Week 8

From the discretuum to the continuum

This week, we spent our efforts on making a transition from systems with a relatively small number of degrees of freedom to those with an essentially infinite number of degrees of freedom. We examined the coupled equations of motion for N masses attached to each other by string which is fixed at either end to walls.

The equations of motion for transverse displacement of the p’th mass was:

$\ddot{y}_p = - \omega_0^2 \left[ 2 y_p - y_{p+1}-y_{p-1} \right]$

where we have $\omega_0^2 = \frac{T}{l m}$ . (Here $l$ is the distance between masses, and $m$ is the mass of each mass).

We found that the spectrum of normal modes is described by amplitudes

$A_p = C \sin \left[ \frac{p n \pi}{N+1} \right]$

and frequencies

$\omega_n^2 = 4 \omega_0^2 \sin^2 \left[ \frac{n \pi}{2 (N+1)}\right]$

With this information, we can write any excitation of the string (sufficiently close to equilibrium) as a superposition over these normal modes! The motion of the p’th mass while the n’th normal mode is excited is:

$y^n_p (t) = C_n \sin \left[ \frac{p n \pi}{N+1} \right] \cos (\omega_n t + \alpha_n)$

finally, the motion of the p’th mass during a completely arbitrary excitation of the string is written as a superposition over all of the normal modes:

$y_p (t) = \sum_n C_n \sin \left[ \frac{p n \pi}{N+1} \right] \cos (\omega_n t + \alpha_n )$

I encourage you all to take a look at this weeks’ mathematica file, which allows you to visualize the normal mode spectrum as you change the number of degrees of freedom, and also as you change the normal mode number, $n$ :

https://jhubisz.expressions.syr.edu/phy360/wp-content/uploads/sites/5/2016/10/Week-8.nb

Physics 360

Vibrations and Waves

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