Damped Harmonic Oscillator Lecture Notes

Overview

• Spring mass system in an external medium can cause energy dissipation due to friction or drag.
• Displacement from equilibrium leads to oscillations that gradually decrease in amplitude over time.
• Ideal harmonic oscillators exhibit simple harmonic motion; damped systems exhibit damped harmonic motion.

Key Concepts

Damped Harmonic Oscillator

• Definition: A harmonic oscillator subject to an external medium causing drag.
• Forces Involved:
  • Restoring force from the spring:
    • Proportional to displacement, acts opposite to it (F = -kx).
  • Drag force from the medium:
    • Proportional to instantaneous velocity (F = -bv).

Equation of Motion

• Using Newton's second law:

  • Net force (F) = mass (m) * acceleration (a)

  • Leads to the equation:

    $$ mx'' + b x' + kx = 0 $$

  • Dividing by m gives:

    $$ x'' + \frac{b}{m} x' + \frac{k}{m} x = 0 $$

Substitutions

• Let:

  • ( \frac{b}{m} = 2\beta )
  • ( \frac{k}{m} = \omega_0^2 )

• Equation becomes:

  $$ x'' + 2\beta x' + \omega_0^2 x = 0 $$

General Solution

• Solutions of the equation take the form:

  $$ x(t) = a_1 e^{\alpha_1 t} + a_2 e^{\alpha_2 t} $$

• Where ( \alpha_1 ) and ( \alpha_2 ) are determined from the characteristic equation.

Scenarios of Damping

1. No Damping (ideal setup, β = 0)

   • Leads to simple harmonic oscillations:

   $$ x(t) = A \cos(\omega_0 t + \phi) $$

2. Underdamping (β² < ω₀²)

   • Oscillatory motion present with exponentially decaying amplitude:

   $$ x(t) = A e^{-\beta t} \cos(\omega_1 t + \phi) $$

   • ( \omega_1 = \sqrt{\omega_0^2 - \beta^2} ) (less than ( \omega_0 )).

3. Critical Damping (β² = ω₀²)

   • The system returns to equilibrium without oscillating:

   $$ x(t) = A t e^{-\beta t} $$

   • Fastest way to equilibrium without oscillations.

4. Overdamping (β² > ω₀²)

   • No oscillations; system returns to equilibrium slowly:

   $$ x(t) = A_1 e^{(-\beta + \sqrt{\beta^2 - \omega_0^2})t} + A_2 e^{(-\beta - \sqrt{\beta^2 - \omega_0^2})t} $$

Summary of Motion Types

• Without damping: Simple harmonic motion (sinusoidal variation).
• With slight damping: Oscillations with exponentially decaying amplitude (underdamping).
• Critical damping: Quickest return to equilibrium without oscillations.
• Overdamping: Slower return to equilibrium without oscillations.

Visual Representation

• Graphs depict the behavior of each damping scenario with respect to time.
• Amplitude behavior and return to equilibrium characterize each scenario.

Conclusion

• Damped harmonic oscillators demonstrate a variety of motions based on drag forces present in the medium.
• Analysis of damping effects is crucial for understanding real-world oscillatory systems.