VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MECHANICS o0o NGUYEN NGOC LINH ANALYSING NONLINEAR RANDOM VIBRATION BY THE EQUIVALENT LINEARIZATION METHOD Major: Engineering Mechanics Code: 62 52 01 01 SUMMARY OF DOCTORAL THESIS HA NOI – 2015 Supervisors: 1. Prof.Dr.Sc. Nguyen Dong Anh 2. Dr. Luu Xuan Hung Reviewer 1: Reviewer 2: Reviewer 3: The thesis submitted to the thesis committee of Institute of Mechanics for the doctoral degree, at 264 Doi Can - Ba Dinh - Ha Noi, on Date Month Year Hardcopy of the thesis can be found at 1. Vietnam National Library 2. Library of Institute of Mechanics 1 INTRODUCTION Random vibrations are popular in engineering problems such as structures subjected to wind or wave load, bearings or shafts in vechicle components. Under such a loading case, the corresponding vibration model is based on statistical and stochastic process theory. The analysing of that model often leads to the construction and solving the stochastic nonlinear differential equations. When the excitation loading varies arbitrarily in time, the corresponding response will also be arbitrary in time. Therefore, the response process and its characteristic properties can be expressed by averaging quantities in probability sense. The requirement of analytical expression of response is very important because of two main reasons, firstly it is allowed to confirm the accuracy of the model in comparing with the measured data, secondly we can estimate the adjustable and controlable paramaters in preliminary, exact or check design stages. However, the exact analytical solutions have been found for only a few nonlinear random vibration problems. Although numerical methods make nonlinear problems to be solvable, but a nonlinear system with numerical solution is unlikely to satisfies the analysing requirements in practice. For example, it takes a long time to construct exact calculating model not only of a multi-degree-of-freedom system but also of single-degree- of-freedom system with many input parameters. The approximate analytical methods are therefore important in study of stochastic nonlinear dynamical systems. Among the approximate analytical methods, the equivalent linearization method is one of the most popular one because of its 2 simplicity, it can be applied to both single-degree-of-freedom and multi-degree-of-freedom systems as well as stationary and nonstationary or hysteresis systems. In Proppe’s estimation (2003), there had been over 400 papers published on the subject of statistical linearization till 1998, and over 200 papers in journals and conference proceedings have been appeared during the next 15 years following Socha’s statistics (2008). However, the main disavantage of the equivalent linearization method is the decreasing of accuracy when the nonlinearity increases, over 20%. The study on this method focus on improving the accuracy of approximate solution. Hence, this thesis concentrates on solving the above mentioned disavantage of the equivalent linearization method with follow specific framework. Objective of the study is to construct a weighted dual criterion of the equivalent linearization method to analysis the second moment response of nonlinear vibration subjeted to stochastic excitation with various nonlinearities, the error of approximate solution is eliminated to 10%. Range of the study is single-degree-of-freedom systems subjeted to Gaussian white noise with polynomial nonlinearity. Research methods use analytical, analytic geometry, numerical methods. CHAPTER 1. BASIS OF PROBABILITY THEORY AND SEVERAL METHODS OF NONLINEAR RANDOM VIBRATION ANALYSIS 1.1 Concepts of probability A random variable is a finite single valued function X which associates a real numerical value with every outcome of a random 3 experiment. A stochastic process is a family of random variables which depends on time. The main characteristics of a stochastic process are evaluated by deterministic quatities such as probability density function, mean and mean square values, variance, correlation function and power spectral density. The probability density function of a random process   x t is       , , / p x t F x t x t    (1.1) where   , F x t is distribution function.   , p x t has property   , 1 p x t dx     (1.2) The mean value or mathematical expectation is the first moment             ; x E x t m t x t x t p x t dx       (1.3) The mean square value is the second moment           2 2 2 ; E x t x t x t p x t dx      (1.4) The variance is defined as             2 2 2 2 x x x D E x t m t x t x t       (1.5) The covariance is   11 1 2 1 2 1 2 , xx D t t x x x x     (1.6) In statistics, the correlation of two random quantities     1 2 X, Y x t x t   , is evaluated by correlation coefficient   11 XY 1/2 X Y 20 02 , 1 r r          , (1.7) In vector space of random variables X and Y, with  is the angle between two vectors, Rodgers (1988) proved that   cosr   (1.8) 4 In application, both the correlation and squared correlation coefficients r và r 2 are used as a measure of the linear correlation strenght between X and Y, they belong to effect size and are classified into weak, medium and strong levels (Cohen, 1988). 1.2.2 Special random processes Several special random processes are usually used in random vibration such as stationary, Wiener, Markov processes are introduced. The most frequently used process, Gaussian white noise, has probability density function   2 2 1 exp 2 2 x p x           (1.9) 1.3 Fokker-Planck-Kolgomorov equation (FPK) For stationary Markov processes, the stationary probability density function may be determined from the FPK equation         2 1 , 1 1 0 2 x x x x n n i ij i i j i i j a p K p x x x                     (1.10) where   x i a and   x ij K are drift and diffusion coefficients. 1.4 Random vibration subjected to Gaussian white noise Consider single-degree-of-freedom illustrated as figure 1.1     , tt tt pt mx b x k x g x x u t        (1.11) where m is mass, b tt is linear damping coefficient, k tt is linear restoring coefficient,   , pt g x x  is funtion of nonlinear damping and restoring forces,   u t is external excitation. Figure 1.1 Single-degree-of-freedom system Let     0 2 / , / , , , / tt tt pt h b m k m g x x g x x m       , when excitation 5 is Gaussian white noise then       / f t u t m t     , (1.11) can be expressed as bellow     2 2 , o x hx x g x x t           (1.12) The analytical probability density function of response of (1.12) may be determined by the stationary FPK equation (1.10) for specific following cases: - when   , 0 g x x   , or (1.12) is linear system. - when (1.12) has the form         2 0 , x f H x x x x g x t           (1.13) where   g x represents nonlinear restoring force, and     , f H x x  is the function of Hamiltonian total energy. - when (1.12) has nonlinear restoring force and linear damping force     2 0 2 x hx x g x t          (1.14) 1.5 Several approximate methods in nonlinear random vibration analysis Due to the elimination of the FPK equation method, several approximate methods have been developed such as pertubation, stochastic averaging, equivalent linearization, equivalent nonlinearization, using power spectral density. Beside that, some numerical method have been also developed for solving FPK equation or Monte Carlo simulation. Conclusions of chapter 1 In this chapter the basis theory of probability and random vibration are presented which can be used for next chapters. Several approximate methods in nonlinear random vibration analysis are introduced with main advantages and dis advantages. 6 CHAPTER 2. THE STOCHASTIC EQUIVALENT LINEARIZATION METHOD Among the analytical approximate methods, equivalent linearization is very popular in engineering application because of its simplicity and deversification. To introduce the main idea of this method, it is necessary to mention the classical criterion, which is one of the most familiar equivalent linearization criterion, proposed by Caughey in years 1950-1960. Consider the system governed by equation (1.12). Replacing the nonlinear function   , g x x  by coressponding linear ones,   , g x x bx kx     , then one gets the linear system       2 2 o x h b x k x t           (2.1) where b, k are called the equivalent linearization coefficients. Based on the equation error between (1.12) and (2.1)     , , e x x g x x bx kx       (2.2) the classical criterion requires         2 2 , , , , min kd b k S b k e x x g x x bx kx        (2.3) The minimum condition in (2.3) leads to     2 2 , , , xg x x xg x x b k x x      (2.4) Conclusions of chapter 2 In this chapter the stochastic equivalent linearization method and several developments of this method such as minimum potential energy, regulated equivalent linearization, equivalent linearization with non-Gauss distribution, partial equivalent linearization criteria are introduced. 