Lecture on Damped Harmonic Oscillator

Introduction

• Speaker: Mayuri Singh
• Channel: Let's Study India
• Topic: Damped Harmonic Oscillator

Simple Harmonic Oscillator

• Definition: A particle at its mean position displaced slightly will exhibit to and fro motion known as simple harmonic motion (SHM).
• Ideal Case: In SHM, the amplitude remains constant over time.
  • Known as free oscillations.
  • Occurs when there is no energy loss; total energy remains constant.

Free Oscillations

• Amplitude is constant.
• Occurs when no energy loss in the system.
• The system's energy is constant during free oscillations.

Damped Harmonic Oscillator

• Actual Case: Amplitude changes and decreases over time.
  • Occurs due to frictional forces that oppose motion.
  • Causes mechanical energy loss, reducing amplitude.
• Known as damped oscillations.
  • Amplitude decreases over time until the particle comes to rest.

Forces in Damped Harmonic Oscillator

• Two forces acting on the particle:
  1. Restoring Force:
     • Formula: ( F = -ky )
     • Restores particle to equilibrium.
     • ( k ): force constant, ( y ): displacement.
  2. Frictional Force:
     • Proportional to velocity: ( F = -Rv )
     • ( R ): damping constant.
     • Can be expressed as: ( F = -R \frac{dy}{dt} )

Total Force on Particle

• Total force is the sum of restoring and frictional forces.
• Equation: [ m \frac{d^2y}{dt^2} + R \frac{dy}{dt} + ky = 0 ]
• Substitutions:
  • ( 2b = \frac{R}{m} )
  • ( \omega^2 = \frac{k}{m} )

Differential Equation

• Differential equation for damped harmonic oscillator: [ \frac{d^2y}{dt^2} + 2b \frac{dy}{dt} + \omega^2 y = 0 ]

Solution to Differential Equation

• Assume solution ( y = Ae^{\alpha t} )
• Derivatives:
  • First: ( \frac{dy}{dt} = \alpha Ae^{\alpha t} )
  • Second: ( \frac{d^2y}{dt^2} = \alpha^2 Ae^{\alpha t} )
• Substitute into differential equation to find roots:
  • ( \alpha^2 + 2b\alpha + \omega^2 = 0 )
  • Roots: ( \alpha_1 = -b + \sqrt{b^2 - \omega^2} ), ( \alpha_2 = -b - \sqrt{b^2 - \omega^2} )

General Solution

• General solution of the equation: [ y = A_1e^{\alpha_1 t} + A_2e^{\alpha_2 t} ]

Cases Based on Damping

• Heavy Damping (Overdamped): ( b > \omega )
• Critical Damping: ( b \approx \omega )
• Underdamped: ( b < \omega )

Conclusion

• Discussed the differential equation and solution.
• Next lecture will focus on the three cases of damping.
• Key Takeaways:
  • Understanding forces and energy loss in damped oscillations.
  • Differential equation formulation and solving for damped harmonic oscillators.