New Closed-Form Solution for Quadratic Damped and Forced Nonlinear Oscillator with Position-Dependent Mass: Application in Grafted Skin Modeling

Abstract

The paper deals with modelling and analytical solving of a strong nonlinear oscillator with position-dependent mass. The oscillator contains a nonlinear restoring force, a quadratic damping force and a constant force which excites vibration. The model of the oscillator is a non-homogenous nonlinear second order differential equation with a position-dependent parameter. In the paper, the closed-form exact solution for periodic motion of the oscillator is derived. The solution has the form of the cosine Ateb function with amplitude and frequency which depend on the coefficient of mass variation, damping parameter, coefficient of nonlinear stiffness and excitation value. The proposed solution is tested successfully via its application for oscillators with quadratic nonlinearity. Based on the exact closed-form solution, the approximate procedure for solving an oscillator with slow-time variable stiffness and additional weak nonlinearity is developed. The proposed method is named the ‘approximate time variable Ateb function solving method’ and is applicable to many nonlinear problems in physical and applied sciences where parameters are time variable. The method represents the extended and adopted version of the time variable amplitude and phase method, which is rearranged for Ateb functions. The newly developed method is utilized for vibration analysis of grafted skin on the human body. It is found that the grafted skin vibration properties, i.e., amplitude, frequency and phase, vary in time and depend on the dimension, density and nonlinear viscoelastic properties of the skin and also on the force which acts on it. The results obtained analytically are compared with numerically and experimentally obtained ones and show good agreement.