Summary of doctoral thesis on solid mechanics: Analysis of nonlinear vibration by weighted averaging approach

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The thesis has developeda technique by combining the equivalent linearization method and the weighted averagingvalueto analyze the responses of some undamped free nonlinear vibrations.

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Nội dung Text: Summary of doctoral thesis on solid mechanics: Analysis of nonlinear vibration by weighted averaging approach

MINISTRY OF EDUCATION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY ----------------------------- Dang Van Hieu ANALYSIS OF NONLINEAR VIBRATION BY WEIGHTED AVERAGING APPROACH Major: Mechanics of Solid Code: 9440107 SUMMARY OF DOCTORAL THESIS IN SOLID MECHANICS Ha Noi, 2021
The thesis has been completed at: Graduate University Science and Technology – Vietnam Academy of Science and Technology. Supervisors: 1. Assoc. Prof. Dr. Ninh Quang Hai 2. Dr. Duong The Hung Reviewer 1: … Reviewer 2: … Reviewer 3: …. Thesis is defended at Graduate University Science and Technology Vietnam Academy of Science and Technology at … , on …. Hardcopy of the thesis be found at : - Library of Graduate University Science and Technology - Vietnam national library
1 PREFACE 1. The necessity of the thesis Vibration is a common phenomenon in nature and technology. In fact, almost all vibrations of technical systems are nonlinear, linear vibration is just the idealization of a vibration phenomenon that we encounter. Only a very small class of the nonlinear vibration problem has the exact solution. The approximate analytical methods are effective tools to find the solutions of the nonlinear vibration problem. Among the approximate analytical methods, the Equivalent Linearization method [1] is a simple but effective method for analyzing nonlinear vibration problems. However, like other approximate analytical methods, the linearization method equivalent with the classical averaging value often has disadvantages that the obtained results are often inaccurate and sometimes unacceptable when the nonlinearity of the problem increases. Many authors have tried to improve this disadvantage of the equivalent linearization method, in which in 2015, Nguyen Dong Anh [2] proposed a new method for estimating the averaging value of a determistic function instead of using the classical averaging value, which is called the weighted averaging technique. The weighted averaging technique has partially overcome the disadvantages of the linearization method equivalent with the classical averaging. With the above analysis, author has selected the topic "Analysis of nonlinear vibrations by weighted averaging approach" as the research topic for this thesis.
2 2. The objectives of the thesis The thesis has developed a technique by combining the equivalent linearization method and the weighted averaging value to analyze the responses of some undamped free nonlinear vibrations. 3. The content of the thesis The content of the thesis is presented in four chapters. Specifically, Chapter 1 presents an overview of nonlinear vibrations and some approximate analytical methods to solve the problem of nonlinear vibrations. Chapter 2 presents the basic ideas of the equivalent linearization method for the deterministic vibration system, definition of the weighted averaging value and some properties of the weighted averaging value. In Chapter 3, the thesis applies the proposed method to analyze the responses of undamped nonlinear free vibration of single-degreeof-freedom systems. Chapter 4 presents two problems of the nonlinear vibration of microbeams resting on an elastic foundation and the nonlinear vibration of the nanbeams subjected to electrostatic force. CHAPTER 1. OVERVIEW This chapter presents an overview of nonlinear vibrations and some approximate analytical methods to solve the problem of nonlinear vibrations; the domestic and foreign regime on the analysis of the nonlinear vibrations. The reviews show that the equivalent linearization method is an effective tool for analyzing nonlinear vibration problems. Proposed for the purpose of analyzing the responses of the nonlinear
