Dynamics Performance and Abrasive Wear of the Automotive Drive Shaft

Abstract

An automotive drive shaft is used in a front drive transmission vehicle for transmitting rotary movement from the gearbox output shaft to the wheels with a constant velocity. This paper presents the kinematic model of ball joint and tripod joint with consideration of clearance and proposes a simplified dynamics model of the drive shaft. The simulation of the dynamics model is given by the measured data. Also, the abrasive wear of the ball type joint is tested on the experimental setup with consideration of full loading. The results show that the dynamics of the motion will be affected by the clearance when the loading changes its direction. The motion will be stable due to the damping. It can be found that the abrasive wear of ball joint largely depends on the relative sliding velocity from the experiments. This will probably affectthe dynamics performance and the impact force, which will also cause the other type of failures.

1. Introduction

An automotive drive shaft is used in a front drive transmission vehicle for transmitting rotary movement from the gearbox output shaft to the wheels with a constant velocity. In general, it is composed of three parts: a fixed type joint, plunging type joint, and intermediate shaft. A ball joint is designed as the fixed type joint for vehicle steering on the wheel side without axial shaft movement. And a tripod joint is designed for allowing the axial movement or plunging of the shaft in order to absorb the axial vibrations initiated from the engine and transmission side. The plunging tripod joint is close to the gearbox, while the ball joint is close to the wheel.

The ball type joint consists of outer and inner races, a cage, and six balls which are equally arranged at 60° intervals fitting into matching half grooves on the two races. The tripod joint consists of an outer housing, which is formed with 3 track grooves extended along the shaft's axis, and a central spider, which is composed of 3 trunnions equally spaced at 120° circumferential intervals supporting needle and spherical roller assemblies, as shown in Figure 1.

Figure 1:

A typical drive shaft for front drive transmission vehicles.

It has been found that the ball type joint is usually failure caused by abrasive wear. This is the motivation to conduct this study. A ball type joint comprises outer and inner races, a cage, and six balls equally arranged at 60° intervals fitting into matching half grooves on the two races, as shown in Figure 2.

Figure 2:

A ball typical joint in a front drive shaft.

Kimata et al. [1] proposed an analytical equation based on dynamics to evaluate the effect of the frictional forces acting on the inside of a fixed ball joint and double offset joint. Hayama [2] started to investigate the dynamic motion and forces generated on a ball joint using the ADAMS program with a consideration of the Hertzian contact theory. However, the friction forces between the internal parts were not taken into consideration. Serveto et al. [3] showed that the sixth component of the secondary torque generated by the ball joint model during torque transmission could produce acoustic beats with the vibration of the engine. He also found that the inertia torque was negligible with respect to the high transmitted torque. Philpott et al. [4] presented a two-phase circular regression algorithm for extracting wear profiles from ball type joint. However, it is quite important to find out the root cause of abrasive wear in theoretical approach. Lee and Polycarpou [5] investigated diverse researches on the frictional characteristics of the roller in a tripod joint through analytical and experimental approaches. Mariot et al. [6] also suggested a mathematical model introducing a viscous and a Coulomb friction approach to the kinematic analysis of a tripod joint. Watanabe et al. [7] developed a closed-loop equation for the end-motion type of tripod universal joints and simulated the normal force acting on the roller groove. Moreover, Lim et al. [8] analyze the drive shaft coupling of the ball and tripod types of constant velocity joints.

However, there has been little investigation on a complete analysis of the ball joint and tripod joint coupling assembly. Therefore, a fully integrated model of the drive shaft is needed to lead a solid foundation for investigating of the front drive shaft. In this paper, the kinematic model of the two types of joint with consideration of the clearance and a coupling dynamics model are proposed in Section 2. The dynamics simulation is given by using the measured clearance in Section 3. In Section 4, the abrasive wear of the ball type joint is experimentally studied. Finally, the conclusions are given in Section 5.

