Damped Harmonic Motion
We can add a dissipation term to the equations of motion of a spring to account for sources of friction. This term resists a change in velocity of the block, while Hooke's term resists a displacement of equilibrium. Newton's law can be written as
where y is the displacement from the equilibrium position y0 (which we take as zero), k is Hooke's spring constant, and c is the damping coefficient.
The type of oscillation generated by this force can be classified into three types:
1. Overdamped, c > 2 mkm; the solution is a sum of decaying exponentials.
2. Critically damped, c = 2 mkm; the solution is the product of a linear term with a minimal-time decaying exponential.
3. Underdamped, c < 2 mkm; the solution is the product of an decaying exponential and a sinusoid.
It is often convenient to write these in terms of a damping factor γ=c2 m, and the natural frequency ω=km. For systems such as car suspensions with a given natural frequency, engineers aim to adjust the damping factor to achieve critical damping, thereby minimizing jolts from bumps in the road.
Applying Newton's second law to the force equation, we obtain
ma=−k y − cⅆyⅆt,
Since we know that acceleration is just the second derivative of position, we can write this as
mⅆ2yⅆt2 =−k y− cⅆyⅆt,
ⅆ2yⅆt2 +2⋅γdyⅆt+ω2 y = 0,
with the damping factor γ=c2 m, and the natural frequency ω=km. The solution to this differential equation can be expressed in one of three ways, depending on the sign of γ−ω :
Critically damped, γ = ω.
When the damping factor matches the natural frequency, the solution is the sum of decaying exponentials:
yt = At+ Be−ω t.
Critical damping is desirable for virtually all applications of oscillatory motion as the solution decays the quickest.
Overdamped, γ > ω.
The solution can be expressed as a sum of decaying exponential functions:
yt = Ae−γ−γ2−ω2t+ Be−γ+γ2−ω2t.
The larger the value of γ, the slower this solution will decay, due to the dominating exponential term Ae−γ−γ2−ω2t.
Underdamped, γ < ω.
In this case the motion is still oscillatory with a decaying amplitude. This is usually the least desirable solution for mechanical systems such as car suspension. The formal solution is
yt = (A sin(ω2−γ2 t + B cosω2−γ2 t) e−γt.
In this case, the smaller the value of γ, the slower this solution will decay.
In all cases, the constants A and B are determined from the initial conditions of the problem.
Try adjusting the damping factor and the natural frequency to see what happens to the motion of the block. See if you can minimize the decay time, that is, the time it takes for the oscillations to decay to 1% of their original amplitudes.
Damping Factor, γ
Natural Frequency, ω
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