Introduction A mud mixer on a sealed household electric furnace production line suffered a complete failure of its WPO-15 worm gear reducer after three months of operation. The motor has a power of 7.5 kW and a speed of 1440 r/min. The motor and worm gear reducer are connected by a belt drive with a transmission ratio of 1:2. The WPO-15 worm gear reducer, with a transmission ratio of 1:40, has an allowable power of 5.5 kW at an input speed of 1500 r/min, and its load cannot exceed 65% during continuous operation. Under heavy production loads, the mixer often operates 24 hours a day for more than 10 days. Under these conditions, it is clear that the allowable power of the worm gear reducer is far less than the motor's power, indicating overload damage. This mixer is a non-standard piece of equipment designed to meet production requirements. The mixing parts are entirely made of stainless steel, and the mechanical transmission parts are installed at the bottom of the mixer. Due to limited space, gear reducers or cycloidal pinwheel reducers are not suitable. If a WPO-20 worm gear reducer with matching power to the motor is selected, it would be too large and too expensive. Therefore, a sprocket reducer was specifically designed for this ceramic mixer. The manufacturing cost of this sprocket reducer is 50% of that of the WPO-15 worm gear reducer, and it is still operating normally after more than a year of use. 1. Structural Design of the Sprocket Reducer The sprocket reducer adopts a two-stage transmission structure, as shown in Figure 1. The first stage transmission consists of a 61A, 8-tooth first-stage sprocket 1 and a 49-section first-stage roller sprocket 2, with a transmission ratio of 8:49; the second stage transmission consists of a 24A, 6-tooth second-stage sprocket 3 and a 39-section second-stage roller sprocket 4, with a transmission ratio of 6:39, and a total transmission ratio of approximately 1:40. 2. Working Principle and Structural Characteristics of the Sprocket Reducer 2.1 Working Principle The working principle of the sprocket reducer is shown in Figure 1. The transmission process is as follows: the shaft of the first-stage sprocket 1 is the input shaft. First-stage reduction is achieved through the meshing of the first-stage sprocket with the first-stage roller sprocket 2. Second-stage reduction is achieved through the meshing of the second-stage sprocket with the second-stage roller sprocket 4. [align=center]1 First-stage sprocket; 2 First-stage roller sprocket; 3 Second-stage sprocket; 4 Second-stage roller sprocket Figure 1 Schematic diagram of the sprocket reducer[/align] 2.2 Innovation A typical chain drive consists of two sprockets mounted on parallel driving and driven shafts, and a closed chain spanning the two sprockets. In the sprocket reducer design, a meshing transmission between the sprocket and the sprocket is used, replacing the chain and driven sprocket with a sprocket, thus creating a new type of sprocket reducer. Both sprocket reducers and gear reducers transmit power and motion through meshing, and their transmission ratios are calculated in the same way. However, the meshing of a sprocket reducer is the meshing of the sprocket teeth and the rollers of the chain disc, which is a non-conjugate meshing; while the meshing of a gear reducer is the meshing between the teeth, which is a conjugate meshing. 2.3 Structural characteristics The sprocket reducer transforms the chain transmission into a meshing transmission between the sprocket and the chain disc, but it does not have the transmission characteristics of a gear. Sprocket reducers and gear reducers are fundamentally different: (1) Since the meshing of a sprocket reducer is a non-conjugate meshing, the processing and installation accuracy requirements of the sprocket reducer are lower, and the requirements for working conditions are not high. (2) The instantaneous transmission ratio of the sprocket reducer is not accurate, but the average transmission ratio is accurate, while the transmission ratio of the gear reducer is a fixed value. Therefore, the transmission of the sprocket reducer is not smooth, generates dynamic load, has a large noise, and has a low transmission speed. (3) The sprocket reducer can only transmit motion and power between parallel shafts. (4) The minimum number of teeth on the sprocket of the sprocket reducer is 6 teeth, while the minimum number of teeth on the sprocket of the gear reducer is 17 teeth. Therefore, under the same power transmission, the sprocket reducer is more compact than the gear reducer. The structural characteristics of the sprocket reducer determine that it cannot be used as widely as the gear reducer. It is only suitable for applications with limited installation space, poor or harsh working conditions, inaccurate instantaneous transmission ratio but accurate average transmission ratio, low requirements for smoothness and noise, low