2. Kinematic Model

As the drive shaft assembly is composed of three parts: a fixed ball type joint, a plunging type tripod joint, and a shaft, it is required to model the assembly in detail for the investigation of the dynamics performance. The models of two joints are presented as follows.

2.1. Ball Type Joint

Qi et al. [9] presented the kinematic model of ball type joint. The typical ball type joint can be illustrated in Figure 3. O is the interaction point of input axis and output axis. The distance between the center point of the ball P and the center of the outer race is r, the same as to inner race. And, therefore, ΔpO₁O₂ and ΔOO₁O₂ are isosceles triangle. C is the midpoint of line O₁O₂, t _s and t _r are unit vectors in the plane of ΔpO₁O₂,n _s is the normal vector of plane ΔpcO. The vector in the direction of C and O₁ is

$\text{and}\hspace{0.17em}\hspace{0.17em}{n}_s=\frac{k+k^{\prime }}{\mathrm{2}\cos \left(\gamma /\mathrm{2}\right)},$

(1)

where k and k′ are the vectors of input shaft and output shaft.

Figure 3:

Illustration of a typical ball type joint.

The vector of Oc is

$t_s=\frac{k-k^{\prime }}{\mathrm{2}\sin \left(\gamma /\mathrm{2}\right)}.$

(2)

The distances between the center of the balls and P are

$r_s=\sqrt{r^{\mathrm{2}}-e^{\mathrm{2}}{\cos }^2\frac{\gamma }{\mathrm{2}}}.$

(3)

It means that the ball is distributed in the circle (P, r _s ). To describe the center of the ball, the coordinator system is set up with the original P (n _s , r _s , t _r ). The vector between the center of the ball and P is

$r_p=r_{\mathrm{2}}{t}_s+r_{\mathrm{3}}{t}_r,$

(4)

where r₂ = esin(γ/2) + r _s cos⁡θ and r₃ = r _s sinθ.

Assume that the input shaft is with the transient angle position of θ₁ and rotating velocity of ${\dot{\theta }}_{1}k$ . Then the output shaft rotates in the direction of t _r and k′ with the angular velocity of ${\dot{\theta }}_2{k}^{\prime }-\dot{\gamma }{t}_r$ . The velocity of ball, ${\dot{r}}_p$ , can be rewritten in

${\dot{r}}_p={\dot{\theta }}_{\mathrm{1}k}\times r_p+{\dot{s}}_{\mathrm{1}},$

(5)

where ${\dot{s}}_{\mathrm{1}}$ is the relative slipping velocity of the ball to the race.

Vector k × r _p is with the same direction of the normal vector of the race. Also, the component of relative slipping velocity ${\dot{s}}_{\mathrm{1}}$ in the direction of the normal vector of race is 0. Therefore, (5) can be rewritten as

$\begin{array}{c}{\dot{r}}_p\left(k\times r_p\right)={\dot{\theta }}_{\mathrm{1}}\left(k\times r_p\right)·\left(k\times r_p\right),\\ {\dot{r}}_p\left(k^{\prime }\times r_p\right)=\left[\left({\dot{\theta }}_{\mathrm{2}}{k}^{\prime }-\dot{\gamma }{t}_r\right)\times r_p\right]·\left(k^{\prime }\times r_p\right).\end{array}$

(6)

For a given angle to the input shaft θ₁ and a constant angle of γ, the variables of the output angle θ₂ and the ball position angle θ can be determined by the above equations.

