13) can be obtained accordingly but it is not performed here for brevity. 4.2. Non-Gaussian random excitation models Various non-Gaussian random excitation processes can be found in [4,5], for example. Of course, there are many non-Gaussian excitation models that are available in the literature. However, for direct comparison to results of a tdof asymmetric nonlinear
Msτ x¨ sτ + (Csτ )˙xsτ + (Ksτ )xsτ = Psτ ,
(26)
where the over-dot and double over-dot now denote, respectively the first and second order derivatives with respect to dimensionless time τ which is related to Ω by τ = Ω t. The second subscript τ refers to the quantity in the τ -domain. Thus, for example, Ksτ is the stiffness matrix at dimensionless time step τs , so that Ksτ = Ks /Ω 2 . The remaining matrices and vector in Eq. (26) are Msτ = Ms , Csτ = Cs /Ω , and Psτ = Ps /Ω 2 . In other words, in Eq. (26) the highest dimensionless natural frequency based on the matrix Ms−τ 1 Ksτ is always equal to unity. Eqs. (13) and (20) can now be applied to compute the recursive mean square matrix of the generalized displacement vector of the discretized system in the dimensionless time domain defined by
C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
79
Eq. (26). It should be noted that the time step size 1t in Eqs. (13) and (20) now should be replaced with the dimensionless time step size 1τ while the mass matrix Ms in Eqs. (13) and (20) should be replaced with Msτ , for example. Of course, ⟨Rs ⟩x is now ⟨Rsτ ⟩x and so on. Once the recursive mean square matrices of displacements in the τ -domain, ⟨Rsτ ⟩x are obtained using Eqs. (13) and (20), they are converted back to the original t-domain by the following relation [7,8],
⟨Rs ⟩x = Ω ⟨Rsτ ⟩x .
(27)
4.4. Adaptive time schemes The basic idea of these schemes is that for nonlinear systems whose natural frequencies are functions of time and therefore at every time step these natural frequencies have to be updated. In other words, the nonlinear stiffness matrix has to be updated at every time step. The updating strategies are called the adaptive time schemes (ATS). Three ATS have been studied by the author and his associate, and presented in [12] for the response computation and analysis of highly nonlinear mdof systems subjected to Gaussian random excitations. In the latter reference it was found that in terms of computational effectiveness, efficiency and accuracy, the SCDTATS was a better strategy for stiff systems. This was further confirmed in [13]. Therefore, in the presently proposed approach only the SCD-TATS is adopted. The SCD-TATS denotes that in the application of the SCD method the TCT is applied once at the beginning of the computation process before the application of the ATS. It may be mentioned that strictly speaking, the ATS used in the present investigation is different from that in [12,13] in that no spatially stochastic averaging was applied in the latter references. The ATS applied in the present investigation is therefore called the modified ATS [7,8]. But for conciseness and without confusion here it is simply referred to as the ATS. 5. Asymmetric nonlinear stiffness system with uncertain natural frequencies
Fig. 1. Two-dof nonlinear system with base excitation.
random disturbances have been verified by the Monte Carlo simulation (MCS) data which were presented in [14]. This indicated that Eqs. (15) and (16) are correct. Thus, in the present studies computed results of the same system with and without uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations are obtained for comparison. In Section 5.1 system parameter matrices and other pertinent data are provided while computed results and comparison studies are included in Section 5.2. 5.1. Two-dof uncertain system with asymmetric stiffness When the uncertainties are disregarded and the excitations are Gaussian the tdof nonlinear system examined here reduces to that considered in [9,14]. The sketch for this system is shown in Fig. 1. For completeness, the matrix equation of motion in [9,14] is included in the following, M x¨ + C x˙ + Kx = P where the parameter matrices are defined as
[ The nonlinear mdof system with uncertain properties and subjected non-Gaussian random excitations considered in Section 4 is general in the sense that nonlinearities in the damping and restoring forcing terms can take a variety of different forms, the uncertain properties can be of large variations, and the non-Gaussian random excitations can include many other types. However, for demonstration of application of the approach developed above and availability of results for direct comparison to those presented in [9], a tdof nonlinear asymmetric system with uncertain natural frequencies and under a non-Gaussian nonstationary random excitation is investigated. It should be emphasized that this system is employed to demonstrate the approach presented in the foregoing since the same system without uncertainty have already been studied previously and these results can be used for direct comparison. It is not employed here to explain nonlinear uncertainty of soil-structure interaction. The uncertain natural frequencies are translated into uncertain stiffness matrix of the system which is directly related to the spatially stochastic variable r in Eq. (10). The non-Gaussian nonstationary random excitations defined in Section 4.2 are applied in this section. It may be appropriate to note that mean square responses from the nonlinear tdof system without uncertain properties and under a non-Gaussian nonstationary random excitation was studied and reported in [9]. These responses have been compared with those of the same system under Gaussian nonstationary random excitations. The results by applying Eqs. (15) and (16) for the same tdof nonlinear system excited by Gaussian
]
1 0
M =
0 , 1
(28a)
2ζ1 W C = −2ζ1 W
] −2µζ2 , 2(1 + µ)ζ2
[
[
k11 k21
K =
]
k12 , k22
(28b)
k11 = W 2 ,
(28c)
k12 = −µ + µηx2 − µε x22 , k21 = −W 2 , K = KL + KNL ,
k22 = 1 + µ + (1 − µ)ηx2 + (1 + µ)ε x22 ,
x = x1
x2
T
,
in which the subscripts L and NL denote, respectively the linear and nonlinear stiffness matrices such that W2 −W 2
[ KL =
] −µ , 1+µ
] µ(η − ε x2 )x2 , [(1 − µ)η + (1 + µ)εx2 ]x2 m22 ω1 µ= , W = , τ¯ = ω2 t , m11 ω2 [
KNL =
ω1 = ci mii
0 0
k1
,
ω2 =
= 2ζi ωi ,
i = 1, 2;
m11
k2
m22
,
P =
θ1 (τ¯ )w(τ¯ ) 0
,
(29)
C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
Ms x¨ s + Cs x˙ s + (Kse + r Φse )xs = Ps
(30)
