Design and Testing of a New Type of Planetary Traction Drive Bearing-Type Reducer

Abstract

This paper presents the design and development of a new type of planetary traction drive bearing-type reducer. In this design, the transmission outer ring is replaced with an elastic ring. The design constructs a circular arc at the axial end of the rolling body’s contour line. This ensures that the contact point of this arc with the reducer’s outer ring and the inner ring’s axial end face is maintained on the radial traction contact line. As a result, it can replace the thrust bearing and provide an axial support function. It has the advantages of simple structure, easy processing, smooth transmission, and low noise. This paper first introduces the design and development process of this bearing-type reducer and presents systematic research on its transmission principle and dynamics. Subsequently, in response to the edge effect phenomenon of the outer ring contact line, the contour line of the outer ring is refined by adopting the shaping method used for bearing rollers, establishing a full circular arc profile shaping method, which significantly improves its edge effect. Finally, in our investigations, combined with experimental tests, a prototype of the bearing-type reducer was fabricated, and the speed ratio, torque, and transmission efficiency of the reducer were studied. The results demonstrate that the bearing-type reducer can achieve high transmission accuracy and efficiency. The transmission performance varies significantly under different lubrication conditions, with the peak efficiency reaching as high as 99.97% when using Santotrac 50 traction oil. The results verify the feasibility of the proposed design method and have the potential to be applied in wheel hub motors and robot joints.

1. Introduction

Currently, an automotive electric wheel and robot joint reducer may be used in the form of a planetary gear reducer, harmonic reducer, or helical gear reducer, among others [1]. However, in whatever form, as the core component of the electric drive reducer has a complex structure, high cost, and long processing cycle, this restricts the development of electric drive integration. Therefore, it is necessary to conduct in-depth studies on the electric wheel and robot reducer to reduce its manufacturing cost, improve the product qualification rate and transmission efficiency, and realize scale development [2].

Planetary gearing has advantages over classical gearing in terms of compactness, high ratios, high utilization, and uniform load distribution. It also has reduced weight compared to classical gearing and makes good use of space within the drive [3]. The main disadvantage of planetary gearing, however, is the need for high precision in the manufacturing and assembly of the drive elements. Planetary branches are often subjected to varying degrees of force, resulting in uneven load-sharing between the planets. There is also the possibility of dynamic imbalance when using a large number of satellites [4]. Furthermore, planetary gearing requires gear meshing, which brings about non-negligible vibration and noise, as well as efficiency losses. Planetary gearing is mainly used in automotive automatic transmissions, aircraft engines, joints and drive systems in robots, etc. Compared with gearing systems, due to the different transmission principles, planetary traction drives have higher contact pressure and smaller capacity, making them less suitable for low-speed and heavy-duty applications [5]. However, they possess inherent advantages in high speed, miniaturization, high-efficiency performance, transmission accuracy, and low vibration. These features have made traction drives a research hotspot in the transmission community [6,7,8,9]. They have been applied in industrial robots, medical equipment, automation equipment, aerospace, and electric vehicles. With the ongoing development of planetary traction drives, it is foreseeable that well-made planetary traction drives will see widespread use in the fields of pure electric vehicles [10] and robot manufacturing.

Planetary traction drive has been studied by scholars worldwide, who have designed various clever drives based on generating positive pressure. Katayama [11] designed a cylindrical roller traction drive using an interference fit to generate compression, eliminating spin between the traction components. This device requires precise interference calculation and high-quality manufacturing. However, it can suffer serious wear if overloaded, leading to reduced compression and transmission performance. Qian [12,13] proposed a hollow roller high-speed planetary traction mechanism for ultra-high-speed transmission above 100,000 rpm with a rigid outer ring. This mechanism is suitable for a large preload initially but may lose transmission capacity over time due to wear. Chen [14] designed an overloaded planetary traction reducer for electric scooter motors, calculating overload using finite element methods to ensure there was sufficient compression force. Hewko [15] created a planetary traction device with a tapered raceway for automatic compression loading, but it produces a spin effect, lowering efficiency. Ai and Rybkoski [16] developed a zero-spin planetary conical roller traction drive, avoiding spin with two rows of back-to-back planetary rollers. Yan [17] created a coplanar, symmetrically loaded planetary traction reducer with three planetary wheels and two tapered sleeves, featuring a helical face auto-loader with up to 30 ratios. Finally, Pan [18] designed a planetary cylindrical roller traction drive with a hollow sun wheel connecting three segments of logarithmic helical profiles for self-adaptive loading.

Ai [19,20], meanwhile, designed a zero-spin traction drive mechanism using tapered rollers to achieve adaptive loading with one loading roller and two support rollers. This improves the wear performance and increases the mechanical efficiency. Their finite element simulations and tests showed that the device operated smoothly with a peak efficiency of 97%. Flugrad [8] developed a traction drive that functions as both a clutch and a reducer, using a floating lever for adaptive loading. The rollers of the device can automatically apply normal force to realize the combination of different reduction ratios. The theoretical transmission efficiency calculated by the empirical formula is 98.4%, ranging from 94% to 99% with good lubrication and 91% to 96% without. Ai [21] designed a traction transmission device that utilizes the wedge effect to realize adaptive loading. The experimental study shows that it has the characteristics of a “wide range of high efficiency”. Shen [22,23] designed a hydraulic compression-type traction drive. Through numerical analysis, they studied the characteristics of double cone traction drive, revealing the relationship between normal loading force, minimum oil film thickness, transmission efficiency, and the transmission ratio. Yuan [24] designed a cone-ring-type traction transmission device and studied its self-assisted speed change mechanism through numerical analysis. This revealed the relationship of parameters such as speed change angle and slip–roll ratio with the traction force in the direction of speed change. Wei Wang [25] studied a special form of zero-rotation, high-constant-ratio (15:1) traction drive reducer for use with a high-speed motor in automotive applications. They plotted the efficiency of the transmission, which was up to 96.4%.

