Implicit Elastoplastic Finite Element Analysis of a Wheel Bearing Shaft Clinching Process Using the Multi-Body Function

Abstract

An implicit, elastoplastic, finite element method (FEM) with multi-body treatment function was applied to accurately analyze the real-world shaft clinching of a duplex-pair tapered roller (DPTR) wheel-bearing unit (WBU) under minimal assumptions during modeling. The inner races were viewed as elastoplastically deformable and were fitted to the hub shaft before clinching by imposing a thermal load reflecting the mechanical load of press-fitting. The forming roller (i.e., the power source) was considered to be force-prescribed, similar to the approach on real shop floors. The predictions focused on the homogenizing stage, during which the two inner races bear the preload. At this time, local plastic deformation occurred at the end of the hub shaft and in the armpit area and the cavity was either maintained or enlarged. The predicted cavity size in case of force-prescribed forming roller increased, compared with the velocity-prescribed forming roller. The residual stress became axisymmetric and was divided into two parts by the cavity. These findings allow engineers to control the pre-stresses imparted to the inner races of tapered roller bearing assemblies.

1. Introduction

Rotary forging [1,2,3,4,5] (a typical incremental forming technology) is useful when assembling wheel-bearing units (WBUs) because of its cost-effectiveness and production of high-quality parts, as well as because of the low forming load; this low load is essential to prevent bearing damage. It has pretty different features in terms of metal flow lines, contact surfaces, and forming loads from the conventional forging.

Assembly is the final step of WBU fabrication. Any design or engineering failure is thus expensive, emphasizing the need for finite element (FE) simulation [6]. The assembly process of a WBU is complex [7] and needs to be optimized [8]. The required computational time is considerable. Thus, the process has rarely been scientifically studied. Changes are made by means of traditional trial-and-error. A clinching process of assembling the WBUs thus belongs to one of the complicated mechanical or structural engineering problems involving large elastoplastic deformation, springback, material inhomogeneity, etc. [9,10,11,12].

One of the most important factors during development of a WBU clinching process is control of the cavity formed near the shoulder of the upper inner race and/or the hub shaft armpit (Figure 1). The present numerical and experimental data indicate that if the cavity around the bent shaft corner disappears, the bearing inner race experiences considerable stress and may fracture during shaft clinching. This problem tends to be more serious for the DPTR bearings. During assembly, stress may be concentrated on the bearing inner races, triggering extensive elastic deformation [13,14].

Several practical advances have been made over the past two decades. Toda et al. [15] were the first to predict WBU formation. Kajihara [16] used computer-aided engineering to develop bearings for automobile wheel assembly and to simulate shaft clinching. In the early years of FE analysis of WBU rotary forging/assembly, Moon et al. [13,14], Cho et al. [17] and Munshi et al. [18] made some assumptions to achieve engineering solutions using hexahedral or tetrahedral Fes. Moon et al. focused on the cavity (Figure 1) formed near the shoulder when predicting a qualitatively acceptable hub shaft deformation, although they assumed that forming roller velocity varied with time. Notably, in all early research works regarding FE simulation of WBU assembly, the velocity constraints were imposed on the forming roller [19,20,21].

WBU assembly simulation requires extensive computational time. Many efforts have thus been made to reduce this time. Moon et al. [13,14] simulated only the major deformation region. Cho et al. [17] and Munshi et al. [18] used coarse mesh systems to reveal unacceptable deformations of the hub shaft end in terms of the cavity and/or the end itself. Many researchers employed explicit elastoplastic FEMs that were numerically vulnerable. Shu et al. [19] performed explicit elastoplastic FE analysis of shaft clinching, focusing on the radial deformations of the inner rings; the results optimized the loading path of the forming die. They simulated a duplex pair ball bearing but did not deal with the contact problem between the inner races. Kang et al. [22] simulated axisymmetric, three-dimensional orbital formation using an explicit FEM. Lee et al. [23] used a simplified axisymmetric model for this analysis. Moon et al. [24] performed FE analysis using a minimized volume; this assumed that the mount-side degrees of freedom were constrained. Cho and Koo [25] performed axisymmetric and explicit three-dimensional FE analysis of the orbital forming process of automotive WBUs, showing the severely oscillating forming load-time curves regardless of mesh refinement. During the past decade, more realistic models have been developed. An et al. [26] used an implicit elastoplastic FEM to simulate orbital assembly of a single ball bearing race onto the WBU hub shaft, then compared their deformation predictions with experimental data. However, they analyzed shaft clinching (using a velocity-prescribed forming roller) of a single ball bearing race in the absence of cavity formation. Qu and Zhang [27] predicted WBU assembly using an elastoplastic FEM that was experimentally validated. Nam et al. [28] reduced the computational time pretty much using artificial planes of symmetry (by one or two) when obtaining engineering solutions. They assumed that the contact/plastic deformation regions were small.