where the equivalent linear stiffness matrix is obtained as Kse = Ks0 + Ks0 = KL =
[
0 0
[
x
2µη⟨⟨x2s ⟩⟩ − 3µε⟨⟨ ⟩⟩ x , 2(1 − µ)η⟨⟨x2s ⟩⟩x + 3(1 + µ)ε⟨⟨x22s ⟩⟩ x
2
W −W 2
x22s
]
−µ , 1+µ
xs =
x1s x2s
]
(Responses of a nonlinear 2dof system) 0.04 0.035 Mean square of X1
where η and ε are the nonlinear parameters of the system; k1 and c1 are the stiffness and damping coefficients between the first mass and base where the excitation is applied; and k2 and c2 are the stiffness and damping constants between the first and second masses. In Eq. (29), θ1 (τ¯ ) is the deterministic modulating function in dimensionless time as defined in Eq. (22) based on Eq. (6). With reference to Fig. 1, x1 = y1 − y0 and x2 = y2 − y1 are the relative displacements which are known as the inter-story drifts in the field of earthquake engineering while y0 , y1 and y2 are the absolute displacements. Note that the above system nonlinear stiffness matrix is asymmetric. The uncertain properties studied in the present investigation are the ratios of natural frequencies which are translated into the uncertain stiffness matrix of the system and will be illustrated in the following. As the nonlinearities of the present system are explicitly defined, it is convenient and relatively simple to apply the timestepwise statistical linearization (SL) technique of [14]. The SL scheme applied in the present study is identified as Scheme IV in the latter reference. In other words, the discretized equivalent linear matrix equation for the system defined by Eq. (11) becomes
0.03 0.025 0.02 0.015 0.01 0.005 0
0
10
20
30
Two starting strategies for implementing the computations of recursive mean squares of responses can be applied. The one adopted in the present investigation is the spatially and temporally stochastic ensemble average or EADCD of displacement vector similar to that defined by Eq. (12),
⟨⟨xs+1 ⟩⟩x = (1t )2 N1 [⟨Ps ⟩ + ⟨⟨Qs ⟩⟩x ] + N¯ 2 ⟨⟨xs ⟩⟩x + N3 ⟨⟨xs−1 ⟩⟩x , (31) in which N¯ 2 = N1 [2Ms − (1t )2 Ks0 ],
⟨⟨Qs ⟩⟩x x µη[2(⟨⟨x2s ⟩⟩x )2 − ⟨⟨x22s ⟩⟩ ] − 2µε(⟨⟨x2s ⟩⟩x )3 = , x (1 − µ)η[2(⟨⟨x2s ⟩⟩x )2 − ⟨⟨x22s ⟩⟩ ] + 2(1 + µ)ε(⟨⟨x2s ⟩⟩x )3 where the remaining symbols have been defined in the foregoing. Note that, Eq. (31) is different from Eq. (12) in two aspects. First, the parameter matrix N2 in Eq. (12) is replaced by N¯ 2 . Second, the pseudo-random loading vector term ⟨⟨Qs ⟩⟩x is not present in Eq. (12). The term ⟨⟨Qs ⟩⟩x reduces to Eq. (7) of [14] for the nonlinear system without uncertain properties and under Gaussian random excitations. Eq. (31) is required to account for the effect of asymmetric nonlinearities of the system on the responses. 5.2. Computed results and comparison For the system studied in the present investigation, the parameters in Section 5.1 are: W = 1.0, ζ1 = ζ2 = 0.10, µ = 1.0, S0 = 0.0012, α11 = 0.125, α21 = 0.250, Er1 = 4.0, η = −1.0, and ε = 1.5, indicating the system is highly nonlinear. The mean squares of uncertain natural frequencies of the system through the spatially stochastic quantity, r are ⟨r 2 ⟩x = 0.00, 0.05, 0.10, and 0.20 which implies very large variation of uncertain natural frequencies. Applying Eqs. (15) and (20), computed mean squares of responses x1 and x2 with and without uncertain natural frequencies and under Gaussian as well as non-Gaussian nonstationary random
80
90 100
(Responses of a nonlinear 2dof system) 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0
0
10
20
30
,
Φse = Kse /W 2 .
40 50 60 70 Dimensionless time
Fig. 2. Comparison of mean squares of displacement response x1 .
Mean square of X2
80
40 50 60 70 Dimensionless time
80
90 100
Fig. 3. Comparison of mean squares of displacement response x2 . (Responses of system with uncertainties) 4 3.5 3 2.5 2 1.5 1 0.5 0
0
10
20
30
40 50 60 70 Dimensionless time
80
90 100
Fig. 4. Comparison of mean squares of displacement response x1 .
excitation have been obtained. Representative computed results for the case without uncertainty and under Gaussian nonstationary random excitation of the proposed approach are included in Figs. 2 and 3. They are denoted by SCD in the plots. For direct comparison, computed data from MCS, labeled as MCS, are also plotted in Figs. 2 and 3. For the cases with uncertainties and under nonGaussian nonstationary random excitations, representative mean squares of x1 and x2 are included in Figs. 4 and 5. The ensemble averages of x1 x2 are given in Fig. 6. It may be appropriate to mention that the above system was originally defined in the dimensionless time domain [14] and therefore all results in the foregoing figures are also expressed in the dimensionless time domain. In the MCS the mean squares and other statistical moments, which are not presented here for brevity, were evaluated by taking an ensemble average of 150 realizations, every one of which is represented by 25,600 points. That is, the total number of points used in determining the simulated results is 150 × 25,600 = 3840,000.
C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81 (Responses of system with uncertainties) 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
10
20
30
40
50
60
70
80
90
100
Dimensionless time
Fig. 5. Comparison of mean squares of displacement response x2 .
2.5 2 1.5
81
the restoring forcing terms can include asymmetric stiffness. It combines the application of the SCD method, TCT, and ATS. It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For direct comparison and demonstration of its use, a tdof nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitation was investigated. Computed results obtained by the proposed approach for the system with and without uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations were presented. It was observed that the inclusion of uncertainty in natural frequencies reduces the magnitudes of mean squares of responses. The stiffness of the system increases with increasing uncertainty of natural frequencies. The dimensionless time corresponding to the peak of the mean squares of responses and ensemble average of responses reduces slightly the periods of oscillations. Finally, the proposed approach is relatively very simple, accurate and efficient to apply. It is believed that the developed approach can be applied to the analysis and design of many engineering dynamic systems, particularly in the fields of aerospace, offshore, shipbuilding and vehicle engineering industries.
1
References 0.5 0
0
10
20
30
40
50
60
70
80
90 100
Fig. 6. Comparison of ensemble averages of x1 x2 .