speed, non-drastic load changes, and parallel rotation of two shafts, such as mixing and material conveying. The meshing of the sprocket reducer is the meshing of the sprocket teeth and the chain rollers, which is the same working principle as the meshing of the sprocket teeth and the chain rollers in chain drive. Therefore, the sprocket reducer has most of the advantages of chain drive: (1) No slippage loss, and the transmission efficiency can reach 98%-10%. (2) Allows for a large transmission ratio. (3) Can transmit a large amount of power at low speed. (4) Can work under high temperature or other harsh conditions (less affected by changes in climate conditions). (5) Compact structure, smaller profile size for transmitting the same power. Sprocket reducers also have some of the disadvantages of chain drives: (1) The transmission ratio is not constant, the circumferential speed fluctuates and is not stable (the fewer the number of sprocket teeth, the greater the fluctuation), and it is easy to generate large impact loads at high speeds. (2) The transmission is noisy. (3) It can only be used for transmission between parallel shafts. Figure 2 The chain disc structure is shown in Figure 2. After the chain is changed into a chain disc, there are no chain plates and sleeves, only pins and rollers between the two spokes. There is no failure mode caused by the increased pitch after the chain plate is stretched. Under the same specifications of sprocket, its pin diameter can be made larger than that of the chain pin. The sprocket tooth width is not limited by the chain width. Therefore, sprocket reducers have advantages that chain drives do not have: (1) The structure is more compact and transmits more power under the same sprocket specifications. (2) The maintenance cost is low. After the sprocket reducer wears and fails, only the small sprocket, pin and roller need to be replaced. (3) The sprocket reducer can be made into an open transmission or can be easily designed into a closed device. 3. Factors to Consider in Sprocket Reducer Design 3.1 Number of Teeth and Pitch of the Small Sprocket The number of teeth on the small sprocket has a significant impact on the service life of the sprocket reducer. If the number of teeth is too small, the unevenness of the transmission and the dynamic load increase. Simultaneously, the sprocket diameter is small, the sprocket shaft diameter is also small, and the allowable power of the sprocket shaft is small. The circumferential force transmitted by the sprocket increases as the number of teeth on the sprocket decreases, accelerating the wear of the sprocket and sprocket. Increasing the number of teeth on the small sprocket reduces the circumferential force transmitted by the sprocket, reduces the polygonal effect, and decreases the angle of sprocket engagement with the sprocket pitch, thus reducing wear. However, it increases the size and weight. Sprocket reducers are designed for more compact transmission space; therefore, the minimum number of teeth on the sprocket can be zmin = 0.6. For the first stage of the sprocket reducer, due to the high speed of the small sprocket, the number of teeth on the small sprocket can be 8-12. For the second stage of the sprocket reducer, due to the relatively low speed of the small sprocket, the number of teeth on the small sprocket is generally 6-8. If the installation space of the sprocket reducer allows, a larger number of teeth on the small sprocket can be selected to improve the service life of the sprocket reducer. Sprocket reducers are suitable for single-piece production. For ease of machining, sprockets generally use an even number of teeth, while the sprocket disc uses an odd number of teeth to ensure even wear. The pitch ρ determines the size of the sprocket disc and sprocket. A larger pitch results in larger dimensions of all parts of the sprocket reducer and increased load-bearing capacity, but also more severe uneven transmission and dynamic load. To select a suitable pitch, the design power of the sprocket reducer must first be calculated. After calculating the design power, select the corresponding sprocket pitch according to Figure 3. If the rated power of the selected sprocket pitch deviates significantly from the design power, the number of teeth on the sprocket can be changed to recalculate the design power, and then a corresponding sprocket pitch can be selected. The goal is to obtain the smallest possible sprocket pitch while still meeting the power transmission requirements, resulting in the most compact sprocket reducer structure. The small sprocket tooth number coefficient Kz = (z1/19)1.08. Therefore, the fewer the small sprocket teeth, the greater the design power of the sprocket reducer, and the larger the required pitch. Different tooth numbers for the small sprocket need to be selected and calculated repeatedly to obtain a smaller number of small sprocket teeth and a smaller sprocket pitch. 