George A. Jaffe
Executive Vice President
Steinmeyer Inc.
Bedford, Mass.
Alexander F. Beck
President
Steinmeyer GmbH
Germany

Stiffness or rigidity is a key factor in evaluating ballscrew 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 "Kvalue" to describe stiffness, from which 20% is typically subtracted to get an actual value. Still other ballscrew 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, R_{nu, ar}represents the most realistic stiffness value because it is verified by actual measurements. Regardless of the definition used, certain design parameters — in particular, preload — control ballscrew 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, ballnut 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, threadprofile 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 (twopoint contact) are more rigid than single nuts (fourpoint contact) with the same number of loadcarrying 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 ballscrew 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 fixedfree support method, shaft stiffness is given by:
A fixedfixed support gives substantially higher shaft stiffness:
Note that the equation for R_{s1}considers 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, R_{s2}is only for those configurations with duplex bearings on both shaft ends.
A realworld example
Now, look at how the choice of nut and supportbearing 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 fixedfixed bearing arrangement more than doubles system stiffness with the same nut.
ORIGINAL CONFIGURATION
Ball screw:
32 X 10 mm 2 X 3 double nut
Accuracy class T3
R_{nu,ar}= 490 N/mm
Fixedsupported bearing with an axial stiffness = 2,300 N/mm
A = 605 mm ^{2}
l_{s}= 1,000 mm
R_{s1}= 127 N/mm
R_{t}= 97 N/mm
UPDATED CONFIGURATION
Fixedfixed arrangement with same bearing
R_{s2}= 508 N/mm
R_{t}= 225 N/mm
NOMENCLATURE AND DEFINITIONS
F_{pr}= Preload force, N
R_{s}= Shaft axial stiffness, N/mm
R_{b}= Support bearing axial stiffness, N/mm
R_{nu,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.
R_{b/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 realworld applications, but often the only number given by some manufacturers. It is typically a large value and therefore may appear advantageous.
R_{nu}= Nut stiffness (N/mm), including deformations described under R_{b/t}, plus nut deformation. This number is also not particularly useful because it is theoretical.
A = Shaft crosssectional area, mm ^{2 }
E = Young's modulus (about 2.1 X 10 ^{5 }N/mm ^{2 })
l_{s} = Maximum distance from thrust bearing to nut, mm
l_{s2} = Distance between nut and bearings when nut centered