With reference to the computed results presented in Figs. 2 through 6, four main observations can be made. First, with reference to Figs. 2 and 3, agreement between the SCD and MCS plots is excellent. Second, the ratio of computational time for the MCS to that of the presently proposed procedure is more than 400. This indicates that the presented procedure is much more efficient than the MCS. Third, consideration of uncertainty in natural frequencies reduces the mean squares of responses. In other words, inclusion of uncertainty in natural frequencies amounts to increasing the stiffness of the system. Fourth, the dimensionless time corresponding to the peak of the mean squares of responses and ensemble average of responses shifts slightly to the left. That is, the periods of the system are slightly reduced with the addition of uncertainty in natural frequencies. This is, of course, consistent with the observation that inclusion of uncertainty in natural frequencies increases the stiffness of the system. 6. Conclusion In the foregoing, an approach for response analysis of relatively general mdof nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations has been presented. It is general in that damping in the system is not limited to Rayleigh or proportional type while
[1] MIL-STD-810F. Department of Defense Test Method Standard for Environmental Engineering Considerations and Laboratory Tests, USA. 2000, pp. 514.5–11. [2] Defence Standard 00-35. Environmental handbook for defence material, Part 5: induced mechanical environments. Ministry of Defence, United Kingdom. 1999. [3] Steinwolf A. Shaker simulation of random vibrations with a high kurtosis value. J Inst Environ Sci 1997;XL(3):33–43. [4] Grigoriu M. Applied non-Gaussian processes. Englewood Cliffs, New Jersey: Prentice-Hall; 1995. [5] Lutes LD, Sarkani S. Stochastic analysis of structural and mechanical vibrations. Englewood Cliffs, New Jersey: Prentice-Hall; 1997. [6] To CWS. On computational stochastic structural dynamics applying finite elements. Arch Comput Methods Eng 2001;8(1):2–40. [7] To CWS. Nonlinear random vibration: computational methods. Columbus, Ohio: Zip Publishing; 2010. [8] To CWS. Large nonlinear responses of spatially nonhomogeneous stochastic shell structures under nonstationary random excitations. Trans ASME J. Comput. Inform. Sci. Eng. 2009;9: 041002-1–8. [9] To CWS. Response analysis of nonlinear multi-degree of freedom systems to non-Gaussian random excitations. In: Zhu WQ, Lin YK, Cai GQ, editors. Proceedings of the international union of theoretical and applied mechanics symposium on nonlinear stochastic dynamics and control. Springer; 2011. p. 117–26. [10] To CWS. Response of multi-degree-of-freedom systems with geometrical nonlinearity under random excitations by the stochastic central difference method. Paper presented at the fourth Conf. Nonlinear Vib., Stab., Dyn. Struct. Mech., Virginia Polytechnic Institute and State University; Blacksburg, Virginia, June 7–11, 1992. [11] To CWS, Liu ML. Random responses of discretized beams and plates by the stochastic central difference method with time co-ordinate transformation. Comput Struct 1994;53(3):727–38. [12] Liu ML, To CWS. Adaptive time schemes for responses of non-linear multidegree-of-freedom systems under random excitations. Comput Struct 1994; 52(3):563–71. [13] To CWS, Liu ML. Large nonstationary random responses of shell structures with geometrical and material nonlinearities. Finite Elem Anal Des 2000;35: 59–77. [14] To CWS, Liu ML. Recursive expressions for time dependent means and mean square responses of a multi-degree-of-freedom nonlinear system. Comput Struct 1993;48(6):993–1000.
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Probabilistic Engineering Mechanics journal homepage: www.elsevier.com/locate/probengmech
Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations Cho W. Solomon To Department of Mechanical Engineering, University of Nebraska, N104 Scott Engineering Center, Lincoln, NE 68588-0656, USA
article
info
Article history: Available online 8 May 2011 Keywords: Nonlinear Multi-degree-of-freedom systems Uncertain properties Non-Gaussian Nonstationary random excitations
abstract The investigation reported in this paper is concerned with the development of an approach for response analysis of multi-degree-of-freedom (mdof) nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The developed approach makes use of the stochastic central difference (SCD) method, time co-ordinate transformation (TCT), and adaptive time schemes (ATS). It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations is considered. Computed results obtained for the system with and without uncertain natural frequencies, and under Gaussian and nonGaussian nonstationary random excitations are presented. It is concluded that the approach is relatively simple, accurate and efficient to apply. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction For safety and economic reasons, many modern structural systems such as those that house nuclear reactors, tall buildings, naval and aerospace installations, and their components have to be designed to withstand various intensive and complicated loadings which can only be realistically modeled as random processes. Until very recently, these latter processes have generally been treated as Gaussian random processes. In practice, particularly in the designs and analysis of space shuttles and many other vehicles, there is a need to deal with the excitation processes as non-Gaussian ones. Specifically, the MIL-STD-810F DOD Test Method Standard [1], and Defence Standard 00-35 of the Ministry of Defence [2] require consideration of the non-Gaussian behavior in simulation and testing environments. In offshore structure design such as the tension-leg platform the response has been known to be nonGaussian. Wind-generated waves in finite water depth have long been recognized as non-Gaussian random processes. In addition, in the context of structural dynamics and particularly in the dynamic analysis of laminated composite plate and shell structures, uncertain system properties such as Young’s modulus of elasticity and damping have presented a formidable challenge to the designer. While the analysis of non-Gaussian random processes [3–5] and dynamic systems with uncertain properties [6–8] have generated
E-mail address: [email protected]. 0266-8920/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.probengmech.2011.05.010
considerable amount of interests in recent years in the field of random vibration and it is understood that there are approaches available in the literature for the analysis of nonlinear systems excited by non-Gaussian random disturbances it should be emphasized that these approaches are not applicable to the efficient analysis of geometrically complicated nonlinear systems represented by the FEM and under non-Gaussian nonstationary random excitations. Those entirely or partially based on the application of Monte Carlo simulation (MCS) are relatively inefficient for nonlinear systems with a large number of dof, if not computationally infeasible. Consequently, there is a definite need to provide an approach for the efficient, relatively general, and accurate analysis of nonlinear multi-degree-of-freedom (mdof) systems with uncertain properties or parameters of large variations and subjected to non-Gaussian random excitations. The investigation being reported here was therefore concerned with addressing such a need to provide a means of efficient and accurate analysis, and predicting responses of highly nonlinear mdof systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The emphasis of the investigation was on the formulation and application of mdof nonlinear systems idealized by the FEM. This is a natural extension of the work reported recently by the author [9]. Owing to their abilities to provide efficient and accurate responses of mdof nonlinear systems and discretized nonlinear shell structures under Gaussian random excitations, the stochastic central difference (SCD) method [6], the time co-ordinate transformation (TCT) [10,11], and the adaptive time schemes (ATS) [12] were combined to form a novel procedure that has been developed in the investigation.