Xiong et al. [29] employed implicit and explicit FEMs with hexahedral FEs to assemble a WBU. They showed an acceptable agreement between the predictions and the experimental data. However, Xiong et al. analyzed shaft clinching (using a velocity-prescribed forming roller) of a single ball bearing race alone. Wang et al. [30] conducted a numerical and experimental study on the rotary forging process for shaft clinching of a hub bearing unit considering the interference assembly, loading, and unloading of the rotary forging process and exhibited the possibility of determining and optimizing its processing parameters using the simulation technology.

It is noteworthy that most researchers have imposed velocity-prescribed boundary conditions because the corresponding constrained dies are amenable to numerical analysis. However, in the real world, dies that engage in rotary forging are controlled by force. The case of mechanically coupled inner races needs a precision FE simulation because of the importance of the preload exerting between the two races. Most researchers (with the exception of Toda et al. [15]) studied the assembly of a single ball bearing race onto the WBU hub shaft.

Here, a rotary forging process with a force-prescribed forming roller for fabricating DPTR WBU was analyzed by the implicit elastoplastic FEM with a multi-body treatment scheme. The finite element analysis was conducted with an emphasis on accurate predictions of the tight fitting of the bearing races with the hollow shaft before rotary forming and cavity formation around the armpit of the bent shaft during rotary forming, as well as the effects of this cavity formation on the residual stresses in the bearing inner races.

2. DPTR WBU Assembling Process and Analysis Model

Figure 1 shows a cross-sectional view of a DPTR WBU [31] assembled by means of rotary forging (orbital forming, also known as shaft clinching). A distinct cavity is evident in the armpit of the bent shaft. Note that the cavity shown is a little different from that before its cutting, because springback after the cutting changed the cross-sectional plane. Thus, the cavity shown in Figure 1 is only qualitative in nature. The cavity greatly affects both assembly and service reliability. Cavity prediction is thus important. The DPTR WBU in Figure 1 was manufactured using a process that differs greatly from the process used in the case of mechanically separable bearing [26,29]; DPTR WBU manufacture requires preloading of the inner races and careful control of the forming load. The inner DPTR races exhibit high-level elastic deformation and stresses that become concentrated at weak points (Figure 1). Moon et al. [24] predicted the mechanical non-contact interface between the hub shaft and the inner race during rotary forging using a rigid-plastic FEM and a simplified system (with a fixed forming roller velocity). The predicted cavity width and length were smaller than the experimental measurements.

The forming roller downward velocity profile is important in terms of shaft clinching, because springback becomes greater than plastic deformation at the final sizing (or homogenizing) stage. This aids cavity formation and moves the target preload to the inner races of the bearing (without causing damage). This reduces the forming load. Therefore, FE predictions should focus on both cavity formation and the preload, both of which are greatly affected by the forming load on the material.

Figure 2 shows a schematic of DPTR WBU assembly. The tilted upper die (i.e., the forming roller) rotates with respect to the tilted axis and moves down. The lower die (or tool) clamps the material and rotates with it. Note that the downward motion of the geometrically axisymmetric forming roller is controlled by the force imparted over time when operating the machine (Figure 3), rather than by velocity. The forming load-time curve indicates that the process can be divided into three parts: (1) lateral joining of the inner races and the shaft by means of press-fitting; (2) early forming or bending of the shaft end; and (3) continued homogenizing. The early bending stroke seeks to bulge the hollow shaft end (and bend it roughly); the homogenizing stage imparts local plastic deformation to the end of the shaft and assigns the preload to the inner races.

In contrast, for the analysis shown in Figure 4, the lower part of the material was fixed to ensure numerical reliability. Idling material that rotates without deformation can render numerical volumetric changes inaccurate. Instead, the upper die was allowed to also rotate with respect to the y-axis, such that the relative motions of dies or tools with respect to the material were consistent with real-world motion. Figure 4 shows the FE analyses of mesh systems [32] featuring three materials and various boundary conditions. The bearing inner races and the WBU hub shaft were assumed to be elastoplastic and the dies/tools were assumed to be rigid. The tribological conditions at the interface between materials were established, based on the experiences [33,34], by reference to the Coulomb friction law with a frictional coefficient of 0.1. The interfaces between the tools and materials were treated similarly.

A general-purpose implicit rigid- or elastic-thermoviscoplastic FE package AFDEX (Altair APA) [35,36,37,38] was used, based on the tetrahedral MINI-element scheme [39] with automatic adaptive meshing or remeshing [32]. The multi-body scheme [31] was employed to deal with the bearing inner races, which gives an accurate prediction of the stress and springback of the material during the homogenizing stage.

