Construction of approximate analytical solutions to strongly nonlinear damped oscillators

Abstract

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton's method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.

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.…”

“…where the coefficients ] 1 , , and can be determined by using the expressions (10). Notice that we have introduced a slight modification in the damping coefficient terms of (29) to take into account the influence of the cubic nonlinear damping term, , in the dynamical system response.…”

“…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.…”

Application of AG method and its improvement to nonlinear damped oscillators

“…They reported that the combination of extended KBM and harmonic balance methods led to the desired results. In another work, Newton's method and harmonic balance method were combined by Wu and Sun [17] for obtaining the analytical approximate solutions of strongly nonlinear damped oscillators. The proposed method can approach very accurate solutions in a few iterations even if damping and nonlinearities are high in the system.…”