1. 2. System with One Degree of Freedom


  1. A one-degree-of-freedom (1-DOF) system requires only one independent coordinate to describe its motion.
  2. A typical 1-DOF mechanical oscillator consists of a mass, spring, and damping element.
  3. The equation of motion (EOM) describes the dynamic behavior of the system.
  4. For a mass–spring system without damping or external excitation, the equation is .
  5. Free motion occurs when the system oscillates without continuous external excitation.
  6. The natural angular frequency of an undamped mass–spring system is .
  7. The natural frequency depends on the mass and stiffness of the system.
  8. Damped motion occurs when a resistive force opposes the motion and dissipates mechanical energy.
  9. For viscous damping, the damping force is proportional to velocity: .
  10. The equation of motion of a damped free oscillator is .
  11. The damping ratio is .
  12. A system is underdamped when ; it oscillates with a decreasing amplitude.
  13. A system is critically damped when ; it returns to equilibrium without oscillating and as quickly as possible.
  14. A system is overdamped when ; it returns to equilibrium without oscillating more slowly than a critically damped system.
  15. Damping causes mechanical energy to be dissipated, usually in the form of heat.
  16. In an undamped system, the total mechanical energy remains constant.
  17. Forced oscillation occurs when a system is subjected to an external periodic force.
  18. The steady-state response of a forced oscillator depends on the excitation frequency, damping, mass, and stiffness.
  19. The amplitude response describes how the steady-state amplitude varies with the excitation frequency.
  20. Resonance occurs when the excitation frequency approaches the system's natural frequency, potentially producing a large vibration amplitude.
  21. Increasing the damping generally reduces the maximum amplitude near resonance.
  22. The study of 1-DOF systems provides the fundamental basis for understanding vibration control, resonance, damping, and more complex mechanical systems.