In summary, scholars have continually sought to optimize the traction transmission structure. They have aimed to achieve low or zero spin and a structure that is adaptive to loading. In addition, they have studied the effects of material, traction oil, and compression force on transmission performance. This has led to the development of many valuable methods and conclusions. Let us consider a couple of the main methods: Although the automatic pressurization method is adaptive, it has a complex structure. This makes it unsuitable for compact spaces. Meanwhile, the constant pressure pressurization method uses the rolling body’s elastic deformation to generate the required positive pressure. However, this approach is prone to contact fatigue failure of the rolling body. Furthermore, most of its outer ring is a rigid ring, which also needs to be used alongside bearings in practical applications.

This paper introduces a novel planetary traction drive bearing-type reducer, developed from a self-designed electric wheel structure and integrating traction transmission characteristics. Utilizing a lubricated elastic flow film, this design achieves mechanical transmission without gear meshing, providing benefits such as simple processing, smooth transmission, high efficiency, and low noise [10]. Innovatively, it replaces the thrust bearing for axial and radial support, thereby simplifying the transmission device’s structure and reducing costs. Ingeniously reshaping the bearing outer ring reduces the edge effect of the contact line. The use of an elastic outer ring, replacing a traditional rigid one, effectively prevents rolling elements’ contact fatigue failure. Compared to other planetary reducers, this bearing-type reducer features a simpler structure and higher efficiency. Furthermore, the design optimizes the internal structures of hub motors and robot joints, decreasing the manufacturing complexity and cost of high-performance reduction devices, and thereby offering significant economic and industrial application value.

2. Theoretical Analysis of Bearing-type Reducer

2.1. Structure of the Bearing-type Reducer

As shown in Figure 1, the bearing-type reducer mainly consists of an inner ring, an outer ring, rolling elements, and a planetary carrier. The outer ring is coupled to the external parts by screws, and the inner ring surrounds three rolling bodies. The planetary frame and planetary cover determine their relative positions. By applying interference fit to the inner ring of the bearing-type reducer, adequate compression force is generated through the interference fit among the inner ring, rolling elements, and the elastic outer ring. This ensures that the contact points between the rolling body arc and the reducer’s outer ring and inner ring shaft end face are maintained at the radial traction contact line, allowing it to replace the thrust bearing and provide axial support. The adoption of three planetary wheels enables a more compact, space-saving planetary gear train design, reducing manufacturing costs and complexity and achieving good load distribution and balance. By adding a planetary cover, not only can the position of the planetary wheels be fixed jointly with the planetary frame but also a contact-type seal can be installed on the planetary cover. This ensures the protection of the bearing-type reducer and guarantees adequate lubrication of the internal traction oil.

2.2. Mechanism of Traction Oil Film Formation

This bearing-type reducer utilizes a traction drive, similar to a wet friction drive [26,27]. It primarily transmits power through the positive pressure between two tightly fitted metal rollers. This design allows the friction wheels to generate a roll suction action during operation, drawing lubricant into the friction interface. Power is transmitted through a shear high-pressure oil film. Therefore, this mechanism can also be referred to as an elastohydrodynamic film traction drive or elastohydrodynamic lubricated wet friction traction drive.

The mechanism proceeds as follows. In traction drives, power is transmitted through the oil film between two metal rollers. There is relative slippage, meaning that accurate ratios cannot be guaranteed beyond the rated torque. Nonetheless, it can be said that a very thin oil film forms between the two smooth metal cylinders. The close fit between these rollers generates enormous pressure, forming a high-pressure oil film. This film can separate the contact surfaces and provide stable support for the rolling body, giving the reducer a radial support function.

The lubrication state of the traction drive directly affects its service life, and usually, a elastohydrodynamic lubrication state is chosen as the ideal lubrication state. Yet, to realize the traction drive, it is necessary to establish a certain thickness of oil film. However, the oil film is often very thin, and its surface roughness is at the micron level. When the oil film thickness is less than the surface roughness, the lubrication performance will be reduced [28]. Therefore, modifications have been introduced so that the planetary traction reduction bearings designed in this study have excellent transmission performance and full oil film lubrication (a film thickness ratio greater than 3 is required [29]) under rated operating conditions, to ensure sufficient service life.

