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

Reading: Main 5.1, 6.1Taylor 5.5, 5.6

THE DRIVEN, DAMPED HARMONIC OSCILLATOR

2

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

3

Natural motion of damped, driven harmonic oscillator

L

R

CI

Vocost

Apply Kirchoff’s laws

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

4

underdamped

large if is small compared to 0

Damping time or "1/e" time is = 1/(>> 1/if is very small)

How many T0 periods elapse in the damping time? This number (times π) is the Quality factor or Q of the system.

LCR circuit obeys precisely the same equation as the damped mass/spring.

LRC circuit

L

R

CI

Natural (resonance) frequency determined by the inductor and capacitor

Damping determined by resistor & inductor

Typical numbers: L≈500µH; C≈100pF; R≈50≈106s-1 (f≈700 kHz)=1/≈2µs; (your lab has different parameters)

Q factor:

0LQ

R

8

Measure the frequency!“ctrl-alt-del” for osc

Put cursor in track mode, one to track ch1, one for ch2

Menu off button “push”=enter

save to usb drive

measure Vout across R Vin to func gen

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 9

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.

10

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

11

ChargeAmplitude

|q0|

ChargePhase q

Driving Frequency------>

"Resonance"

0

-π

-π/2

12

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.

13

CurrentAmplitude|I0|

CurrentPhase

Driving Frequency------>

“Resonance”

14

0

π/2

-π/2

ChargeAmplitude

|q0|

Driving Frequency------>

“Resonance”

15

CurrentAmplitude|I0|

0

0

CurrentPhase

Driving Frequency------>

16

ChargePhase q

0

-π

-π/2

0

π/2

-π/2

0

0