Chapter 5, Problem 5.5CP

Does an automobile radio antenna vibrate in resonance due to vortex shedding? Consider an antenna of length L and diameter D. According to beam vibration theory [see [34] or [35, p. 401]], the first mode natural frequency of a solid circular cantilever beam is ${\omega }_n=3.516{\left[EI/\rho A{L}^4\right]}^{1/2}$ , where E is the modulus of elasticity, I is the area moment of inertia, $\rho$ is the beam material density, and A is the beam cross-section area. (a) Show that ${\omega }_n$ is proportional to the antenna radius R. (b) If the antenna is steel, with L = 60 cm and D = 4 mm, estimate the natural vibration frequency, in Hz. (c) Compare with the shedding frequency if the car moves at 65 mi/h.