Analysis of nonlinear vibration by weighted averaging approach

ation: ( , ) ( )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:   2 2 , ( , ) ( , )e X X g X X X X Min        (2.3) Thus, we obtain: 2 2 2 2 ( , ) ( , )g X X X X g X X X XX X X XX     (2.6) 2 2 2 2 ( , ) ( , )g X X X X g X X X XX X X XX     (2.7) 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]: 0 ( ) ( ) ( ) w x t h t x t dt    (2.12) where h(t) is a time-dependent function, which is called the weighted coefficient function, it is satisfied: 0 ( ) 1h t dt   (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]: 2 2( ) , 0s th t s te s   (2.15) in which, the coefficient s is called the adjustment parameter. The weighted averaging value has some properties, specifically, when 0s  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: 3 5 7 2 1 1 3 5 7 2 1 0 n nX X X X X X            (3.4) in which, 1 3 5 7 2 1, , , , , n      are constants, n is a natural number, equation (3.4) satisfies the initial conditions: (0) , (0) 0X A X  (3.5) Applying the equivalent linearization method, the square of the approximate frequency of oscillation can be found: 4 6 8 2 2 2 1 3 5 7 2 12 2 2 2 n n X X X X X X X X              (3.9) Based on the harmonic solution of the linear equation cos( )X A 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: 3 1 3 0X X X    (3.15) The approximate frequency of oscillation is given in: 8 6 4 2 2 1 34 2 2 2 28 248 416 1536 ( 2 8)( 16) s s s s A s s s             thesis (3.16) 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 1s  and 2s  . 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% 0 1 2 3 4 5 0 5 1 0 1 5 2 0 R el at iv e er ro r (% ) Initial amplitude, A EBM Thesis, s=2 Thesis, s=1 Luận án, s=3 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%. 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 ( 4.1672Exact  ) [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 4.1 4.15 4.2 4.25 4.3 4.35 4.4 4.45 0 2 4 6 8 1 0 F re q u en cy , ω Parameter, s Thesis EBM Exact 9 When n = 2, equation (3.4) becomes a quintic Duffing oscillator: 3 5 1 3 5 0X X X X      (3.19) With s = 2, the approximate frequency of the quintic Duffing oscillator can be get as: 2 4 Thesis 1 3 50.72 0.575A A      (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. 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 0 0.5 1 1.5 2 2.5 0 5 1 0 1 5 2 0 R el at iv e er ro r (% ) Initial amplitude, A EBM Thesis, s=2 10 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 1s  và 2s  . 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: 0. n m p u u u u u           (3.27) with the initial conditions: (0) , (0) 0.u A u  (3.28) in which,  ,  ,  ,  and  are constants; m, n and p are the positive exponents. 700 750 800 850 900 950 1000 0 1 2 3 4 5 6 7 8 9 1 0 F re q u en cy , ω Parameter, s Thesis EBM Exact 11 Based on the equivalent linearization method, the approximate frequency of the oscillation is given in: 2 2 2 2 1 1 1 1 1 1 2 2 2 2 cos ( ) cos ( ) cos ( ) cos ( ) cos ( ) . cos ( ) cos ( ) p p w w m m m p m p w w n n w p p w w A t A t A t A t A t A t A t                                            (3.35) where 2 2 2 0 0 cos ( ) cos ( ) cos ( ) .k s t k s k w t s te t dt s e d             (3.36) 3.2.1. Duffing-harmonic oscillator With 1  , 1  , 1  , 1  , 1  , 3m  , 1n  and 2p  ; equation (3.27) becomes: 3 2 0. 1 u u u u u      (3.37) The approximate frequency of this oscillator can be found: 2 2 1 1 0.72 . 1 0.72 Thesis A A      (3.38) 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 3m  , we have Duffing oscillator with double-well potential: 3 0.u u u   (3.40) The solution of equation (3.40) depends on the initial conditions. The period ThesisT obtained in the thesis, the period achieved by Momeni et al. [36] using the energy balance method EBMT and the exact period ExactT [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 ( 2A  ) A ExactT [26] EBMT [36] Error (%) ThesisT 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 2A  ) A ExactT [26] EBMT [36] Error (%) ThesisT 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: 0,u u u u    (3.63) Case 2: 3 0,u u u u    (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 4th 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 10A  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:   24 2 2 4 2 0 2 2 3 2 2 2 . L L P NL w EA w w EI Al dx L xx x w w k w k k w A q x t                           (4.29) Figure 4.1. Model of a microbeam resting on a nonlinear elastic foundation For convenience, the following dimensionless variables are introduced: 4 2 2 44 2 4 6 , , , , , 1 , 1 , , , .NLL PL P NL x w I AL l x w r t t L r A EI h k r Lk L k L qL K K K q EI EI EI EIr                  (4.33) Using equation (4.33), the equation of motion (4.29) is rewritten in dimensionless form: 214 2 2 2 3 4 2 2 2 0 1 . 