1. USSC3002 Oscillations and Waves Lecture 6 Forced 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. FORCED OSCILLATIONS 2 1 degree of freedom (DOF) systems Mechanics Question 1. What do F and E model ? Question 2. What happens to energy ? Question 3. Is u determined by F, E ? Question 4. Can they describe > 1 DOF systems ? Electronics

3. ENERGY 3 The mechanical equation so energy can increase, decrease depending on F. where can be rewritten as

4. GENERAL SOLUTION 4 Definingand gives hence

5. GREEN’s FUNCTION AND STEADY STATE 5 Therefore, the solution u satisfies where Question 1. What is the form of G ? For F bounded on the steady state solution is and where the Green’s Function G is defined by is a solution of the homogeneous eqn

6. GREEN’s FUNCTION 6 Since and ( from andDirac’s Delta ‘function’. ) Therefore

7. GREEN’s FUNCTION 7 Theorem Ifthen ProofCompute the indefinite integral twice so G is continuous atand the result follows since

8. GREEN’s FUNCTION FOR DISTINCT EIGENVALUES 8 Sinceand Complex Roots Real Roots let

9. GREEN’s FUNCTION FOR CRITICAL DAMPENING 9 Then hence and

10. SINUSOIDAL RESPONSE 10 and observe that solves defineWe choose iff is maximized by choosing resonant frequency to be the Define (>.5 iff u. c. d.)

11. TRANSIENT RESPONSE 11 If then so doeswhereis any solution of the homogeneous equation. Therefore, we can obtain the solution for any nonhomogeneous initial value problem for sinusoidal F as the sum of a sinusoidal solution and a transient solution of the homogeneous initial value problem given by solves

12. INHOMOGENEOUS WAVE EQUATION 12 The inhomogeneous initial value problem for u admits the solution Proof. Pages 23-24 in Coulson and Jeffrey.

13. TUTORIAL 6 1.Show that the Green’s function for any damped harmonic oscillator (that we derived in these notes) satisfies G(t)  0 for large t 13 2. Show the equivalence on the bottom of page 6. 3. Use the Greens functions to derive the steady state solution to the nonhomogeneous oscillation equation with complex sinusoidal forcing term. 4. Outline an approach to solve the general multidim. nonhomogeneous oscillation equation. 5. Verify that the solution on page 11 satisfies the initial values and the nonhomogeneous equation. You do not need to use Green’s Theorem !