Contact Analysis of EMB Actuator Considering Assembly Errors with Varied Braking Intensities

Abstract

Differential planetary roller lead screw (DPRS) serves as a quintessential actuating mechanism within the electromechanical braking (EMB) systems of vehicles, where its operational reliability is paramount to ensuring braking safety. Considering different braking intensities, how assembly errors affect the contact stress in DPRS was analyzed via the finite element method. Firstly, the braking force of the EMB system that employed DPRS was verified by the braking performance of legal provisions. Secondly, a rigid body dynamics model of DPRS was established to analyze the response time, braking clamping force, and axial contact force of DPRS under varied braking intensities. Finally, a finite element model of DPRS was constructed. The impact of assembly errors in the lead screw and rollers on the contact stress were investigated within the DPRS mechanism based on this model. The results indicate that as braking intensity increases, the deviation of the lead screw exerts a greater influence on the contact stress generated by the engagement between the lead screw and rollers compared to that between the nut and rollers. The skewness of the rollers also affects the contact stress generated by the engagement of both the lead screw with rollers and the nut with rollers. When assembly errors reach a certain threshold, the equivalent plastic strain is induced to exceed the critical value. This situation significantly impairing the normal operation of DPRS. This study provides guidance for setting the threshold of assembly errors in DPRS mechanisms. It also holds significant implications for the operational reliability of EMB systems.

1. Introduction

The automotive industry is rapidly transitioning toward new energy vehicles, fully electric control systems, and intelligent technologies. Autonomous driving, as a core component of intelligent systems, drives and embodies this technological evolution. The electronic chassis control system is a critical subsystem in intelligent vehicles, forming a foundational technology for realizing autonomous driving functions [1]. Among electronic braking systems, the electromechanical braking (EMB) system—enabling precise control via electrical signals—offers advantages such as rapid braking response, stable brake force distribution, and seamless integration with anti-lock braking systems (ABSs) and electronic stability programs (ESPs), thereby simplifying structures and reducing spatial requirements [2].

The EMB actuator comprises three primary components: torque transmission units, speed-reduction torque-amplification mechanisms, and motion conversion devices [3]. The motion conversion device transforms the rotational motion of motor gears into linear motion of friction pads, compressing the brake disc to convert motor-generated torque into clamping force during braking. Due to the high-speed, high-precision, and high-load operational intensities of EMB actuators, the reliability of motion conversion mechanisms is critically constrained.

Currently, the mainstream motion conversion devices in EMB systems include ball screw pairs [4] and planetary roller screw pairs [5]. Ball screw pairs have been widely used as EMB actuators in passenger vehicle due to their smooth motion, minimal wear, and high transmission efficiency. Planetary roller screw mechanisms have existed for a long time, with primary types including standard planetary roller screws, inverted planetary roller screws, and differential planetary roller screws (DPRS) [6]. Among these, the differential planetary roller screw pair not only possesses all the advantages of ball screw pairs but also offers higher load capacity, longer lifespan, more compact structure, and smaller lead [7]. Therefore, adopting this motion conversion device in EMB systems can enhance braking response speed, improve control precision, and increase braking force, better meeting the requirements of intelligent vehicles for braking systems.

Significant progress and achievements have been made in EMB research both domestically and internationally. In terms of EMB structural design and integration optimization, Continental Teves developed an EMB prototype integrating a planetary gear reduction mechanism with a ball screw [8]. Zhao Chunhua et al. [9] designed an EMB structure that combined bevel gears and planetary gears, while the motion conversion device employed a hybrid ball screw and sliding screw mechanism, reducing the overall axial dimensions while increasing the transmission ratio and torque.

In system performance testing and validation, Ji Houhua et al. [10] verified the effectiveness of the system in braking control by constructing an EMB test bench and establishing a corresponding simulation model, providing experimental foundations for electromechanical braking system development. Shen Chen et al. [11] conducted relevant tests on a self-developed EMB prototype, determining the relationship between pedal signals and generated braking force.