From (1) and (3), the vector of input and output shaft can be rewritten as

$\begin{array}{c}k=\cos \frac{\gamma }{\mathrm{2}}{n}_s+\sin \frac{\gamma }{\mathrm{2}}{t}_s,\\ k^{\prime }=\cos \frac{\gamma }{\mathrm{2}}{n}_s-\sin \frac{\gamma }{\mathrm{2}}{t}_s.\end{array}$

(7)

Because the vector of input shaft k is fixed, the output shaft is rotating along vector t _r with the velocity of $-\dot{\gamma }{t}_r$ . Therefore, the changing rate of k′ is

$k^{\prime }=-\dot{\gamma }{t}_r\times k^{\prime }=-\dot{\gamma }\cos \frac{\gamma }{\mathrm{2}}{t}_s-\dot{\gamma }\sin \frac{\gamma }{\mathrm{2}}{n}_s.$

(8)

Substituting (1) and (2), we can get

$\begin{array}{c}{\dot{t}}_s=\frac{\dot{\gamma }}{\mathrm{2}}{n}_s,\\ {\dot{n}}_s=-\frac{\dot{\gamma }}{\mathrm{2}}{t}_s.\end{array}$

(9)

Substituting the above equations into (5), the velocity of the ball can be rewritten as

${\dot{r}}_p={\dot{r}}_{\mathrm{2}}{t}_s+{\dot{r}}_{\mathrm{3}}{t}_r+r_{\mathrm{2}}\frac{\dot{\gamma }}{\mathrm{2}}{n}_s.$

(10)

From the geometric relation, the distance between the ball and input shaft is

$r_1^2=\sqrt{r_2^2{\cos }^2\frac{\gamma }{\mathrm{2}}+r_3^2}.$

(11)

The component velocity of the ball in the direction of t _s and t _r can be obtained, respectively, as follows:

$\begin{array}{c}{\dot{r}}_{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}e\dot{\gamma }\cos \frac{\gamma }{\mathrm{2}}+{\dot{r}}_s\cos \theta -r_s\dot{\theta }\sin \theta ,\\ {\dot{r}}_{\mathrm{3}}={\dot{r}}_s\sin \theta +r_s\dot{\theta }\cos \theta .\end{array}$

(12)

The changing rate of the distance of the ball and C is

${\dot{r}}_s=\frac{e^{\mathrm{2}}\dot{\gamma }\sin \gamma }{\mathrm{4}{r}_s}.$

(13)

Therefore, we can get

$\begin{array}{c}{\dot{\theta }}_{\mathrm{1}}{r}_1^2-\frac{\dot{\gamma }}{\mathrm{2}}{r}_{\mathrm{3}}{r}_{\mathrm{2}}\sin \frac{\gamma }{\mathrm{2}}=\cos \frac{\gamma }{\mathrm{2}}\left({\dot{r}}_{\mathrm{3}}{r}_{\mathrm{2}}-r_{\mathrm{3}}{\dot{r}}_{\mathrm{2}}\right),\\ {\dot{\theta }}_{\mathrm{2}}{r}_1^2-\frac{\dot{\gamma }}{\mathrm{2}}{r}_{\mathrm{3}}{r}_{\mathrm{2}}\sin \frac{\gamma }{\mathrm{2}}=\cos \frac{\gamma }{\mathrm{2}}\left({\dot{r}}_{\mathrm{3}}{r}_{\mathrm{2}}-r_{\mathrm{3}}{\dot{r}}_{\mathrm{2}}\right).\end{array}$

(14)

From (14), we can easily prove that ${\dot{\theta }}_{\mathrm{1}}={\dot{\theta }}_{\mathrm{2}}$ . It means that the constant velocities of the input shaft and output shaft are achieved by this design conditions. However, if we consider the clearance, the relation of the angle position of input shaft and the output shaft can be modeled as

${\theta }_{\mathrm{2}}={\theta }_{\mathrm{1}}\pm C_{\mathrm{1}},$

(15)

where C₁ is the clearance of the ball joint.

