Research on non-Linear random vibration by the global – local mean square error criterion

The equivalent linearization method (ELM) is one of the most

commonly used methods and is an effective method for nonlinear

systems with weak nonlinear coefficients. For nonlinear systems

with larger nonlinear coefficients, the accuracy of this method is

significantly reduced. The dissertation focuses on researching and

developing EL method to improve errors when analyzing nonlinear

oscillations.

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covariance. 1.3 Some special stochastic processes There are definitions of: Stationary random processes and Ergodic process; Normal random process or Gaussian process; White noise process; colored noise process; Wiener process and Markov process. 1.4 Some approximately analytical methods for analyzing random oscillation Numerical methods, approximately analytical methods are very popular methods. In detail, there are some useful methods in this dissertation [29-31]: - Perturbation technique. - Fokker-Planck-Kolmogorov (FPK) equation technique. - Stochastic averaging technique. - Statistical linearization technique. 1.5 Fokker-Planck-Kolmogorov (FPK) equation technique and Stochastic averaging technique 1.6 Overview of studies on random oscillations The problem of random vibrations has been studied and presented in many textbooks [26–33]. Oscillation analysis based on nonlinear mathematical models requires appropriate methods. In the theory of random oscillation, the stochastic equivalent linearization method (ELM) replacing the nonlinear system by an equivalent linear system is a common method because it preserves some essential properties of the original nonlinear system. This method has been described in many review papers [42, 43] and summarized in monographs [29] and [44]. 4 Although the accuracy of EL methods may not be high, this is overcome by improved techniques [43]. Canor et al. [45] also wrote: Thanks to its easy and fast implementation technique, the equivalent linearization method has become a universal probability approach for analyzing large nonlinear structures. The EL method has been used in many research papers. A EL based on analytic method was developed in [46, 47] to analyze nonlinear energy extraction systems. The nonlinear oscillation system of the wing profile was studied in [48, 49] by using EL method. Silva - Gonzlez et al. [52] used the stochastic EL method to study elastic non-linear structure of seismic load. In Vietnam, the dissertation of Nguyen Ngoc Linh [4] analyzed the nonlinear random oscillation of systems of 1 degree of freedom by the stochastic EL method according to the weighted duality criteria. The dissertation of Nguyen Nhu Hieu [5] has developed the duality criterion in EL method for multi-degree-of-freedom nonlinear systems with random excitation. Nguyen Minh Triet has conducted a doctoral dissertation on the analysis of the response of the airplane wing profile according to the duality approach, in which the study of nonlinear periodic oscillation by EL method [6]. In his PhD dissertation in 2002 [7] Luu Xuan Hung developed the Local Mean Square Error Criterion (LOMSEC) based on the idea of replacing integral over infinite domain (-∞ , + ∞) equals the integral over a finite domain [-rx, + rx] where the system's response is concentrated. Continuing development of this research direction, the dissertation carried out by the PhD student will develop the Global- Local Mean Square Error Criterion (GLOMSEC) of the EL method to nonlinear MDOF systems subjected to random excitation. In this development, a dual approach is used to address the finite domain of integration [-rx, + rx]. Conclusion of chapter 1 Chapter 1 introduced some basic concepts and formulas for probability theory and random processes, and some methods of nonlinear random oscillation analysis. Some research results on nonlinear random oscillations related to the dissertation have also been reviewed and analyzed as the basis for the next chapters. 