WO2018086160A1 - 基于粗糙表面的直齿轮三维接触刚度计算方法 - Google Patents
基于粗糙表面的直齿轮三维接触刚度计算方法 Download PDFInfo
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- the invention belongs to the field of gear mechanics analysis and dynamics research, and particularly relates to a method for calculating a three-dimensional contact stiffness of a spur gear based on a rough surface.
- the internal excitation of the gear pair is generated inside the system during the gear pair meshing process, and is also the focus and difficulty in the research of gear system dynamics.
- the internal excitation of the gear system is mainly divided into three types: time-varying stiffness excitation, time-varying error excitation and meshing impact excitation.
- the time-varying stiffness excitation is mainly composed of three parts: the bending stiffness, shear stiffness and contact stiffness of the gear teeth.
- the current application of the IOS6336 meshing stiffness calculation formula and the stiffness excitation of the gear are equivalent to Fourier expansion.
- the calculation of contact stiffness is based on the Hertz contact theory of smooth surfaces, so it is not possible to accurately simulate the deformation of the contact state of the actual tooth surface. Therefore, in order to establish a more effective gear dynamics model, it is necessary to study the calculation method of three-dimensional contact stiffness of spur gear based on rough surface.
- the object of the present invention is to provide a calculation method for three-dimensional contact stiffness of a spur gear based on a rough surface, which can effectively and accurately determine the accuracy of the gear contact stiffness model, and fully considers factors such as assembly error and manufacturing error.
- the problem of uneven distribution of the tooth engagement load is to provide a calculation method for three-dimensional contact stiffness of a spur gear based on a rough surface, which can effectively and accurately determine the accuracy of the gear contact stiffness model, and fully considers factors such as assembly error and manufacturing error.
- S1 uses a hexahedral mesh, requiring each square mesh area A 0 on the tooth surface to be equal;
- S2 calculates the smooth tooth surface contact pressure distribution based on the finite element method, and extracts the contact zone node pressure P i ;
- the S1 specifically includes the following steps:
- the step 2) specifically includes the following steps:
- the step 3) specifically includes the following steps:
- the mesh used in the finite element analysis is already small enough to treat the extracted contact zone node pressure P i as the contact pressure in the square mesh on the tooth surface, so the contact pressure of the different square mesh on the tooth surface is expressed. for:
- n is the frequency index
- z is the height of the rough surface profile
- D is the fractal dimension
- ⁇ is the size parameter of the spectral density
- G is the roughness parameter.
- x represents the sampling length coordinate. The same material and processing method are used to obtain the gear sample block, and the coordinates of the tooth surface topography are obtained on the three-dimensional shape measuring instrument. Then the power spectral density method is used to fit the WM function to obtain two parameters of D and G.
- the contact of the two rough surfaces in the contact model of the gear pair is simplified to the contact of a rigid surface with the rough surface.
- the dotted line indicates that the neutral surface indicates the contact position of the ideal smooth gear pair, and the portion where the rough surface above the neutral surface protrudes is defined as a microprotrusion.
- the rough surface is composed of a myriad of circular microprotrusions of various sizes, assuming that each microprotrusion is separated from each other and the interaction is negligible.
- a' is the cross-sectional area where the micro-convex body intersects the rigid surface before denaturation
- a the contact area of the micro-convex body after deformation with the rigid surface is defined as the true contact area.
- the deformation of the micro-convex is divided into elastic and plastic deformation, and the critical cross-sectional area that determines the elastic or plastic deformation of the micro-convex is determined by the material properties of the gear pair:
- the lower corners e and p represent elastic and plastic deformation, respectively.
- H represents the hardness of the softer material
- H 2.8Y
- R and ⁇ represent the curvature and normal deformation of the single microprotrusion, respectively.
- the total elastic deformation force F E and the total plastic deformation force F P of a single cube mesh are expressed as:
- the step 3) further includes the following steps:
- the deformation is divided into two stages: full elastic deformation and complete plastic deformation, wherein the plastic deformation stage has a stiffness of zero.
- the single microprotrusion normal contact stiffness where complete elastic deformation occurs can be expressed as
- the normal contact stiffness in a single square mesh is obtained by integrating in the fully elastic region.
- ⁇ denotes the coefficient of static friction
- ⁇ 1 , ⁇ 2 , G 1 , G 2 represent the Poisson's ratio and shear modulus of the two materials, respectively
- t and f represent the tangential and normal loads of a single micro-convex, respectively
- r represents the true contact radius of a single micro-convex
- a indicates the true contact area.
- T ⁇ b A r
- ⁇ b the shear strength of the softer material
- a r the actual contact area of the square grid
- the step 4) specifically includes the following steps:
- the tangential contact stiffness K 2 of the contact tooth surface is
- ⁇ represents the pressure angle of the gear.
