Whether or not the shaft or just the nut rotates can make a critical difference, and not just in terms of critical speeds.

**Alexander F. Beck**

President**George A. Jaffe**

Executive Vice President

Steinmeyer GmbH

www.steinmeyer.com

Engineers working with ball screws usually use conventional life-expectancy equations for nuts with two-point contact. When it comes to four-point contact nuts, however, many manufacturers that sell them do not discuss how to calculate their operational life. But the difference between single and double-nut life calculations lies in the preload. When applied external loads are low, for example, preload determines as much as 90% of the life.

Classic L_{10} lifetime equations are based solely on material fatigue. But although material fatigue may be the most important, it is not necessarily the only factor in determining ball-screw life. Adverse conditions such as contamination or lack of lubricant, for example, make other factors more important.

When engineers determine the life of a ball screw, they first break down its typical duty cycle into distinct steps, each with specified (external) loads or thrusts, speeds, and travel duration. The goal is to determine how many revolutions the ball screw will make under each load. These forces, designated F_{i}, are converted into an equivalent load *F _{m}** that results in the same life expectancy as the actual duty cycle.

The next equation calculates life expectancy from a ball screw's equivalent load, *F _{m}**, and dynamic load capacity,

*C*. The answer, L

_{a}_{10}, in million revolutions or inches, respectively, depends on which load capacity definition or industry standard is used.

So far, this is nothing unusual. But, what happens if the ball screw is just used for positioning with light, or even negligible thrust (i.e., *F _{m}* goes to zero), and there is still significant preload for repeatability? Plug the numbers into the equation, and life goes to infinity, which certainly cannot be true. After all, the balls carry the nut's internal load or preload, so they must affect life.

At this point, a graph will help explain the situation. The preload graph consists of two curves, each representing a force-deflection curve of one ball nut in a double nut configuration. It shows deflections increasing with greater force. It also displays how forces in the opposite nut decrease as soon as deflection causes axial displacement. Or, just imagine two springs acting against each other. As you push against one of them, it is compressed or loaded, while the other is unloaded.

Where the two curves intersect represents a state of equilibrium with zero outside thrust. The distance between the graph's *X* axis and the crossing lines is the preload force. Now consider an external thrust in either direction on the ball nut. The load on one nut increases, while the load on the other decreases because the ball nut moves slightly in the direction of the thrust, giving the opposite nut more room. The resulting thrust vector is the difference (or distance) between the two curves.

At some point, as the load increases, the load on one nut becomes zero.

The higher thrust vector is the important one for life calculations. (Assume the curves are linear rather than curved, which is not exactly true but a close enough approximation.) The modified load on the ball nut during step *i*, is then equal to preload, *F _{pr}*, (move from the crossing point horizontally to the right) plus half the outside thrust,

*F*, because the horizontal line splits the outside thrust vector approximately in half.

_{i}This remains true until thrust is enough to completely unload one ball nut (approximately 2.83 times the preload), and then thrust equals the total load on the ball nut. By observing recommended guidelines this situation should never be reached, but the approximation is useful in calculating total load on the preloaded nut now designated the modified load *F _{i}**.

This approximation is adequate because the inputs to the equation, the applied forces, are typically estimates so more accurate calculations usually aren't necessary.

As noted above, this considers only the nut carrying the higher thrust. Obviously, in double nuts more thrust is carried by one nut and it will fail first. So properly using load equations means entering outside thrust as a plus or minus value, depending on its direction, and determining which nut will be critical in the duty cycle. To do so, calculate modified loads for each step in the duty cycle, determine the equivalent modified load, *F _{m}** by using equation 3 for each nut. Then use the higher of the two values to calculate life in equation 2.

But what about single nuts with four-point contact? Is the approach to preload in equation 3 applicable? Yes, but with some modification. Per our definition, single nuts have four contact points instead of two, so there are twice as many load/unload cycles for any given spot on the ball surface. Thus material fatigue failure should happen in only half the distance traveled by a double nut before it fails (given the same applied load). This is true under recommended operating conditions, namely anytime there is a mean load less than 2.83 times the preload. If applied forces exceed this value, four-point contact will approach two-point contact and a single nut's lifetime becomes similar to that of a double nut. Therefore, the difference in life between the two nut types under the same conditions cannot be expressed in load capacities. In fact, single and double nuts generally have identical load capacities, except when a single nut uses spacer balls to smooth out running and reduce the number of load-carrying balls. So, the answer lies in how preload is handled.

Again, as a reasonable approximation, the difference between one and two-point contact can be reflected in a modified version of equation 3:

Note the 25% "surcharge" to the preload. This yields a 50% reduction in life if there is no other load, because of the cubic relationship in equation 2. Also note that external thrust, *F _{i}*, is always positive for single nuts regardless of thrust direction because the same set of balls supports the load.

So the actual penalty for four-point contact can be as high as a 50% reduction in operational life if the only force the nut sees is preload, or as low as zero if there is zero preload. In most applications, the life penalty of single nuts falls somewhere between and can be easily compensated for by reducing preload or adding extra ball circles for higher load capacity. Eliminating the second nut of a double nut eliminates three or more ball circles, so there is room to add one to the remaining single nut. And replacing a 3 X 2 double nut with a 4 X 1 single nut makes it more compact and more economical.

As noted earlier, one nut half in a double nut can be unloaded if thrust is enough to cause enough deflection in the other half. This means one set of balls runs unloaded, and this is generally unacceptable. (It's also potentially catastrophic when reversing direction under heavy loads.) So preload has to prevent unloading. This may make it necessary to select preloads on the high side which, in turn, increases modified equivalent loads and decreases life expectancy.

This is not the case for single nuts. There is no ball set to be unloaded, thus lower preloads can be used, increasing ball-screw life.