In the study of motion conversion device mechanisms, existing research focuses on load characteristics analysis, parameter optimization, and error impact. Lisowski F. et al. [12] established a finite element model for single-thread contact in standard planetary roller screws, obtaining stress and thread deformation curves, investigating the influence of random pitch errors on load distribution, and analyzing contact stiffness using the finite element method. Yang Chao [13] developed a theoretical load distribution model for differential planetary roller screws, analyzing the effects of roller count and varying loads on load distribution and validating the model via finite element analysis. Cui Gaoshang [14] studied the contact deformation, dynamic characteristics, and load distribution of DPRS, using the finite element method to simulate the influence of helix angle on lead accuracy. Ma Shangjun et al. [15] established a transmission accuracy model for planetary roller screws considering load distribution, clarifying the critical impact of assembly errors. Zheng Zhengding et al. [16] developed a geometric model for DPRS load capacity calculation, analyzed its load characteristics, and proposed methods to mitigate accuracy loss and extend service life. Yao Qin et al. [17] determined the contact positions of planetary roller screw mechanism (PRSM) components based on coordinate transformation and meshing equations, deriving the relative motion velocities and contact stress distributions between the screw–roller and roller–nut interfaces. Hu Rui et al. [18] proposed a PRSM load distribution optimization method based on roller taper modification, determining the optimal grinding angle through deformation coordination equations and force balance analysis, providing design solutions for PRSM optimization.

In summary, as a mature actuation mechanism, the application of DPRS in EMB has become an important development trend. However, current research on assembly errors of DPRS in EMB systems remains relatively scarce. With the gradual adoption of EMB in the automotive sector, the impact of assembly errors on actuator performance requires in-depth investigation, particularly as the coupling mechanisms of these errors under different braking intensities have not been sufficiently studied. Therefore, it is significantly important to analyze assembly errors of DPRS in the EMB system for improving the reliability and braking safety of automobiles.

This paper establishes a performance analysis framework for DPRS under the coupled effects of braking conditions and assembly errors, proposes a method for determining dynamic load boundaries of DPRS based on braking regulations, and reveals the influence patterns of assembly errors on contact stress. First, the braking force of the DPRS-based electromechanical braking system was verified according to braking performance regulations. Then, a rigid-body dynamics model of DPRS was established to analyze response time, clamping force, and axial contact force under different braking intensities, with validation against theoretical calculations. Finally, a finite element model of DPRS was developed, applying the obtained axial contact force as the load boundary to analyze the influence of screw and roller assembly errors on contact stress distribution. This provides guidance for setting DPRS assembly error thresholds and holds significant importance for the operational reliability of EMB systems.

2. Principle and Validation of EMB Actuator

2.1. Working Principle

2.1.1. Composition of EMB Actuator

The subject of this study is the EMB actuator, as illustrated in Figure 1a, comprising a motor, a two-stage helical gear transmission, a differential planetary roller lead screw mechanism (DPRS), and a brake calliper body. The structural composition of the DPRS is detailed in Figure 1b, primarily consisting of a screw, rollers, and a nut.

2.1.2. Kinematic Analysis of DPRS

As shown in Figure 2, the kinematic diagram of the DPRS mechanism includes the following parameters: ω_s denotes the angular velocity of the lead screw (rad/s), r_s represents the engagement radius between the lead screw and roller (4.55 mm), R_n indicates the engagement radius between the nut and roller (7.05 mm), r_n is the smaller radius of the roller (1.05 mm), r_r is the larger radius of the roller (1.35 mm), n_s is the thread count of the lead screw (2 threads), and P₀ is the pitch of lead screw (0.75 mm).

The core transmission principle of the DPRS is explained as follows: the lead screw, acting as the driving component, rotates about its axis. Its threaded surface engages with the annular roller raceway of the roller. During rotation, the roller undergoes a composite motion: revolving around the lead screw axis (orbital motion) while simultaneously rotating about its axis (self-rotation), accompanied by axial displacement along the lead screw. The nut, functioning as the driven component, transmits the axial displacement of the roller to itself through engagement with the annular roller raceway. In the EMB actuator, the nut is constrained to pure axial motion via a limiting mechanism. Consequently, the rotational motion of the lead screw is converted into axial linear displacement of the nut through differential transmission. This realizes efficient conversion between rotational and linear motion.