The film thickness ratio is defined in Equation (1):

$$\lambda =\frac{h_{\mathit{min}}}{\sqrt{H_{\mathit{jf}1}^2+H_{\mathit{jf}2}^2}}$$

(1)

where $\lambda$ is the traction sub-film thickness ratio; $H_{\mathit{jf}1}$ and $H_{\mathit{jf}2}$ are the root-mean-square (RMS) values of the roughness of the traction sub-surface; and $h_{\mathit{min}}$ is the minimum oil film thickness of the traction sub-surface. Considering the material parameters, speeds, and loads tested in a previous study [30], the oil film thickness can be more accurately determined using Dowson’s empirical formula for line contact elastic flow film thickness [31] when the contact surfaces involve metal-to-metal contact and the lubricant is mineral oil [29]:

$$h_{\mathit{min}}=1.6{\alpha }^{0.6}{\left(\mu V\right)}^{0.7}{R}^{0.43}{E}^{0.03}{\left(\frac{L}{Q}\right)}^{0.13}$$

(2)

where $h_{\mathit{min}}$ is the minimum thickness for full-film lubrication, $\mu$ is the lubricant viscosity, $V$ is the coil suction speed of the moving pair, $R$ is the equivalent radius of curvature, $E$ is the equivalent modulus of elasticity, $L$ is the length of the contact line, and $Q$ is the external load.

2.3. Transmission Ratio and Transmission Efficiency Analysis

If the bearing-type reducer is regarded as a gear transmission, according to the division of the gear wheel system [32], it contains rotating planetary carriers, so it can be regarded as a turnover wheel system. Since there are no gear teeth in the rolling elements and there are inner and outer rings in the bearing-type reducer, the ratio’s formula needs to be rederived. When the bearing-type reducer is used individually, the inner ring (N₂) serves as the input, the outer ring (N₁) is fixed, and the rolling elements (N₃) are connected to the planetary carrier for the output, as illustrated in Figure 2.

The ratio of the angular velocities of the two rotating members in a bearing-type reducer is converted into the ratio of the radii [33], as expressed in Equation (3):

$$i_{1n}=\frac{N_1}{N_n}=\frac{{\omega }_1}{{\omega }_n}=\frac{r_n{v}_1}{r_1{v}_n}=\frac{r_n}{r_1}$$

(3)

where $n_n$, ${\omega }_n$, $v_n$, and $r_n$ are the rotational speed, angular speed, linear speed, and radius of the nth member, respectively, so the transmission ratio $i_{13}$ is obtained by the following derivation:

$$i_{12}^3=\frac{N_1-N_3}{N_2-N_3}=-\frac{N_1-N_3}{N_3}=-\frac{r_2}{r_1}$$

(4)

The transmission ratio $i_{13}$ can be found after obtaining the relationship between $n_1$ and $n_3$ through Equation (4):

$$i_{13}=\frac{N_1}{N_3}=\frac{r_1+r_2}{r_1}$$

(5)

Then, the transmission efficiency is calculated based on the ratio of output power to input power. When the bearing-type reducer is fixed on the outer ring, and the inner ring serves as the input while the planetary carrier serves as the output, the transmission efficiency can be derived from the following derivation:

$$\eta =\frac{P_{\mathit{output}}}{P_{\mathit{input}}}=\frac{F_s{V}_s}{F_p{V}_p}=\frac{T_s{r}_{p}i}{T_p{r}_s}$$

(6)

where $F_s$ and $F_p$ are the tangential forces of the inner ring and the planetary carrier, respectively; $V_s$ and $V_p$ are the circumferential velocities of the inner ring and the planetary carrier, respectively; $T_s$ and $T_p$ are the torques of the inner ring and the planetary carrier, respectively; and $r_s$ and $r_p$ are the radii of the inner ring and the planetary carrier, respectively.

3. Development and Design of Bearing-type Reducer

3.1. Overall Structure of Bearing-type Reducer

Figure 3 depicts the actual structure of the bearing-type reducer utilized in the electric wheel. This illustrates a specific application where two different-sized bearing-type reducers are combined to enhance the reduction ratio and torque. In this transmission system, the motor drives the small reducer and the inner ring of the large reducer rotation to ensure their angular velocities are synchronized. To ensure the same angular speed, the outer ring of the small reducer is affixed through the screw and the left end of the magnet steel set, while the rolling bodies of the small and large reducer are connected through a duplex planetary frame. Additionally, the outer ring of the large reducer is screwed to the rim to output the torque. In other words, the inner ring of the small reducer serves as the input and the outer ring of the large reducer serves as the output.

3.2. Geometric Design and Analysis at the Flange

As shown in Figure 4, when taking a single bearing-type reducer as an example, at the axial end of the rolling body profile (as shown by the red line in the figure), a circular arc is formed to ensure that the contact extensions of the rolling body with the outer ring and inner ring (as shown by the blue dashed line in the figure), as well as the contact points at the flange, are all located on the same traction line. Through this structure, the bearing-type reducer can realize axial and radial support, and it can be automatically centered during operation to prevent collisions caused by the eccentricity of the rolling body. This simplifies the structure while increasing the traction transmission capability. Therefore, it is especially suitable for compact, lightweight, and small-sized low-cost hub motors, such as automated guided vehicles and the reduction drive system of robot front-end joints.

To ensure the design model of the outer ring–rolling body, inner ring–rolling body at the flange intersection with the outer ring–rolling body, and inner ring–rolling body contact line in the same traction line, the following theoretical analysis is presented.

