Abstract: By giving the rigid or flexible wheel in a harmonic friction drive a tapered hole or tapered outer surface, the relative displacement between the flexible and rigid wheels along the axial direction is changed, allowing the circumference difference 2R" (R-r) between the flexible and rigid wheels to change continuously. This enables stepless speed change in radial inner wave harmonic drives, radial outer wave harmonic drives, end-face harmonic drives, closed harmonic drives, and multi-stage harmonic drives. A continuously variable harmonic drive device was designed based on this method. Calculations show that continuously variable harmonic drives can achieve a large speed ratio; the larger the maximum achievable speed ratio, the higher the requirements for the dimensional accuracy of the rigid and flexible wheels. Keywords: Continuously variable harmonic drive, flexible wheel, rigid wheel Introduction Major countries producing mechanical continuously variable transmissions (CVTs) include Japan, the United States, Italy, and Russia. Products come in over 30 structural forms, including friction, chain, belt, and pulse types. Mechanical CVTs have the following main characteristics: stable speed, low slip ratio, reliable operation, constant power mechanical characteristics, high transmission efficiency, simple structure, convenient maintenance, and relatively low price. However, they have high requirements for component processing and lubrication, low load-bearing capacity, and poor resistance to overload and impact. They are generally suitable for medium and small power transmissions. China started developing CVTs around the 1960s and currently can initially meet the requirements of social production. [sup][1] [/sup] Harmonic drive technology is a new type of transmission technology that emerged in the mid-20th century with the development of space science and aerospace technology, based on the theory of elastic thin shells. It has advantages such as high motion accuracy, large transmission ratio, light weight, small size, high load-bearing capacity, and the ability to operate normally in confined spaces and under conditions of medium radiation. It has been successfully applied in fields such as space technology, instrumentation, robotics, printing machinery, and medical devices. It consists of three main components, commonly known as the "three major components." Currently, for harmonic friction drives using steel flexible wheels, the single-stage transmission ratio range is approximately 20-1000, and for harmonic gear drives using steel flexible wheels, the single-stage transmission ratio is approximately 30-320. Typically, a single-stage harmonic drive device can only provide a fixed transmission ratio. Its kinematics can be briefly described as follows [sup][2] [/sup]. As shown in Figure 1, in a harmonic drive with a built-in wave generator, a flexible, thin-walled elastic component with a circular cross-section undergoes elastic deformation under the action of the wave generator. Along the long axis of the wave generator, the outer wall of the flexible wheel is pressed tightly against the inner wall of a rigid wheel with a circular cross-section. When the wave generator rotates, the flexible wheel rolls along the inner wall of the rigid wheel. If it is non-slip rolling, ignoring the effect of the flexible wheel wall thickness, the angle through which the flexible wheel and the rigid wheel rotate relative to each other is determined by the difference in their circumferences. Assume the radius of the flexible wheel's outer wall before deformation is r, the radius of the rigid wheel's inner wall is R, and their circumference difference is , where w[sub]o[/sub] is the radius difference between the rigid wheel and the flexible wheel. For each revolution of the wave generator, the angle through which the two wheels rotate relative to each other is: when the rigid wheel is fixed, the angle through which the flexible wheel rotates in the opposite direction of the wave generator's rotation; when the flexible wheel is fixed, the angle through which the rigid wheel rotates in the direction of the wave generator's rotation. In ordinary friction transmission, the transmission ratio is equal to the ratio of the diameters of the two friction wheels. In harmonic friction transmission, the transmission ratio is the ratio of the radius of the driven wheel to the difference between the radii of the two wheels. If the contact surfaces of the flexible wheel and the rigid wheel are respectively made with teeth, the above ratio should be replaced by the ratio of the number of teeth to the difference in the number of teeth. By reasonably selecting the meshing parameters and structural parameters [E3], the harmonic friction transmission is transformed into a harmonic gear transmission. People have been studying how to fully utilize the excellent performance of harmonic drives and achieve stepless speed change of harmonic drives. For example, reference [4] introduces a structure that achieves stepless speed change by changing the circumference of the flexible wheel with an additional device, using a truncated cone flexible wheel, and utilizing the different speeds at different points in the circumferential direction of the deformed flexible wheel. In the structure using a truncated cone flexible wheel, the ring rigid wheel can move axially, the position of the wave generator is fixed, and during the movement of the rigid wheel, the distance between it and the position of the wave generator changes, the rigidity of the deformed part of the flexible wheel changes, so the pressure at the contact point between the rigid wheel and the flexible wheel also changes. The closer to the position of the wave generator, the