A steel leaf spring is clamped at one end like a cantilever

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 steel leaf spring is clamped at one end like a cantilever. Its thickness

h = 2.0\,\mathrm{mm}

is uniform along its length, but its width tapers linearly from

b_0

at the clamp (

x=0

) to zero at the free tip (

x=L

), so

b(x) = b_0(1 - x/L)

with

L = 50\,\mathrm{mm}

. Take

E = 200\,\mathrm{GPa}

and

\rho = 7800\,\mathrm{kg\,m^{-3}}

for the steel, and treat the vibration as thin-beam bending with no shear deformation or rotary inertia. ((a)) Using the trial shape

w(x) = x^2

, which satisfies the clamped conditions

w(0) = 0

and

w'(0) = 0

, derive the Rayleigh quotient for the fundamental angular frequency

\omega_1

in terms of

E

\rho

h

and

L

only (the width

b_0

should cancel). ((b)) Evaluate

f_1 = \omega_1/2\pi

for the given dimensions.

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