# Fuzzy Stochastic Vibrations of Double-Beam Complex System as Model Sandwich Beam with Uncertain Parameters

Authors: Krystyna Mazur-Śniady; Katarzyna Misiurek; Olga Szyłko-Bigus; Paweł Śniady
Journal: ISRN Applied Mathematics
 (2013)

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## Abstract

The dynamic behavior of a double Euler-Bernoulli beam system with uncertain parameters (fuzzy random variables) under a fuzzy stochastic excitation and axial compression is being considered. The beams are identical and parallel, one is above the other, and they are continuously coupled by a linear two-parameter (Pasternak subsoil) elastic element. This double Euler-Bernoulli beam system can be also treated as a theoretical model of a sandwich beam. The load process is fuzzy random both in space and time. The top beam carries a fuzzy stochastic load. The solution of the problem was found thanks to the fuzzy random dynamic influence function. The aim of the paper is to find the solution for the membership function of the probabilistic characteristics of the response of the structure.


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## Body

## 1. Introduction

Uncertainty modeling in computational mechanics has received significant attention in recent years. Dynamic analysis of structures often involves two kinds of uncertainty. One of them is referred to as aleatory uncertainty (randomness, stochastic variability) and the other as epistemic uncertainty which describes, among others, imprecision, vagueness, ambiguity, and lack of the data. The random variability is described by the use of the probability theory and the imprecision by the use of fuzzy sets. Very often sufficient statistical data is not available; in this case a fuzzy function (fuzzy process) or a fuzzy random variable (fuzzy stochastic process) is suitable for the modeling purposes. The basis of the probabilistic methods is primarily statistical data while in the theory of fuzzy sets we refer also to the human intuition and experience. Using fuzzy random variables or fuzzy stochastic processes we combine these two approaches.  The classical problem of the dynamic response of structures under stochastic excitation has been presented in many monographs. In most cases it is assumed that the parameters of the structure are deterministic. On the other hand, the structural parameters like geometry characteristics and material and damping properties might be uncertain to some extent. Their uncertainty may have a strong influence on the reliability of a structure in the dynamic context. The idea of fuzzy set theory was initiated by Zadeh [1]. The concept of fuzzy random variables was introduced by Kwakernaak [2] and Puri and Ralescu [3] and combines both randomness and imprecision. The application of the uncertain forecasting in engineering and computational mechanics based on fuzzy stochastic processes is presented in the monograph [4]. Various types of structures and girders like beams, plates, shells, and frames have been considered. An important technological extension of a single string, beam, or plate is a double-string, double-beam, or string-beam system. Various aspects of the dynamic response of a double-string and double-beam system have been considered by Oniszczuk [5–9]. Free and forced vibrations of a double-beam system have been considered among others in [10, 11]. The problem of the vibration and buckling of a double-beam system under compressive axial loading is presented in [12]. Vibrations of a complex system under moving force have been studied in [13–15].In this paper the dynamic behavior of a double Euler-Bernoulli beam system with uncertain parameters (fuzzy random variables) under a fuzzy stochastic excitation and axial compression is being considered. The beams are identical, parallel, one above the other and continuously coupled by a linear two-parameter (Pasternak subsoil) elastic element. This double Euler-Bernoulli beam system can be also treated as a theoretical model of a sandwich beam, in which the shear and compression of core are taken into account. The load process is fuzzy random both in space and time. The solution of the problem was found thanks to the fuzzy random dynamic influence function [16–18]. The aim of the paper is to find the solution for the membership function of the probabilistic characteristics of the response of the structure. The probabilistic characteristics of the response of the structure are sought in the form of the first two probabilistic moments, that is, the expected value and the correlation (covariance) function. The stochastic response of structure with random and uncertain parameters has been considered, among others, in [18–27]. Fuzzy stochastic finite element method based a spectral approach to analyze complex engineering structures under  dynamic excitation has been presented in [28].


