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 velocity
and in the opposite direction hence
The equation for damped vibrations is
or
(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 factor
to give
This is a quadratic equation in
with solutions
and
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 partsand
where
is 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)