5 CHAPTER 2. EQUIVALENT LINEARIZATION METHOD AND GLOBAL-LOCAL MEAN SQUARE ERROR CRITERION 2.1. The classical criterion of equivalent linearization We present the EL method for 1 DOF nonlinear random oscillator systems of type [9, 29, 44]: 2 02 ( , ) ( )x hx x g x x t        (2.10) Where, x , x and x are displacement, velocity and acceleration, respectively; h is damping coefficient, ( , )g x x is a nonlinear function, ( )t is Gaussian white noise excitation with intensity 2 ; 0 is the natural frequency for 0h  , ( , ) 0g x x  . The equivalent linearization equation of (2.10) is as follows: 2 02 ( )x hx x bx kx t         (2.11) where b, k are equivalent linearization coefficients. The error between (2.10) và (2.11) satisfies the Mean Square Error Criterion proposed by Caughey [10]:   2 d , ( , ) mink b k S g x x bx kx    (2.14) Hence: kd kd0; 0 S S b k       (2.15) Assuming that the solution is a random process, the response x , x should be independent, that is 0xx  , solving the system of equations (2.15) we obtain   2 ,xg x x b x     ,   2 ,xg x x k x   (2.17) 6 Equations (2.11) và (2.17) forms a system of equations for 3 unknowns x(t), b, k. Iterative algorithms are usually applied proposed by Atalik and Utku [59] as follows: a) Assign the initial value to the second order moments 2 2,x x . b) Use (2.17) to determine the linear coefficients. c) Solve equation (2.11) to find the new instantaneous second order moments 2 2,x x . d) Repeat b) and c) until the specified accuracy is reached. We consider the nonlinear system with multi degrees of freedom under random excitation    , ,Mx + Cx + Kx + Φ x,x x = Q t    (2.20) where x , x , x are displacement, velocity and acceleration vectors, respectively , ,M C Kij ij ijn n n n n nm c k                are mass, damping and stiffness matrices;  ,Φ x,x x  - nonlinear function vector,  Q t is Gaussian white noise excitation vector with zero intensity and spectral density matrix    S ij n nS     where  ijS  is cross spectral density function of iQ and jQ . The equivalent linearization system is as follows:         ,t   e e eM + M q C + C q K + K q Q  (2.21) where e e eM , C , K are the equivalent matrices of mass, damping and stiffness. In equation (2.21) we use the notation  q t to show that this is only an approximation to  x t in the original nonlinear equation (2.20). The error between the system (2.20) and the system (2.21) is    .  e e ee Φ q, q, q M q + C q + K q    (2.22) 7 The error e satisfies the Mean Square Error Criterion upon e e eM , C , K : min .E     e e e T M ,C ,K e e (2.24) Where the expectation in the left side of (2.24) is calculated by the probability density function of (2.21). Atalik and Utku (1976) [59] show that the criterion (2.24) leads to the following equation:   , TT e e e Tzz M C K zΦ zE E           (2.25) 2.2. Some improved equivalent linearization criteria For decades, many studies on equivalent linearization criteria have been proposed to improve the accuracy of the equivalent linearization method [11-24, 20-24, 67, 68]. 