As can be seen from Figure 5, the rotary axis of the bearing-type reducer is defined as the x-axis, the line between the midpoint of the contact line of the outer ring–rolling body and the axis of the rotary axis is defined as the y-axis, $r_0$ is the radius of the chamfering radius of the rolling body flange, $R_0$ is the distance from the chamfering center of the rolling body to the x-axis, $R_P$ is the distance of the contact line of the outer ring–rolling body to the x-axis, $\theta$ is the angle of the flange of the outer ring with the x-axis, and Δ is the radial distance between the point of intersection of the outer ring–rolling body at the flange and the contact line. If the rolling body radius is denoted as $R$, the rolling body rotation angular speed as ${\omega }_1$, the rolling body rotation angular speed as ${\omega }_2$, the relative linear speed of the contact line as $v_1$, and the relative linear speed of the intersection of the flange and the extension of the contact line as $v_2$, combined with the parameters in Figure 5, the following can be obtained:

$$\mathsf{\Delta }=R_P-R_0-r_0\cos \theta$$

(7)

$$v_1={\omega }_{1}R+{\omega }_2{R}_P$$

(8)

$$v_2={\omega }_1(R+\mathsf{\Delta })+{\omega }_2(R_P+\mathsf{\Delta })$$

(9)

3.3. Bearing-type Reducer Design Program

The specific design requirements of the motorized wheel are shown in

Table 1. Motorized wheel design requirements.

Parameters	Value
Maximum axial size/mm	100
Maximum radial dimension/mm	75
Target ratio	20
Motor torque/Nm	2.29
Motor power/kW	1.2
Motor speed/(r/min)	5000

.

3.3.1. Basic Dimensions’ Design

With comprehensive consideration of the space occupied by the controller and the motor, the outer ring diameter of the large bearing-type reducer is set at 71 mm and the bearing-type reducer axial size is set at 25 mm. At this time, without considering the size of the interference, the design parameters of the large and small bearing-type reducers are shown in

Table 2. Preliminary parameters of bearing-type reducer.

Parameters	Value
Small inner ring outer diameter/mm	11.5
Large inner ring outer diameter/mm	17
Small inner ring axial length/mm	19.5
Large inner ring axial length/mm	19.5
Small rolling element axial length/mm	17.2
Large rolling element axial length/mm	17.2
Small outer ring axial length/mm	27
Large outer ring axial length/mm	25
Small reduction transmission ratio	5.28
Large reduction transmission ratio	4.57
Combined ratio	26.6

.

3.3.2. Design of Contact Line Length

The axial dimensions of the bearing-type reducer have already been initially set in the basic size design, and the contact line length must be smaller than its axial dimensions. Furthermore, when designing a bearing-type reducer, it is necessary to ensure that the contact line length is designed so that the contact stress does not exceed the permissible value of 3000 MPa for bearing steel. A minimum contact stress standard of 1000 MPa is selected as the basis for designing the contact line length. When we consider two cylinders’ traction drive, its equivalent modulus of elasticity $E$ is

$$\frac{1}{E}=\frac{1}{2}\left(\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}\right)$$

(10)

where $E_1$ and $E_2$, respectively, are the two cylinders’ elastic moduli; and $v_1$ and $v_2$, respectively, are the two cylinders’ Poisson ratios.

When two elastic cylinders are extruded into each other in convex–convex contact, the resulting contact area can be regarded as a narrow rectangle. This configuration generates contact stresses on the contact surfaces, with the maximum contact stresses appearing at the centerline of the contact rectangle. Assume that the squeezing force is $T$, the width of the rectangle is $b$, and the length is $l$. The following formulas are based on the Hertz theory [34], and when at rest, the bearing-type reducer satisfies these assumptions of the Hertz theory.

When two elastic cylinders are in convex–convex contact, the expression for the contact stress $\sigma$ on the contact midline is given by Equation (11):

$$\sigma =\sqrt{\frac{T\left(\frac{1}{R_1}+\frac{1}{R_2}\right)}{\pi h\left(\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}\right)}}=\sqrt{\frac{\mathit{TE}}{2\pi \mathit{Rh}}}$$

(11)

where $E$ is the equivalent modulus of elasticity; and $v_1$ and $v_2$ are the two cylinders’ Poisson ratios. Similarly, when the two cylinders come in convex–concave contact, then the contact stress $\sigma$ expression on the contact center line becomes

$$\sigma =\sqrt{\frac{T\left(\frac{1}{R_1}-\frac{1}{R_2}\right)}{\pi h\left(\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}\right)}}=\sqrt{\frac{\mathit{TE}}{2\pi \mathit{Rh}}}$$

(12)

Similarly, in the case of convex–concave contact between the two cylinders, the expression for the rectangular width $b$ of the contact surface is

$$b=\sqrt{\frac{T\left(\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}\right)}{\pi h\left(\frac{1}{R_1}+\frac{1}{R_2}\right)}}=\sqrt{\frac{2\mathit{TR}}{\pi \mathit{hE}}}$$

(13)

Comparably, the expression for the rectangular width $b$ when the two cylinders are in convex–concave contact is given by

$$b=\sqrt{\frac{T\left(\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}\right)}{\pi h\left(\frac{1}{R_1}-\frac{1}{R_2}\right)}}=\sqrt{\frac{2\mathit{TR}}{\pi \mathit{hE}}}$$

(14)

In the case of the contact between the outer ring and the rolling body, the contact form for the two cylinders is convex–concave contact. Similarly, between the inner ring and rolling body, the contact form for the two cylinders is convex–convex contact. Their contact characteristics can be calculated with the equivalent modulus of elasticity, using Equation (10). Subsequently, the dimensions of the bearing-type speed reducer outer ring and the rolling body, the rolling body, and the equivalent radius of curvature of the inner ring can be calculated. Finally, combined with Equations (11) and (12), the contact lengths of a reasonable range of contact lines can be planned based on the contact stress.