## 2. Formulation of the Problem: General Solution

Let us consider the vibrations of a connected double Euler-Bernoulli beam continuous system connected by two-parameters elastic element under axial compressionN excited as is shown Figure 1.Figure 1
Double-beam system.The differential equations of motion of the beams system have the form(1)E0I∂4w1(x,t)∂x4+N∂2w1(x,t)∂x2-k1[∂2w1(x,t)∂x2-∂2w2(x,t)∂x2]+k0[w1(x,t)-w2(x,t)]+c∂w1(x,t)∂t+m∂2w1(x,t)∂t2=p(x,t),(2)E0I∂4w2(x,t)∂x4+N∂2w2(x,t)∂x2-k1[∂2w2(x,t)∂x2-∂2w1(x,t)∂x2]+k0[w2(x,t)-w1(x,t)]+c∂w2(x,t)∂t+m∂2w2(x,t)∂t2=0,
where E0I is the flexural rigidity of the beam, E0 is Young’s modulus of elasticity, I is the moment of inertia of the cross-section area of the beam, m is the mass of the beam, c is the damping coefficient, k0 is the elastic stiffness (Winkler’s parameter), and  k1 is the shear stiffness of the elastic element.The boundary conditions of the simply supported beam are(3)w1(0,t)=w1(L,t)=0,∂2w1(x,t)∂x2|x=0=∂2w1(x,t)∂x2|x=L=0,w2(x,0)=w2(x,L)=0,∂2w2(x,t)∂x2|x=0=∂2w2(x,t)∂x2|x=L=0.
The above double-beam complex system can be treated also as a theoretical model of a sandwich beam which consists of two bendable outer layers and a sheared middle layer, which is also compressed.The following system parameters are assumed to be uncertain: the Young’s modulusE0=E0α(e),    the mass of the beam m=mα(e), the damping coefficient c=cα(e), and the parameters of the elastic element k0=k0α(e), k1=k1α(e). The symbol e denotes an elementary event, and for simplicity we will skip it in the paper. The symbol α denotes the fuzziness of the parameters. The uncertain parameters of the system are assumed to be fuzzy random variables. The load process p(x,t)=pα(e,x,t) is assumed to be a fuzzy stochastic process. The responses of the system w1(x,t)=w1α(e,x,t) and w2(x,t)=w2α(e,x,t) are also fuzzy stochastic processes as well. The aim is to find the solution for the membership function of the probabilistic characteristics of the response, w1α(x,t) and w2α(x,t) of the double-beam complex system. The probabilistic characteristics of the response of the structure are sought in the form of the first two probabilistic moments, that is, the expected value and the correlation (covariance) function.In order to decouple (1) and (2), let us introduce two new functions
(4)wIα(x,t)=w1α(x,t)+w2α(x,t),wIIα(x,t)=w1α(x,t)-w2α(x,t).From (1) and  (2), we obtain two new differential equations for functions wIα(x,t) and wIIα(x,t):
(5)E0αI∂4wIα(x,t)∂x4+N∂2wIα(x,t)∂x2+cα∂wIα(x,t)∂t+mα∂2wIα(x,t)∂t2=pα(x,t),(6)E0αI∂4wIIα(x,t)∂x4+(N-2k1α)∂2wIIα(x,t)∂x2+2k0αwIIα(x,t)+cα∂wIIα(x,t)∂t+mα∂2wIIα(x,t)∂t2=pα(x,t).Equation (5) describes vibrations of a single beam, while (6) describes vibrations of a single beam resting on an elastic Pasternak support with parameters 2k0 and 2k1. From (4), it follows that
