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Tom1
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A driven harmonic oscillator is a physical system that undergoes periodic motion or oscillation due to the application of an external driving force or input. Examples of driven harmonic oscillators include a simple pendulum, a mass-spring system, or an LC circuit.
The equation of motion for a driven harmonic oscillator is given by:
m·x″ + k·x = F_{0} sin(ωt)
where m is the mass of the oscillator, k is the spring constant, x is the displacement from equilibrium, F_{0} is the amplitude of the driving force, and ω is the angular frequency of the driving force.
The driving force can either amplify or dampen the oscillation of a driven harmonic oscillator, depending on the frequency of the driving force. If the frequency of the driving force matches the natural frequency of the oscillator, resonance occurs and the amplitude of the oscillation increases. However, if the frequency of the driving force is significantly different from the natural frequency, the amplitude of the oscillation decreases due to damping.
The resonance frequency of a driven harmonic oscillator is equal to the natural frequency of the oscillator and is given by:
ω_{r} = √(k/m)
where k is the spring constant and m is the mass of the oscillator.
Damping is a force that opposes the motion of a driven harmonic oscillator and causes the oscillation to gradually decrease in amplitude. The amount of damping is determined by the damping coefficient, which can be either underdamped, critically damped, or overdamped. Underdamped oscillators continue to oscillate with decreasing amplitude, critically damped oscillators return to equilibrium the fastest, and overdamped oscillators take the longest to return to equilibrium.