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.
Highlights
Minorsky [1], Caughey [2], Iwan [3,4], Sinha and Srinivasan [5], Agrwal and Denman [6], Yuste and Sánchez [7,8], Cai [9], Farzaneh and Tootoonchi [10], and many others have reported approaches that replace linear and nonlinear dynamic systems by equivalent ones with known solutions that are closed to the original system, which produce the same oscillations that appear in the original equations of motion
One can conclude that the improvement of the numerical prediction obtained the equivalent representation will depend physical system, its nonlinearities, A from transformation approach has beenforms proposed to find on thethe equivalent representation form in the and its initial conditionofvalues
By studying the dynamic responses of three nonlinear dynamic systems, the validity of the proposed conjecture has been numerically examined by comparing the Lyapunov characteristic exponents (LCEs) of the original equations with those obtained from their equivalent uncoupled
Summary
Minorsky [1], Caughey [2], Iwan [3,4], Sinha and Srinivasan [5], Agrwal and Denman [6], Yuste and Sánchez [7,8], Cai [9], Farzaneh and Tootoonchi [10], and many others have reported approaches that replace linear and nonlinear dynamic systems by equivalent ones with known solutions that are closed to the original system, which produce the same oscillations that appear in the original equations of motion. From the normal form representation of the system (1) it is assumed that the modal system’s generalized coordinates can be equivalently expressed as a power series expansion [18,19,20,21] to decouple each normal mode equation into a forced, damped, nonlinear differential equation of the Duffing type, whose approximate solution has been widely discussed in the literature [22,23,24]. The aim of this paper is to verify the previously mentioned conjecture by introducing a nonlinear approach to replace the canonical normal form of a dynamical two-degree-of-freedom system by two equivalent decoupled expressions of the Duffing type. The steps involved in the proposed transformation approach to decouple nonlinear equations are introduced
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