2 L P NL w w w w w dx K w K K w q xx x x t                         (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 4 2 23( ) 0.72 . 4 4 NL P L NLK K K                (4.55) - For clamped-clamped microbeam: 4 4 2 216 4 350.72 . 3 3 3 48 NL P L NLK K K                    (4.56) Table 4.1. Comparison of the frequency ratio of macrobeams Initial amplitude α Pinned-Pinned Clamped-Clamped Azrar et al. [49] Simsek [65] (error %) Thesis (error %) Azrar et al. [49] Simsek [65] (error %) Thesis (error %) 1 1.0891 1.0897 (0.06) 1.0863 (0.26) 1.0221 1.0231 (0.09) 1.0222 (0.01) 2 1.3177 1.3228 (0.39) 1.3114 (0.48) 1.0856 1.0897 (0.37) 1.0862 (0.06) 3 1.6256 1.6393 (0.84) 1.6186 (0.43) 1.1831 1.1924 (0.79) 1.1853 (0.19) 4 - 2.0000 (-) 1.9697 (-) 1.3064 1.3228 (1.26) 1.3115 (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 19 The effect of the flexural rigidity ratio ( 2Al EI  ) on the frequancy ration of the microbeams is presented in Figure 4.10. It can be seen that the flexural rigidity ratio reduces the frequency ratio of the microbeams, which is similar to the effect of the material length scale parameter. Figure 4.10. The variation of of the frequency ration of microbeams to the the flexural rigidity ratio with KL = 10, KP = 10, KNL = 10 and S=20; (a) pinned-pinned, (b) clamped-clamped Figure 4.14. The variation of the frequency ratio of microbeams to the slenderness ratio with KL = 30, KP = 50, KNL = 30 and θ = 6; (a) pinned-pinned, (b) clamped-clamped Figure 4.14 depicts the variation of the frequency ratio of microbeams to the slenderness ratio ( 2 /S AL I ) for some values of the initial amplitude. It can observe that the frequency ration increases 20 monotonously with the slenderness ratio. It can be estimated that when the slenderness ratio increases 33%, the frequency ratio increases about 31% for pinned-pinned microbeam and about 33% for clamped-clamped microbeam. 4.2. Nonlinear vibration of nanobeam subjected to electrostatic force A clamped-clamped nanobeam is placed between two fixed electrodes is considerd in Figure 4.18. The nanobeam has a length L, across-section b h , mass density  , Young’s modulus E and moment of inertia I. The nanobeam is compressed by an axial force P0. The initial gap between two stationary electrodes and nanobeam is g0, two electrodes is applied by a voltage V0. Figure 4.18. A model of nanobeam is placed between two electrodes Based on the nonlocal strain gradient theory and Euler-Bernoulli beam model, the equation of motion for the nanobeam is: 21 22 4 2 4 2 2 02 4 2 4 0 2 4 2 2 2 2 2 2 2 1 ( ) 2 ( ) ( ) . L w EA w w w EI l P dx ea L xx x x x w w f A ea f ea t x t x                                              (4.89) in which, f is the electrostatic force given in:     2 0 2 2 0 0 1 1 ( , ) . 2 vbVf x t g w g w           (4.76) where 8.85 /v pF m  is the vacuum permittivity. Applying the Galerkin technique, the equation of motion for the nanobeam is converted into the following ordinary differential equation:  2 4 6 81 2 3 4 5 3 5 7 9 11 6 7 8 9 10 11 0, q c c q c q c q c q c q c q c q c q c q c q            (4.96) Using the equivalent linearization method and the weighted averaging technique, the approximate frequency of the nanobeam is expressed as: 2 4 6 6 7 8 9 8 10 10 11 8 6 5 4 4 2 3 2 1 0.72 0.575 0.4836 0.4198 0.3722 . 0.4198 0.4836 0.575 0.72 thesis c c c c c c c c c c c                               (4.124) When / 0l L   and / 0ea L   , the model examined in the thesis becomes the model which is studied by Fu et al. [78], Qian et al. [79] based on the classical elasticity theory. The approximate frequencies obtained by various analytical methods and exact frequencies are listed in Table 4.8 for some values of the initial 22 amplitude  , axial compressive force P and applied voltage V. We can observe the accuracy of the approximation solution obtained by the thesis thesis compared with the approximate solution obtained by the energy balance method EBM and the variational method VA . Table 4.8. Comparison of the approximate frequencies with the exact frequency α P V exact [79] EBM [78] thesis VA 0.3 10 0 