An iterative procedure for determining limit cycles using Lagrangian mechanics
Abstract
In many nonlinear autonomous mechanical systems, determination of the amplitudes and periods of limit cycles by direct numerical integration of the system differential equations can be very timeconsuming computationally. In this investigation, an efficient iterative algorithm which converges to limit cycles of singledegreeoffreedom systems is presented. It is based on the work done by nonconservative forces in the system. Numerical results for several illustrative examples are given, including the van der Pol equation, a feedback control system with hysteresis, a system having an infinite number of stable and unstable limit cycles, and an oscillator with nonlinear dry friction.
 Publication:

AIAA Journal
 Pub Date:
 March 1976
 DOI:
 10.2514/3.7099
 Bibcode:
 1976AIAAJ..14..320P
 Keywords:

 Equations Of Motion;
 EulerLagrange Equation;
 Iterative Solution;
 Mechanical Oscillators;
 Nonlinear Systems;
 Numerical Integration;
 Algorithms;
 Autonomy;
 Convergence;
 Degrees Of Freedom;
 Differential Equations;
 Dry Friction;
 Dynamic Characteristics;
 Feedback Control;
 Hysteresis;
 Physics (General)