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C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitation is considered. This same nonlinear tdof system without uncertain properties and subjected to a non-Gaussian nonstationary random excitation was studied and reported by the author recently [9]. In Section 2, the SCD method for the mdof nonlinear systems under nonstationary Gaussian random excitations is included for completeness. Section 3 is concerned with the SCD method for the mdof nonlinear systems with uncertain properties and under Gaussian nonstationary random excitations. The theoretical development of the approach for nonlinear mdof systems with uncertain properties and under non-Gaussian nonstationary random excitations is presented in Section 4. Computed results and comparisons between those obtained for systems with and without uncertain properties of large variations and under Gaussian and non-Gaussian nonstationary random excitations are included in Section 5. Conclusion is presented in the final section, Section 6. 2. Nonlinear systems under Gaussian excitations For completeness and in order to provide a foundation of the formulation and presentation of the systems with uncertain properties of large variations as well as subjected to non-Gaussian random excitations, to be presented in subsequent sections, analysis of mdof nonlinear systems under Gaussian random excitations is included in this section. The governing matrix equation of motion is given by M x¨ + C x˙ + Kx = P
(1)
where M , C and K are, respectively the nonlinear assembled mass, damping and stiffness matrices of the system; P is the external random excitation vector which, in general, includes Gaussian nonstationary random forces; x is the generalized displacement vector; the over-dot and double over-dot denote, respectively the first and second derivatives with respect to time t. A typical example for P is the product of modulating function vector and a zero-mean Gaussian white noise process. Discretizing Eq. (1) in the time domain, one has Ms x¨ s + Cs x˙ s + Ks xs = Ps ,
(2)
where the subscript s denotes the time step; for instance, xs is the value of x at time step ts such that the time step size 1t = ts+1 − ts and t0 = 0. While the recursive equations for mean squares of displacement vector have been obtained and presented in [6], for example, in order to provide a better understand in the derivation of recursive mean squares of displacements for nonlinear mdof systems subjected to nonstationary non-Gaussian random excitations and for completeness detailed steps are presented in the following. By Taylor’s theorem, the displacement vector at the next time step ts+1 is xs+1 = x(t + 1t ) = xs + (1t )˙xs +
1
xs−1 = x(t − 1t ) = xs − (1t )˙xs +
1
(1t )2 x¨ s + · · · .
2 Similarly, the displacement vector at the previous time step is
By substituting the last two equations into Eq. (2), and eliminating the velocity and acceleration terms, one has xs+1 = (1t )2 N1 Ps + N2 xs + N3 xs−1 , where the coefficient matrices are
[
1
, [ ] 1 N3 = N1 (1t )Cs − Ms . N1 = Ms +
2
(1t )Cs
] −1
N2 = N1 2Ms − (1t )2 Ks ,
2
The last recursive displacement vector is generally known as the central difference algorithm. Taking the transpose of the last recursive displacement vector, one obtains xTs+1 = (1t )2 PsT N1T + xTs N2T + xTs−1 N3T , where the superscript T designates ‘‘the transpose of’’. Performing the matrix operation xs+1 xTs+1 , taking the ensemble average or mathematical expectation of the resulting matrix, and re-arranging terms, one can show that the mean square matrix of generalized displacement vector becomes Rs+1 = N2 Rs N2T + N3 Rs−1 N3T + (1t )4 N1 Bs N1T + N2 Ds N3T
+ N3 DTs N2T + (1t )2 N2 ⟨xs PsT ⟩N1T + (1t )2 N1 ⟨Ps xTs ⟩N2T + (1t )2 N3 ⟨xs−1 PsT ⟩N1T + (1t )2 N1 ⟨Ps xTs−1 ⟩N3T ,
(3)
in which Rs = ⟨xs xTs ⟩, Ps PsT
Bs = ⟨
(4a)
⟩,
(4b)
xs xTs−1
(4c)
Ds = ⟨
⟩,
⟨xs+1 ⟩ = (1t )2 N1 ⟨Ps ⟩ + N2 ⟨xs ⟩ + N3 ⟨xs−1 ⟩,
(4d)
where the angular brackets denote the mathematical expectation or ensemble average. Eq. (3) contains the recursive covariance matrix for three consecutive time steps. The terms containing Ps xTs , Bs , Ds and their transposes on the right-hand side (RHS) of Eq. (3) require further algebraic manipulation and expansion. Without loss of generality, the external random excitation vector P in Eq. (1) may be defined by p = Θ (t )w(t ),
(5)
where Θ (t ) is the vector of deterministic modulating functions in time t, every element or entry of the deterministic modulating function vector can be written as
θi (t ) = Eri (e−α1i t − e−α2i t ) (6) in which α1i , i = 1, 2, 3, . . ., and α2i are positive constants satisfying the condition, α1i ≤ α2i , and Eri is a constant applied to normalize θi (t ) such that max{θi (t )} = 1.0; w(t ) is the zero-mean Gaussian white noise whose discrete variance,
⟨ws2 (0)⟩ = 2π S0 δ(0) (7) where δ(·) is the Kronecker delta function such that δ(0) = 1; S0 is
xs+1 − xs−1 = 2(1t )˙xs ,
the spectral density of the discrete Gaussian white noise (DGWN) process. It may be appropriate to note that while the excitation process is assumed to be Gaussian at every time step the nonlinear responses at every time step are not necessary Gaussian. For the case with non-Gaussian random excitations, it will be considered in Section 4. Of course, for stationary random excitations θi (t ) = 1 which are just special cases of Eq. (6). Applying Eqs. (5) through (7) to Eq. (3) and after some algebraic manipulation, one can show that
with a leading error of order (1t )2 .
Rs+1 = R(s1) + R(s2) + R(s3) ,
(1t )2 x¨ s − · · · .
