Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 1 (2016), pp. 49 – 62 DOI:10.15625/0866-7136/38/1/6629 VIBRATION ANALYSIS OF BEAMS SUBJECTED TO RANDOM EXCITATION BY THE DUAL CRITERION OF EQUIVALENT LINEARIZATION Nguyen Nhu Hieu1,∗, Nguyen Dong Anh1, Ninh Quang Hai2 1Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam 2Hanoi Architectural University, Vietnam ∗E-mail: nhuhieu1412@gmail.com Received July 29, 2015 Abstract. In this paper responses of beams subjected to random loading are analyzed by the dual approach of the equivalent linearization method. The external random loading is assumed to be a space-wise and time-wise white noise in which the exact solutions of the modal equations can be found. A system of nonlinear algebraic equations for linearization coefficients of the modal linearized system is obtained in a closed form and is solved by the fixed-point iteration method. Results obtained from the proposed dual criterion are compared with the exact solution and those obtained from other approaches including energy method, and conventional linearization method. It is observed that the solution obtained by the dual criterion is in good agreement with the exact solution, especially, in the case of strong nonlinearity of beam. Keywords: Random vibration, equivalent linearization, dual criterion, modal response, nonlinear beam. 1. INTRODUCTION Over decades, the equivalent linearization (EQL) is one of the most extensively used methods in investigating mechanical systems. The earliest researches on the EQL method were carried out by Booton [1], Kazakov [2] and Caughey [3,4]. The fundamental idea of the method lies on replacing the original nonlinear system under a random exter- nal excitation by a linearized system under the same excitation in which linearization co- efficients are found from a specified optimal criterion, for example, the mean-square error criterion [4], spectral criterion [5]. This method has developed and been applied success- fully to find approximate responses of nonlinear system subjected random excitation. For discrete systems with single- and multi-degree-of-freedom, studies using the equivalent c© 2016 Vietnam Academy of Science and Technology 50 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai linearization can be found in some works [6–11], review articles [12–16], and in mono- graphs by Crandall and Mark [17], Lin [18], Roberts and Spanos [19], Socha [5] with ref- erences therein. For continuous systems, the method of EQL is also applied [14, 20–26]. In vibration analysis of beam structures under random excitations, the EQL method is studied by several authors. In Ref. [21], Herberts considered the effect of the mem- brane force on the stresses in a simply supported Bernoulli-Euler beam by the method of EQL. Seide [22] investigated nonlinear mean-square multimode responses of beams subjected to uniform pressure uncorrelated in time. Using EQL method, he obtained mean-square stresses and displacements of beams with arbitrary end conditions. Iwan and Whirley [23] developed a version of the EQL that can be applied to continuous sys- tems under non-stationary random excitations. Their technique allows the replacement of the original nonlinear system with a time-varying linear continuous system. In [24,25], a new technique of equivalent linearization method is proposed based on the energy ap- proach. Unlike the traditional replacement of EQL method [3, 4], the energy method requires that the mean-square error between the potential energy of the original nonlin- ear system and that of corresponding linearized system must be minimum. In [26], Anh et al. extended the approach of regulated equivalent linearization (RGEL) in studying