In the practical operation of the DPRS mechanism, the orbital angular velocity (ω_c) and self-rotational angular velocity (ω_d) of the roller can be mathematically expressed as follows:

$${\omega }_{\mathrm{c}}={\displaystyle \frac{r_{\mathrm{s}}{r}_{\mathrm{n}}}{(r_{\mathrm{s}}+r_{\mathrm{r}})(r_{\mathrm{n}}+r_{\mathrm{r}})}}{\omega }_{\mathrm{s}}$$

(1)

$${\omega }_{\mathrm{d}}=-{\displaystyle \frac{r_{\mathrm{s}}{\omega }_{\mathrm{s}}}{r_{\mathrm{n}}+r_{\mathrm{r}}}}$$

(2)

For the nut displacement velocity (v_ns) in the DPRS mechanism and the lead (L) of the DPRS, they can be mathematically expressed as follows:

$$v_{\mathrm{ns}}={\displaystyle \frac{{\omega }_{\mathrm{s}}-{\omega }_{\mathrm{c}}}{2\pi }}{n}_{\mathrm{s}}{P}_0$$

(3)

$$L=v_{\mathrm{rs}}{\displaystyle \frac{2\pi }{{\omega }_{\mathrm{s}}}}$$

(4)

2.1.3. Working Principle of the EMB Actuator

The power transmission of the EMB actuator is illustrated in Figure 3. The motor generates braking torque, which is amplified through a two-stage helical gear transmission. Subsequently, the planetary roller screw mechanism converts the rotational motion into linear motion of the nut and piston, driving the friction pads to eliminate the brake clearance. The friction pads then exert clamping force on the brake disc to initiate vehicle braking. While the braking is completed, the motor reverses its rotation to recreate the brake clearance, preparing for the next braking cycle.

2.2. Design Parameters

The target performance parameters of the EMB actuator are summarized in

Table 1. Performance parameter of EMB.

Parameter Name	Numerical Value
Maximum pressure of brake FR (N)	17,163
Gap elimination time t (s)	0.12
Brake Clearance (unilateral) s (mm)	0.1

. Based on these targets, the selected motor has a maximum stalling torque of 1.5 N·m and a peak speed of 800 rpm. The two-stage helical gear train achieves a reduction ratio of 13.21, while the differential planetary roller screw mechanism (DPRS) is designed with a lead of 0.977 mm.

2.3. Braking Force Verification

The force analysis of a vehicle during braking on optimal road intensities is illustrated in Figure 4.

The brake torque T_μ is expressed as follows:

$$T_{\mu }=2F_{\mathrm{R}}Rf$$

(5)

where F_R is the maximum clamping force of the brake, _R is the effective radius of the brake disc, defined as 129.5 mm, and f represents the friction coefficient between the friction pad and brake disc, taken as 0.4 according to component materials and operational conditions with reference to literature [19]. Here, the brake torque T_μ is calculated as 1,778,086 N∙mm.

Neglecting deceleration inertial forces, inertial torque, and rolling resistance torque, the brake force F_μ is derived from the moment equilibrium as follows:

$$F_{\mu }={\displaystyle \frac{T_{\mu }}{r}}$$

(6)

where r represents the rolling radius of the wheel, set to 304 mm in the above equation. Here, the brake force F_μ is calculated as 5848.97 N.

The maximum ground adhesion force, F_{Xb max}, and F_{Xb max1}, the maximum adhesion force per wheel, are calculated as follows:

$$F_{\mathrm{X}\mathrm{b}\mathrm{ }\mathrm{m}\mathrm{a}\mathrm{x}}=F_{\mathrm{z}}\phi$$

(7)

$$F_{\mathrm{X}\mathrm{b}\mathrm{ }\mathrm{m}\mathrm{a}\mathrm{x}1}={\displaystyle \frac{F_{\mathrm{X}\mathrm{b}\mathrm{ }\mathrm{m}\mathrm{a}\mathrm{x}}\beta }{2}}$$

(8)

where F_z—normal force exerted by the ground on the wheel, set to 15,000 N, β—braking force distribution coefficient, set to 0.69. $\phi$ is the peak adhesion coefficient of the road surface, taken as 0.8. Here, the maximum ground adhesion force, F_{Xb max}, is calculated as 12,000 N, and the maximum adhesion force per wheel, F_{Xb max1}, is calculated as 4140 N.