2 By adding and subtracting the above two equations, one has, respectively xs+1 + xs−1 = 2xs + (1t )2 x¨s , and
(8)
C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
where R(s1) = N2 Rs N2T + N3 Rs−1 N3T ,
R(s2) = N2 Ds N3T + N3 DTs N2T ,
R(s3) = (1t )4 N1 Bs N1T . Eq. (8) is the recursive mean square of generalized displacement vector of the nonlinear system under discrete Gaussian nonstationary random excitations. It is noted that the parameter matrices N1 , N2 and N3 have to be updated at every time step since they are time-dependent for nonlinear systems. If the system is linear these parameter matrices are constant. The expression for the recursive covariance matrix of the generalized displacement vector can be obtained from Eq. (8) as Us+1 = Us(1) + Us(2) + Us(3) ,
(9)
where (1)
Us
=
77
where the angular brackets with a superscript x designate the ensemble average in the spatial domain while the inner angular brackets denote the ensemble average in the time domain. Eq. (12) is referred to as the extended average deterministic central difference (EADCD) scheme [7] and is used in the updating process for systems with finite deformations. Superficially, if Ps is Gaussian the updating process cannot be performed since ⟨Ps ⟩ = 0. However, starting strategies are available for this case and have been considered in [8]. By taking the ensemble average of Eq. (8) in the spatial domain, the recursive mean square matrix of the generalized displacement vector, or simply referred to as mean square matrix, becomes
⟨Rs+1 ⟩x = ⟨Rs+1 ⟩(1) + ⟨Rs+1 ⟩(n) , x
N2 Us N2T 4
N3 Us−1 N3T N1 Bs N1T xs xs T Am s
+
Us(3) = (1t )
,
(2)
Us
=
T N2 Am s N3
+
(
)
T T N3 Am N2 s
,
,
Us = Rs − ⟨ ⟩⟨ ⟩ ,
Consider the matrix equation of motion for a general nonlinear mdof system with uncertain properties and under Gaussian random excitations [7,8], (10)
where the symbols have already been defined in the last section except that the term r Φ is the associated stiffness with uncertain properties and whose definition will be presented later in this section. The temporally discretized form of Eq. (10) becomes (11)
in which the subscript s is an integer denoting the time step as defined in Section 2 above. For simple illustration, consider the uncertain property or spatially homogeneous stochastic modulus of elasticity which can be expressed as E˜ = E + r, where E is the deterministic component of Young’s modulus of elasticity and r is the spatially random component with zero ensemble average in the spatial domain. A practical example is the soil–structure interaction problem in which Young’s modulus of elasticity of soil varies stochastically in the spatial domain due to various rock formations. A further practical example can be found in laminated composite plate and shell structures where Young’s modulus of elasticity of every layer may vary spatially due to small air pockets created during the manufacturing process. This, in turn, causes the system natural frequencies to vary spatially. Then Φs in Eq. (11) is equal to Ks /E, in which Ks is the assembled tangent stiffness matrix for the system at ts without the spatial uncertainty. In the general case, the assembled tangent stiffness matrix at ts , Ks in the context of nonlinear finite element analysis (FEA) can readily be obtained in most, if not all, commercially available packages. It should be emphasized that many stochastic properties other than Young’s modulus of elasticity can be dealt with in a similar manner. Further, the uncertain properties considered above are of large variations. By taking the temporal and spatial ensemble averages of the central difference algorithm that was employed to the derivation of Eq. (3) in the foregoing, one obtains
⟨⟨xs+1 ⟩⟩x = (1t )2 N1 ⟨Ps ⟩ + N2 ⟨⟨xs ⟩⟩x + N3 ⟨⟨xs−1 ⟩⟩x ,
⟨Rs+1 ⟩
= =
x
N2 Rs N2T N2 Ds x N3T
⟨ ⟩
(12)
and
+ N3 ⟨Rs−1 ⟩x N3T ,
N3 DTs x N2T x xs Ps T N1T 1t 2 N1 x xs−1 Ps T N1T
⟨ ⟩
+
⟨
+ (1t ) N2 ⟨⟨ ( ) ⟩⟩ 2
3. Nonlinear systems with uncertain properties and under Gaussian excitations
Ms x¨ s + Cs x˙ s + (Ks + r Φs )xs = Ps
(1)
⟨Rs+1 ⟩
Before leaving this section it is noted that an efficient route of computing recursive covariances and mean squares of generalized displacement vector of the nonlinear system is to first apply Eq. (9) for the covariance matrix Us+1 and then evaluate the mean square matrix Rs and so on.
M x¨ + C x˙ + (K + r Φ )x = P
⟨Rs+1 ⟩x = ⟨⟨xs+1 xTs+1 ⟩⟩ , (n)
= Ds − ⟨xs ⟩⟨xs−1 ⟩T .
(13)
in which the expression on the left-hand side (LHS) is
+ (1t )2 N3 ⟨⟨
(14a)
⟩
+(
)
x
⟨⟨(Ps )xTs ⟩⟩ N2T
( ) ⟩⟩ x
+ (1t ) N1 ⟨⟨(Ps )xTs−1 ⟩⟩ N3T + (1t )4 N1 ⟨Bs ⟩x N1T . 2
(14b)
Eq. (13) is similar to (3) except that application of the ensemble averaging in the spatial domain has been made. The RHS of Eq. (13) consists of two terms. Mathematically, the first term ⟨Rs+1 ⟩(1) is not affected by the non-Gaussian property of the excitations while the second term includes those depend on non-Gaussian characteristics of the excitation processes. For the case with nonGaussian excitations Eq. (14b) has to be further evaluated. This latter case will be considered in Section 4. Returning to the present case, if Ps is a zero-mean Gaussian random excitation vector Eqs. (12) and (13) can further be simplified. In particular, after some lengthy algebraic manipulation, Eq. (13) reduces to
⟨Rs+1 ⟩x = N2 ⟨Rs ⟩x N2T + N3 ⟨Rs−1 ⟩x N3T + (1t )4 N1 ⟨Bs ⟩x N1T + N2 ⟨Ds ⟩x N3T + N3 ⟨DTs ⟩x N2T ,
(15)
where
⟨Bs ⟩x = Brs + Bx2 Φs ⟨Rs ⟩x ΦsT ,
(16a)
x
⟨Ds ⟩x = ⟨⟨xs xTs−1 ⟩⟩ = N2 ⟨Rs−1 ⟩x + N3 ⟨DTs−1 ⟩x ,
(16b)
Brs
= ⟨Ps (Ps ) ⟩,
(16c)
BX2
= ⟨r ⟩ .
(16d)
T
2 x
Eq. (15) is the recursive mean square of generalized displacements of the spatially and temporally stochastic nonlinear systems under temporally stochastic excitations. Note that Eq. (15) can be easily extended to cases with temporally and spatially stochastic excitations but it is not pursued here for brevity. The expression for the recursive covariance matrix associated with the recursive mean square matrix of Eq. (15) can be simply obtained, after some algebraic manipulation, as [7,9]
⟨Us+1 ⟩x = ⟨Us+1 ⟩(1) + ⟨Us+1 ⟩(2) + ⟨Us+1 ⟩(3) ,
(17)
where
⟨Us+1 ⟩(1) = (1t )4 N1 [Brs + Bx2 Φs (⟨Us ⟩x + ⟨⟨xs ⟩⟩x ⟨⟨xTs ⟩⟩ )ΦsT ]N1T , x
⟨Us+1 ⟩(2) = N2 ⟨Us ⟩x N2T + N3 ⟨Us−1 ⟩x N3T , x T m T x T ⟨Us+1 ⟩3 = N2 ⟨Am s ⟩ N3 + N3 ⟨(As ) ⟩ N2 , x x m T x ⟨Am s ⟩ = N2 ⟨Us−1 ⟩ + N3 ⟨(As−1 ) ⟩ x x T x = ⟨Ds ⟩ − ⟨⟨xs ⟩⟩ ⟨⟨xs−1 ⟩⟩ , x
⟨Us ⟩x = ⟨Rs ⟩x − ⟨⟨xs ⟩⟩x ⟨⟨xTs ⟩⟩ .
