Damped and Forced Vibrations
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
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
Substitute these into (5) to get
Equate the coefficients ofand
We solve these equations simultaneously
Then from (6)
Then from (8)