Vibration of plates

The vibration of plates is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two. This suggests that a two-dimensional plate theory will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.^[1]

There are several theories that have been developed to describe the motion of plates. The most commonly used are the Kirchhoff-Love theory^[2] and the Mindlin-Reissner theory. Solutions to the governing equations predicted by these theories can give us insight into the behavior of plate-like objects both under free and forced conditions. This includes the propagation of waves and the study of standing waves and vibration modes in plates.

1 Kirchhoff-Love plates
- 1.1 Isotropic plates
- 1.2 Free vibrations
  - 1.2.1 Circular plates
  - 1.2.2 Rectangular plates
2 References
3 See also

Kirchhoff-Love plates

The governing equations for the dynamics of a Kirchhoff-Love plate are

$\begin{align} N_{\alpha\beta,\beta} & = J_1~\ddot{u}_\alpha \\ M_{\alpha\beta,\alpha\beta} - q(x,t) & = J_1~\ddot{w} - J_3~\ddot{w}_{,\alpha\alpha} \end{align}$

where $u_\alpha$ are the in-plane displacements of the mid-surface of the plate, $w$ is the transverse (out-of-plane) displacement of the mid-surface of the plate, $q$ is an applied transverse load, and the resultant forces and moments are defined as

$N_{\alpha\beta}�:= \int_{-h}^h \sigma_{\alpha\beta}~dx_3 \quad \text{and} \quad M_{\alpha\beta}�:= \int_{-h}^h x_3~\sigma_{\alpha\beta}~dx_3 \,.$

Note that the thickness of the plate is $2h$ and that the resultants are defined as weighted averages of the in-plane stresses $\sigma_{\alpha\beta}$ . The derivatives in the governing equations are defined as

$\dot{u}_i�:= \frac{\partial u_i}{\partial t} ~;~~ \ddot{u}_i�:= \frac{\partial^2 u_i}{\partial t^2} ~;~~ u_{i,\alpha}�:= \frac{\partial u_i}{\partial x_\alpha} ~;~~ u_{i,\alpha\beta}�:= \frac{\partial^2 u_i}{\partial x_\alpha \partial x_\beta}$

where the Latin indices go from 1 to 3 while the Greek indices go from 1 to 2. Summation over repeated indices is implied. The $x_3$ coordinates is out-of-plane while the coordinates $x_1$ and $x_2$ are in plane. For a uniformly thick plate of thickness $2h$ and homogeneous mass density $\rho$

$J_1�:= \int_{-h}^h \rho~dx_3 = 2\rho h \quad \text{and} \quad J_3�:= \int_{-h}^h x_3^2~\rho~dx_3 = \frac{2}{3}\rho h^3 \,.$

Isotropic plates

For an isotropic and homogeneous plate, the stress-strain relations are

$\begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix} = \cfrac{E}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & 1-\nu \end{bmatrix} \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{12} \end{bmatrix} \,.$

where $\varepsilon_{\alpha\beta}$ are the in-plane strains. The strain-displacement relations for Kirchhoff-Love plates are

$\varepsilon_{\alpha\beta} = \frac{1}{2}(u_{\alpha,\beta}%2Bu_{\beta,\alpha}) - x_3\,w_{,\alpha\beta} \,.$

Therefore, the resultant moments corresponding to these stresses are

$\begin{bmatrix}M_{11} \\ M_{22} \\ M_{12} \end{bmatrix} = -\cfrac{2h^3E}{3(1-\nu^2)}~\begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & 1-\nu \end{bmatrix} \begin{bmatrix} w_{,11} \\ w_{,22} \\ w_{,12} \end{bmatrix}$

If we ignore the in-plane displacements $u_{\alpha\beta}$ , the governing equations reduce to

$D\nabla^2\nabla^2 w = -q(x,t) - 2\rho h\ddot{w} \,.$

Free vibrations

For free vibrations, the governing equation of an isotropic plate is

$D\nabla^2\nabla^2 w = - 2\rho h\ddot{w} \,.$

Circular plates

For freely vibrating circular plates, $w = w(r,t)$ , and the Laplacian in cylindrical coordinates has the form

