CN111967096A - Design method of diamond roller and worm grinding wheel - Google Patents

Design method of diamond roller and worm grinding wheel Download PDF

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CN111967096A
CN111967096A CN202010519002.7A CN202010519002A CN111967096A CN 111967096 A CN111967096 A CN 111967096A CN 202010519002 A CN202010519002 A CN 202010519002A CN 111967096 A CN111967096 A CN 111967096A
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grinding wheel
worm grinding
diamond roller
worm
coordinate system
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CN111967096B (en
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周元生
唐进元
石贤林
张无忌
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Central South University
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Abstract

本申请涉及面齿轮磨削加工领域，尤其涉及一种金刚滚轮及蜗杆砂轮的设计方法，包括：步骤一、通过插齿刀齿面方程得出蜗杆砂轮齿面方程；步骤二、由步骤一的蜗杆砂轮齿面方程得出蜗杆砂轮轴截面齿廓并以蜗杆砂轮轴截面齿廓作为金刚滚轮齿廓；步骤三、建立金刚滚轮包络蜗杆砂轮的坐标系，得出金刚滚轮包络出的蜗杆砂轮。本申请采用新的齿形式金刚滚轮齿廓来加工蜗杆砂轮，通过对加工蜗杆砂轮的齿形式金刚滚轮齿廓进行合理设计，使得加工出的蜗杆砂轮齿面精度更高，从而保证了蜗杆砂轮磨削出的面齿轮具有较高的精度。

Figure 202010519002

The present application relates to the field of face gear grinding, and in particular, to a design method of a diamond roller and a worm grinding wheel, comprising: step 1, obtaining the tooth surface equation of the worm grinding wheel through the tooth surface equation of a gear shaper; The tooth profile of the worm grinding wheel is obtained from the tooth surface equation of the worm grinding wheel, and the tooth profile of the worm grinding wheel shaft is used as the tooth profile of the diamond roller; Step 3: Establish the coordinate system of the diamond roller enveloping the worm grinding wheel, and obtain the worm enveloped by the diamond roller. Grinding wheel. This application adopts the new tooth profile of diamond roller to process the worm grinding wheel. By rationally designing the tooth profile of the diamond roller with the tooth form for processing the worm grinding wheel, the precision of the tooth surface of the processed worm grinding wheel is higher, thus ensuring the grinding of the worm grinding wheel. The chipped face gear has high precision.

Figure 202010519002

Description

金刚滚轮及蜗杆砂轮的设计方法Design method of diamond roller and worm grinding wheel

【技术领域】【Technical field】

本申请涉及面齿轮磨削加工领域，尤其涉及一种金刚滚轮及蜗杆砂轮的设计方法。The present application relates to the field of face gear grinding, in particular to a design method of a diamond roller and a worm grinding wheel.

【背景技术】【Background technique】

面齿轮传动具有结构紧凑、安装调试方便、传动比与重合度大的优点。目前主要通过蜗杆砂轮磨削的方式来实现面齿轮的高精高效加工，而蜗杆砂轮又主要通过锥面金刚滚轮或者齿形式金刚滚轮加工得到。锥面金刚滚轮加工蜗杆砂轮主要包括以下三个步骤：The face gear transmission has the advantages of compact structure, convenient installation and debugging, and large transmission ratio and coincidence. At present, the high-precision and high-efficiency machining of face gears is mainly achieved by worm grinding wheel grinding, and worm grinding wheels are mainly processed by conical diamond rollers or toothed diamond rollers. The processing of worm grinding wheel with tapered diamond roller mainly includes the following three steps:

1)根据蜗杆砂轮的齿面表达式，计算在蜗杆砂轮齿面上的一系列分布规律的齿面离散点；1) According to the tooth surface expression of the worm grinding wheel, calculate a series of tooth surface discrete points with regular distribution on the tooth surface of the worm grinding wheel;

2)在锥面金刚滚轮轴截面内选一点(对应锥面金刚滚轮齿面上的一条线)，使得该点与蜗杆砂轮齿面上的每一点相切。从而计算出锥面金刚滚轮在加工蜗杆砂轮齿面时的各刀位点。2) Select a point (corresponding to a line on the tooth surface of the tapered diamond roller) in the cross section of the tapered diamond roller so that the point is tangent to each point on the tooth surface of the worm grinding wheel. Thus, each tool position of the tapered diamond roller when machining the tooth surface of the worm grinding wheel is calculated.

