3.3.2. Mechanical Transmission

For what concerns the mechanical part of the EMA, particular attention was paid to the mechanical transmission between the rotatory motion of the electric motor and the linear displacement required to move the aerodynamic surface, i.e., the ball screw. It was mainly composed of a screw shaft, 1 or 2 nuts, and a set of spheres that allowed the mechanical efficiency of this component to reach extremely high values due to the replacement of sliding with rolling friction. To accurately describe its behavior and performance, a 3-dimensional modeling approach was necessary. In fact, in order to transmit the motion, the spheres rolled between the screw shaft and the nut's grooves along a helical path. Moreover, a recirculating insert was present to maintain them within the nut body. As a result, the motion of each sphere was governed by an intrinsically 3D contact pattern.

**Figure 5.** Effect of an occurring turn-to-turn short in a BLAC drive. (**a**) Healthy condition. (**b**) Developing short.

The developed dynamic model, whose main screen is shown in Figure 6, considers all these peculiarities of the mechanism through a multibody approach, integrated with a highly detailed description of the contact conditions. In fact, a generic gothic arch profile of the raceways was considered; hence each sphere could enter in contact with each groove in a maximum of 2 points simultaneously, 1 for each circular side of the profile. The contact force and footprint extent were calculated at each time step by means of a penalty method with variable contact stiffness, dependent also on the contact angle. The contact parameters were calculated according to the approximated Hertzian theory [29]. The latter allows sensibly reducing the computational time with a precise closed form solution and without the need to run implicit iterative calculi at each integration time step. The normal contact force was obtained by the extent of the contact constraint violation and was composed by an elastic and a dissipative term, dependent on the approaching speed, related to energy dissipation that occurs during the deformation of the material.

**Figure 6.** Main screen of the Simscape Multibody high-fidelity model of the ball screw component.

To avoid unrealistic discontinuities in the contact force, the elastic component was also used as a saturation value for the damping part.

From the evaluation of the relative sliding speed in each contact point and of its direction, the friction forces were calculated. Although sliding friction was replaced with rolling friction, a little amount of slippage always occurs due to the elastic deformations of the bodies in the contact area and to the kinematics of the mechanism itself. Therefore, ball screws are usually lubricated with grease to create a proper separation of the mating bodies to avoid wear on the rolling surfaces and fatigue damage. The developed high-fidelity model took into account the presence of grease lubrication, considering the effect of its base oil, which is the main actor in the surface separation. The lubricating media rheology was considered by means of Roeland's model for the viscosity dependence on shear stress and temperature, and of the Eyring model, which describes the nonlinear dependence of the shear stress from the shear rate in the lubricant film [30]. It depends on the film height that, following the Grubin approach [31] for the sake of simplicity, was obtained by the solution of the Nijembanning approximated formulation [32].

Generally, as in rolling bearings, elastohydrodynamic lubrication regime takes place in the contact regions; therefore hydrodynamic rolling resistance forces, rolling hysteresis torques, lubricant pressure components, and microslip losses were taken into account according to [33], as well as the spinning friction torque. Finally, for low entrainment speeds and/or high normal loads, the 2 surfaces had direct contacts: the model considered the variability of the lubrication regime, calculating the sliding friction force as a weighted sum of the dry and full film lubrication forces based on the Tallian parameter [34]. When the mating surfaces of the spheres and the grooves are sufficiently separated, almost no wear phenomena occur, while in the mixed and boundary lubrication regimes, wear becomes not negligible. The grooves' wear is one of the most important issues in ball screws and is detrimental to their positioning accuracy as it causes preload loss, increased vibration, and, eventually, complete backlash [35]. Indeed, it was included as one of the ball screw degradation models through the well-known Archard equation, calculating the volume worn in each sphere/groove contact point at each time step and summing all the contributions to obtain a uniform mean wear of each raceway. The Archard coefficient was assumed to be dependent on the lubrication regime, a function of the Tallian parameter [36].

The probability of contact of the mating surfaces increases with the component's usage. In fact, grease aging involves a degradation of the lubricity property of the base oil in time, i.e., viscosity reduction, meaning a decrease of the load-bearing capacity and a thinner film thickness, with increased direct contact probability and wear. Furthermore, particularly at high speeds, film thinning is also caused by lubricant starvation, which was considered following the approach presented in [37,38]. The adopted lubricant aging model calculated the reduction of viscosity as a function of the entropy generated within the lubricant due to internal mechanical shearing [21,39].

Figure 7 depicts the results of a study executed with the ball screw high-fidelity model on a ball screw with a 5 mm lead and a nominal diameter of 16 mm, directly connected to the electric motor and the grease, lubricated and subjected to different external loads levels while operating at various rotational speeds. Lubricant starvation, wear of the grooves, and lubricant aging were considered. The analyses were aimed at investigating the effect of the latter phenomenon on the macroscopic behavior of the component. It can be seen from the plot that higher external forces, combined with low speeds, lead to higher wear rates and, ultimately, to a shorter operative life. In the case of excessive wear, backlash can occur, as shown in Figure 8, where the effects of an excessive backlash are depicted for what concerns the speed and position. The left graph depicts the linear position of the nut and the equivalent linear position of the screw obtained considering the ideal transmission ratio, i.e., the 2 lines should overlap. The position of the screw shaft, being speed controlled, follows a sinusoidal trend; however, because of the presence of backlash, the nut fails to replicate its positioning and stops close to the motion reverse points. The internal spheres lose contact and engage again when the relative position reaches the backlash gap. This can be more easily observed in Figure 8b, which shows the equivalent rotational speed of the screw and the nut. At motion reversal points, due to the friction on the linear guides and

the presence of backlash, the nut speed settles at 0 and remains steady until the backlash is closed; at this time, the screw's speed also oscillates because of the new engagement.

**Figure 7.** Wear rate versus aging level for different speeds and loads for a ball screw with 5 mm lead and 16 mm nominal diameter (© 2022 IEEE. Reprinted, with permission, from [21]).

**Figure 8.** Simulation results of a ball screw with a high backlash of 0.3 mm. (**a**) Comparison between the nut position and the equivalent linear position of the screw shaft; (**b**) Comparison between the equivalent angular speeds of the screw shaft and the nut (© 2020 ASME. Reprinted, with permission, from [33]).
