1. Introduction
The usage of a drive mechanism with a high-speed reduction ratio is ubiquitous in various industries and has to be carefully selected for different engineering applications. For example, the design of a joint drive module used in a robotic system has to ponder its working speed, force/torque load capacity, rigid transmission, space allowance, positioning accuracy, and relatively long service life. To provide a design of reducers with a large reduction ratio in a small space, harmonic reducers [
1], cycloidal pinwheel reducers [
2,
3] (RV reducer), and planetary gear reducers [
4] (PGR reducer) having a common structural feature of coaxial input and output shafts are often considered. Harmonic drives with high-speed reduction ratios and relatively small volumes have been employed in machinery for light-load applications and high precision. Because the elastic spline of a harmonic reducer is a thin, flexible part, it may avoid backlash by flexible transmission [
5]. However, its expected lifespan may be deteriorated due to high loads.
Elias and Nader [
6] propose a compact and high-torque gear mechanism that can be used in joint drive systems for space robots. It can provide a high reduction ratio of 1:2116. Two sets of planetary gears are used to prevent the output gear set from swinging and can effectively improve the output load capacity. This gear mechanism requires high manufacturing accuracy and assembly accuracy because this mechanism is composed of Profile-shifted gears. This article replaces the gears by using a 2D Lobe cam which is easier to manufacture. Due to the high contact ratio of the cam, the attachment capacity of the mechanism is further improved.
A cycloidal reducer can be classified into three types: single-stage cycloidal speed reducer [
7,
8], two-stage cycloidal speed reducer, and RV reducer. Botsiber and Kingston [
9] proposed the basic structure, operation, and synthesis principle of the cycloid reducer and compared it with other types of reducers. Malhotra and Parameswaran [
10] calculated the force acting on each element of the cycloid reducer, evaluated the theoretical transmission efficiency, and conducted the optimal design of this type of reducer. Litvin and Feng [
11] analyzed the geometry of planar cycloidal gearings and ameliorated the design to avoid geometric singularities. Blagojevic et al. [
12] presented a two-stage cycloidal speed reducer and the rollers were assembled on an intermediate disk that had transmission motion from the input shaft to two cycloidal disks. In [
13,
14], pins were used to transfer movement from the first cycloidal disk to the second cycloidal disk. This design can obtain a more compact configuration by simplifying the intermediate disk from [
12]. However, the two cycloidal disks still have a relative speed difference during the transfer of the intermediate disk.
At the same time, there is a sliding contact between the cycloidal discs and the rollers in the two sections. When two sliding contact situations occur during the transmission process, it will be more easily affected by changes in speed and load [
15]. For a modified cycloid reducer like a planocentric drive, Jang et al. [
16] showed a procedure for producing an epitrochoidal gear contour without using pin-rollers and designed the internal and external gears based on varying tooth thickness ratios as well as the distance from the center of the rolling circle of the epitrochoid. Li et al. [
17] studied the effect of ring pin position deviation on the distributed load, contact stress, load transfer error, and instantaneous transmission ratio of mismatched cycloidal pinwheel pairs. However, the action of the lobe cam reducer in this study is rolled by the conjugate cam on the two sets of rollers. The operation of the lobe cam reducer is different from that of the RV reducer [
18,
19].
The lobe cam reducer proposed in this paper is based on our concept design shown in Taiwan invention patent TW I431209 [
20]. The needle roller bearings are used instead of pins used in two-stage cycloidal speed reducers [
12] so that this cam reducer can ensure a pure rolling motion between the cam and the roller without sliding contact. Compared to a two-stage cycloidal speed reducer [
13], the proposed conjugate cam design is a single unit component with two lobe cams at both ends without any assembly for pins required, as shown in
Figure 1.
The main advantage of this design is that, with a low number of components, the design can achieve a rigid transmission with a relatively compact volume. As a result, the design for rigid transmission with a relatively compact volume can be achieved. The lobe cam proposed in this manuscript is not a cycloidal shape. The cam reducer design is based on the relative motion relationship between the output shaft and its input shaft to determine the paths of the rollers. Additionally, since the cam profile is synthesized by the rigid body transformation method, the output rotating turret motion can be accurately defined.
