Determination of geometric parameters for flexible splines of wave gearing

The paper considers the influence of a number of flexspline geometric parameters on the stresses acting in it. The influence of the most significant parameters such as the diameter of the flexible spline and the thickness of its rim under the gear rim is considered. As a result of analysis of the flexspline geometric parameters influence on its load capacity, the recommendations are given for assigning the main dimensions of the spline at the stage of design.


Introduction
The main criteria for the wave gearing (WG) operational capability are the flexspline fatigue strength and the efficiency of the wave generator. To prevent fatigue failure of a flexible spline, it is necessary to correctly assign its basic geometric parameters, which determine the stresses arising in it. This should be done at the design stage. The main parameter of the flexible spline is its internal diameter, so first of all it is necessary to determine the internal diameter of flexspline depending on the transmitted torque. Then it is important to assign the remaining dimensions of a flexible spline, which significantly affect the stress in it, especially in the gear rim, where, as a rule, the fatigue crack initiation and flexible spline destruction occur. These are such dimensions as the shell thickness under the gear rim, thickness of the cylindrical shell, length of the shell, length of the gear rim and others. Several different stresses must be considered when analyzing them in the flexible spline of the wave gearing. These are the bending stresses in the spline that arise from its deformation by the wave generator [1]. A very significant influence is exerted by stresses arising from the load on the teeth [2]. In such a case it is appropriate to take into account the distribution of these stresses [3]. Besides, the interaction of a flexible spline with a wave generator is also investigated. Some researchers consider a spatial model of a flexible spline [4]. Also theoretical studies of the effect of various parameters on the wave gearing limit torque are known [5]. In the study of wave gearing operational capability by Ivanov M. N.  The design of the flexspline of a "glass" type adopted for the study is shown in figure 1. This design is most common and is used in real harmonic drive units. Figure 1 represents the design and main dimensions of the flexible spline as well as load F from the wave generator. Total or equivalent stresses are determined by the maximum shear stress theory To determine the value of the actual stresses ߪ .௦ through the stress in ߪ ௗ௦ we introduce a correction coefficient This coefficient should be estimated according to the results of testing.
To determine the stresses ɡ V arising from the load on the teeth, the dependence was used Nominal stresses t.l. V are defined according to [3,7] on the dependence The coefficient th Ʉ was obtained as a result of flexible splines testing on a real gear in operation.
According to the paper [4] the rated torsional stresses are defined as The greatest local shear stresses ɬQ W from the action of torque were specified ɬQ ɬ ɬQ K W W Here the revising coefficient ɬ K was also determined experimentally.

Experiment results
In the experimental part, we tested the wave gearing for fatigue resistance in the first series without load on the teeth, and in the second series with a load in a real wave gearing. The following geometrical parameters of the flexible spline were varied in the experiments (Figure 1): m is the module of gear; S is the width of the teeth cavity by the reference diameter; R = d/2 is the radius of the inner surface of the flexible spline; į LV WKH WKLFNQHVV RI WKH IOH[LEOH VSOLQH XQGHU WKH JHDU ULP L is the length of the flexible spline; b is the width of the gear rim; į 0 is the thickness of the cylinder part of the flexspline; R is the tooth pitch; r is the transition radius from the gear rim to the smooth part; l is the length of the smooth part in front of the gear rim.

Results and discussion
As a result of experimental studies the dependences (8)  A particularly important geometric parameter that determines the flexible spline stress, and, FRQVHTXHQWO\ LWV GXUDELOLW\ DQG ORDG FDSDFLW\ LV WKH WKLFNQHVV RI WKH VKHOO į XQGHU WKH JHDU ULP 7KH stresses in the flexible splines with a small thickness į (į = 0.01 d) are less in the absence of load on the teeth, and with the increase in the transmitted torque they grow rapidly. Stresses in flexible splines with a large thickness į (į = 0.02 d), in the absence of load are greater than in thin splines. However, when the load on the teeth increases, the stresses increase more slowly. There is a certain torque value at which the stresses are equal in splines with a small value į and a large value. As an example, the calculation of stresses in the region of the flexspline gear rim on the used dependences with allowance for the obtained factors was performed. Flexible splines with the following parameters were considered in calculations.

Conclusion
The analysis of the results allows one to draw recommendations on the design of flexible splines of WG. 1. 7KLFNQHVV į LV WKH PRVW VLJQLILFDQW GLPHQVLRQ that affects stresses in the flexible spline under the gear rim and its assignment depends on the transmitted load.