Question 10.10: The aircraft shown in Fig. 10.19, of which only one side is ......
The aircraft shown in Fig. 10.19, of which only one side is shown, is fitted with a device which applies a known force time history, F(t). This force is recorded, together with the responses z_{1}(t), z_{2}(t), z_{3}(t), etc., which are derived from accelerometers.
The aircraft is assumed to be in flight, and subjected to unknown random forces due to turbulent airflow, in addition to the known force F(t).
Show how the FRFs between the applied force F(t) and each response location can be found, enabling the dynamic properties of the aircraft wing to be determined.
Consider first one typical response, z(t), and let its Fourier transform be Z(f). Also let the Fourier transform of the input force, F(t), \mathrm{be} F(f). Then the FRF, H_{zF}(f), between the force F(t) and the response z(t) is given by:
H_{zF}(f) = \frac{Z(f)}{F(f)} (A)Multiplying numerator and denominator by F^{*}(f), the complex conjugate of F(f), we have
H_{zF}(f) = \frac{Z(f) F^{\ast}(f)}{F(f) F^{\ast}(f)} = \frac{S_{zF} (f)}{S_{F} (f)} (B)Thus, the FRF, H_{zF}(f), between the applied force and the response is given by dividing the cross-power spectrum between input and response, S_{zF}(f), by the input power spectrum S_{F}(f). The unknown forcing due to the turbulent flow is, of course, present, and the structure will respond to it, but since it is uncorrelated with F(t), it will not affect the measurement of S_{zF}(f), if the latter is obtained by averaging over a sufficiently long period of time.
The FRF, H_{zF}(f), obtained from this procedure is, in theory, exactly the same as the FRF that would have been obtained if F(t) had been a sinusoidal force, applied at the same series of frequencies, and contains all the information needed to find the main dynamic properties of the structure, such as its natural frequencies and damping coefficients.
If the exercise described above is carried out with the single force, F(t), but several responses,z_{1}(t), z_{2}(t), z_{3}(t), etc., Eq. (B) can be used to find all the FRFs between the force and the response points, enabling approximations to the normal mode shapes to be plotted also.