Higher-order approximation of cubic–quintic duffing model

We apply an Artificial Parameter Lindstedt-Poincaré Method (APL-PM) to find improved approximate solutions for strongly nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. This approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution which makes it a unique solution. It is demonstrated that this method works very well for the whole range of parameters in the case of the cubic-quintic oscillator, and excellent agreement of the approximate frequencies with the exact one has been observed and discussed. Moreover, it is not limited to the small parameter such as in the classical perturbation method. Interestingly, This study revealed that the relative error percentage in the second-order approximate analytical period is less than 0.042% for the whole parameter values. In addition, we compared this analytical solution with the Newton– Harmonic Balancing Approach. Results indicate that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems. Utter simplicity of the solution procedure confirms that this method can be easily extended to other kinds of nonlinear evolution equations

A Note on He’s Parameter-Expansion Method of Coupled Van der Pol–Duffing Oscillators

This paper presents the analytical and approximate solutions of the coupled chaotic Van der Pol-Duffing systems, by using the He\u27s parameter-expansion method (PEM). One iteration is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. From the obtained results, we can conclude that the suggest method, is of utter simplicity, and can be easily extended to all kinds of non-linear equations

Higher accuracy approximate solution for oscillations of a mass attached to a stretched elastic wire by rational harmonic balance method

A second-order modified rational harmonic balance method is used for approximately solve the nonlinear differential equation that governs the oscillations of a system typified as a mass attached to a stretched elastic wire for which the restoring force for this oscillator has an irrational term with a parameter lambda that characterizes the system. A frequency-amplitude relation is constructed and this frequency is valid for the complete range of oscillation amplitudes A and parameter lambda, and excellent agreement of the approximate frequencies with the exact one is demonstrated and discussed. The discrepancy between the approximate frequency and the exact one never exceed 0.12%. This error corresponds to lambda = 1. while for lambda < 1 the relative error is much lower. For example, its value is lower than 0.017% for lambda = 0.5

Periodic Solution of Nonlinear Conservative Systems

Conservative systems represent a large number of naturally occurring and artificially designed scientific and engineering systems. A key consideration in the theory and application of nonlinear conservative systems is the solution of the governing nonlinear ordinary differential equation. This chapter surveys the recent approximate analytical schemes for the periodic solution of nonlinear conservative systems and presents a recently proposed approximate analytical algorithm called continuous piecewise linearization method (CPLM). The advantage of the CPLM over other analytical schemes is that it combines simplicity and accuracy for strong nonlinear and large-amplitude oscillations irrespective of the complexity of the nonlinear restoring force. Hence, CPLM solutions for typical nonlinear Hamiltonian systems are presented and discussed. Also, the CPLM solution for an example of a non-Hamiltonian conservative oscillator was presented. The chapter is aimed at showcasing the potential and benefits of the CPLM as a reliable and easily implementable scheme for the periodic solution of conservative systems

SIMPLIFIED HAMILTONIAN-BASED FREQUENCY-AMPLITUDE FORMULATION FOR NONLINEAR VIBRATION SYSTEMS

The Hamiltonian-based frequency formulation has been hailed as an unprecedented success for it gives a straightforward insight into a complex nonlinear vibration system with simple calculation. This paper gives a systematical analysis of the formulation, and two simplified formulations are suggested.  The cubic-quintic Duffing oscillator is used as an example to show extremely simple calculation and remarkable accuracy. It can be used as a paradigm for many other applications, and the one-step solving process has cleaned up the road of the nonlinear vibration theory.

A semi-analytical approach for the response of nonlinear conservative systems

This work applies Parameter expanding method (PEM) as a powerful analytical technique in order to obtain the exact solution of nonlinear problems in the classical dynamics. Lagrange method is employed to derive the governing equations. The nonlinear governing equations are solved analytically by means of He’s Parameter expanding method. It is demonstrated that one term in series expansion is sufficient to generate a highly accurate solution, which is valid for the whole domain of the solution and system response. Comparison of the obtained solutions with the numerical ones indicates that this method is an effective and convenient tool for solving these types of problems

Analysis of Nonlinear Dynamic Behaviour of Nanobeam resting on Winkler and Pasternak Foundations

Dynamic modeling of nanobeam under stretching and two-parameter foundation effects result in nonlinear equations that are very difficult to find exact analytical solutions. In this study, variation iteration method is used to develop approximate analytical solutions to nonlinear vibration analysis of nanobeam under the effects of stretching and Winkler and Pasternak foundations. The governing equation of motion for the nanotube was derived based on Euler-Bernoulli beam theory. The developed approximate analytical solutions for the governing equation are validated the results of other methods of analysis, are also used to carry out effects of some model parameters on the dynamic behaviour of the nanobeam.  These analytical solutions can serve as a starting point for a better understanding of the relationship between the physical quantities in the problems as it provides clearer insights to understanding the problems in comparison with numerical methods

Higher-order approximation of cubic–quintic duffing model

We apply an Artificial Parameter Lindstedt-Poincaré Method (APL-PM) to find improved approximate solutions for strongly nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. This approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution which makes it a unique solution. It is demonstrated that this method works very well for the whole range of parameters in the case of the cubic-quintic oscillator, and excellent agreement of the approximate frequencies with the exact one has been observed and discussed. Moreover, it is not limited to the small parameter such as in the classical perturbation method. Interestingly, This study revealed that the relative error percentage in the second-order approximate analytical period is less than 0.042% for the whole parameter values. In addition, we compared this analytical solution with the Newton– Harmonic Balancing Approach. Results indicate that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems. Utter simplicity of the solution procedure confirms that this method can be easily extended to other kinds of nonlinear evolution equations