Probabilistic Analysis of Combined Dynamic Responses

Approximate solutions are developed for the extreme-value distribution of a stationary Gaussian process with spectral density function that exhibits two well-separated or closely spaced modes. These approximate solutions are based on the properties of two envelope processes.

The above results are extended to nonstationary processes representing the transient responses of 1-dof systems and 2-dof systems with well-separated or closely-spaced modal frequencies. For the 1-dof system and for the 2-dof system with closely spaced frequencies, two subtle effects of nonstationarity are identified and treated; these effects are the tendency of the envelope's up- and down-crossing rates to be different and the tendency of some 2-dof systems to have their peak responses after the excitation has ceased. These effects are important in some circumstances.

The foregoing extreme-value results are then used to investigate the accuracy of the SRSS rule and proposed modifications thereto. Analysis of these rules is done for two different situations: conditionally (i.e., given the random-vibration characteristics of the responses) and marginally. Combinations of two modal responses (with common excitation) and combinations of responses to two loads are considered. For responses to two loads, a new criterion is introduced for the accuracy of SRSS. This criterion is analogous to the first Kennedy-Newmark criterion, but is more convenient, especially if the responses are characterized as random processes.

In the last chapter, approximate procedures are investigated for the estimation of upcrossing rates for sums of dependent (and not necessarily gaussian) random processes.