$\nabla^2 w \equiv \frac{1}{r}\frac{\partial }{\partial r}\left(r \frac{\partial w}{\partial r}\right) \,.$

Therefore, the governing equation for free vibrations of a circular plate of thickness $2h$ is

$\frac{1}{r}\frac{\partial }{\partial r}\left[r \frac{\partial }{\partial r}\left\{\frac{1}{r}\frac{\partial }{\partial r}\left(r \frac{\partial w}{\partial r}\right)\right\}\right] = -\frac{2\rho h}{D}\frac{\partial^2 w}{\partial t^2}\,.$

Expanded out,

$\frac{\partial^4 w}{\partial r^4} %2B \frac{2}{r} \frac{\partial^3 w}{\partial r^3} - \frac{1}{r^2} \frac{\partial^2 w}{\partial r^2} %2B \frac{1}{r^3} \frac{\partial w}{\partial r} = -\frac{2\rho h}{D}\frac{\partial^2 w}{\partial t^2}\,.$

To solve this equation we use the idea of separation of variables and assume a solution of the form

$w(r,t) = W(r)F(t) \,.$

Plugging this assumed solution into the governing equation gives us

$\frac{1}{\beta W}\left[\frac{d^4 W}{dr^4} %2B \frac{2}{r}\frac{d^3 W}{dr^3} - \frac{1}{r^2}\frac{d^2W}{dr^2} %2B \frac{1}{r^3} \frac{d W}{dr}\right] = -\frac{1}{F}\cfrac{d^2 F}{d t^2} = \omega^2$

where $\omega^2$ is a constant and $\beta�:= 2\rho h/D$ . The solution of the right hand equation is

$F(t) = \text{Re}[ A e^{i\omega t} %2B B e^{-i\omega t}] \,.$

The left hand side equation can be written as

$\frac{d^4 W}{dr^4} %2B \frac{2}{r}\frac{d^3 W}{dr^3} - \frac{1}{r^2}\frac{d^2W}{dr^2} %2B \frac{1}{r^3} \cfrac{d W}{d r} = \lambda^4 W$

where $\lambda^4�:= \beta\omega^2$ . The general solution of this eigenvalue problem that is appropriate for plates has the form

$W(r) = C_1 J_0(\lambda r) %2B C_2 I_0(\lambda r)$

where $J_0$ is the order 0 Bessel function of the first kind and $I_0$ is the order 0 modified Bessel function of the first kind. The constants $C_1$ and $C_2$ are determined from the boundary conditions. For a plate of radius $a$ with a clamped circumference, the boundary conditions are

$W(r) = 0 \quad \text{and} \quad \cfrac{d W}{d r} = 0 \quad \text{at} \quad r = a \,.$

From these boundary conditions we find that

$J_0(\lambda a)I_1(\lambda a) %2B I_0(\lambda a)J_1(\lambda a) = 0 \,.$

We can solve this equation for $\lambda_n$ (and there are an infinite number of roots) and from that find the modal frequencies $\omega_n = \lambda_n^2/\beta$ . We can also express the displacement in the form

$w(r,t) = \sum_{n=1}^\infty C_n\left[J_0(\lambda_n r) - \frac{J_0(\lambda_n a)}{I_0(\lambda_n a)}I_0(\lambda_n r)\right] [A_n e^{i\omega_n t} %2B B_n e^{-i\omega_n t}] \,.$

For a given frequency $\omega_n$ the first term inside the sum in the above equation gives the mode shape. We can find the value of $C_1$ using the appropriate boundary condition at $r = 0$ and the coefficients $A_n$ and $B_n$ from the initial conditions by taking advantage of the orthogonality of Fourier components.

Rectangular plates

Consider a rectangular plate which has dimensions $a\times b$ in the $(x_1,x_2)$ -plane and thickness $2h$ in the $x_3$ -direction. We seek to find the free vibration modes of the plate.

