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 2 1 degree of freedom (DOF) systems Mechanics 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 ? Electronics

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

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

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

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

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

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

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

10. MULTIDIMENSIONAL SYSTEMS 10 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

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

12. TUTORIAL 5 1.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. 12 2. Computeon p 7 from 3. Compute the matrices E on p 6 and E on page 9. 4. Show directly thatsatisfies if 5. Plot some solutions of the equation above for under, critically, and over damped systems.