Lecture 8: Damped and Forced Oscillations

Damped Oscillations

• Example of the damping is action of the friction force proportional to velocity F = - b v_x . Friction coefficient b has dimensionality kg/s and is positive, with friction working against the motion.
• The equation of motion is
• The mechanical energy is not conserved, changing at the rate
• leading to a loss of mechanical energy

Weak damping

• We are in the regime of weak damping if b^2 < 4 k m
• In this case damped oscillator is described by

Strong damping

• Strong damping occurs when b^2 > 4 k m
• In this case damped oscillator is described by
• Interesting feature: strongly damped oscillator cannot pass equilibrium point more than once

Critical damping

• Critical damping occurs when the friction is exactly equal critical value b^2 = 4 k m

Forced Oscillations, Resonance

• Consider an oscillator driven from equilibrium by the action of the periodic force F = F_max cos(&omega_d t )
• The amplitude the oscillator reaches depends on the
  • Amplitude of the driving force F = F_max
  • Frequency of the driving force &omega_d
  • Natural frequency of the oscillator &omega
  • damping b
• The amplitude is highest when the frequency of the driving force is equal to natural frequency of the oscillator, i.e when the force is in resonance
  &omega_d = &omega
• In resonance A_res = F_max/(b &omega_d) Resonance amplitude is formally infinite if there is no damping ! (well in real system it may reach large values before saturation by other effects)