3 vibrations subject to random excitation, the equivalent linearization method is easy to apply to deterministic nonlinear vibration systems. However, the error of the equivalent linearization method is sometimes quite large, especially for strong nonlinear vibration systems. The weighted averaging value overcomes the disadvantages of the classical averaging one in improving the accuracy of the equivalent linearization method. Based on the overall evaluation, the thesis has selected the research topic and proposed research issues in details. CHAPTER 2 THE EQUIVALENT LINEARIZATION METHOD FOR DETERMINISTIC NONLINEAR SYSTEM AND WEIGHTED AVERAGING VALUE In this Chapter, the thesis presents the basic ideas of the equivalent linearization method for the deterministic nonlinear vibration system, the definition of the weighted averaging value and some properties of the weighted averaging value. 1.1. The equivalent linearization for deterministic nonlinear vibration system Consider the deterministic nonlinear vibration of single-degree- of-freedom system described by the following nonlinear differential equation: X  g ( X , X )  F (t ) (2.1) where X , X and X are displacement, velocity and acceleration, respectively; g ( X , X ) is a nonlinear of displacement and velocity; and
4 F (t ) is an external excitation. The linearized form of Eq. (2.1) is introduced as: X   X   X  F (t ) (2.2) The coefficients  and  are determined from the minimum mean-square deviation criterion:  g( X , X )   X   X  2 e2 ( X , X )   Min (2.3)  , Thus, we obtain: g( X , X ) X X 2  g( X , X ) X XX  2 (2.6) X2 X 2  XX g( X , X ) X X 2  g( X , X ) X XX   2 (2.7) X 2 X 2  XX In Eqs. (2.3)-(2.7), the symbol is the averaging operator over time. 2.2. The weighted averaging Definition: The weighted avaraging value of of an integrable deterministic function x(t) is defined as [2]:  x(t ) w   h(t ) x(t )dt (2.12) 0 where h(t) is a time-dependent function, which is called the weighted coefficient function, it is satisfied:   h(t )dt  1 0 (2.13)
5 For the vibration problems, we will consider only ω-periodic functions x(t ) , a form of the weighted coefficient function is considered as follows [2]: h(t )  s 2 2te st , s  0 (2.15) in which, the coefficient s is called the adjustment parameter. The weighted averaging value has some properties, specifically, when s  0 the weighted averaging value becomes the classical averaging value; the weighted averaging value of the periodic function can be calculated via the Laplace transformation; the weighted averaging value preserves the linear characteristic of the classical averaging value; and the weighted averaging value contains more information about the periodic function than the classical averaging one. Conclusion of Chapter 2 Chapter 2 presents the basic ideas of the equivalent linearization method for the deterministic nonlinear vibration system and the definition of the weighted averaging value. The analysis shows that the weighted averaging value has advantages over the classical averaging one, which should create positive signals when using the weighted averaging value to analyze the nonlinear vibration systems. Some results of Chapter 2 have been published in the article [T1] in the section "List of works related to the thesis" in clarifying some properties of the weighted averaging value and its advantages compared to with the classical averaging one.
6 CHAPTER 3. NONLINEAR VIBRATION OF SINGLE- DEGREE-OF-FREEDOM SYSTEMS In Chapter 3, the equivalent linearization method with the weighted averaging value will be applied to analyse undamped nonlinear free vibration of single-degree-of-freedom systems. The obtained results are compared with the published ones, the exact ones and the numerical ones. 3.1. Duffing oscillator In this section, the thesis considers a nonlinear generalized Duffing oscillator given in: X  1 X  3 X 3  5 X 5  7 X 7    2n1 X 2n1  0 (3.4) in which, 1 , 3 , 5 , 7 , , 2n1 are constants, n is a natural number, equation (3.4) satisfies the initial conditions: X (0)  A, X (0)  0 (3.5) Applying the equivalent linearization method, the square of the approximate frequency of oscillation can be found: X4 X6 X8 X 2n 2   1   3 2  5  7     2 n 1 (3.9) X2 X2 X2 X2 Based on the harmonic solution of the linear equation X  A cos(t ) , the averaging values in expression (3.9) can be calculated by using the definition of the weighted averaging value in Chapter 2 and the Laplace transform. 3.1.1. Cubic Duffing oscillator
7 When n = 1, we have a cubic Duffing oscillator: X  1 X  3 X 3  0 (3.15) The approximate frequency of oscillation is given in: s8  28s 6  248s 4  416s 2  1536 thesis  1  3 A2 (3.16) ( s  2s  8)( s  16) 4 2 2 2 5 4 EBM Relative error (%) Thesis, s=2 3 Thesis, s=1 Luận án, s=3 2 1 0 0 5 10 15 20 Initial amplitude, A Hình 3.1. The variation of the relative error of the approximate frequencies to the initial amplitude A of the cubic Duffing oscillator with α1=10 and α3=10 Figures 3.1 presents the variation of the relative error of the frequency obtained in the thesis ( thesis ) and the frequency obtained by the energy balance method ( EBM ) [35] to the initial amplitude A. Some values of the parameter s have been selected (s = 1, 2 và 3). From this Figure, it can see that the approximate frequency obtained in the thesis is much better than the approximate frequency obtained by the energy balance method for two values of the parameter s that are s  1 and s  2 . Specifically, when the initial amplitude A increases, the relative error of the energy balance method reaches to 2.2%, while the relative error of the method used in the thesis is only 1.18% for s = 1 and 0.15%
8 for s = 2. However, with the larger values of the parameter such as s = 3, the relative error of the method used in the thesis is up to 4.4%. 4.45 Thesis 4.4 EBM 4.35 Exact Frequency, ω 4.3 4.25 4.2 4.15 4.1 0 2 4 6 8 10 Parameter, s Hình 3.3. The variation of the frequencies to the parameter s of the cubic Duffing oscillator with α1=10, α3=10 and A=1 With α1 = 10, α3 = 10 and A = 1, Figure 3.3 represents the variation of the approximate frequencies to the parameter s. For this figure, it can see that the approximate frequency abtained in the thesis is equal to the exact frequency ( Exact  4.1672 ) [43] corresponding to the two values of the parameter s = 0.5 and s = 2.5. Through the survey, the optimal value of the parameter s varies according to each system, with the desire to choose s as a natural number, the thesis used the value s = 2 for comparison purposes. Furthermore, from this figure it can be seen that as the parameter s increases, the approximate frequency of the oscillation will increase and this leads to a decrease in the accuracy of the obtained solution. 3.1.2. Quintic Duffing oscillator
9 When n = 2, equation (3.4) becomes a quintic Duffing oscillator: X  1 X  3 X 3  5 X 5  0 (3.19) With s = 2, the approximate frequency of the quintic Duffing oscillator can be get as: Thesis  1  0.723 A2  0.5755 A4 (3.20) Comparing the approximate frequency obtained in the thesis and the approximate frequency obtained by the energy balance method with the exact frequency of the quintic Duffing nonlinear oscillator is shown in Figure 3.5. We see that when the initial amplitude A increases, the relative error of the energy balance method [35] reaches to 2.26%, while the relative error of the method used in the thesis is only 1.52% with selected values of the parameters of the system as shown in Figure 3.5. 2.5 2 Relative error (%) 1.5 1 EBM 0.5 Thesis, s=2 0 0 5 10 15 20 Initial amplitude, A Figure 3.5. The variation of the relative error of the approximate frequencies to the initial amplitude of the quintic Duffing oscillaotr with α1=1, α3=10 and α5=100
10 1000 950 Thesis EBM Frequency, ω 900 Exact 850 800 750 700 0 1 2 3 4 5 6 7 8 9 10 Parameter, s Figure 3.6. The variation of the approximate frequencies of the quintic Duffing oscillator to the parameter s with α1=10, α3=100, α5=100 and A=10 For α1 = 1, α3 = 100, α5 = 100 and A = 10, the variation of the approximate frequencies to the parameter s is presented in Figure 3.6. From this figure, we see that the approximate frequency obtained in the thesis is equal to the exact frequency ( Chính xác  751.6951 ) corresponding to two values of the parameter s  1 và s  2 . The relative error of the obtained approximate solution increases with larger values of the parameter s. 3.2. Generalized nonlinear oscillator In this section, a generalized nonlinear oscillator is considered as:  un u  u   um   0. (3.27)  up with the initial conditions: u(0)  A, u(0)  0. (3.28) in which,  ,  ,  ,  and  are constants; m, n and p are the positive exponents.