3)根据2)的原理，建立实际加工的机床坐标系，计算加工的各个时刻机床各个轴的位置，生成NC代码。3) According to the principle of 2), establish the machine tool coordinate system for actual processing, calculate the position of each axis of the machine tool at each time of processing, and generate NC code.

上述三个步骤中，步骤1)2)主要包括以下四个步骤：In the above-mentioned three steps, step 1) 2) mainly includes the following four steps:

a)建立插齿刀包络蜗杆砂轮的坐标系，通过坐标变换和啮合原理，由插齿刀齿面方程rs(us,uz)及齿面法向量ns(us,uz)推导蜗杆砂轮齿面方程rws(us,φs)及齿面法向量nws(us,φs)。a) Establish the coordinate system of the gear shaper enveloping the worm grinding wheel, through the coordinate transformation and meshing principle, the gear shaper tooth surface equation rs (u s , u z ) and the tooth surface normal vector n s (u s , u z ) ) to derive the worm grinding wheel tooth surface equation r ws (u s , φ s ) and the tooth surface normal vector n ws (u s , φ s ).

b)通过离散蜗杆砂轮齿面方程rws(us,φs)和齿面法向量nws(us,φs)的参数us,φs，求得蜗杆砂轮一系列的齿面点以及齿面点所对应的法向量。b) By discretizing the worm grinding wheel tooth surface equation r ws (u s , φ s ) and the parameters u s , φ s of the tooth surface normal vector n ws (u s , φ s ), obtain a series of tooth surface points of the worm grinding wheel And the normal vector corresponding to the tooth surface point.

c)锥面金刚滚轮的齿面方程以及齿面法向量可以分别表示为：rc(ux,θs)和nc(ux,θs)。在锥面金刚滚轮轴截面齿廓上选取一点rc *(ux *，θs *)，该点对应的齿面法向量为nc *(ux *，θs *)。使该点与蜗杆砂轮齿面上某一点相切，从而加工出这一点，循环往复直至加工完整个蜗杆砂轮齿面。c) The tooth surface equation and tooth surface normal vector of the tapered diamond roller can be expressed as: rc (u x , θ s ) and n c ( u x , θ s ), respectively. Select a point rc * (u x * , θ s * ) on the tooth profile of the tapered diamond roller shaft section, and the tooth surface normal vector corresponding to this point is n c * ( u x * , θ s * ) . Make this point tangent to a certain point on the tooth surface of the worm grinding wheel, so as to process this point, and cycle back and forth until the entire tooth surface of the worm grinding wheel is processed.

锥面金刚滚轮加工蜗杆砂轮主要存在以下问题：The main problems of worm grinding wheel processing with tapered diamond rollers are as follows:

采用锥面金刚滚轮加工蜗杆砂轮时，需要规划大量的加工路径去加工一个高精度的蜗杆砂轮，加工效率低。When using the tapered diamond roller to process the worm grinding wheel, it is necessary to plan a large number of processing paths to process a high-precision worm grinding wheel, and the processing efficiency is low.

步骤a)中，通过插齿刀包络方法计算出的蜗杆砂轮齿面可能会出现奇异点，无法求解刀具在奇异点处的刀位点与刀轴矢量。In step a), a singular point may appear on the tooth surface of the worm grinding wheel calculated by the enveloping method of the gear shaping tool, and the tool position point and the tool axis vector of the tool at the singular point cannot be solved.

