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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.

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USSC3002 Oscillations and Waves Lecture 5 Dampened Oscillations

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  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

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