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Joint Mechanism and Prediction of Strength for a Radial Knurling Connection of Assembled Camshaft Using a Subsequent Modeling Approach.

Introduction

Assembled camshaft has tremendous advantage compared to conventional manufactured camshaft (like casting and forging) in the aspects of manufacturing accuracy, production cost, production efficiency, materials collocation, weight reduction, improved fuel economy, and emission-reduction etc. [I]. However, it is challenging to achieve high reliability of an assembled joint. De Abreu Duque et al. [2] summarized common connection methods applied in the assembled camshaft including shrink fit, welding, bonding, powder sintering, mechanical expanding joint, hydraulic expanding joint and knurling connection. Shrink fit, welding, and powder sintering would lead to undesired thermal expansion of cam lobe which would make it different to control the assembly accuracy and result in uncertainty of the joint reliability. Bonding, mechanical expanding, and hydraulic expanding joint could not supply enough joint strength for high requirements of reliability and uprated engines. Thus, the knurling joint has been known as the most applicable connection method in manufacturing assembled camshafts [3].

Knurling interference connection method for assembled camshaft was proposed in 1970s for military applications. With the development of assembled processes, the knurling connection was divided into axial and radial knurled methods. Since knurling assembling joint is the main joint method for assembled camshaft in automobile industry, it is important to have a systematic understanding of the joint mechanism and reliability.

Press-fit load is a determinant for designing the assembly equipment. The magnitude of deformation for shafts and cam lobes is highly relevant to assembled accuracy. The torque strength and the accuracy of cam lobe profile are important measurement indexes for assembled reliability [4]. In classical experiments, process parameters optimization is time-consuming, high cost, and tedious trial-and-error process. Finite element method (FEM) along with experimental verification could act as a powerful tool to summarize universal disciplines for assembled processes.

Therefore, the objectives of this study are to build a sequence modeling approach to predict the press-fit and strength of the connection assembled joint by simulating the procedures of shaft knurling, press-fit, and static torque test; and to study the influence of knurling parameters on press-fit and connection strength and provide technical support for knurling interference assembly. In this approach, the surface residual stress field was effectively transferred through the whole assembled joint manufacturing chain. The regularities of stress and strain of tubes and cam lobes were analyzed systematically. The regularities and affect mechanism of press-fit load, static torque strength and deformation of external contour of cam lobe were studied. The simulation results were validated by experimental measurements.

Joining Mechanism of the Radial Knurling Assembled Camshaft

As shown in Figure 1a, the radial knurling interference connection includes two subsequent processes which are teeth knurling and press-fit. First, the surface of a hollow tube was rolled by knurling tools and a series of protractile teeth with a direction perpendicular to axial direction of the tube were formed. Thus, the diameter in the knurled area was increased due to the protrusion from the surface of tube. The protractile tooth led to an interference fit between the tube with enlarged diameter and bulges along inner wall of the cam, as shown in Figure 1a, which is pre-lathed before assembled. Then the knurled tube was pressed into the inner hole of the cam lobe at room temperature to accomplish a joint. The interference fit joint between the cam lobe and the tube was not only strengthened by lower elastic deformation of the tube due to lower Young's modulus value, but also by the concave and convex material mosaic caused by plastic flow of the tube material into the quasi-spline interspace as illustrated in Figure 1b. The multiple joining mechanisms result in superior torque resistance which is consisted of both the friction and the shear load.

During the manufacturing process of radial knurling assembled camshaft, especially the press-fit process, the hollow tube was undertaken large amount of multiaxial impact load. An elastic deformation in radial direction of the tube during press-fit process was highly possible to occur due to the tube-to-cam positioning error and the large length-to-diameter ratio of the tube. This formation would further decrease the accuracy of axial and phase angle of the pressed-in cam lobes. Furthermore, the radial load on the cam lobe would increase as the increase of press-fit load. And this would lead to more radial displacement of the outer contour of the cam lobe which is not acceptable for an engine gas distribution system [5]. When the press-fit load is beyond certain threshold, the cam lobe would be at risk of increasing fracture probability in the regions of stress concentration.