2.3 Criterion of global-local mean square error In this section, we propose a new equivalent linearization criterion called the global-local mean square error criterion. We consider a nonlinear random oscillation of one degree of freedom: 2 02 ( , ) ( )x hx x g x x t        (2.47) Where, the notation is defined as above. The equivalent linearization equation of (2.47) takes the form: 2 02 ( )x hx x x x t           (2.48) where λ, μ are linearization coefficients. The error between (2.47) và (2.48) is:     xxxxgxxe    ,, (2.49) The classical criterion gives [29, 44] 2 , ( , ) ( , ) mine x x P x x dxdx            (2.51) Where ( , )P x x is the probability density function (PDF) of x and x : 8 2 2 ( , ) ( , ) , . g x x x g x x x x x        (2.53) Since the integration domain in (2.51) is (  , ), the criterion (2.51) is called the global mean square error criterion. With the assumption that the integration should focus for a more accurate solution, Anh and Di Paola proposed the local mean square error criterion (LOMSEC) [15]: 0 0 0 0 2 , ( , ) ( , ) min x x x x e x x P x x dxdx              (2.54) Where, 00, xx  are positive values. The integration (2.54) is transformed into the non-dimension one by introducing 0 0,x xx r x r    where r is a positive value, x and x  are standard deviations of x and x : 2 2 , [ ( , )] ( , ) ( , ) min x x x x r r r r e x x e x x P x x dxdx                    (2.55) where [.] denotes: [ . ] ( . ) ( , ) x x x x r r r r P x x d xd x                (2.56) We have analogously:     2 2 ( , ) ( , ) ( ) , ( ) . g x x x g x x x r r x x               (2.57) It is seen from (2.57) the local equivalent linearization coefficients (LOMSEC) are functions of r, ( ), ( )r r     .Using the dual viewpoint, we can propose that r change across the non- negative domain and the EL coefficients ,  can be chosen as the following[24]: 9 0 0 1 ( ) ( ) , 1 ( ) ( ) . s s s s r Lim r dr s r Lim r dr s                           (2.60) We have found from LOMSEC a new EL criterion called the global-local mean square error criterion (GLOMSEC). Next we develop the global-local mean square error criterion (GLOMSEC) to MDOF systems:    tfzgz  (2.61)  1 2, ,..., T nz z z z are vector of state variables, n is a natural number, g is a nonlinear function of z, f(t) is a Gausian stochastic process with zero mean. Denote:      tfzgzze   (2.62) Introduce new linearization terms into (2.62):      tfzgAzAzzze   (2.64) where  ijaA  is matrix n×n. Let y is a stationary solution of:   0 tfAyy (2.65) From (2.64) and (2.65) we have:    ygAyye  (2.66) Denote p(y) probability density function (PDF) of y of (2.65). According to the LOMSEC we have:          ,min 0 0 1 0 1 1 0 1 22 ij ynn ynn y y a y y i y y i dyypyenye           i,j = 1,,n It gives:      1  TT yyyygA (2.72) 10 The iterative algorithm is applied similarly to the one proposed by Atalik and Utku [59]. According to the GLOMSEC, the EL coefficients aij can be chosen as the following 00 1 0 0 0 1 2 0 0 0 1 2 0 0 0 0 0 0 1 2 1 20 0 0, ,..... 1 2 0 0 ( , , ..... ) 1 .... ( , , ..... ) ..... ..... n n ij ij n yy ij n n y y y n a a y y y Lim a y y y dy dy dy y y y             Conclusion of chapter 2 The second chapter deals with the development of the criterion of Global – local mean squared error (GLOMSEC) for the systems of one and multi degrees of freedom. The results in chapter 2 are presented in the articles [1,6] of the List of publication of dissertation. CHAPTER 3. APPLICATION OF GLOMSEC IN ANALYSIS OF RANDOM ONE DEGREE OF FREEDOM SYSTEMS 3.1. Analysis of domain of concentred response of nonlinear systems 3.1.1. Duffing oscillation system subjected to white noise excitation 3.1.2 Nonlinear damped oscillation system subjected to white noise excitation We consider a nonlinear damped oscillation system subjected to white noise excitation:     2 21x x x x x d t          (3.6) The exact PDF two-dimensional probability density function of the system [29, 44]:       22 2 2 2, exp 0.5p x x C x x x x d              (3.7) 11 where C is the normalized constant. If  Prob a x a   is chosen then the domain  aa, will be detemined by:     Prob ,aaa x a p x x dx dx            (3.8) Suppose we choose  Prob 0.98a x a    and consider the parameter d = 2 while the nonlinear parameter  changes. Then the values a will be obtained (Table 3.2). From Table 3.2 we also find that the finite domain [-a, a] in which the responses are concentrated with probability 0.98. Observations show that the response domain shrinks as the nonlinear parameter  increases as shown in Table 3.2. and Figure 3.2 as follows: Tab. 3.2. Values of a depending on   0.1 0.5 1 5 10 30 50 80 100 a 2.92 2.04 1.78 1.36 1.26 1.15 1.11 1.08 1.07 -4 -2 0 2 4 x -4 -2 0 2 4 x 0 0.02 0.04 p -1 0 1 x -1 0 1 x 0 0.1 0.2 0.3 0.4 p a.  =0.1 b.  =100 Fig.3.2. Graphics of PDF of nonlinear damped system ( =0.1; 100) 12 3.2. Application examples of global-local mean square error criterion (GLOMSEC) 3.2.1 Vibration with 3-order nonlinear damping Consider a nonlinear damped oscillation system subjected to white noise excitation:    3 22 ox h x x x t         (3.11) where , , ,oh    are positive values. The corresponding linearization system is as follows:    22 ox h b x x t       (3.12) where b is the linear coefficient. The mean square response of (3.12) is:   2 2 22 2 o x h b     (3.13) The coefficient b defined by the classical criterion is: 26b h x   (3.15) By GLOMSEC is: 2,2 2 1,0 1 ( ) 2 2.4119* 2 s r s r T b b r h x Lim dr h x s T                   (3.22) Substituting (3.22) into (3.13) gives: 2 2 2 2 2.4119 2*2.4119GL o h h h x h       (3.23) To evaluate approximate solutions we use the solution 2 ENL x [29]. The relative errors of 2 GL x , 2 kd x compared to the solution 2 ENL x are defined by (3.24): 13 2 2 2 2 ( ) ( )2 2 *100%, *100%kd cx GL cxC GL cx cx x x x x Err Err x x     (3.24) In Table 3.4, the results show that the solution 2 GL x has better accuracy than the solution 2 kd x , in particular the largest error of GLOMSEC is only 1.93%. Table 3.4. The second moment of the response of the nonlinear damping oscillator system 0.05, 1, 4oh h    , and γ changes γ 2 ENL x 2 kd x ( ) % CErr 2 GL x ( ) % GLErr 1 0.4603 0.4342 5.61 0.4692 1.93 3 0.3058 0.2824 7.65 0.3090 1.05 5 0.2479 0.2270 8.32 0.2495 0.77 8 0.2025 0.1844 8.99 0.2032 0.35 10 0.1835 0.1667 9.16 0.1839 0.22 3.2.2. Van der Pol system to white noise Consider Van der Pol system     2 2 ox x x x t          (3.25) where , , , ,o     are positive values,  t is white noise with unit intensity. We replace   2,g x x x x  by the linear one bx , where b is the linear coefficient:    2ox b x x t         (3.26) 14 The coefficient b defined by the classical criterion is: 2b x   (3.29) By GLOMSEC is: 1,2 2 0,0 1 ( ) 0.8371 s r s r T b b r x Lim dr x s T                  (3.34) Mean square response 2 GL x of Van der Pol system (3.25) by GLOMSEC is: 2 2 2 2 2 1 1,6742 1,6742GL o x                   (3.36) To evaluate approximate solutions, we use the Monte Carlo simulation solution, [29]. The relative error between the approximate solutions 2 GL x , 2 kd x , compared to the simulation solutions 2 MC x is calculated by the formula (3.24). Table 3.5. Mean square responses of Van der Pol oscillator with α*ε=0.2; 0 =1;  =2; σ 2 changes 2 2 MC x 2 kd x ( ) % CErr 2 GL x ( ) % GLErr 0.02 0.2081 0.1366 34.33 0.1574 24.32 0.20 0.3608 0.2791 22.46 0.3113 13.52 1.00 0.7325 0.5525 24.58 0.6095 16.79 2.00 1.0310 0.7589 26.40 0.8349 19.02 4.00 1.4540 1.0513 27.70 1.1544 20.61 15 In Table 3.5, the results 2 GL x have better accuracy than 2 kd x , in which the largest error values respectively are 24.32% compared to 34.33% 3.2.3 Vibration in Duffing system to random excitation Consider Duffing system subjected to white noise excitation:  2 32 ox hx x x t        (3.37) The notation is the same as in the previous example. The exact solution is [29, 44] 2 2 2 4 2 2 x 2 2 4 2 4 1 1 exp 2 4 4 1 1 exp 2 4 o c o h x x x dx x h x x dx                                  (3.39) The equivalent liner system is:  22 ox hx x kx t      (3.40) The linearization coefficient by GLOMSEC is: 2, 2 1,0 0 2,2 2 1,0 1 1 ( ) ( ) 1 2.4119 s s r s s r s r s r T k k r Lim k r dr Lim x dr s s T T x Lim dr x s T                                       (3.48) Mean square response 2 GL x of Duffing system (3.37) by GLOMSEC: 2 2 2 41 2.4119 2* 2.4119 o oGL x h                (3.49) 16 The relative error between the approximate solutions 2 GL x , 2 kd x with the exact one 2 xc x defined by (3.24) and presented in Tab. 3.6. Table 3.6 Mean square responses of Duffing system, 1, 0.25, 1o h    ;  changes  2 xc x 2 kd x ( ) % CErr 2 GL x ( ) % GLErr 0.1 0.8176 0.8054 1.49 0.8327 1.857 1.0 0.4680 0.4343 7.194 0.4692 0.263 10 0.1889 0.1667 11.768 0.1839 2.626 100 0.0650 0.0561 13.704 0.0624 4.076 The results show that the approximation determined by the classical criterion has good accuracy with small nonlinear elastic coefficient , the error increases to over 13% as the nonlinear elastic coefficient increases. Accuracy of GLOMSEC criterion is better with maximum error of 4.1%. 3.2.4. Duffing system with nonlinear damping to white noise 3.2.5. Vibration of ship The rolling motion of the ship in random waves has been considered by [55], [56], [57]. The equation of the ship's motion is of the form [56-57] 2 2 ( )D t           (3.63) The system (3.63) is replaced by the linear one 2 ( ) ec D t        (3.66) 17 The linearization coefficient ec by GLOMSEC is: 3 ,2 1/ 2 2 1/2 1,0 0 1 1 ( ) ( ) { } 1.49705 { } s s t re e e s s r T c c r Lim c r dr E Lim dr E s s T                         Mean square response by GLOMSEC is:     2/3 2 2 2 1/2 0.76415 1.49705 { }eGL GL D D D E E c E                  Mean square response by the classical criterion is:     2/3 2 2 2 1/2 0.7323 1.5958 { }eC C D D D E E c E                  Mean square response by the nonlinear equivalent linearization method is:     2/3 2 2 0.765 ENL ENL D E E            The relative error between the approximate solutions 2 GL x , 2 C x , compared to the nonlinear equivalent linearization method, is calculated by the formula (3.24). We have: ( ) ( )4.314%; 0.130%C GLErr Err  The results show that GLOMSEC gives the good agreement with the ENL solution and GLOMSEC improves the accuracy of the classical criterion. Conclusion of chapter 3 In Chapter 3, the GLOMSEC was applied to analyze the mean square responses for a number of 1-order-freedom random oscillating systems. The examples applied confirmed the outstanding advantages proposed in the GLOMSEC. The results are presented in [1,3,5] of List of publications of the dissertation. 18 CHAPTER 4. APPLICATION OF GLOMSEC TO THE ANALYSIS OF RANDOM MDOF SYSTEMS 4.1. Two-degree-freedom nonlinear oscillation system Consider the two-degree-freedom nonlinear oscillation system described by:     332 1 1 1 21 1 1 1 11 2 33 2 1 2 2 2 22 2 2 2 1 0 ( )1 0 0 ( )0 1 x b x xx x x w ta x x x w ta x b x x                                                              (4.1) where: , , , ,i i ia b   (i=1, 2) are constants. 