When showing graphs displaying angular velocities and having associated discussions in terms of acceleration it is quite possible that people are mixing the two and perhaps thinking that when seeing a peak in velocity assuming that this implies also maximum acceleration. This is not quite true and is illustrated in Fig 1.

When position is measured and velocity and acceleration are subsequently derived through differentiation, there is an undesirable side effect, i.e., high frequency noise signals, frequently present, are being amplified. This is because differentiation accentuates signals with rapid changes. This is illustrated schematically in Fig2.

In Fig2a the noise signal is barely noticeable in the position signal. Differentiation to obtain the velocity, Fig2b,  the noise signal is much more prominent. Subsequent differentiation to get the acceleration the noise signal dominates the acceleration signal, Fig2c.

Low pass filtering is usually incorporated to attenuate as much as possible undesirable noise components. However this is usually a compromise since smoothing too much results in loosing also useful features of the desired signal.

If signal and noise have basically similar frequency components one can’t do much to eliminate the noise nuisance. However acquiring the same signal several times and averaging reduces the noise, the signals being coherent and the noise being random like.

So what is actually differentiation?  This process gives an indication proportional to the slope of a signal, i.e., its rate of change. If the slope is zero  differentiation produces a zero amplitude signal and largest signal when the slope is maximum.

Some examples of signals and their differentiated counter parts are given in Fig3 and Fig4.  At impact various angular velocities are peaking or close; the associated accelerations/torques are hence small.

Mandrin