In some circumstances, we will be interested in signals of very different magnitudes. Generally, all that is required to do so is to use appropriate units. But, we may also want to scale the signal by a factor, $S$.

$$ x_2=Sx_1\Rarr\dot x_2=S\dot x_1\Rarr\ddot x_2=S\ddot x_1 $$

If we are interested in signals with very different rates of dynamics, we may scale the time signal.

$$ t_2=\omega_St_1 $$