1. 1 Waves 2 Lecture 2 - Background from 1A Revision: Resonance and Superposition D Aims: ëContinue our review of driven oscillators: > Velocity resonance; > Displacement resonance; > Power absorption > Impedance matching (electrical circuits). ëSuperposition of oscillations: > Same frequency; > Different frequency (beats). ëTransient response of a driven oscillator

2. 2 Waves 2 Impedance. D Mechanical impedance (Section 1.3.1) ë Last lecture we had: ëZ = force applied / velocity response ëMagnitude:  Minimum value is b, when  m = s / . ëPhase:   = 0: phase = -  / 2   =  o : phase = 0  : phase = +  / 2 Q =2 Q = 5 Q = 15

3. 3 Waves 2 Displacement resonance D Velocity resonance  Occurs when  =  o. ëThe lower the damping the greater the “response”. (the lower the damping, the greater the amplitude of the velocity response). D Displacement resonance algebra is a little more complicated: ësolution of eq.[1.3] (last lecture) gave: ëMaximum when magnitude of denominator is smallest i.e.  Resonance frequency is always less than  o. (But usually only by a small amount)

4. 4 Waves 2 Velocity resonance D Magnitude and phase vs frequency ëCurves for Q=2; Q=5; Q=15.  Note: maximum velocity response at  =  o.

5. 5 Waves 2 Displacement response D Magnitude and phase vs frequency ëCurves are for Q=2; Q=5; and Q=15.  Note: max displacement response at  <  o. Phase curves shifted by -  /2 but otherwise the same as for velocity resonance.

6. 6 Waves 2 Violin D Violin bridge ëreal-life mechanical system: ëRef: “The physics of the violin”, L Cremer, MIT Press, (1983). Impedance

7. 7 Waves 2 Power absorption D Mean power absorbed: (sect. 1.1.3) ëfrom fig. D Notes:  Power absorption -> 0 as  -> 0, and as  -> , (since Z ->  ).  Power absorption is maximum when  =  o. The max value is

8. 8 Waves 2 Impedance matching, I D Power transmission from source to load: ëElectrical circuit: Source impedance Z s Load impedance Z l ëPower dissipated

9. 9 Waves 2 Impedance matching, II D Notes:  R s and R l are always positive, X s and X l may be positive or negative. ëMaximum power transmitted when: ëImpedance of the load must be equal to the complex conjugate of the impedance of the source. i.e. when there is an impedance match.

10. 10 Waves 2 1.4 Superposition of oscillators D Linearity:  Our equations are linear in z. Thus solutions can be superposed. D Vibrations with equal frequency: ëTwo forcing terms, with different amplitude and phase. ëCoherent excitation: const. > interference ëIncoherent excitation: Energy is simply the sum of energies of the two excitations. Interference term A 2  energy

11. 11 Waves 2 Superposition cont…... D Vibrations of different frequency ë (for simplicity) take D Beats: ëWhen there are many rotations of A o before the length changes significantly.  Time between successive maxima in amplitude is 2  /(  1 -  2 ). ëThe beat frequency is the frequency difference.

12. 12 Waves 2 Transients D Full solution for the forced oscillator. D Full solution for the forced oscillator. Sum of two parts: ëParticular integral: i.e. solution of ëComplementary function: i.e. solution of (decays with time, and oscillates for a lightly damped system).