Lecture 1 - Background from 1A

1. Lecture 1 - Background from 1A Revision: Resonance and Superposition • Aims: • Continue our review of driven oscillators: • Velocity resonance; • Displacement resonance; • Power absorption • Impedance matching (electrical circuits). • Superposition of oscillations: • Same frequency; • Different frequency (beats). • Transient response of a driven oscillator

2. Q =2 Q = 5 Q = 15 Impedance. • Mechanical impedance (Section 1.3.1) • Last lecture we had: • Z = force applied / velocity response • Magnitude: • Minimum value is b, when wm = s/w. • Phase: • w = 0: phase = -p / 2 • w = wo: phase = 0 • : phase = +p / 2

3. Displacement resonance • Velocity resonance • Occurs when w = wo. • The lower the damping the greater the “response”. (the lower the damping, the greater the amplitude of the velocity response). • Displacement resonance algebra is a little more complicated: • solution of eq.[1.3] (last lecture) gave: • Maximum when magnitude of denominator is smallest i.e. • Resonance frequency is always less than wo.(But usually only by a small amount)

4. Velocity resonance • Magnitude and phase vs frequency • Curves for Q=2; Q=5; Q=15. • Note: maximum velocity response at w=wo.

5. Displacement response • Magnitude and phase vs frequency • Curves are for Q=2; Q=5; and Q=15. • Note: max displacement response at w< wo.Phase curves shifted by -p/2 but otherwise the same as for velocity resonance.

6. Violin • Violin bridge • real-life mechanical system: • Ref: “The physics of the violin”, L Cremer, MIT Press, (1983). Impedance

7. Power absorption • Mean power absorbed: (sect. 1.1.3) • from fig. • Notes: • Power absorption -> 0 as w -> 0, and as w -> ¥, (since Z -> ¥). • Power absorption is maximum when w = wo.The max value is

8. Impedance matching, I • Power transmission from source to load: • Electrical circuit:Source impedance ZsLoad impedance Zl • Power dissipated

9. Impedance matching, II • Notes: • Rs and Rl are always positive, Xs and Xl may be positive or negative. • Maximum power transmitted when: • Impedance 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. 1.4 Superposition of oscillators • Linearity: • Our equations are linear in z. Thus solutions can be superposed. • Vibrations with equal frequency: • Two forcing terms, with different amplitude and phase. • Coherent excitation: const. • interference • Incoherent excitation:Energy is simply the sum of energies of the two excitations. A2 µ energy Interference term

11. Superposition cont…... • Vibrations of different frequency • (for simplicity) take • Beats: • When there are many rotations of Ao before the length changes significantly. • Time between successive maxima in amplitude is 2p/(w1-w2) . • The beat frequency is the frequency difference.

12. Transients • Full solution for the forced oscillator.Sum of two parts: • Particular integral: i.e. solution of • Complementary function: i.e. solution of(decays with time, and oscillates for a lightly damped system).