Improved recursive formula for calculating shock response spectra

NOMENCLATURE x(t) = base input displacement of a single-degree-of-freedom system y(t) = response displacement of a single-degree-of-freedom system) t (x & & = base input acceleration) t (y & & = response acceleration z(t) = relative displacement y(t) – x(t) ζ = fraction of critical damping n ω = natural frequency of a single-degree-of-freedom system, rad/sec s = complex variable H = transfer function L[ ] = Laplace transform L-1 [ ] = Inverse Laplace transform Z = z transform Z-1 = Inverse z transform d ω = damped natural frequency, 2 1 n ζ − ω T = sample interval) t (δ = delta function;) t (δ = 1 for t = 0 ,) t (δ = 0 elsewhere d m = digital delta function; d m = 1 for m = 0 , d m = 0 all other m SDOF = single degree of freedom u(t) = units step function; u(t) = 1 for t > 0 , u(t) = 0 for t < 0 t = time