1. Hour 12 Driven Harmonic Oscillators
Physics 321
Hour 12
Driven Harmonic Oscillators

2. Bottom Line
Starting from rest, the system tries to oscillate at the natural frequency
In time, it oscillates at the driving frequency
The equation for a driven, damped oscillator:
𝑥 +2𝛽 𝑥 + 𝜔0 2 𝑥= 𝐹(𝑡) 𝑚 ≡ 𝑓 0 cos 𝜔𝑡
The steady state solution is:
𝑥 𝑡 =𝐴 cos (𝜔𝑡−𝛿)
FWHM of the resonance curve is approximately 2β
So – what information can we obtain from the potential energy well?
Bound vs. unbound, turning points, T(r).

3. Driven Oscillator The equation: 𝑥 +2𝛽 𝑥 + 𝜔0 2 𝑥= 𝐹(𝑡) 𝑚 ≡𝑓(𝑡)
Let 𝑓 𝑡 = 𝑓 0 cos 𝜔𝑡
The oscillator wants to oscillate at 𝜔 0 but the driver forces it to oscillate at 𝜔. This leads to transient vs steady state behavior!

4. Example
Driven_Osc.nb

5. Driven Oscillator 𝑥 +2𝛽 𝑥 + 𝜔0 2 𝑥 =𝑓 0 cos 𝜔𝑡
We assume a solution something like
𝑥 𝑡 =𝐴 cos (𝜔𝑡−𝛿)
But 𝑥 (𝑡)=−𝐴𝜔 sin (𝜔𝑡−𝛿)
So we employ a trick…
The driving force is the real part of
𝑓 0 𝑒 𝑖𝜔𝑡

6. Driven Oscillator 𝑧 +2𝛽 𝑧 + 𝜔0 2 𝑧 =𝑓 0 𝑒 𝑖𝜔𝑡
𝑧 +2𝛽 𝑧 + 𝜔0 2 𝑧 =𝑓 0 𝑒 𝑖𝜔𝑡
We assume a solution of the form
𝑧 𝑡 =𝐴 𝑒 −𝑖𝛿 𝑒 𝑖𝜔𝑡 =𝐶 𝑒 𝑖𝜔𝑡
This gives:
(− 𝜔 2 +2𝑖𝛽𝜔+ 𝜔0 2 )𝐶 𝑒 𝑖𝜔𝑡 =𝑓 0 𝑒 𝑖𝜔𝑡
𝐶= 𝑓 0 𝜔0 2 − 𝜔 2 +2𝑖𝛽𝜔

7. Driven Oscillator Conclusion 1:
|𝐶| 2 = 𝐴 2 = 𝑓 ( 𝜔0 2 − 𝜔 2 ) 2 +4 𝛽 2 𝜔 2

8. Driven Oscillator (− 𝜔 2 +2𝑖𝛽𝜔+ 𝜔0 2 )𝐴 𝑒 −𝑖𝛿 𝑒 𝑖𝜔𝑡 =𝑓 0 𝑒 𝑖𝜔𝑡
(− 𝜔 2 +2𝑖𝛽𝜔+ 𝜔0 2 )𝐴 𝑒 −𝑖𝛿 𝑒 𝑖𝜔𝑡 =𝑓 0 𝑒 𝑖𝜔𝑡
(− 𝜔 2 +2𝑖𝛽𝜔+ 𝜔0 2 )𝐴 =𝑓 0 𝑒 𝑖𝛿
𝜔0 2 − 𝜔 2 𝐴+2𝑖𝛽𝜔𝐴 =𝑓 0 cos 𝛿 +𝑖 𝑓 0 sin 𝛿
Real parts: 𝜔0 2 − 𝜔 2 𝐴 =𝑓 0 cos 𝛿
Imaginary parts: 2𝛽𝜔𝐴= 𝑓 0 sin 𝛿
tan 𝛿= 𝑓 0 sin 𝛿 𝑓 0 cos 𝛿 = 2𝛽𝜔 𝜔0 2 − 𝜔 2

9. Driven Oscillator Conclusion 2: tan 𝛿 = 2𝛽𝜔 𝜔0 2 − 𝜔 2
And finally the steady state solution is:
𝑥 𝑡 =𝐴 cos (𝜔𝑡−𝛿)