3.2 Sprocket Speed ​​Due to the chain drive characteristics of the sprocket reducer, the limiting speed of the sprocket is limited by the dynamic load. Figure 3 shows the rated power curve of the roller chain, which is the rated power curve of the 01 model single-row bushing roller chain. It can be seen from the figure that the rated power of the bushing roller chain increases with the increase of the small sprocket speed. When the sprocket speed reaches a certain value, the rated power of the bushing roller chain decreases rapidly with the increase of the small sprocket speed. Therefore, the speed of the sprocket in the sprocket reducer should not exceed the speed corresponding to the peak value in the rated power curve of the bushing roller chain. 3.3 Sprocket Tooth Width: Since sprockets are not limited by the chain, theoretically, they can be made with any width. As the sprocket tooth width increases, wear on the sprocket and chain disc decreases, lifespan increases, and power transmission increases. However, the pin also lengthens with the sprocket tooth width; an excessively long pin will reduce its rigidity, making the pin and rollers of the chain disc prone to fatigue failure. Because sprockets in sprocket reducers have few teeth and small diameters, they often need to be machined as a single unit with the sprocket shaft, as shown in Figure 4. To extend the shaft's service life, it is generally recommended that the sprocket tooth width of a sprocket reducer be twice the standard sprocket tooth width. 3.4 Sprocket Tooth Profile: The meshing between the sprocket and chain disc in a sprocket reducer is similar to the meshing between the sprocket and roller chain in chain drives; it is a non-conjugate meshing. The design of its tooth profile allows for considerable flexibility. BG/T1243-1997 does not specify a specific sprocket tooth profile, only the maximum and minimum tooth groove shapes and their limiting parameters. In sprocket reducers, the sprocket is a wear part and is often integrated with the shaft. Therefore, it is recommended to use the three-circular-arc straight-line tooth profile specified in GB/T1243-58. It has the advantages of low contact stress, light wear, low impact, and high tooth tip. 4 Design calculation of sprocket reducers 4.1 Failure modes of sprocket reducers (1) Under good lubrication, fatigue failure is the main failure mode of the pins and rollers on the sprocket. When the lubrication is improper or the speed is too high, scuffing failure occurs. (2) Excessive wear on the tooth surface of the sprocket (3) Under low speed heavy load or impact load, the pins and rollers of the sprocket break. 4.2 Rated power of sprocket reducers Compared with the failure modes of chain drives, the failure modes of sprocket reducers are the same except for the failure mode of the pitch lengthening caused by chain wear. Therefore, the rated power of the sprocket reducer can be determined by the rated power curve of the roller chain shown in Figure 3. According to the rotational speed of the small sprocket, select the corresponding sprocket pitch as shown in Figure 3, so that the rated power of each stage of the sprocket reducer is greater than or equal to the design power. Pd = KaP / KzKm (1) Where, Pd is the design power, kW; P is the transmitted power, kW; Kz is the operating condition coefficient; Kz is the number of teeth of the small sprocket Kz=(z1/19)1.08; Km is the sprocket tooth width coefficient. When the sprocket tooth width is the standard sprocket tooth width: Km=1; when the sprocket tooth width is twice the standard sprocket tooth width: Km=1.7; when the sprocket tooth width is three times the standard sprocket tooth width: Km=2.5; when the sprocket tooth width is other multiples, Km can be obtained by interpolation. 4.3 Checking the shear strength of the sprocket pin circumferential speed of the sprocket 5 Another structural form of the sprocket reducer In some low-speed transmission mechanisms, the structure shown in Figure 5 can be adopted, in which the driving wheel is designed as a sprocket and the driven wheel is designed as a sprocket. The biggest advantage of this structure is the low maintenance cost. The pin and roller of the driving sprocket are wear parts, and once damaged, they can be quickly replaced with simple tools. However, since only one roller of this sprocket reducer meshes with the chain teeth of the sprocket, once the pin breaks, the driven sprocket will lose power, which must be considered in the design. 6 Promotional value of sprocket reducers (1) Sprocket reducers are suitable for some situations where the installation space is limited, the speed is low and the load is heavy, and the working environment is poor. (2) Sprocket reducers are simple to design, easy to process, easy to maintain, and have low cost, and are suitable for single-piece production. (3) New sprocket reducer series can be formed.