1. PH421: Oscillations; do not distribute
11/12/07
Summary so far:
• Free, undamped, linear (harmonic) oscillator
• Free, undamped, non-linear oscillator
• Free, damped linear oscillator
Starting today:
• Driven, damped linear oscillator
• Laboratory to investigate LRC circuit as example of driven, damped oscillator
• Time and frequency representations
• Fourier series
Lecture 5/6 - damped oscllations

2. PH 421: Oscillations - do not distribute
11/12/2018
THE DRIVEN, DAMPED HARMONIC OSCILLATOR
Reading:
Main 5.1, 6.1
Taylor 5.5, 5.6
Lecture 6 - Driven oscillations

3. PH 421: Oscillations - do not distribute
Natural motion of damped, driven harmonic oscillator
11/12/2018 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

4. PH 421: Oscillations - do not distribute
Natural motion of damped, driven harmonic oscillator
11/12/2018
Apply Kirchoff’s laws L R C I
Vocoswt
Lecture 6 - Driven oscillations

5. PH 421: Oscillations - do not distribute
11/12/2018
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
11/12/2018
Assume V0 real, and constant
Task #2: In the lab, you'll actually measured 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
11/12/2018
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
11/12/2018
“Resonance”
Charge
Amplitude
|q0| w0
Driving Frequency------>
Current
Amplitude
|I0| w0
Lecture 6 - Driven oscillations

9. PH 421: Oscillations - do not distribute
11/12/2018
Task #2: In the lab, you actually measured 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
11/12/2018
“Resonance”
Current
Amplitude
|I0|
Driving Frequency------>
π/2
Current
Phase
-π/2
Lecture 6 - Driven oscillations

11. PH 421: Oscillations - do not distribute
11/12/2018
Charge
Phase fq
-π/2 -π w0
Driving Frequency------>
π/2
Current
Phase
-π/2 w0
Lecture 6 - Driven oscillations