The observed phase of the rotation in stopped image, i.e. the rotational angle at the moment of the flash,
is purely accidental. Through brief changes in of the flash frequency however the desired phase position
can be adjusted approximately. In the same way, RPM fluctuations can cause a change in phase position.
Exact phase stability, i.e. sharply stopped image, can be achieved when the flash frequency is controlled
externally by the moving object (does not apply for MINISTROB series 2000).
6.2 Measurement of RPM and frequencies
To measure the RPM v either the highest flash frequency f1 = v which results in a stopped image of the object
can be determined, or two neighbouring flash frequencies fn and fn+1 can be determined and from these
the rotational frequency computed. For the periods for f and fn+1 in the flash frequency the equation is:
r = Tn+1 - Tn
From this we derive the frequencies:
v = fn • fn+1 ÷ fn - fn+1
6.3 Slow-motion cycle
If the period T of the flash frequency deviates slightly from a whole-number multiple Tn = nr of the rotation
time r of the object, i.e.
T = (n + e) r with /e/ < 1
then the object no longer appears stopped, but has rotated through the angle 2e between two succeeding flashes.
If /e/ is sufficiently small the eye perceives a constant slow-motion cycle. Angular speed w’, at which the object
appears to rotate, is given by:
w’ = 2 v’ = 2 π e = 2 π e 2 π e
T (n+e) r nr
If we compare this with the true angular speed of the object, we obtain:
w’ = (e ÷ n ) • w
For e > 0 (i.e. T > Tn and/or f > fn) w and w’ have the same sign, so that true and apparent rotation are
in the same direction.
The opposite holds for e < 0. With increasing /e/ the angular speed w’ of the apparent rotation rises.
Finally the angle 2πe becomes so large that the mark on the rotating disc appears at two different places
during two succeeding flashes. Other phenomena (described below) also occur.
6.4 Stopped images of phantom objects
Stopped images of rotating objects results from flash frequency periods Tn = nr, and also at other flash frequencies.
However, the latter represent phantom objects, not the real object. Using the example of the rotating disc with an
eccentric mark , it is obvious that stopped images also occur when:
T = ( n ÷ k ) r and / or f = ( k ÷ n ) v,
whereby n and k are whole relatively-prime numbers. The stopped image shows k marks, which are arranged in the
corner of a regular k-angle. Only a very few of the theoretically infinite number of flash frequencies result in observable
images, since at each corner of the k-angle there is only one mark for k sequential flashes, but (k - 1) times no marks.
As k increases then the images have less and less contrast. The images of the real object (k =1) always appear sharpest.
In addition, the images become more and more faint at a given k with increasing n. The interval in which the mark
is illuminated at one corner of the k-angle amounts to n rotation periods. In conclusion , the k mark images must not
overlap. Altogether we may expect observable images only with low values of n and k.
In objects with a complicated texture the phantom objects mostly disappear in an untextured background.