Stability of multi-degree-of-freedom Duffing oscillators

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.