Ball-screw rigidity is a matter of definition

June 2, 2005
Not all ball-screw makers spec products the same way.

George A. Jaffe
Executive Vice President
Steinmeyer Inc.
Bedford, Mass.

Alexander F. Beck
Steinmeyer GmbH

This stiffness test rig uses a force sensor to measure axial thrust. A bracket clamped onto the drive journal prevents backdriving of the screw. The nut block has exchangeable inserts and hydraulically clamps to the cast-iron base. Three LVDT sensors measure relative motion between a shaft and nut. They attach to the unloaded portion of the screw and their tips touch the nut flange from behind. The arrangement averages all deformations of the nut body but does not include changes in shaft length. Forces in the rig can reach several thousand Newtons, though the shaft elastic displacement is only a few microns.

Stiffness or rigidity is a key factor in evaluating ball-screw drives for specific needs. Rigidity, in this context, describes a ball screw's elastic deformation under load and directly affects the precision of an axis drive, especially in machine tools. Unfortunately, not all screw manufacturers quantify rigidity based on DIN/ISO standards, which can cause confusion.

Japanese manufacturers, for example, often use a theoretical "K-value" to describe stiffness, from which 20% is typically subtracted to get an actual value. Still other ball-screw makers don't specify which value is being used at all, and the numbers given may well be theoretical and therefore useless for purposes of application engineering.

In contrast, Rnu, arrepresents the most realistic stiffness value because it is verified by actual measurements. Regardless of the definition used, certain design parameters — in particular, preload — control ball-screw rigidity. Rigidity scales with preload, though higher preloads raise friction (linearly), heat generation, and equivalent load, shortening service life. Catalog rigidity specs should give the corresponding preload value. Use the following formula to calculate rigidity at a different preload: (Note: Nomenclature and term definitions are at the end of this article.)

For example, ball-nut rigidity at 5% preload is about 20% less than at 10% preload. And a nonpreloaded nut has about half the stiffness of the same nut in a preloaded condition. Other factors as well control rigidity, including ball count and diameter, thread-profile geometry, and nut type.

The use of more balls boosts rigidity but also increases nut length and cost. Smaller diameter balls as well increase rigidity but lower load capacity. More balls are then needed to maintain capacity, which also increases nut length and cost. Manufacturers can change what is called track conformity to raise rigidity. But they are often reluctant to do so because such changes also boost friction, heat generation, and wear. As a rule, double nuts (two-point contact) are more rigid than single nuts (four-point contact) with the same number of load-carrying balls. However, a single nut with one additional ball circle typically has higher rigidity and equivalent calculated life than a double nut, yet occupies a smaller envelope and costs less.

Overall system stiffness considers not only nut stiffness but also that of the ball-screw shaft and support bearing:

Shaft stiffness keys to Young's modulus of the material, shaft cross section and length, and of course to the support arrangement. When using the fixed-free support method, shaft stiffness is given by:

A fixed-fixed support gives substantially higher shaft stiffness:

Note that the equation for Rs1considers screw arrangements where the shaft is completely unsupported on one end and also where only a single bearing is used on the rear of the shaft. In other words, Rs2is only for those configurations with duplex bearings on both shaft ends.

A real-world example

Now, look at how the choice of nut and support-bearing arrangement can change rigidity. Suppose that a ball screw system with a total stiffness = 97 N/mm needs to increase by at least 50%. Why not just raise nut stiffness? As it turns out, that will have little effect. For example, consider the switch from a 2 X 3 double nut with 490 N/mm rigidity to a 2 X 5 double nut with 750 N/mm rigidity.

Running the numbers with the same shaft stiffness gives a 104 N/mm rigidity, a negligible improvement. Why? Because of the equation's inverse nature, the shaft (weakest link) governs total stiffness. However, switching to a fixed-fixed bearing arrangement more than doubles system stiffness with the same nut.

Ball screw:
32 X 10 mm 2 X 3 double nut

Accuracy class T3
Rnu,ar= 490 N/mm

Fixed-supported bearing with an axial stiffness = 2,300 N/mm
A = 605 mm 2
ls= 1,000 mm
Rs1= 127 N/mm
Rt= 97 N/mm

Fixed-fixed arrangement with same bearing
Rs2= 508 N/mm
Rt= 225 N/mm


Fpr= Preload force, N

Rs= Shaft axial stiffness, N/mm

Rb= Support bearing axial stiffness, N/mm

Rnu,ar= Nut stiffness (N/mm) and includes a reduction for machining tolerances. This value is most meaningful for calculating stiffness of a drive system and is verified by actual measurements.

Rb/t = Stiffness of a ball contact zone (N/mm) including all deformations incorporated in the balls and ball race. It is not particularly useful for real-world applications, but often the only number given by some manufacturers. It is typically a large value and therefore may appear advantageous.

Rnu= Nut stiffness (N/mm), including deformations described under Rb/t, plus nut deformation. This number is also not particularly useful because it is theoretical.

A = Shaft cross-sectional area, mm 2

E = Young's modulus (about 2.1 X 10 5 N/mm 2 )

ls = Maximum distance from thrust bearing to nut, mm

ls2 = Distance between nut and bearings when nut centered

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