Question 10.6: A proposed environmental vibration test on a radar pod, atta......
A proposed environmental vibration test on a radar pod, attached to the wing of a high performance aircraft, involves applying a single lateral random force to the pod by means of an exciter and rigid connecting rod, as shown in Fig. 10.11 at (a). The pod,considered rigid, is attached to the wing by a flexible pylon, and the combination has a lateral mode of vibration at f_{n} = 15 Hz, with a non-dimensional viscous damping coefficient equal to γ = 0.05 of critical. Its lateral vibration behavior at the exciter attachment point can be represented by the single-DOF lumped equivalent system shown in Fig. 10.11 at (b), where the equivalent mass m is 100 kg, and the natural frequency and damping coefficient are as given above. The applied force time history is F(t) and the resulting displacement history is y(t). The force power spectral density, defined at the exciter attachment rod, is held constant at 4000 N²/Hz in the range 7.5–50 Hz, and is zero outside these limits. The force supplied by the exciter has a Gaussian probability density,with zero mean, but is limited at source to 3 times the RMS value in each direction.
Find
(a) the RMS (or standard deviation) value, σ_{F}, of the force required from the exciter,and its maximum and minimum values.
(b) the RMS (or standard deviation) displacement, \sigma_{\mathrm{y}}, of the point on the pod where the exciter rod is attached, and the total travel of the exciter rod.
Part (a)
From Eq. (10.30), the mean square value of a random time history (equal to the variance if the mean value is zero), between the frequency limits f_{1} \mathrm{and} f_{2}, is given by integrating the PSD function between these limits. Applied to this case:
\mathrm{If} S_{F}(f) \mathrm{has the constant value} S_{F}= 4000 N²/Hz between 7.5 and 50 Hz:
\sigma_{F}^{2} = \int_{7.5}^{50}{4000 df} = \left[4000f\right]_{7.5}^{50} = 170 \times 10^{3} \mathrm{N² and} \sigma _{F} = 412 NThe RMS force required from the exciter is 412 N, and since the force is limited to ± 3 times the RMS value in this case, the maximum and minimum values are ± 1236 N.
Part (b)
The natural frequency of the system is 15 Hz, and the half-power bandwidth of the single mode is 2 \gamma f_{n} = (2 × 0.05 × 15) = 1.5 Hz. The excitation PSD is constant in the band from 7.5 Hz to 50 Hz, and thus extends from 5 times the half-power bandwidth below the natural frequency, to over 23 times above it. The error in the RMS value due to using Eq. (10.52) will therefore be negligible, at less than 3%. The RMS displacement at the exciter attachment point (also equal to the standard deviation, since the mean value is zero) is therefore given by Eq. (10.52):
where m = 100 kg; S_{F} = 4000 N²/Hz; f_{n} = 15 Hz; γ = 0.05
The RMS value, or standard deviation, of the point on the pod where the exciter is attached is
\sigma_{\mathrm{y}} =1.093 × 10^{-3} m = 1.093 mm
Since the input force was assumed to have a Gaussian probability density, and the system is assumed linear, then the displacement response will also be Gaussian. Assuming that this is limited, like the input force, to 3 times the standard deviation in each direction, then the maximum exciter travel is 6\sigma_{\mathrm{y}} \mathrm{or} 6 × 1.093 = 6.59 mm peak to peak.