In traction drive, the oil film pressure of the contact line needs to reach GPa level. This is achieved with the help of the shear force from oil film shear deformation to transfer motion and power. When subjected to the same compression force, a shorter contact line results in increased oil film pressure and viscosity. This leads to a reduction in sliding power loss and an enhancement in transmission efficiency. However, if the contact line is excessively short, this can cause high line loads and contact stresses, potentially leading to frictional wear and contact fatigue damage. Therefore, under the premise of ensuring full oil film lubrication and contact fatigue strength, the contact line length should be reasonably reduced. This improves transmission efficiency and system power density and makes the structure more compact.

This paper adopts the Hertz contact theory and Dowson line contact elastic flow film thickness empirical formula, which are used to calculate the maximum contact stress and lubrication state of the inner and outer contact line of the bearing-type reducer’s underrated working conditions. Subsequently, a more reasonable value for the inner and outer contact line’s length is determined. Considering the establishment of film thickness and processing and manufacturing costs, the average arithmetic deviation $R_a$ of the surface roughness of all transmission parts in this study is designed as 0.1 $\mathsf{\mu }\mathrm{m}$. Consequently, the roughness of the surface of the transmission parts is 0.125 $\mathsf{\mu }\mathrm{m}$, and the integrated roughness of the inner and outer contact line is 0.1768 $\mathsf{\mu }\mathrm{m}$. Therefore, the critical film thickness for the establishment of the elastic-flow full-film lubrication state at the inner and outer contact line is 0.530 $\mathsf{\mu }\mathrm{m}$.

It can be seen from Figure 6a,b that the film thickness ratio of the inner and outer contact line increases with the length of the contact line. The upward amplitude gradually decreases, and the maximum contact stress decreases with the increase in the contact line length. The downward amplitude gradually decreases, as well. According to Figure 6 and the basic structure of the bearing-type reducer, the contact line length of the inner ring–rolling body is equal to that of the outer ring–rolling body. Combined with the range of the contact line lengths calculated in the previous section, it was finally decided that the contact line length should be set to be 14 mm. At this length, the film thickness ratios were 3.18 and 5.33, respectively.

3.3.3. Minimum Positive Pressure Design

When the bearing-type reducer is operating, we take the rolling body as the object of study to conduct force analysis, ignoring the friction of the planetary frame on the rolling body. Through force analysis, the following can be obtained:

$$F_N=\frac{\mathit{KT}}{\mathit{nf}{R}_s}$$

(15)

where $F_N$ is the minimum positive pressure required for the rolling element; $K$ is the safety factor; $T$ is the input torque; $n$ is the number of rolling elements; $f$ is the traction coefficient; and $R_s$ is the outer diameter of the inner ring.

In this paper, the safety factor K is set at 1.5, and the traction coefficient $f$ is set at 0.1. By substituting these values into Equation (15), it can be determined that the minimum positive pressure exerted by the outer ring of the small bearing-type reducer on the rolling element is 996.5 N, and for the large bearing-type reducer, it is 674.1 N.

3.3.4. Interference Design

In the context of cylindrical bodies with a finite length in contact, if they are compressed against an infinitely large plane, then the contact deformation from the axis of the cylinder to any point on the compressed surface can be calculated using Palmgren’s formula [35]:

$$\mathsf{\Delta }=3.81{\left(\frac{1-{\nu }_1^2}{\pi E_1}+\frac{1-{\nu }_2^2}{\pi E_2}\right)}^{0.9}\frac{Q^{0.9}}{l^{0.8}}$$

(16)

where Δ indicates the contact deformation of the cylinder; Q indicates the external load; and l indicates the length of the contact line.

Applying the relevant design parameters to calculate the contact deformation of the bearing-type reducer yields approximately 0.00232 mm for the small bearing-type reducer and 0.00163 mm for the large bearing-type reducer, and so does the contact of the outer ring–rolling body as well as the inner ring–rolling body. The final surplus for the small bearing-type reducer is 0.06 mm, and for the large bearing-type reducer, it is 0.08 mm. Considering the structural complexity and machining difficulty of the rolling elements and the outer ring, the interference fit is applied to the conveniently machinable inner ring of the reducer. Consequently, the final size of the outer diameter of the inner ring for both the large and small bearing-type reducers is determined.

Table 3. Final design parameters for bearing-type reducer.

Parameters	Value/mm
Small inner ring outer diameter	11.56
Large inner ring outer diameter	17.08
Small rolling element outer diameter	19
Large rolling element outer diameter	22
Small inner diameter of the outer ring	49.5
Large outer diameter of the outer ring	61
Small outer ring thickness	4.5
Large outer ring thickness	4.5
Small outer diameter	59.5
Large outer diameter	71
Length of the contact line	14

shows the final parameters of the large and small bearing-type reducers.

4. Dynamic Analysis of Bearing-type Speed Reducer

4.1. Analysis of Contact Stress at the Contact Line of the Bearing-type Reducer

The object under study here is a large bearing-type reducer, and our investigation focuses on the contact stresses between the inner ring and the rolling element as well as between the outer ring and the rolling element. Therefore, the planetary carrier is not included in the model. The model consists of three main parts: the inner ring, the rolling element, and the outer ring. The data in

Table 4. Finite element model material property settings.

