| Spiral bevel gear is widely used in aviation transmission system for its advantages of high loading capacity,smooth transmission and compact structure,and nowadays is becoming much lighter as well as bearing heavier loads.As a crucial part of the transmission system,it has an evidently influence on the performance of the aircraft.Therefore,many researches focused their attention on the mechanical behavior of spiral bevel gear transmission system.The widely used Lumped-mass Method(LM)could hardly show the meshing characterictics which usually depend on the Finite Element Method(FEM);and also some parameters of LM rely on experiences and some hypothesis which jeopardize the credibility,authority and accuracy of the method.On the other hand,the FEM needs a refined mesh model to achieve a good result which increases the computing cost sharply and it also has a bad convergence to deal with high-speed or strong non-linear problems.These two problems make FEM unsuitable in huge and complex dynamic analysis.To show the meshing characteristics in dynamics and reduce the influence of subjective parameters,we introduce the Vector Form Intrinsic Finite Element Method(VFIFE Method)to deal with gear meshing analysis which could also has a good computing efficiency and accuracy.Firstly,we derived the formula of hexhedral element which supplement the traditional VFIFE method and make it suitable for 3D problems.During the process,we modified the process of solving pure nodal deformation of the VFIFE Method,which simplified the solving process and increase efficiency.Secondly,the matching contact algorithm for 3D of the VFIFE method had been derived,which make the method possible to solve contact problems between 3D complicated surfaces.Then the computing accuracy and efficiency of the method was verified by some numerical examples whose results were compared to FEM and theoretical method.Finally,we made some modifications of the damping ratio in dynamic analysis and then the modified VFIFE method was used in gear static and dynamic meshing analysis.We conducted the static analysis of spiral bevel gear using VFIFE Method and FEM,compared the results again and then discussed the meshing performances with edge contact and installation errors.In the dynamics,the changing rules of dynamic response are concluded.The proposed method conquer the subjective parameters shortages of the LM as well as has a good computing speed.In this paper,we introduce the VFIFE Method from the civil engineering and make some modifications to improve the computing efficiency,which broadens the fields of the method and lays a foundation for the method applied in complex transmission systems and even in mechanical systems.The main contents and results are as followings:(1)We proposed a new thought to deal with the reverse movement,sicne the traditional method is more complicated and time consuming.The modifications and improvement of the traditional method could reduce computing time,especially in large-scale problems.(2)We derived the formulas of the eight-node haxhedral element dependently.The literatures and references are usually focused on plane and truss structures in civil engineering.There are few discussions of the widely used haxhedral element.Therefore,we derived the formulas of the eight-node haxhedral element according to that process of plane element,membrane element and truss element.The modifications of pure nodal deformation also has been introduced,which expand the use of the method.(3)We conduct the formulas of contact algorithm for the meshing analysis using the mater-slave algorithm and inside-outside algorithm.According to the features of gear meshing and contact algorithms,we used the mater-slave algorithm for the global search and inside-outside algorithm for the local search.The combination of the contact algorithm and VFIFE Method could lay the foundation of gear meshing analysis.(4)The static meshing performance was analyzed using the modified VFIFE Method which was compared to the results of FEM.According to the theory and formula of spiral bevel gear,the accurate mesh model was established and analyzed by batch process.The performance indexes,such as contact stress curves,bending stress curves,contact pattern and transmission error,had been extracted and the edge contact phenomenon and influences of installation error were discussed.A pair of spiral bevel gear with different mesh densities was analyzed by modified VFIFE Method and FEM;the results show that the proposed method has a good convergence and accuracy while it is also less sensible to mesh conditions,which means it could get a good result with a rather coarse mesh.This characteristic makes it suitable for large-scale or strong non-linear problems,even though it cost much more time than FEM in static analysis.(5)Dynamic responses were discussed based on the static analysis,and the results were compared to that of LM.According to the theory of VFIFE Method,we modified the expression of damping ratio to improve the speed of convergence.The results show the rationality and accuracy of proposed method,and the computing time is also acceptable.Then the affects of speed,load and friction are discussed and the changing rules are summarized.The proposed method could avoid some subjective parameters and make the results more convincing and reasonable.Meanwhile,the results and changing rules also provide a reference for the gear tooth surface design. |
PDF Full Text Request