Summary of Mechanical engineering and Engineering mechanics doctoral dissertation: Research on non-linear random vibration by the global – local mean square error criterion

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Propose the global – local mean square error criterion (GLOMSEC) for Gaussian equivalent linearization method (GEL) for randomly excited MDOF nonlinear system subjected to white noise or color noise excitation. The mean square of response solution will be concentrated to evaluate the accuracy of the proposed criterion by comparison with exact solution or other accepted solutions.

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Nội dung Text: Summary of Mechanical engineering and Engineering mechanics doctoral dissertation: Research on non-linear random vibration by the global – local mean square error criterion

MINISTRY OF EDUCTION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ----------------------------- Nguyen Cao Thang RESEARCH ON NON-LINEAR RANDOM VIBRATION BY THE GLOBAL – LOCAL MEAN SQUARE ERROR CRITERION Major: Engineering Mechanics Code: 9 52 01 01 SUMMARY OF MECHANICAL ENGINEERING AND ENGINEERING MECHANICS DOCTORAL DISSERTATION HANOI - 2019
The dissertation has been completed at: Graduate University of Science and Technology – Vietnam Academy of Science and Technology Supervisors : Dr. Luu Xuan Hung Prof. Dr. Sc. Nguyen Dong Anh Reviewer 1: … Reviewer 2: … Reviewer 3: …. Dissertation is defended at Graduate University of Science and Technology – Vietnam Academy of Science and Technology at …, on date … month … 2019. Hardcopy of the dissertation can be found at: - Library of Graduate University of Science and Technology - Vietnam National Library
1 INTRODUCTION 1. Rationale of the dissertation The analysis, design and control of vibration play an important role in improving effectiveness and performance of structures, vehicles and engines. In recent years, the multi degree of freedom system is used for most of engineering applications. Accordingly, it is necessary to develop Gaussian equivalent linearization method (GEL) for randomly excited MDOF nonlinear system, based on dual concept to develop Global local mean square error criterion (GLOMSEC) for MDOF nonlinear system. 2. Object of the dissertation Apply dual concept to solve the limited area [-rx , + rx] in the local mean square error criterion (LOMSEC). By that way, propose the global – local mean square error criterion (GLOMSEC) for Gaussian equivalent linearization method (GEL) for randomly excited MDOF nonlinear system subjected to white noise or color noise excitation. The mean square of response solution will be concentrated to evaluate the accuracy of the proposed criterion by comparison with exact solution or other accepted solutions. 3. Research methodology In the dissertation, analyse method, numerical method, Monte – Carlo Simulation method are considered. The analyse method is considered to create the error criterion: base on dual concept in analyse response of nonlinear systems (consider two different approaches to a problem) to obtain the linearized coefficients by close analysis method. The numerical method is considered to program by Matlab software to compute and simulate random nonlinear vibration MDOF systems. The Monte Carlo simulation is considered to find simulation solution for determination the accuracy of linearization method. 4. Scientific and practical application - Develop Gaussian equivalent linearization method (GEL) – one of most popular method used in Random vibration. Particularly, the
2 Global Local Mean Square Error Criterion – GLOMSEC is generalized for MDOF random nonlinear system. - Develop close equation system to determine mean square of responses. Investigate and evaluate the accuracy of proposed criterion for MDOF nonlinear random systems subjected to white noise or color process. - The results of the dissertation are applied to analyse technical nonlinear random systems. 5. Structure of the dissertation The structure of the dissertation includes: the introduction, 4 chapters, the conclusions, a list of publications, the references and the appendix. CHAPTER 1. INTRODUCTION TO PROBABILITY THEORY AND SOME METHODS ANALYSING NON-LINEAR RANDOM VIBRATION 1.1. Random variable and its probabilistic properties Define probability of a random event [29], [69]: Perform n experiments, if the outcome M occurs m times, than probability of outcome M, denote P(M) is the limitation of frequence f(M) = m/n when the number of experiments n increases to infinity: lim f ( M )  P ( M ) (1.1) n  Random variable X is a quantity that links each outcome r of an experiment with a real number X(r) satisfies: a) Set X  x is called an event M for each real number x, b) probability of event X =   equal zero: PX =  = 0 (1.2) The cumulative distribution function (cfd) of the random variable X is defined for any real number x by:
