Response of linear, viscous damped systems to excitations having time-varying frequency

Abstract

The response of linear, viscous damped systems to excitations having time-varying frequency is the subject of exact and approximate analyses, which are supplemented by an analog computer study of single degree of freedom system response to excitations having frequencies depending linearly and exponentially on time. The technique of small perturbations and the methods of stationary phase and saddle-point integration, as well as a novel bounding procedure, are utilized to derive approximate expressions characterizing the system response envelope -- particularly near resonances -- for the general time-varying excitation frequency. Descriptive measurements of system resonant behavior recorded during the course of the analog study--maximum response, excitation frequency at which maximum response occurs, and the width of the response peak at the half-power level--are investigated to determine dependence upon natural frequency, damping, and the functional form of the excitation frequency. The laboratory problem of determining the properties of a physical system from records of its response to excitations of this class is considered, and the transient phenomenon known as "ringing" is treated briefly. It is shown that system resonant behavior, as portrayed by the above measurements and expressions, is relatively insensitive to the specifics of the excitation frequency-time relation and may be described to good order in terms of parameters combining system properties with the time derivative of excitation frequency evaluated at resonance. One of these parameters is shown useful for predicting whether or not a given excitation having a time-varying frequency will produce strong or subtle changes in the response envelope of a given system relative to the steady-state response envelope. The parameter is shown, additionally, to be useful for predicting whether or not a particular response record will exhibit the "ringing" phenomenon.