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, 6.1 Taylor 5.5, 5.6 THE DRIVEN, DAMPED HARMONIC OSCILLATOR 2

Natural motion of damped, driven harmonic oscillator x m m k k viscous medium F 0 cos  t Note  and  0 are not the same thing!  is driving frequency   is natural frequency 3

Natural motion of damped, driven harmonic oscillator L R C I V o cos  t Apply Kirchoff’s laws 4

underdamped large if  is small compared to  0 Damping time or "1/e" time is  = 1/   (>> 1/   if  is very small) How many T 0 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 C I Natural (resonance) frequency determined by the inductor and capacitor Damping determined by resistor & inductor Typical numbers: L≈500µH; C≈100pF; R≈50     ≈10 6 s -1 (f   ≈700 kHz)  =1/  ≈2µs; (your lab has different parameters) Q factor:

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 V out across R V in to func gen

V 0 real, constant, and known But now q 0 is complex: This solution makes sure q(t) is oscillatory (and at the same frequency as F ext ), 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 |q 0 | and  q  in 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 V 0 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 V 0 real, and constant Task #1: Substitute this assumed form into the equation of motion, and find the values of |q 0 | 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

Charge Amplitude |q 0 | Charge Phase  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

Current Amplitude |I 0 | Current Phase Driving Frequency------> “Resonance” 14 0 π/2 -π/2

Charge Amplitude |q 0 | Driving Frequency------> “Resonance” 15 Current Amplitude |I 0 | 00 00

Current Phase Driving Frequency------> 16 Charge Phase  q 0 -π -π/2 0 π/2 -π/2 00 00