Hence the press-fit load should be kept as low as possible. At least a safety factor of two for the static torque strength of the joint should be considered when designing a reliable radial knurling assembled camshaft for an internal combustion engine gas distribution system [6]. The typical torque strength should be more than 200 Nm for a typical automotive camshaft [3].

Experimental Procedure and Subsequent Simulation Methodology

In order to efficiently predict the reliability of the knurling assembled joint, a modeling approach which includes three subsequent models has been developed. The manufacturing chain of a knurling assembled joint along with its torsion strength was simulated, using the general finite element software Abaqus, and validated with measurement. Figure 2 gives an overview of the subsequent simulation methodology. First, a simplified knurling process model was created in which the formation of the teeth on the tube surface, residual stress field of the tube was obtained. Second, the residual stress field calculated from the knurling process model was imported as initial condition into the press-fit model which simulates the assembly manufacturing process. In this model, the press-fit load was recorded and validated with experimental data. The residual stress field of the joint was also recorded. Third, the residual stress of the joint was then transferred to the joint strength model to predict the torsion strength of the joint.

Experiment Details

The material for hollow tube used in this study is modulated 1045 steel and manufactured from cold extruded seamless steel tube. The cam material is GCr15 steel. GCr15 steel has a Vickers hardness of 780, which is much higher than that of 1045 steel. Geometrical dimensions of the tube and cam are listed in Table 1. The dimensions of cam are shown in Figure 1a. The knurling tool used in the experiment has a tooth height of 0.68 mm, addendum angle of 80[degrees], and the number of teeth is ten. Same geometry of the knurling tool was applied in creating the simulation model. Knurling experiments were carried out on an in-house developed CNC knurling machine as shown in Figure 3a. The hollow tube was clamped to the spindle of the CNC machine, and the rotation of the tube was driven by the spindle which was CNC controllable. Three knurling tools which were evenly distributed by 120[degrees] are drive by a hydraulic system. When the tube was rotating, the knurling tools were pressed against the tube surface. Then the tools were driven by the rotation of the tube due to the contact force between tube and knurling tools. A series of teeth in radial direction were generated on the surface of the tube. After the teeth are shaped, the positions of the knurling tools were then reset by the hydraulic system. As illustrated in Figure 3a, a cam lobe was clamped by a customized fixture which was capable to precisely adjust the phase angle of the cams. During the press-fit experiment, the tube was driven again by the spindle and move downwards to the cam at a speed of 2 mm/s. A pressure sensor was installed into the CNC machine to dynamically measure the axial load applied on the cam lobe during the press-fit processes. All the eight press-fit experiments were performed on a single hollow tube. Then the assembled sample was cut into four pieces for torsion experiment. The torsion experiment setup is shown in Figure 3b.

Modeling Assumptions

Several reasonable assumptions have been made to reduce the computational cost:

* The assembly processes for different cam lobes are exactly the same and have very limited influence on each other. For this consideration, the assembly process of only one cam lobe was included in the model.

* Based on the experimental observation that very limited amount of heat was generated during the knurling, press-fit, and torsion failure experiment, all the simulation models were purely mechanical analysis. Coupling with thermal analysis was not considered in this study.

* Only half of the tube was modeled in all three models due to the fact that the hollow tube is a rotational symmetric body. In the knurling process model, the knurling tools were modeled as rigid body because the tool material (GCr15) has much higher hardness than that of the tube material (1045 steel).

* In the press-fit and knurling process models, acute material flow on the tube surface existed. The classical Lagrange analysis which leads to mesh distortion and higher computational cost was not feasible. A coupled Euler-Lagrange (CEL) which is suitable for large deformation was applied instead [7].

