Motion System Design

Screw stiffness

Ballscrews are essentially long beams, so there's no way to prevent them from flexing and elastically deforming under load. That said, they can be made stiffer for better axis drive precision. First off, modifications can be made to the internal workings of the nut to make it more rigid. The shaft itself can be made thicker or shorter, or be made out of a material with a higher Young's modulus. Finally, supporting the ballscrew at more points can make the system less yielding.

Stiffness is defined several ways. Some manufacturers use a theoretical K value. The most systematic definition is outlined in DIN/ISO standards by three parameters. Rb/t is stiffness of the ball contact zone including all ball and ballrace deformations. This is theoretical and not particularly useful, but is often listed in catalogs because it is normally a large value and makes almost any ballscrew appear stiff. Rnu is the stiffness of the nut, which includes deformations described under Rb/t plus nut deformation. This number is also theoretical. The final measure is Rnu, ar, which accounts for nut stiffness but also includes a reduction to adjust for machining tolerances. Verified by actual measurements, this value is the most useful measure when calculating stiffness of ballscrew drives.

This month's handy tips provided by George A. Jaffe, vice president of Steinmeyer Inc., Burlington, Mass. For more information, call (781)273-6220 or e-mail the editor at [email protected].


How do supports increase stiffness?

When a shaft is supported as a fixed/free member, its stiffness is:

where Rs = Axial stiffness of the shaft

Rb = Stiffness of support bearing

A = Shaft cross section

E = Young's modulus, about 2.1.10-3

ls = Maximum distance from thrust bearing to nut

ls2 = Distance from thrust bearing to nut, when latter is centered

Rs1 is where the shaft is completely unsupported on one end, and with only a single bearing on the shaft rear.

Rs2 is for those configurations where duplex bearings are used on both ends of the shaft. Notice how if the support is fixed/fixed, screw assembly stiffness is increased substantially.

How can stiffness be increased in the nut?

There are several ways. As a general rule, double nuts (with two-point contact) have higher rigidity than single nuts (with four-point contact) for the same number of load-carrying balls. A single nut with one additional ball circle often has higher rigidity and equivalent life, yet in a smaller envelope and at reduced cost.

Preload is another approach, in some cases, nearly doubling nut stiffness. However, increasing preload linearly increases friction and heat generation, and increases the equivalent load, reducing life.

Increasing a nut's preload from 5 to 10% increases stiffness by about 20%.

More load-carrying balls or smaller-diameter balls require a longer, more costly nut, but also increase rigidity. A final approach is to tighten thread profile geometry — so-called track conformity. The penalty here is increased friction and wear.

What affects ballscrew stiffness more?

Suppose we have a ballscrew with some stiffness, but it's not transmitting power accurately enough. Just increase the nut stiffness, right? Actually, the supports have more of an effect on total system stiffness.

To illustrate, suppose we replace a 2 × 3 double nut with 490 N/μm rigidity with a 2 × 5 double nut with 750 N/μm rigidity. Assuming the same shaft stiffness, we arrive at 104 N/μm — a negligible improvement. Why? Total stiffness is governed by the weakest link here — the shaft. Now suppose we change to a fixed/fixed bearing arrangement while keeping the same nut.

System stiffness is doubled.

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