Page 268 - Introduction to Continuum Mechanics

252 The Elastic Solid

Example 5.11.1

(a) Find the thickness -stretch vibration of a plate, where the left face (xj = 0) is subjected
e
to a forced displacement u = (acos o>0 i and the right face (*i = / ) is fixed.
(b) Determine the values of co that give resonance.
Solution, (a) Using Eq. (5.11.3) and the first boundary condition, we have

Therefore

The second boundary condition gives

Therefore

and the vibration is given by

(b) Resonance is indicated by unbounded displacements. This occurs for forcing frequencies
corresponding to tan a)l/ci = 0, that is, when

Example 5.11.2
(a) Find the thickness-shear vibration of an infinite plate which has an applied surface
on tne
traction t = -(ficQW)i)*2 plane jq = 0 and is fixed at the plane jcj = /.

t These values of frequencies correspond to the natural free vibration frequencies with both faces fixed.