single-degree-of-freedom system to random vibration analysis of beams under random loadings. The effective of the RGEL method is recorded by its excellent performance in calculating approximate modal responses of the beam. Recently, Anh et al. [27] have proposed a dual criterion of the EQL method for non- linear single-degree-of-freedom systems under random excitations. The authors showed that the accuracy of the mean-square response obtained by the dual criterion is signif- icantly improved when the nonlinearity is increasing. This dual approach is then ex- tended to cases of multi-degree-of-freedom systems [28]. Naturally, the dual approach may be extended to random vibration analysis of many continuous systems. In this research, a version of dual criterion of the EQL method is developed for analyzing modal responses of beams subjected to random loading. A nonlinear algebraic system of linearization coefficients obtained in a closed form is solved by an iteration method. To elucidate the dual approach, the obtained results are com- pared with those of the conventional linearization and energy methods. 2. THE GOVERNING EQUATION OF BEAM Consider the governing equation of a beam on elastic foundation, restrained at its ends and subjected to a space-wise distributed time-dependent loading p (x, t) [24, 25] EI ∂4w ∂x4 − N ∂ 2w ∂x2 + µA ∂2w ∂t2 + β ∂w ∂t + K fw = p (x, t) , (1) where the axial force N is given by N = EA 2L L∫ 0 ( ∂w ∂x )2 dx. (2) Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 51 Here, A and I are the area and inertia moment of the cross-section, respectively; E is the elastic modulus, µ the mass density, β the viscous damping coefficient, L the length of the beam, K f the stiffness of the elastic foundation; w (x, t) is the deflection of the beam. In this paper, a space-wise and time-wise white noise loading p (x, t) is considered. In order to solve (1) one expands p (x, t)in series [24] p (x, t) = ∞ ∑ n=1 qn (t) φn (x), (3) where qn (t) (n = 1, 2, . . .) are zero-mean Gaussian white noise stationary random pro- cesses with corresponding correlations E [qm (t) qn (t+ τ)] = 2piSmδmnδ (τ) , (m, n = 1, 2, . . .) (4) in which δmn is the Kronecker-delta notation, δ (τ) is the Dirac-delta; the quantities Sn (n = 1, 2, . . .) are constant spectral density values of random processes qn (t). The func- tions φn (x) (n = 1, 2, . . .) in the series (3) are modal shapes satisfying the following rela- tionships d4φn dx4 = µA EI ω2nφn, (5) 1∫ 0 φmφndξ = δmn, ξ = x L (6) where ωn (n = 1, 2, . . .) are natural frequencies associated with free vibration of the sys- tem (1) without the axial force, viscous damping, elastic foundation and external excita- tion for a simply supported boundary condition ω2n = n4EIpi4 µAL4 . (7) Let the deflection function w (x, t) of the beam be expended in terms of an appro- priate orthogonal set of modal shapes φn (x) as follows w (x, t) = ∞ ∑ n=1 wn (t) φn (x), (8) where wn (t) is the modal contribution corresponding to nth-mode. Substituting Eq. (8) into the expression of the axial force N in Eq. (2) yields N = EA 2L2 ∞ ∑ n=1 ∞ ∑ m=1 Knmwnwm, (9) where it is denoted Knm = 1∫ 0 dφn dξ dφm dξ dξ = Kmn. (10) 52 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai In view of Eqs. (5) and (6), the governing equation (1) takes the form ∞ ∑ n=1 ( µAw¨n + βw˙n + K fwn + µAω2nwn ) φn − EA2L2 ∞ ∑ n=1 ∞ ∑ i=1 ∞ ∑ j=1 Kijwiwjwn d2φn dx2 = p (x, t) . (11) Multiplication of Eq. (11) by φm, integration over the length of the beam, and then use of orthogonality conditions given in Eq. (6) yield a set of coupled nonlinear differen- tial equations for modal amplitudes wm (t) w¨m + β µA w˙m +ω2mwm + K f µA wm + E 2µL4 ∞ ∑ n=1 ∞ ∑ i=1 ∞ ∑ j=1 KijKnmwiwjwn = bm (t) , (12) where the random function bm (t) is given by bm (t) = 1 µAL L∫ 0 p (x, t) φm (x) dx. (13) Further in the expansion (8), it is assumed that only the first M modes of the beam significantly contribute to formulate responses. Crandall and Yildiz [29] shown that if the infinite series that are representing quantities such as displacement, mean-square stresses, etc., converge, then the results can be made as accurate as desired by taking sufficiently large M. For this reason, Eq. (12) can be taken by the following finite form with the first M modes w¨m + β µA w˙m +ω2mwm + K f µA wm + Gm (w1,w2, . . . ,wM) = bm (t) , (14) in which nonlinear components Gm (m = 1, 2, . . . , M) are functions of M variables w1, . . . ,wM Gm = Gm (w1,w2, . . . ,wM) = E 2µL4 M ∑ n=1 M ∑ i=1 M ∑ j=1 KijKnmwiwjwn. (15) Our objective is to find approximate mean-square modal responses of the beam from the modal system (14). In the next section, the dual criterion of stochastic lineariza- tion method will be applied to this nonlinear system. 3. A DUAL CRITERION FOR MODAL EQUATIONS OF BEAM The dual criterion of stochastic linearization method appears from an idea that the original nonlinear system can be replaced by an equivalent linearization system, and then this equivalent system is replaced by another nonlinear system that belongs to the same class of the original nonlinear system. Some obtained results using the dual criterion are presented in works of Anh et al. [27,28] for single- and multi-degree-of-freedom systems subjected to random excitations. Naturally, the dual criterion of stochastic linearization needs to be developed in investigating continuous systems under random excitation. For Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 53 the governing equation of the beam in the modal form (14), one can make a linearization version as follows w¨m + β µA w˙m +ω2mwm + K f µA wm +ω2mkeq,mwm = bm (t) , (16) where keq,m (m = 1, 2, . . . , M) are non-dimensional linearization coefficients determined from a specified criterion of stochastic linearization. We here utilize the dual criterion [27, 28] for determining coefficients keq,m (m = 1, 2, . . . , M). In the first step, the original nonlinear term Gm is replaced by a linearized one ω2mkeq,mwm, and then the linear term ω2mkeq,mwm is replaced by another nonlinear quantity, λmGm, that can be considered as a term belonging to the same class of the original function Gm, where the coefficients keq,mand λm are determined from the following proposed criterion for beam vibration e1 = E [( Gm −ω2mkeq,mwm )2] + ρE [( ω2mkeq,mwm − λmGm )2]→ min keq,m,λm , (17) with the detuning parameter ρ taking two values 0 or 1. When the parameter ρ is equal to zero, the criterion (17) becomes the conventional mean-square error criterion which can be found in the literature. On the other hand, as the parameter ρ is taken to be 1, the criterion (19) is so-called dual one. In this criterion, the first expectation can be understood as a component of the conventional replacement, whereas the second one describes a dual replacement of the linearization problem. Similar to the conventional linearization (see [6]), the criterion (17) leads to that partial derivatives of the expression e1 with respect to variables keq,m and λm are equal to zero ∂e1 ∂keq,m = 0, ∂e1 ∂λm = 0, (m = 1, 2, . . . , M). (18) The system (18) yields a set of algebraic equations of variables keq,m and λm, (m = 1, 2, . . . , M) as follows( (1+ ρ)ω2mE [ w2m ]) keq,m = (E [wmGm]) (1+ ρλm) , λm = ( ω2m E [wmGm] E [G2m] ) keq,m. (19) Solving the system (19) for unknowns keq,m and λm, we arrive at keq,m = 1 ω2m 1 1+ ρ− ρηm E [wmGm] E [w2m] , (20) λm = ηm 1+ ρ− ρηm , (21) where ηm = (E [wmGm]) 2 E [w2m] E [G2m] . (22) It is observed that, using the dual criterion (17), the original nonlinear Eq. (14) of modes of beam vibration is replaced by its linearization version (16), in which lineariza- tion coefficients keq,m (m = 1, 2, . . . , M) are determined from expressions (20)-(22). In the 54 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai framework of this article, the dual criterion (17) is elucidated for random vibrations of a simply supported beam under a space-wise and time-wise white noise loading. 