(7)w1α(x,t)=wIα(x,t)+wIIα(x,t)2,w2α(x,t)=wIα(x,t)-wIIα(x,t)2.In the case when the parameters of (5) and  (6) are random, we can solve the problem if the right-hand side is deterministic. To overcome these difficulties, we introduce the space-time fuzzy random dynamic influence functions HIα(x,t) and HIIα(x,t) (FRDIF) which satisfy the following equations with a deterministic right-hand side:
(8)E0αI∂4HIα(x,ξ,t)∂x4+N∂2HIα(x,ξ,t)∂x2+cα∂HIα(x,ξ,t)∂t+mα∂2HIα(x,ξ,t)∂t2=δ(t)δ(x-ξ),E0αI∂4HIIα(x,ξ,t)∂x4+(N-2k1α)∂2HIIα(x,ξ,t)∂x2+2k0αHIIα(x,ξ,t)+cα∂HIIα(x,ξ,t)∂t+mα∂2HIIα(x,ξ,t)∂t2=δ(t)δ(x-ξ).After solving (8) one obtains
(9)H1α(x,ξ,t)=HIα(x,ξ,t)+HIIα(x,ξ,t)2,H2α(x,ξ,t)=HIα(x,ξ,t)-HIIα(x,ξ,t)2.If the FRDIFH1α(x,ξ,t) and H2α(x,ξ,t) are known, the response of the system w1α(x,t) and w2α(x,t) can be presented in the following form:
(10)wiα(x,t)=∫0L∫t0tHiα(x,ξ,t-τ)pα(ξ,τ)dτdξ,
where i=1,2, and if t0=0, then one considers transition vibrations, and for t0=-∞, one considers steady-state vibration case.Thus, in order to determine the probabilistic characteristics of the displacement of the beams, one can apply the expectancy operator to (10) and consequently obtain the expected values as follows:
(11)E[wiα(x,t)]α=∫0L∫t0tE[Hiα(x,ξ,t-τ)]E[pα(ξ,τ)]dτdξ.
Using α-level optimization procedure [4] for arbitrary α=αk∈[0,1] or the max-min operator in the extension principle [1], the smallest and the largest expected values at an established point x and time t can be found.Due to relationship (11), one obtains
(12)E[wiα(x,t)]αl=min{∫0L∫t0tE[Hiα(x,ξ,t-τ)]E[pα(ξ,τ)]dτdξ},E[wiα(x,t)]αr=max{∫0L∫t0tE[Hiα(x,ξ,t-τ)]E[pα(ξ,τ)]dτdξ}.Taking into account (10), the covariance functions of the beams displacement on the fuzziness level α have the form
(13)Cwiα(x1,x2,t1,t2)=∫0L∫0L∫t0t1∫t0t2E[Hiα(x1,ξ1,t1-τ1)Hiα(x2,ξ2,t2-τ2)]×C(pp)α(ξ1,ξ2,τ1,τ2)dτ1dτ2dξ1dξ2+∫0L∫0L∫t0t1∫t0t2C(HiHi)α(x1,x2,ξ1,ξ2,t1-τ1,t2-τ2)×E[pα(ξ1,τ1)]×E[pα(ξ2,τ2)]dτ1dτ2dξ1dξ2,
where
(14)C(HiHj)α(x1,x2,ξ1,ξ2,t1,t2)=E[Hiα(x1,ξ1,t1)Hjα(x2,ξ2,t2)]α-E[Hiα(x1,ξ1,t1)]αE[Hjα(x2,ξ2,t2)]α,C(pp)α(ξ1,ξ2,τ1,τ2)=E[pα(ξ1,τ1)pα(ξ2,τ2)]α-E[pα(ξ1,τ1)]αE[pα(ξ2,τ2)]α.The lower and upper endpoints of the covariance could be defined using (13) as
(15)Cwiαl(x1,x2,t1,t2)=min{∫0L∫0L∫t0t1∫t0t2E[Hiα(x1,ξ1,t1-τ1)×Hiα×Hiα(x2,ξ2,t2-τ2)]C(pp)α×(ξ1,ξ2,τ1,τ2)dτ1dτ2dξ1dξ2+∫0L∫0L∫t0t1∫t0t2C(HiHi)α×(x1,x2,ξ1,ξ2,t1-τ1,t2-τ2)×E[pα(ξ1,τ1,)]×E[pα(ξ2,τ2,)]dτ1dτ2dξ1dξ2∫0L∫0L∫t0t1∫t0t2E[Hiα(x1,ξ1,t1-τ1)},(16)Cwiαr(x1,x2,t1,t2)=max{∫0L∫0L∫t0t1∫t0t2E[Hiα(x1,ξ1,t1-τ1)×Hiα×Hiα(x2,ξ2,t2-τ2)]C(pp)α×(ξ1,ξ2,τ1,τ2)dτ1dτ2dξ1dξ2+∫0L∫0L∫t0t1∫t0t2C(HiHi)α×(x1,x2,ξ1,ξ2,t1-τ1,t2-τ2)×E[pα(ξ1,τ1,)]×E[pα(ξ2,τ2,)]dτ1dτ2dξ1dξ2∫0L∫0L∫t0t1∫t0t2E[Hiα(x1,ξ1,t1-τ1)}.In order to find the expected valuesE[Hiα(x,ξ,t)] and E[Hiα(x1,ξ1,t1)Hiα(x2,ξ2,t2)], we can use the perturbation method or Monte Carlo method. In this paper the perturbation method has been used for a particular solution of the expected values and variances and is presented in Section 3 for the response of the double-beam system.


