Free undamped, unforced vibrations in simple harmonic motion obey the equationor Typically the vibration will be subject to a resistive term R, which for low speeds is proportional to velocityand in the opposite direction henceThe equation for damped vibrations isor(1)

We can solve this equation as with any constant coefficient second order linear differential equation by assuming that

Substitution of this into (1) givesWe can divide by the none zero factorto giveThis is a quadratic equation inwith solutionsand

There are three possibilities.:

Ifthen there is heavy damping and the vibration decays to zero without oscillation.

The vibrations obey the equation(2)

Ifthen there is light damping and the vibration oscillates while decaying to zero.

The vibrations obey the equation(3)

Ifthen there is critical damping and the vibration decays to zero without oscillation.

The vibrations obey the equation(4)

If there is a forcing termthen the equation (1) becomes

(5)

The solution will consist of two partsandwhereis one of (2), (3) or (4) depending on the values of

To findassume a solution of (5) of the form

and

Substitute these into (5) to get

Equate the coefficients ofand

(6)

(7)

We solve these equations simultaneously

From (7)(8)

Then from (6)

Then from (8)