4. RESPONSES OF A SIMPLY SUPPORTED BEAM AT BOTH ENDS 4.1. Equivalent linearization coefficients For a simply supported beam, one has the following expression for the modal shape φm [25] φm (ξ) = √ 2 sin (mpiξ) . (23) Using the expressions (10) and (23), one obtain Knm = Kmn = 1∫ 0 dφn dξ dφm dξ dξ = { pi2m2 if m = n, 0 if m 6= n. (24) Eq. (14) becomes w¨m + β µA w˙m +ω2m ( 1+ α m4 ) wm + ω2m 2R2m2 M ∑ n=1 n2w2nwm =bm (t) , (25) where ω2m = ω 2 0m 4, ω20 = EIpi4 µAL4 , α = K f µAω20 , R = √ I A . (26) The nonlinear functions Gm (m = 1, 2, . . . , M) in Eq. (25) take the form Gm = ω2m 2R2m2 M ∑ n=1 n2w2nwm. (27) Substituting expressions Gm (m = 1, 2, . . . , M) from Eq. (27) into Eq. (20) yields keq,m = 1 2R2m2 1 1+ ρ− ρηm M ∑ n=1 n2E [ w2nw2m ] E [w2m] , (28) where ηm = M ∑ i=1 M ∑ j=1 i2 j2E [ w2i w 2 m ] E [ w2jw 2 m ] E [w2m] M ∑ i=1 M ∑ j=1 i2 j2E [ w2i w 2 jw 2 m ] . (29) It is noted that, to calculate higher-order moments in Eqs. (28) and (29), we em- ploy the following generalized formula expressed in terms of second-order moments of Gaussian random processes with zero-mean [30] E [z1z2 . . . z2m] = ∑ all independent pairs ( ∏ j 6=k E [ zjzk ]) , (30) Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 55 where the number of independent pairs is equal to (2m)! / (2mm!). Particularly, in view of the present dual method for beam vibration, the following higher-order moment terms will appear E [ w2mw 2 n ] = E [ w2m ] E [ w2n ] + 2 (E [wmwn]) 2 = ymmynn + 2y2mn, E [ w2jw 2 jw 2 m ] = E [ w2i ] E [ w2j ] E [ w2m ] + 2 ( E [ wiwj ])2 E [w2m] + 2 ( E [ wjwm ])2 E [w2i ]+ 2 (E [wiwm])2 E [w2j ] + 8E [ wiwj ] E [ wjwm ] E [wiwm] = yiiyjjymm + 2y2ijymm + 2y 2 jmyii + 2y 2 imyjj + 8yijyjmyim, (31) where the notation ymn for the second moment of wm is introduced ymn = ynm = E [wmwn] . (32) Substituting expressions (31) in Eq. (28), we arrive at keq,m = 1 2R2m2 1 1+ ρ− ρηm M ∑ n=1 n2 ynnymm + 2y2mn ymm , (33) where ηm = M ∑ i=1 M ∑ j=1 i2 j2 ( yiiymm + 2y2im ) ( yjjymm + 2y2jm ) ymm M ∑ i=1 M ∑ j=1 i2 j2 ( yiiyjjymm + 2y2ijymm + 2y 2 jmyii + 2y 2 imyjj + 8yijyjmyim ) . (34) For purpose of comparing the present dual criterion with other methods, in this paper, we also present two known results of the conventional linearization [21, 22], and energy method [24, 25]. From the linearized system (16), the following expressions of the equivalent linearization coefficients keq,m (m = 1, 2, . . . , M) are obtained using the conventional linearization method keq,m conventional = 1 2R2m2 M ∑ n=1 n2 ynnymm + 2y2nm ymm . (35) It is observed that the expression (35) is also obtained from (33) by taking the de- tuning parameter ρ be zero. The linearization method based on energy criterion gives the equivalent coefficients keq,m (m = 1, 2, . . . , M) obtained from the following system [24,25]{ 1+ α+ k1,eq, 24 ( 1+ α 24 + k2,eq ) , . . . , M4 ( 1+ α M4 + kM,eq )}T = 2 ω20 A−1 { E [ w21U ] E [ w22U ] . . . E [ w2MU ]}T , (36) 56 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai where the matrix A and potential energy U of the system are determined as follows A =  E [ w21w 2 1 ] E [ w21w 2 2 ] . . . E [ w21w 2 M ] E [ w22w 2 1 ] E [ w22w 2 2 ] . . . E [ w22w 2 M ] . . . . . . . . . . . . E [ w2Mw 2 1 ] E [ w2Mw 2 2 ] . . . E [ w2Mw 2 M ]  , (37) U = ω20 2  M∑ m=1 ( α+m4 ) w2m + 1 4R2 ( M ∑ m=1 m2w2m )2 . (38) To get a closed form of the equivalent linearization coefficients keq,m (m = 1, 2, . . . , M) in Eq. (33), we utilize responses of the linearized (16) via spectral density of the external excitations. 