Dynamic stiffness method for exponentially varying harmonic excitation of continuous systems

Abstract

The dynamic stiffness method relating the amplitudes of applied forces and responses of a harmonically vibrating continuum has received wide attention. It enables the infinite number of natural modes to be represented by a finite number of nodal co-ordinates for continuous structures of beams and folded plates. However, the method has been applied almost exclusively to harmonic, or periodic, oscillations. This is due mainly to the rather misleading intuition that only harmonic vibrations can be described by solutions with separate time- and space-dependent factors. It is shown here that a much wider class of problem of exponentially varying harmonic excitations can also be analyzed by the dynamic stiffness method. The extension is achieved simply by using complex frequency parameters. The forced response (that is, the part of the response which is independent of the initial conditions) can be obtained directly by solution of linear equations. A single degree of freedom system is considered first, as an illustrative example. It is shown that the present method is equivalent to the usual Duhamel integral method except that integration is completely avoided and the transient effects due to the initial conditions can be considered separately. The method is then applied to undamped straight beam members and is modified so that damped vibration can be covered as well. Distributed loads are then considered and explicit formulae are introduced. Finally, for completeness of presentation, the responses are compared with those obtained by using modal analysis. The method is proved to be equivalent to modal analysis and has the advantages over the latter that (i) integrations in the time variable are completely avoided; (ii) the forced response can be obtained directly; (iii) decomposition into generalized forces is not required; and (iv) the force-response relation is easily visualized.