(18a) (18b)
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C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
Before leaving this section it should be noted that the foregoing procedure has been generalized to include deterministic components in the excitation vector and applied to the nonlinear random response analysis of shell structures represented by the FEM [7,13]. Further, an efficient route of computing recursive mean squares and covariances of generalized displacement vector is to first apply Eq. (17) for ⟨Us+1 ⟩x and then determine the mean square matrix ⟨Rs ⟩x by Eq. (18b). 4. Nonlinear systems with uncertain properties and under nonGaussian random excitations The SCD method for nonlinear mdof systems with uncertain properties of large variations and subjected to temporally Gaussian random excitations presented in Section 3 is extended to include the case with non-Gaussian random excitations in Section 4.1. The non-Gaussian random excitation model, TCT technique, and ATS are briefly introduced in Sections 4.2 through 4.4, respectively.
system without uncertain property presented in [9], the following specific non-Gaussian nonstationary random excitation process is employed. Consider the discretized excitation vector
Ps = P1s
P2s
P3s
···
Pns
T
,
(21)
where Pis is the ith element or entry of the non-Gaussian random excitation vector at time step ts . In the present investigation,
w 2 (ts ) w2 Pis = Pis (ts ) = [θi (ts )]2 √ = (θis )2 √ s . 6π S0 6π S0
(22)
The corresponding ensemble average is defined by
⟨Pis ⟩ =
2π S0 /3θis2 .
(23)
The corresponding mean square becomes
⟨Pis2 ⟩ = 2π S0 θis4 .
(24)
Finally, the corresponding variance, however, is obtained as 4.1. Uncertain systems under non-Gaussian random excitations
⟨σis2 ⟩ = (4π S0 /3)θis4 .
In this case, the recursive expressions for the mean square matrix of generalized displacements are identical to those defined in Eq. (13) except that the second term on its RHS, that is Eq. (14b), has to be evaluated differently. As, in general, ⟨Ps ⟩ and ⟨xs ⟩ are jointly non-Gaussian such that the terms on the RHS of Eq. (14b) have to be dealt with as in the following. Firstly, let one writes
With the above results, Eqs. (13) and (20) can now be evaluated. It should be noted that the stochastic loading vector P, whose discrete form is defined by Eq. (21), represents the random excitations applied at the discrete masses in the case of lumped parameter models. In the case of distributed parameter models considered in the FEA, for example, P is the assembled applied nodal stochastic excitation vector. While the development of the present approach can easily be extended to cases involving spatially correlated excitations it is beyond the scope of the present investigation and will not be pursued in this report.
⟨Rs ⟩(n1) = (1t )2 N2 ⟨⟨xs ⟩⟩x ⟨PsT ⟩N1T , (n2)
⟨Rs ⟩
(n3)
⟨Rs ⟩
(n4)
⟨Rs ⟩
= (1t ) N3 ⟨⟨xs−1 ⟩⟩ ⟨ ⟩ 2
x
x
= N2 ⟨Ds ⟩
N3T
x
in which ⟨Ds ⟩ = ⟨⟨ N3 ⟨DTs−1 ⟩x , and
N1T
(19a)
,
(19b)
,
= (1t ) N1 ⟨Bs ⟩ 4
PsT
x
N1T
xs xTs−1
, x
⟩⟩ = (1t ) N1 ⟨Ps ⟩ ⟨⟨ 2
xTs−1
x
(19c)
4.3. Time co-ordinate transformation
(19d)
For stiff mdof nonlinear systems or systems with large numbers of dof, typically encountered in the FEA, a computational strategy, known as the TCT technique has been proposed by the author [10] to deal with computational instability [7] and efficiency, and its use in the FEA has been demonstrated by the author and his associate [11,13], for example. Another advantage of applying the TCT technique in the FEA is the fact that the finite element size or mesh dimensions can be relatively much coarser for accurate response computations since the dimensionless highest natural frequency is always equal to unity. Why this is so will be explained later in this subsection. For completeness, the TCT is briefly introduced in the following. In the TCT it is assumed that the linear counter part of the stiff nonlinear system governed by Eq. (2) or (11) has its highest natural frequency Ωs or simply written as Ω . Within every time step the system can be considered as linear. Therefore, one can apply the following matrix equation
x
⟩⟩ + N2 ⟨Rs−1 ⟩ +
⟨Bs ⟩x = Brs + Bx2 Φs ⟨Rs ⟩x ΦsT , where Brs and Bx2 have already been defined in Eqs. (16c) and (16d), respectively. Secondly, with the definitions in Eq. (19) the term on the LHS of Eq. (14b) becomes
⟨Rs+1 ⟩(n) = ⟨Rs ⟩(n) + (⟨Rs ⟩(n) )T + ⟨Rs ⟩(n4) ,
(25)
(20)
where
⟨Rs ⟩(n) = ⟨Rs ⟩(n1) + ⟨Rs ⟩(n2) + ⟨Rs ⟩(n3) . With Eqs. (20) and (13), and the particular definition of nonGaussian random excitations Ps the recursive mean square matrix for a mdof nonlinear system with uncertain properties of large variations and under non-Gaussian random excitations can be determined. Similar to the derivation of Eq. (17), the recursive covariance matrix associated with the mean square matrix in Eq. (13) can be obtained accordingly but it is not performed here for brevity. 4.2. Non-Gaussian random excitation models Various non-Gaussian random excitation processes can be found in [4,5], for example. Of course, there are many non-Gaussian excitation models that are available in the literature. However, for direct comparison to results of a tdof asymmetric nonlinear
Msτ x¨ sτ + (Csτ )˙xsτ + (Ksτ )xsτ = Psτ ,
(26)
where the over-dot and double over-dot now denote, respectively the first and second order derivatives with respect to dimensionless time τ which is related to Ω by τ = Ω t. The second subscript τ refers to the quantity in the τ -domain. Thus, for example, Ksτ is the stiffness matrix at dimensionless time step τs , so that Ksτ = Ks /Ω 2 . The remaining matrices and vector in Eq. (26) are Msτ = Ms , Csτ = Cs /Ω , and Psτ = Ps /Ω 2 . In other words, in Eq. (26) the highest dimensionless natural frequency based on the matrix Ms−τ 1 Ksτ is always equal to unity. Eqs. (13) and (20) can now be applied to compute the recursive mean square matrix of the generalized displacement vector of the discretized system in the dimensionless time domain defined by
C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
79
Eq. (26). It should be noted that the time step size 1t in Eqs. (13) and (20) now should be replaced with the dimensionless time step size 1τ while the mass matrix Ms in Eqs. (13) and (20) should be replaced with Msτ , for example. Of course, ⟨Rs ⟩x is now ⟨Rsτ ⟩x and so on. Once the recursive mean square matrices of displacements in the τ -domain, ⟨Rsτ ⟩x are obtained using Eqs. (13) and (20), they are converted back to the original t-domain by the following relation [7,8],
⟨Rs ⟩x = Ω ⟨Rsτ ⟩x .
