1 Questions

How to make equivalence from real structures?

2 Vibration mode

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}

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.

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}.

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

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.

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}