Ball screws, a mainstay of linear motion, advance to meet the demanding dynamics of today’s applications.

Ball screws are ranked and rated according to factors like speed, noise, accuracy, and life expectancy. To qualify as “high performance,” they must achieve certain minimum levels across all such variables. As with most engineered systems, there is some give and take between various properties, but these can be adjusted to favor the application. Insight into some of the dynamics at work can help you figure out the best design.

### In the fast lane

Faster machines save time, and time can be essential. “Fast,” for a ball screw, is anywhere between 100 and 200 m/min, the higher end of these speeds occurring under test-lab conditions. Of course, what we now consider quick will probably become commonplace in a year or two.

The angular velocity required to produce a given traverse speed depends on the screw lead. While greater leads lower the necessary rotation speed, they also require more driving torque, often necessitating a larger, more expensive motor. Furthermore, increasing the lead compromises accuracy. In practice, leads are best kept within 62.5 to 100% of the screw shaft diameter.

Ball screws are efficient at converting from rotary to translational motion (efficiency is upwards of 90%), but they also require various inventive means to transfer (circulate) the ball bearings from one end of the ball nut to the other. The impact of ball circulation on speed can be emphasized by a quick comparison with rotary roller bearings. The speed capability in bearings and ball screws is commonly given by the mean shaft diameter multiplied by the rotating speed (*d _{m}n*). Where a 40-mm shaft running at 5,000 rpm might not be a big deal with precision bearings, the 200,000

*d*product is significant for a ball screw, presenting a considerable challenge to the recirculation system.

_{m}nShaft whirling and vibration at critical frequencies is a very imposing factor when it comes to ball screw speed. In a conventional arrangement, the screw spins while the nut translates. Long, slender screws are most susceptible to whirl; they have lower lateral natural frequencies (correlating to lower critical speeds) and tend to behave in a flexible manner.

To minimize constraints of critical speed, some ball screws employ hollow screw shafts. Pound-for-pound, hollow shafts are more rigid and have a higher natural frequency than solid shafts. However, the higher shaft inertia can impair performance in other areas of the system and may require larger drive motors. As travel length increases, the large shaft diameters needed to limit whirling can put a real strain on the rest of the system, including the nut, and there have been extreme cases where inertial effects led to failure after a few hundred hours.

To get around shaft whirl in some highspeed applications, the nut may be rotated instead of the shaft. While running a motor to the shaft is pretty straightforward, driving nuts can require a bit more arrangement. For example, in one method the motor is offset from the ball screw axis and turns the nut through a timing belt.

### Put it right there

Accuracy and repeatability of a linear motion system relies not only on the ball screw but on the larger assembly to which it mounts. As for the ball screw, accuracy is instilled by closely controlling the lead and the ball circle diameter during manufacture. Normally, the minimum specification would be ANSI class 5, ISO class 3, or JIS class C5. Under these specifications, the lead accuracy would be between 12 and 18 μm per 300 mm of track.

To ensure ball screw repeatability, zerobacklash or a light preload might be instated between the nut and screw. There are various methods to generate a preload or control backlash. Preloads are often stated in terms of the ball screw’s dynamic capacity and for high-speed units they are normally set in the range of 0.5 to 5%.

Among some high-performance screws, a four-point contact principle is popular. This relies on two contact points between the ball and screw thread, and two contact points between the ball and nut thread. With this technique the degree of preload can be varied by changing the ball bearings’ diameter by a few μm.

Another typical preload mechanism uses a split-nut design. The nuts are axially loaded, either pulled apart or together, creating ball contact that’s “mirrored” across the split. Holding the components to their outer or inner extremes of contact reduces the possibility of nuts or screws displacing axially when there’s no rotation.

Another way to improve the stiffness of the system is to apply pretension to the shaft itself. Not only does this take some of the slack out of the assembly, it also helps accommodate thermal expansion. Shaft expansion varies with the preload and the overall work being done by the ball screw; these factors affect heat generation. With hollow-shaft screws, external cooling is sometimes employed. For non-cooled ball screws, the amount of shaft strain applied from pretension is usually between 12 and 25 μm/m.

