Physics 105A
Analytical Mechanics
Coupled Oscillators
Normal modes
28 June 2016
Manuel Calderón de la Barca Sánchez
Coupled Oscillations: Normal Modes
 Consider two masses m connected to each other and to
two walls by three springs each with constant k. Find the
most general solution for the positions of the masses as a
function of time.
Can there be a solution where both masses oscillate with
identical frequencies?
28 June 2016
MCBS
Normal Modes: One method
1.
2.
3.
4.
5.
6.
7.
8.
9.
Find relevant coordinates (identify degrees of freedom).
Apply 2nd Law (or Lagrangian method) to find coupled equations
of motion.
Guess that there will be a solution eiat that makes all coordinates
oscillate with identical frequency.
Take derivatives of solution, plug into Eq. of Motion.
Divide out eiat, put remaining linear eqs. in matrix form.
Non-trivial solution when determinant = 0.
Solve for a, Use quadratic, cubic, etc…
Each a solution will give a relationship between amplitudes,
plug into linear eqs. from step 5, obtain eigenvectors.
Eigenvectors and eigenfrequencies define a given normal mode,
for a given normal coordinate.
28 June 2016
MCBS
Normal modes: Springs on a Circle
 4.12 Springs on a circle
 Two identical masses m are
constrained to move on a
horizontal hoop. Two identical
springs with spring constant k
connect the masses and wrap
around the hoop. Find the normal
modes.
28 June 2016
MCBS