1 / 58

CLASSICA DYNAMIC OF PARTICLES AND SYSTEMS— CHAPTER 3 Oscillations

CLASSICA DYNAMIC OF PARTICLES AND SYSTEMS— CHAPTER 3 Oscillations. Associate Professor: C. H.L IAO. Contents:. 3.1 Introduction 99 3.2 Simple Harmonic Oscillator 100 3.3 Harmonic Oscillations in Two Dimensions 104 3.4 Phase Diagrams 106 3.5 Damped Oscillations 108

dwight
Download Presentation

CLASSICA DYNAMIC OF PARTICLES AND SYSTEMS— CHAPTER 3 Oscillations

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. CLASSICA DYNAMIC OF PARTICLESAND SYSTEMS—CHAPTER 3 Oscillations Associate Professor: C. H.LIAO

  2. Contents: • 3.1 Introduction 99 • 3.2 Simple Harmonic Oscillator 100 • 3.3 Harmonic Oscillations in Two Dimensions 104 • 3.4 Phase Diagrams 106 • 3.5 Damped Oscillations 108 • 3.6 Sinusoidal Driving Forces 117 • 3.7 Physical Systems 123 • 3.8 Principle of Superposition-Fourier Series 126 • 3.9 The Response of Linear Oscillators to Impulsive Forcing129

  3. 3.1 Introduction • If the particle is displaced from the origin (in either direction), a certain force tends to restore the particle to its original position. An example is an atom in a long molecular chain. • The restoring force is, in general, some complicated function of the displacement and perhaps of the particle's velocity or even of some higher time derivative of the position coordinate. • We consider here only cases in which the restoring force F is a function only of the displacement: F = F(x).

  4. 3.2 Simple Harmonic Oscillator

  5. the kinetic energy The incremental amount of work dW

  6. ωo represents the angular frequency of the motion, which is related to the frequency νo by

  7. Ex. 3-1 Find the angular velocity and period of oscillation of a solid sphere of mass m and radius R about a point on its surface. See Figure 3-l. Sol.: The equation of motion for θ is

  8. 3.3 Harmonic Oscillations in 2-D (2- Dimensions) where the restoring forceis

  9. Conclusions:

  10. 3.4 Phase Diagrams We may consider the quantities x(t) andto be the coordinates of a point in a two-dimensional space, called phase space.

  11. 3.5 Damped Oscillations • The motion represented by the simple harmonic oscillator is termed a free oscillation;

  12. The general solution is:

  13. Underdamped Motion The envelope of thedisplacement versus time curve is given by

  14. Ex. 3-2 Sol.:

  15. Critically Damped Motion

  16. Overdamped Motion

  17. Ex. 2-3 Sol.:

  18. 3.6 Sinusoidal Driving Forces The simplest case of driven oscillation is that in which an external driving forcevarying harmonically with time is applied to the oscillator where A = F0/m and where w is the angular frequency of the driving force

  19. Resonance Phenomena

  20. 3.7 Physical Systems • We stated in the introduction to this chapter that linear oscillations apply to more systems than just the small oscillations of the mass-spring and the simple pendulum. • We can apply our mechanical system analog to acoustic systems. In this case,the air molecules vibrate.

  21. The hanging mass-spring systemV.S. equivalent electrical circuit.

  22. Ex.3-4 Sol.:

  23. Ex. 3-5 Sol.:

  24. 3.8 Principle of Superposition-Fourier Series The quantity in parentheses on the left-hand side is a linear operator, which wemay represent by L.

  25. Periodic function In the usual physical case in which F(t) is periodic with period τ = 2π /ω F(t) has a period τ *Fourier's theorem for any arbitrary periodic function can be expressed as:

  26. Ex. 3-6 Sol.:

  27. Problem discussion. Thanks for your attention.

  28. Problem discussion. • Problem: • 3-1, 3-6, 3-10, 3-14, 3-19, 3-24, 3-26, 3-29, 3-32, 3-37, 3-43

  29. 3-1

  30. 3-6

  31. 3-10

  32. 3-14

  33. 3-19

  34. 3-24

  35. 3-26

  36. 3-29

More Related