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On the profile design of transmission splines and keys
Daniel Z.Li
Abstract: Splines and keys are machinery components placed at the interface between shafts and hubs of power-transmitting elements. A spline (or key) is usually machined (or attached) onto the shaft of a power-transmitting pair, and the corresponding groove is cut into the hub. The influence of spline profiles on the performance of power transmission is investigated in this paper. The optimal design of spline profiles for three different design criteria is presented. The method of calculus of variation is used to determine profile functions for maximum value. Analytical results are successfully obtained. They show that the splines with involute profiles lead to uniform deformation on the hub, in addition they can carry the maximum transmission load capacity. On the other hand, radial straight profiles result in optimum transmission efficiency. We think that these findings are worthy reporting and also believe that this approach could be used for the spline design with other performance criteria imposed.
1 Introduction
A key is a machinery component placed at the interface between a shaft and the hub of a power-transmitting element such as gear and sprocket . A spline performs the same function as a key in transmitting torque from the shaft to the mating element . The main difference between splines and keys is that splines are integral with the shaft but keys are inserted between shaft and hub. As compared with one or two keys used for load transmission, there are usually four or more splines on a shaft. Therefore, the transmission torque is more uniform and the loading for each spline is lower. Splines play an important role in transmitting torque and their profiles do have the influence on the performance of power transmission. Unlike the conjugate profiles, the shaft with splines and hub have the same rotation axis and they are in surface contact without relative motion, they are connected together and have the same angular velocity. Therefore, it seems that any profiles except the shaft surface can be used for the design of splines. However, the load between the spline and hub is not evenly distributed over the entire contact surface in practice. The load may always concentrate on a small portion of contact surface and deformthe hub surface. This results in undesired clearance between the shaft and hub and will lead to serious damage of hub surface as the working cycles increase. To solve these problems, how the profiles of splines affect the torque transmission needs to be further investigated to find out the suitable design of spline profiles.
Currently there are two main types of splines used, namely, straight-sided and involute splines. The involute splines provide the mating element with self-centering and can be machined with standard hob cutter used to cut gear teeth. To date, the related research work focuses on conjugate profiles and gear design as well as the design of profile curvatures for reducing the wear of contact surfaces. However, none of them can be applied to the profiles of splines directly due to different working conditions. Also, there is no research work on how to design spline profiles under given requirements. In this paper, the basic equations for spline profiles are established and used to synthesize desired profiles for different design objectives. Three design objectives, uniform deformation, maximum torque transmission, and optimum efficiency, are used to determine spline profiles. Analytical solutions are successfully obtained.
2 Problem description and basic assumptions
As shown in fig .1, The hub is driven by the shaft and the spline is fixed on the shaft. The radius of the shaft, the height of the spline, and the number of spline teeth are determined by the design requirements and cannot be altered. Only the spline profile can be modified to improve the performance of transmission. To simplify the design problem for analysis, the following assumptions were made:
(1) The spline is a rigid body.
Compared with the hub, the spline is made of hard material and assumed no deformation after applying the load.
(2) The hub is under elastic deformation
The surface deformation of the hub is within the range of elasticity and the surface stress is proportional to the normal deformation.
(3) There is no beam deformation on the spline.
For spline keys, usually the height of tooth shape is small relative to its width. Therefore, we assume there is no accumulated deformation at the free end. The only deformation is the normal deformation on the hub surface.
(4) There is no clearance between the spline and hub when they are in contact. (Surface contact)
The profile of the spline is exactly the same as that of the hub without considering manufacturing errors. They are in surface contact without clearance.
3 Spline profile for uniform hub deformation
The first design objective is to have the uniform deformation on the surface of the hub, which also implies the uniform stress on the hub. This design can ensure the surface stress is evenly distributed and avoid the failure of material at some weak points. Referring to fig.2, Let denote the radius of shaft and denote a small rotation angle of spline. Since we assume that the spline is a rigid body, the change between two spline positions will be the deformation of the hub.
4 It’s simply to confirmed the dangerous sections
Prerequisite that traditional design method considered whether pair influence part design variable of working state, for instance stress , intensity , safety coefficient , load , environmental factor , material performance , part size and structural factor ,etc., deal with the single value variable confirmed. Describe part mathematical model of state , i.e. variable and relation of variable , to go on single value vary and win the dangerous section through deterministic function.
There are several methods that usually the dangerous sections are determined:
4.1 Minimum diameter of the spline
Spline dangerous sectional reliability very getting high, this to confirm according to traditional design experience because of diameter of spline. If require appropriate reliability value, then the diameter of the axle can select smaller value for use .
4.2 Safety coefficient law of dependability
While adopting the safety coefficient law design of dependability , must know the distribution types of stress and intensity and be distributed estimated value of the parameter . And the accumulation of dependability data is a long-term job, therefore we must utilize the existing data materials , it is (such as the terminal theorem in the centre and " 3 rules " to use relevant theorems and rule ), to confirm the distribution types of a lot of random variables involved of design process and is distributed the parameter. In the safety coefficient of dependability is calculated , deal with all design parameters involved a random variable, link the concept of safety coefficient to concept of dependability , thus set up corresponding probability model. Because of considering the uncertainty (randomness ) of the phenomenon taking place in project reality and sign parameter, therefore can announce the original appearances of the things even more. Theory analysis and practice indicate , the dependability design is designed more than traditional machinery , can punish some problem of the design , raise product quality , reduce part size effective, thus save the raw materials , lower costs .
5 Concluding remarks
The mechanical reliability design is one kind of modern design theory and the method which in the recent several dozens years develop, it take improves the product quality as the core, take the theory of probability, the mathematical statistic as the foundation, synthesizes using the engineering mechanics, the system engineering, the operations research and so on the multi-disciplinary knowledge studies the mechanical engineering most superior design question. At present, the reliability design theory tended to the consummation, but uses in the machine parts design project actual very being actually few truly. When uses the reliable security method of correlates design, must know the stress and the intensity distributed type and the distributed parameter estimated value. But the reliable data accumulation also is a long-term work, thus we must use the existing data material, the utilization related theorem and the principle, determined in the design process involves many random variable distributed types and distributed parameter. In this paper the optimal design of spline (or key) profiles for three different design criteria is presented. The method of calculus of variation is used to determine profile functions for maximum value. Analytical results are successfully obtained. It shows that the splines with involute profiles lead to uniform deformation on the hub, in addition they can carry the maximum transmission load capacity. On the other hand radial straight profiles result in optimum transmission efficiency. We believe similar approach could be used to determine other spline profiles when new performance criteria are imposed.
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