Elsevier

Mechanism and Machine Theory

Volume 104, October 2016, Pages 1-16
Mechanism and Machine Theory

Diaphragm stress analysis and fatigue strength evaluation of the flex-spline, a very thin-walled spur gear used in the strain wave gearing

https://doi.org/10.1016/j.mechmachtheory.2016.05.020Get rights and content

Highlights

  • Experimental and theoretical studies on diaphragm stresses

  • Fatigue strength evaluation of the diaphragm of the flex-spline

  • Contact analysis of the strain wave gearing

Abstract

This paper deals with diaphragm stress analysis and fatigue strength calculation problems of the flex-spline, a thin-walled spur gear used in the strain wave gearing (SWG). Firstly, a mechanics model and theoretical method used for contact analysis of SWG are presented in this paper. Then finite element method (FEM) software is self-developed through very long time efforts in order to realize the contact analysis and stress calculations of SWG. With the developed FEM software, bending and shear stresses of the diaphragm of the flex-spline are analyzed successfully. Secondly, in order to confirm the FEM software, an experimental method and device are presented in this paper. A three-directional, strain gauge is stuck on a straight surface of the tapered diaphragm of the flex-spline next to the corner of the diaphragm with the boss. Diaphragm strains are measured under the conditions of zero and 110 Nm torques respectively and the measured stresses are compared with the calculated ones with the developed FEM software under the same conditions. It is found that the two results are in agreement with each other well. This means that the presented mechanics model and theoretical method are correct and effective. Finally, a method used to evaluate fatigue failure strength of the diaphragm is suggested based on the findings in this paper.

Graphical abstract

Diaphragm stresses of the flex-spline, a thin-walled spur gear used in the strain wave gearing are investigated both in theory and experiment.

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Introduction

The strain wave gearing (SWG), often called the harmonic drive device, is a gear reducer with many advantages, such as lightweight, small size, high speed ratio, and low backlash. So, it is widely used in industrial robots (as joints), semi-conductor devices and space-exploring machines. Though so many units of this gear reducer are used in industry, strength calculation problems of this gear device have not been solved completely since its invention in 1959 [1], [2]. This is because elastic deflection of the flex-spline, a very thin-walled spur gear, is used to generate tooth engagement movement, a completely different type of tooth engagement movement from other gears such as spur gears.

Shen [3] and Nbahob [4] studied strength calculation problems of the tooth root of SWG. Alfutov [5] and Le [6], [7] studied tooth load calculation problems of SWG using a method of material mechanics. Cui [8] studied tooth load analysis and root stress calculations of SWG using a two-dimensional (2D), FEM. Ishida [9] and Sentoku [10] studied tooth load distribution evaluation using an experimental method. Some recent papers on SWG and static load sharing in internal involute spur gearing with thin rimmed pinion can also be found in the studies of Maiti et al. [11], [12]. Though so many papers can be found, diaphragm stress analysis and fatigue strength calculation problems of the flex-spline could not be investigated in these papers.

Li [13], [14] also studied strength calculation problems of SWG. Though the phenomenon of fatigue breakage failure of the diaphragm was introduced in Li's research, problems of stress analysis and fatigue strength calculation of the diaphragm could not be investigated deeply. The reality is that the problems of stress analysis and fatigue strength calculation of the diaphragm still remained as unsolved problems. This paper tries to solve the problems completely.

Breakage at the corner of the diaphragm with the boss of the flex-spline is a typical fatigue failure pattern of SWG. In order to avoid this breakage, it is necessary to find the reasons why the breakage happened at the corner. Also, it is necessary to find a suitable method that can calculate stress levels of the corner and evaluate fatigue failure strength of the corner. To realize this aim, this paper presents a mechanics model and principle used for contact analysis of SWG based on the mathematical programming method [15], [16]. Special FEM software development is conducted to realize the aim through a very long time effort. With the self-developed FEM software, it becomes possible to analyze diaphragm stress calculations of the flex-spline. An experimental method and device are also presented in the paper to confirm the mechanics model and principle presented in this paper. It is found that the calculation results are in agreement with the experiment ones well. This means that the mechanics model and method presented in this paper are reasonable. Finally, it is suggested that the maximum shear stress at the corner should be used to evaluate fatigue strength of the diaphragm.

Section snippets

Structure of SWG

Fig. 1 is a so-called cup-type of SWG. It mainly consists of four components: (1) the flex-spline (FS), a very thin-walled spur gear taking the shape of a cup; (2) the circular spline (CS), an internal spur gear; (3) the wave generator (WG), an elliptical cam that can change the shape of FS from a round shape into an elliptical one; and (4) the flexible ball bearing (FB), a thin-walled ball bearing mounted on the outside of WG. Since walls of the outer and inner rings are very thin, this

Fatigue breakage failure of the diaphragm at the corner

Fig. 2 is a section drawing of FS structure with fatigue breakage at the corner (the joint) of the diaphragm with the boss. As shown in Fig. 2, the diaphragm has the fatigue breakage at the part of stress concentration of FS structure. This failure pattern is very dangerous for industrial robots when SWG is used as joints. So, it is necessary to avoid this fatigue failure in the stage of designing SWG.

In order to avoid the fatigue breakage of the diaphragm, it is necessary to know the position

SWG used as research object

CSF40-100 made by Harmonic Drive Systems Inc. [17] is purchased and used as research object in this paper. Speed ratio and rated torque of this SWG are i = 100 and T = 265 Nm respectively. The main structural dimensions of this SWG can be available through the homepage of the maker. Other dimensions and gearing parameters of this SWG that cannot be available from the maker are measured and guessed using the purchased CSF40-100. Since it is still a trade secret for the maker, design parameters of

Theoretical investigations on the diaphragm stresses

It is quite a difficult thing to conduct stress analysis and strength calculations of SWG in theory. This is because a cup-shaped thin-walled spur gear (the flex-spline) is used in the device and elastic deflection of the thin-walled flex-spline is used to generate tooth engagement movement, a completely different type of tooth engagement movement from usual spur gears. This gearing principle makes the problems of stress analysis and strength calculations of SWG very difficult in theory. Though

Conclusions

  • 1)

    An experimental method and apparatus are presented in this paper to measure diaphragm strains of the flex-spline, a thin-walled spur gear used in the strain wave gearing. Diaphragm stresses next to the corner of the diaphragm with the boss are investigated experimentally.

  • 2)

    Three-dimensional, mechanics model and finite element method are presented in this paper to conduct contact analysis and stress calculations of the strain wave gearing in theory through very long time efforts. Special FEM

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