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Modified Lindstedt-Poincaré method for obtaining resonance periodic solutions of nonlinear non-autonomous oscillators

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Abstract

A modified Lindstedt-Poincaré (LP) method for obtaining the resonance periodic solutions of nonlinear non-autonomous vibration systems is proposed in this paper. In the modified method, nonlinear non-autonomous equations are converted into a group of linear ordinary differential equations by introducing a set of simple transformations. An approximate resonance solution for the original equation can then be obtained. The periodic solutions of primary, super-harmonic, sub-harmonic, zero-frequency and combination resonances can be solved effectively using the modified method. Some examples, such as damped cubic nonlinear systems under single and double frequency excitation, and damped quadratic nonlinear systems under double frequency excitation, are given to illustrate its convenience and effectiveness. Using the modified LP method, the first-order approximate solutions for each equation are obtained. By comparison, the modified method proposed in this paper produces the same results as the method of multiple scales.

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Authors and Affiliations

  1. School of Mechanical Engineering, Tianjin University, Tianjin, 300072, China

    Kangkang Guo  (郭抗抗) & Shuqian Cao  (曹树谦)

  2. Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin, 300072, China

    Kangkang Guo  (郭抗抗) & Shuqian Cao  (曹树谦)

Authors
  1. Kangkang Guo  (郭抗抗)
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  2. Shuqian Cao  (曹树谦)
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Corresponding author

Correspondence to Shuqian Cao  (曹树谦).

Additional information

Supported by the National Natural Science Foundation of China (No. 11172199) and the Key Project of Tianjin Municipal Natural Science Foundation (No. 11JCZDJC25400).

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Guo, K., Cao, S. Modified Lindstedt-Poincaré method for obtaining resonance periodic solutions of nonlinear non-autonomous oscillators. Trans. Tianjin Univ. 20, 66–71 (2014). https://doi.org/10.1007/s12209-014-2126-9

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  • DOI: https://doi.org/10.1007/s12209-014-2126-9

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