3 F(x) = P[X  x] (1.3) 1.2 Stochastic processes There are definitions of: Probability density function; High order moment; Mathematical expectation; Mean square; Variance; Auto- correlation; 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]: x  2hx  02 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 h  0 , g ( x, x )  0 . The equivalent linearization equation of (2.10) is as follows: x  2hx  02 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]: S kd   g ( x, x )  bx  kx  2  min (2.14) b, k Hence: Skd S (2.15)  0; kd  0 b k Assuming that the solution is a random process, the response x , x should be independent, that is xx  0 , solving the system of equations (2.15) we obtain   x, x  xg xg  x, x  b ,k (2.17) x 2 x2
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 x 2 , x 2 . b) Use (2.17) to determine the linear coefficients. c) Solve equation (2.11) to find the new instantaneous second order moments x 2 , x 2 . d) Repeat b) and c) until the specified accuracy is reached. We consider the nonlinear system with multi degrees of freedom under random excitation  + Cx + Kx + Φ  x, x ,  Mx x  = Q t  , (2.20) where x , x , x are displacement, velocity and acceleration vectors, respectively M   mij  , C   cij  , K   kij  are n n n n n n 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  Sij  where Sij  is nn cross spectral density function of Qi and Q j . The equivalent linearization system is as follows:  M + M  q   C + C  q   K + K  q  Q  t  , e e e (2.21) where M e , Ce , K e 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  q e  Φ  q, q,     M e q  + Ce q + K e q  . (2.22)
7 The error e satisfies the Mean Square Error Criterion upon M , Ce , K e : e E  e T e   emin . (2.24) e e M ,C , K 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: T E  zz T   M e Ce K e   E  zΦT  z   , (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: x  2hx  02 x  g ( x , x)  (t )  (2.47) Where, the notation is defined as above. The equivalent linearization equation of (2.47) takes the form: x  2hx  02 x   x   x  (t )  (2.48) where λ, μ are linearization coefficients. The error between (2.47) và (2.48) is: e x , x   g  x , x    x  x (2.49) The classical criterion gives [29, 44]   2 ( x , x )P ( x , x ) dxdx  min (2.51)  e    , Where P( x, x) is the probability density function (PDF) of x and x :
8 g ( x , x ) x g ( x , x ) x   ,  . (2.53) x 2 x 2 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]:  x0  x 0 e 2 ( x , x )P ( x , x ) dxdx  min (2.54)    x0  x0  , Where, x0 , x0 are positive values. The integration (2.54) is transformed into the non-dimension one by introducing x0  r x , x0  r x where r is a positive value,  x and  x are standard deviations of x and x :  r x  r x [ e 2 ( x , x )]  e 2 ( x , x ) P ( x , x ) dxdx  m in (2.55) r  r   , x x where [.] denotes:  r x  r x [.]  ( . ) P ( x , x ) d x d x (2.56) r  r  x x We have analogously:  (r )   g ( x , x ) x  ,  ( r )   g ( x , x ) x  . (2.57)  x 2   x 2  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 s 1     ( r )  Lim    ( r ) dr  , s s 0 (2.60) s 1     ( r )  Lim    ( r ) dr  . s  s 0 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: z  g z   f t  (2.61) T z   z1 , z2 ,..., z n  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: e z   z  g z   f t  (2.62) Introduce new linearization terms into (2.62): ez   z  Az  Az  g  z   f t  (2.64) where A  a ij  is matrix n×n. Let y is a stationary solution of: y  Ay  f t   0 (2.65) From (2.64) and (2.65) we have: e y   Ay  g  y  (2.66) Denote p(y) probability density function (PDF) of y of (2.65). According to the LOMSEC we have:  y 01 y 1  y n0 yn e 2 i  y    y  n  y  0 1 y1 0 n yn ei2  y  p y dy  min , i,j = 1,…,n aij It gives:  A  g  y yT  yy  T 1 (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 aij  aij ( y10 , y 20 ,..... y n0 )   y10 y n0  1  0 0 Lim 0   ....  aij ( y10 , y 20 ,..... y n0 ) dy10 dy 20 .....dy n0  y1 , y 2 ,..... y n   y1 y 2 ..... y n0 0 0   0 0  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: x     1  x 2  x 2  x  x  d   t   (3.6) The exact PDF two-dimensional probability density function of the system [29, 44]:  2  (3.7) p  x, x   C exp   x 2  x 2   0.5  x 2  x 2     d   
