Nonlinear Oscillations of a Spring Pendulum at the 1:1:2 Resonance by Normal Form Methods

Abstract

The object of this research is the process of small oscillations of a three-dimensional elastic pendulum, tuned to a 1:1:2 resonance to vertical and horizontal oscillations. The purpose is to develop a symbolic algorithm for calculating small oscillations of the pendulum. The primary efforts are aimed at building a software package that generates formulas that approximate the movements of the pendulum with sufficient accuracy. The algorithms of this work are developed based on the resonant normal form method. The importance of the study is due to the wide range of applicability of the method of normal forms for constructing approximations of periodic and conditionally periodic local families of solutions of ODEs. For high-dimensional resonance systems, the technique is a generalization of the Poincare–Linstedt method, and for coarse systems, the Carleman linearization method. The reliability of the results is confirmed by comparisons with the results of numerical solutions. Results can be useful for specialists working at the interface of computational mathematics and continuum mechanics. Approaches and methodologies developed in this paper can be applied to solve various modeling problems.