7 CHAPTER 3. THE DUAL CRITERION OF STOCHASTIC EQUIVALENT LINEARIZATION METHOD 3.1. Basic idea of the general weighted dual criterion Based on the dual approach, N.D.Anh (2010) suggested the criterion       2 2 2 3 3 3 3 1 2 , min k S x kx p kx x p x x             (3.1) where 1 2 , p p are weighting parameters. 3.2. The dual criterion 3.2.1. Basic concept of the dual criterion The dual replacement of N.D. Anh A kB A    (3.2) where denotes A is nonlinear function, B is coressponding linear function, k is the linearization coefficient,  is return coefficient. Based on the dual replacement, consider the dual criterion     2 2 , 1 1 min 2 2 dn dn dn dn dn dn k S A k B k B A        (3.3) Using minimum condition in (3.3), one gets 2 1 2 dn AB k B    (3.4) 2 dn      (3.5) where 2 2 2 AB A B   (3.6) µ is dimensionless quatity, follow the inequation Schwarz has 0 1    (3.7) 3.2.2 The linear dependence of the dual criterion Under the assumption that A and B has zero means, 2 2 A A   , 2 2 B B   , AB AB   , then their correlation coefficient is 8 1/2 1/2 2 2 AB r A B  (3.8) Combining (3.6) and (3.8) yields 2 r   (3.9) Hence µ is called the linear dependence level. 3.2.3 Geometric meaning of the dual criterion Consider two dimension Hilbert space H of random functions     , u x v x with zero mean, finite second moment and the probability density function   p x R  , the inner product, norm and distance of this space are defined as, recspectively                       1/2 2 2 1/2 2 2 , , u v uv u x v x p x dx u u u u u x p x dx u v u v u x v x p x dx                        (3.10) where     , u x v x are represented by vectors u, v. Following perpendicular projection theorem, there exists only projection vector , p u  which sastifies , inf p u u u v     (3.11) Vector , p u  , the correlation coefficient and the angle θ between u and v have relationship   1/2 1/2 2 2 ( , ) cos uv u v r u v u v     (3.12) ,p v u r u v   (3.13) [...]... nonstationary and hysteresis systems PUBLISHED PAPERS RELATED TO THE THESIS 1 Anh N.D., Hieu N.N., Linh N.N (2012), “A dual criterion of equivalent linearization method for nonlinear systems subjected to random excitation”, Acta Mech March 2012, Volume 223, Issue 3, pp 645-654 (DOI 10.1007/s00707-011-0582-z2011) 2 N.D Anh, L.X Hung, N.N Linh (2012), “On the equivalent linearization method using dual... 2nd International Conference on Engineering Mechanics and Automation (ICEMA2), August 16-17 3 N.D Anh, N.N Linh (2014), “The effective range of the dual criterion of equivalent linearization method”, National conference on Engineering Mechanics, Volumn 1, ISBN 978-604-913-233-9, pp 465-468 4 Nguyen Dong Anh, Nguyen Ngoc Linh (2014), “ A weighted dual criterion of equivalent linearization method for... Symposium on Vibration and Control of Structures under Wind Action, February 5 Nguyen Dong Anh, Nguyen Ngoc Linh, Ninh Quang Hai (2014), “A Weighted Dual Criterion for the Problem of Equivalent”, Proceedings of the ASCE-ICVRAM-ISUMA conference, Institute for Risk and Uncertainty - University of Liverpool, UK, July 13-16 6 N.D Anh, N.N Linh (2014), “A weighted dual criterion for stochastic equivalent linearization... 0.527 11.33 0.30 5.340 0.568 0.386 32.07 0.465 18.22 0.20 6.633 0.536 0.326 39.20 0.386 28.06 0.10 14.681 0.451 0.166 63.18 0.181 59.78 0.001 Conclusions of chapter 3 Based on the dual replacement of N.D .Anh, the dual criterion and the concept of dual projection are proposed Main features and geometric properties of this criterion are constructed Examples of applying show that the approximate solutions . SUMMARY OF DOCTORAL THESIS HA NOI – 2015 Supervisors: 1. Prof.Dr.Sc. Nguyen Dong Anh 2. Dr. Luu Xuan Hung Reviewer 1: Reviewer 2: Reviewer 3: The thesis submitted. METHOD 3.1. Basic idea of the general weighted dual criterion Based on the dual approach, N.D .Anh (2010) suggested the criterion       2 2 2 3 3 3 3 1 2 , min k S x kx p kx x p x x  . 3.2. The dual criterion 3.2.1. Basic concept of the dual criterion The dual replacement of N.D. Anh A kB A    (3.2) where denotes A is nonlinear function, B is coressponding linear function,