Component	Material	Density (kg/m2)	Young’s Modulus (MPa)	Poisson’s Ratio
Inner ring	GCr15	7900	207,000	0.3
Rolling element	GCr15	7900	207,000	0.3
Outer ring	GCr15	7900	207,000	0.3

were entered into the ABAQUS model and the corresponding material section properties were assigned for these parts and set as homogeneous solids.

Figure 7 shows that the contact stresses at the outer ring–roller contact location have obvious edge effects [36], and the stress distribution at the inner ring–roller contact is uneven. The stress singularity at the endpoints of the two ends of the contact line is caused by a finite element model based on an erroneous mathematical model, i.e., the stress is theoretically infinite at the sharp corners [37]. Therefore, in the subsequent analysis, data at 0.2 mm from the ends were selected for extraction.

In Figure 8, the blue line represents the contact stress of the most central contact line in the stress distribution cloud diagram. In the cloud diagram, the redder the color, the higher the stress value, and the bluer the color, the lower the stress. Figure 8a shows a schematic diagram of the contact stress of the outer ring–rolling body, and from the diagram, it can be seen that the contact stress in the middle is approximately 1100 MPa but fluctuates, the contact stress at about 1 mm from the end is approximately 1400 MPa, and the contact stress at the two ends is approximately 2000 MPa. This indicates that the contact stress fluctuates by approximately 300 MPa in the contact line of 10 mm in the center, while it fluctuates by up to 900 MPa from the center to the two ends, displaying an obvious edge effect.

In Figure 8b, the contact stress diagram for the inner ring–rolling body shows that the stress in the middle is around 2450 MPa but fluctuates. Moreover, the contact stress at about 1 mm from the end reaches approximately 2600 MPa, and it reaches around 2700 MPa at the two ends. This indicates that the contact stress fluctuates by around 200 MPa in the central 10 mm of the contact line, and the contact stress from the center to the two ends also fluctuates by around 250 MPa.

According to Figure 8, there is a serious stress concentration phenomenon in the outer ring–roller contact line, with the contact stress difference between the middle and ends reaching as high as 900 MPa. This phenomenon will inevitably lead to a decrease in the life and load-carrying capacity of the parts [38]. The edge effect of the inner ring–rolling body contact line is not obvious; on the other hand, only the outer ring of the bearing-type reducer needs to be further contoured to reduce the stress concentration. However, there is no unified modification method for convex modification of the outer ring raceway. To proceed, it is possible to refer to the shaping method of cylindrical rollers, which may be used to shape the corresponding parts of the bearing-type reducer, and the most appropriate shaping method can be selected through finite element contact stress analysis.

4.2. Bearing-type Reducer Contour Line Modification

Currently, widely used convexity repair methods for bearing roller design include [39] full arc prime line repair, intersecting arc prime line repair, tangent arc prime line repair, and logarithmic prime line repair. However, there is no systematic repair method for the edge effect of traction drive. In this study, the outer ring of the bearing-type reducer was contoured based on the shaping method of the bearing roller, and then the contact stress distribution of the contact line under different shaping conditions was studied by using ABAQUS software.

Figure 9a shows the full arc prime trimming method with the following trimming equation:

$$\delta \left(x\right)=R-\sqrt{R^2-x^2}, |x|\in \left[0,\frac{L}{2}\right]$$

(17)

where $\delta$ is the trim length; $R$ denotes the length of the roller busbar radius; x denotes the distance to the roller end face; and L is the roller axial length.

The repair method utilizing the intersecting arc prime is illustrated in Figure 9b, accompanied by the corresponding repair equation:

$$\delta \left(x\right)=\left\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}|x|\in \left[0,\frac{L_1}{2}\right]\\ \sqrt{R^2-{\left(\frac{L_1}{2}\right)}^2}-\sqrt{R^2-x^2}\hspace{1em}\hspace{1em}|x|\in \left[\frac{L_1}{2},\frac{L}{2}\right]\end{array}\right.$$

(18)

where $L_1$ is the length of the middle straight part of the roller.

The method of trimming the tangent arc prime line is shown in Figure 9c, with the following trimming equation:

$$\delta \left(x\right)=\left\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}|x|\in \left[0,\frac{L_1}{2}\right]\\ R-\sqrt{R^2-{\left(\left|x\right|-\frac{L_1}{2}\right)}^2}\hspace{1em}\hspace{1em}|x|\subseteq \left[\frac{L_1}{2},\frac{L}{2}\right]\end{array}\right.$$

(19)

Figure 9d displays the trimming method for the tangent arc prime, accompanied by the corresponding trimming equations:

$$\left\{\begin{array}{c}\delta \left(L_0\right)=\frac{k_{0}Q}{L}\ln \frac{1}{1-{\left(\frac{2L_0}{L}\right)}^2}\\ \delta \left(L_0\right)=\frac{k_{0}Q}{L}\left(1.1932+\ln \frac{L}{2a}\right)\end{array}\right.$$

(20)

$$\left\{\begin{array}{c}{k}_0=\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}\\ b=\sqrt{\frac{2k_0\mathrm{QD}}{L}}\end{array}\right.$$

(21)

where $k_0$ is the material constant; and $b$ is the contact half-width.