1 2( ), ( )w t w t are white noise processes with zero mean and  ( ) ( ) 2 ( )i i iE w t w t S     (i=1, 2), ( )  is Delta Dirac function, 1 2,S S = const. The equivalent linear system is: 2 1 1 1 11 11 12 1 11 12 2 2 2 2 221 1 2 22 21 2 22 ( )1 0 ( )0 1 e e e e e e e e x x x w tc c k a k x x x w tc c a k k                                                        (4.4) where , ; ( , 1,2)e eij ijc k i j  are equivalent linear coefficients. The equation error is: ( , ) e eC X K Xx x     (4.5)     33 1 1 1 21 33 2 2 2 2 1 ( , ) x b x x x x x b x x                     111 12 11 12 221 22 21 22 ; ; ;e eC X K e e e e e e e e xc c k k xc c k k                     1 2 ;X x x        (4.6) To simplify the calculation we suppose that 1 2,x x are independent. Using the Appendix of dissertation and noting 2 1 2 1 0 ( )n mi jE x x i j       GLOMSEC gives: 19   2,211 11 1 1 1,0 1 ( ) s re e s r T c c r E x Lim dr s T             ,   2,222 22 2 2 1,0 1 ( ) s re e s r T c c r E x Lim dr s T                2, 1,2 211 11 1 2 1, 0,0 0 1 1 ( ) 3 . s s r re e s s r r T T k k r b E x Lim dr E x Lim dr s T s T                            2, 1,2 212 12 2 1 1, 0,0 0 1 1 ( ) 3 . s s r re e s s r r T T k k r b E x Lim dr E x Lim dr s T s T                             2, 1,2 221 21 1 2 1, 0,0 0 1 1 ( ) 3 . s s r re e s s r r T T k k r b E x Lim dr E x Lim dr s T s T                             2, 1,2 222 22 2 1 1, 0,0 0 1 1 ( ) 3 . s s r re e s s r r T T k k r b E x Lim dr E x Lim dr s T s T                         (4.11) The limits in (4.11) equal to: 2, 1,0 1 lim 2.41189 s r s r T dr s T         , 1, 00 1 lim 0.83706 s r s r T dr s T        (4.12) To evaluate the approximate solution while the original nonlinear system does not have an exact solution, we use the approximate probability density function by the equivalent nonlinear method (ENL) [77]. Table 4.1 presents the approximate mean square responses and their relative errors compared to ENL solutions. 20 Table 4.1. The mean square responses of 1 2,x x follow 1 2  với 1 2 1 2 0 1a b S          . 1 2 ,   21 ENLE x  21 CE x ( ) % CErr  21 GLE x ( ) % GLErr   2 2 ENL E x  22 CE x ( ) % CErr  22 GLE x ( ) % GLErr 0.1 1.573 1.216 22.68 1.407 10.54 1.573 1.151 26.83 1.327 15.64 1 0.496 0.422 15.07 0.488 1.59 0.496 0.370 25.51 0.419 15.50 5 0.253 0.220 13.19 0.254 0.268 0.253 0.205 19.19 0.234 7.573 10 0.194 0.171 12.07 0.197 1.533 0.194 0.162 16.48 0.186 4.178 It is seen that GLOMSEC gives good improvements for the accuracy of approximate solutions when the nonlinearity increases. 4.2. Nonlinear oscillation systems subjected to color noise The introduction of the 1-DOF system subjected to color noise excitation in Chapter 4 is because the color noise random process is described as a white noise process passing through a second-order differential filter. The oscillation equation is solved with the filter equation so it can be considered as a system of multi-degrees-of- freedom. 4.2.1. Extend GLOMSEC to the case of random color noise excitation 4.2.2. Duffing system to color noise excitation Consider Duffing system to random color noise excitation: 2 3( )z z z z f      (4.41) where f is the color noise random process 2 2 f ff f f w       . (4.22) The nonlinear system is replaced by the linear one x cx kx f    (4.27) 21 The linear coefficients by GLOMSEC are: 2 2 22.41189 , .xk c     (4.45) By the classical criterion: 2 2 23 ,xk c      (4.46) By the Energy criterion: 2 2 22.5 ,xk c     (4.50) The relative errors between approximate solutions 2,x GL , 2 ,x C compared to 2,x E are presented in Table 4.3. The results show the solution 2,x GL is much better accurate then the solution 2 ,x C , namely with the bigest errors 2.392% compared to 11.398%, respectively. Tab. 4.3. Mean square response with 2 2f, ,S, , 1      changes.  2 ,x E 2 ,x C %CErr 2 ,x GL %GLErr 0.1 1 10 100 1.86038 0.66376 0.16687 0.03720 1.75024 0.60015 0.14855 0.03296 5.920 9.583 10.979 11.398 1.88195 0.67688 0.17072 0.03809 1.159 1.977 2.307 2.392 4.2.3. Duffing system with nonlinear damping to color noise excitation Consider Duffing syste

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