Lyapunov-Perron Transformation for Quasi-Periodic Systems and Its Applications

Susheelkumar C. Subramanian, Sangram Redkar

Research output: Contribution to journalArticlepeer-review

Abstract

This paper depicts the application of symbolically computed Lyapunov-Perron (L-P) transformation to solve linear and nonlinear quasi-periodic systems. The L-P transformation converts a linear quasi-periodic system into a time-invariant one. State augmentation and the method of normal forms are used to compute the L-P transformation analytically. The state augmentation approach converts a linear quasi-periodic system into a nonlinear time-invariant system as the quasi-periodic parametric excitation terms are replaced by "fictitious" states. This nonlinear system can be reduced to a linear system via normal forms in the absence of resonances. In this process, one obtains near identity transformation that contains fictitious states. Once the quasi-periodic terms replace the fictitious states they represent, the near identity transformation is converted to the L-P transformation. The L-P transformation can be used to solve linear quasi-periodic systems with external excitation and nonlinear quasi-periodic systems. Two examples are included in this work, a commutative quasi-periodic system and a non-commutative Mathieu-Hill type quasi-periodic system. The results obtained via the L-P transformation approach match very well with the numerical integration and analytical results.

Original languageEnglish (US)
Article number041015
JournalJournal of Vibration and Acoustics, Transactions of the ASME
Volume143
Issue number4
DOIs
StatePublished - Aug 2021
Externally publishedYes

Keywords

  • Dynamics
  • Mechatronics and electro-mechanical systems
  • Nonlinear vibration
  • Stability
  • Vibration control

ASJC Scopus subject areas

  • Acoustics and Ultrasonics
  • Mechanics of Materials
  • Mechanical Engineering

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