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Oscillations

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**1. **1 Oscillations SHM review
Analogy with simple pendulum
SHM using differential equations
Auxiliary Equation
Complex solutions
Forcing a real solution
The damped harmonic oscillator
Equation of motion
Auxiliary equation
Three damping cases
Under damping
General solution
Over damping
General solution
Solution in terms of initial conditions
Critical Damping
Break down of auxiliary equation method and how to fix it
General solution
Solution in terms of initial conditions
Over damping as ideal damping
Phase diagrams
Un-damped phase diagram
Obtaining phase equations directly
The under-damped logarithmic spiral
Critical damping example
Harmonic oscillations in two dimensions
Lissajous figures

**2. **2 What will we do in this chapter?

**3. **3 Simple harmonic motion

**4. **4 SHM solution by DE methods

**5. **5 DE methods (continued)

**6. **6 Damping cases

**7. **7 An under-damped case

**8. **8 Over damped case

**9. **9 Critical damping

**10. **10 Example of Critical Damping

**11. **11 Phase Diagram

**12. **12 Phase Diagrams (continued)

**13. **13 Critically damped phase diagram

**14. **14 Harmonic motion in two directions

**15. **15 Lissajou figure with unequal frequency

**16. **16 A cool trig identity

**17. **17 Lissajous figures with different frequencies

**18. **18 Lissajous with irrational frequency ratios