1 / 12

Lecture 2 - Background from 1A

This lecture covers the concepts of resonance, superposition of oscillations, and impedance matching in driven oscillators. It explores velocity and displacement resonance, power absorption, and the transient response of a driven oscillator. The lecture also explains the concepts of mechanical impedance and power transmission in electrical circuits.

kendrick
Download Presentation

Lecture 2 - Background from 1A

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lecture 2 - 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).

More Related