- the three-dimensional contact stiffness calculation method of the spur gear based on the rough surface proposed by the invention further corrects the gear contact stiffness calculation model, solves the disadvantages of the smooth contact according to the Hertz theory in the original calculation process, and reveals the roughness parameter during the rough tooth surface contact process.
- the influence law and calculation method of the stiffness characteristics of the gear are based on the finite element three-dimensional contact stiffness calculation model, which is more accurate than the original two-dimensional calculation model or calculation formula, and takes into account the influence of errors in the actual assembly and manufacturing process of the gear, so that the gear contact stiffness is more accurate, The basis of gear dynamics analysis.
- the established spur gear contact stiffness model contains roughness-related parameters for the first time, and indirectly describes the relationship between machining mode and gear contact stiffness.
- the fractal theory is applied to the different positive direction grids of the contact area for the first time.
- the three-dimensional contact stiffness of the gears is obtained, and the gear pair assembly is avoided.
- the influence of the eccentric load caused by the manufacturing error on the meshing stiffness is avoided.
- S1, S2, and S3 are equally applicable to gears other than spur gears.
- Figure 1 is a simplified contact model diagram of a gear pair.
- Figure 2 is a diagram of a single microprotrusion contact area.
- Figure 3 is a schematic diagram of the spur gear contact stiffness transformation.
- a method for calculating a three-dimensional contact stiffness of a spur gear based on a rough surface comprising the following steps:
- the step 1) specifically includes the following steps:
- the step 2) specifically includes the following steps:
- the step 3) specifically includes the following steps:
- the mesh used in the finite element analysis is small enough to consider the extracted contact zone node pressure P i as the contact pressure in the square mesh on the tooth surface, so the contact pressure of the different square mesh on the tooth surface It can be expressed as:
- the outline of the rough flank can be represented by the W-M function as:
- n is the frequency index
- z is the height of the rough surface profile
- D is the fractal dimension
- ⁇ is the size parameter of the spectral density
- G is the roughness parameter
- x Indicates the sampling length coordinate.
- the contact of the two rough surfaces in the contact model of the gear pair is simplified to the contact of a rigid surface with the rough surface, as shown in FIG.
- the dotted line is the neutral surface indicating the contact position of the ideal smooth gear pair, and the portion where the rough surface above the neutral surface is convex is defined as a micro-convex, as shown in FIG.
- Rough surface is made by A plurality of circular microprotrusions of different sizes are formed, assuming that each microprotrusion is separated from each other, and the interaction is negligible.
- a' is the cross-sectional area where the micro-convex body intersects the rigid surface before denaturation
- a the contact area of the micro-convex body after deformation with the rigid surface is defined as the true contact area.
- the deformation of the micro-convex is divided into elastic and plastic deformation, and the critical cross-sectional area that determines the elastic or plastic deformation of the micro-convex is determined by the material properties of the gear pair:
- the lower corners e and p represent elastic and plastic deformation, respectively.
- E 1 E 2 v 1 v 2 represents the elastic modulus and Poisson's ratio of the two contact tooth faces, respectively
- H represents the hardness of the softer material
- H 2.8Y
- R and ⁇ respectively represent a single microconvex The curvature of the body and the amount of normal deformation.
- the total elastic deformation force F E and the total plastic deformation force F P of a single cube mesh can be expressed as:
- the step 3) specifically includes the following steps:
- the deformation is divided into two stages: full elastic deformation and complete plastic deformation, wherein the plastic deformation stage has a stiffness of zero.
- the single microprotrusion normal contact stiffness where complete elastic deformation occurs can be expressed as
- the normal contact stiffness in a single square mesh is obtained by integrating in the fully elastic region.
- ⁇ denotes the coefficient of static friction
- ⁇ 1 ⁇ 2 G 1 G 2 represents the Poisson's ratio and shear modulus of the two materials, respectively
- t and f represent the tangential and normal loads of a single micro-convex, respectively
- r represents the true contact area radius of a single micro-convex and Where a represents the true contact area.
- T ⁇ b A r
- ⁇ b the shear strength of the softer material
- a r the actual contact area of the square grid
- the step 4) specifically includes the following steps:
- the tangential contact stiffness K 2 of the contact tooth surface is
- the contact stiffness of the tooth surface can be expressed as
- ⁇ represents the pressure angle of the gear.
