Forced oscillator

A ratio of response amplitude to imposed excitation amplitude in a vibration problem. One must specify the input and output: displacement transmissibility and force transmissibility are different measurements. For a base-excited oscillator whose spring and viscous damper both act relative to its support, complex displacement amplitudes obey $X/Y=(k+i\gamma \omega )/(k-m{\omega }^2+i\gamma \omega )$. Its magnitude is the displacement transmissibility. A damper fixed to the laboratory rather than the support has a different numerator, so the attachment geometry matters.

A base-excited oscillator has a moving support rather than a directly applied periodic force. If its damping depends on velocity relative to the support, $z=x-y$ obeys $m\ddot{z}+b\dot{z}+kz=-m\ddot{y}$. A model with damping against absolute velocity instead obeys $m\ddot{x}+b\dot{x}+kx=ky$ after subtracting its static equilibrium. These conventions give different transfer functions.

For a damped linear oscillator driven periodically, the steady-state response is the periodic particular solution left after homogeneous transients decay.