Symbolic stability boundaries and root locus plots in time-periodic systems

Research output: Contribution to conferencePaperpeer-review


The recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems is used to symbolically compute stability boundaries as an explicit function of the system parameters and to construct root locus plots. By evaluating this matrix at lhe end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincare map, is obtained. The subsequent use of well-known criteria enables one to obtain the equations for the stability boundaries in the parameter space as polynomials of the system parameters. The symbolic nature of the method also allows one to obtain root locus plots in the complex plane as a function of the system parameters. The roots of the FTM (Floquet multipliers) must lie within the unit circle for stability. Further, the technique can successfully be applied to periodic systems whose internal excitation is strong. The symbolic software Mathematica is used here to perform all symbolic calculations. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.

Original languageEnglish (US)
Number of pages9
StatePublished - 2001
Event18th Biennial Conference on Mechanical Vibration and Noise - Pittsburgh, PA, United States
Duration: Sep 9 2001Sep 12 2001


Other18th Biennial Conference on Mechanical Vibration and Noise
Country/TerritoryUnited States
CityPittsburgh, PA

ASJC Scopus subject areas

  • Modeling and Simulation
  • Mechanical Engineering
  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design


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