An inverse shock response spectrum

Abstract

The shock response spectrum (SRS) is a tool commonly used by application engineers that characterizes the severity of a transient acceleration. Due to the definition of the SRS, neither an analytical nor a unique inverse exists for an arbitrary function. An SRS presented without any temporal information makes creating a corresponding acceleration time history for an experimental or numerical study prohibitively difficult without a rigorous method to determine an inverse of the SRS (a corresponding time history). The present work develops a method to calculate an inverse of an arbitrary SRS using three sets of well characterized basis functions: an impulse function, a sine function/damped sine function, and a modified Morlet wavelet. These three basis functions are specifically chosen for the properties of their transformations: the impulse introduces a constant increase to the SRS above a given frequency, the sine wave introduces a narrow peak at a given frequency, and the Morlet wavelet introduces a plateau with an adjustable width and relative height. Using the definition of the SRS, the transformations of the basis functions are calculated and these expressions are used to derive a methodology for calculating an inverse SRS. The effectiveness of the method is demonstrated by several examples. The quality of an inverse SRS is evaluated by comparing the SRS of the inverse to the target SRS. This method is developed in order to provide a quick estimate of a corresponding time history; in applications where a higher fidelity representation of the SRS is needed than can be provided by the method developed, a genetic algorithm is used to optimize the coefficients of the basis functions. Given a sufficient number of basis functions for the optimization, the resulting SRS can almost exactly match a randomly generated target SRS that is nonzero over the frequency range considered. For applications in which the permissable basis functions are limited (such as for an experimental test apparatus), an extension of the genetic algorithm method is discussed.