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Response to a Single Random Loading
Response Involving CrossCorrelations
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_ 2 _ a(if amplitude applied forces assumed beam complete complex conjugate component convenient correlation corresponding cross spectral density cross-correlation functions curve defined degrees of freedom denote describe determine displacement distribution function Duhamel integral dxA dxB equal equations of motion example exciting force expectation expressed filter force P(t Fourier transform Gaussian distribution generalised coordinates given giving the response harmonic analysis hysteretic integral interval Lagrange's equations large number mean value mean-square value natural frequencies normal coordinates normal mode number of points obtained plotted possible probability distribution problem product terms quantity x(t random loading random process random vibration randomly varying force randomly varying quantity receptance relationship Rx(x Rxy(x Section shown in Fig simply simulation single randomly varying Sp(xA spectral density SP(f spectrum SPQ(f spring-mass system SQ(f SQP(f SR(f stationary statistical stress Sx(f Sxy(f theory tion trials value of x(t variables white noise wr(x xB;f zero