Stability of multi-degree-of-freedom Duffing oscillators

Abstract

Different methods of examining the stability of periodic solutions of non-linear multi-degree-of-freedom systems are compared in this paper. The methods are particularly suited for solutions computed with the harmonic balance method and the non-linear systems considered are represented by systems of coupled Duffing-type equations; that is, with cubic non-linear expressions. The latter systems appear frequently in models of thin-walled structures, particularly in straight beams and flat plates. In the first method reviewed in this paper the harmonic balance procedure is applied to define an eigenvalue problem, from which the characteristic exponents are determined. If the real part of any of these characteristic exponents is greater than zero, then the solution is unstable. The second method relies on the sign of the determinant of a Jacobian matrix; a matrix that is also often used to solve the equations of motion. The last method is based on a perturbation procedure and with it a closed-form expression for stable regions is achieved. The advantages and shortcomings of the different methods are discussed. INTRODUCTIONA non-linear dynamic system excited by a harmonic force can have periodic and non-periodic solutions. With regard to the periodic solutions, it is possible that more than one solution exists for a single excitation frequency and the initial conditions determine to which solution the system converges. An unstable steady-state solution does not occur in practice: if a solution is unstable, the system will evolve to another attractor. Hence, it is important to determine whether or not a solution is stable. The local stability can be determined by investigating the evolution of a perturbed solution that results from the addition of a small disturbance. A perturbation near an unstable equilibrium condition leads to a departure from this condition and the inverse occurs near a stable equilibrium condition. Several types and orders of instability can occur and this is a subject widely analysed 1-33 but still of considerable interest.