Generating “idealised” impulse response functions to improve or repair single degree of freedom system measurements

A single degree of freedom (SDOF) system can be characterised using transient or continuous time signals and both should lead to the same frequency response function. As commonly defined, the equation of the displacement response of a SDOF system to impulse excitation alone is ambiguous leading to two possible interpretations: the impulse response (IR) and one corresponding to solely the free oscillation of the system (FR). These are analysed in detail for idealised and practical cases and it is found that the IR and FR yield two distinct acceleration spectra, and that only the phase distinguishes the normalised FR between displacement, velocity and acceleration alleviating the need to perform differentiation and integration to generate the response in other parameters. An unambiguous mathematical definition of the IR is suggested for the velocity response. Finally, it is shown that an idealised version of the IR can be artificially generated from the FR. This can be used to repair acceleration signals that have a truncated or distorted impulsive part, alleviate spectral windowing effects and generally allow cleaner response spectra. This is demonstrated experimentally.