greater the pressure, and the farther away from the position of the wave generator, the smaller the pressure. This causes the load-bearing capacity of the stepless transmission device to change according to this law. The structural feature of the flexible wheel is that both ends are closed. Due to the limitation of the input shaft, the length of the flexible wheel cannot be changed. Therefore, the flexible wheel cannot achieve the required radial deformation. There is also a method of adding a planetary conical wheel device between the wave generator and the input shaft, and relying on the axial displacement of the planetary conical wheel to achieve stepless speed regulation of the input speed of the wave generator [sup][5]. Alternatively, a clutch or other transmission mechanism can be used to control the speed change. The latter methods are characterized by a constant transmission ratio in the harmonic drive section. This paper mainly studies the case of obtaining continuously variable speed (CVS) by changing the circumference difference between the flexible and rigid wheels in a harmonic friction drive. By giving the rigid or flexible wheel in the harmonic friction drive a tapered hole or tapered outer surface, the circumference difference 2a-(R-r) between the flexible and rigid wheels changes continuously as the relative displacement along the axial direction is continuously changed. Under the action of the floating wave generator, the flexible and rigid wheels remain in close contact. According to the aforementioned kinematic analysis, CVS achieves CVS. 1 Principle of CVS Harmonic Drive The principle of continuously variable harmonic drive can be illustrated by Figure 2. As shown in Figure 2, the rigid wheel has a conical inner hole. Under the action of the floating wave generator, the flexible wheel undergoes elastic deformation. In the long axis direction of the wave generator, the outer wall of the elastically deformed flexible wheel is pressed tightly against the inner wall of the rigid wheel. At position A-A, the flexible wheel does not undergo elastic deformation, and its radius is r_o, which is equal to the radius R_o of the rigid wheel at this position. When the wave generator rotates, the difference in circumference between the flexible wheel and the rigid wheel is . At this time, the transmission ratio of the continuously variable harmonic drive is . When the flexible wheel moves axially relative to the rigid wheel to position B-B, the wave generator, under the action of the clamping force, always keeps the flexible wheel and the rigid wheel tightly pressed together. At this position, the radius of the inner wall of the rigid wheel is , while the circumference of the flexible wheel does not change. Let W_oR represent the radius of the inner wall of the rigid wheel. 2. Simplified diagrams of several mechanisms The limit for the increase in circumference difference is the maximum value allowed by the strength of the flexible wheel material [sub]min[/sub], at which point the transmission ratio of the continuously variable harmonic drive is at its minimum. For metallic materials, the current minimum transmission ratio is approximately 20. If non-metallic materials such as plastics or nylon are used to manufacture the flexible wheel, the minimum transmission ratio can be less than 10. Continuously variable harmonic drives have advantages such as small size, light weight, and a wide output speed range. For harmonic friction drives with steel flexible wheels, the achievable speed range is approximately 20 to 1000, with a speed ratio exceeding 50. If non-metallic flexible wheels such as plastics or nylon are used, the speed ratio can reach over 100. This broadens the application field of harmonic drives. In this continuously variable harmonic drive, if a closed harmonic drive flexible wheel structure is used, it also has the characteristic of being able to output continuously variable transmission in special environments such as closed spaces and media radiation. Figure 3 shows simplified diagrams of the mechanisms for achieving continuously variable transmission using radial harmonic drive, end-face harmonic drive, and closed harmonic drive, respectively. Figure 3a is a simplified diagram of a continuously variable harmonic drive (CVT) used for radial inner-wave harmonic transmission. The flexure is connected to the output shaft, and the wave generator is the input. A rigid wheel with a tapered inner hole, capable of only axial sliding relative to the housing, can move axially relative to the flexure and the wave generator under the adjustment of the speed regulator, achieving stepless speed regulation. Figure 3b is a simplified diagram of a CVT used for radial outer-wave harmonic transmission. The flexure is connected to the output shaft, and the wave generator is the input. A rigid wheel with a tapered outer wall, capable of only axial sliding relative to the housing, can move axially relative to the flexure and the wave generator under the adjustment of the speed regulator, achieving stepless speed regulation. Under the adjustment of the speed controller, the flexible wheel and wave generator can move axially relative to each other, achieving stepless speed change. Figure 3c is a simplified diagram of the mechanism for continuously variable harmonic drive used in end-face harmonic drive. The flexible wheel is connected to the output shaft, and the wave generator input is a rigid wheel with a conical outer wall that can only slide axially relative to the housing. Under the adjustment of the speed controller, it can move axially relative to the flexible wheel and wave generator, achieving stepless speed change. Figure 3d is a simplified diagram of the mechanism for continuously variable harmonic drive used in closed harmonic drive. The conical flexible wheel is connected to the sealing wall 2, and the outer edge of the connecting wall is connected using elastic components such as corrugated plates, allowing the flexible wheel to move axially relative to the wall. The wave generator input is a rigid wheel with an inner hole connected to the output shaft. Under the adjustment of the speed controller, the flexible wheel and the connected sealing wall 2 can move axially relative to the rigid wheel and wave generator, achieving stepless speed change. 