Within these equations, F_μ > F_{Xb max1}: the brake force exceeds the maximum ground adhesion force per wheel. According to braking regulations [20], the minimum braking deceleration for a vehicle with an initial velocity of 80 km/h is 4.87 m/s². The corresponding minimum ground adhesion force F_{Xb min} required at this point is

$$F_{\mathrm{X}\mathrm{b}\mathrm{ }\mathrm{m}\mathrm{i}\mathrm{n}}=Ma$$

(9)

where M—curb weight of the target vehicle, set to 1500 kg. Here, the corresponding minimum ground adhesion force F_{Xb min} is calculated as 7305 N.

From Equations (2), (3), and (5), it follows that 4 F_μ > F_{Xb max} > F_{Xb min}, indicating that the brake force provided by the system exceeds the maximum ground adhesion force while satisfying the minimum braking requirements. Thus, the designed EMB brake actuator meets the braking regulations.

3. Simulation of Braking Clamping Force for the Actuator

A dynamic model of the EMB actuator is established to carry out simulations and validated with the results compared to those of the theoretical model. Then, the axial contact forces of the DPRS under different braking intensities are obtained, providing the boundary loads for subsequent mechanics analysis considering assembly errors for DPRS.

3.1. Dynamic Model

The simplified dynamic model of the EMB actuator is shown in Figure 5. Since the focus is on dynamic simulation of the power transmission system, the designed EMB actuator is simplified to retain only the two-stage helical gear transmission, the differential planetary roller screw mechanism (DPRS), the inner and outer friction pads, and the brake disc.

3.1.1. Application of Kinematic Pairs

Kinematic pairs are applied to the retained components of the EMB actuator, The kinematic pairs among the screw, roller, and nut in the core DPRS mechanism are summarized in

Table 2. DPRS movement secondary relationship.

Components	Kinematic Pair
Ground and lead screw	Revolute joint
Cage and lead screw	Roller joint
Nuts and lead screws	Prismatic joint
Nut and piston	Fixed joint

.

3.1.2. Contact Force and Load Application

Considering the influence of friction, the friction parameters between the screw and rollers, as well as between the nut and rollers, are determined from reference [21]. Specifically, the static friction coefficient is taken as 0.3, the dynamic friction coefficient as 0.25, the static slip velocity as 0.1 mm/s, and the dynamic slip velocity as 10 mm/s.

The initial braking speed of the vehicle is set to 80 km/h. In the clearance elimination phase, the motor shaft gear speed is configured to the peak speed. In the clamping force tracking phase, the demanded torque corresponding to a braking intensity of 1.3 g is applied to the motor shaft gear.

3.2. Simulation of the Brake Clearance Elimination Phase

A total of 12 rollers in the DPRS are evenly distributed along the screw’s circumference. Due to structural symmetry, the simulation results of any single roller represent those of all rollers. The simulated angular velocity of the DPRS components during the brake clearance elimination phase is shown in Figure 6. It is evident that within 0.12 s, the screw angular velocity, roller self-rotation angular velocity, and roller orbital angular velocity exhibit certain fluctuations, with mean values of 6.25 rad/s, 11.5 rad/s, and 2.1 rad/s, respectively. The deviations may arise from thread gaps and potential sliding between the screw and roller threads.

The simulated displacement results of the inner friction pad during the brake clearance elimination phase are shown in Figure 7. It can be seen that when the motor operates at a peak speed of 800 rpm for 0.12 s, the axial displacement of the inner friction pad reaches 0.115 mm. It is clear that a 0.1 mm brake clearance can be eliminated within 0.12 s, satisfying the design requirements.