采用锥面金刚滚轮加工高精度的蜗杆砂轮对机床的控制精度以及机床各轴的行程要求非常高。The use of tapered diamond rollers to process high-precision worm grinding wheels has very high requirements on the control accuracy of the machine tool and the travel of each axis of the machine tool.

齿形式金刚滚轮加工蜗杆砂轮主要包括以下两个步骤：The machining of worm grinding wheels with toothed diamond rollers mainly includes the following two steps:

4)定义齿形式金刚滚轮的齿廓并建立齿形式金刚滚轮加工蜗杆砂轮的模型；4) Define the tooth profile of the toothed diamond roller and establish a model for the toothed diamond roller to process the worm grinding wheel;

5)根据4)的原理，建立实际加工的机床坐标系，计算加工的各个时刻机床各个轴的位置，生成NC代码。5) According to the principle of 4), establish the machine tool coordinate system for actual processing, calculate the position of each axis of the machine tool at each time of processing, and generate NC code.

齿形式金刚滚轮加工蜗杆砂轮存在的最主要问题是：The main problems existing in the machining of worm grinding wheels with toothed diamond rollers are:

齿形式金刚滚轮的齿廓难以精确定义，目前主要采用插齿刀齿廓或等效齿轮齿廓作为金刚滚轮齿廓。从而导致加工出的蜗杆砂轮难以满足磨削精密面齿轮的要求。The tooth profile of the toothed diamond roller is difficult to define precisely. At present, the tooth profile of the gear shaper cutter or the equivalent gear tooth profile is mainly used as the tooth profile of the diamond roller. As a result, it is difficult for the processed worm grinding wheel to meet the requirements of grinding precision face gears.

【发明内容】[Content of the invention]

为解决现有的齿形式金刚滚轮的齿廓难精确定义，导致加工出的蜗杆砂轮难以满足磨削精密面齿轮要求的问题，本申请提供一种齿形式金刚滚轮和蜗杆砂轮的设计方法。In order to solve the problem that the tooth profile of the existing tooth-shaped diamond rollers is difficult to precisely define, which makes the processed worm grinding wheel difficult to meet the requirements of grinding precision face gears, the present application provides a design method of a tooth-shaped diamond roller and a worm grinding wheel.

本申请为解决其技术问题所采用的技术方案：The technical solution adopted by this application to solve its technical problems:

金刚滚轮及蜗杆砂轮的设计方法，包括：Design method of diamond roller and worm grinding wheel, including:

步骤一、通过插齿刀齿面方程得出蜗杆砂轮齿面方程；Step 1. Obtain the tooth surface equation of the worm grinding wheel through the tooth surface equation of the gear shaper;

步骤二、由步骤一的蜗杆砂轮齿面方程得出蜗杆砂轮轴截面齿廓并以蜗杆砂轮轴截面齿廓作为金刚滚轮齿廓；In step 2, the tooth profile of the worm grinding wheel shaft section is obtained from the tooth surface equation of the worm grinding wheel in step 1, and the tooth profile of the worm grinding wheel shaft section is used as the tooth profile of the diamond roller;

步骤三、建立金刚滚轮包络蜗杆砂轮的坐标系，得出金刚滚轮包络出的蜗杆砂轮，金刚滚轮与蜗杆砂轮的齿廓包络满足Step 3: Establish the coordinate system of the diamond roller enveloping the worm grinding wheel, and obtain the worm grinding wheel enveloped by the diamond roller. The tooth profile envelope of the diamond roller and the worm grinding wheel satisfies

Figure BDA0002531244090000031
Figure BDA0002531244090000031

其中，

Figure BDA0002531244090000032
表示从金刚滚轮固联坐标系Sg到蜗杆砂轮固联坐标系Sw的齐次坐标变换矩阵；in,
Figure BDA0002531244090000032
Represents the homogeneous coordinate transformation matrix from the fixed coordinate system S g of the diamond roller to the fixed coordinate system S w of the worm grinding wheel;