2.2. Tripod Type Joint

Tripod type joint was invented in 1910, and many researches were focused on studying its kinematic model. Kinematics of the tripod joint has already been illustrated in Figure 4 [10]. Using Euler angles Ψ, δ, and θ, the orientation matrix of the tripod with respect to the tulip is given by

${{}^{1}A}_2^{}=\mathrm{rot}(z,\psi )\mathrm{rot}(x,\delta )\mathrm{rot}(z,\theta )$

(16)

and due to the 120° respective orientation of the trunnions in the tripod plane, it has been shown that ψ = − θ. Since the joint angle δ always keeps small, ${{}^{1}A}_2^{}$ becomes

${{}^{1}A}_2^{}=\left(\begin{array}{ccc}\mathrm{1}-\frac{{\delta }^{\mathrm{2}}}{\mathrm{2}}\text{si}{\text{n}}^{\mathrm{2}}\theta & -\frac{{\delta }^{\mathrm{2}}}{\mathrm{4}}\sin \mathrm{2}\theta & -\delta \sin \theta \\ -\frac{{\delta }^{\mathrm{2}}}{\mathrm{4}}\sin \mathrm{2}\theta & \mathrm{1}-\frac{{\delta }^{\mathrm{2}}}{\mathrm{2}}{\cos }^2\theta & -\delta \cos \theta \\ \delta \sin \theta & \delta \cos \theta & \mathrm{1}-\frac{{\delta }^{\mathrm{2}}}{\mathrm{2}}\end{array}\right).$

(17)

Figure 4:

Illustration of tripod type joint.

Lee and Akbil [11], and the authors also [10], have given the kinematics closure equations between input angle φ and output angle θ for a given angle δ:

$\begin{array}{c}e\sin \mathrm{3}\theta =-d\sin (\phi -\theta ),\\ e\cos \delta \cos \mathrm{3}\theta =d\cos \left(\phi -\theta \right)-L\sin \delta ,\end{array}$

(18)

where e is the offset between tripod center and tulip shaft projected in the tripod plane and

$e=\frac{r}{\mathrm{2}}\left(\frac{\mathrm{1}}{\cos \delta }-\mathrm{1}\right).$

(19)

Since the joint angle δ and the deviation φ − θ from a CVJ have R₃ behavior, they can be developed in a Fourier series limited to the first term:

$\begin{array}{c}\delta ={\delta }_m-{\delta }_{\mathrm{0}}\cos \mathrm{3}\phi ,\\ \phi -\theta ={\theta }_{\mathrm{0}}\sin \mathrm{3}\phi .\end{array}$

(20)

The constants δ _m , δ₀, and θ₀ can be analytically or geometrically determined [8]. Since δ₀ is negligible versus δ _m , we can write the following:

$e=\frac{r}{\mathrm{2}}\left(\frac{\mathrm{1}}{\cos {\delta }_m}-\mathrm{1}\right).$

(21)

Approximations of δ₀ and θ₀ lead to

$\begin{array}{c}{\delta }_{\mathrm{0}}=\arctan \left(\frac{e}{L}\right),\\ {\theta }_{\mathrm{0}}=\arctan \left(\frac{e}{d}\right),\\ d=L\sin {\delta }_m.\end{array}$

(22)

Based on the theory mentioned above, the kinematic formula can be rewritten as follows:

$\phi =\theta +\frac{r_{\mathrm{1}}}{\mathrm{2}L}tg\delta t{g}^{\mathrm{2}}\frac{\delta }{\mathrm{2}}\cos \mathrm{3}\theta .$

(23)

However, this model is without the consideration of clearance. If we take the clearance into account, the nonlinear model can be described as

$\phi =\theta +\frac{r_{\mathrm{1}}}{\mathrm{2}L}tg\delta t{g}^{\mathrm{2}}\frac{\delta }{\mathrm{2}}\cos \mathrm{3}\theta \pm C_{\mathrm{2}},$

(24)

where C₂ is clearance of the tripod joint.

2.3. Simplified Dynamics Model of the Front Drive Shaft

It has been reported that the dynamics performance of the joint can be neglected when considering the dynamics performance of the front shaft. Then the front shaft can be illustrated in Figure 5.

Figure 5:

Simplified dynamics model of a front drive.

