Driven OscillatorIf a damped oscillator is driven by an external force, the solution to the motion equation has two parts, a transient part and a steadystate part, which must be used together to fit the physical boundary conditions of the problem.
In the underdamped case this solution takes the form The initial behavior of a damped, driven oscillator can be quite complex. The parameters in the above solution depend upon the initial conditions and the nature of the driving force, but deriving the detailed form is an involved algebra problem. The form of the parameters is shown below. Details of parameter evaluationExamples of driven oscillators

Index Periodic motion concepts  

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Underdamped Driven OscillatorThe expanded expressions for the underdamped oscillator in terms of the mass, spring constant, damping, and driving force. They are also affected by the initial values of position and velocity.

Index Periodic motion concepts Reference Barger & Olsson Marion & Thornton, Ch 3  

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Transient Solution, Driven OscillatorThe solution to the driven harmonic oscillator has a transient and a steadystate part. The transient solution is the solution to the homogeneous differential equation of motion which has been combined with the particular solution and forced to fit the physical boundary conditions of the problem at hand. The form of this transient solution is that of the undriven damped oscillator and as such can be underdamped, overdamped, or critically damped.Examples of driven oscillators 
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SteadyState Solution, Driven OscillatorThe solution to the driven harmonic oscillator has a transient and a steadystate part. The steadystate solution is the particular solution to the inhomogeneous differential equation of motion. It is determined by the driving force and is independent of the initial conditions of motion.Examples of driven oscillators 
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