3 Design Examples Figure 4 shows a continuously variable harmonic drive device. The cone disc 15, bearings 17 and 18, support shaft 19, and roller bracket 20 constitute the wave generator. The speed regulating wheel 6 and speed regulating screw 10 constitute the speed regulating mechanism. The pressure spring 13, cone disc 15, and nut 12 constitute the pressure device. Adjusting the speed regulating wheel 6 drives the speed regulating screw to rotate. The rigid wheel 9, driven by the speed regulating screw 10, can move axially, continuously changing the circumference of the working contact point between the flexible wheel 16 and the rigid wheel 9, thus achieving continuously variable speed. Throughout the operation and speed adjustment process, the pressure spring 13, through the bearings 17 on the cone disc 15 and support shaft 19, tightly presses the working parts of the flexible wheel 16 and the rigid wheel 9 together, ensuring normal transmission. Figure 5 shows two structures of the working contact part between the flexible wheel and the rigid wheel. These two structures, placed inside the wheel, form a high-speed external harmonic drive. Figure 5a shows the structure of the working part on the flexible wheel that contacts the rigid wheel. The arc-shaped working part 16' on the flexible wheel 16 that contacts the rigid wheel 9 must smoothly transition with the adjacent part of the flexible wheel. Figure 5b shows a structure where the working part of the flexible wheel and the rigid wheel is made into a separate part, which is assembled on the flexible wheel 16. Figure 6 shows several floating internal wave generator structures that can be used in the design: Figure 6a shows a wave generator structure with pressure cones on both sides of the generator roller support. Figure 6b shows a planetary wave generator structure where the generator roller support does not rotate with the input shaft. Figure 6c shows a wave generator structure with two pressurizing devices and rollers mounted on the roller support. Figure 6d shows a wave generator structure with four pressurizing devices and rollers mounted on the roller support. Figure 6b shows the structure of a planetary wave generator with roller support 20 rotating with the input shaft 22. Referring to Figure 4, in this structure, the roller support 21) holds the support shaft 19 and the working rollers on the long shaft. The conical discs 15 on both sides of the roller support 20 are connected to the input shaft 22 by keys 14. The two conical discs 15 rotate together with the input shaft 22. The two conical discs 15 drive the support shaft 19 and the working roller surface 19 to roll along the inner surface of the flexible wheel 16 via the conical disc 25, thus forming a continuously variable harmonic drive planetary wave generator structure. In this structure, since the working contact 25 of the planetary drive part moves up and down with the left and right movement of the rigid wheel 9 during the rotation process, the transmission ratio of the planetary drive part also changes steplessly within a certain range. The transmission ratio of the entire continuously variable harmonic drive is the product of the transmission ratio of the harmonic drive and the transmission ratio of the planetary drive part. 4. Transmission Ratio Calculation As shown in Figure 7a, the rigid wheel of the continuously variable harmonic drive is circumferentially fixed, with a wave generator as input and a flexible wheel as output. The radius of the flexible layer of the undeformed flexible wheel is r_o, and the length of the flexible wheel is L_o. The radii of the upper and lower ends of the rigid wheel with a conical inner hole are b₂ and b₁, respectively. The length of the conical inner hole of the rigid wheel is H. The intersection point of the extended line of the flexible wheel BA and the extended line of the rigid wheel BA is C_o. The distance from point C to the lower end of the rigid wheel is a_o. The distance from the bottom of the flexible wheel to the lower end of the rigid wheel is a_o. The contact area between the flexible wheel and the rigid wheel is arc-shaped, with a radius of r_s and a distance from the center of the arc to the neutral layer of the flexible wheel of c_s. After the flexible wheel deforms, point E becomes point E_s, which contacts the inner wall MN of the rigid wheel. The distance from point E to the foot of the perpendicular F is shown in Figure 7b. The main dimensions of the designed continuously variable harmonic drive are as follows: the radius of the neutral layer of the undeformed flexible wheel is r_o = 50 mm, and the length of the flexible wheel is L = 100 mm. The radii of the large and small ends of the conical inner hole of the rigid wheel are b₂ = 57 mm and b₁ = 53 mm, respectively, and the length of the conical inner hole L of the rigid wheel is H = 50. The distances from the bottom of the flexible wheel to the small end of the rigid wheel are a[sub]min[/sub]=42mm and a[sub]max[/sub]=74mm respectively. Since the intersection point c of the extended line of the inner wall MN of the conical hole and the flexible wheel AB is located to the left of point A, a is substituted with