3.2.3 Plate vibration

The governing equation for a vibrating plate can be derived by a similar method to that used in section 3.2.1 for the beam: consider a small element, look at the shear forces, bending moments and twisting moments on all the edges, then take a limit as the element size tends to zero. We will not go through that in detail here: instead we will give a plausibility argument which leads to the correct equation, by analogy with the bending beam. Recall from section 3.2.1 that the governing equation for beam vibration was

$$\rho A \dfrac{\partial^2 w}{\partial t^2}+EI \dfrac{\partial^4 w}{\partial x^4}=0. \tag{1}$$

for displacement $w(x,t)$. For the plate, we will have displacement $w(x,y,t)$. What should we expect to be the equivalent of the fourth derivative in $x$? A natural candidate is found by replacing $\frac{\partial^2}{\partial x^2}$ with $\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$, an operator called the Laplacian that is known to be invariant under change of coordinate system.

This leads us to expect an equation of the form

$$\rho h \dfrac{\partial^2 w}{\partial t^2}+EK \left[\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} \right] \left[\dfrac{\partial^2 w}{\partial x^2}+\dfrac{\partial^2 w}{\partial y^2}\right]=0\tag{2}$$

so that

$$\rho h \dfrac{\partial^2 w}{\partial t^2}+EK \left[\frac{\partial^4 w}{\partial x^4}+2\frac{\partial^4 w}{\partial x^2 \partial y^2} +\frac{\partial^4 w}{\partial y^4} \right]=0\tag{3}$$

for a plate of thickness $h$, with some constant $K$. $E$ is Young’s modulus and $\rho$ is density as before: we are assuming for the moment that our plate is made of an isotropic material like a metal, with the same properties in all directions. Explicit calculation reveals that

$$K=\frac{h^3}{12(1-\nu^2)} \tag{4}$$

where $\nu$ is Poisson’s ratio. This Poisson’s ratio factor arises because of an assumption that the plate is in a state of plane stress, because it is thin enough for any through-thickness stresses to relax to zero.

We are interested in modes, so as usual we assume $w(x,y,t)=u(x,y) e^{i \omega t}$. By far the simplest special case arises if we assume a rectangular plate of dimensions $a \times b$, with simply-supported boundary conditions on all four edges. This requires zero displacement, and zero bending moment about each edge line. It is easily shown that equation (3) and these boundary conditions are all satisfied by the shapes

$$u_{nm}=\sin(n \pi x/a) \sin(m \pi y/b) \tag{5}$$

where $n$ and $m$ can take any integer values 1,2,3,… The corresponding natural frequencies are given by

$$\omega_{nm} = \sqrt{\frac{EK}{\rho h}} \left[\frac{n^2 \pi^2}{a^2}+\frac{m^2 \pi^2}{b^2} \right] .\tag{6}$$

Later, especially when we look at the soundboards of stringed instruments, we will be interested in the vibration of wooden plates. Soundboards are usually made of high-quality softwood: typically varieties of spruce or cedar. Such materials are by no means isotropic: the Young’s modulus along the grain of the wood can be at least an order of magnitude greater than the Young’s modulus in the radial direction of the growing tree. That case is governed by a similar equation, but instead of a single constant $E$ we will need different constants for each of the three terms in the square bracket in eq. (3). A fourth elastic constant may also enter the problem if we have more complicated boundary conditions than the ones assumed here. We will go into this in more detail in section ??.