An aluminium alloy diving-board blank is a uniform bar of rectangular cross-section

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

An aluminium alloy diving-board blank is a uniform bar of rectangular cross-section, length

L_0 = 3.0\,\mathrm{m}

, width

b = 0.50\,\mathrm{m}

and thickness

h = 30\,\mathrm{mm}

, density

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

. Before it is mounted, quality control suspends it freely (free at both ends) and taps it: its fundamental free-free bending frequency is measured to be

f_0 = 17.4\,\mathrm{Hz}

. ((a)) Using the free-free bar frequency formula with its first eigenvalue

\beta_1L_0 = 4.730

, find the Young's modulus

E

of the alloy from this measurement. ((b)) The blank is then clamped at a fulcrum, leaving a free (overhanging) length

L = 1.5\,\mathrm{m}

that acts as a cantilever. A diver of mass

M = 75\,\mathrm{kg}

stands at the free tip; treat the diver as a rigid point mass with no rotational inertia of its own, and ignore damping. Using Rayleigh's method with the static cantilever deflection under a tip point load,

w(x) = 3Lx^2 - x^3

for

0 \le x \le L

(an overall constant amplitude cancels), show that the fundamental angular frequency of the loaded board satisfies

\omega^2 = \frac{12EI}{\tfrac{33}{35}\rho A L^4 + 4ML^3},

where

I = bh^3/12

is the second moment of area of the cross-section and

A = bh

, and evaluate the fundamental frequency in hertz. ((c)) Evaluate the same formula with

M = 0

to find the frequency the board alone (with no diver) would have at that free length, and comment briefly on how much the diver's mass changes the fundamental frequency.

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