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One should differentiate between ideal membrane vibration, in which the vibration is described by a wave equation, and plate/shell theory (in which the vibration is described by more complex approaches, such as Kirchhoff–Love plate theory and Mindlin–Reissner plate theory). As @wetsavannaanimal-aka-rod-vance pointed out, the main difference between a ...

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From a mathematical point of view, an ideal drum means that it is governed by the wave equation, in which the wave speed depends only on the tension at the boundary, the thickness and the density of the membrane (wikipedia). From a physical point of view, it means that the restoring force is generated by the tension at the boundary, and have the same ...

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You cannot use that formula for a stiff plate - you have a fundamentally different situation. In the stiff plate, the restoring elastic force comes from the stress arising from the strain induced elastic deformation of the plate's material. This is related to the materials' modulusses (Young's, Shear and Bulk Modulusses) and Poisson ratio. See how these ...

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$F_u=\tau\left(\frac{\partial^2 u}{\partial x^2}\Delta x+\mathcal{O}(\Delta x^2)\right)$ is a Taylor expansion of $F_u$ around $x$. The $\mathcal{O}(\Delta x^2)$ means that there are further terms, which all include a factor of $\Delta x^n$ for $n>1$, i.e. for $\Delta x \ll 1$ these terms are negligable. Generally, for a function $f(y)$ the Taylor ...

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It comes from the definition of the (forward) derivative, it's probably only notation that is holding you back here. It might be more clear if you consider: $\frac{\partial^2{u}}{\partial{x}^2} = \lim_{\Delta x->0} \frac{(\frac{\partial{u}}{\partial{x}}(x+\Delta x)-\frac{\partial{u}}{\partial{x}}(x))}{\Delta x}$ Just think of ...

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You simply need to define a few state variables. Here is the standard, control theory recipe that should get you going: Let $x_1(t) = x(t)$ and $x_2(t) = \dot{x_1}(t)$. Then your vibrating string equation becomes $M \mathrm{d}_t x_2(t) + C x_2(t) + K x_1(t) = b(t)$. That is: \frac{\mathrm{d}}{\mathrm{d}\,t} ...

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