## Damped and Forced Vibrations

Free undamped, unforced vibrations in simple harmonic motion obey the equation or 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) gives We can divide by the none zero factor to give This is a quadratic equation in with solutions and There are three possibilities.:

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

The vibrations obey the equation (2)

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

The vibrations obey the equation (3)

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

The vibrations obey the equation (4)

If there is a forcing term then the equation (1) becomes (5)

The solution will consist of two parts and where is one of (2), (3) or (4) depending on the values of To find assume a solution of (5) of the form  and Substitute these into (5) to get Equate the coefficients of and  (6) (7)

We solve these equations simultaneously

From (7) (8)

Then from (6) Then from (8)   