A new fast algorithm for the calculation of response of a single-degree-of-freedom system to arbitrary load in time

The typical response calculation of a single-degree-of-freedom system uses a recursive algorithm for evaluating the relative displacement and velocity {$\text{x[n + 1],}\text{dot}\text{x}\text{[n + 1]}}$ from {$\text{x[n],}\text{dot}\text{x}\text{[n]}}$ and the acceleration values {a[n], a[n + 1]}. Each time step of the iteration requires eight multiplications and six additions. A new algorithm using the digital impulse invariant and step invariant simulation of a continuous system has been derived which requires only two multiplications and three additions. This amounts to a 75% saving in multiplication time and 50% saving in addition time. The accuracy of the new algorithm is guaranteed by the sampling theorem in digital signal analysis. Calculation of the response spectra using the new algorithm is as accurate as that using the old algorithm.