1. Real Oscillators
… constant forces  integrate EOM  parabolic trajectories.
… linear restoring force  guess EOM solution  SHM
… nonlinear restoring forces  ?
linear spring
nonlinear spring? F F x x

2. WTF! The spring of air : use Ideal Gas Law: PV=NRT Patm
chamber volume: V=Ax
WTF!
(whoa there, fella) m
EOM A +x
Stable Equilibrium at
xeq = NRT / (mg + APatm)
P, V

3. Taylor Series Expansions:
Turns a function into a polynomial near x = a
Example:

4. Expand around x = -3:
2nd order
0th order
1st order

5. Expand around x = 2:
0th order
1st order
2nd order

6. Expand NRT/x around xeq:
Is it safe to linearize it? Better check a unitless ratio. How about:
(Yes, excellent choice Dr. Hafner!)

7. .. Displacement 5% of xeq: 0 .05 .0025 …. Perhaps you would prefer….
SHM with

8. Beware of assumptions and approximations!

9. Clerk = person who derived an approximation
Clouseau = You
Clerk = person who derived an approximation
Dog = domain of interest
Clerk’s dog = domain of his approximation
Bite = result of mistaken assumption that you are working within the Clerk’s domain

10. Simple Pendulum: Stable Equilibrium: Length: L Mass: m Q
Displace by Q:
mg cosQ T
mg cosQ sinQ -x
mg cosQ
EOM: mg
Expand it! mg

11. Derivatives:

12. Now express as a unitless ratio of the dependent variable and some parameter of the system:
Displacement 5% of length: …
SHM with

13. The world is not linear. However, one can use a Taylor expansion to linearize an EOM by assuming only small perturbations around a point of stable equilibrium (which may not be the origin).