1. Generalities on Vibrations

  1. A vibration is an oscillatory motion around an equilibrium position.
  2. The equilibrium position is the reference position of the vibrating system.
  3. Vibrations can be free, forced, damped, or undamped.
  4. A free vibration occurs when a system oscillates without continuous external excitation.
  5. A forced vibration results from an external periodic excitation.
  6. Damping causes the vibration amplitude to decrease with time.
  7. An undamped system can theoretically vibrate indefinitely.
  8. The amplitude represents the maximum displacement from equilibrium.
  9. The period is the time required to complete one oscillation.
  10. The frequency is the number of oscillations completed per unit time.
  11. Frequency and period are related by .
  12. The angular frequency is given by .
  13. Simple harmonic motion can be described using sine or cosine functions.
  14. The restoring force acts toward the equilibrium position.
  15. A stable equilibrium corresponds to a minimum of the potential energy.
  16. For stable equilibrium, and .
  17. The kinetic energy depends on the velocity of the vibrating body.
  18. The potential energy depends on its displacement from equilibrium.
  19. In an ideal undamped oscillator, the total mechanical energy remains constant.
  20. The study of vibrations provides the basis for understanding waves, resonance, damping, and oscillatory systems.