Modeling Procedure and Boundary Conditions

In the knurling process model as shown in Figure 4a, two knurling tools are horizontally placed against the tube axis.

The radial movement and rotation of knurling tools were driven by the friction force with tube. The simplified 3D models of press-fit process model are shown in Figure 4b. Base on the press-fit model, the positioning fixtures for cam lobes were removed. And the torque was applied on the reverse mold to implement torsion simulation. The torsion model is shown in Figure 5a. Considering the severe deformation for the tube material, the Euler mesh was introduced to discrete the tube, with the type of C3D8R. The mapped mesh for the cam lobe is shown in Figure 5b. Because the part described by the Euler method was not allowed to have any displacement, the movement in each step depended on the displacement of knurling tool and cam instead of tube. As shown in Figure 5c. the boundary conditions in the torsion model were described as follows: (1) the degree of freedom of radial was restricted for inner surface of tube in each step ([U.sub.R] = 0); (2) the longitudinal symmetry surface was kept restrained along axial direction ([U.sub.z] = 0); (3) two side surfaces were restrained along circumference direction(([U.sub.T] = 0). The symmetric surfaces of cam should be restrained in circumference degrees of freedom ([U.sub.T]= 0) in the process of press-fit.

Base on the press-fit model, the positioning fixtures for cams are removed and a torsion mold is pushed to the position of cam, while the torque is imposed on the reverse mold to implement torsion simulation, as shown in Figure 4.

Simulation Conditions and Johnson-Cook Material Model

High tangential and normal force was applied on tube from the knurling tools and cam lobes, which led to significant elastic-plastic deformation on the surface of the tube. A shear stress-based friction model was used in the knurling and press-fit simulation. Many new surfaces would be generated on the knurling region of the tube, so a friction coefficient of 0.28 was chosen [8]. The torsion model was created under the working condition of interference assembles without lubricant, and the coefficient of friction was fitted to 0.2 [8].

Due to the significant discrepancy of stress state and severe deformation of material through knurling, press-fit and torsion analysis, the Johnson-Cook (J-C) constitutive model and failure criterion were chosen for the simulation. Chen [9] studied the plastic stress-strain behavior of 45 steel, using MTS material experiment machine and SHPB test, in range of 20[degrees]C-75[degrees]C in temperature and [10.sup.-4]/s-[10.sup.3]/s in strain-rate. Moreover, the materials are the same in this article compare to Chen's work. Thus, the parameters of constitutive models of J-C are suitable for simulating the quasi-static plastic flow and failure of the material in this research. The influence of temperature was ignored because of little calorific value in each step. The J-C model could be simplified and given as:

[[sigma] = (A + B[[epsilon].sup.n]) x (1 + Cln[epsilon]*)] Eq. (1)

[[epsilon]* = [[epsilon]/[[epsilon].sub.0]]] Eq. (2)

Where A is the yield strength; B is Material constant, n stands for parameters of strain hardening; C is the sensitivity parameter of strain rate; e represents the equivalent plastic strain; [epsilon] stands for strain rate; [[epsilon].sub.0] stands for reference strain rate.

J-C failure model is based on damage accumulation theory to describe the variations of load history with strain rate, temperature, and stress triaxiality [10]. The strength and stiffness of material are affected by damage, in case the degree of damage up to critical value the press set as zero, and the degree of damage scale with damage strain [epsilon]'. The model without the influence of temperature could be revealed as:

[[epsilon]' = [[D.sub.1] + [D.sub.2]exp([D.sub.3][sigma]*)][1 + [D.sub.4] ln[epsilon]]] Eq. (3)

Where [D.sub.1]-[D.sub.4] are failure parameters of material; [SIGMA]* = p/[[sigma].sub.e] stands for parameter of stress state. Chen [11] studied the quasi static and impact dynamics features of 1045 steel in order to fitting the parameters of 1045 steel for constitutive and failure model, which were shown in Table 2.

