Let
.
Then solutions of the form qj = Re(Ajeiw
t) can be found. We can find the w2
from det(kij-w 2Tij)
= 0. For a system with n
degrees of freedom, n characteristic frequencies
wa can be found. Some frequencies may be
degenerate.
For a particular frequency wa
we solve
to find the Aja.
[While the secular equation det(kij-w 2Tij) = 0 can in principle always be solved, it is often simpler to find the normal modes by using physical insight and noting the symmetries of the system.]
The most general solution for each coordinate qj is a sum of simple harmonic oscillations in all of the frequencies wa.
.
Simple Harmonic Motion: |
x(t) = Acos(wt+f),
v(t) = dx(t)/dt = -wAsin(wt+f), a(t) = d2x(t)/dt2 = -w2Acos(wt+f) = -w2x |
| Energy: | K = (1/2)mv2, U = (1/2)kx2, E = K+U = (1/2)kA2 |
| A mass on a spring: | w = (k/m)1/2, T = 2p(m/k)1/2, f = (1/(2p))(k/m)1/2 |
| A simple pendulum: | q(t) = qmaxcos(wt+f), w2 = g/L (small oscillations) |
Mechanical waves: |
wave equation: d2y(x,t)/dx2 = (1/v2)d2y(x,t)/dt2. |
| Sinusoidal waves: | y = Asin(kx ± wt + f), k = 2p/l, w = 2p/T = 2pf, v = l/T = lf. |
| Waves in a string: | v = (F/m)1/2, F = tension in the string, m = mass per unit length. |
| Standing waves: | String and tube with two open ends: fn = nv/(2L) = nf1 Tube with one closed end: fn = nv/(4L) = nf1, n = odd |
| Doppler effect: | f = f0(v+vo)/(v-vs), f = observed frequency, f0 = frequency of source, v = speed of wave, vo = velocity of observer towards source, vs = velocity of source towards observer. |