Figure BDA0002531244090000033
表示金刚滚轮齿面在蜗杆砂轮固联坐标系Sw中形成的曲面族；
Figure BDA0002531244090000033
Represents the surface family formed by the tooth surface of the diamond roller in the fixed coordinate system S w of the worm grinding wheel;

rg(us，θs)表示金刚滚轮齿面方程；r g (u s , θ s ) represents the tooth surface equation of the diamond roller;

Figure BDA0002531244090000034
表示金刚滚轮和蜗杆砂轮在Sp坐标系下的啮合方程；
Figure BDA0002531244090000034
Represents the meshing equation of diamond roller and worm grinding wheel in Sp coordinate system;

np表示金刚滚轮在Sp坐标系下的单位法向量； n p represents the unit normal vector of the diamond roller in the Sp coordinate system;

vp wg表示金刚滚轮和蜗杆砂轮在Sp坐标系下的相对速度。v p wg represents the relative speed of the diamond roller and the worm grinding wheel in the Sp coordinate system.

如上所述的金刚滚轮及蜗杆砂轮的设计方法，步骤一得出的蜗杆砂轮齿面方程满足The above-mentioned design method of diamond roller and worm grinding wheel, the worm grinding wheel tooth surface equation obtained in step 1 satisfies

Figure BDA0002531244090000041
Figure BDA0002531244090000041

其中，

Figure BDA0002531244090000042
表示从插齿刀固联坐标SS到蜗杆砂轮固联坐标系SW的齐次坐标变换矩阵；in,
Figure BDA0002531244090000042
Represents the homogeneous coordinate transformation matrix from the fixed coordinate S S of the gear shaper cutter to the fixed coordinate system SW of the worm grinding wheel;

Figure BDA0002531244090000043
表示插齿刀齿面在蜗杆砂轮固联坐标系Sw中形成的曲面族；
Figure BDA0002531244090000043
Represents the surface family formed by the tooth surface of the gear shaper in the fixed coordinate system S w of the worm grinding wheel;

rs(us，uz)表示插齿刀齿面方程；r s (u s , u z ) represents the tooth surface equation of the gear shaper;

Figure BDA0002531244090000044
表示插齿刀和蜗杆砂轮在Ss坐标系下的啮合方程；
Figure BDA0002531244090000044
Represents the meshing equation of the gear shaper and the worm grinding wheel in the S s coordinate system;

ns(us)表示插齿刀在Ss坐标系下的单位法向量；n s (u s ) represents the unit normal vector of the gear shaper in the S s coordinate system;

Figure BDA0002531244090000045
表示插齿刀和蜗杆砂轮在Ss坐标系下的相对速度。
Figure BDA0002531244090000045
Indicates the relative speed of the gear shaper and the worm grinding wheel in the S s coordinate system.

如上所述的金刚滚轮及蜗杆砂轮的设计方法，步骤二的蜗杆砂轮轴截面齿廓通过将zwowyw平面绕zw轴旋转某个角度

Figure BDA0002531244090000046
*<360)，求旋转后的平面与蜗杆砂轮齿面交线的方法来求取，其满足The design method of the diamond roller and the worm grinding wheel as described above, the tooth profile of the worm grinding wheel shaft section in step 2 is rotated by a certain angle around the z w o w y w plane around the z w axis
Figure BDA0002531244090000046
* <360), find the intersection of the rotated plane and the tooth surface of the worm grinding wheel, and it satisfies

Figure BDA0002531244090000051
Figure BDA0002531244090000051

Figure BDA0002531244090000052
Figure BDA0002531244090000052

其中，pi表示蜗杆砂轮的轴截面齿廓点。Among them, pi represents the tooth profile point of the shaft section of the worm grinding wheel.

