Nonlinear vibration of microbeams based on the nonlinear elastic foundation using the equivalent linearization method with a weighted averaging

2019 ◽  
Vol 90 (1) ◽  
pp. 87-106
Author(s):  
Van-Hieu Dang ◽  
Dong-Anh Nguyen ◽  
Minh-Quy Le ◽  
Quang-Hai Ninh
Keyword(s):  
Nonlinear Vibration ◽  
Elastic Foundation ◽  
Linearization Method ◽  
Weighted Averaging ◽  
Equivalent Linearization ◽  
Nonlinear Elastic Foundation ◽  
Equivalent Linearization Method ◽  
Nonlinear Elastic

scholarly journals An Approximate Solution for a Nonlinear Duffing – Harmonic Oscillator

2019 ◽  
pp. 1-14
Author(s):  
Van Hieu – Dang
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Harmonic Oscillator ◽  
Analytical Methods ◽  
Nonlinear Oscillations ◽  
Linearization Method ◽  
Weighted Averaging ◽  
Equivalent Linearization ◽  
Exact Results ◽  
Equivalent Linearization Method

The equivalent linearization method introduced by Caughey is a powerful tool for analyzing random oscillations. The method is also easy to apply for deterministic oscillations. However, with strong nonlinear systems, the error of this method is usually quite large and even not acceptable. In conjunction with a weighted averaging, the equivalent linearization method has shown more accuracy than the classical one in which the conventional averaging value is used. Combining advantages of the classical equivalent linearization method and accuracy of the weighted averaging, the proposed method has shown that it is a useful tool for analyzing nonlinear oscillations including strong nonlinear systems. In this paper, the proposed method is applied to analyze a nonlinear Duffing – harmonic oscillator. The present results are compared with the results obtained by using other analytical methods, exact results and numerical results.

scholarly journals Postbuckling and Free Nonlinear Vibration of Microbeams Based on Nonlinear Elastic Foundation

2018 ◽  
Vol 2018 ◽  
pp. 1-17 ◽  
Author(s):  
Dang Van Hieu
Keyword(s):  
Nonlinear Vibration ◽  
Elastic Foundation ◽  
Scale Parameter ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Nonlinear Elastic Foundation ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Nonlinear Elastic ◽  
Material Length

In this paper, post-buckling and free nonlinear vibration of microbeams resting on nonlinear elastic foundation subjected to axial force are investigated. The equations of motion of microbeams are derived by using the modified couple stress theory. Using Galerkin’s method, the equation of motion of microbeams is reduced to the nonlinear ordinary differential equation. By using the equivalent linearization in which the averaging value is calculated in a new way called the weighted averaging value, approximate analytical expressions for the nonlinear frequency of microbeams with pinned–pinned and clamped–clamped end conditions are obtained in closed-forms. Comparisons with previous solutions are showed accuracy of the present solutions. Effects of the material length scale parameter and the axial compressive force on the frequency ratios of microbeams; and effect of the material length scale parameter on the buckling load ratios of microbeams are investigated in this paper.

scholarly journals The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
D. V. Hieu ◽  
N. Q. Hai ◽  
D. T. Hung
Keyword(s):  
Perturbation Method ◽  
Harmonic Balance Method ◽  
Variational Iteration Method ◽  
Linearization Method ◽  
Balance Method ◽  
Weighted Averaging ◽  
Equivalent Linearization ◽  
Runge Kutta Method ◽  
Practical Engineering ◽  
Equivalent Linearization Method

The Equivalent Linearization Method (ELM) with a weighted averaging is applied to analyze five undamped oscillator systems with nonlinearities. The results obtained via this method are compared with the ones achieved by Parameterized Perturbation Method (PPM), Min–Max Approach (MMA), Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Energy Balance Method (EBM), Harmonic Balance Method (HBM), 4th-Order Runge-Kutta Method, and the exact ones. The obtained results demonstrate that this method is very convenient for solving nonlinear equations and also can be successfully applied to a lot of practical engineering and physical problems.

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