1. PH 421: Oscillations - do not distribute
4/25/2017
THE DRIVEN, DAMPED HARMONIC OSCILLATOR
Reading:
Main 5.1, 6.1
Taylor 5.5, 5.6
Lecture 6 - Driven oscillations

2. PH 421: Oscillations - do not distribute
Natural motion of damped, driven harmonic oscillator
4/25/2017 x m k
viscous medium
F0coswt
Note w and w0 are not the same thing!
is driving frequency
w0 is natural frequency
Lecture 6 - Driven oscillations

3. PH 421: Oscillations - do not distribute
Natural motion of damped, driven harmonic oscillator
4/25/2017
Apply Kirkoff’s laws L R C I
Vocoswt
Lecture 6 - Driven oscillations

4. PH 421: Oscillations - do not distribute
4/25/2017
What is the response of the system? x(t), q(t), or in general, y(t)?
Qualitative questions first:
What is the basic form of the system response after long times?
Sinusoidal. y(t) = ymaxcos(wt+f) (after times longer than 1/b)
Is the frequency of the system response the same, smaller, or larger than the driving frequency?
The same - it must be!
How does the magnitude of the response depend on the driving frequency?
It is large close to the natural frequency w0, and small at lower and at higher frequencies (this is called resonance)
How does the phase of the response depend on the driving frequency?
We'll have to see.
“Response” can be displacement/charge OR velocity/current – what effect on the above?
Lecture 6 - Driven oscillations

5. PH 421: Oscillations - do not distribute
4/25/2017
V0 real, constant, and known
Let's assume this form for q(t)
But now q0 is complex:
This solution makes sure q(t) is oscillatory (and at the same frequency as Fext), but may not be in phase with the driving force.
Task #1: Substitute this assumed form into the equation of motion, and find the values of |q0| and fq in terms of the known quantities. Note that these constants depend on driving frequency w (but not on t – that's why they're "constants"). How does the shape vary with w?
Lecture 6 - Driven oscillations

6. PH 421: Oscillations - do not distribute
4/25/2017
Assume V0 real, and constant
Task #2: In the lab, you'll actually measure I (current) or dq/dt. So let's look at that: Having found q(t), find I(t) and think about how the shape of the amplitude and phase of I change with frequency.
Lecture 6 - Driven oscillations

7. PH 421: Oscillations - do not distribute
4/25/2017
Assume V0 real, and constant
Task #1: Substitute this assumed form into the equation of motion, and find the values of |q0| and f in terms of the known quantities. Note that these constants depend on w (but not on t - that’s why they’re “constants”). How does the shape vary with w?
Lecture 6 - Driven oscillations

8. PH 421: Oscillations - do not distribute
4/25/2017
“Resonance”
Charge
Amplitude
|q0|
Driving Frequency------>
Charge
Phase fq
-π/2 -π
Lecture 6 - Driven oscillations

9. PH 421: Oscillations - do not distribute
4/25/2017
Task #2: In the lab, you’ll actually measure I (current) or dq/dt. So let’s look at that: Having found q(t), find I(t) and think about how the shape of the amplitude and phase of I change with frequency.
Lecture 6 - Driven oscillations

10. PH 421: Oscillations - do not distribute
4/25/2017
“Resonance”
Current
Amplitude
|I0|
Driving Frequency------>
π/2
Current
Phase
-π/2
Lecture 6 - Driven oscillations

11. PH 421: Oscillations - do not distribute
4/25/2017
“Resonance”
Charge
Amplitude
|q0| w0
Driving Frequency------>
Current
Amplitude
|I0| w0
Lecture 6 - Driven oscillations

12. PH 421: Oscillations - do not distribute
4/25/2017
Charge
Phase fq
-π/2 -π w0
Driving Frequency------>
π/2
Current
Phase
-π/2 w0
Lecture 6 - Driven oscillations