Under the same conditions, the shaping length δ and the radius length R of the roller generatrix vary with different shaping methods. The repair method using full arc plain rollers shows limited stability under light loads. However, stability improves and load capacity increases once full-length contact is established and edge effects are eliminated, resulting in a more uniform distribution of contact stresses. In contrast, the intersecting arc plain repair and tangent arc plain repair methods display a stress concentration phenomenon. While using the logarithmic plain roller repair method, there is no stress concentration, and contact stress is minimized, but it demands a high level of precision in the manufacturing process [40]. The bearing-type reducer transmission type is traction transmission, working under heavy load conditions. In this section, analogous to the repair method of cylindrical roller bearings, the outer ring of the bearing-type reducer is repaired with a full arc vein. The convexity size is, respectively, set as 0, 5 $\mathsf{\mu }\mathrm{m}$, 10 $\mathsf{\mu }\mathrm{m}$, 15 $\mathsf{\mu }\mathrm{m}$, 20 $\mathsf{\mu }\mathrm{m}$, and further classified by whether the ends are chamfered under different convexity sizes. Finite element models are established and simulated for the aforementioned ten groups of trimming methods, respectively.

Figure 10a shows the variation in contact stresses in the inner ring–rolling body of the unchamfered outer ring under different convexity modifications. As the convexity increases, the contact stress increases, but the degree of stress fluctuation in the middle of the contact line is not significantly different at each convexity, indicating that the convexity of the outer ring has little effect on the contact stress in the middle of the inner ring–rolling body. Figure 10b reveals that the contact stress of the outer ring–rolling body changes significantly under different convexity modifications of the unchamfered outer ring. When the convexity is 10 $\mathsf{\mu }\mathrm{m}$ and 15 $\mathsf{\mu }\mathrm{m}$, the stress difference between the end and the middle of the contact line is about 500 MPa, and the stress fluctuation in the middle of the line is significantly improved. This indicates that the end stress concentration can be effectively reduced by rationally modifying the convexity through the full arc vein line modification method.

Figure 11a illustrates the inner ring–rolling body contact stress changes for an outer ring with a 1 mm end chamfer and varying convexity. Increasing convexity leads to a rise in middle contact stress, with a consistent difference of about 300 MPa between the end and middle. The end chamfer results in a more uniform stress distribution in the middle compared to unchamfered ends, reducing stress fluctuation. Figure 11b shows the outer ring–rolling body contact stress changes with a 1 mm end chamfer under different convexities. An increase in convexity from 10 $\mathsf{\mu }\mathrm{m}$ to 15 $\mathsf{\mu }\mathrm{m}$ results in a contact stress difference of about 500 MPa between the end and middle, with uniform stress distribution at the center. At 20 $\mathsf{\mu }\mathrm{m}$, the stress difference narrows, but the distribution becomes uneven, indicating improved stress fluctuation due to end chamfering.

The contact stress cloud diagram of the outer ring–roller contact line before and after the repair is shown in Figure 12. In Figure 12a,b, it can be seen that when only the convexity of the outer ring of the bearing-type reducer is repaired, the contact stress of the outer ring–rolling body is uniform, but there is still a serious edge effect. Meanwhile, Figure 12b,c show that when only the outer ring of the bearing-type reducer is chamfered, the outer ring–rolling element contact stress uniformity is slightly improved. Then, the combination of Figure 12b,d shows that when the outer ring of the bearing-type reducer is simultaneously cambered and chamfered, the outer ring–rolling body contact stress is more uniform and the edge effect is not obvious.

Comprehensive analysis of the above, compared with the simulation analysis of ten groups of modification modes, revealed the necessary bearing-type reducer outer ring profile modification method. This method includes full arc vein line modification with a convexity of 10 μm and outer ring chamfering of 1 mm. Without considering the edge effect at the ends, the maximum contact stress does not exceed 2600 MPa. Moreover, the contact stress difference between the middle and ends of the outer ring contact line has been reduced from the original 900 MPa to 400 MPa, meeting the fatigue strength requirements of contact stress.

5. Application Examples and Experimental Testing

In this section, a bearing-type reducer prototype is presented, as shown in Figure 13. Furthermore, concerning the test method of the traction drive and gear reducer, the test bench of the bearing-type reducer is described, as shown in Figure 14. This may be used to carry out tests of the transmission efficiency and transmission ratio under different speeds, torques, and traction oils, and to analyze the change rules and reasons for variations in transmission efficiency and transmission ratio under different speeds and torques.

Figure 15 gives the real-time variation curves of input torque and output torque during a no-load condition at 1000 rpm. It can be observed that both input and output torques exhibit significant fluctuations due to vibrations caused by insufficient assembly precision and coaxiality of the test stand, as well as electromagnetic interference from the motor. In this study, data falling within 30% above and below the median were considered valid. The final result was determined by calculating the average of these valid data points.

Taking the large bearing-type reducer as the research object, several commercially available traction oils were selected for use in this experiment to explore their effects on transmission efficiency. As shown in Figure 16, four types of traction oils, including the Santotrac 50 and Ub series, were used in this experiment. The experimental results show that Santotrac 50 traction oil performed best in terms of transmission efficiency when the input speed was 1500 rpm. It not only had a higher transmission efficiency but could also maintain this high efficiency over a wider range of output torques. Therefore, Santotrac 50 traction oil was preferred in the subsequent experiments.