11 where C is the normalized constant. If Proba  x  a is chosen then the domain  a, a will be detemined by: Prob a  x  a   a a    p  x, x  dx dx  (3.8) Suppose we choose Proba  x  a  0.98 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 0.04 0.4 p 0.3 4 p 0.02 0.2 1 2 0.1 0 0 0 x 0 -4 x -1 -2 -2 0 0 x 2 x -1 -4 4 1 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: x  2 h  x   x 3    o2 x    t   (3.11) where h ,  ,  o ,  are positive values. The corresponding linearization system is as follows: x   2h  b  x  o2 x    t   (3.12) where b is the linear coefficient. The mean square response of (3.12) is: 2 x2  2  2h  b  o2 (3.13) The coefficient b defined by the classical criterion is: b  6 h  x 2  (3.15) By GLOMSEC is: 1 s T  b  b(r )  2h  x 2  Lim   2, r dr   2.4119 * 2h  x 2  (3.22) s   s T   0 1,r  Substituting (3.22) into (3.13) gives: h  h 2  2.4119h 2 x2  (3.23) GL 2 * 2.4119ho2 To evaluate approximate solutions we use the solution x2 [29]. The relative errors of x 2 , x 2 compared to the ENL GL kd solution x 2 are defined by (3.24): ENL
13 x2  x2 x2  x2 kd cx GL cx Err( C )  *100%, Err( GL )  *100% x2 x2 cx cx (3.24) In Table 3.4, the results show that the solution x2 has better GL accuracy than the solution x 2 , in particular the largest error of kd GLOMSEC is only 1.93%. Table 3.4. The second moment of the response of the nonlinear damping oscillator system h  0.05, o  1,   4h , and γ changes Err( C ) Err(GL ) γ x2 x2 x2 ENL kd GL % % 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 x      x 2  x  o2 x    t   (3.25) where  ,  ,  , o ,  are positive values,   t  is white noise with unit intensity. We replace g  x , x    x 2 x by the linear one bx , where b is the linear coefficient: x     b  x  o2 x    t   (3.26)
14 The coefficient b defined by the classical criterion is: b    x 2  (3.29) By GLOMSEC is: 1 s T  b  b(r )    x 2  Lim   1, r dr   0.8371  x 2  (3.34)    s 0 T0, r  s  Mean square response x 2 of Van der Pol system (3.25) by GL GLOMSEC is: 1   2 2 1,6742 2   x2         (3.36) GL 1,6742   o2   To evaluate approximate solutions, we use the Monte Carlo simulation solution, [29]. The relative error between the approximate solutions x 2 , x 2 , compared to the simulation solutions GL kd x2 is calculated by the formula (3.24). MC Table 3.5. Mean square responses of Van der Pol oscillator with α*ε=0.2;  0 =1;  =2; σ2 changes Err( C ) Err(GL ) 2 x2 x2 x2 MC kd GL % % 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 x 2 have better accuracy than x 2 , in GL kd 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: x  2hx  o2 x   x 3    t   (3.37) The notation is the same as in the previous example. The exact solution is [29, 44]  2  4h  1 1  x exp  2  o2 x 2   x 4   dx    2 4  x2   (3.39) cx  4h  1 2 2 1 4    exp   2  2 o x  4  x  dx The equivalent liner system is: x  2hx  o2 x  kx    t   (3.40) The linearization coefficient by GLOMSEC is: 1 s  1 s T  k  k ( r )  Lim   k ( r ) dr   Lim    2, r  x 2  dr     s 0  s 0 T1, r s s     1 s T     x 2  Lim   2, r dr   2.4119  x 2  s  s   0 T1, r  (3.48) Mean square response x2 of Duffing system (3.37) by GL GLOMSEC: 1  2  2  x2   o  o4  2.4119  (3.49) GL 2 * 2.4119  h   
16 The relative error between the approximate solutions x2 , GL x 2 with the exact one x 2 defined by (3.24) and presented in kd cx Tab. 3.6. Table 3.6 Mean square responses of Duffing system, o  1, h  0.25,   1 ;  changes Err( C ) Err(GL )  x2 cx x2 kd x2 GL % % 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   c e    2 D  ( t ) (3.66)
17 The linearization coefficient c e by GLOMSEC is: 1 s   1 s Tt 3 , r  c e  c e (r )  Lim   c e (r )dr    E{ 2 }1/ 2 Lim   dr   1.49705 E{ 2 }1/ 2 s  s s   s T   0   0 1, r  Mean square response by GLOMSEC is: 2/3 D D D E  2   E  2     0.76415   GL GL c 1.49705 E{ } e 2 1/ 2   Mean square response by the classical criterion is: 2/3 D D D E  2   E  2     0.7323   C C c e 1.5958 E{ 2 }1/2   Mean square response by the nonlinear equivalent linearization method is: 2/3 D E  2   E  2   0.765   ENL ENL   The relative error between the approximate solutions x 2 , GL x2 , compared to the nonlinear equivalent linearization method, is C calculated by the formula (3.24). We have: Err(C )  4.314%; Err(GL )  0.130% 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: 3 1 0   x1   1 0   x1  12 a   x1   1x13  b  x1  x2    w1 (t )  0 1        2      x2   0 1  2   x2   a 2   x2   2 x 32  b  x2  x1    w2 (t )  3  (4.1) where: i , a, b, i , i (i=1, 2) are constants. w1 (t ), w2 (t ) are white noise processes with zero mean and E wi (t ) wi (t   )  2 Si ( ) (i=1, 2),  ( ) is Delta Dirac function, S1, S2 = const. The equivalent linear system is: 1 0    x1   1  c11e c12e   x1  12  k11e a  k12e   x1   w1 (t )  0 1          e      x2   a  k21 2  k22   x2   w2 (t )  e e e 2    x2   c21 1  2  c22 (4.4) where cije , kije ; (i, j  1,2) are equivalent linear coefficients. The equation error is:    ( x , x )  C e X  K e X (4.5) 3     1 x1  b  x1  x2   3 ( x, x )   1    3   2   2 x 32  b  x2  x1   ce c12e    x1   k11e k12e  x  (4.6) Ce   11e e  ; X   x  ; K e   e ; X   1 ;  c21 c22   2  k21 k22e   x2  To simplify the calculation we suppose that x1 , x2 are independent. Using the Appendix of dissertation and noting E  xi2 n 1 x 2j m 1   0 (i  j ) GLOMSEC gives: