page_title: Harmonic oscillator
section_title: Damped harmonic oscillator
hierarchical_section_title: Harmonic oscillator / Damped harmonic oscillator
caption_attribution_description: English: Step response for two-pole feedback amplifier Русский: Отклик линейного осциллятора с затуханием на ступенчатое возмущение
context_page_description: In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:
  where k is a positive constant.
  If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency.
  If a frictional force proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
  Oscillate with a frequency lower than in the undamped case, and an amplitude decreasing with time.
  Decay to the equilibrium position, without oscillations.
  The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called critically damped.
  If an external time-dependent force is present, the harmonic oscillator is described as a driven oscillator.
  Mechanical examples include pendulums, masses connected to springs, and acoustical systems.
context_section_description: In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases in proportion to the acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic oscillator there is in addition a frictional force which is always in a direction to oppose the motion. In many vibrating systems the frictional force Ff can be modeled as being proportional to the velocity v of the object: Ff = −cv, where c is called the viscous damping coefficient.
  The balance of forces (Newton's second law) for damped harmonic oscillators is then
  which can be rewritten into the form
  where
   is called the "undamped angular frequency of the oscillator",
   is called the "damping ratio".
  The value of the damping ratio ζ critically determines the behavior of the system. A damped harmonic oscillator can be:
  Overdamped (ζ > 1): The system returns (exponentially decays) to steady state without oscillating.  Larger values of the damping ratio ζ return to equilibrium more slowly.
  Critically damped (ζ = 1):  The system returns to steady state as quickly as possible without oscillating (although overshoot can occur).  This is often desired for the damping of systems such as doors.
  Underdamped (ζ < 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. The angular frequency of the underdamped harmonic oscillator is given by  the exponential decay of the underdamped harmonic oscillator is given by 
  The Q factor of a damped oscillator is defined as 
  Q is related to the damping ratio by the equation
caption: Step response of a damped harmonic oscillator; curves are plotted for three values of μ = ω1 = ω0√1 − ζ2. Time is in units of the decay time τ = 1/(ζω0).
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