The kinematic parameters of the DPRS are calculated using Equations (1)–(4). The comparison between the simulation results and the theoretical model of the DPRS during the brake clearance elimination phase is presented in

Table 3. Theoretical and simulation values.

Parameter	Lead Screw Angular Speed ω (rad/s)	Roller Angular Velocity ω (rad/s)	Nut Displacement s (mm)
Simulated results	6.25	11.50	0.115
Theoretical value	6.28	11.91	0.117
Relative error	0.4%	3%	2%

. The overall small relative errors indicate that the established dynamic model is reliable and accurate.

3.3. Simulation of the Clamping Force Tracking Phase

After the brake clearance elimination phase, the braking process enters the clamping force tracking phase. The braking torque generated by the motor is transmitted to the DPRS through a two-stage helical gear transmission, which reduces speed and amplifies torque. The screw engages with 12 rollers via thread interaction, generating axial contact forces. These forces are transferred to the rollers and subsequently to the nut meshed with the 12 rollers, driving the nut into linear motion. The nut is rigidly connected to the piston, which ultimately pushes the friction pads against the brake disc to generate the clamping force. Thus, the axial contact force produced by the DPRS is finally converted into the braking clamping force.

Under the 1.3 g braking intensity, the angular velocity of the brake disc is shown in Figure 8. The simulation results indicate that the vehicle decelerates from an initial speed of 80 km/h to stationary and completes the braking process within 1.7 s. Furthermore, the braking deceleration reached 13 m/s² during this process. This value exceeded the regulatory requirement of 4.87 m/s², thus complying with the design specifications.

Under the 1.3 g braking intensity, the axial contact forces between the inner/outer friction pads and the brake disc (i.e., the braking clamping force) and the axial contact forces among the DPRS components are shown in Figure 9. It is observed that the braking clamping forces exerted by both the inner and outer friction pads on the brake disc are 17,700 N, with a 3% deviation from the target design value of 17,163 N, indicating a relatively small error. Meanwhile, the axial contact forces between the rollers and the screw, as well as between the rollers and the nut, are equal in magnitude at 1475 N, which corresponds to one-twelfth of the total braking clamping force.

The clamping force between the inner friction pad and the brake disc under different braking intensities was calculated according to reference [22]:

$$F={\displaystyle \frac{2\pi T_{\mathrm{m}}i{\eta }_p}{{\mathrm{L}}_0}}$$

where T_m represents the motor output torque; i denotes the total transmission ratio of the reduction mechanism; ${\eta }_p$ is the total system efficiency; and L₀ is the lead of the screw.

The clamping force between the inner friction pad and the brake disc under varying braking intensities was calculated. The comparison between the theoretical and simulation results of the clamping force is shown in Figure 10, where the overall error remains relatively small. This further validates the accuracy of the established dynamic model.

Finally, the axial contact forces corresponding to the motor drive torque of the DPRS under different braking intensities are obtained, as shown in

Table 4. Axial contact force of DPRS.

Braking Intensities a (m/s2)	Axial Load F (N)
0.2 g	216
0.3 g	306
0.5 g	550
0.8 g	1127
1.0 g	1300
1.3 g	1475

. These values will serve as boundary loads for subsequent assembly error analysis of the screw and rollers.

4. Mechanics Simulation Analysis Considering Assembly Errors

The assembly relationship of the three-dimensional model of the DPRS during dynamic analysis is considered ideal, also serving as the ideal assembly model for finite element analysis. In actual assembly process, inevitable assembly errors occur, necessitating an analysis of the impact of screw and roller assembly errors on contact stress in the DPRS. Due to the symmetry of the DPRS mechanism, one-twelfth of the DPRS model is retained following the methodology in reference [13]. Based on this, a finite element model of the DPRS is constructed considering assembly error by varying the positions of the screw and roller to analyze mechanics behaviour of DPRS.

