Vibrations in Beams
============================
```{image} ../WV/beam.png
:alt: free body diagram
:width: 600px
:align: center
```

The Euler Bernoulli beam equation 

$$
{\color{blue}{q}} - EI\frac{\partial ^4 u}{\partial x^4} =0
$$

For ${\color{blue}{q}}= -\rho A \ddot{u}$  we arrive at:

$$
\frac{\partial ^2 u}{\partial t^2} =-\frac{h^2}{12}{\color{blue}{C_L}}^2\frac{\partial ^4 u}{\partial x^4}
$$

here 

$$
f_1 = 0.586\frac{{\color{blue}{C_L}}h}{2L^2}
$$

```{admonition} Fracture Mechanics Clip-On Gages 
A sensor (e.g. a clip-on gauge) based on bending strain measurement with $L=8mm$ and $h=0.5mm$ has a natural frequency of $f_1 = 7.325KHz$ 

The bridge in the load frame used for the same measurement ($L=1m,h=2.5cm$) has a natural frequency of $f_1 = 36.6Hz$
```