Correlation between torsional vibration and translational vibration

Abstract

This paper presents theoretical investigation on the cross correlation between torsional vibration (<TEX>$u_{\theta}$</TEX>) and translation vibration (<TEX>$u_x$</TEX>) of asymmetrical structure under white noise excitation. The formula reveals that the cross correlation coefficient (<TEX>${\rho}$</TEX>) is a function of uncoupled frequency ratio (<TEX>${\Omega}={\omega}_{\theta}/{\omega}_x$</TEX>), eccentricity, and damping ratio (<TEX>${\xi}$</TEX>). Simulations involving acceleration records from fifteen different earthquakes show correlation coefficients results similar to the theoretical correlation coefficients. The uncoupled frequency ratio is the dominating parameter to <TEX>${\rho}$</TEX>; generally, <TEX>${\rho}$</TEX> is positive for <TEX>${\omega}_{\theta}/{\omega}_x$</TEX> > 1.0, negative for <TEX>${\omega}_{\theta}/{\omega}_x$</TEX> < 1.0, and close to zero for <TEX>${\omega}_{\t</TEX><TEX>heta}/{\omega}_x$</TEX> = 1.0. When the eccentricity or damping ratio increases, <TEX>${\rho}$</TEX> increases moderately for small <TEX>${\Omega}$</TEX> (< 1.0) only. The relation among <TEX>$u_x$</TEX>, <TEX>$u_{\theta}$</TEX> and corner displacement are best presented by <TEX>${\rho}$</TEX>; a simple way to hand-calculate the theoretical dynamic corner displacements from <TEX>$u_x$</TEX>, <TEX>$u_{\theta}$</TEX> and <TEX>${\rho}$</TEX> is proposed as an alternative to dynamic analysis.