1 Questions
How to make equivalence from real structures?
2 Vibration mode
2.1 Longitudinal vibration
Area: $A$, Length: $L$.
Governing equation:
\[\rho u_{tt} = E u_{xx}\]Frequency of lowest mode under fixed-fixed, free-free condition:
\[\omega = \sqrt{\frac{E}{\rho}} \frac{\pi}{L}\]  |
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Frequency of lowest mode under fixed-free condition:
\[\omega = \sqrt{\frac{E}{\rho}} \frac{\pi}{2L}\]Rao, S. S. (2019). Vibration of continuous systems. John Wiley & Sons.
2.2 Flexural vibration
The governing equation is
\[\frac{\partial^{2}}{\partial x^{2}}\left(E I \frac{\partial^{2} w}{\partial x^{2}}\right)+\rho A \frac{\partial^{2} w}{\partial t^{2}}=f(x, t)\]where
\[I=I_{y}=\iint_{A} z^{2} d A\]Lowest mode for beam simply supported at both ends
\[\omega=\left(\beta_{1} l\right)^{2}\left(\frac{E I}{\rho A l^{4}}\right)^{1 / 2}=\pi^{2}\left(\frac{E I}{\rho A L^{4}}\right)^{1 / 2}.\]
2.3 Torsional Vibration of Shafts
The governing equation is
\[\frac{\partial}{\partial x}\left(G J \frac{\partial \theta(x, t)}{\partial x}\right)+m_{t}(x, t)=I_{0} \frac{\partial^{2} \theta(x, t)}{\partial t^{2}}\]where $I_0 = \rho J$, $J = \int_A r^2 dA$. Uniform and without attached mass $m_t$, we have
\[G \frac{\partial^2 \theta(x, t)}{\partial x^2}=\rho \frac{\partial^{2} \theta(x, t)}{\partial t^{2}}\]Frequency of lowest mode under fixed-fixed, free-free condition:
\[\omega = \sqrt{\frac{G}{\rho}} \frac{\pi}{L}\]Frequency of lowest mode under fixed-free condition:
\[\omega = \sqrt{\frac{G}{\rho}} \frac{\pi}{2L}\]where
2.4 Shear vibration
Lowest frequency for shear mode is
\[\omega=\frac{\pi}{2 b} \sqrt{\frac{\mu}{\rho}}\]where $b$ is half of the thickness $d$. Or
\[\omega=\frac{\pi}{4 d} \sqrt{\frac{\mu}{\rho}}\]where $d$ is the thickness.
2.5 Bending vibration of the plate
Governing equation is
\[D \nabla^{4} w+\rho h \ddot{w}-f=0\]in which $D$ represents the flexural rigidity of the plate:
\[D=\frac{E h^{3}}{12\left(1-\nu^{2}\right)}\]Solution for a simply supported rectangular plate is
\[\omega_{m n}=\lambda_{m n}^{2}\left(\frac{D}{\rho h}\right)^{1 / 2}=\pi^{2}\left[\left(\frac{m}{a}\right)^{2}+\left(\frac{n}{b}\right)^{2}\right]\left(\frac{D}{\rho h}\right)^{1 / 2}\]The lowest mode (if $a>b$) is
\[\omega=\pi^{2}\left(\frac{1}{a}\right)^{2}\left(\frac{D}{\rho h}\right)^{1 / 2}\]Solution for circular plate is
\[\omega_{m n}=\lambda_{m n}^{2}\left(\frac{D}{\rho h}\right)^{1 / 2}\]where $\lambda_{mn}$ is the solution of the following equation
\[I_{m}(\lambda a) J_{m-1}(\lambda a)-J_{m}(\lambda a) I_{m-1}(\lambda a)=0, \quad m=0,1,2, \ldots\]The lowest root is $\lambda_{01} a=3.196$. Therefore, the frequency of lowest mode is
\[\omega=\left(\frac{3.196}{a}\right)^{2}\left(\frac{D}{\rho h}\right)^{1 / 2}\]