Reading: Main 5.1, 6.1Taylor 5.5, 5.6

THE DRIVEN, DAMPED HARMONIC OSCILLATOR

1

Natural motion of damped, driven harmonic oscillator

x

m

mk

kviscous medium

F0cost

Note and 0 are not the same thing! is driving frequency is natural frequency

2

Natural motion of damped, driven harmonic oscillator

L

R

CI

Vocost

Apply Kirkoff’s laws

http://www.sciencejoywagon.com/physicszone/lesson/otherpub/wfendt/accircuit.htm

3

What is the response of the system? x(t), q(t), or in general, (t)?

Qualitative questions first:

•What is the basic form of the system response after long times?Sinusoidal(t) = maxcos(t+) (after times longer than 1/)

•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 0, 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?

4

V0 real, constant, and known

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 qin terms of the known quantities. Note that these constants depend on driving frequency (but not on t – that's why they're "constants"). How does the shape vary with 5

Let's assume this form for q(t)

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.

6

Assume V0 real, and constant

Task #1: Substitute this assumed form into the equation of motion, and find the values of |q0| and in terms of the known quantities. Note that these constants depend on (but not on t - that’s why they’re “constants”). How does the shape vary with

7

ChargeAmplitude

|q0|

ChargePhase q

Driving Frequency------>

“Resonance”

0

-π

-π/2

8

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.

9

CurrentAmplitude|I0|

CurrentPhase

Driving Frequency------>

“Resonance”

10

0

π/2

-π/2

ChargeAmplitude

|q0|

Driving Frequency------>

“Resonance”

11

CurrentAmplitude|I0|

0

0

CurrentPhase

Driving Frequency------>

12

ChargePhase q

0

-π

-π/2

0

π/2

-π/2

0

0