Results and Discussions

Residual Stress and Tooth Formation in Knurling Process

The maximum principal plastic strain fields during the knurling process are shown in Figure 6. In these experiments and numerical simulation, the knurling knives were all with 0.68 mm tooth height and 80[degrees] addendum angle. The plastic strain concentration regions were observed locating in the subsurface with a depth no more than 0.2 mm. No obvious plastic strain was found at the depth of 1 mm from the surface of the tube. One could state that severe plastic deformation of the tube material only occurred at the top layer of the tube. As shown in Figure 6b, highest plastic strain was found in the contact area against the knurling tools. It was mainly caused by high contact force, and the material in the contact area tends to flow towards the free surfaces. With the increase of knurling time, the highest plastic strain spots began to move to the core of each "tooth" and the value increased to 0.5 as illustrated in Figure 6c. As shown in Figure 6d. the highest strain concentration regions were located at the core of each tooth. This was due to the fact that the tube materials which physically contacted with the knurling tool had the highest flow sufficiency. As the gap between the teeth and knurling tools became smaller, the tube materials located at the top of the teeth were under higher compressive stress. The previously contact area which is the high plastic strain spot was gradually pushed towards the core of the teeth. In this way, a high plastic residual strain area was formed in the core of each tooth. A final profile of the processed teeth is shown in Figure 6f. Pile-up at the edge of the pressed area was observed in the predicted profile which was validated by the experimental results as illustrated in Figure 6e. As shown in Figure 7. obvious compressive residual stress was found in the on the surface of the tube knurling region, the maximum equivalent residual and average stress were 210.9 MPa and 175.3 MPa respectively. According to the radial residual stress, the top of tooth maintains tensile, which is up to 233.8 MPa, on the contrary, average compression stress of 260.3MPa is at the bottom of tooth. Because of the stress singularities, the value of max residual compression was inaccuracy, instead the average value is more reliable. The residual stress field could significantly affected the following simulation results of press-fit and torsion, so that residual stress along with the displacement field were then exported and applied as initial conditions for the press-fit model which will be discussed in detail in the next session.

Press-Fit Process

The residual stress and displacement field of the tube calculated from knurling process were imported as initial condition in the press-fit model. The contours of Mises stress during the press-fit process were shown in Figure 8. Because of splined profile for the inner wall of the cam lobe, the contact between the tube teeth and the inner wall of the cam lobe was discontinuous during the whole press-fit process. The portion of the teeth which is directly contacted with the inner wall of the cam was pushed down towards the cam movement direction. This severe plastic deformation of tube was caused by high shear stress. As illustrated in Figure 8c, the highest Mises stress was observed on the top portion of the teeth which was up to 665 MPa. This value was close to the ultimate strength of the tube material, and fracture was possible to be observed on the top of the formed teeth. A detailed electron microscopy observation would be conducted to confirm the existence of the fracture in a future study. Part of teeth were squeezed by inner wall of cam to "flat" surfaces, and a protrusion was formed between those two "flat" surfaces as shown in Figure 8d. The "flat" surfaces kept interference with the inner wall of cam. The protrusion part of the teeth was actually embedded into the groove structure of the cam inner wall, which is very similar to a key connection (see Figure 10a). This specific key connection would provide additional radial torque strength to the camshaft under service.

