Calculating time-correlated gust loads using matched filter and random process theories

Abstract

This paper describes and illustrates two ways of performing time-correlated gust-load calculations. The first is based on matched filter theory, the second on random process theory. Both approaches yield theoretically identical results, represent novel applications of the theories, are computationally fast, and may be applied to other dynamic-response problems. A theoretical development and example calculations using both matched filter theory and random process theory approaches are presented. Nomenclature G(s) = transfer function of vertical velocity component of von Karman turbulence HJ(U>) = Fourier transform of h f (t) (frequency responsefunction of output variable /) hi(t) = time response of output variable / to unit impulse K = arbitrary constant L = scale of turbulence RIJ(T) = cross-correlation function between quantities / and j; autocorrelation function if / equals j Sy(<*>) = cross-spectrum between quantities / andy; autospectrum if / equals j s = Laplace variable t = time to = arbitrary time shift V = velocity of aircraft W g ((jci) = Fourier transform of w g (t) w g (t) = vertical velocity component of gust X(co) = Fourier transform of x(t) x(t) = "matched" excitation waveform Y(u>) -Fourier transform of y(t) y(t) = time response of output quantity y y y (t) = time response of output quantity y to excitation matched to y y z (t) = time response of output quantity y to excitation matched to z Z(co) = Fourier transform of z(t) z(t) = time response of output quantity z z y (t) -time response of output quantity z to excitation matched to y Z z (t) -time response of output quantity z to excitation matched to z oi = root-mean-square of quantity / T = time argument for auto-and cross-correlation functions

Prediction of Time-Correlated Gust Loads Using an Incremental Stochastic Search

“…(2)] and the maximum responsefunctional[Eq. (4)], we can now formally state the problem addressed in this paper to be "Find the global maximum of f when x.t / is restricted to the set of inputs that satisfy the energy constraint U .x/ D U .…”

“…3¡9 In stress analysis, 1;2 a study is made of the various responseloads, referred to as time-correlated, or balanced, loads of a system when one particular load, known as the reference load, is maximal. When the system is linear, the values of each output can be calculated either by a cross-correlationtechnique 2 or by a technique based upon matched lter theory (MFT) 2 ; the results of both methods being theoretically identical. 2 When the system is nonlinear, such techniques are no longer applicable.…”

“…When the system is linear, the values of each output can be calculated either by a cross-correlationtechnique 2 or by a technique based upon matched lter theory (MFT) 2 ; the results of both methods being theoretically identical. 2 When the system is nonlinear, such techniques are no longer applicable. We therefore consider an existing generalization 5 of the MFT method that extends to cover nonlinear systems.…”

Matched Filter Concept and Maximum Gust Loads

“…In Ref. 3, this sample is referred to as the matched excitation waveform, since it is derived from, or matched to, the impulse response of the combined turbulence/aircraft-load lter through Eq. (3).…”

“…NASA conducted the evaluation of the SDG method, focusing on its overlap with the existing linear methods. 2 In the course of these studies, it was found that the concept of the matched lter from signal processing theory 3,4 could be applied to the analysis of time-correlated, worst-case gust loads for mathematically linear aircraft models. Much of the work on the matched lter approach to gust loads analysis that followed had been geared toward a fundamental understanding of the implications of the matched lter theory interpretation of the gust loads analysis problem, as well as various attempts at extending the techniques to nonlinear aircraft models.…”