Nonstationary response of structures with closely spaced frequencies

The response statistics are examined for linear structures subjected to nonstationary stochastic input. Modal expansions are used and attention is on the mean square, mean zero-upcrossing rate, and bandwidth of the response. The analysis begins with an exact derivation of the covariances of the response and its Hilbert transform in terms of stationary statistics. The resulting expressions are found to be lengthy double integrals. Then, approximations for these statistics are developed for lightly damped structures with closely spaced natural frequencies. These approximations, which are in terms of simple convolution integrals, are generalizations of previous results for oscillator response. It is found that the expressions for the modal cross-covariances have a different form from those of the modal auto-covariances. It is also shown that the nonstationary responses can have timevarying mean zero-upcrossing rates and mean-square responses that are highly dependent on the modal participation factors. A conclusion is that nonstationary response of structures with closely spaced frequencies is fundamentally different from that of oscillator systems.