## 3. Particular Solutions

The solutions of (8) for boundary conditions (3) are assumed to be in the form of the sine series:
(17)HJα(x,ξ,t)=∑n=1∞yJnα(t,ξ)sinnπxL,
where J=I,II.By substituting expression (17) into (8) and using orthogonalization method, one obtains the following set of uncoupled ordinary differential equations:
(18)y¨Jnα(t,ξ)+2βαy˙Jnα(t,ξ)+ωJnα2yJnα(t,ξ)=2mLδ(t)sinnπξL,
where, for J=I, ωJnα2=ωInα2=(nπ/L)2[E0αI(nπ/L)2-N]/m and for J=II, ωJnα2=ωIInα2=((nπ/L)4E0αI-(nπ/L)2(N-2k1α)+2k0α)/m and 2βα=cα/mα.The dots denote differentiation with respect to the time. These functions fulfill the initial conditions(19)yJnα(0,ξ)=0,y˙Jnα(0,ξ)=0.
The solution of  (18) taking into account the initial conditions (19) has  the form
(20)yJnα(t,ξ)=2mLω-Jnαe-βαtsinω-JnαtsinnπξL,
where ω-Jnα2=ωJnα2-βα2.Taking into account (9), (17), and (20) we have
(21)H1α(x,ξ,t)=1mLe-βαt∑n=1∞(sinω-Inαtω-Inα+sinω-IInαtω-IInα)sinnπξLsinnπxL,H2α(x,ξ,t)=1mLe-βαt∑n=1∞(sinω-Inαtω-Inα-sinω-IInαtω-IInα)sinnπξLsinnπxL.Let us consider the steady-state vibration(t0=-∞), assuming  that the excitation load  is a fuzzy weakly stationary “white noise” stochastic process both in time and space. In this case we have E[pα(x,t)]α=E[pα]α=const. and C(pp)α(x1,x2,t1,t2)=σ(p)α2δ(t1-t2)δ(x1-x2). We assume that the Young’s modulus E0α is a fuzzy random variable. The other system parameters are assumed to be deterministic. In order to find the probabilistic characteristics the function of the random variables has been expanded into Taylor series around the mean value and restricted to three items of the expansion.  The expected value for steady-state case is equal to
(22)E[w1α(x,∞)]=E[pα]αL∑n=1∞(1-(-1)nnπ/L)×{×(nπL)2+2k0}2)-1)1E[E0α]I(nπ/L)4-N(nπ/L)2×[1+(nπ/L)8I2σE02{E[E0α]I(nπ/L)4-N(nπ/L)2}2]+1E[E0α]I(nπ/L)4-(N-2k1)(nπ/L)2+2k0×[×(nπL)2+2k0}2)-1)1+((nπL)8I2σE02)××(+2k0(nπL)4}2{E[E0α]I(nπL)4-(N-2k1)(nπL)2××××+++2k0(nπL)4}2)-1]}sinnπxL,E[w2α(x,∞)]=E[pα]αL∑n=1∞(1-(-1)nnπ/L)×{+2k0(nπL)4}2)-1)1E[E0α]I(nπ/L)4-N(nπ/L)2×[1+(nπ/L)8I2σE02{E[E0α]I(nπ/L)4-N(nπ/L)2}2]-1E[E0α]I(nπ/L)4-(N-2k1)(nπ/L)2+2k0×[×(nπL)2+2k0}2)-1)1+((nπL)8I2σE02)××××(+2k0(nπL)4}2{E[E0α]I(nπL)4-(N-2k1)(nπL)2××+++2k0(nπL)4}2)-1]}sinnπxL,
