ENCODER DUPLEX BEARING RADIAL LOAD CALCULATIONS

Copyright (c) 2013 Donald E. Barnett

All rights reserved.

Ball bearing rating life for radial contact bearings is calculated per ANSI standard and AFBMA standard 9-1978. It is known as the L10 rating and is the life rating of apparently identical bearings in millions of revolutions. For a single bearing, it is the life associated with 90 percent reliability. The calculations that follow are outlined in Machinery's Handbook 25.

The encoder under study uses a R6 series bearing (52100 chrome steel) of standard construction. The bore is 0.375 inches, .0005 inch maximum radial play, .0007 inch maximum axial play, and having 13 degrees maximum contact angle.

D, ball diameter, inches

a, nominal contact angle, degrees

dm, mean pitch diameter of balls

fq, quality factor rating

For a unit consisting of two similar, single row, radial contact ball bearings, in a duplex mounting, the pair is considered as one, double row, radial contact ball bearing. i equals 2 in such a case and fc is from the duplex bearing column in table 25.

fc, quality factor per Machinery's Handbook 25, page 2204, table 25 is based on factor fq.

Z, number of bearing balls in a row

i, number of rows of balls in the bearing

C, basic load rating, pounds

Encoder Unit Under Study (UUS):

Given axial load of 40 pounds and a radial load of 35 pounds, we can calculate the L10 life rating.

Fa, applied axial load in pounds

fl, ratio of axial load to static load

Fr, applied radial load in pounds

fr, ratio per specification

ratio to compare with e if finding X and Y per table 26

Therefore, per table 26, if fr is greater than e, then:

X, radial load factor

Y, axial load factor

P, magnitude of equivalent radial and axial loads

L10, ball bearing life in millions of revolutions

Benchmark encoder (BMK):

BEI H25, axial load 40 pounds, radial load 35 pounds, published L10 rating 2 x 108 revolutions.

Assuming the load factors are the same, we can calculate the basic bearing load rating (C) for the benchmark encoder.

L10, ball bearing life in millions of revolutions

C, basic load rating, pounds

Using the benchmark value for L10 and the unit under study value for C, we can calculate the improvement in load for the conditions in P.

P, magnitude of equivalent radial and axial loads

Fr, applied radial load in pounds

Therefore, given the same bearing life and axial load as the benchmark, the unit under study having a greater value for C has a greater radial load specification.

Since the maximum load condition specification is a combination of the axial and radial loads, a graph more appropiately describes the bearing load specification.

L10, ball bearing life in millions of revolutions

Fa, applied axial load in pounds

Fr, Unit Under Study applied radial load in pounds

P, magnitude of equivalent radial and axial loads

Fr, Benchmark applied radial load in pounds

Next we consider the unit shaft loading in certain applications. These applications fall into two basic classifications. In the first, the unit shaft is attached to a coupling which is then attached to a drive shaft. In the second case, the unit shaft is driven though an attached wheel, pulley or gear. When the unit shaft is coupled to another shaft, the loads can be minimized by careful alignment of the two shafts. Maximum loads are not likely to be present and in most of these applications, a light duty bearing will serve. However, in applications that join the encoder to the mechanical system through wheels, pulleys, gears, etc., the radial bearing load in likely to be significant. The axial shaft load is not much more than the bearing preload plus some axial vector due to misalignment or wear.

Shaft encoders generally use two like bearings to position the encoder shaft. The design can be modeled as a lever to determine true radial bearing loads. A force applied to the end of the encoder shaft is distributed by the two shaft bearings porportional to the distances between the forces. Because shaft lengths can and do vary, we will use a standard distance of 0.5 inches from the face of the bearing to the applied radial load.

Another variable is the distance between the bearings. Larger distances between bearings result in better distribution of the shaft load.

Fa, axial bearing load in pounds

FrA, radial load on near bearing in pounds

A, distance between shaft load and first bearing race in inches

B, nominal distance between bearing races in inches

Fs, radial shaft load in pounds

As the graph above illustrates, the bearing load is much less in the UUS encoder due to the greater distance between bearings.

Further, if we hold the near bearing radial load to a specified 35 pounds and move the shaft load out to a greater distance along the shaft, we see that the allowable shaft load increases while maintaining the same bearing life.

A, distance between shaft load and first bearing race in inches

Fs, radial shaft load in pounds

Fs, radial shaft load in pounds

Let us now compare the two designs in a given application to see just how dramatic the increase in bearing life can be.

Fs, radial shaft load in pounds

P, magnitude of equivalent radial and axial loads

L10, ball bearing life in millions of revolutions

Conclusion

The design of the Unit Under Study is superior to the Benchmark encoder. Its specification sheet should outline the vast difference in allowable radial load and longevity.