Relative Motion to Absolute Motion

by ⋅ Aside

rel_mot_abs_mot_diagramJust a quick reminder about converting relative motion to absolute motion.  Say we already have the transfer function for input acceleration to relative motion:

sys_{rel} = \frac{\delta}{\ddot{x}_1} = \frac{x_2-x_1}{\ddot{x}_1}

The absolute motion of the second mass can be found as

\frac{x_{2}}{\ddot{x}_{1}} = \frac{\delta}{\ddot{x}_{1}} + \frac{1}{s^{2}}

The derivation is as follows:

sys_{rel} = \frac{\delta}{\ddot{x}_{1}}= \frac{x_{2}-x_{1}}{\ddot{x}_{1}}=\frac{x_2}{\ddot{x}_1}-\frac{x_1}{\ddot{x}_1}

\frac{x_2}{\ddot{x}_1} = \frac{\delta}{\ddot{x}_{1}}+\frac{x_1}{\ddot{x}_1}=\frac{\delta}{\ddot{x}_{1}}+\frac{1}{s^2}