The predicted deformation of the tube teeth was compared with the experiment and summarized in Figure 9. The connection region between the tube and the cam in the joint was a series of rectangular strips (illustrated by arrows shown in Figure 9a and b) distributed in the axial direction. Those rectangular strips were referred to the "flat" surfaces mentioned in the previously session. Residual stress was over 300 MPa for most of the regions on the tube surface. Axial joint cross-section for both simulation and experiment are shown in Figure 10a and b. Certain amount of tube materials were pushed into the groove of the cam as shown in Figure 9c. This predicted results further proved that a key-connection was formed in the press-fit process. Comparison of two results shown good agreement in terms of connection morphology of simulation and experiment, thus the simulation results to some extent could act as a good substitute to experiments to study processes of assembly and torsion. Both the predicted and experimental observed radial cross-sections of the joint are shown in Figure 10a and b. Tube materials were pushed into the valley of the teeth as pointed out by arrows in Figure 9c and d. The stuffing of the material into the valley increased the contact area between the tube and the cam which further increased the joint strength. The high compressive residual stress region was identified on the surface of the tube which was corresponded to interference connection. The generated high compressive residual stress could be beneficial for an improved fatigue performance since it tends to restrain the propagation of the crack growth under cyclic loading.

Simulated press-fit load was plot in Figure 11a and compared with experiment measurement. Similar fluctuation trend of the two curves was observed and both the peak and valley values for two curves were approximately the same with an error less than 9%. The cam contacted with the first tooth when the tube was pressed into the cam. The load gradually increased up to 6 kN as the movement of the cam. After the first tooth was completely pushed down as shown in Figure 8a, the contact force began to decrease down to 1.5 kN because of a gap existed between two neighboring teeth which lead to a decrease in resistance from material in the tooth and friction between cam and tube teeth was 1.5 kN. It was noted that the peak load was increasing as the increase of displacement of the cam. This was caused by the increasing contact area and friction between the cam and tube teeth during the press-fit process. In general, the simulation results overestimated the axial load with an average error of less than 9% except for the first peak load. The highest detected axial load applied on the cam was approximately 18.7 kN while the predicted value was 20.5 kN. And it was considered as a good estimation. However, the error could be sourced from simulation assumptions. As shown in Figure 11a. there is visible offset of peak load location between predicted and experimental values. The reason is that mapped mesh of cam in contact region with tube is only three layers otherwise the element is too small, which leads to less deformation than press-fit experiment in contact region of cam. The positioning accuracy of the cam and tube could be another major error source. If the cam and the tube was not coaxial, higher radial load could be generated which lead to a decrease of axial load.

Dimensional tolerance of the camshaft base circle for a typical internal combustion engine was kept within [+ or -] 0.03 mm. The radial displacement of cam presented as [U.sub.1] in Abaqus, is one of the main criterion of dimensional precision. The radial displacement contour of cam after the press-fit process is shown in Figure 11b. The maximum radial displacement on the outer profile of the cam was 0.0425 mm and it was located in the region with thinner wall thickness due to the lower stiffness of the thinner wall area. The displacement in most of the region was in the range from 0.0268 mm to 0.0346 mm. A follow-up precision grinding process with a depth of cut within 0.013 mm would satisfy the technical requirements of the camshaft which could ensure a superior accuracy of the cam profile, little material removal, and limited thermal damage to the cam for a reliable radial knurling joint.

Torsion Strength Prediction and Validation

Torsion analysis specimens were cut from the assembled camshaft as shown in Figure 12. Torsion experiments were carried out in the QD-B1 static torsion test stand. In the experiment, a piece of cam was fixed by clamp and the other piece was torqued till a relative-rotation between two pieces of cam. The computational and experimental curves of static torsion strength are shown in Figure 12. The simulation results agreed with the measured torsion strength very well with an error less than 5%. The curves could be divided into three stages: linear rising (elastic), nonlinear rising (plastic), and decline (failure). The maximum predicted torsion strength is 427.2 Nm while 392.3 Nm for the measurement, both of which are much higher than the requirements of typical static torsion strength for assembled camshaft for passenger can engines e.g. V6, 4.0 L about 200 Nm [3]. The corresponding torsion angles were 1.15[degrees] and 1.26[degrees] for predicted and experimental values respectively. The gap between the testing equipment and the sample fixtures in the torsion experiment could be responsible for the underestimation of rotational angle.

