# Vibration of annular sector plate

The deflection of the sector is assumed in the following form But I don't know why they assumed G(θ) in the following form and how to compute η.Can somebody help me? Thanks.

• The cos and sin terms represent standing waves, and I think the cosh and sinh terms represent decay at free and clamped edges. Your textbook should have an explanation. Mar 23, 2017 at 11:58

The governing equation for free vibrations of a circular plate is (see Wikipedia article on vibrating Kirchhoff plates) $$\frac{1}{r}\frac{\partial }{\partial r}\left[r \frac{\partial }{\partial r}\left\{\frac{1}{r}\frac{\partial }{\partial r}\left(r \frac{\partial w}{\partial r}\right)\right\}\right] = -\frac{2\rho h}{D}\frac{\partial^2 w}{\partial t^2}\,.$$ Separation of variables leads to a general solution of the form $$w(r,t) = \sum_{n=1}^\infty C_n\left[J_0(\lambda_n r) - \frac{J_0(\lambda_n a)}{I_0(\lambda_n a)}I_0(\lambda_n r)\right] [A_n e^{i\omega_n t} + B_n e^{-i\omega_n t}] \,.$$ The term $$A_n e^{i\omega_n t} + B_n e^{-i\omega_n t}$$ can be expressed as a sum of sine, cosines, and hyperbolic sines and cosines when we take the real part of $w$. That's why the expression for $G(\theta)$ in your book has that form.
For a annular sector of a circular plate, things are a bit more complicated and you will have to do the algebra yourself (or look it up from a book) to find out what $\eta$ is.
Biswajit covered most of it, but to build on his answer a little: when you are solving these types of problems, your answer has to satisfy both the partial differential equation and the boundary conditions. $G(\theta)$ as given in your question is the most general solution to the PDE. But it will not satisfy the boundary condition for all values of $\eta$, A, B, C, and D. e.g. a clamped boundary at 0 would be expressed as $G(0)=0$, $G'(0)=0$. If you picked $A=B=D=0$, and $C=1$, you can see that this does not satisfy the boundary conditions, because $cos(0)=1$. But there is some other combination of parameters that will satisfy it. To solve for the 4 unknowns, you will need to apply 4 boundary conditions, 2 at each end of your sector.