Page 269 - Introduction to Continuum Mechanics

Vibration of an Infinite Plate 253

(b) Determine the resonance frequencies.
1
Solution. The traction on jcj = 0 determines the stress T 12l Cl=0 = ^3cosftJ t. This shearing
stress forces a vibration of the form
=
«2 (/4cosA Xi + BsinAjqXCcoscT-A t + DsincjA t).
Using Hooke's law, we have

or,

Thus,

The boundary condition at x\ =1 gives

Thus,

and

(b) Resonance occurs for

or

We remark that these values of (a correspond to free vibration natural frequencies with one
face traction-free and one face fixed.