Lecture 3 Coupled Oscillators - Harvard University

What Did We Le

Linear algebra gives you a rec
by solving an eigenvalue prob

  We saw how it worked for our sim

It guarantees that the normal m

  There will be n normal modes if w
  Subtlety: all eigenvalues −ω2 mus

  This is true for any system nea
there were a positive eigenval
expands exponentially with tim

We can now extend our proble
mode, constant frequency solu

  This is enough information to let u
many many coupled oscillators

earn?

cipe for finding normal modes
blem

mple problem

modes exist

we couple n pendulums
st be negative
ar a stable equilibrium, because if
lue, it will give us a motion that
me

em knowing that the normal-
utions exist

us attack the next problem – many

Many Coupled P

Connect N pendulums with sp

  Displacement of the n-th pendulu

  Equation of motion:

m d2 xn = − gm xn − k(
dt 2 L

  Assume that L is very long  Ign

m d2 xn = −k(xn − x
dt 2

Pendulums

prings

um is xn (n = 1, 2, … N)

(xn − xn−1) − k(xn − xn+1)

nore the gm/L term

xn−1) − k(xn − xn+1)

Going Continuou

Now we make N very large, w
spring smaller and smaller

It starts to look like a spring wi

  Good model for mechanical wave

us

while making the mass and the

ith distributed mass

es such as sound

Summary

Studied coupled oscillators

  General solution is an linear com
= patterns of oscillation with cons

  Surprising pattern shows up – Be
  Linear algebra guarantees that th

  Eigenvalues  Normal freque
  Eigenvectors  Normal mode

We are ready to extend couple
mass-spring transmission line
Next: continuous waves

mbination of normal modes
stant frequencies
eats
he normal modes exist
encies
es

ed oscillators into