1. Reading: Main 3.1, 3.2, 3.3 Taylor 5.4 Giancoli 14.7, 14.8 THE DAMPED HARMONIC OSCILLATOR

2. x m m k k Free, undamped oscillators - summary No friction  mg m T L C I q Common notation for all

3. Natural motion of damped harmonic oscillator How do we choose a model? Physically reasonable, mathematically tractable … Validation comes IF it describes the experimental system accurately x m m k k

4. Natural motion of damped harmonic oscillator  and   (rate or frequency) are generic to any oscillating system This is the notation of TM; Main uses  = 2 . inverse time Divide by coefficient of d 2 x/dt 2 and rearrange:

5. Natural motion of damped harmonic oscillator Substitute: Now p is known (and there are 2) C, p are unknown constants Try Must be sure to make x real!

6. Natural motion of damped HO underdamped overdamped critically damped Can identify 3 cases time --->

7. underdamped Keep x(t) real complex amp/phase System oscillates at “frequency”  1 but in fact there is not only one single frequency associated with the motion as we will see. time --->

8. 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.

9. L (inductance), C (capacitance), cause oscillation, R (resistance) causes damping LRC circuit L R C I

10. 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; Q≈45 (your lab has different parameters) Q factor:

11. Does the model fit?

13. Summary so far: Free, undamped, linear (harmonic) oscillator Free, undamped, non-linear oscillator Free, damped linear oscillator Next Driven, damped linear oscillator Laboratory to investigate LRC circuit as example of driven, damped oscillator Time and frequency representations Fourier series