4.2. Responses of the linearized system For the linearized system (16) under the random excitation b, one can obtain second- order moments E [wmwn] of the responses wm as follows (see [19] for details) E [wmwn] = ∞∫ −∞ Hm (−ω) Bmn (ω)Hn (ω) dω, (39) where Bmn (ω) = Smδmn (µA)2 , (40) and the frequency-response function Hm (ω) is given by Hm (ω) = 1( 1+ αm4 + keq,m ) ω2m −ω2 + i βµAω . (41) Because Bmn (ω) given by (40) are constants, moments E [wmwn] can be rewritten as E [wmwn] = Bmn ∞∫ −∞ Hm (−ω)Hn (ω) dω. (42) Introducing (41) into the right-hand side of the expression (42) and employing residual theorem in theory of complex variable functions, we get ymn = E [wmwn] = 4piβBmn µA {[( 1+ α m4 + keq,m ) ω2m − ( 1+ α n4 + keq,n ) ω2n ] 2 +2 ( β µA )2 [( 1+ α m4 + keq,m ) ω2m + ( 1+ α n4 + keq,n ) ω2n ]}−1 . (43) It is seen that a closed system of nonlinear algebraic equations for unknowns keq,m (m = 1, 2, . . . , M) is obtained by substituting (43) in to the right hand side of Eq. (33). As noted before, because the contribution of the first modes of the system is significant, we here restrict our calculations for modal responses of the beam in two cases: single-, and Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 57 two-mode using three approaches: the conventional linearization, energy method, and present dual criterion. 5. NUMERICAL RESULTS AND DISCUSSIONS It is seen that, in Eq. (25) for vibrational modes, as R tends to infinity, the effect of nonlinear terms Gm(m = 1, 2, . . . , M) disappear. Therefore, one can view the magnitude of the quantity 1/R as the parameter related to the magnitude of nonlinearity of the original nonlinear system (25). Assume that spectral densities (m = 1, 2, . . . , M) of the stochastic processes qm (t) have the same value, i.e. S1 = S2 = . . . = SM = S0. For this assumption, from the Fokker-Planck equation corresponding to the system (25), the exact expression of the probability density function can be obtained [24] P (w1,w2, . . . ,wM) = 1 C exp −βµAω202piS0  M∑ m=1 ( α+m4 ) w2m + 1 4R2 ( M ∑ m=1 m2w2m )2 = 1 C exp { −βµA piS0 U (w1,w2, . . . ,wM) } , (44) where U = U (w1,w2, . . . ,wM) is the potential energy of the system (25) given by (38), and C is the normalization constant C = ∞∫ −∞ . . . ∞∫ −∞︸ ︷︷ ︸ M− f old exp −βµAω202piS0  M∑ m=1 ( α+m4 ) w2m + 1 4R2 ( M ∑ m=1 m2w2m )2 dw1 . . . dwM. (45) The exact modal mean-square response of Eq. (25) is evaluated by the following multiple integral with M-fold E [ w2m ] exact = 1 C ∞∫ −∞ . . . ∞∫ −∞︸ ︷︷ ︸ M−fold w2mexp − βµAω202piS0  M∑ m=1 ( α+m4 ) w2m+ 1 4R2 ( M ∑ m=1 m2w2m )2 dw1 . . . dwM. (46) In general, the multiple integral (46) must be calculated using a numerical method. In the following computation, we use the exact solution (46) in the case of single-mode (M = 1) and of two-mode (M = 2) to elucidate the accuracy of the proposed dual criterion method (33), and other methods for comparison purpose. 