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Abstract
Description
Claims (6)
- 基于粗糙表面的直齿轮三维接触刚度计算方法,其特征在于:本方法包括如下步骤,S1采用六面体网格划分,要求齿面上每个正方形网格面积A0相等;S2基于有限元法计算光滑齿面接触压力分布,提取接触区节点压强Pi;S3基于分形理论计算单个正方形网格法向接触刚度KN,切向接触刚度KT;S4计算齿面接触刚度K。
- 根据权利要求1所述的基于粗糙表面的直齿轮三维接触刚度计算方法,其特征在于:所述S1具体包括以下步骤:齿轮对采用六面体网格划分,约束六面体网格的边长为lmm,因此得到齿轮对的接触面上均是标准的正方形网格,网格面积A0=l2mm2。
- 根据权利要求1所述的基于粗糙表面的直齿轮三维接触刚度计算方法,其特征在于:所述步骤2)具体包括以下步骤:将网格划分好的齿轮对导入有限元仿真软件中,设置静力学分析环境,通过定义每个齿轮的材料属性、接触对、固定被动齿轮的中心孔,然后在主动齿轮轴线上施加扭矩T,以载荷步的方式得到当前齿轮特定接触面的应力和应变云图,此时提取特定接触面上接触区节点压强Pi。
- 根据权利要求1所述的基于粗糙表面的直齿轮三维接触刚度计算方法,其特征在于:所述步骤3)具体包括以下步骤:在有限元分析中使用的网格已经足够小,将提取到的接触区节点压强Pi看作是齿面上正方形网格内的接触压强,因此齿面上不同的正方形网格的接触压力表示为:Fi=PiA0 (1)由于机械加工的齿面具有分形特征,存在局部与整体自相似的特性;粗糙齿面的轮廓由W-M函数表示为:其中,n为频率指数,n=0和nmax是最低及最高截止频率对应的序列,z表示粗糙表面轮廓高度,D表示分形维数,γ表示谱密度的尺寸参数,G表示粗糙度参数,x表示采样长度坐标;采用相同的材料和加工方法得到齿轮样品方块,在三维形貌测量仪上获得齿面形貌点坐标,再采用功率谱密度方法拟合W-M函数得到D和G两个参数;在齿轮对的接触模型中两个粗糙表面的接触被简化为一个刚性表面与粗糙表面的接触;虚线是中性面表示理想光滑齿轮对的接触位置,将中性面的上方的粗糙表面凸出的部分定义为微凸体;粗糙表面是由无数多个大小不一的圆形微凸体组成,假设每个微凸体之间彼此分立,相互作用忽略不计;当粗糙表面与刚性表面相互接触时,不同微凸体在压力的作用下发生弹性或塑性形变,则微凸体横截面积a′的分布规律满足:式中,a′为微凸体变性前与刚性表面相交的横截面积,a表示微凸体变形后与刚性面的接触面积定义为真实接触面积,当微凸体发生弹性变形时,a′=2a,当发生塑性形变时,a′=a,D表示分形维数,a′l表示最大横街面积,表示域拓展因子,可由超越方程(4)求得:[ψ(2-D)/2-(1+ψ-D/2)-(2-D)/D]/[(2-D)/D]=1 (4)根据微凸体横截面积的不同,将微凸体的变形分为弹性和塑性变形,则决定微凸体发生弹性或塑性变形的临界横截面积是由齿轮对的材料属性决定的:凸体的横截面积a′>a′c,这发生弹性变形,若横截面积a′≤a′c,这发生塑性变形;粗糙表面单个微凸体法向变形量由W-M函数中峰-谷间幅值来表示,单个微凸体曲率R为,对于单个微凸体的弹性或塑性形变,其法向载荷f与横截面积a′满足如下关系,fp=Ha′ (9)单个正方体网格总的弹性变形力FE和总塑性变形力FP表示为:由此得到单个正方形网格内总的接触压力,Fi=FE+FP (12)。
- 根据权利要求4所述的基于粗糙表面的直齿轮三维接触刚度计算方法,其特征在于:所述步骤3)还包括以下步骤:根据式(1)和(12)可以得到单个正方形网格当中最大横截面积a′l;对于单个微凸体,其变形分为完全弹性变形及完全塑性变形两个个阶段,其中塑性变形阶段刚度为0;根据刚度定义,发生完全弹性变形的单个微凸体法向接触刚度可表示为,因此通过在完全弹性区域进行积分,得到单个正方形网格内法向接触刚度研究表明单个微凸体法切向变形量δt表示为式中μ表示静摩擦系数,G′表示结合面的等效剪切模量且满足1/G′=(2-ν1)/G1+(2-ν2)/G2,其中ν1、ν2、G1、G2分别表示两材料的泊松比和剪切模量,t和f分别表示单个微凸体的切向和法向载荷,r表示单个微凸体的真实接触区域半径且a表示真实接触面积;则单个微凸体的切向刚度可以表示为,对于单个微凸体切向载荷t与法向载荷f满足如下关系,T=τbAr,τb表示较软材料的剪切强度,Ar表示正方形网格的实际接触面积则单个正方形网格的切向接触刚度KT表示为,
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