26.8372 26.3867 26.7577 26.3644 0.3 10 20 16.6486 16.3829 16.5865 16.3556 0.6 10 10 28.5382 26.5324 28.2199 26.1671 0.6 10 20 18.5902 17.5017 18.5507 17.0940 Figure 4.21. Variation the nonlinear frequency (a) and the frequency ratio (b) to the nonlocal parameter  at given initial amplitude  The influence of the nonlocal parameter ( /ea L  ) on the nonlinear frequency and frequency ration of the nanobeam is presented in Figure 4.21 for fixed values of 0.3  , 5P  , 0.2  , 10V  and 40  . As can see that the nonlocal leads to a decreasing in the nonlinear frequency and an increasing in the frequency ratio. This observation is completely consistent with the nonlocal elasticity theory [59, 60] because the nonlocal parameter reduces the stiffness of the 23 nanobeam, thus the nonlinear frequency decreases with increasing value of the nonlocal parameter. Figure 4.24. Variation of the nonlinear frequency (a) and the frequency ratio (b) to the material length scale parameter for some values of the initial amplitude The effect of the material length scale parameter ( /l L  ) on the nonlinear frequency and the frequency ratio of the nanobeam is shown in Figure 4.24 for fixed values of 0.2  , 5P  , 0.2  , 10V  and 60  . The material length scale parameter enhances the stiffness of the nanobeam, thus the nonlinear frequency increases when the material length scale parameter increase; this issue is completely consistent with the strain gradient theory [61-63]. Furthermore, the increase in the linear frequency due to the material length scale parameter is larger than the increase in the nonlinear frequency, thus the frequency ratio of the nanobeam decreases when the material length scale parameter increases. .. For fixed values of 0.2  , 0.2  , 10P  , 0.1  and 10V  , Figure 4.26 shows the variation of the nonlinear frequency and frequency ratio of the nanobeam to the slenderness ration for some values of the initial amplitude. From this figure, we can see that both 24 the nonlinear frequency and frequency ration of the nanobeam increase when the slenderness ratio increases. Figure 4.26. Variation of the nonlinear frequency (a) and the frequency ratio (b) to the slenderness ration for some values of the initial amplitude Figure 4.28. Variation of the nonlinear frequency (a) and the frequency ratio (b) to the axial compressive force for some values of the initial amplitude Figures 4.28 and 4.30 depict the effects of the axial compressive force P and applied voltage V on the nonlinear vibration response of the nanobeam, respectively. In which, Figure 4.28 is plotted with 0.2  , 0.2  , 0.2  , 30  and 15V  ; while Figure 4.30 is plotted with 0.2  , 0.2  , 0.2  , 30  and 20P  . It can be concluded that both the axial compressive force and applied voltage 25 reduce the nonlinear frequency and increase the frequency ratio of the nanobeam. Figure 4.30. Variation of the nonlinear frequency (a) and the frequency ratio (b) to the applied voltage for some values of the initial amplitude Conclusion of Chapter 4 The equivalent linearization method and the weighted averaging technique were applied to analyze the nonlinear vibration response of the microbeams resting on elastic foundation based on the modified couple stress theory and nonlinear vibration behavior of the nanobeam subjected to electrostatic force based on the nonlocal strain gradient theory. Comparing obtained analytical solutions with analytical solutions using other methods, the exact solutions showed the accuracy of the obtained results. The content of Chapter 4 has been published in the articles[T6], [T7], [T8] and [T9] in "List of works related to the thesis". 26 CONCLUSIONS AND RECOMMENDATIONS With the aim of applying the equivalent linearization method in combination with the weighted averaging value in analysis of undamped nonlinear vibrations. The new results obtained by the thesis include: - Developed a method by combining the equivalent linearization method and weighted averaging value to analyze the undamped nonlinear vibration of single-degree-of-freedom systems. - Applying the proposed method to analyze the undamped nonlinear vibration of single-degree-of-freedom systems and continuous systems such as micro- and nano-beams. Accordingly, the main results obtained from the thesis include: - Clarifying the properties of the weighted averaging value, the relationship between the weighted averaging value and the classical averaging value, the relationship between th