A marimba maker starts from a uniform rectangular aluminium bar

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

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

Problem

A marimba maker starts from a uniform rectangular aluminium bar, free at both ends, with length

L = 0.400\,\mathrm{m}

, thickness (in the bending direction)

h = 0.024\,\mathrm{m}

, Young's modulus

E = 70\,\mathrm{GPa}

and density

\rho = 2700\,\mathrm{kg\,m^{-3}}

. The first free-free eigenvalue is

\beta_1L = 4.730

, and the fundamental mode has its two internal nodes at

0.224L

from each end. ((a)) Find the fundamental frequency

f_1

of the uniform bar. ((b)) State, in millimetres from each end, where the maker should drill the support holes so that the fundamental rings with as little damping as possible from the supports. ((c)) The maker now undercuts the middle third of the bar (from

x = L/3

x = 2L/3

), reducing its thickness there to

0.6h

while leaving the rest of the bar unchanged. Using Rayleigh's method with the trial shape taken to be the unchanged fundamental mode shape of the uniform bar, estimate the new fundamental frequency. You are given that, for this mode shape,

38\,\%

of the integral

\int\rho AW^2\,\mathrm{d}x

and

72\,\%

of the peak bending energy lie in the central third before undercutting.

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