However, the motion synthesis and dynamic analysis of such a reducer driven by a conjugate lobe cam have not yet been reported. Based on the idea above, this study is focused on developing a systematic design procedure for the lobe cam reducer. As shown in
Figure 1, one of the obvious advantages of this design is the use of bearing rollers instead of pin rollers to decrease wearing. High reduction ratios can be achieved with a smaller number of rollers. At the same time, a smaller number of assembly elements makes assembly easier.
To provide a systematic design tool, a lobe cam reducer is synthesized, analyzed, made, and tested for the feasibility of the proposed procedure, as shown in the flowchart (
Figure 2).
In the motion synthesis of the reducer, first, the kinematic parameters for a desired speed reduction ratio given by the number of cam lobes and rollers shown in
Figure 1b are determined. Then, for determining the conjugate lobe cam profiles, it is described by the relative position between the lobe cam and its eccentric input shaft and the rigid body transformation method [
21]. For a given application, the minimum size that a conjugate lobe cam can have must be the minimum size that does not undercutting. To characterize the needed input torque of the reducer, the transmission force derived from Newton’s second law is proportional to the actuating arm length of each engaged roller on the fixed and rotating output turrets by its geometric vector. As a result, for a specific application, the lobe cam diameter, roller diameter, eccentricity, number of cam lobes, limiting load of rollers, input/output torques, and contact stresses can be conveniently observed and adjusted during the design process.
3. Geometric Design of the Lobe Cam
Generally, a disk cam with a roller follower is driven by its own camshaft, and a variety of processes can be analytically used to find the cam contour [
5]. The profile determination of a lobe cam differs significantly from that of a traditional disk cam with a roller follower since the lobe cam is actuated by an eccentric input shaft, and its rotational center becomes variable. Moreover, at an instant time, all the rollers on a turret are simultaneously in contact with their cam contour. To overcome these two difficulties, a convenient approach based on the relative position between the lobe cam and its eccentric input shaft, as well as the rigid body transformation method [
22], is proposed to generate the trajectory of the lobe cam’s roller and then to find its offset for the conjugate lobe cam profiles. For brevity, the process introduced in [
21] for determining a 3D cam contour can be referenced in a similar way for producing the lobe cam profiles.
3.1. Lobe Cam Profile
The lobe cam axis is not fixed and is rotated around the axis of an eccentric input shaft in an opposite direction, as shown in
Figure 1b. To generate the lobe cam profiles, the relative position between the lobe cam and its eccentric input shaft must be considered. Since the lobe cam rotates counterclockwise, based on the design parameters shown in
Figure 1 and the rigid body transformation method shown in
Figure 3a, the trajectory of the roller center of the fixed turret (
PrA) in the fixed
xc-yc coordinate system can be produced by rotating the roller counter clockwise. It can be formulated as
where
rdA is the radius of the pitch circle, and
OrA, as well as
OT, are the positions of the roller center on the fixed turret and the input shaft center, respectively.
To obtain the offset of
PrA, the unit normal vector with the input cam axial direction vector (
k) can be written as
where the components are
Then, the input lobe cam profile can be found as an offset of the roller center trajectory with a distance equal to the roller radius (
rrA) along the positive direction of the unit normal vector. Therefore, it can be derived as
Similarly, the output lobe cam profile is
where
rdB is the radius of the pitch circle and
OrB is the position of the roller center on the rotating output turret. The trajectory of the roller center of the rotating output turret (
PrB) in the fixed
xc-
yc coordinate system is shown in
Figure 3b and can be described as
The components of the unit normal vector are
The angle between the transmitted force (the common normal at the cam-roller contact point) and the tangential direction of rotation of the turret is defined as the pressure angle [
5]. The transmitted force direction (the unit normal vector
of the input lobe cam) of a resisting roller of the fixed turret for the input lobe cam is from the roller center (
OrA) to the contact point (
PpA). The pressure angle (
ψA) between each roller of the fixed turret and the input lobe cam is shown in
Figure 4a. The pressure angle (
ψB) between each roller of the rotating output turret and the output lobe cam is illustrated in
Figure 4b,c and can be seen similarly with the pressure angle (
ψA).
