高速直齿轮齿面有限元网格精准离散方法

A precise finite element mesh discretization method of the tooth surface of high speed spur gears

为解决航空高速直齿圆柱齿轮齿面有限元网格精准离散的问题,提出一种可以适应轮齿啮合过程中齿面滚滑速度以及曲率半径变化的确定齿面网格尺寸的方法。根据齿轮啮合过程中的几何关系和主从动轮齿廓速度矢量关系,通过理论推导得出主从动轮齿廓上任意啮合点处的切向速度、相对滑移速度、曲率半径以及法向载荷的计算公式。以某航空发动机中的传动齿轮为例,利用公式计算得出航空发动机高速齿轮齿面曲率半径、切向速度、相对滑移速度以及载荷的大致范围。基于此范围,建立多组曲率半径适宜,转速和接触面网格尺寸不同且转速低及载荷恒定的圆柱滚子曲面接触有限元模型。利用两圆柱滚子曲面接触摩擦产生的滑移能损失模拟轮齿接触产生的滑移能损失,最终以滑移能损失仿真值与理论值之间的误差小于10%为前提,对比分析得出在不同齿面切向速度下,齿面网格尺寸与齿面曲率半径之间的最佳倍数关系。研究结果表明:选用合理的齿面网格尺寸能大大提高高速直齿圆柱齿轮有限元仿真结果的准确性。

Finite element,as a common gear mesh simulation analysis method,is widely used in the study of gear dynamics performance.When the gear tooth surface is discretized by finite element grids,the grid size has a great influence on the correctness of the gear mesh finite element simulation results.In the gear meshing process,the contact between the gear teeth belongs to the contact between the two surfaces.The basic idea of surface finite element discretization is to use a finite number of grids to simulate a continuous smooth surface body,and to achieve the same discretization accuracy.The smaller the radius of curvature of the surface,the smaller the grid size is required.In addition,high linear speed spur gears have a high rotational speed,and the mesh of the tooth surface finite element has large centrifugal strength,which leads to impact vibration between meshes when the tooth surface finite element model is involved in meshing.The model has some error regarding the actual tooth surface contact.Therefore,in order to establish an accurate tooth contact finite element model,the key problem is to find an optimal multiplicative relationship between the tooth face finite element grid size and the tooth face radius of curvature at different rotational speeds.To solve the precise discretization problem of the finite element mesh of a high-speed spur gear tooth surface,this paper presents a method for selecting the tooth surface grid size.This method can adapt to the variation of the tooth surface sliding speed and the curvature radius during tooth meshing.Formulas of the curvature radius,tangential velocity,relative slip velocity and loads at any meshing point on the tooth profile are acquired according to the geometric and velocity vector relations to the main and slave gear tooth profile in the meshing process.Taking the transmission gear in an aero-engine under research as an example,the approximate range of parameters such as radius of curvature,tangential velocity,relative slip velocity,and loads on the high-speed gear tooth surface of the aero-engine is obtained through calculation.Based on this range,several sets of cylindrical roller surface contact finite element models are established with suitable curvature radius,different rotational speeds,contact surface grid size,and constant relative slip velocity and loads.The sliding energy generated by the contact of the two cylindrical rollers is used to simulate the sliding energy generated by the tooth contact.Finally,under the condition that the error between the simulation value and the theoretical value of the sliding energy is less than 10%,the optimal relationship between the tooth surface grid size and the tooth surface curvature radius at different tooth surface tangential speeds is obtained through analysis.The research results show that,when the maximum tooth surface tangential velocity is 0-25 m/s,the tooth surface grid size should be one fortieth of the minimum curvature radius of the tooth surface;when the maximum tooth surface tangential velocity is 25-40 m/s,the tooth surface grid size should be one seventieth of the minimum curvature radius of the tooth surface;when the maximum tooth surface tangential velocity is 40-55 m/s,the tooth surface grid size should be one hundred and tenth of the minimum curvature radius of the tooth surface.The simulation results of the finite element model obtained by using the above-mentioned tooth surface finite element grid discretization method have an error of less than 10%of the theoretical value.