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Results tagged with normal-modes
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user 333454
Normal modes refer to fundamental patterns of motion of a system which oscillate at fixed, well defined frequencies. They may be used as building blocks for more complicated motions.
6
votes
Accepted
Initial conditions in an infinite string of masses
Yes, however there are some details to iron out. Essentially, you have the discrete analogue of Fourier transform:
$$
\phi(k)=\sum_{n=-\infty}^{+\infty}\psi_ne^{in k} \\
\psi_n=\int_0^{2\pi}\frac{dk}{ …
- 15.1k
2
votes
Accepted
Normal Modes: Is the mass matrix always diagonal?
In general, you can always perform a change of basis and make $M$ non diagonal. It is more a question of whether the physically natural variables make $M$ diagonal.
A classic example where it is not t …
- 15.1k
0
votes
Accepted
Diagonalizing the tridiagonal matrix for finding the normal mode
You can try to solve the eigenvalue problem directly, the calculations are not too hard. You just solve the second order induction and match the boundary conditions.
A faster method is to revert to th …
- 15.1k
0
votes
Exciting a normal mode of $N$ coupled oscillator with driving force
Given:
$$
\omega^2=2(1-\cos k)
$$
(Setting $a=1$ and $T/ma=1$ up to a change of length and time scale) you want to find the $\omega$ poles of:
$$
A=\frac{1}{\sin(k(N+1))}
$$
which is the prefactor of …
- 15.1k