Bending, Ringing, Breaking and Flowing, Ch Vibrations of rods, beams and plates
Problem
A slender steel rod of circular cross-section, diameter d=8mm and length L=0.60m, stands like a short vertical mast: rigidly clamped at its base and free at its top. Take E=200GPa and ρ=7800kgm−3 for the steel.
((a)) Treating the rod as a fixed-free column (effective-length factor Ke=2), find its Euler buckling load Pcr.
((b)) Treating the same rod instead as a cantilever beam vibrating transversely, find its fundamental bending frequency f1, using the clamped-free eigenvalue β1L=1.8751.
((c)) The fundamental mode has an effective flexural wavenumber k1=β1, so a flexural wavelength λ1=2π/β1. Compute λ1/d and say, using the fact that thin-beam (Euler-Bernoulli) theory overestimates the frequency by only about 1.7% at a wavelength of 20 thicknesses and is essentially exact for much longer wavelengths, whether thin-beam theory can be trusted for this mode.