The continuous simulations including three steps contributed to transfer residual stress field from one step to the next one, for instance from knurling to press-fit. Residual stress had significant impacts on processing hardening and strain hardening. Moreover, the parameters of model of J-C including n and c, [epsilon] separately stand for processing hardening and strain hardening of material of tube teeth during the continuous steps. The influence of processing hardening and strain hardening in the plasticity model on press-fit load and torsion strength was studied. With the same tooth dimensions, the material model with strain hardening was named A while without strain hardening was named B. The comparison of results is shown in Figure 13. The strain hardening material had higher yield strength which led to larger press-fit load and torsional strength. The press-fit load and torsion strength values for model A were 11.8% and 10.7% higher than that of model B respectively, which means the effect of strain hardening had great impact on the assembled press-fit load and torsional strength.

Table 3 gives a summary of the press-fit loads and static torsion strengths for both simulation and experiment with different knurling tool geometries and feed percentage and changeless connection area between tube and cam. The feed percentage was defined as the ratio of the actual feed distance against the maximum feed. The discrepancies between the prediction and measurement for the press-fit load and the static torsion strength were within 10.5% and 10% respectively. It can be concluded that the proposed subsequent modeling approach is capable to sufficiently predict the reliability of the knurling assembled joint with high accuracy. As summarized in Table 3, a higher maximum press-fit load always leads to higher torsion strength, because of the larger interference the higher press-fit, raising the value of torsion strength. The highest maximum press-fit load was up to 24.8 kN with the tooth height of 0.68 mm, and addendum angle of 55[degrees]. In the table, the maximum static torque strength is 468.2 Nm which is more than twice as much as the technical requirement of a typical camshaft in internal combustion engines. For a single tooth of tube, a higher tooth height tends to need higher press-fit load and will create a stronger joint, while as the increase of addendum angle of the tooth, the press-fit load and torsion strength will significantly increase too. However, with the limited and confirmed connection area, the number of teeth decreased with the increase of the tooth height and addendum angle. So the values of press-fit and torsion strength are not the positive correlation with tooth height and addendum angle. A balance between high torsion strength and low press-fit must be reached. The combination #1 shown in Table 3 is beneficial for improving of the camshaft performance and minimizing potential damage to the cam lobe. The proposed modeling approach could provide a powerful numerical tool to optimize the knurling tool design and estimate the joint performance.

Conclusion

With a subsequent modeling approach, the reliability of a radial knurling assembled joint was accurately predicted. The proposed approach included the knurling process model, press-fit process model, and the torsion failure analysis model. Residual stress and displacement field was successfully transferred from knurling process model to the torsion model. The influences of geometrical dimensions of knurling tool and amount of feed on joining mechanism and the torsion strength were quantitatively analyzed and validated by corresponding experiments. The key conclusions of this study can be drawn as follows:

1. The residual stress generated during the knurling process was found to have significantly effects on the press-fit load and static torsion strength. Only the material on the surface of tube severe plastically formed after knurling. An average compressive residual stress of 227 MPa was predicted on the surface of teeth after knurling. Equivalent residual stress was found higher than 300 MPa in most regions in the connecting area of cam lobe and tube after press-fit process.

2. The impact of knurling tool dimensions to press-fit and connection strength were discussed as follows: tooth height of 0.68 mm and addendum angle of 80[degrees] for reasonably low press-fit load and high torsion strength. The predicted press-fit load curve presented a fluctuating rising trend with the maximum press-fit load up to 18.5 kN which was strongly agreed by the experimental validation. The predicted radial deformation of the cam lobe was in the range from 0.0268 mm to 0.0346 mm which ensured a high assembling accuracy.

3. A reliable joint was obtained through knurling assembly. The predicted and experimental maximum static torque strength were 438.2 Nm and 406.5 Nm which were all two times higher than the requirement of a typical camshaft. The simulation results of static torsion strength for all cases were highly consistent with the experiment, with errors less than 10%.

