Damped Simple Harmonic Motion

Damped Simple Harmonic Motion - Damped Simple Harmonic...

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Damped Simple Harmonic Motion Up to now we have discussed systems in which the force is proportional to the displacement, but pointed in an opposite direction. In these cases, the motion of the system can be described by simple harmonic motion. However, if we include the friction force, the motion will not be simple harmonic anymore. The system will still oscillate, but its amplitude will slowly decrease over time. Suppose the total force acting on a mass is not only proportional to its displacement, but also to its velocity. The total force can be represented in the following way In this formula, b is called the damping constant . Substituting the expression for the force in terms of the acceleration we obtain the following differential equation The general solution of this differential equation will have the form Substituting this expression in the differential equation we obtain This equation can be rewritten as and the solutions for [omega] are Substituting this in the expression for x(t) we obtain

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Damped Simple Harmonic Motion - Damped Simple Harmonic...

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