On the distribution of peaks and ranges of dynamic models of the linear and nonlinear responses of plates

The focus of this work is on the response of aircraft panels to random acoustic excitations and possible steady temperature effects. More specifically, the statistical distribution of the rainflow ranges of the response is analyzed in detail for an ensuing fatigue life estimation. A special emphasis is on high sound pressure levels situations for which the panel exhibits a strong, Duffing-type nonlinearity and significant membrane stresses inducing a quadratic relationship between displacements and stresses.;Two semi-empirical models are first proposed for the probability density function of the rainflow ranges which are based on the distributions of the peaks of displacement and stresses, respectively, of a one-mode model of the panel response. It is demonstrated that the simpler model, i.e. the one based on the peaks of displacements, provides an excellent fit of rainflow ranges obtained in simulation studies in a broad set of situations, except when the bending and membrane stresses are of comparable magnitudes. In such situations, the more complex model, based on peak of stresses, is shown to be accurate.;This modeling effort emphasized the need to better understand and predict the occurrence of small rainflow ranges. In the particular case of the response of a linear oscillator to white noise input, it was shown that there exist very close similarities between the distributions of the rainflow ranges and all peak-to-valley ranges both occurring in a given time interval τ, provided that τ is less than half a period. An approximate distribution of the values of τ is also proposed which displays the correct basic trends.;The peaks of the response envelope which could be expected to lead to particularly large rainflow ranges are finally studied. The envelope definition used primarily here is the amplitude process appropriately smoothed to remove any high frequency content. Relying on the stochastic averaging technique, an approximate expression is derived for the distribution of the peaks of the envelope for linear oscillators and a class of nonlinear ones. The excellent match between the expressions derived for both linear and Duffing oscillators and the corresponding results from Monte Carlo simulations validates the approach.