Question 10.9: The aircraft wing shown in Fig. 10.18 is excited by two rand......
The aircraft wing shown in Fig. 10.18 is excited by two random forces, P(t) and Q(t), assumed to have zero mean values. Show how the variance of the displacement at the wing tip can be found in the following cases:
(a) When the forces P(t) and Q(t) are correlated, say, because they are derived from turbulence in the same airstream.
(b) When the forces are uncorrelated, because they are derived from entirely different sources, such as different jet engines.
The following standard result relates the displacement power spectrum at the wing tip to the power and cross-power spectra of the applied forces:
S_{z} = H^{\ast}_{zP}H_{zP}S_{P} + H^{\ast}_{zP}H_{zQ}S_{PQ} + H^{\ast}_{zQ}H_{zP}S_{QP} + H^{\ast}_{zQ}H_{zQ}S_{Q} (A)where all the quantities are functions of frequency, f, and are defined as follows:
S_{z} = power spectral density of the wing tip displacement z(t),
S_{P} = power spectral density of the force P(t),
S_{Q} = power spectral density of the force Q(t),
S_{PQ} = complex cross-power spectral density between force P(t) and force Q(t),
S_{QP} = complex cross-power spectral density between force Q(t) and force P(t),
H_{zP} = complex frequency response function giving z(t) per unit P(t),
H_{zQ} = complex frequency response function giving z(t) per unit Q(t).
Case (a):
When there is correlation between the applied forces P(t) and Q(t), Eq. (A) applies in full, and all the quantities on the right side of Eq. (A) must be known.
The variance of the wing tip displacement, \sigma^{2}_{z}, is then given by Eq. (10.30):
\sigma^{2}_{f_{1},f_{2}} = \int_{f_{1}}^{f_{2}}{S(f) df} (10.30) \\ \sigma^{2}_{z} = \int_{f_{1}}^{f_{2}}{S_{z}(f) df} (B)that is by integrating (or summing) the displacement PSD at the wing tip over an appropriate frequency range, f_{1} \mathrm{to} f_{2}
Case (b)
If the forces P(t) and Q(t) are completely uncorrelated, the cross-power spectra, S_{PQ} \mathrm{and} S_{QP} are zero, and Eq. (A) becomes
Thus, the response spectrum can be found by simply adding the response power spectra due to each force acting separately. The variance of the displacement at the wing tip is given by Eq. (B), as before.