A stubby aluminium-alloy cantilever ( , ) has a square cross-section of side and length

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Vibrating bodies, musical and architectural acoustics · Vibrating bodies, musical and architectural acoustics

Bending, Ringing, Breaking and Flowing, Ch Vibrations of rods, beams and plates

Problem

A stubby aluminium-alloy cantilever (E=70 GPaE = 70\,\mathrm{GPa}, ρ=2700 kg m−3\rho = 2700\,\mathrm{kg\,m^{-3}}) has a square cross-section of side h=10 mmh = 10\,\mathrm{mm} and length L=30 mmL = 30\,\mathrm{mm}, clamped at one end and free at the other. The Rayleigh wave speed of this alloy is cR=2900 m s−1c_R = 2900\,\mathrm{m\,s^{-1}} (Chapter 5). ((a)) Using the clamped-free eigenvalue β1L=1.8751\beta_1 L = 1.8751, find the fundamental bending frequency f1f_1 of this cantilever. ((b)) The mode's effective flexural wavenumber is k1=β1k_1 = \beta_1. Find the phase velocity vp=ω1/k1v_p = \omega_1/k_1 of this mode and compare it with cRc_R. ((c)) Find the ratio λ1/h\lambda_1/h, where λ1=2π/β1\lambda_1 = 2\pi/\beta_1, and use the fact that thin-beam theory overestimates the bending frequency by about 6.4%6.4\% at a wavelength of ten thicknesses to say whether thin-beam theory can still be trusted for this mode, even though vpv_p is well below cRc_R.

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