Lyapunov Equivalent Representation Form of Forced, Damped, Nonlinear, Two Degree-of-Freedom Systems

Abstract

Featured Application: The qualitative and quantitative dynamic behavior of two degree-offreedom nonlinear systems can be studied by using their corresponding decoupled one degree of freedom Duffing type equivalent representation forms in the sense of Lyapunov, with the advantage of capturing amplitude-dependent nonlinear mode shapes.Abstract: The aim of this paper focuses on finding equivalent representation forms of forced, damped, two degree-of-freedom, nonlinear systems in the sense of Lyapunov by using a nonlinear transformation approach that provides decoupled, forced, damped, nonlinear equations of the Duffing type, under the assumption that the driving frequency and the external forces are equal in both systems. The values of Lyapunov characteristic exponents (LCEs), Lyapunov largest exponents (LLE), and time-amplitude and frequency-amplitude curves computed from numerical integration solutions, indicate that the decoupled Duffing-type equations are equivalent, in the sense of Lyapunov, to the original dynamic system, since both set of motion equations tend to have the same qualitative and quantitative behaviors.

He's frequency–amplitude formulation for nonlinear oscillators using Jacobi elliptic functions

“…1,2 Furthermore, these Duffing-type oscillators can be used to transform nonlinear oscillators that have rational or irrational restoring forces into polynomial ones. 32–40…”

“…1,2 Furthermore, these Duffingtype oscillators can be used to transform nonlinear oscillators that have rational or irrational restoring forces into polynomial ones. [32][33][34][35][36][37][38][39][40] Table 1 illustrates the errors attained by considering different values of the nonlinear terms a 1 , a 2 , a 22 , and a 3 in equation (18). For comparison purposes, Table 1 lists the frequency values obtained by using the approach introduced by Ren and Hu to derive the frequency-amplitude equation 41…”

Recent strategy to study fractal-order viscoelastic polymer materials using an ancient Chinese algorithm and He’s formulation

“…To find the frequency-amplitude expression for equation (9) using the ancient Chinese algorithm Ying Bu Zu Shu, we first use the approach proposed in Refs. [67][68][69][70][71][72][73][74] to find its power-form equivalent representation, and then, we introduce a coordinate transformation to eliminate the damping term.…”

“…Following the transformation approach introduced in Refs. [67][68][69][70][71][72][73][74] , equation ( 7) can be written in the following form…”

Lyapunov Exponent‐Based Study of Chaotic Mechanical Behavior of Concrete under Compression

“…Secondly, the chaotic tools can be applied to predict data series using the method of chaos theory. Lyapunov exponents could characterize the chaotic phenomena quantitatively [14]. Several researchers implemented this index to analyze and predict the data series of structures [15] and acoustic signals [16].…”