11 Based on the equivalent linearization method, the approximate frequency of the oscillation is given in:  A2 cos 2 (t )   A p  2 cos p  2 (t )   w w     Am 1 cos m 1 (t )   Am  p 1 cos m  p 1 (t )   w w   An 1 cos n 1 (t )     w . (3.35)  A2 cos 2 (t )   A p  2 cos p  2 (t ) w w where   cos k (t ) w   0 s 2 2te st cos k (t )dt   s 2 e s cos k ( )d . (3.36) 0 3.2.1. Duffing-harmonic oscillator With   1 ,   1 ,   1 ,   1 ,   1 , m  3 , n  1 and p  2 ; equation (3.27) becomes: u u  u  u3   0. (3.37) 1 u2 The approximate frequency of this oscillator can be found: 1 Thesis  1  0.72 A2  . (3.38) 1  0.72 A2 Comparing the approximate solution obtained in the thesis with the numerical solution using the 4th order Runge - Kutta method is shown in Figure 3.12. The accuracy of the obtained approximate solution for the Duffing-harmonic oscillator can be observed in this figure.
12 Figure 3.12. Comparison of the analytical solution with the numerical solution of the Duffing-harmonic oscillator 3.2.2. Duffing oscillator with double-well potential For   1 ,   1 ,   0 and m  3 , we have Duffing oscillator with double-well potential: u  u  u3  0. (3.40) The solution of equation (3.40) depends on the initial conditions. The period TThesis obtained in the thesis, the period achieved by Momeni et al. [36] using the energy balance method TEBM and the exact period TExact [26] are listed in Tables 3.3 and 3.4 for some values of the initial amplitude A. Table 3.3. Comparison of the approximate frequencies with the exact frequency of the Duffing oscillator with double-well potential potential ( A 2) A TExact [26] TEBM [36] Error (%) TThesis Error (%) 1.42 15.0844 8.7784 41.8047 9.3477 38.0306 1.45 11.2132 8.2725 26.2253 8.7656 21.8278 1.5 9.2237 7.5778 17.8442 7.9797 13.4869 1.7 6.3528 5.8150 8.4655 6.0438 4.8639 2 4.6857 4.4429 5.1817 4.5825 2.2024
13 5 1.5286 1.4914 2.4335 1.5239 0.3074 10 0.7471 0.7304 2.2353 0.7457 0.1873 50 0.1484 0.1451 2.2237 0.1481 0.2021 100 0.0742 0.0726 2.1563 0.0741 0.1347 100 0.0074 0.0073 1.3513 0.0074 0.0000 Table 3.4. Comparison of the approximate frequencies with the exact frequency of the Duffing oscillator with double-well potential potential ( 1 A  2 ) A TExact [26] TEBM [36] Error (%) TThesis Error (%) 1.05 4.3061 4.3045 0.0373 4.3349 0.0067 1.1 4.1781 4.1748 0.0781 4.2309 0.0126 1.15 4.0582 4.0530 0.1267 4.1309 0.0179 1.2 3.9460 3.9384 0.1923 4.0347 0.0225 1.25 3.8417 3.8303 0.2961 3.9420 0.0261 1.3 3.7468 3.7282 0.4964 3.8529 0.0283 1.35 3.6688 3.6316 1.0139 3.7671 0.0268 1.4 3.6897 3.5399 4.0576 3.6845 0.0014 1.41 3.8506 3.5222 8.5261 3.6684 0.0473 1.412 3.9755 3.5164 11.548 3.6652 0.0781 3.3. Nonlinear oscillator with discontinuity In this section, two nonlinear oscillators with discontinuity are considered: Case 1: u   u   u u  0, (3.63) Case 2: u   u3   u u  0, (3.70)
14 The approximate solution obtained in the thesis and the approximate solution found by using the homotopy perturbation method [11] are compared with the numerical solution using the 4 th order Runge-Kutta and shown in Figure 3.20 (Case 1) and Figure 3.23 (Case 2). Figure 3.20. Comparison of the analytical solutions with the numercal solution of the nonlinear oscillator with discontinuity for   10 ,   100 and A=1 Figure3.23. Comparison of the analytical solutions with the numercal solution of the nonlinear oscillator with discontinuity for   10 ,   10 and A  10 Conclusion of Chapter 3 In Chapter 3, the thesis has applied the equivalent linearization method and the weighted averaging technique to analyse undamped
15 nonlinear free vibration of single-degree-of-freedom systems. The accuracy of the approximate analytical solutions obtained by the thesis has been verified by comparing the obtained results with the exact results, published results using other approximate methods and numerical results. The obtained results confirm that the weighted averaging value overcomes the disadvantages of the linearization method equivalent in which the classical averaging value is used. The method used in the thesis is not only valid for weak nonlinear systems, but also for strong and medium nonlinear systems. The results of Chapter 3 have been published in articles [T1], [T2], [T3], [T4], [T5] in the section "List of works related to the thesis" CHAPTER 4 NONLINEAR VIBRATION OF MICRO- /NANO-BEAMS In this Chapter, nonlinear vibration of microbeams resting on elastic foundation based on the modified couple stress theory and nonlinear vibration of nanobeam under electrostatic force based on the nonlocal strain gradient theory are investigated. 4.1. Nonlinear vibration of microbeams resting on elastic foundation Considering an isotropic microbeam with length L and cross- section b  h as shown in Figure 4.1. The microbeam rests on a nonlinear elastic foundation with the foundation parameters kL, kP and kNL corresponding to the Winkler linear layer, Pasternak layer and the nonlinear layer.
