Applying Newton's second law to the force equation, we obtain
Since we know that acceleration is just the second derivative of position, we can write this as
or
with the damping factor and the natural frequency The solution to this differential equation can be expressed in one of three ways, depending on the sign of :
1.
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Critically damped, γ = ω.
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When the damping factor matches the natural frequency, the solution is the sum of decaying exponentials:
Critical damping is desirable for virtually all applications of oscillatory motion as the solution decays the quickest.
The solution can be expressed as a sum of decaying exponential functions:
The larger the value of , the slower this solution will decay, due to the dominating exponential term .
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
In this case, the smaller the value of , the slower this solution will decay.
In all cases, the constants A and B are determined from the initial conditions of the problem.