Acknowledgments

This research strain was supported by The People's Republic of China ministry of science and technology, the "Advanced CNC Machine Tools and Basic Manufacturing Equipment" National Science and Technology Major Project (No.2012ZX0409011).
Definitions/Abbreviations
CEL -   Coupled Euler-Lagrange
J-C -   Johnson-Cook


References

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[4.] Zhang, P., Kou, S.Q., Lin, B.J., and Wang, Y.M., "Optimization for Radial Knurling Connection Process of Assembled Camshaft Using Response Surface Method," Int J Adv Manuf Technol 77(1-4):653-661, 2015, doi:10.1007/S00170-014-6486-Z.

[5.] Londhe, A., Yadav, V., and Patil, A., "Evaluation of a 4-stroke 4-cylinder Diesel Engine Valve-train for Replacement of a Solid Camshaft with a Hollow Camshaft," SAE Int. J. Engines 2(1):254-262, 2009, doi:10.4271/2009-01-0405.

[6.] Blanchard, P., Nigarura, S., Trasorras, J., and Wordsworth, R., "Assembled Camshaft with Sintered Cam Lobes: Torsional Fatigue Strength and Wear Performance," SAE Technical Paper 2000-01-0397, 2000, doi:10.4271/2000-01-0397.

[7.] Qiu, G., Henke, S., and Grade, J., "Application of a Coupled Eulerian-Lagrangian Approach on Geomechanical Problems Involving Large Deformations," Computers and Geotechnics 38(1):30-39, 2011, doi:10.1016/i.compgeo.2010.09.002.

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Peng Zhang, Taiyuan University of Technology

Shuqing Kou and Chao Li, Jilin University

Zimin Kou, Taiyuan University of Technology

History

Received: 09 Oct 2017

Revised: 23 Feb 2018

Accepted: 14 Mar 2018

e-Available: 25 Jun 2018

Keywords

Assembled camshaft, Radial knurling joint, Assembled mechanism, Press-fit, Connection strength, Numerical simulation

Citation

Zhang, P., Kou, S., Li, C., Kou, Z., "Joint Mechanism and Prediction of Strength for a Radial Knurling Connection of Assembled Camshaft Using a Subsequent Modeling Approach," SAE Int. J. Engines 11(3):301-309, 2018, doi:10.4271/03-11-03-0020
TABLE 1 Dimension parameters of shaft and cam.

      Inner diameter [[PHI].sub.1] (mm)        24.8
Cam   Base circle diameter [[PHI].sub.2] (mm)  34.4
      Thickness L(mm)                          12.0
      Outer diameter [d.sub.1] (mm)            24.6
      Inner diameter [d.sub.2] (mm)            19.0
Tube  Length L (mm)                            40.0

TABLE 2 Parameters of J-C constitutive and failure of 1045 steel [11].

A/MPa      507
B/MPa      320
C            0.28
n            0.064
[D.sub.1]    0.15
[D.sub.2]    0.72
[D.sub.3]    1.66
[D.sub.4]    0.005

TABLE 3 Press-fit load and static torsion strength of simulation and
experiment.

                     Tooth height (mm)             0.68     0.45
Variables            Addendum angle ([degrees])   80       80
                     Feed amount (%)              90       90
                     Simulation                   18.7     14.4
Press-fit load (kN)  Experiment                   20.5     15.9
                     Error                         8.8%    10.1%
                     Simulation                  427.2    283.5
Static torque        Experiment                  392.3    258.3
strength             Error                         8.9%     9.3%

                       0.68    0.68
Variables             80      55
                      80      90
                      15.5    22.9
Press-fit load (kN)   17.1    24.8
                       9.4%    7.7%
                     355.0   468.2
Static torque        323.6   428.5
strength               9.7%    8.6%
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Author:Zhang, Peng; Kou, Shuqing; Li, Chao; Kou, Zimin
Publication:SAE International Journal of Engines
Article Type:Technical report
Date:Jun 1, 2018
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