5.1. The case of single-mode For the single-mode, M = 1, the governing equation of the simply supported beam (25) takes the form w¨1 + β µA w˙1 +ω20 (1+ α)w1 + ω20 2R2 w31 = 1 µA q1 (t) . (47) 58 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai This is the well-known Duffing oscillator subjected to random excitation [4, 5]. From Eq. (46), one can get an exact solution of mean-square response of w1 in Eq. (47) as follows (see also [22, 26]) E [ w21 ] exact, 1 = ∞∫ −∞ w21 exp { − βµApiS0 ( 1 2ω 2 0 (1+ α)w 2 1 + ω20 8R2 w 4 1 )} dw1 ∞∫ −∞ exp { − βµApiS0 ( 1 2ω 2 0 (1+ α)w 2 1 + ω20 8R2 w 4 1 )} dw1 = (1+α)R2 [ K3/4 ( (1+α)2 4 R2 R201 ) −K1/4 ( (1+α)2 4 R2 R201 )]/ K1/4 ( (1+α)2 4 R2 R201 ) , (48) where Kν (y) is the modified Bessel function of the second kind of order ν of the variable y, and the quantity R01 is given by R01 = √ piS0 β (µA)ω20 . (49) Using Eqs. (43) and (28) for M = 1, we get the following single-mode approxi- mate mean-square response E [ w21 ] depending upon the nonlinear parameter 1/R in the following form E [ w21 ] dual,1 = 2R201 1+ α+ √ (1+ α)2 + 307 R201 R2 . (50) Similarly, approximate mean-square responses of w1 corresponding to the conven- tional linearization (35) and energy method (36) in the case of single-mode are obtained, respectively, E [ w21 ] conventional,1 = 2R201 1+ α+ √ (1+ α)2 + 6R 2 01 R2 , (51) E [ w21 ] energy,1 = 2R201 1+ α+ √ (1+ α)2 + 5R 2 01 R2 . (52) Numerical results for the first mode of beam vibration in the case of single-mode using four methods, including the exact solution (48), conventional linearization (51), en- ergy method (52), and dual criterion method (50) are illustrated in Tabs. 1 and 2. The sys- tem parameters are ω0 = 1, β = 0.1, µA = 1. Tab. 1 shows a comparison of relative errors between results obtained from approximate and exact solution methods. The stiffness pa- rameter α of the system is fixed at 1, whereas the nonlinearity parameter 1/R varies from 0.01 to 10.0. It is seen that, for small values of 1/R, for example 1/R = 0.01, 0.02, 0.05, the conventional linearization yields quite small errors, about 0.05%, whereas errors of the energy and dual criterion methods are larger. In the range [1, 10] of 1/R, the error of conventional linearization becomes larger 10% while that of the energy method and Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 59 Table 1. Mean-square response of w1 of the simply supported beam in case of single-mode with ω0 = 1, α = 1, β = 0.1, µA = 1, S0 = 1 and various values of 1/R (CL: Conventional Linearization; EM: Energy Method; DM: Dual Criterion Method) 1/R E [ w21 ] exact,1 E [ w21 ] CL,1 Error (%) E [ w21 ] EM,1 Error (%) E [ w21 ] DM,1 Error (%) 0.01 15.6895 15.6895 0.0001 15.6926 0.0195 15.6948 0.0335 0.02 15.6349 15.6346 0.0014 15.6468 0.0761 15.6554 0.1317 0.05 15.2778 15.2707 0.0466 15.3403 0.4086 15.3907 0.7388 0.10 14.2613 14.1964 0.4549 14.4101 1.0437 14.5706 2.1690 0.20 11.9200 11.6419 2.3324 12.0674 1.2370 12.4086 4.0991 0.50 7.3952 6.8668 7.1448 7.3248 0.9518 7.7220 4.4191 1.00 4.4131 3.9581 10.3102 4.2767 3.0909 4.5614 3.3618 2.00 2.4261 2.1276 12.3034 2.3146 4.5968 2.4842 2.3928 5.00 1.0294 0.8890 13.6400 0.9712 5.6599 1.0463 1.6388 10.00 0.5251 0.4510 14.1093 0.4934 6.0421 0.5322 1.3564 Table 2. Mean-square response of w1 of the simply supported beam in case of single-mode with ω0 = 1, β = 0.1, µA = 1, S0 = 1, R = 1 and various values of α α E [ w21 ] exact,1 E [ w21 ] CL,1 Error (%) E [ w21 ] EM,1 Error (%) E [ w21 ] DM,1 Error (%) 1.0 4.4131 3.9581 10.3102 4.2767 3.0909 4.5614 3.3618 2.0 4.0310 3.6844 8.5978 3.9549 1.8889 4.1930 4.0181 3.0 3.6976 3.4334 7.1448 3.6624 0.9518 3.8610 4.4191 4.0 3.4056 3.2038 5.9244 3.3975 0.2382 3.5629 4.6202 5.0 3.1489 2.9944 4.9075 3.1581 0.2921 3.2960 4.6714 6.0 2.9224 2.8036 4.0656 2.9422 0.6756 3.0573 4.6147 7.0 2.7218 2.6300 3.3715 2.7475 0.9442 2.8438 4.4840 8.0 2.5433 2.4721 2.8008 2.5719 1.1241 2.6528 4.3056 9.0 2.3840 2.3284 2.3324 2.4135 1.2370 2.4817 4.0991 10.0 2.2412 2.1975 1.9480 2.2703 1.3000 2.3281 3.8785 dual criterion are remaining about 6%. For large values of 1/R, for instance 1/R = 5, 1/R = 10, however, the error of the dual criterion method is smallest (about 2%). In Tab. 2, the parameter 1/R is taken to be 1, the stiffness parameter α varies from 1.0 to 10.0, other parameters have the same values as in Tab. 1. It is observed that, the dual criterion method gives a good prediction on response errors (about 5%) as α varies, and the error of energy method is smallest. 