Response and Stability of Linear Dynamic Systems With Many Degrees of Freedom Subjected to Nonconservative and Harmonic Forces

Abstract

Dynamic systems whose response can be characterized by a set of linear differential equations with harmonic coefficients which are proportional to a small parameter ε, are considered. These systems are such that the corresponding autonomous sets of equations which are obtained by setting ε = 0, are defined by nonself-adjoint linear differential operators; i.e., they correspond to dynamic systems subjected to nondissipative nonconservative forces. For these systems, general asymptotic solutions are developed and their stability is examined. An interesting feature of these solutions is that, when the exciting frequency is close to, say, twice of a suitable eigenfrequency, or when it is close to the sum or the difference of two suitable frequencies of the autonomous system, then the asymptotic solution will involve negative fractional powers of ε. Hence, the nonsecular asymptotic solution, in general, may not reduce to the solution of the autonomous system as ε goes to zero. Another interesting feature of the present results is that the addition of small suitable harmonic forces does indeed stabilize an inherently unstable nondissipative nonconservative dynamical system; except when the frequency of the harmonic force resonates with one or several of the frequencies of the autonomous system in either a subharmonic or a combinational-type oscillation.