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### Endurance

The expected life of a ball screw is based on the statistical probability of metal fatigue failure in either the ball bearings or the raceways. While the math that explains this is fairly complex, the derived formulas are straightforward and easily applied. When application details are known along with the rotation speed and loading of the ball screw, the dynamic capacity required for a specific life is given by.

where*C _{a}* = required dynamic capacity (kN)

*F*= mean applied load (kN)

_{m}*L*= expected life (hr)

_{10}*n*= mean rotational speed (rpm)

_{m}*f*= operation factor:

_{x}for fully-damped smooth running, *f _{x}* = 1.0 to 1.3

for normal operation with hollow shafts, *f _{x}* = 1.4 to 1.6

for normal operation with solid shafts, *f _{x}* = 1.7 to 2.5

Manufacturers provide tables giving their ball screws’ dynamic capacities.

Required values of *L _{10}* are usually between 20,000 and 25,000 hr, but there are stringent applications that may call for abilities outside the standard catalog offerings. For instance, a router manufacturer required an

*L*of 26,500 hr, and the load range indicated a dynamic capacity of 85 kN for a 63-mm shaft diameter and a 40- mm lead. Special variations needed to be designed into the ball screw, including triple-start threads.

_{10}In following a common theme in engineering and physics, there are compromises in ball screw design, and fortifying one area may detract from another. For example, using relatively large ball bearings provides high dynamic capacity and long life, but smaller ones give lower deflection, and their reduced mass contributes to lower noise levels. Now, if a designer tries to compensate for a loss of dynamic capacity by using more circuits of smaller ball bearings, he will have set against himself a “law of diminishing returns,” as the dynamic capacity is not linearly related to the number of circuits; rather, the number of circuits is raised to a power of 0.8. So doubling the circuits will increase the dynamic capacity by less than two – and the shortcoming is compounded with greater numbers of circuits.

For high-speed screws between 32 and 63 mm in diameter, smaller balls (from 4.76 to 6.36 mm diameter) with a close ball-to-track conformity tend to provide good overall performance. (The conformity is simply the track or groove radius divided by the ball diameter.)

### Shake, rattle, and roll

To achieve high performance levels, including low noise and vibration, ball screws need to be straight and well-aligned, with consistent raceways. Manufacturers have similar specifications for these parameters, but differences exist in the way they are measured. For instance, shaft straightness might be specified at 25 μm of maximum runout, but one manufacturer might restrict this runout allowance to the center point of the shaft span, while another, using a more lenient guideline, approves of the same 25 μm of runout occurring across a much larger section of shaft.

Good ball screw alignment goes hand in hand with noise reduction. And, the quality of raceway manufacture is particularly important during high rates of acceleration and deceleration, such as 1 to 2 *g*.

Shaft support and overall configuration of the assembly also affect noise level. Good suppression of noise results when shaft ends are mounted completely fixed – with no axial or lateral freedom – and the shaft is under some amount of pretension.

While there are no standard international rules for calculating ball screw noise, analytical and empirical data is available. According to researchers Kajita and Ishikawa in *Noise Level of Precision Ball Screws*:

*dBA* = 25.2(log_{10}*D _{w}d_{m}n*10

^{-5}) + 63.9

where*dBA* = predicted noise*D _{w}* = steel ball diameter

*d*= ball pitch diameter

_{m}*n*= rotational speed

The tests accounted for therein were carried out at much lower speeds than those considered here, but it was found that the predicted values agreed well with other test results.

Critical speed and natural frequency, of course, must be addressed for the potentially violent effect such excitations can have on the ball screw structure. The lateral natural frequency of vibration of a ball screw shaft, fixed at both ends, is given by.

where*N* = natural frequency (Hz)*E* = modulus of elasticity*I* = moment of inertia*W* = mass per unit length of the shaft*L* = length of the shaft between supports

Therefore, to design shafting with a high natural frequency, maximum diameter and minimum mass are key, but don’t forget that the higher shaft inertia inherent in such a design can be detrimental in other ways. Having a shaft bore of around 48% of the outer diameter has been proven to work well for many systems.

Not all shafts are limited to speeds below the critical speed. In some cases they can be quickly pushed past the first natural frequency. Ball screw behavior above the first critical speed is therefore of interest to manufacturers wanting to minimize the noise and vibration resulting from such operation.

*Kelvin Kellond is Technical Manager at Thomson IBL Co., Barnstaple, U.K.*

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