Efficient algorithm for computing displacement, velocity and acceleration responses of an oscillator to arbitrary ground motion

Abstract

The response calculations of a single-degree-of-freedom system typically use a recursive algorithm for computing the relative displacement and velocity responses {x[n+1], x[n+1]} from {x[n], x[n]} and the acceleration input {a[n], a[n+1]}. A fast algorithm using digital impulse and step invariant simulations of a continuous system has previously been derived for the same calculations using only 2 multiplications and 3 additions per time step1. A new improved algorithm is presented here. Besides calculating the relative displacement and velocity responses, it provides simultaneous computation of the acceleration responses with only one more addition per time step. The improvement resulted in the digital system functions for relative displacement, velocity and acceleration responses, all showing good agreement with the corresponding continuous transfer functions in both amplitudes and phases in a specified frequency range (0.07 to 25 Hz, say).