# Maximum Assembly Stress of Flexible Wheel for Involute Harmonic Gear

Harmonic gear reducer is widely used in robotics, aerospace and other fields due to its high transmission ratio, high load-carrying capacity and high transmission accuracy. Flexible wheels fluctuate and deform twice every revolution of wave generator. The amplitude of alternating stress is usually twice the maximum assembly stress. Therefore, as a key component of harmonic gear reducer, the fatigue strength of flexible wheels becomes the decisive factor for harmonic gear reducer.The main factor of working life is to reduce the highest assembling stress of flexible wheel, which is the main means to improve the service life of harmonic gear. It is very important to study its variation law and influencing factors.

Based on the shell theory with equal thickness, a simplified shell model with equal thickness for deformation and stress of flexible wheel cylinder under assembly state is proposed; a ring model is proposed by using the crank beam theory. Based on these two models, the stress and deformation of flexible wheel under assembly state are studied in reference ; the influence of various flexible wheel structural parameters on stress distribution under no-load is analyzed. The stress distribution under no-load is calculated by using displacement restraint deformation and meshing force distribution.Calculate the deformation of flexible wheel along different sections in axial direction under transmission state.Considering the action of transmission load on the teeth, a flexible wheel model combining shell unit and beam unit is established to calculate the deformation and stress of the neutral layer of the flexible ring gear and the deformation of the tooth root and tip.A solid unit flexible wheel model is established to calculate the stress of the flexible ring gear and the cylinder.A self-made finite element analysis software is used to analyze the stress of the flexible ring gear and the cylinder.The cup bottom stress of the flexible wheel in harmonic gear is studied. The stress distribution of the flexible wheel on several sections along the axial direction is studied. The above study reflects the increase of stress of the flexible wheel ring caused by the gear teeth through the equivalent thickness (1.673 times the thickness of the aperture without the gear teeth). It only reflects the influence of the gear teeth on the average stress of the flexible wheel ring, but it can not reflect the maximum stress of the groove in the dangerous section of the gear ring.

In order to reflect the influence of gear teeth on local stress of alveolus, the trapezoidal tooth model is put forward to estimate the influence coefficient of gear teeth, which can be roughly 1.4~2.5, and the stress influence coefficient of gear teeth is 2.3~2.6 by experiment. A rectangular tooth model is put forward and its change trend with wall thickness is given. Although the experimental research is more practical, it is difficult to accurately obtain the maximum stress at the alveolus of harmonic gear with small module.Fitting the empirical formula of the highest stress of engineering structure based on the finite element results becomes a calculation method for structural design, but the ring stress of continuous groove can not be calculated from the look-up table.In this paper, a 2D contact model of flexible wheel with involute tooth profile is established, and the influence of tooth angle on the stress at the groove of flexible wheel is analyzed.The validity of finite element model needs theoretical verification. At the same time, the calculation amount of contact analysis is large and the research on stress regularity is time-consuming and laborious.

The influence of gear teeth on the maximum stress of ring gear is reflected in the increase of internal force and stress concentration, which is not only related to the wall thickness of ring gear, but also to the width of the alveolus and radius of chamfer of the root. A more accurate calculation model is needed to study its variation law. For this purpose, a solid unit finite element model of ring gear is established, which contains actual tooth profile information. A more fine mesh is used to partition the high-stress area and to calculate it.Calculate the circumferential stress of the ring gear under the action of four-roller wave generator.To verify the validity of the finite element model in this study, establish the ring model for gear-cutting, solve the roller thrust and maximum circumferential stress, and compare with the theoretical solution to verify the validity of the finite element model in this paper.By analyzing the variation rule of the maximum circumferential stress with the radius of the root chamference, the corresponding minimum maximum groove is obtained.Optimum radius of root chamfer for circumferential stress; define the influence coefficient of tooth stress as the ratio of maximum circumferential stress to maximum circumferential stress of the ring gear finite element model of the actual tooth profile in this paper, and the ratio of roller thrust to theoretical value of roller thrust of the finite element model with teeth as the increase coefficient of bending stiffness; calculate the stress concentration factor at the groove numerically. Obtain the increase of bending stiffness.Coefficient and stress concentration coefficient vary with the thickness of the ring gear and the width of the groove. 