| Abstract: | A simple rule is derived to combine, within the framework of a complex mode superposition, the maximum modal responses of systems such as soil-structure and structure-equipment systems, for which closely spaced natural frequencies are likely, and for which, because of the large difference in the damping values of their various components, the assumption of an orthogonal damping matrix may lead to significant errors. The rule constitutes the generalization of Rosenblueth's rule for systems with closely spaced natural frequencies and classical modes, and is expressed in terms of their complex mode shapes and natural frequencies. Its derivation is based on the theory of a complex modal analysis for systems with non-classical modes of vibration and on Rosenblueth's original derivation. As in this original derivation, earthquake ground motions are modelled as a stationary white noise process, but the formulae obtained under this assumption are modified later on to account for the transient nature of actual earthquakes. A numerical example is presented to illustrate the application of the rule, and a comparative study with numerical integration solutions is performed to assess its accuracy. In this comparative study, it predicts the numerical integration solutions with an average error of 0.3 per cent. |
