In this presentation, you will get the detailed information and some application about Damped harmonic oscillator.

1. Damped Harmonic
Oscillator
Kamran Ansari02–01-2018

2. Content
• Simple harmonic oscillator (SHM),
• Damped harmonic oscillator
• Over damped,
• Under damped,
• Critical damped
• Application of damped oscillator.

3. Simple harmonic oscillator
The spring exerts a restoring force for small
displacement x,
𝑭 = −𝑘𝒙
The equation of motion,
𝑥 +
𝑘
𝑚
𝑥 = 0
and the solution is,
𝑥 = 𝐴 𝑐𝑜𝑠(𝜔0 𝑡 + 𝜑)
𝜔0 = natural frequency =
𝑘
𝑚
𝜑 = phase difference

4. Damped harmonic oscillator
The spring’s restoring force and
frictional force exert opposite
force,
𝑭 = −𝑘𝒙 − 𝑏 𝒙
𝑚 𝒙 = −𝑘𝒙 − 𝑏 𝒙
The equation of motion,
𝒙 + 𝛾 𝑥 + 𝜔0
2
𝒙 = 0
γ = damping coefficient =
𝑏
𝑚
−𝒃 𝒙

5. Solution of the equation,
𝑥 𝑡 = 𝑐1 𝑒
−
𝛾
2
+
𝛾
2
2
−𝜔0
2 𝑡
+ 𝑐2 𝑒
−
𝛾
2
−
𝛾
2
2
−𝜔0
2 𝑡
(a) Over damped,
𝜸
𝟐
> 𝝎 𝟎
The system comes to its equilibrium state without oscillating,
𝒙
𝒙 𝟎

6. (b) Under damped,
𝜸
𝟐
< 𝝎 𝟎
The system comes to tis equilibrium state after some oscillations
𝑥 𝑡 = 𝑥0 𝑒−
𝛾
2
𝑡
cos(𝜔′ 𝑡 + 𝜑) 𝑤ℎ𝑒𝑟𝑒 𝜔′
= 𝜔0
2 −
𝛾
2
2
𝑒−
𝛾
2 𝑡

7. (c) Critically damped,
𝜸
𝟐
= 𝝎 𝟎
The system comes to its equilibrium state most quickly,
𝑥 𝑡 = (𝑐1 + 𝑐2 𝑡)𝑒−
𝛾
2 𝑡
𝒙
𝒙 𝟎

9. Application of Damped harmonic
oscillator
• Car suspension system

11. Thank you