For machining the lobe cam profiles with a tool diameter (
dt) not greater than the diameter of roller (
dr), the tool path (
Pt) can be defined by Equations (6) and (7) with the roller center trajectory (
Pr) and its unit normal vector (
nc) as
3.2. Contact Point
As described earlier, the conjugate rigid lobe cam is rotated around the eccentric input shaft and all of the rollers of the turrets are simultaneously in contact with the lobe cams during operation. For locating the contact points needed for dynamic analysis between the lobe cams and their turret rollers, a computation method is developed to determine the distance between the profile position of the lobe cam and its roller center. When this distance is equal to the radius of the roller, the corresponding point of the lobe cam is then the contact point.
To find the distance for locating contact points, the view of the lobe cam profile from the input rotating center (
O) shown in
Figure 3a, Equation (6) for
PcA must be transformed from the lobe cam center to the input rotating center and rotated clockwise about the axis of the input shaft by an angle
θc of the cam rotation. Then, it can be denoted as
Hence, the roller center of the fixed turret is
and the angular position of the
i-th roller on the fixed turret is
,
. As a result, the distance between the roller center (
OrA) and the transformed input lobe cam profile (
PcTA) is
When the distance (LA) equals the radius of the roller, the point of the cam profile (PcTA) is the contact point (PpAi) corresponding to i-th roller of the fixed turret.
In a similar way, the output lobe cam profile,
PcB shown in Equation (7), can be transformed from the lobe cam center (
Oc) to the input rotating center (
O) shown in
Figure 3b and rotated clockwise about the axis of the input shaft by an angle
θc of the cam rotation. Then, it can be represented as
Since the
j-th roller center position of the rotating output turret can be described as
and
,
, its rotated center position can be expressed as
, when the output turret is rotated at an angle of
θo. As a result, the distance between the roller center (
OrTB) and the transformed output lobe cam profile (
PcTB) is
When the distance (LB) is the same as the radius of the roller, the point of the cam profile (PcTB) is then the contact point (PpBj) corresponding to j-th roller of the rotating output turret.
3.3. Actuating Angle
In this lobe cam reducer, the engagement resistance and drive of the cam and roller are not accomplished by the engagement of a single roller, but by the simultaneous engagement action of multiple rollers. Before the kinematic analysis, the number of actuating rollers on the turrets should be determined. In this study, the geometric-vector method is used to evaluate the number of actuating rollers. As illustrated in
Figure 4a, the angle between the reverse direction of the tangential force of the eccentric input shaft (−
ti) and the direction of the resisted cam transmission force (
ncAi) is the actuating angle of the input lobe cam, and it can be written as
where the direction of the tangential force of the eccentric input shaft is represented as
.
Similarly, the angle between the direction of the tangential force of the input shaft (
ti) and the direction of the driving cam transmission force (
ncBj) is the actuating angle of the output lobe cam, which can be expressed as
If the actuating angle is acute, the contact between the roller and its lobe cam will exert a force to rotate the conjugate lobe cam in a reverse direction to the eccentric input shaft. When the actuating angle becomes obtuse, no force will be transmitted between the roller and its lobe cam. As a result, the number of turret rollers that will exert transmission forces can be counted.
3.4. Actuating Arm Length
The transmission force is proportional to the actuating arm length which is the normal distance between the direction of the transmission force and the center of the input eccentric shaft. The actuating arm length of the
i-th actuating roller is
where
is the actuating arm angle that is equal to the angle between the direction of the transmitted force and the line connecting the centers of the input shaft and the roller center. As a result, the actuating arm angle can be found as
The actuating arm length of the
j-th actuating roller is
where
is the actuating angle that is equal to the angle between the direction of the transmitted force and the line connecting the centers of the input shaft and the roller center. Again, the actuating arm angle can be computed as
5. Application Examples and Experimental Tests
In this section, three LCRs I, II, and III separately corresponding to three different speed reduction ratios of −399, −90, and −24 are presented. By adjusting the radii of the turrets, roller diameter, and eccentricity, the maximum outer diameter of the LCR was 92 mm. The fundamental design parameters for these three lobe cam reducers are listed in
Table 2. As shown in
Figure 5, the cam profile of the three LCRs can be easily synthesized by the aforementioned method and analyzed as not undercutting. Under suitable design parameters, the LCR can have the characteristics of a large reduction ratio with a small volume.
