Abstract
A new analytic algorithm called ‘continuous piecewise linearization method’ (CPLM) is developed to obtain periodic solutions of freely vibrating Duffing-type oscillators. This simple analytic algorithm is based on continuous piecewise linearization of the nonlinear stiffness with respect to displacement and was shown to produce very accurate results for few iterations. The algorithm is valid for Duffing-type oscillators possessing strong nonlinearity and/or undergoing large-amplitude oscillations. Studies conducted on Duffing oscillators with cubic, cubic-quintic and trigonometric sine stiffness nonlinearities showed that the CPLM results match standard numerical solutions and is more accurate than the popular energy balance method (EBM). Additionally, the present analysis shows that the CPLM is capable of predicting the quasi-linear behaviour observed in the oscillation history of Duffing-type oscillators with strongly nonlinearity and/or large-amplitude oscillations. This quasi-linear behaviour cannot be predicted by the EBM to which the CPLM is compared.
Highlights
Conservative systems, which are naturally oscillating undamped systems, are common in nature (e.g. vibration of human eardrum (He, 1999b)) and in artificially designed systems
The recent perturbation methods (He, 1999a; He, 2000; He, 2001; Lui, 2005; El-Naggar & Ismail, 2016) attempt to deal with the issue of small parameter in order to find solutions that are valid for both small- and large-amplitude oscillations, and weak and strong nonlinearity
The algorithm is based on the idea of continuous piecewise linearization of the nonlinear stiffness and is capable of accurately predicting the oscillation histories and nonlinear frequency of Duffing-type oscillators characterized by strong nonlinearity and/or large-amplitude oscillations
Summary
Conservative systems, which are naturally oscillating undamped systems, are common in nature (e.g. vibration of human eardrum (He, 1999b)) and in artificially designed systems (e.g. simple pendulum motion). The recent perturbation methods (He, 1999a; He, 2000; He, 2001; Lui, 2005; El-Naggar & Ismail, 2016) attempt to deal with the issue of small parameter in order to find solutions that are valid for both small- and large-amplitude oscillations, and weak and strong nonlinearity. The present study proposes a new iterative analytic algorithm to address the issues of solution accuracy, solution complexity and limited range of validity observed in existing perturbation and non-perturbation methods. The algorithm is based on the idea of continuous piecewise linearization of the nonlinear stiffness and is capable of accurately predicting the oscillation histories and nonlinear frequency of Duffing-type oscillators characterized by strong nonlinearity and/or large-amplitude oscillations
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