Assume a displacement field of the form

$w(x_1,x_2,t) = W(x_1,x_2) F(t) \,.$

Then,

$\nabla^2\nabla^2 w = w_{,1111} %2B 2w_{,1212} %2B w_{,2222} = \left[\frac{\partial^4 W}{\partial x_1^4} %2B 2\frac{\partial^4 W}{\partial x_1^2 \partial x_2^2} %2B \frac{\partial^4W}{\partial x_2^4}\right] F(t)$

and

$\ddot{w} = W(x_1,x_2)\frac{d^2F}{dt^2} \,.$

Plugging these into the governing equation gives

$\frac{D}{2\rho h W}\left[\frac{\partial^4 W}{\partial x_1^4} %2B 2\frac{\partial^4 W}{\partial x_1^2 \partial x_2^2} %2B \frac{\partial^4W}{\partial x_2^4}\right] = -\frac{1}{F}\frac{d^2F}{dt^2} = \omega^2$

where $\omega^2$ is a constant because the left hand side is independent of $t$ while the right hand side is independent of $x_1,x_2$ . From the right hand side, we then have

$F(t) = A e^{i\omega t} %2B B e^{-i\omega t} \,.$

From the left hand side,

$\frac{\partial^4 W}{\partial x_1^4} %2B 2\frac{\partial^4 W}{\partial x_1^2 \partial x_2^2} %2B \frac{\partial^4W}{\partial x_2^4} = \frac{2\rho h \omega^2}{D} W =: \lambda^4 W$

where

$\lambda^2 = \omega\sqrt{\frac{2\rho h}{D}} \,.$

Since the above equation is a biharmonic eigenvalue problem, we look for Fourier expansion solutions of the form

$W_{mn}(x_1,x_2) = \sin\frac{m\pi x_1}{a}\sin\frac{n\pi x_2}{b} \,.$

We can check and see that this solution satisfies the boundary conditions for a freely vibrating rectangular plate with simply supported edges:

$\begin{align} w(x_1,x_2,t) = 0 & \quad \text{at}\quad x_1 = 0, a \quad \text{and} \quad x_2 = 0, b \\ M_{11} = D\left(\frac{\partial^2 w}{\partial x_1^2} %2B \nu\frac{\partial^2 w}{\partial x_2^2}\right) = 0 & \quad \text{at}\quad x_1 = 0, a \\ M_{22} = D\left(\frac{\partial^2 w}{\partial x_2^2} %2B \nu\frac{\partial^2 w}{\partial x_1^2}\right) = 0 & \quad \text{at}\quad x_2 = 0, b \,. \end{align}$

Plugging the solution into the biharmonic equation gives us

$\lambda^2 = \pi^2\left(\frac{m^2}{a^2} %2B \frac{n^2}{b^2}\right) \,.$

Comparison with the previous expression for $\lambda^2$ indicates that we can have an infinite number of solutions with

$\omega_{mn} = \sqrt{\frac{D\pi^4}{2\rho h}}\left(\frac{m^2}{a^2} %2B \frac{n^2}{b^2}\right) \,.$

Therefore the general solution for the plate equation is

$w(x_1,x_2,t) = \sum_{m=1}^\infty \sum_{n=1}^\infty \sin\frac{m\pi x_1}{a}\sin\frac{n\pi x_2}{b} \left( A_{mn} e^{i\omega_{mn} t} %2B B_{mn} e^{-i\omega_{mn} t}\right) \,.$

To find the values of $A_{mn}$ and $B_{mn}$ we use initial conditions and the orthogonality of Fourier components. For example, if

$w(x_1,x_2,0) = \varphi(x_1,x_2) \quad \text{on} \quad x_1 \in [0,a] \quad \text{and} \quad \frac{\partial w}{\partial t}(x_1,x_2,0) = \psi(x_1,x_2)\quad \text{on} \quad x_2 \in [0,b]$

we get,

$\begin{align} A_{mn} & = \frac{4}{ab}\int_0^a \int_0^b \varphi(x_1,x_2) \sin\frac{m\pi x_1}{a}\sin\frac{n\pi x_2}{b} dx_1 dx_2 \\ B_{mn} & = \frac{4}{ab\omega_{mn}}\int_0^a \int_0^b \psi(x_1,x_2) \sin\frac{m\pi x_1}{a}\sin\frac{n\pi x_2}{b} dx_1 dx_2\,. \end{align}$

References

^ Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis.
^ A. E. H. Love, On the small free vibrations and deformations of elastic shells, Philosophical trans. of the Royal Society (London), 1888, Vol. série A, N° 17 p. 491–549.