1. Will Bergman and Mike Ma
Harmonic Oscillators
Will Bergman and Mike Ma

2. Overview Simple Harmonic Motion Driven Simple Harmonic Oscillators
Spring and Mass
Periodic Driving Force
General Form
Resonance
Damped Simple Harmonic Oscillators
Tacoma Bridge Example
Conclusion and Further Applications
Underdamped Case
Overdamped Case
Critically Damped Case

3. Spring and Mass 𝐹=𝑚𝑎 𝑚 𝑑 2 𝑥 𝑑 𝑡 2 =−𝑘𝑥 Guess: 𝑥 𝑡 = 𝑒 𝑟𝑡
𝑚 𝑑 2 𝑥 𝑑 𝑡 2 =−𝑘𝑥
Guess: 𝑥 𝑡 = 𝑒 𝑟𝑡
𝑟 1 =+𝑖 𝑘 𝑚 , 𝑟 2 =−𝑖 𝑘 𝑚
𝑥 𝑡 = 𝐶 1 𝑥 1 𝑡 + 𝐶 2 𝑥 2 𝑡
𝑥 𝑡 = 𝐶 1 𝑒 𝑟 1 𝑡 + 𝐶 2 𝑒 𝑟 1 𝑡
(Taylor, 2003)

4. General Form 𝑚 𝑑 2 𝑥 𝑑 𝑡 2 =−𝑘𝑥
𝑚 𝑑 2 𝑥 𝑑 𝑡 2 =−𝑘𝑥
𝑥 𝑡 = 𝐶 1 + 𝐶 2 𝑐𝑜𝑠 𝑤𝑡 +𝑖 𝐶 1 − 𝐶 2 𝑠𝑖𝑛 𝑤𝑡
𝑥 𝑡 =𝐵 1 𝑐𝑜𝑠 𝑤𝑡 + 𝐵 2 𝑠𝑖𝑛 𝑤𝑡
(Taylor, 2003)

5. Damped Simple Harmonic Oscillators
B – damping constant 𝑤 natural frequency
Relationship between B and 𝑤 0 determine different cases of damping
Solution form: 𝑥 𝑡 = 𝑒 𝑟𝑡

6. Underdamped Case (𝐵< 𝑤 0 )
𝐵 2 − 𝑤 =𝑖 𝑤 0 2 − 𝐵 2 =𝑖 𝑤 1
𝑥 𝑡 =𝑒 −𝐵𝑡 ( 𝐶 1 𝑒 𝑖 𝑤 1 𝑡 + 𝐶 2 𝑒 −𝑖 𝑤 1 𝑡 )
Amplitude of oscillations decrease exponentially
(Taylor, 2003)

7. Overdamped Case (𝐵> 𝑤 0 )
𝑥 𝑡 = 𝑒 −𝐵𝑡 ( 𝐶 1 𝑒 𝐵 2 − 𝑤 𝑡 + 𝐶 2 𝑒 − 𝐵 2 − 𝑤 𝑡 )
No Oscillations!
(Taylor, 2003)

8. Critically Damped Case (𝐵= 𝑤 0 )
Repeated Eigenvalues
𝐵= 𝑤 0 is a bifurcation value
𝑥 𝑡 = 𝑒 −𝐵𝑡
𝑥(𝑡)=𝑡𝑒 −𝐵𝑡
𝑥 𝑡 = 𝐶 1 𝑒 −𝐵𝑡 + 𝐶 2 𝑡𝑒 −𝐵𝑡
(Blanchard et al., 2012)
(Taylor, 2003)

9. Driven Simple Harmonic Oscillators
𝑑 2 𝑥 𝑑 𝑡 2 +𝐵 𝑑𝑥 𝑑𝑡 + 𝑤 0 2 𝑥=𝑓(𝑡)
Solution = general solution of homogeneous equation (unforced) + one particular solution to nonhomogeneous equation (forced)
𝑥 𝑡 = 𝐶 1 𝑥 1 𝑡 + 𝐶 2 𝑥 2 𝑡 + 𝑥 𝑝 (𝑡)
Resonance- the frequency of the driving force is equal to the natural frequency of the oscillating system

10. Conclusion and Further Applications of Theory

11. References
Taylor, John R. "Chapter 5: Oscillations." Classical Mechanics. Sausalito, CA: U Science, Print.
Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. "Chapter 2.3: The Damped Harmonic Oscillator." Differential Equations. Boston, MA: Brooks/Cole, Cengage Learning, Print.