where σE02is the variance of the Young modulus.Formula (13) gives the variance of the system displacement for the steady-state vibrations (t→∞) in the form
(23)σwiα2(x,∞)=Cwiα(x,x,∞,∞)=σpα2∫0L∫0∞E[Hiα2(x,ξ,τ)]dτdξ+E2[pα]α×∫0L∫0L∫0∞∫0∞C(HiHi)α××××××××(x,x,ξ1,ξ2,τ1,τ2)dτ1dτ2dξ1dξ2,
where
(24)σpα2∫0L∫0∞E[H1α2(x,ξ,τ)]dτdξ=σpα2L2(mL)2×∑n=1∞E{JInα+JIInα+JI,IInα}sin2nπxL,σpα2∫0L∫0∞E[H2α2(x,ξ,τ)]dτdξ=σpα2L2(mL)2×∑n=1∞E{JInα+JIInα-JI,IInα}sin2nπxL,JInα=14βE[1ωInα2]=m4β{(nπ/L)4E[E0α]I-(nπ/L)2N}×[×([(nπL)4I-(nπL)2NE[E0α]]2)-1)1+((nπL)8I2vE0α2)I××([(nπL)4I-(nπL)2NE[E0α]]2)-1],JIInα=14βE[1ωIInα2]=m4β{(nπ/L)4E[E0α]I-(nπ/L)2(N-2k1)+2k0}×[+2k0E[E0α]}2)-11+((nπL)8I2vE0α2)(nπL)11×(+2k0E[E0α]}2{(N-2k1)E[E0α](nπL)4E[E0α]I-(nπL)2(nπL)++1+×××(N-2k1)E[E0α]+2k0E[E0α](nπL)4}2)-1],JI,IInα=8βE[1(2β2+ωInα2+ωIInα2)2-4ω-Inα2ω-IInα2].
The numerical calculations have been done under the following assumptions.(i)
The mean valuesE[E0α] and E[pα] are fuzzy numbers with triangular membership function (Figures 2 and 3),(ii)
The variation coefficientsvE0 are constant and do not depend on the level of fuzziness.(iii)
Other quantities describing the structure are deterministic.Membership function of  the mean value  forN=0.
(a)
(b)Membership function of  the variance forN=0.
(a)
(b)The following values of the parameters are used in the following numerical calculations:L=10m, E[E01]=1·1010Nm-2, E[E00l]=0,8·1010Nm-2, E[E00r]=1,2·1010Nm-2, I=3,8·10-4m4, k0=1·105Nm2, k1=2·104Nm2, N=0 and N=0,4(π2/L2)E[E01]I, vE0=0,2, E[p1]=const., E[p0l]=0,8E[p1], and E[p0r]=1,2E[p1].In (23) it has been assumed that E2[pα]α=0.In Figures2, 3, 4, 5, 6, and 7 the membership of the expected value and variance of the beams response are presented.Expected value and variance of the response of the beams.
(a)
(b)
(c)
(d)
(e)Membership function of  the mean value forN=0,4(π2/L2)E[E01]I.
(a)
(b)Membership function of  the mean  variance forN=0,4(π2/L2)E[E01]I.
(a)
(b)Expected value and variance of the response of the beams.
(a)
(b)
(c)
(d)
(e)


