Damped harmonic motion is all about how oscillations change over time due to resistance. It's like pushing a kid on a swing - eventually, they slow down because of air resistance and friction.

There are three types of damping: underdamped (slow decay), overdamped (no oscillation), and critically damped (fastest return to equilibrium). Damping affects amplitude but not frequency, and causes energy loss through non-conservative forces like friction.

Damped Harmonic Motion

Types of damped oscillations

• Underdamped systems
  • Damping force is less than the critical damping force causes the system to oscillate with gradually decreasing amplitude over time (lightly damped pendulum)
• Overdamped systems
  • Damping force is greater than the critical damping force prevents oscillation and returns the system to equilibrium position without overshooting (heavily damped door closer)
• Critically damped systems
  • Damping force equals the critical damping force allowing the system to return to equilibrium position in the shortest possible time without oscillating
  • Represents the boundary between underdamped and overdamped systems (carefully designed car suspension system)

Effects of damping on motion

• Period and frequency
  • Damping forces do not significantly affect the period or frequency of oscillation which are primarily determined by the system's mass and spring constant
• Amplitude
  • Damping forces cause the amplitude of oscillation to decrease over time at a rate dependent on the strength of the damping force
    • Stronger damping leads to faster amplitude decay
  • In underdamped systems, the amplitude decreases exponentially with each oscillation
  • In overdamped systems, the amplitude decreases rapidly without oscillation
  • In critically damped systems, the amplitude decreases to zero in the shortest possible time without oscillation
• The damping ratio is a measure of how quickly oscillations decay in a system

Energy loss in damped systems

• Non-conservative forces (friction or air resistance) dissipate energy from the system by converting mechanical energy into other forms (heat or sound)
  • The work done by non-conservative forces is path-dependent and not recoverable
• In damped harmonic systems, non-conservative forces cause the total energy of the system to decrease over time at a rate dependent on the strength of the damping force
• As energy is removed from the system, the amplitude of oscillation decreases
  1. In underdamped systems, the energy is gradually dissipated over multiple oscillations
  2. In overdamped systems, the energy is rapidly dissipated without oscillation
  3. In critically damped systems, the energy is dissipated in the shortest possible time without oscillation

Forced and Free Oscillations

• Free oscillations occur when a system is displaced from equilibrium and allowed to oscillate without external forces
• Forced oscillations happen when an external periodic force is applied to the system
  • The system's response depends on the frequency of the applied force
  • When the driving frequency matches the system's natural frequency, resonance occurs
• The quality factor is a dimensionless parameter that describes how under-damped an oscillator is
• The resonance frequency is the natural frequency at which a system oscillates with maximum amplitude