Based on the same output torque of 80 Nm, the dynamic analysis results are shown in
Table 3. If LCR has more numbers of actuating rollers, the needed force of a single roller will be less. LCR III has only three actuating rollers, so each roller needs to load a considerable force. Even if the radius of curvature is small, the contact stress will be too large due to the excessive force of each roller. Although the LCR can be designed with a small volume mechanism with a large reduction ratio, an excessively small volume will reduce the efficiency, just like the LCR I. After considering the allowable analysis results, the design of LCR II can be applied in practice. In addition, a prototype based on the design parameters of LCR II together with a test bed is manufactured for experimental tests to verify the feasibility of the developed design procedure and to investigate the efficiency of LCR II.
Regarding the analyses of actuating angles and transmission forces of LCR II with an output load of 80 Nm, they are illustrated in
Figure 6 separately for the fixed turret rollers and the rotating output turret rollers. As shown in
Figure 6b, the computed maximum transmission force is 960 N, which is less than the bearing limit load of 1050 N [
24]. At the output lobe cam in
Figure 6d, the maximum transmission force is 720 N, which is lower than the maximum transmission force of the input lobe cam. Referring to
Figure 6a,c, we can see that each turret has seven actuating rollers (
kA =
kB = 7) with various actuating angles to transmit different magnitudes of contact forces along with the rotating input angles.
The contact stress analysis of LCR II for its two lobe cams and their turret rollers is shown in
Figure 7. The calculated maximum contact stress plotted in
Figure 7a is 185 MPa, which is less than the allowable compression stress (
σa = 480 MPa) of the medium carbon steel (S45C). As illustrated in
Figure 7b, the evaluated maximum contact stress of the output lobe cam is 161 MPa, which is smaller than that of the input lobe cam. Based on Equations (32) and (33) for the needed input torque with a friction coefficient of 0.05 for the input shaft and efficiency of the LCR II, the computed efficiency varies in the range from 60% to 70% with an average efficiency of about 66.6%.
To further investigate the reducer efficiency, a prototype of LCR II (
Figure 8a) without lubricant inside was made for experimental tests. In addition, for conducting the experiments, a test bed photographed in
Figure 8b,c was also built. The operational speed was maintained at 900 rpm and the output load torques were regulated at 5.71, 16.66, and 24.71 Nm. The test results presented in
Table 4 show that the average reduction ratio is 89.31, and the reducer efficiency is in the 49%~52.6% range. Compared with the theoretical values of input torques and efficiency, the major deviation may be reasonably attributed to the extra loads of installed equipment and the absence of lubricant inside the housing of the conjugate lobe cam reducer.
6. Conclusions
As described above, this study mainly focuses on the systematic methodology of kinematic synthesis, geometric design, and dynamic analysis for conjugate lobe cam reducers. Three application examples have been displayed to illustrate the synthesis of reducer motions. Moreover, a case with its real prototype has been provided to show its kinematic and dynamic characteristics together with experimental tests. The feasibility and effectiveness of the developed approach have been demonstrated through computational and experimental results.
The conjugate lobe cam profile can be easily generated by using the rigid body transformation method of the eccentric rotating shaft, and the conjugate lobe cam can be manufactured with a reasonable effort. Furthermore, the lobe cam reducer is compact, and its cam roller is in rolling contact. The engagement actions between both the lobe cams and their turret rollers have been identified through the analysis of multiple actuating rollers. Based on Newton’s second law in dynamic analysis, the resisting and driven rollers of the turrets of the reducer have been analyzed for correct torque transmissions. Employing the developed procedure can significantly reduce the time needed for a specific application design of the conjugate lobe cam reducer.