Most of us want to just instinctively squeeze a belt between
a pair of pulleys to test the belt tension. What is not as instinctive
is just how much force such a procedure can put
on the shaft—often significantly past the manufacturer’s
rated limits for small motors. This can cause damage to
both the shaft and the bearings.
First a diagram (Fig.1) and a little math. The force calculations
are not dependent on the diameter or spacing
of the pulleys, but let’s assume the pulleys at a spacing of
4inches (100mm) for other parts of the discussion where
we calculate deflection.

    For a 5-pound squeeze (22N) and a 5 degree angle (0.139
radians), the tangent is 0.132. (or
approximately the angle in radians,
if you still remember your
old-math, small-angle approximations).
This says the tangent force
from each side is 7.6× larger than
the applied squeeze, and both
sides contribute to the pull on the
shaft; so both shafts see 15.2× the
squeeze (half of which is counteracted
by the belt to the left and half
to the right)—or about 38pounds"
(170N) of compression. The 5
degrees deflection corresponds to
approximately 0.26inch (6.6mm)
of deflection on each side.
But you say I would not squeeze
that hard, more like one pound
(4.4N). So, let’s linearize and estimate
that the deflection would be
about 1
?5; so the angle would be only 1 degree. The deflection of each belt would then be about
.05" (1.3mm)—that should not be so bad.

    Well, the deflection is now .0262 radians, and Tan
(.0262)=.0262 (to 3 significant places), so the force multiplier
is now 1
?.0262 or 38.2×. When accounting for both sides
of the belt, this is now a multiplier of 38. The unexpected
result is that with this lighter force and the smaller deflection,
it is still putting almost the same enormous force
between the two shafts!

    While there was an assumption of linearity of deflection
to simplify the calculations—and this is not a perfect
assumption and the width of the force application is
also not infinitesimally small—but this order of magnitude
calculation (given the force and the deflection) is
still fairly close. This should cause one to pause to think
before you pinch!—especially if you are working with
small motors having ?inch or smaller shaft diameters.
A typical NEMA 17-frame motor specifies no more
than 6.2 lbs at .78inches (28N 20mm) from the flange,
while a NEMA 23 calls out 17pounds" at .78inches
(75N 20mm) from mounting flange. In both of these
cases, the 38pounds" (170N) would be well in excess
of the motor ratings, bending shafts and or damaging
bearings.

    A much better way to actually set a calibrated belt tension
is to provide a metal “sled” holding the motor to
which you can attach a fish scale or a known hanging
weight (around a pulley if you want a side force). Set
the tension, tighten the screws; belt tension is now set to a known and repeatable level. This gives your servo systems
more repeatable servo tuning, while not accidently damaging
shafts or bearings. An example drawing of such a motor
“sled” is shown at www.quicksilvercontrols.com/SP/TD/QCI-TD076_Belt_Tightening.pdf.

    
QuickSilver Controls produces compact, high-performance
integrated servo motors and controllers and provides
assistance in applying them to customer systems.
Disclaimer: As usual with free advice, do not depend on
these calculations for your critical designs. Perform these
calculations yourselves and/or make the appropriate
measurements. The author takes no responsibility for any
errors or omissions. The information is only provided for
your entertainment and insight into possible issues.