5.2. The case of two-mode Numerical computations used the fixed-point iteration method [25, 26] to find ap- proximate mean-square response of the first mode of beam vibration are carried out for 60 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai the nonlinear algebraic system (33) and (43) with unknowns km,eq in the case of two-mode. The exact solution is obtained from the multiple integral (46) with M = 2. The obtained numerical results are presented in Tabs. 3 and 4. Tab. 3 shows that, the error of dual cri- terion method is in good agreement with that of the energy method. In Tab. 4, in general, the dual criterion and energy method yield values that are close to the exact solutions with different value of the stiffness parameter α. Table 3. Mean-square response of w1of the simply supported beam in case of two-mode with ω0 = 1, α = 1, β = 0.1, µA = 1, S0 = 1 and various values of 1/R 1/R E [ w21 ] exact,1 E [ w21 ] CL,1 Error (%) E [ w21 ] EM,1 Error (%) E [ w21 ] DM,1 Error (%) 0.01 15.6865 15.6866 0.0007 15.6901 0.0232 15.6921 0.0358 0.02 15.6233 15.6232 0.0006 15.6372 0.0887 15.6449 0.1385 0.05 15.2125 15.2050 0.0494 15.2847 0.4744 15.3290 0.7658 0.10 14.0574 13.9915 0.4688 14.2343 1.2587 14.3672 2.2039 0.20 11.4744 11.2121 2.2857 11.6951 1.9235 11.9358 4.0207 0.50 6.7777 6.3387 6.4772 6.8873 1.6175 7.0652 4.2423 1.00 3.9033 3.5576 8.8571 3.9627 1.5219 4.0385 3.4636 2.00 2.0923 1.8799 10.1519 2.1284 1.7243 2.1529 2.8951 5.00 0.8714 0.7764 10.9030 0.8891 2.0276 0.8936 2.5492 10.00 0.4414 0.3923 11.1312 0.4510 2.1828 0.4522 2.4529 Table 4. Mean-square response of w1of the simply supported beam in case of two-mode with ω0 = 1, β = 0.1, µA = 1, S0 = 1, R = 1 and various values of α α E [ w21 ] exact,1 E [ w21 ] CL,1 Error (%) E [ w21 ] EM,1 Error (%) E [ w21 ] DM,1 Error (%) 1.0 3.9033 3.5576 8.8571 3.9627 1.5219 4.0385 3.4636 2.0 3.5878 3.3203 7.4557 3.6862 2.7440 3.7299 3.9603 3.0 3.3112 3.1035 6.2725 3.4338 3.7025 3.4526 4.2703 4.0 3.0677 2.9058 5.2784 3.2038 4.4375 3.2038 4.4375 5.0 2.8526 2.7256 4.4511 2.9947 4.9812 2.9808 4.4929 6.0 2.6616 2.5616 3.7582 2.8047 5.3755 2.7807 4.4741 7.0 2.4915 2.4121 3.1849 2.6321 5.6419 2.6010 4.3960 8.0 2.3393 2.2760 2.7071 2.4752 5.8105 2.4394 4.2811 9.0 2.2026 2.1517 2.3090 2.3326 5.9016 2.2938 4.1422 10.0 2.0793 2.0383 1.9738 2.2027 5.9355 2.1623 3.9922 Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 61 6. CONCLUSIONS The method of equivalent linearization is one of effective tools in solving random vibration problems of mechanical systems. In this study, the modal responses of a simply supported beam subjected to a space-wise and time-wise white noise loading are carried out by the dual criterion of equivalent linearization. Our calculations are restricted in two cases of single- and two-mode of beam vibrations. The exact solutions of the original modal equation system obtained by Fokker-Planck equation are available for both cases. A closed form of nonlinear algebraic system is obtained by the dual approach associated with the frequency-response function method for the linearized modal system. In the case of single-mode, the analytical solution of the first mode of the beam is easy to solve explicitly for four methods considered (the exact solution, energy method, conventional linearization and dual criterion method). 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