1 / 12

USSC3002 Oscillations and Waves Lecture 5 Dampened Oscillations

USSC3002 Oscillations and Waves Lecture 5 Dampened Oscillations. Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543. Email matwml@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6874-2749. 1. FREE OSCILLATIONS.

alyn
Download Presentation

USSC3002 Oscillations and Waves Lecture 5 Dampened 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. USSC3002 Oscillations and Waves Lecture 5 Dampened Oscillations Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6874-2749 1

  2. FREE OSCILLATIONS 1 degree of freedom (DOF) systems Mechanics Electronics Question 1. What do these equations model ? Question 2. What is the energy in these systems ? Question 3. Is the energy preserved ? Question 4. Can they describe > 1 DOF systems ? 2

  3. ENERGY The mechanical equation can be rewritten as where Question 1. What does this imply about the rate of energy change ? Where does the energy go in the mechanical and electrical systems ? 3

  4. GENERAL SOLUTION Defining gives where hence Question 1. How is this A different from before ? Question 2. What are the eigenvalues of A ? Question 3. How can we compute w and hence u ? 4

  5. EIGENVALUES of are Question 1. How do these differ from before ? Question 2. When can A be diagonalized ? Question 3. Then how can u(t) be expressed ? 5

  6. DISTINCT EIGENVALUES A has distinct e.v. iff Then there exist a 2 x 2 matrix E whose columns are the corresponding eigenvectors hence Question 1. When are these non real ? 6

  7. NON REAL EIGENVALUES Clearly and u(t) is real and therefore where 7

  8. DISTINCT REAL EIGENVALUES Clearly then If where and where else if Question 1. What is u if ? 8

  9. CRITICAL DAMPING Then and there exists a nonsingular (Jordan form) matrix E with therefore so u(t) is a linear combination of 9

  10. MULTIDIMENSIONAL SYSTEMS The most general are where all coefficient matrices are positive definite and M and K are symmetric (using Lagrange Equations). Hence the solution equals so u(t) is a linear combination of terms where the eigenvalues of B satisfy since 10

  11. DAMPENED WAVES The generalized equation of telegraphy is with p, q nonnegative. If with k real then If then we obtain a relatively undistorted wave 11

  12. TUTORIAL 5 • Derive the equation of motion for a falling particle if the force due to air resistance is –pv where v is its velocity. Then solve this equation. 2. Compute on p 7 from 3. Compute the matrices E on p 6 and E on page 9. 4. Show directly that satisfies if 5. Plot some solutions of the equation above for under, critically, and over damped systems. 12

More Related