4.1 ELASTICITY EQUATIONS FOR PLATE VIBRATION

In this chapter, we will analyze plate vibration. After reviewing the general plate equations, we will consider two situations separately: (a) axial vibration of plates; and (b) flexural vibration of plates.

Because the plate surface is free, the z-direction stress is assumed to be zero ( ? _zz = 0). Also zero are assumed to be the surface shear stresses ? _yz = 0 and ? _zx = 0. Hence, the 3-D elasticity relations given in Appendix B reduce to

In our analysis, we are only interested in the strains ? _xx, ? _yy, ? _xy. Solution of Eq. (1) yields

4.2 AXIAL VIBRATION OF RECTANGULAR PLATES

Axial vibrations are related to extension/compression motion in the plate, i.e., the motion is in-plane polarized.

4.2.1 GENERAL EQUATIONS FOR AXIAL VIBRATION OF RECTANGULAR PLATES

Assume in-plane displacements u( x, y, t) and v( x, y, t) which are uniform across the plate thickness. The strains of interest are

The strains are constant across the plate thickness. Substitution of Eq. (3) into Eq. (2) yields

Note that the stresses are also constant across the plate thickness. Integration of stresses across the plate thickness gives the stress resultants (forces per unit width) N _x, N _y, and N _xy shown in Fig. 4.1, i.e.,

Figure 4.1: Infinitesimal plate element in Cartesian coordinates for the analysis of in-plane vibration.

Newton...