15.5 - Damped Simple Harmonic Motion

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Last updated 5:44 PM on 6/4/26
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Damped Harmonic Motion

Damped harmonic motion is a type of oscilaltory motion in which a dampening factor is introduced; this dampening factor results in the motion eventually ceasing after a set amount of time. The dampening factor can be due to drag, friction, air resistance, etc, but regardless of its source, the end result is always the same: motion must cease eventually.

This is why pendulums eventually come to a stop and do not swing forever; furthermore, this is why pendulums also reduce in the maximum magnitude of their angular displacement on each swing: drag reduces the maximum angular displacement per swing by some factor.

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Motion for Damped Harmonic Motion

The way that we determine the equation of motion for a damped harmonic oscillator is by the following:

  1. Realize that there is a drag force present that acts opposite of the direction of motion that is also proportional to the velocity of the object; imagine a mass-spring system in which the mass is attached to a vane in a jar of water. The vane moves downwards (as does the mass-spring system), but the drag from the water resists the downwards motion and pushes upwards. Regardless of the motion of the mass-spring system itself, the drag force will always be directed in the direction opposite of the motion. So we have Fd = -bv; where b is the dampening factor and v is the velocity of the mass-spring system and vane (since they are all a part of the same system after all).

  1. We also have the spring force that acts as a restoring force and thus opposes the direction of motion: Fs = -kx.

  1. Finally we sum the forces in the y direction to obtain Fy,net = -bv -kx. We then set this sum equal to ma (Newton’s 2nd Law) to obtain: -bv -kx = ma.

We omit gravity for simplicity’s sake.

We then realize that each of the terms can be written with respect to position so as to obtain a 2nd-order, linear, differential equation:

v(t) = dx(t)/dt

a = dx²(t)/d²(t)

Once we solve the 2nd-order differential equation, we obtain the solution as necessary.

<p>The way that we determine the equation of motion for a damped harmonic oscillator is by the following:</p><p></p><ol><li><p>Realize that there is a drag force present that acts opposite of the direction of motion that is also proportional to the velocity of the object; imagine a mass-spring system in which the mass is attached to a vane in a jar of water. The vane moves downwards (as does the mass-spring system), but the drag from the water resists the downwards motion and pushes upwards. Regardless of the motion of the mass-spring system itself, the drag force will always be directed in the direction opposite of the motion. So we have F<sub>d</sub> = -bv; where b is the dampening factor and v is the velocity of the mass-spring system and vane (since they are all a part of the same system after all). </p></li></ol><p></p><ol start="2"><li><p>We also have the spring force that acts as a restoring force and thus opposes the direction of motion: F<sub>s</sub> = -kx. </p></li></ol><p></p><ol start="3"><li><p>Finally we sum the forces in the y direction to obtain F<sub>y,net</sub> = -bv -kx. We then set this sum equal to ma (Newton’s 2nd Law) to obtain: -bv -kx = ma. </p></li></ol><p></p><p>We omit gravity for simplicity’s sake. </p><p></p><p>We then realize that each of the terms can be written with respect to position so as to obtain a 2nd-order, linear, differential equation: </p><p></p><p>v(t) = dx(t)/dt</p><p>a = dx²(t)/d²(t)</p><p></p><p>Once we solve the 2nd-order differential equation, we obtain the solution as necessary. </p><p></p>
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Differential Equation for Damped Harmonic Motion

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Mechanical Energy of a Damped Harmonic Oscillator

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