Exact solutions for oscillators with quadratic damping and mixed-parity nonlinearity

Abstract

Exact vibration modes of a nonlinear oscillator, which contains both quadratic friction and a mixed-parity restoring force, are derived analytically. Two families of exact solutions are obtained in terms of rational expressions for classical Jacobi elliptic functions. The present solutions allow the investigation of the dynamical behaviour of the system in response to changes in physical parameters that concern nonlinearity. The physical significance of the signs (i.e. attractive or repulsive nature) of the linear, quadratic and cubic restoring forces is discussed. A qualitative analysis is also conducted to provide valuable physical insight into the nature of the system.

Investigation of the Equivalent Representation Form of Strongly Damped Nonlinear Oscillators by a Nonlinear Transformation Approach

“…Here, = , = 2 10 , and = 4 10 . In accordance with our proposed nonlinear transformation approach, we first replace the restoring force ( ,) = 2]̇+ + 3 + 5 by an equivalent cubic-like polynomial expression by using (8), (9), and (10). This provides the following restoring force expression:…”

“…Notice that and are the parameter values that must satisfy (9) and (10). Figure 1 illustrates the numerical integration solutions of (12) and 15 Figure 1.…”

“…Here, 1 ( 10 ) and 2 ( 10 ) are the first and second-order Bessel functions of the first kind, respectively. Once again, and are the fitting parameters that must satisfy (8) and (9). We next illustrate in Figure 2 the numerical integration solutions of (19) and 20 To further assess the applicability of our proposed approach, we next derive the equivalent cubic-like representation form of a damped oscillator with a mass attached to two stretched elastic springs.…”

“…The main motivation to find a nonlinear transformation form of (1) is based on the fact that exact or approximate solutions of damped nonlinear oscillators of the Duffing type can be found in the literature. See, for instance, [7][8][9][10] and references cited therein. Therefore, if we can transform (1) into the damped Duffing equation, one could find its dynamical response in an easier way.…”

Equilibrium Configurations of the Noncircular Cross-Section Elastic Rod Model with the Elliptic KB Method

“…Reference [20] developed a nonlinear transformation approach to obtain the equivalent representation form of conservative two-degree-of-freedom nonlinear oscillators. Lai and Chow [21] used Jacobi elliptic Krylov-Bogoliubov (KB) method to find two families of exact solutions for oscillators with quadratic damping and mixedparity nonlinearity. Motivated by the above literatures review, this paper focuses on accurate solutions for the reduced Kirchhoff equations and undertakes a qualitative analysis of the topological configuration of DNA segments.…”