Fast solution for deformation characteristics of flexible bearing of robot based on thin-walled ring theory

Abstract: Agricultural robots are an important part of the modern agriculture, and its function execution is mainly accomplished by robotic arm. Flexible bearing of the harmonic driver is a key part of the joint of the robots' arm. At present, there is little research on the design of flexible bearings. Different from most rolling bearings which are used as supporting components in ordinary, the flexible bearings are used as transmission components. A non-circular cam which is called wave generator is assembled into the inner ring of the bearing before working and causes a greatly pre-deformation of the flexible bearing. Previous theories of rolling bearing will not applicable because of this deformation. Meanwhile, a pair of radial force in the opposite direction is applied to both ends of the long axis of deformation of the flexible bearing in the transmission process. Therefore, more complicated shape state and bending stresses are generated. Those unusual working conditions will lead the flexible bearing more prone to damage than ordinary bearing. Although FEA simulation can get relatively accurate results, it usually takes several hours to solve finite element model, and the calculation process is difficult to converge. Therefore, it is necessary to establish a new calculation method to obtain the performance of flexible bearings. In this paper, the stress and deformation characteristics of the inner and outer rings of the flexible bearing were solved separately by the following methods: 1) First, the outer ring was equivalent to a statically indeterminate structure. The deformation coordination equation of the outer ring was established according to the equivalent model and solved by Mohr's integral theory. Overall bending moment of the outer ring of the flexible bearing formed by deformation was obtained. 2) Combined the theory of thin-walled ring with the bending moment equation which was obtained above, the radial and circumferential deformation characteristics of the outer ring of the flexible bearing were obtained. 3) According to the theory of multi span beam which was introduced in mechanics of materials, the loading model of the outer ring of the flexible bearing was built into three moment equations. The maximum stress was obtained by combining the three moment equation and the loading formula which was summarized by Ivanov's experiment. Above theoretical equations were compiled by MATLAB programs. Finally, a finite element simulation model of flexible bearing was established by ANSYS Workbench. The time consumed by simulation was about 4 - 5 hours while the calculation of theoretical equations only needed 5-8 min. By comparison, the maximum error between simulation value and theoretical value was only within 7%, it proved the correctness of the theoretical model calculation. Through analysis of bending stress and deformation of the flexible bearing, conclusions can be drawn as the following: 1) the force state of the outer ring was different from the inner one during rotation of flexible bearings in harmonic drive, cyclic deformation of the outer ring caused a large alternating bending stress and prone to fatigue failure; 2) stress of the outer ring formed by deformation increased sharply with the thickness while stress caused by external load would decline. And width only affected stress caused by external load. Increase of thickness was beneficial to carrying capacity of flexible bearing. However, increase of thickness lead to increasing of bending stress formed by deformation. The total bending stress had a minimum value in optimal thickness. 3) Width was an important parameter which had a greater effect on the carrying capacity of the bearing, but it was constrained by external structural and cannot be too large. Calculation results would provide a theoretical reference for the design and optimization of flexible bearings.