Transient Behavior

1
Transient Behavior
We know the solution to this! So there is an
initial behavior that depends on the initial
conditions and decays away
2
Steady-State Solution
3
Matlab
Experiment with different possibilites
Consider, the underdamped driven harmonic
oscillator.
4
Nonlinear Oscillator
We need to do some sort of numeric solution or an
approximation. (Youve already done a numeric
nonlinear oscillator)
Method of successive approximations
Lets do a power series
Plug it into our power series
5
Nonlinear Oscillator
Use Eulers identity and our definition of the
natural frequency to simplify
6
Nonlinear Oscillator
Do some algebra
For small g3
7
Nonlinear Oscillator
For small g3
If
Then we have a solution
8
Nonlinear Oscillator
Note the frequency depend on the amplitude!
To improve, notice that we ignored a term of 3w,
so try
This suggests that nonlinearity gives rise to
harmonics