The hub shaft and inner races were made of S50C (AISI 1050 equivalent) and SUJ2 (AISI 52100 equivalent), respectively. The input data for the FE analysis model including material properties and process conditions were summarized in

Table 1. Material properties and process conditions.

Parameter	Value
Rotational speed of upper die $N_0$	$200 \mathrm{rpm}$
Inclination angle of upper die $\alpha$	5°
Flow stress of shaft	$\overline{\sigma }=520{\left(1+\overline{\epsilon }/0.001\right)}^{0.135} \mathrm{MPa}$ [20]
Flow stress of the two inner races	$\overline{\sigma }=1960{\left(1+\overline{\epsilon }/0.009\right)}^{0.08} \mathrm{MPa}$ [37]
Young’s modulus of S50C	200 GPa
Young’s modulus of SUJ2	212 GPa
Poisson’s ratio of S50C	0.29
Poisson’s ratio of SUJ2	0.3
Coefficient of thermal expansion of SUJ2	0.0000125/°C
Number of tetrahedral elements	300,000

. All working conditions were based on real-world manufacturing on industry shop floors. Note that the major working conditions (or parameters) of shaft clinching include the push-down force of the forming roller (Figure 3) and the tilt angle α. Both greatly affect deformation of the hub shaft and the pressure on the shaft. The shaft and inner races were made of S50C and SUJ2, respectively. The penalty constant was assumed to be 100,000,000.0 when contact was maintained between the hub shaft and the inner races. When the distance between the nodes of different materials was less than 0.01 mm, the nodes were considered to be a part of the contact interface if compression stress was imparted to that interface. The initial strains were assumed to be zero because no cold metal-forming had taken place. Note that the inner races were tightly press-fitted (in a mechanical sense) before assembly. The gap before the press-fitting was 0.01 mm, which was reflected by the shrink-fitting of the upper and lower inner races by the equivalent thermal loads of −35.5 °C and −32.7 °C, respectively. Thus, these thermal loads were applied to the respective inner races during shrink-fitting simulation before assembly simulation.

Figure 5 shows the effective stresses (Figure 5a) and radial stresses (Figure 5b) of the ready-to-be-formed material, preloaded along the interface by the thermal load equivalent of the initially negative gap after press-fitting. The radial stress component along the interface of Figure 5b (which satisfies Newton’s third law of motion) shows that the predictions were valid.

3. FE Predictions and Discussion

An implicit elastoplastic FEM [30,32,37,38] with a MINI-tetrahedral element mesh system was used to run the model defined above. No remeshing was required. The multi-body simulation employed a penalty method to manage the contact mechanics between the two inner bearing races and the hub shaft. In most real-world rotary forging processes, the stroke is controlled by force. However, all earlier works [7,19,20,21,27] imposed velocity profiles on the forming tool because it is more difficult to numerically model force-prescribed forming rollers than to numerically model velocity-prescribed forming rollers. However, force imposition is inevitable when seeking to predict the mechanics of the preloading or homogenizing stage at the final stroke (where springback predominates).

In real-world processes, the push-down force imposed on the material by the forming roller is hydraulically controlled. The forming roller velocity, which minimized the difference in push-down force between the predicted forming load and the real force, was calculated. It was assumed that the forming roller could not move backward during rotary forging or homogenizing, regardless of whether a forming load greater than the predicted force was imparted to the forming roller.

Figure 6a–c shows the effective stresses at the beginning of homogenizing, at the final stroke, and immediately after that stroke, respectively. Figure 6c thus shows the final residual stress. Figure 6 indicates that the effective stresses before and after the forming roller separates from the deformed hub shaft are distinctly different. Thus, the residual stress of Figure 6c becomes near-axisymmetric after the forming roller moves up.

Figure 7 shows the forming load variations over time and the input forming load–time curve. The predicted (controlled) load demonstrated acceptable agreement with the real forming load. The maximum and mean differences were 11.1% and 5.2%, respectively. The experimental forming load–stroke curve was similar to the predicted curve. It is noteworthy that the forming load–time curve in Figure 7 is almost the same with the observation of Toda et al. [15].

Figure 8 clearly shows the cavity between the clinched hub shaft and the shoulder of the bearing inner race at the final stroke. A similar cavity was observed in the related experiments in Figure 1. The new findings suggest that excessive stress would be imparted to the bearing inner races that form the hub shaft end if the cavity disappeared. Thus, cavity prediction is important because it greatly affects the mechanics of the inner races. However, no paper that predicted a cavity near the shoulder of the bearing inner races was found.

The predicted shapes of the hub shaft end of Figure 8a were compared with the experimental shapes of the hub shaft end shown in Figure 1. The predictions of the velocity-prescribed forming roller case of Figure 8b were also compared, revealing that the force-prescribed die example was able to better predict the cavity size. As noted above, quantitative comparisons between predictions and experimental data are impossible because of the extensive springback during cutting for preparing the cross-sectional view in Figure 1.

