Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

alt="bold-italic upper G left-parenthesis omega right-parenthesis equals bold-italic upper H left-parenthesis omega right-parenthesis dot 1"/> (1.179)

The inverse Fourier transform of the unit spectrum ejωt0=1 for t0=0 is the delta function δ(t) according to (1.32). In the time domain this reads as the following convolution integral

h left-parenthesis t right-parenthesis equals integral Subscript negative normal infinity Superscript normal infinity Baseline h left-parenthesis t minus tau right-parenthesis delta left-parenthesis tau right-parenthesis d tau (1.180)

We see that the transfer function in time domain is the response of the system to a delta function. Consequently, this function is called the impulse response of the system. When we calculate the inverse Fourier transform of the frequency response function of the damped harmonic oscillator we get as impulse response function

h left-parenthesis t right-parenthesis equals StartFraction 1 Over StartRoot 1 minus zeta squared EndRoot omega Subscript n Baseline m EndFraction e Superscript minus zeta m omega Super Subscript n Superscript t Baseline sine left-parenthesis StartRoot 1 minus zeta squared EndRoot omega Subscript n Baseline t right-parenthesis (1.181)

This is the solution of the following equation of motion

m ModifyingAbove u With two-dots plus c Subscript v Baseline ModifyingAbove u With dot plus k Subscript s Baseline u equals delta left-parenthesis t right-parenthesis (1.182)

which corresponds to the solution of the homogeneous version in (1.1) for the following initial conditions

u left-parenthesis 0 right-parenthesis equals 0 v Subscript x Baseline 0 Baseline equals ModifyingAbove u With dot left-parenthesis 0 right-parenthesis equals 1 slash m (1.183)

Finally, with the definition of system response functions in time and frequency domain, there is a powerful description available. This will be used in Section 1.6.3 in order to describe the system response to random signals.

1.6.3 Systems Excited by Random Signals

We know from section 1.5.4 that the random signal cannot be Fourier transformed. As a consequence we apply the correlation methods from above to the output of a system excited by random signals. We start with the autocorrelation of the system output g from excitation by random input f

(1.184)

The impulse response can by taken out from the expected value operation, because it does not depend on the time. Hence we get

(1.185)

When we assume a retarded time argument of τ′=τ+τ1−τ2 the expected value can be interpreted as the autocorrelation of f(t). With this assumption we get:

(1.186)

The term in the rectangular brackets can be seen as the convolution with argument (τ+τ1).

upper R Subscript g g Baseline left-parenthesis tau right-parenthesis equals integral Subscript negative normal infinity Superscript normal infinity Baseline h left-parenthesis tau 1 right-parenthesis left-bracket h left-parenthesis t plus tau 1 right-parenthesis asterisk upper R Subscript f f Baseline left-parenthesis t plus tau 1 right-parenthesis right-bracket d tau 1 (1.187)

By replacing τ₁ by −u this reads as

(1.188)

using that Rff(τ) is symmetric in τ. This equation can be converted into frequency domain using the fact that time reversal in time corresponds to complex conjugate spectra:

(1.189)

So we know now the autospectrum of the system excited by random response. Next we investigate the cross correlation between input and output.

(1.190)

StartLayout 1st Row equals integral Subscript negative normal infinity Superscript normal infinity Baseline h left-parenthesis tau 1 right-parenthesis upper E left-bracket f left-parenthesis t right-parenthesis f left-parenthesis t plus tau minus tau 1 right-parenthesis right-bracket d tau 1 EndLayout (1.191)

This expression can be simplified by assuming the expected value to be the auto-convolution with argument τ−τ1 to:

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