Well-characterized bearing fatigue mechanisms make it easy to understand trade offs between load and lifetime for ball-screw actuators.

By: Kristin Lewotsky, Contributing Editor
(posted 04/12/2012)

Linear actuators such as ball-screw actuators (see figure 1) represent an economical, effective method for converting the rotational motion of a rotary servo motor or stepper motor into linear motion. Just as a motor must be sized correctly for the application, so must an actuator be sized correctly to ensure that it will both support the load and operate for the desired lifetime of the application.

Screw-type actuators encompass so-called lead-screw actuators, which consist of a threaded screw and a nut (see figure 2); ball-screw actuators, in which ball bearings placed in the raceways between the nut and the screw act to reduce friction and wear; and planetary roller screws, which feature an array of secondary screws placed around a central shaft in a planetary configuration. Lead screw actuators tend to be most economical but suffer where elements lifetime, while planetary roller screws provide high-reliability operation with even higher loads. Ball-screw actuators provide a good compromise solution.

A combination of factors can impact the lifetime of screw-type actuators. The material of the actuators can be compromised by friction-induced wear, corrosion, contamination, intrinsic flaws, and material fatigue. In the case of lead-screw actuators, the sliding friction increases wear as well as generating heat that might cause speed material fatigue. The combination of factors makes predicting the lifetime of lead-screw actuators challenge. In contrast, the lifetime and sizing of ball-screw actuators and roller-screw actuators are well characterized. The effects of fatigue, for example, can be described by a simple set of equations that simplify the task of sizing a component. By leveraging these equations, users can easily make tradeoffs between load and lifetime, for example choosing an additional support bearing for the load, if necessary.

Determining Lifetime
Depending on the standard, we can define the lifetime L of a ball screw or roller screw as either the number of revolutions (ISO¹/DIN) or the length of travel (ANSI) that screw can endure prior to fatigue. To help us calculate this figure, we need to establish the dynamic axial load rating C_d, which specifies the load a screw can tolerate without reaching fatigue while still achieving a standard performance level such as the 10⁶ revolutions defined by ISO-3804. The quantity C_d is determined by testing and supplied by the manufacturer.

The performance of real systems does not meet theoretical predictions. For purposes of sizing an actuator, it's more useful to think of L₁₀, which is defined as the lifetime that 90% of a group of representative actuators would survive to 10⁶ revolutions. We can determine L₁₀ by:

           (1)

where F_app is the application load and 10⁶ is the number of revolutions specified for the L₁₀ lifetime by the ISO standard. It is important to note that the application load F_app bears an inverse-cube relationship to the L₁₀ lifetime—modifying the application load can have a profound effect, pro and con, on the lifetime. With equation 1, we can determine how long an actuator will last in a given application. Conversely, we can solve for C_d, using the application load and desired lifetime to arrive at a dynamic axial load rating to guide the search for the right hardware.

Accounting for real-world conditions
Equation 1 describes an ideal situation, but engineers don't design in an ideal world (remember that old joke about the farmer's daughter's fiancé and his plan to up production assuming a spherical cow radiating milk isotropically?) To account for factors such as bearing vibration and collision during operation of the actuator, which are effects that change depending on speed; and also factors like corrosion and contamination, manufacturers introduce a quantity known as dynamic load factor or safety factor f_s.

With the introduction of f_s, we can define a service lifetime L_10h as:

     (2)

The dynamic axial load rating is an ideal quantity. In reality, applications often involve moving varying loads over different distances. We can represent that using a quantity called equivalent load (F_e). We define F_e as the mean load that produces the same amount of fatigue as a collection of varying loads applied over different time frames:

     (3)

where x is total travel, x_1…N  represent the incremental distances that make up the total travel, and F_1…N  represent the incremental forces that the ball screw must move over the course of those incremental distances. It is important to remember that systems do not exert force identically in both directions of travel. When performing these calculations, calculate F_e in both directions and apply whichever of the two values is largest.

Now we can restate the L_10h lifetime as:

     (4)

        (5)

For cases in which the speeds vary significantly, we can express equation 3 more granularly as:

      (6)

In which case, equation 5 becomes:

    (7)

Where n₁…n_N are speeds in rpm traveled during the time periods q₁…q_N, and n_m and q_t are the average speed and total time, respectively.

At very high speeds, actuators typically do not achieve the lifetimes predicted by equation 2, especially as loads increase. It's very important to use an appropriate safety factor.

Equation 2 does not hold at very low speeds (less than 10 rpm), either. In these cases, system designers must base their selection on the static axial load rating (C_s), which is defined as the static load that permanently deforms the ball/ball track at point of highest load over a distance of 0.0001d, where d is ball diameter.

Calculating time to fatigue as shown provides a useful estimate of actuator life. Fatigue is not the only failure mode for all-screw actuators, however. To build a robust system, it is important to evaluate all aspects of the application and adjust lifetime estimates to take into account all effects that may degrade lifetime.

Acknowledgements
Thanks go to Igor Glikin, senior mechanical engineer at Tolomatic for useful conversations.

References
ISO-3804