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

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  • $\begingroup$ 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. $\endgroup$ – Carl Witthoft Mar 23 '17 at 11:58
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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.

P.S. The correct term for these orthogonal functions is basis functions (and not basic functions) - see Wikipedia basis function.

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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.

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