The fundamental transverse vibration frequency of a uniform cantilever beam

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

Scientific Computing for Physicists, Ch The numerical toolkit closed: backward error and the audit of a library routine

Problem

The fundamental transverse vibration frequency of a uniform cantilever beam, clamped at one end and free at the other, is fixed by the smallest positive root z1z_1 of the transcendental equation cos⁡z cosh⁡z=−1,\cos z\,\cosh z = -1, where z=β1Lz=\beta_1 L is dimensionless. The root lies in the bracket z∈[1,3]z\in[1,3] (the function changes sign there and nowhere below z=1z=1). ((a)) Using Brent's method on the bracket [1,3][1,3], find z1z_1 to five decimal places. ((b)) A steel cantilever has length L=0.50 mL=0.50\,\mathrm{m}, Young's modulus E=200 GPaE=200\,\mathrm{GPa}, second moment of area I=8.0×10−9 m4I=8.0\times10^{-9}\,\mathrm{m^4}, cross-sectional area A=1.0×10−4 m2A=1.0\times10^{-4}\,\mathrm{m^2} and material density ρ=7850 kg m−3\rho=7850\,\mathrm{kg\,m^{-3}}. Its fundamental angular frequency is ω1=z12EI/(ρAL4)\omega_1=z_1^2\sqrt{EI/(\rho A L^4)}. Evaluate ω1\omega_1.

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