4.1. Finite Element Model of the DPRS

A finite element model of the DPRS under ideal assembly is shown in Figure 11. To reduce computational cost, local mesh refinement was applied only at the contact regions between roller–nut and roller–screw. The mesh sizes were set as 0.02 mm at the nut contact zone, 0.03 mm at the roller contact zone, and 0.02 mm at the screw contact zone. The element type employed was temperature–displacement coupled C3D8T (8-node thermally coupled brick element with trilinear displacement and temperature). The model contains 80,327 elements for the nut, 147,490 elements for rollers, and 21,024 elements for the screw. As a result, the total mesh count was 248,841 elements.

All components of the DPRS are made of GCr15 steel, with a critical equivalent stress threshold for plastic deformation at 1617 MPa. The plastic strain values generated under different braking intensities are presented in

Table 5. Plastic strain–stress relationship of GCr15.

Plastic Strain	Stress σe (MPa)
0	1617
0.001587	1699
0.002983	1894
0.015157	2107
0.025784	2120

. The maximum allowable equivalent plastic strain during normal operation is set to 0.001, as defined in reference [14]. The boundary loads are derived from the axial contact forces obtained through dynamic simulation of the DPRS.

For ideal assembly model, the overall half-sectional stress contour of the DPRS and the stress contours of the screw and its adjacent roller under a braking intensity of 1.0 g are shown in Figure 12, Figure 13 and Figure 14. It is observed that the maximum equivalent stress of 1453 MPa occurs on the screw, with nearly identical equivalent stress values on the screw and its adjacent roller. Furthermore, the stress distribution along the thread flanks of the screw is non-uniform, decreasing progressively along the axial direction. This phenomenon arises from uneven load distribution between each contacting thread flank pair of the screw and roller, a pattern consistent with the finite element simulation results in reference [13]. As a result, it confirms the validity of the established finite element model.

As shown in Figure 15, the load distribution unevenness coefficient (ratio of actual to theoretical load) for each thread segment of the screw and roller, as well as the corresponding results from reference [13], is presented. The finite element solution aligns closely with the load distribution unevenness coefficient reported in reference [13]. This demonstrates that the established finite element model is sufficiently accurate for conducting mechanical simulation analyses of assembly errors in the screw and roller.

4.2. Mechanical Analysis of Screw Offset Error

Screw offset error in the DPRS refers to the displacement of the screw from its original installation centerline along a specific direction. This error reduces the contact area between the screw and rollers, thereby increasing the contact stress under same loading intensities. A screw offset along the radial direction of the roller is representative [23], as illustrated in Figure 16. The impact of varying radial offsets on the contact stress within the DPRS mechanism is investigated.

Under different braking intensities (0.2 g, 0.3 g, 0.5 g, 0.8 g, and 1.3 g), the maximum equivalent stress and equivalent plastic strain results for the contact regions between the screw and rollers, as well as between rollers and the nut, are presented in Figure 17, Figure 18, Figure 19 and Figure 20. It is observed that the maximum equivalent stress on both sides of the rollers increases with higher braking intensity for the ideal assembly model. At 1.3 g braking intensity, the maximum equivalent stress on both sides exceeds the critical threshold, resulting in measurable plastic strain. However, the maximum equivalent plastic strain remains below the critical equivalent plastic strain threshold (0.001, as defined in reference [14]) for all braking intensities. Therefore, the DPRS operates reliably under all tested intensities when assembled ideally.

From Figure 17 and Figure 18, it can be observed that as the screw offset distance increases, the maximum equivalent stress on the roller–screw interface rises more sharply with increasing braking intensity compared to that on the roller–nut interface. When the screw offset reaches 0.06 mm, the maximum equivalent stress on the screw side exceeds 2000 MPa at 0.8 g braking intensity, whereas the maximum equivalent stress on the nut side remains below 2000 MPa across all braking intensities. With a screw offset of 0.1 mm, the maximum equivalent stress on the screw side reaches 2450 MPa under 1.3 g braking intensity, while that on the nut side is 1980 MPa.

From Figure 19 and Figure 20, it can be seen that when the screw offset reaches 0.06 mm, the maximum equivalent plastic strain on the screw side exceeds the critical equivalent plastic strain at a braking intensity of 0.8 g. In contrast, the maximum equivalent plastic strain on the nut side remains below the critical value under all braking intensities. When the screw offset increases to 0.1 mm, the maximum equivalent plastic strain on the screw side reaches 0.026 under 1.3 g braking intensity, while that on the nut side is 0.009, which does not exceed the critical threshold.

