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Foundation Vibration Analysis Using Simple Physical Models by John P. Wolf

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Appendix C. Recursive Evaluation of Convolution Integral

Overview

A typical convolution integral relating the input time history x(τ) 0 ≤ τ≤ t to the output y(t) at time t is written as

Equation C-1. 

where h(t) is the unit-impulse response function. In a flexibility or Green’s function formulation (Duhamel’s integral), the x(τ) corresponds to the force and y(t) to the displacement. In a stiffness formulation involving a convolution of the velocity, the x(τ) is this velocity and y(t) represents (the regular part of) the interaction force.

The evaluation of the convolution integral, which has to be performed for each time station, from its definition ...

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