Notably, the formed cavity separated the residual stresses at the shoulder of the upper inner race into two parts (as shown in Figure 8c), which is mechanically valuable. One part prevents play between the hub shaft and the inner race of the bearing, while the other part maintains the duplex pair (i.e., by joining the two inner races of the tapered roller bearing).

The motion of point M in Figure 9 was observed, which oscillated between the lowest point of –0.22 mm and the highest point of 0.06 mm (with respect to the final position) when the forming roller was removed during homogenizing. Figure 9 emphasizes that the deformation of hub shaft approaches a steady-state condition in gradual before the homogenizing operation ends.

Figure 10 shows how the shaft end is subjected to plastic deformation. Although the amplitude was considerable, plastic deformation (evaluated using the effective strain rate) was locally concentrated at the right end of the shaft (Figure 10), implying that the force-prescribed die triggered plastic deformation only at the end of the shaft and in the armpit. This deformation created a larger cavity that usefully assigned the desired preload to the inner race and thus minimized the forming load. Notably the preload and forming load belong to the critical indicators of the process and product [40,41]. Such repetitive action and reaction associated with local plastic deformation and extensive springback are the essence of the active mechanics when homogenizing the material before the process concludes. During homogenizing, the cavity persisted and no stress was directly imparted to the convex corner (i.e., the shoulder of the bearing inner race). This renders it possible to manufacture WBUs by exploiting the plastic deformation associated with shaft clinching.

After homogenizing, residual stress was evident, which was near-axisymmetric. The maximum effective stress exerted on the upper inner race (1660 MPa at commencement of homogenizing) decreased to 1210 MPa when springback occurred as the forming roller separated from the material.

Figure 11 shows the deformations at (1) 0.95, (2) 1.17, (3) 1.50, and (4) 3.31 s; the strains and stresses are shown in Figure 11a,b, respectively. Figure 10 indicates no plastic deformation occurring in the bearing inner races since there was no residual strain. However, non-negligible effective stress (residual stress, Figure 6c) was imparted to the bearing corners. The predicted increase in the inner diameter of the lower bearing race was 0.02 mm, attributable to pre-straining, which is a measure of the integrity of contact between the inner hub shaft and the bearing races.

4. Conclusions

An implicit elastoplastic FE analysis of a DPTR WBU assembly (known as hub shaft clinching), including the tight fitting of two inner races with the hub shaft and rotary forging of the shaft end was performed. The multi-body treatment scheme based on the penalty method was adopted to accurately deal with the elastic deformation of bearing races. The hub shaft, two inner races, and forming roller were considered to be elastoplastic, elastic, and rigid, respectively. The force-prescribed forming roller with unknown velocity was utilized, which was similar to the forming roller employed in the real world.

Rotational motion of the material was constrained but the relative motion of the material of the forming roller was maintained by imposing the numerical motion on the forming roller. This reduces numerical error because the idling rotational motion of undeformed body could be removed. Idling usually compromises numerical convergence of the solution and volumetric change of the material. When the present approach was compared with the approaches in previous works featuring the velocity-prescribed forming roller, it became clear that even well-approximated velocities might not accurately predict deformation of the hub shaft end, particularly during homogenizing, because of the large springback. The present model is both practical and reliable, reflecting real-world situations by means of numerical simulation. It was shown that the predicted forming load-time (or stroke) curve was acceptable.

The FE predictions focused on the shape of the cavity between the upper inner race and hub shaft end, as well as the instant/residual stresses imposed on the inner races. It was possible to predict the extent of the cavity between the hub shaft and inner bearing race around the shoulder of the race; this separated the contact area into two discrete parts in terms of stress. Two different stresses were thus imparted to the inner radius of the inner race and the sidewall of the tapered roller bearing. The stress to the inner radius of the inner race prevents any play between the hub shaft and the bearing inner race; the stress to the sidewall of the tapered roller bearing tightly interconnects the duplex pair. It was found that the predicted shape of the cavity formed when a force-prescribed forming roller that was employed was a little greater than the predicted shape of the cavity created when a velocity-prescribed roller was employed, implying that a precision simulation of the homogenizing process was accomplished.

The key FE predictions of WBU assembly are greatly affected by homogenizing; currently, local plastic deformation may occur at the tip of the shaft. This greatly affects cavity formation and the preload imparted to the inner races when the forming load is minimized. Thus, forming load minimization allows engineers to control the preload. The predicted residual stresses become near-axisymmetric when the rotating forming roller moves up; the present predictions are thus reliable. Notably, because springback is high, a well-approximated velocity–time curve may not reliably predict plastic deformation in the region of the hub shaft end.