In this paper, a new analysis technique in the study of dynamical systems with periodically varying parameters is presented. The method is based on the fact that all linear periodic systems can be replaced by similar linear time-invariant systems through a suitable periodic transformation known as the Liapunov-Floquet (L-F) transformation. A general technique for the computation of the L-F transformation matrices is suggested. In this procedure, the state vector and the periodic matrix of the linear system equations are expanded in terms of the shifted Chebyshev polynomials over the principal period. Such an expansion reduces the original differential problem to a set of linear algebraic equations from which the state transition matrix (STM) can be constructed over the period in closed form. Application of Floquet theory and eigenanalysis to the resulting STM yields the L-F transformation matrix in a form suitable for algebraic manipulations. The utility of the L-F transformation in obtaining solutions of both linear and nonlinear dynamical systems with periodic coefficients is demonstrated. It is shown that the application of L-F transformation to free and harmonically forced linear periodic systems directly provides the conditions for internal and combination resonances and external resonances, respectively. The application of L-F transformation to quasilinear periodic systems provides a dynamically similar quasilinear systems whose linear parts are time-invariant and the solutions of such systems can be obtained through an application of the time dependent normal form theory. These solutions can be transformed back to the original coordinates using the inverse L-F transformation. Two dynamical systems, namely, a commutative system and a Mathieu type equation are considered to demonstrate the effectiveness of the method. It is shown that the present technique is virtually free from the small parameter restriction unlike averaging and perturbation procedures and can be used even for those systems for which the generating solutions do not exist in the classical sense. The results obtained from the proposed technique are compared with those obtained via the perturbation method and numerical solutions computed using a Runge-Kutta type algorithm.
Skip Nav Destination
Article navigation
April 1996
Research Papers
Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems
S. C. Sinha,
S. C. Sinha
Nonlinear Systems Research Laboratory, Department of Mechanicai Engineering, Auburn University, Auburn, AL 36849
Search for other works by this author on:
R. Pandiyan,
R. Pandiyan
Nonlinear Systems Research Laboratory, Department of Mechanicai Engineering, Auburn University, Auburn, AL 36849
Search for other works by this author on:
J. S. Bibb
J. S. Bibb
Nonlinear Systems Research Laboratory, Department of Mechanicai Engineering, Auburn University, Auburn, AL 36849
Search for other works by this author on:
S. C. Sinha
Nonlinear Systems Research Laboratory, Department of Mechanicai Engineering, Auburn University, Auburn, AL 36849
R. Pandiyan
Nonlinear Systems Research Laboratory, Department of Mechanicai Engineering, Auburn University, Auburn, AL 36849
J. S. Bibb
Nonlinear Systems Research Laboratory, Department of Mechanicai Engineering, Auburn University, Auburn, AL 36849
J. Vib. Acoust. Apr 1996, 118(2): 209-219 (11 pages)
Published Online: April 1, 1996
Article history
Revised:
February 1, 1994
Online:
February 26, 2008
Citation
Sinha, S. C., Pandiyan, R., and Bibb, J. S. (April 1, 1996). "Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems." ASME. J. Vib. Acoust. April 1996; 118(2): 209–219. https://doi.org/10.1115/1.2889651
Download citation file:
Get Email Alerts
Nonlinear damping amplifier friction bearings
J. Vib. Acoust
Application of Embedded Thermophones Toward Tonal Noise Cancelation in Small-Scale Ducted Fan
J. Vib. Acoust (April 2025)
Related Articles
On the Proper Orthogonal Modes and Normal Modes of Continuous Vibration Systems
J. Vib. Acoust (January,2002)
A Python Implementation of a Robust Multi-Harmonic Balance With Numerical Continuation and Automatic Differentiation for Structural Dynamics
J. Comput. Nonlinear Dynam (July,2023)
A High Precision Direct Integration Scheme for Nonlinear Dynamic Systems
J. Comput. Nonlinear Dynam (October,2009)
On the Global Analysis of a Piecewise Linear System that is excited by a Gaussian White Noise
J. Comput. Nonlinear Dynam (September,2016)
Related Proceedings Papers
Related Chapters
Conclusions
Introduction to Finite Element, Boundary Element, and Meshless Methods: With Applications to Heat Transfer and Fluid Flow
Cellular Automata: In-Depth Overview
Intelligent Engineering Systems through Artificial Neural Networks, Volume 20
Numerical Computation of Singular Points on Algebraic Surfaces
International Conference on Computer Technology and Development, 3rd (ICCTD 2011)