Theoretical ratio verification tests and transmission efficiency tests were carried out for the large bearing-type reducer at different speeds. Figure 17 gives the ratio change curves when the rotational speed ranged from 500 rpm to 3000 rpm and the load was changed from no load to 3 Nm.

In Figure 17, it is evident that within the range of rated torque, the magnitude of speed has a minimal impact on the transmission ratio of the reducer. Moreover, the actual reduction ratio is almost consistent with the theoretical reduction ratio of 4.57. As the load increases, the slip rate between the rolling body and the inner and outer ring also rises, resulting in an increased transmission ratio. When the load approaches the rated torque, a significant increase occurs. This phenomenon results since when the load gradually approaches the rated torque, the rolling element and the outer and inner rings will produce violent slippage, leading to an increase in the transmission ratio. In particular, after a load of 2.7 Nm is applied at an input speed of 500 rpm, the bearing-type reducer slips purely, resulting in an infinite transmission ratio. Based on the transmission ratios under different loads in this figure, the slip ratio under different loads was calculated and then compared with the transmission efficiencies in the subsequent tests.

Our test data were organized to produce transmission efficiency curves as the output torque varied from 0 Nm to 11 Nm at speeds ranged from 500 rpm to 3000 rpm, as shown in Figure 18.

Figure 18 reveals that the transmission efficiency of the bearing-type reducer was extremely high, reaching up to 99.97% within the rated torque. Although an increase in rotational speed decreased the efficiency slightly, the transmission efficiency was still maintained at above 90% for a certain range of input torque. However, we found that the transmission efficiency of bearing-type gearboxes was significantly reduced under heavy loads, so they can only be used for high speeds and low-torque applications. Curves of slip rate versus transmission efficiency versus output torque were plotted at 2500 rpm, as shown in Figure 19.

As depicted in Figure 19, the slip rate increased as the load increased, determined by the characteristics of the traction drive, which also offered overload protection to the structure.

It can be seen from the above experiments that the presented bearing-type speed reducer can realize high transmission efficiency and power density. This performance is incomparably greater than those of harmonic and cycloidal speed reducers. In particular, its peak efficiency is higher compared to the traction drive reduction mechanisms designed by Flugrad [19], Ai [17,18], Wang [24], and other scholars. Moreover, its structure is even simpler, fully demonstrating the feasibility of this design. However, compared to previous designs, the torque transmitted by the present bearing-type reduction mechanism is on the low side. Pure sliding occurs at 11Nm, leaving it currently suitable only for small motorized wheels and robot front-end joints. In the future, the transmitted torque will be increased by enlarging the interference amount and changing the material of the elastic outer ring to increase the preload.

6. Conclusions

In this paper, a new type of planetary traction drive bearing-type reducer has been presented. This reducer has a simple structure, easy processing, smooth and efficient operation, and very low noise. It transmits power through the shear force of the elastic flow film. The reducer not only provides speed reduction and torsion increase but also offers axial and radial support functions similar to a bearing. Our research primarily focused on the transmission principle, geometrical design, and applying a systematic method of dynamic analysis of this bearing-type reducer. In this article, we have included an example of the actual prototype, illustrating its kinematics and dynamics. Experimental tests were conducted, selecting an appropriate traction oil for the device, verifying its high efficiency within the range of rated torque, and demonstrating its strong applicability in high-speed, low-torque scenarios. The main conclusions are drawn as follows:

• Based on the target speed, torque, and transmission ratio, the compression force and interference of the bearing-type reducer were designed. By choosing an appropriate oil film thickness formula and incorporating contact stress, the contact line length was designed, subsequently determining the basic dimensions of the bearing-type reducer.
• A strategic design enables the planetary wheel, the outer ring, and the contact line of the sun wheel, as well as the axial and radial contact of the flange, to align on a traction line, and the structure allows the bearing-type reducer to achieve bi-directional bearing (in both axial and radial directions). Additionally, it automatically adjusts to prevent planetary wheel eccentricity during work. The implementation of centering, to prevent planetary wheel eccentricity caused by collision or other factors, simplifies the structure while enhancing the traction transmission capacity.
• Drawing on the shaping method used for bearing rollers, the contour line of the outer ring of the bearing-type reducer was refined. A method of using a full circular arc profile to shape the outer ring was determined to be suitable. When the outer ring’s convexity was set to 10 μm with a 1 mm chamfer, the stress difference between the middle and ends of the outer ring contact line reduced from the original 900 MPa to 400 MPa. This resulted in a more uniform stress distribution and a less notable edge effect.
• By designing a bearing-type reducer test rig and performing experiments on transmission efficiency and transmission ratio, their variation patterns for the bearing-type reducer with changes in the load and speed were acquired. It was verified to be highly efficient in transmission, with a peak efficiency of 99.97%. Moreover, no wear or significant heat loss was observed during the experiments.

In this paper, we have described the feasibility and effectiveness of our design, which has been validated through simulation calculations and experimental results. Our work has yielded valuable experience and knowledge of optimization methods for the design of bearing-type reducers, providing significant reference value for subsequent research. In the future, efforts will be made to increase the preload of the bearing-type reducer. Furthermore, through analytical methods and test-bed experiments, the specific effects of different lubrication states on the bearing-type reducer will be investigated. Additionally, detailed kinematic analyses, temperature impact analyses, and fatigue life measurements of its specific structure will be conducted to improve the design’s overall performance.