The dynamics model of the system can be described by the following equations:

$\begin{array}{c}{J}_{\mathrm{1}}{\ddot{\theta }}_{\mathrm{1}}+D_{\mathrm{1}}{\dot{\theta }}_{\mathrm{1}}-M_{\mathrm{12}}+M_i=\mathrm{0},\\ J_{\mathrm{2}}{\ddot{\theta }}_{\mathrm{2}}+D_{\mathrm{2}}{\dot{\theta }}_{\mathrm{2}}+M_{\mathrm{12}}+M_{\mathrm{23}}=\mathrm{0},\\ J_{\mathrm{3}}{\ddot{\theta }}_{\mathrm{3}}+D_{\mathrm{3}}{\dot{\theta }}_{\mathrm{3}}+M_{\mathrm{23}}-F_{f}R=\mathrm{0},\\ M_{\mathrm{12}}=\{\begin{array}{cc}{K}_{\mathrm{1}}({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}})\hfill & \hfill \\ \mathrm{0}\hfill & \hfill \end{array}\hspace{1em}\text{while}\hspace{0.17em}\hspace{0.17em}\left({\theta }_{\mathrm{1}}-{\theta }_{\mathrm{2}}\right)<C_{\mathrm{1}},\\ M_{\mathrm{23}}=\{\begin{array}{cc}{K}_{\mathrm{2}}({\theta }_{\mathrm{2}}-{\theta }_{\mathrm{3}})\hfill & \hfill \\ \mathrm{0}\hfill & \hfill \end{array}\hspace{1em}\text{while}\hspace{0.17em}\hspace{0.17em}\left({\theta }_{\mathrm{2}}-{\theta }_{\mathrm{3}}\right)<C_{\mathrm{2}},\end{array}$

(25)

where K₁ and K₂ are the equivalent stiffness of the drive shaft, assuming that K₁ = K₂, D₁, D₂, and D₃ are the damping coefficient, M _i is the torque generated by the engine, R is the radius of the wheel, F _f is the friction force between the wheel and the ground, J₁ is the effective inertia from engine to the gear box, J₂ is the inertia of the drive shaft, and J₃ is the effective inertia of the wheel.

3. Simulation

The models mentioned above are not considered the clearances. The clearances are really related to the dynamics performance of the transmission system. It has been conducted to measure the clearance in the test bench shown in Figure 6. The clearances of the ball joint and the tripod joint have been measured, as shown in Table 1.

Table 1:

The clearance measured by experiments.

Number	Backlash clearance (′)
	Outboard joint	Inboard joint	Assembly
1	7.2	22.6	29.8
2	8.3	21.8	30.1
3	7.6	19.6	27.2
4	8.5	23.5	32.0
5	8.2	24.2	32.4
6	9.0	25.7	34.7
Average	8.13	22.9	31.03

Figure 6:

Experimental setup.

By the above model, the simulation program is developed in MATLAB/Simulink as shown in Figure 7. We simulate the dynamics system of the proposed model with the given coefficients in Table 2. The input torque and the output torque of the driven shaft are presented in Figure 8 as well as the rotating velocities.

Table 2:

Simulation coefficients.

Simulation coefficients	Value
K1/, K2	5.089 × 104/Nm
D1/, D2/, D3	0.1 Nm/rad
M i	262 Nm
R	230 mm
J 1	20.5 Kg·m
J 2	0.1 Kg·m
J 3	1.05 kg·m2

Figure 7:

Numerical program coded in MATLAB/Simulink.

Figure 8:

Simulation results.

