Cylinder Tolerances In Perspective

by Mark Drzewiecki

To completely specify the geometry of a tube or cylinder, three types of geometric specifications are needed. These three specifications; size, roundness, and straightness, are basic factors in the cost and performance of a finished piece. Cylinder size tolerances and roundness are generally specified since their effect on performance and fabrication costs are easily visualized. In fact, in most cases the specification methods are standard as shown in Figures 1 and 2.

However, straightness specifications are far from being standard and are often completely omitted. But, when straightness is called for, the specification method chosen can greatly affect the cost and performance.

There are three methods commonly used to define straightness. The first is straightness per unit length. This is the least costly, but the total effect of this type of tolerance is not easily visualized. Figure 3 shows the worst case for a 4" long cylinder with a tolerance of .001" per inch. This condition of "bow" can have a cumulative effect. On a 48" long cylinder, it amounts to a bow of over 2".

The second method is overall straightness (figure 4). Although the effect on performance is more easily seen, over specification can result in much higher production costs. The third method (Figure 5) is simply a combination of specifying straightness per unit length and overall limit.

The degree of difficulty of reliable inspection is a major reason for variations in production costs. Inspection techniques become extremely critical when the necessary overall straightness approaches .0001". In such cases, even the methods of work piece support for inspection can affect measurements.

Examination of elastic deformation equations for a horizontally supported bar show that the position of the supports can significantly alter the deformation of a "perfectly" straight bar due to its own weight.

For a bar supported between centers or at its ends, as in Figure 6, the deflection at the center is given by the following equation:
ê = (5WL4) / (384EI)
Where:
W= Weight per Inch Length
E= Elastic Modulus
I= Moment of Inertia of Cross Section

For example, a solid steel cylinder 5" diameter, 60" long, has a deflection at the center of .001".

For a bar supported at two points a distance from the ends of .223 times the length, as in Figure 7, the deflection at the center is minimized and equal to the deflection at the ends. These particular points are commonly referred to as the "airy" points or points of minimum deflection, and the deflection is given by the following equation:
ê = (2WL4) / (7431EI)

The same 5" diameter, 60" long bar of the previous example has a deflection of only .000021" when it is supported on its airy points. Stated as ratio of deflections, the same bar supported at the ends will bend 50 times as much as when it is supported at its airy points.

A chart can be constructed to indicate when these deflections become critical. Figure 8 is such a cart using .0001" as a boundary criteria. Similar charts can be constructed for different materials, shapes and boundary limits.

For a tube, the deflections are even greater because of the reduced stiffness. This can be seen by comparing the previous deflection chart for a solid cylinder to one for a tube with a .500" wall thickness (Figure 9).

Two short programs for finding deflections on a programmable calculator (Texas Instruments SR-52) are shown in figure 10. One is for solid steel cylinders and on for steel tubes. Both yield the amount of deflection when supported at the ends given the O.D., I.D., and length.

After the necessary method of support is decided, there are several techniques of measuring the straightness. Since inside diameters of tubes are the most difficult to measure, the several measuring techniques described are for I.D. measurements.

For small I.D.'s there are two methods that will work with the "limited" access on such parts. When the overall straightness called for is critical, an air gage probe of the type illustrated in Figure 11 can be employed. The second method (Figure 12) utilizes a mating plug with a known clearance. This method, although in most cases not as accurate as the air gage probe, is simpler because it is a go or no go situation.

As the I.D. and the length increase, the previous two methods become very expensive because of the cost of the gaging involved. For the large diameter holes, and electronic probe with a remote readout can be used. The probe is mounted on a traverse bar which is supported externally for shorter bores (Figure 13) and internally for longer bores where the deflection of the traverse bar would be too great when supported externally (Figure 14).

When the overall straightness of a long bore is highly critical, an optical method of measurement can be employed. In this set-up a sled with a front surface mirror is set up at one end of the tube. A collimator is zeroed at this point. The sled is then moved in overlapping increments along the I.D. and the angular deviation of the mirror is measured (Figure 15). Using this method, the straightness of any length tube can be plotted at a resolution limited only by the roundness at a given cross section.

All of the methods are subject to the condition that the plane of maximum bow has been located. Finding this plane is illustrated in Figure 16. By measuring the bow in four planes 45ö apart, the zone containing the plane of maximum bow can be found. With three more trials the desired plane can be located within 7.5ö which is sufficient in most cases.