This indicates that screw offset has a more significant impact on the contact stress on the screw side of the DPRS. This is primarily because the offset mainly reduces the meshing contact area between the screw and rollers, with little effect on the contact stress between the nut and roller side. Therefore, in practical assembly, the allowable screw offset error should be limited to within 0.04 mm.

4.3. Analysis of Roller Misalignment Error

Roller misalignment error in the DPRS refers to a scenario where after the screw deviates by a certain distance, the roller rotates about a point and tilts at an angle (θ) in a specific direction, deviating from its original installation centerline. This error simultaneously reduces the meshing contact area between the screw and roller as well as between the roller and nut, thereby increasing the contact stress under constant load intensities. A representative case is when the roller tilts along the tangential direction of the screw [24], as illustrated in Figure 21.

With a screw offset of 0.6 mm, the roller is assumed to rotate about its centre and tilt in the radial direction of the screw at different angles. Under various braking intensities (0.2 g, 0.3 g, 0.5 g, 0.8 g, and 1.3 g), the maximum equivalent stress and equivalent plastic strain at the contact interfaces between the screw and roller, as well as between the roller and nut, are shown in Figure 22, Figure 23, Figure 24 and Figure 25.

From Figure 22 and Figure 23, it can be observed that as the roller misalignment angle increases, the increment in maximum equivalent stress at the meshing contacts between the roller and both the screw and nut sides becomes more pronounced. This is especially true when considering higher braking intensities, as compared to ideal assembly intensities. When the roller has a misalignment angle of 0.04°, the maximum equivalent stress on both the screw side and nut side exceed 2000 MPa at 1.3 g braking intensity. When the misalignment angle is 0.1° with same braking intensity, the maximum equivalent stress on the screw side reaches 2470 MPa, while that on the nut side is 2350 MPa.

From Figure 24 and Figure 25, it can be seen that when the roller has a misalignment of 0.04°, the maximum equivalent plastic strain at both the screw side and nut side exceeds the critical equivalent plastic strain under a braking intensity of 1.3 g. When the misalignment increases to 0.1°, the maximum equivalent plastic strain on the screw side reaches 0.026 and that on the nut side reaches 0.024 under the same braking intensity. Therefore, roller misalignment significantly affects the contact stress on both the screw and nut sides. This is mainly due to the simultaneous reduction caused by the misalignment in the meshing contact area between the screw and roller, as well as between the nut and roller. In practical assembly, the allowable roller misalignment should therefore be limited to within 0.02°.

5. Conclusions

This study investigates the influence of assembly errors in the screw and rollers on the contact stress within the DPRS mechanism of an EMB actuator by using finite element analysis. The findings provide guidance for setting allowable thresholds for assembly errors in DPRS mechanisms. The main conclusions are as follows:

(1)

As braking intensity increases, the axial contact force acting on the DPRS mechanism also increases, resulting in a higher clamping force between the brake disc and friction pad. Furthermore, the maximum braking force should meet regulatory requirements, indicating that the designed DPRS should also satisfy relevant braking standards.

(2)

Both screw offset and roller misalignment errors lead to increased contact stress in the DPRS mechanism. Screw offset has a more significant effect on the contact stress at the screw side, whereas roller misalignment affects the contact stress on both the screw and nut sides.

(3)

The DPRS mechanism will exceed the critical equivalent plastic strain under specific braking intensities (e.g., 0.8–1.3 g) when either the screw’s deviation distance surpasses 0.06 mm or the roller’s inclination angle exceeds 0.04°, thereby compromising the normal operation of DPRS and the operational reliability of the EMB system actuator.

The thermo-mechanical coupling analysis of DPRS for the EMB system will be carried out to more comprehensively evaluate the mechanical behaviour under extreme operating conditions in future work. Additionally, fatigue life assessment of the DPRS within the EMB system will also be investigated, which will contribute to enhancing the long-term reliability of the EMB actuator and provide critical support for practical engineering applications.