(27)
4.4. Adaptive time schemes The basic idea of these schemes is that for nonlinear systems whose natural frequencies are functions of time and therefore at every time step these natural frequencies have to be updated. In other words, the nonlinear stiffness matrix has to be updated at every time step. The updating strategies are called the adaptive time schemes (ATS). Three ATS have been studied by the author and his associate, and presented in [12] for the response computation and analysis of highly nonlinear mdof systems subjected to Gaussian random excitations. In the latter reference it was found that in terms of computational effectiveness, efficiency and accuracy, the SCDTATS was a better strategy for stiff systems. This was further confirmed in [13]. Therefore, in the presently proposed approach only the SCD-TATS is adopted. The SCD-TATS denotes that in the application of the SCD method the TCT is applied once at the beginning of the computation process before the application of the ATS. It may be mentioned that strictly speaking, the ATS used in the present investigation is different from that in [12,13] in that no spatially stochastic averaging was applied in the latter references. The ATS applied in the present investigation is therefore called the modified ATS [7,8]. But for conciseness and without confusion here it is simply referred to as the ATS. 5. Asymmetric nonlinear stiffness system with uncertain natural frequencies
Fig. 1. Two-dof nonlinear system with base excitation.
random disturbances have been verified by the Monte Carlo simulation (MCS) data which were presented in [14]. This indicated that Eqs. (15) and (16) are correct. Thus, in the present studies computed results of the same system with and without uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations are obtained for comparison. In Section 5.1 system parameter matrices and other pertinent data are provided while computed results and comparison studies are included in Section 5.2. 5.1. Two-dof uncertain system with asymmetric stiffness When the uncertainties are disregarded and the excitations are Gaussian the tdof nonlinear system examined here reduces to that considered in [9,14]. The sketch for this system is shown in Fig. 1. For completeness, the matrix equation of motion in [9,14] is included in the following, M x¨ + C x˙ + Kx = P where the parameter matrices are defined as
[ The nonlinear mdof system with uncertain properties and subjected non-Gaussian random excitations considered in Section 4 is general in the sense that nonlinearities in the damping and restoring forcing terms can take a variety of different forms, the uncertain properties can be of large variations, and the non-Gaussian random excitations can include many other types. However, for demonstration of application of the approach developed above and availability of results for direct comparison to those presented in [9], a tdof nonlinear asymmetric system with uncertain natural frequencies and under a non-Gaussian nonstationary random excitation is investigated. It should be emphasized that this system is employed to demonstrate the approach presented in the foregoing since the same system without uncertainty have already been studied previously and these results can be used for direct comparison. It is not employed here to explain nonlinear uncertainty of soil-structure interaction. The uncertain natural frequencies are translated into uncertain stiffness matrix of the system which is directly related to the spatially stochastic variable r in Eq. (10). The non-Gaussian nonstationary random excitations defined in Section 4.2 are applied in this section. It may be appropriate to note that mean square responses from the nonlinear tdof system without uncertain properties and under a non-Gaussian nonstationary random excitation was studied and reported in [9]. These responses have been compared with those of the same system under Gaussian nonstationary random excitations. The results by applying Eqs. (15) and (16) for the same tdof nonlinear system excited by Gaussian
]
1 0
M =
0 , 1
(28a)
2ζ1 W C = −2ζ1 W
] −2µζ2 , 2(1 + µ)ζ2
[
[
k11 k21
K =
]
k12 , k22
(28b)
k11 = W 2 ,
(28c)
k12 = −µ + µηx2 − µε x22 , k21 = −W 2 , K = KL + KNL ,
k22 = 1 + µ + (1 − µ)ηx2 + (1 + µ)ε x22 ,
x = x1
x2
T
,
in which the subscripts L and NL denote, respectively the linear and nonlinear stiffness matrices such that W2 −W 2
[ KL =
] −µ , 1+µ
] µ(η − ε x2 )x2 , [(1 − µ)η + (1 + µ)εx2 ]x2 m22 ω1 µ= , W = , τ¯ = ω2 t , m11 ω2 [
KNL =
ω1 = ci mii
0 0
k1
,
ω2 =
= 2ζi ωi ,
i = 1, 2;
m11
k2
m22
,
P =
θ1 (τ¯ )w(τ¯ ) 0
,
(29)
C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81
Ms x¨ s + Cs x˙ s + (Kse + r Φse )xs = Ps
(30)
where the equivalent linear stiffness matrix is obtained as Kse = Ks0 + Ks0 = KL =
[
0 0
[
x
2µη⟨⟨x2s ⟩⟩ − 3µε⟨⟨ ⟩⟩ x , 2(1 − µ)η⟨⟨x2s ⟩⟩x + 3(1 + µ)ε⟨⟨x22s ⟩⟩ x
2
W −W 2
x22s
]
−µ , 1+µ
xs =
x1s x2s
]
(Responses of a nonlinear 2dof system) 0.04 0.035 Mean square of X1
where η and ε are the nonlinear parameters of the system; k1 and c1 are the stiffness and damping coefficients between the first mass and base where the excitation is applied; and k2 and c2 are the stiffness and damping constants between the first and second masses. In Eq. (29), θ1 (τ¯ ) is the deterministic modulating function in dimensionless time as defined in Eq. (22) based on Eq. (6). With reference to Fig. 1, x1 = y1 − y0 and x2 = y2 − y1 are the relative displacements which are known as the inter-story drifts in the field of earthquake engineering while y0 , y1 and y2 are the absolute displacements. Note that the above system nonlinear stiffness matrix is asymmetric. The uncertain properties studied in the present investigation are the ratios of natural frequencies which are translated into the uncertain stiffness matrix of the system and will be illustrated in the following. As the nonlinearities of the present system are explicitly defined, it is convenient and relatively simple to apply the timestepwise statistical linearization (SL) technique of [14]. The SL scheme applied in the present study is identified as Scheme IV in the latter reference. In other words, the discretized equivalent linear matrix equation for the system defined by Eq. (11) becomes