4. Abrasive Wear of Ball Type Joint

As given above, Qi et al. [9] have proved that constant velocities of the input shaft and output shaft are achieved. It has been found that the “constant speed” largely depends on the geometric shape of the race and the ball. Also, it has significant relative motion between the ball and the races, as well as the ball and the cage in each cycle. The previous studies have shown that it is relative to the abrasive failure. The formula for the calculation of the relative velocity between the tracks and balls has been proposed by Li [12]. The simplified model of the relative velocity between the ball and the outer race track Δω₀ and the relative velocity between the ball and inner race track Δω _i are

$\begin{array}{c}\left|\mathrm{\Delta }{\omega }_{\mathrm{0}}\right|=\omega [\frac{-R\cos \alpha \sin \theta \cos \delta }{\sin \phi }+R\sin \alpha \sin \theta \sin \delta +r\sin \theta \sin \delta ]\\ \left|\mathrm{\Delta }{\omega }_i\right|=\omega [\frac{R\cos \alpha \sin \theta \cos \delta }{\sin \phi }-R\sin \alpha \sin \theta \sin \delta +r\sin \theta \sin \delta \phantom{\frac{-R\cos \alpha \sin \theta \cos \delta }{\sin \phi }}],\end{array}$

(26)

where R is the radius of the ball; r is the distances between the center point of the ball and the center of the outer race; ω is the input/output velocity of the shaft; δ is input angular position of the ball; θ is the angle of the CV joint; φ is defined as the angle of joint; α is the contact angle between the races track and the ball, in general, = 15°~17°.

The experiment setup is illustrated in Figures 9 and 10. It composes two motor divers and two loaders, as well as two torque sensors. Four front drive shafts can be tested in the experimental setup in the meantime. In this experiment, the parameters are given in Table 3.

Table 3:

Experimental parameters.

Number	Parameters	Value
1	R	8 mm
2	r	56 mm
3	Testing torque	262 Nm
4	Testing speed	1000 rpm
5	Joint angle	δ = 4.48°
6	Test cycles	1.12 × 107

Figure 9:

Illustration of the experimental setup.

Figure 10:

Experimental setup.

With the experimental conditions given in Table 3, (26) can be used for calculating the relative slipping velocity of the ball joint. With MATLAB, the simulation could be done easily, as given in Figure 11.

Figure 11:

Relative slipping velocity of the ball joint.

The quantification of wear of the parts was inspected by visualization using a obtical microscope. Experiment result of the ball joints is summarized in Table 4 and the photos of the key components are shown in Figure 12. It can be found that the ball, the outer race track, and the inner race track have high wear rate than cages. This has been also reported in [4]. This phenomenon also reflects the theoretical study of relative speed.

Table 4:

The experiments results.

Number	Grading of ball joints
	OR track	IR track	Cage window		Ball
			F	R
1	6	6	7~8	9	6
2	6	6	7~8	9	6
3	6	6	8	9	6
4	6	6	8	9	6
Rating	Wear of components
10	Unmarked, as new condition
9	Very light superficial polishing
8	Distinctly polished zones
7	Heavily polished zones, light wear, discolored
6	Very heavy polishing, moderate wear
5	Evidence of local heavy wear or brinelling, but less than 2 mm
4	Definite spalling in evidence, more than 2 mm
3	Spalling and severe wear, component still functional, possible NVH problems
2	Large spalled areas, component damage imminent
1	Component damage, local seizures, heat damage

Figure 12:

Photo of the components.

5. Conclusions

This paper presented a simplified dynamics model with the consideration of clearance based on the kinematics motions of the ball type joint and tripod type joint. Also, the clearances are measured by experiments. According to the results, the following conclusions are drawn.

1. The clearances should be considered in the dynamics study, as it has impacted the performance.

2. The presented dynamics model can be used for quick prediction of the front drive shaft. But the accuracy of the model should be further improved in the future.

3. The abrasive wear of the ball type joint largely depends on the relative velocity between the ball and the races/cages with a constant loading. It is suggested to improve the surface hardness of the component that resists the abrasive wear.

4. It has been found that the clearance is significantly larger than its initial value. This is related to the abrasive wear of the components. This will probably affect the dynamics performance and the impact force, which will also cause the other type of failures.

5. Experiments with various relative velocities should be conducted for verification of the theoretical model in the future research.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper. Also there is no conflict of interest regarding the support of State Key Laboratory of Vehicle NVH and Safety Technology (no. NVHSKL-201005).