## 4. Fuzzy Stochastic Moving Load

Let us notice that the general solution presented in Section2 can also be used, after some modifications, if the double-beam system is loaded by a fuzzy stochastic load moving with velocity v (Figure 8).Figure 8
The double-beam system loaded by fuzzy stochastic moving load.In this case the right-hand side in (1) is equal to  pα(x,t)=pα(x-vt).The FRDIFHiα(x,t) (i=1,2) in solutions (10)–(16) should be replaced by fuzzy random dynamic moving influence function (FRDMIF) Himα(x,ξ,t) which satisfies, for 0≤t≤L/v, the equations
(25)E0αI∂4HImα(x,t)∂x4+N∂2HImα(x,t)∂x2+cα∂HImα(x,t)∂t+mα∂2HImα(x,t)∂t2=δ(x-vt),E0αI∂4HIImα(x,t)∂x4+(N-2k1α)∂2HIImα(x,t)∂x2+2k0αHIImα(x,t)+cα∂HIImα(x,t)∂t+mα∂2HIImα(x,t)∂t2=δ(x-vt),H1mα(x,t)=HImα(x,t)+HIImα(x,t)2,H2mα(x,t)=HImα(x,t)-HIImα(x,t)2.The fuzzy random dynamic moving influence functions have the forms(26)HImα(x,t)=2L3EI0α∑n=1∞sinnπ(vt/L)sinnπ(x/L)(nπ)2[(nπ)2-N0-η2]-2L3EI0αη×∑n=1∞sin[(nπ(nπ)2-N0)/η](vt/L)sinnπ(x/L)(nπ)2[(nπ)2-N0-η2](nπ)2-N0,(27)HIImα(x,t)=2L3EI0α∑n=1∞sinnπ(vt/L)sinnπ(x/L)(nπ)4-(nπ)2[N0-2κ1+η2]+2κ0-2L3EI0αη×∑n=1∞(((nπ)sin[(nπ)4-(nπ)2(N0-2κ1)+2κ0ηvtL]×sinnπxL[(nπ)4-(nπ)2(N0-2k1)+2k0η(vt/L)])×((nπ)4-(nπ)2(N0-2k1)+2k0{(nπ)4-(nπ)2[N0-2κ1+η2]+2κ0}×(nπ)4-(nπ)2(N0-2κ1)+2κ0)-1[(nπ)4-(nπ)2(N0-2k1)+2k0η(vt/L)]),
where N0=NL2/EI0α, κ0=k0L4/EI0α, κ1=k1L2/EI0α, and η2=mv2L2/EI0α.The general solutions for moving fuzzy stochastic load after modification of (10)–(16) have the forms
(28)wimα(x,t)=∫0tHimα(x,t-τ)pα(τ)dτ,
where i=1,2, 0≤t≤L/v,
(29)E[wimα(x,t)]α=∫0tE[Himα(x,t-τ)]E[pα(τ)]dτ,(30)E[wimα(x,t)]αl=min{∫0tE[Himα(x,t-τ)]E[pα(τ)]dτ},(31)E[wimα(x,t)]αr=max{∫0tE[Himα(x,t-τ)]E[pα(τ)]dτ},(32)Cwimα(x1,x2,t1,t2)=∫0t1∫0t2E[Himα(x1,t1-τ1)Himα(x2,t2-τ2)]×C(pp)α(τ1,τ2)dτ1dτ2+∫0t1∫0t2C(HiHi)α(x1,x2,t1-τ1,t2-τ2)×E[pα(ξ1,τ1,)]E[pα(ξ2,τ2,)]dτ1dτ2,
where
(33)C(HiHj)α(x1,x2,t1,t2)=E[Himα(x1,t1)Hjmα(x2,t2)]α-E[Himα(x1,t1)]αE[Hjmα(x2,t2)]α,C(pp)α(τ1,τ2)=E[pα(τ1)pα(τ2)]α-E[pα(τ1)]αE[pα(τ2)]α.The lower and upper endpoints of the covariance could be defined using (32) as
(34)Cwiαml(x1,x2,t1,t2)=min{∫0t1∫0t2E[Himα(x1,t1-τ1)Himα(x2,t2-τ2)]×C(pp)α(τ1,τ2)dτ1dτ2+∫0t1∫0t2C(HiHi)α(x1,x2,t1-τ1,t2-τ2)×E×E[pα(ξ1,τ1,)]×E×E[pα(ξ2,τ2,)]dτ1dτ2∫0t1∫0t2E[Himα(x1,t1-τ1)Himα(x2,t2-τ2)]},Cwiαmr(x1,x2,t1,t2)=max{∫0t1∫0t2E[Himα(x1,t1-τ1)Himα(x2,t2-τ2)]×C(pp)α(τ1,τ2)dτ1dτ2+∫0t1∫0t2C(HiHi)α(x1,x2,t1-τ1,t2-τ2)×E[pα(τ1)]E[pα(τ2)]dτ1dτ2∫0t1∫0t2E[Himα(x1,t1-τ1)Himα(x2,t2-τ2)]},
where  0≤t1 and t2≤L/v.For steady-state solutions and it should be assumed thatt1=t2=L/v.


## 5. Conclusion

In the paper, the dynamic behavior of a double Euler-Bernoulli beam complex system with uncertain parameters (fuzzy random variables) under a fuzzy stochastic excitation and axial compression has been studied. The load process is fuzzy random both in space and time. In order to find the solution for the membership function of the probabilistic characteristics of the response of the structure, the idea of the fuzzy random dynamic influence function has been used. The probabilistic characteristics of the response of the structure are sought in the form of the first two probabilistic moments, that is, the expected value and the correlation (covariance) function. This double Euler-Bernoulli beam system can be also treated as a theoretical model of a sandwich beam. The algorithm (similar to the one presented in the paper for the double Euler-Bernoulli beam complex system) can be used in stochastic dynamic analysis of other complex systems like strings, beams, plates, and so on, with uncertain parameters.



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*Source: 340145-2013-10-22.xml*