0.03 0.025 0.02 0.015 0.01 0.005 0
0
10
20
30
Two starting strategies for implementing the computations of recursive mean squares of responses can be applied. The one adopted in the present investigation is the spatially and temporally stochastic ensemble average or EADCD of displacement vector similar to that defined by Eq. (12),
⟨⟨xs+1 ⟩⟩x = (1t )2 N1 [⟨Ps ⟩ + ⟨⟨Qs ⟩⟩x ] + N¯ 2 ⟨⟨xs ⟩⟩x + N3 ⟨⟨xs−1 ⟩⟩x , (31) in which N¯ 2 = N1 [2Ms − (1t )2 Ks0 ],
⟨⟨Qs ⟩⟩x x µη[2(⟨⟨x2s ⟩⟩x )2 − ⟨⟨x22s ⟩⟩ ] − 2µε(⟨⟨x2s ⟩⟩x )3 = , x (1 − µ)η[2(⟨⟨x2s ⟩⟩x )2 − ⟨⟨x22s ⟩⟩ ] + 2(1 + µ)ε(⟨⟨x2s ⟩⟩x )3 where the remaining symbols have been defined in the foregoing. Note that, Eq. (31) is different from Eq. (12) in two aspects. First, the parameter matrix N2 in Eq. (12) is replaced by N¯ 2 . Second, the pseudo-random loading vector term ⟨⟨Qs ⟩⟩x is not present in Eq. (12). The term ⟨⟨Qs ⟩⟩x reduces to Eq. (7) of [14] for the nonlinear system without uncertain properties and under Gaussian random excitations. Eq. (31) is required to account for the effect of asymmetric nonlinearities of the system on the responses. 5.2. Computed results and comparison For the system studied in the present investigation, the parameters in Section 5.1 are: W = 1.0, ζ1 = ζ2 = 0.10, µ = 1.0, S0 = 0.0012, α11 = 0.125, α21 = 0.250, Er1 = 4.0, η = −1.0, and ε = 1.5, indicating the system is highly nonlinear. The mean squares of uncertain natural frequencies of the system through the spatially stochastic quantity, r are ⟨r 2 ⟩x = 0.00, 0.05, 0.10, and 0.20 which implies very large variation of uncertain natural frequencies. Applying Eqs. (15) and (20), computed mean squares of responses x1 and x2 with and without uncertain natural frequencies and under Gaussian as well as non-Gaussian nonstationary random
80
90 100
(Responses of a nonlinear 2dof system) 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0
0
10
20
30
,
Φse = Kse /W 2 .
40 50 60 70 Dimensionless time
Fig. 2. Comparison of mean squares of displacement response x1 .
Mean square of X2
80
40 50 60 70 Dimensionless time
80
90 100
Fig. 3. Comparison of mean squares of displacement response x2 . (Responses of system with uncertainties) 4 3.5 3 2.5 2 1.5 1 0.5 0
0
10
20
30
40 50 60 70 Dimensionless time
80
90 100
Fig. 4. Comparison of mean squares of displacement response x1 .
excitation have been obtained. Representative computed results for the case without uncertainty and under Gaussian nonstationary random excitation of the proposed approach are included in Figs. 2 and 3. They are denoted by SCD in the plots. For direct comparison, computed data from MCS, labeled as MCS, are also plotted in Figs. 2 and 3. For the cases with uncertainties and under nonGaussian nonstationary random excitations, representative mean squares of x1 and x2 are included in Figs. 4 and 5. The ensemble averages of x1 x2 are given in Fig. 6. It may be appropriate to mention that the above system was originally defined in the dimensionless time domain [14] and therefore all results in the foregoing figures are also expressed in the dimensionless time domain. In the MCS the mean squares and other statistical moments, which are not presented here for brevity, were evaluated by taking an ensemble average of 150 realizations, every one of which is represented by 25,600 points. That is, the total number of points used in determining the simulated results is 150 × 25,600 = 3840,000.
C.W.S. To / Probabilistic Engineering Mechanics 27 (2012) 75–81 (Responses of system with uncertainties) 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
10
20
30
40
50
60
70
80
90
100
Dimensionless time
Fig. 5. Comparison of mean squares of displacement response x2 .
2.5 2 1.5
81
the restoring forcing terms can include asymmetric stiffness. It combines the application of the SCD method, TCT, and ATS. It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For direct comparison and demonstration of its use, a tdof nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitation was investigated. Computed results obtained by the proposed approach for the system with and without uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations were presented. It was observed that the inclusion of uncertainty in natural frequencies reduces the magnitudes of mean squares of responses. The stiffness of the system increases with increasing uncertainty of natural frequencies. The dimensionless time corresponding to the peak of the mean squares of responses and ensemble average of responses reduces slightly the periods of oscillations. Finally, the proposed approach is relatively very simple, accurate and efficient to apply. It is believed that the developed approach can be applied to the analysis and design of many engineering dynamic systems, particularly in the fields of aerospace, offshore, shipbuilding and vehicle engineering industries.
1
References 0.5 0
0
10
20
30
40
50
60
70
80
90 100
Fig. 6. Comparison of ensemble averages of x1 x2 .
With reference to the computed results presented in Figs. 2 through 6, four main observations can be made. First, with reference to Figs. 2 and 3, agreement between the SCD and MCS plots is excellent. Second, the ratio of computational time for the MCS to that of the presently proposed procedure is more than 400. This indicates that the presented procedure is much more efficient than the MCS. Third, consideration of uncertainty in natural frequencies reduces the mean squares of responses. In other words, inclusion of uncertainty in natural frequencies amounts to increasing the stiffness of the system. Fourth, the dimensionless time corresponding to the peak of the mean squares of responses and ensemble average of responses shifts slightly to the left. That is, the periods of the system are slightly reduced with the addition of uncertainty in natural frequencies. This is, of course, consistent with the observation that inclusion of uncertainty in natural frequencies increases the stiffness of the system. 6. Conclusion In the foregoing, an approach for response analysis of relatively general mdof nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations has been presented. It is general in that damping in the system is not limited to Rayleigh or proportional type while
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