Damped And Forced Oscillations, Resonance (Copy)

Definition and Nature of Damping

• Damping refers to the process by which the energy of an oscillating system is gradually lost to the surroundings
• It typically happens due to resistive forces, such as:
  • Air resistance
  • Friction
  • Viscous drag
• These forces oppose the motion of the oscillating system and remove energy from it
• As energy is lost, the amplitude of oscillation decreases over time

Types of Damping

1. Light Damping (Underdamping)

• Occurs when damping is present but relatively small
• System continues to oscillate but with gradually decreasing amplitude
• The motion is still oscillatory
• Energy is removed slowly

Displacement–Time Graph Characteristics:

• Graph shows sinusoidal oscillations
• Amplitude decreases exponentially over time

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

2. Critical Damping

• The system returns to equilibrium in the shortest possible time without oscillating
• It is the threshold between oscillatory and non-oscillatory motion
• Used in systems where rapid damping is desirable:
  • Car suspension systems
  • Door dampers

Displacement–Time Graph Characteristics:

• No oscillation is observed
• Displacement falls to zero quickly but smoothly

3. Heavy Damping (Overdamping)

• Damping is so large that the system returns to equilibrium very slowly
• There is no oscillation
• The resistive force is very large compared to the restoring force

Displacement–Time Graph Characteristics:

• A slow, non-oscillatory return to equilibrium
• Much slower than critically damped systems

Comparison Table: Types of Damping

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Definition and Nature of Forced Oscillations

• A forced oscillation occurs when a system is made to vibrate by a periodic external driving force
• The system is not oscillating freely but is responding to the external force
• The driving frequency is the frequency of the external force
• The amplitude of the oscillation depends on:
  • The natural frequency of the system
  • The frequency of the driving force
  • The amount of damping

Natural Frequency

• Every oscillating system has its own natural frequency, which is the frequency it oscillates at when not subjected to an external force
• Given by:
  • For spring-mass system: f = (1 / 2π) × √(k/m)
  • For a simple pendulum: f = (1 / 2π) × √(g/L)
• Resonance occurs when the driving frequency = natural frequency

Resonance

• Resonance is the phenomenon where a system subjected to a periodic driving force vibrates with maximum amplitude when the driving frequency equals the natural frequency
• At resonance:
  • Energy transfer is most efficient
  • Amplitude increases significantly
• Amplitude becomes limited only by the damping present

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Amplitude vs Frequency Graph (Resonance Curve)

• The graph shows how amplitude varies with the driving frequency
• Undamped System:
  • Sharp peak at resonance
  • Infinite amplitude theoretically
• Lightly Damped System:
  • High, sharp peak
  • Narrow bandwidth (sharp resonance)
• Heavily Damped System:
  • Low, broad peak
  • No sharp resonance

Phase Difference in Forced Oscillations

Graphical Summary

• Displacement vs Time for different damping levels:
  • Light damping: decreasing sinusoidal
  • Critical damping: non-oscillatory, sharp fall
  • Heavy damping: non-oscillatory, slow fall
• Amplitude vs Frequency:
  • Peak at resonance shifts and lowers with increasing damping
  • Width of resonance curve increases with damping

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Practical Examples of Resonance

Damping and Resonance Applications

• Desirable resonance:
  • Musical instruments (tuned for resonance)
  • Radios, clocks (need sharp resonance for accuracy)
• Undesirable resonance:
  • Bridges (e.g., Millennium Bridge in London)
  • Machinery (causes fatigue or failure)
  • Buildings during earthquakes
• Damping is used to:
  • Reduce unwanted oscillations
  • Prevent damage in structures
  • Increase system stability

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Key Equations and Concepts Recap

• Natural frequency (spring system):
  f = (1 / 2π) × √(k/m)
• Total energy in SHM:
  E = ½mω²x₀²
• Driving frequency = natural frequency ⇒ Resonance
• Damping reduces amplitude and shifts the resonance peak to a lower frequency