Research On Calculation Method Of Meshing Stiffness Of Spiral Bevel Gears With Tooth Root Crack

Spiral bevel gear transmission system has a wide application prospect in the fields of national defense,aerospace,energy,transportation,shipping,petrochemical and other fields because of its smooth operation and high load capacity.With the continuous development of modern industrial technology,the requirements for high-precision and high-reliability products are becoming higher and higher.Spiral bevel gear has become an indispensable key component in mechanical equipment.However,the spiral bevel gear system is subjected to fatigue load under complex working conditions,which may cause cracks at the stress concentration part of the tooth root.The expansion of tooth root crack may lead to gear failure and further cause serious accidents,which has become a major problem limiting the development of gear transmission system towards long life and high reliability.Therefore,how to effectively judge the tooth root crack state has been a research hotspot in the field of gear structural strength.Although it is difficult to monitor the tooth root crack state in real time by online measurement,the tooth root crack state is closely related to the gear tooth meshing stiffness,which will affect the vibration characteristics of the system.Therefore,the change of meshing stiffness of spiral bevel gear can be judged by measuring the vibration signal,and the crack generation and crack growth status of the tooth root can be further judged.In this technical path,the tooth root crack-meshing stiffness correlation model is the key part to realize the judgment from vibration characteristics to crack state,and is also the research topic of this paper.Aiming at the problem that the tooth root crack of spiral bevel gear presents a three-dimensional complex space surface and it is difficult to calculate the stiffness of the cracked gear,this paper develops a time-variant meshing stiffness calculation method for normal and cracked gears based on the idea of energy equivalence and two-dimensional simplification of the tooth surface.Furthermore,based on the extended finite element method,the distribution law of stress intensity factors at the front edge of the crack at different parts of the gear surface is further studied.Finally,based on the finite element analysis of crack low-cycle fatigue,the fatigue crack growth life under alternating load is calculated,and the remaining load cycles and stiffness changes of the crack under fatigue load are obtained.The main research contents of this paper are as follows:1)First of all,the time-varying meshing stiffness of two dimensional gear with normal plane state and root crack is studied by energy method.The series-parallel model of gear stiffness is studied,and the time-varying meshing stiffness model of normal and cracked gear in plane state is derived.2)In order to solve the problem of calculating the meshing stiffness of spiral bevel gears with spacial curved surface cracks,starting from the tooth surface equation of spiral bevel gears,the normal and cracked slice stiffness including bending,shearing,axial compression and tooth base stiffness are calculated respectively according to the energy method and the series theory of stiffness.Combined with the modified Hertz contact theory and the stiffness parallel theory,the single-tooth meshing stiffness of spiral bevel gears is calculated.Finally,the time-varying meshing stiffness of spiral bevel gear is deduced according to the coincidence degree.3)The mechanism of initial crack propagation at the root of spiral bevel gear was studied.The crack propagation at the root of spiral bevel gear is a typical discontinuous problem.The crack propagation can be directly and effectively simulated by the extended finite element method(XFEM).In this paper,the level set method is used to prefabricate the crack at the root of the tooth,and the extended finite element method is used to simulate the crack propagation and check the crack propagation path.In the extended finite element method,the interaction integral method is used to calculate the stress intensity factor and its distribution law at the static crack tip of each part of the spiral bevel gear.4)Finally,based on the Pairs criterion of energy release rate,the crack propagation life of the remaining spiral bevel gear root is predicted by the low-cycle fatigue propagation finite element method,and the number of cycles of the crack under the constant amplitude fatigue load spectrum is obtained.By calculating the stiffness curve of gear pair with initial crack,the stiffness of gear with crack is coupled with the problem of fatigue crack propagation life,and the remaining number of cycles can be directly determined by the stiffness.