Forced Transverse Oscillations in a Simple Spring-Mass System

• Published in: SIAM journal on applied mathematics Vol. 51; no. 5; pp. 1380 - 1396
• Main Author: Forbes, Lawrence K.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.10.1991
• Subjects: Accuracy; Algorithms; Amplitude; Differential equations; Duffing equation; Eigenvalues; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Newtons method; Numerical analysis; Oscillators; Physics; Resonance; Solid mechanics; Spring constant; Spring mass systems; Structural and continuum mechanics; Symmetry; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Transversal vibration; Forced vibration; Bifurcation; Resonance; Non linear effect; Periodic solution; Floquet method; Spring mass system
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/0151069

An efficient numerical method is developed to solve for the periodic motion of a simple, forced mechanical oscillator. The physical system consists of a mass executing transverse oscillations on the mid-line between two opposing parallel walls, and subject both to a periodic force as well as restraining forces due to Hookean springs attached to the walls. The governing second-order differential equation is therefore nonlinear and nonautonomous. The numerical solutions computed display a wide range of nonlinear behavior, including resonances, folds, and symmetry-breaking bifurcations. Solution profiles and phase-plane orbits of high accuracy are presented, and their stability to infinitesimal perturbations is determined automatically by the solution technique, using a numerical implementation of Floquet theory.