# Driven Oscillator

If a damped oscillator is driven by an external force, the solution to the motion equation has two parts, a transient and a steady-state part, which must be used together to fit the physical boundary conditions of the problem.

### Examples of driven oscillators

 Transient solution Steady-state solution Expansion of above terms
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# Underdamped Driven Oscillator

The expanded expressions for the underdamped oscillator in terms of the mass, spring constant, damping, and driving force.

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# Transient Solution, Driven Oscillator

The solution to the driven harmonic oscillator has a transient and a steady-state 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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The solution to the driven harmonic oscillator has a transient and a steady-state part. The steady-state 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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# Motion Equation Notation

Motion equations for constant mass systems are based on Newton's 2nd Law, which can be expressed in terms of derivatives:

In many advanced mechanics texts, derivatives with respect to time are represented by a dot over the position variable which is being differentiated.

This makes it simpler to write equations where the forces are position or velocity dependent. For example, the damped oscillator has forces:

and the motion equation can be written

This notation is used for the damped oscillator and driven oscillator discussions.

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Periodic motion concepts

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