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rod_vibrations.html
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<title>Sam Bellows, Acoustician</title>
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<center><h2>Vibrations of Rods</h2></center>
For rods, the general form of the eigenfunctions are
$\Psi_n (x) = A\sin k_n x + B\cos k_n x + C \sinh k_nx + D\cosh k_n x$
For a simply suppored rod at both ends
$\Psi_n (x) = \sin k_nx$
<br><br>
<center><img src="./img/simply_supported_rod.gif" width="350" height="350"/></center>
<center>Eigenfunctions for a simply supported rod at both ends.</center>
<br><br>
For a free-free rod at both ends
$\Psi_n (x) = \sin k_n x + \sinh k_nx - \frac{(\sin k_n L- \sinh k_n L)}{(\cos k_n L - \cosh k_n L)}(\cos k_n x + \cosh k_n x)$
<br><br>
<center><img src="./img/free_free_rod.gif" width="350" height="350"/></center>
<center>Eigenfunctions for a vibrating string free at both ends.</center>
<br><br>
For a rod clamped at both ends
$\Psi_n (x) = \sin k_n x - \sinh k_nx - \frac{(\sin k_n L- \sinh k_n L)}{(\cos k_n L - \cosh k_n L)}(\cos k_n x - \cosh k_n x)$
<br><br>
<center><img src="./img/clamped_clamped_rod.gif" width="350" height="350"/></center>
<center>Eigenfunctions for a vibrating rod clamped at both ends.</center>
<br><br>
For a cantilever rod (clamped at one end and free on the other)
$\Psi_n (x) = \sin k_n x - \sinh k_nx - \frac{(\sinh k_n L + \sin k_n L)}{(\cosh k_b L + \cos k_b L)}(\cos k_n x - \cosh k_n x)$
<br><br>
<center><img src="./img/cantilever_rod.gif" width="350" height="350"/></center>
<center>Eigenfunctions for a rod clamped at one end and free on the other.</center>
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