A Transformation Method for Solving Conservative Nonlinear Two-Degree-of-Freedom Systems

Elı́as-Zúñiga, Alex; Trejo, Daniel Olvera; Real, Inés Ferrer; Martínez-Romero, Óscar

doi:10.1155/2014/237234

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…To determine the above integration constants, Equations (11) and (12) are substituted into Equations (7), and (13)- (16) are substituted into Equation (8). This step yields, after performing the corresponding integrations, the following expressions: …”

Section: Computation Of η I η Ii V I and V Ii Valuesmentioning

confidence: 99%

“…On the other hand, a nonlinear transformation approach that is based on a cubication method, was recently introduced to obtain the equivalent representation form of conservative two degree-offreedom nonlinear oscillators [11]. There, the authors developed an approach to replace a two degree-of-freedom homogeneous, undamped system by another equivalent system with known solutions that were closed to the original one.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2018

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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.

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Section: Computation Of η I η Ii V I and V Ii Valuesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2018

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“…Reference [19] applied the enhanced cubication method to develop approximate solutions for the most common nonlinear oscillators and leads to amplitude-time response curves and angular frequency values. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Wang

Zhang

Wang

2015

Mathematical Problems in Engineering

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The mechanical deformation of DNA is very important in many biological processes. In this paper, we consider the reduced Kirchhoff equations of the noncircular cross-section elastic rod characterized by the inequality of the bending rigidities. One family of exact solutions is obtained in terms of rational expressions for classical Jacobi elliptic functions. The present solutions allow the investigation of the dynamical behavior of the system in response to changes in physical parameters that concern asymmetry. The effects of the factor on the DNA conformation are discussed. A qualitative analysis is also conducted to provide valuable insight into the topological configuration of DNA segments.

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Free Vibration Analysis of a Conservative Two-Mass System with General Odd Type Nonlinear Connection

Mirzabeigy

Madoliat

2017

Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.

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A Transformation Method for Solving Conservative Nonlinear Two-Degree-of-Freedom Systems

Cited by 3 publications

References 18 publications

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

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

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

Free Vibration Analysis of a Conservative Two-Mass System with General Odd Type Nonlinear Connection

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