Question 10.11: The aircraft shown in Fig. 10.20 is fitted with transducers ......
The aircraft shown in Fig. 10.20 is fitted with transducers enabling the responses z_{1}(t), z_{2}(t), z_{3}(t), etc. to be recorded, but there is no special provision for a known applied force. The aircraft is, however, excited by random forces due to turbulent airflow, or from the jet engines.
Show how the ambient random excitation can be utilized to find the natural frequencies, damping coefficients and approximate mode shapes of the wing.
Consider a typical response, z(t), of the wing, and let the Fourier transform of z(t) \mathrm{be} Z(f). Although we do not know the input force producing this response, let us call it F(t), and let its Fourier transform be F(f).
We can then write
\left|H_{zF}(f)\right|^{2} = \frac{Z(f) Z^{\ast}(f)}{F(f) F^{\ast}(f)} = \frac{S_{z} (f)}{S_{F} (f)} (A)where \left|H_{zF}(f)\right| is the modulus of the FRF relating z(t) \mathrm{and} F(t), S_{z}(f) the auto power spectrum of z(t), and is known and S_{F}(f) the auto power spectrum of F(t), and is unknown.
Although the input power spectrum S_{F}(f) is unknown, if we can make the assumption that it is reasonably flat in a frequency band surrounding each resonance of the structure, then it can be taken as a constant for any given mode. Thus we know the shape of \left|H_{zF}(f)\right| in that frequency band, but not its overall size. Nevertheless, this is sufficient to enable the natural frequency and damping coefficient of each mode to be found.
If several responses, z_{1}(t), z_{2}(t), z_{3}(t), etc., are measured, at suitable locations, it is possible to find not only the frequencies and damping coefficients of the normal modes, but also the approximate mode shapes, in the following way.
One of the responses, z_{1}(t), say, is designated as the ‘master’ or ‘reference’ response, and the following spectra are calculated:
The auto-spectrum of the master response, z_{1}(t): \\ S_{1}(f) = Z_{1}(f) Z^{*}_{1}(f) (B)
The cross-spectrum between z_{1}(t) \mathrm{and} z_{2}(t): \\ S_{12}(f) = Z_{1}(f) Z^{*}_{2}(f) (C)
The cross-spectrum between z_{1}(t) \mathrm{and} z_{3}(t): \\ S_{13}(f) = Z_{1}(f)_{1} Z^{*}_{3}(f) (D)
and similarly the cross-spectra between z_{1}(t) \mathrm{and} z_{4}(t), z_{1}(t) \mathrm{and} z_{5}(t) and so on, where Z_{1}(f), Z_{2}(f), etc. are the Fourier transforms of z_{1}(t), z_{2}(t), etc.
If all the spectra are now plotted, for example in magnitude and phase form, the resonant frequencies will be identifiable as peaks, and the approximate mode shapes can be found by comparing the magnitudes and phases of the cross spectra with the auto power spectrum at the master location, at each resonant frequency.