Free Harmonic Oscillations 

Machines with rotating components commonly involve mass-spring systems or their equivalents in which the driving force is simple harmonic. The motion of a mass attached to a spring serves as a simple example of vibrations that occur in more complex mechanical systems.

From a teaching point of view it is suitable to consider a body of mass m attached to one end of a spring that resists compression as well as stretching. A rod attached to the mass carries a disk moving in an oil-filled cylinder (a dashpot). The other end of the spring could be attached to a fixed wall and vibrating horizontally. The resultant force on the body is generally the sum of the restoring force and the damping force. is the force constant, is the distance of the body of mass from its equilibrium position, t is time and and r is the damping constant. We take when the spring is stretched. The differential equation of motion is therefore:
 

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If we set in deq1 the motion is undamped. Otherwise the solution of deq1 presents three distinct cases of damping according to whether is greater than, equal to, or less than zero. 

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Let us replace m with a new variable mv in deq1 . 

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With this differential equation, deq2 , the system is overdamped if , critically damped if and underdamped if  

If we request Maple to solve deq1 for the undamped case and deq2 for each of the damped cases,subject to initial condition . 

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we get: