Answer to Question #193737 in Mechanical Engineering for Lemony

Question

The natural frequency of vibration, in bending, of the wing of a military aircraft is found to be 20 Hz. Find the new frequency of bending vibration of the wing when a weapon, weighing 2000 lb, is attached at the tip of the wing, as shown in Figure. The stiffness of the wing tip, in bending, is known to be 50,000 lb/ft.

Answer 1

The natural frequency of the wing itself is $\omega_{11}=20 Hz =(20 *2 \pi)=125.664 rad/s$

The natural frequency of the weapon $\omega _{22}=\sqrt{\frac{k}{m}}=\sqrt{\frac{50000*386.4}{12*2000}}=28.3725 rad/s$

The frequency of vibration of the wing with weapon

$\frac{1}{\omega _1^2}=\frac{1}{\omega _{11}^2}+\frac{1}{\omega _{22}^2}$

$\frac{1}{\omega _1^2}=\frac{1}{125.664^2}+\frac{1}{28.3725^2}$

$\omega _1=\frac{\sqrt{\frac{1}{130.5563*10^{5}}}}{2\pi}=\frac{27.6759}{2*3.14}=4.4047 Hz$