Symbolic Computation of Secondary Bifurcations in a Parametrically Excited Simple Pendulum

A symbolic computational technique is used to study the secondary bifurcations of a parametrically excited simple pendulum as an explicit function of the periodic parameter. This is made possible by the recent development of an algorithm which approximates the fundamental solution matrix of linear time-periodic systems in terms of system parameters in symbolic form. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet transition matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation boundaries in the parameter space. Since this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. By repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution, it is shown how the bifurcation locations as well as the new linearized equations may be obtained in closed form as a function of the periodic parameter. Bifurcation diagrams are constructed and the results are compared with those obtained elsewhere using the point mapping method.

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Approximate Lyapunov-perron Transformations: Computation and Applications to Quasi-periodic Systems

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4050614 ◽

2021 ◽

Author(s):

Ashu Sharma

Keyword(s):

Analytical Solutions ◽

Periodic System ◽

Linear Part ◽

Closed Form Solution ◽

Hill Equation ◽

Periodic Systems ◽

Normal Form Theory ◽

Symbolic Form ◽

Stochastic Perturbations ◽

P Transformations

Abstract A new technique for the analysis of dynamical equations with quasi-periodic coefficients (so-called quasi-periodic systems) is presented. The technique utilizes Lyapunov-Perron (L-P) transformation to reduce the linear part of a quasi-periodic system into the time-invariant form. A general approach for the construction of L-P transformations in the approximate form is suggested. First, the linear part of a quasi-periodic system is replaced by a periodic system with a 'suitable' large principal period. Then, the state transition matrix of the periodic system is computed in the symbolic form using Floquet theory. Finally, Lyapunov-Floquet theorem is used to compute approximate L-P transformations. A two-frequency quasi-periodic system is studied and transformations are generated for stable, unstable and critical cases. The effectiveness of these transformations is demonstrated by investigating three distinct quasi-periodic systems. They are applied to a forced quasi-periodic Hill equation to generate analytical solutions. It is found that the closeness of the analytical solutions to the exact solutions depends on the principal period of the periodic system. A general approach to obtain the stability bounds on linear quasi-periodic systems with stochastic perturbations is also discussed. Finally, the usefulness of approximate L-P transformations to nonlinear quasi-periodic systems is presented by analyzing a quasi-periodic Hill equation with cubic nonlinearity using time-dependent normal form theory. The closed-form solution generated is found to be in good agreement with the exact solution.

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Analytical stability analysis of periodic systems by Poincaré mappings with application to rotorcraft dynamics

Mathematical Problems in Engineering ◽

10.1155/s1024123x97000598 ◽

1997 ◽

Vol 3 (4) ◽

pp. 329-371

Author(s):

Henryk Flashner ◽

Ramesh S. Guttalu

Keyword(s):

Stability Analysis ◽

Degrees Of Freedom ◽

Mapping Method ◽

Periodic Systems ◽

Resonance Problem ◽

Point Mapping ◽

Analytical Stability ◽

Functional Relations ◽

Ground Resonance ◽

The Stability

Apoint mappinganalysis is employed to investigate the stability of periodic systems. The method is applied to simplified rotorcraft models. The proposed approach is based on a procedure to obtain an analytical expression for the period-to-period mapping description of system's dynamics, and its dependence on system's parameters. Analytical stability and bifurcation conditions are then determined and expressed as functional relations between important system parameters. The method is applied to investigate the parametric stability of flapping motion of a rotor and the ground resonance problem encountered in rotorcraft dynamics. It is shown that the proposed approach provides very accurate results when compared with direct numerical results which are assumed to be an “exact solution” for the purpose of this study. It is also demonstrated that the point mapping method yields more accurate results than the widely used classical perturbation analysis. The ability to perform analytical stability studies of systems with multiple degrees-of-freedom is an important feature of the proposed approach since most existing analysis methods are applicable to single degree-of-freedom systems. Stability analysis of higher dimensional systems, such as the ground resonance problems, by perturbation methods is not straightforward, and is usually very cumbersome.

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Symbolic Stability Boundaries and Root Locus Plots in Time-Periodic Systems

Volume 6B: 18th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc2001/vib-21449 ◽

2001 ◽

Author(s):

Eric A. Butcher

Keyword(s):

Linear Time ◽

Linear Part ◽

Periodic Systems ◽

Root Locus ◽

Stability Boundaries ◽

System Parameters ◽

Symbolic Calculations ◽

The Stability ◽

Parameter Dependent ◽

Time Periodic

Abstract 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 the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincaré map, is obtained. The sub sequent 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.

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