16 Based on the modified couple stress theory and Euler-Bernoulli beam model, the equation of motion for microbeam in transverse displacement w is given by:  4 w  EA  w    2 w L 2  EI   Al  x4   2L   x  dx  x2  2  0  (4.29) 2 w 2 w k L w  k P 2  k NL w3   A 2  q. x t Figure 4.1. Model of a microbeam resting on a nonlinear elastic foundation For convenience, the following dimensionless variables are introduced: x w I  AL4 l 6 2 x , w , r  ,tt ,   ,   1 , L r A EI h 1  (4.33) k L4 k L2 k r 2 L4 qL4 K L  L , K P  P , K NL  NL ,q . EI EI EI EIr Using equation (4.33), the equation of motion (4.29) is rewritten in dimensionless form:  4 w  1  w   2 w 2 w 2 w 1 2 x 4  2 0  x       dx   K w  K  K w3   q.  x x 2 t 2 2 L P NL (4.34)
17 Employing the equivalent linearization method and the weighted averaging technique, the approximate frequencies of microbeams can be found: - For pinned-pinned microbeam: 4 3  NL  ( 4    2 K P  K L )  0.72   K NL   2 . (4.55)  4 4  - For clamped-clamped microbeam:  16 4    4 35  NL    4    2 K P  K L   0.72   K NL   2 . (4.56)  3 3   3 48  Table 4.1. Comparison of the frequency ratio of macrobeams Initial Pinned-Pinned Clamped-Clamped amplitude Azrar et Simsek Thesis Azrar et Simsek Thesis α al. [49] [65] (error al. [49] [65] (error (error %) %) (error %) %) 1 1.0891 1.0897 1.0863 1.0221 1.0231 1.0222 (0.06) (0.26) (0.09) (0.01) 2 1.3177 1.3228 1.3114 1.0856 1.0897 1.0862 (0.39) (0.48) (0.37) (0.06) 3 1.6256 1.6393 1.6186 1.1831 1.1924 1.1853 (0.84) (0.43) (0.79) (0.19) 4 - 2.0000 1.9697 1.3064 1.3228 1.3115 (-) (-) (1.26) (0.39) Comparison of the frequency ratio NL / L (ratio of the nonlinear frequency NL to the linear frequency L ) of the macrobeams using different methods is shown in Table 4.1. It can be seen that the results obtained by the thesis are better than those of Şimşek (especially for clamped-clamped macrobeams).
18 The effect of the material length scale parameter on the nonlinear vibration response of the microbeams is shown in Figures 4.5 and 4.6. We can see that the material length scale parameter reduces the frequency ratio of the microbeams, while both the linear frequency and the nonlinear frequency of the microbeams increase as the material length scale parameter increases. Figure 4.5. The variation of the nonlinear frequency and frequency ration of the pinned-pinned to the material length scale parameter with KL = 50, KP = 30 and KNL=50 Figure 4.6. The variation of the nonlinear frequency and frequency ration of the pinned-pinned to the material length scale parameter with KL = 50, KP = 30 and KNL=50