a negative value in the calculation. From equation (4), a[sub]o[/sub] is 37.5mm, a=4.5739[sup]. The radius r[sub]s[/sub] of the arc of the contact part between the flexible wheel and the rigid wheel is 15mm, and the distance C[sub]s[/sub] from the center o[sub]s[/sub] to the neutral layer of the flexible wheel is 11.18mm. Using the above formula, calculations show that when n continuously changes from 42 to 73, as shown in Figure 8, the transmission ratio continuously varies between 19.4 and 545.0, with a gear ratio of 28. When n=73.5, the transmission ratio is 963.6, the gear ratio is 49.6, and the circumferential radius of the contact point of the flex wheel is 53.773 mm. The difference in the contact arc radius between the flex wheel and the rigid wheel is 0.0519 mm, equivalent to a manufacturing tolerance of grade 8 for this dimension. When a=74, the transmission ratio is 4155.1, and the gear ratio is 214.2. The circumferential radius of the contact point of the flex wheel is 53.772 mm, and the difference in the contact arc radius is 0.0120 mm, equivalent to a manufacturing tolerance of grade 5 for this dimension. Therefore, the larger the maximum transmission ratio that the continuously variable harmonic drive can achieve, the higher the requirements for the machining accuracy of the rigid wheel and the flex wheel. Conclusion 1) By giving the rigid or flexible wheel in the harmonic drive a conical hole or conical outer surface, the relative displacement between the flexible and rigid wheels along the conical axis is changed, allowing the circumference difference between the flexible and rigid wheels to change continuously, thus achieving stepless speed regulation harmonic drive. During the continuous change of the relative displacement between the flexible and rigid wheels along the conical axis, the flexible and rigid wheels remain in close contact under the action of the floating wave generator. This stepless speed regulation harmonic drive method can be applied to achieve stepless speed regulation of radial inner wave harmonic drive, radial outer wave harmonic drive, end-face harmonic drive, closed harmonic drive, and multi-stage harmonic drive. 2) In continuously variable harmonic drives, the wave generator structure can be a conventional floating structure with the same rotational speed as the input shaft, or it can be a floating planetary wave generator. In this structure, because the working contact surface of the planetary transmission part moves up and down with the left and right movement of the rigid wheel during speed regulation, the transmission ratio of the planetary transmission part also changes steplessly within a certain range. The transmission ratio of the entire continuously variable harmonic drive is the product of the transmission ratio of the harmonic drive and the transmission ratio of the planetary transmission part. 3) The speed regulating mechanism can be a nut and screw mechanism, a ball screw drive mechanism, a worm gear drive mechanism, a rack and pinion drive mechanism, a gear drive mechanism, etc., or a combination of them with a motor to achieve automatic speed regulation. 4) The structural design of the working part where the flexible wheel contacts the rigid wheel is shown in Figure 5. It includes a housing and an input shaft housed within the housing. A horizontal arc-shaped cam and a lifting arc-shaped cam are fixed on the input shaft. A horizontal driven plate and a lifting driven plate are also located at the lower ends of the horizontal and lifting arc-shaped cams, respectively. The horizontal and lifting driven plates are engaged with the horizontal and lifting arc-shaped cams via rolling bearings with handles. The horizontal driven plate is slidably connected to the output arm via a horizontal swing arm. A slidable slider is mounted on the output arm, and the slider is also mounted on a guide rod. The lifting driven plate is hinged to the slider via a lifting swing arm and a connecting rod. The horizontal and lifting arc-shaped cams rotate uniformly with the input shaft and drive the output arm in a horizontal reciprocating linear motion via the rolling bearings with handles, the horizontal driven plate, and the horizontal swing arm. The lifting arc-shaped cam drives the swing arm via the rolling bearings with handles, which in turn drives the slider via the connecting rod, thus driving the output arm in a vertical reciprocating linear motion. The horizontal and vertical travel of the output arm is adjustable within a certain range. It is controlled by a PLC, allowing control of its dwell position and time. Pre-tightening can be achieved during assembly by fine-tuning the center distance, enabling backlash-free meshing. This design fills a domestic gap and improves upon similar foreign products, and has been granted a national patent, patent number ZL 2004200415567. 3 Conclusion The arc-face cam manipulator, with its multi-point meshing and rolling friction, is a highly efficient and energy-saving automated device with broad development prospects. Its design and manufacturing are still in the research stage in China, requiring further expansion and in-depth study to achieve the production of serialized and standardized products. References: 1. Ge Zhenghao. Modular Design and Application Research of Cam Linkage Mechanism System [Doctoral Dissertation]. Xi'an: Xi'an University of Technology, 2001. 2. Peng Guoxun, Xiao Zhengyang. Cam Mechanism Design of Automatic Machinery. Beijing: Machinery Industry Press, 1990. 3. D. M. Tsay, H. M. Wei. A general Approach to the Determination of Planar and Spatial Cam Profiles. Trans. ASME, J. of Mech. Des, 1996, 118: 259-265. Continuously Variable Speed ​​Harmonic Drive Principle and Transmission Ratio Calculation: PDF