如上所述的金刚滚轮及蜗杆砂轮的设计方法，蜗杆砂轮的转角

Figure BDA0002531244090000053
与金刚滚轮的摆角
Figure BDA0002531244090000054
满足关系
Figure BDA0002531244090000055
其中，Ns和Nw分别对应插齿刀齿数和蜗杆砂轮头数。The design method of the diamond roller and the worm grinding wheel as described above, the angle of rotation of the worm grinding wheel
Figure BDA0002531244090000053
Swing angle with diamond roller
Figure BDA0002531244090000054
Satisfaction relationship
Figure BDA0002531244090000055
Among them, N s and N w correspond to the number of gear shaper teeth and the number of worm grinding wheel heads, respectively.

与现有技术相比，本申请采用新的齿形式金刚滚轮齿廓来加工蜗杆砂轮有如下优点：Compared with the prior art, the present application adopts the new tooth form diamond roller tooth profile to process the worm grinding wheel, which has the following advantages:

1、加工效率高，相对于锥面金刚滚轮加工蜗杆砂轮的方法，采用齿形式金刚滚轮加工蜗杆砂轮需要规划的刀具路径大大减少。1. The processing efficiency is high. Compared with the method of processing the worm grinding wheel with the tapered diamond roller, the tool path required to be planned is greatly reduced by using the toothed diamond roller to process the worm grinding wheel.

2、不存在因为蜗杆砂轮齿面奇异性而导致无法规划刀具路径的问题。2. There is no problem that the tool path cannot be planned due to the singularity of the tooth surface of the worm grinding wheel.

3、大大降低了对机床轴行程以及控制精度的要求。3. It greatly reduces the requirements for the machine tool axis travel and control accuracy.

4、通过对加工蜗杆砂轮的齿形式金刚滚轮齿廓进行合理设计，使得加工出的蜗杆砂轮齿面精度更高，从而保证了蜗杆砂轮磨削出的面齿轮具有较高的精度。4. By rationally designing the tooth profile of the worm grinding wheel, the tooth profile of the diamond roller can make the tooth surface of the worm grinding wheel more accurate, thereby ensuring the surface gear ground by the worm grinding wheel has a high precision.

【附图说明】【Description of drawings】

为了更清楚地说明本申请实施例中的技术方案，下面将对实施例描述中所需要使用的附图作简单地介绍，显而易见地，下面描述中的附图仅仅是本申请的一些实施例，对于本领域普通技术人员来讲，在不付出创造性劳动的前提下，还可以根据这些附图获得其他的附图。In order to illustrate the technical solutions in the embodiments of the present application more clearly, the following briefly introduces the drawings that are used in the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present application. For those of ordinary skill in the art, other drawings can also be obtained from these drawings without creative effort.

图1是插齿刀包络蜗杆砂轮坐标系的示意图；Fig. 1 is the schematic diagram of the coordinate system of the gear shaper enveloping the worm grinding wheel;

图2是齿形式金刚滚轮加工蜗杆砂轮坐标系的示意图；Fig. 2 is the schematic diagram of the coordinate system of the toothed diamond roller processing the worm grinding wheel;

图3是蜗杆砂轮轴截面齿廓求解及比较示意图；Figure 3 is a schematic diagram of the solution and comparison of the tooth profile of the worm grinding wheel shaft section;

图4是例1采用本申请的加工方法加工出的蜗杆砂轮齿面与现有齿形式金刚滚轮加工出的蜗杆砂轮齿面偏差对比图；4 is a comparison diagram of the deviation of the tooth surface of the worm grinding wheel processed by the processing method of the present application and the tooth surface of the worm grinding wheel processed by the existing tooth form diamond roller;

图5是例2采用本申请的加工方法加工出的蜗杆砂轮齿面与现有齿形式金刚滚轮加工出的蜗杆砂轮齿面偏差对比；Fig. 5 is the worm grinding wheel tooth surface deviation contrast of the worm grinding wheel tooth surface processed by the processing method of the present application and the worm grinding wheel tooth surface processed by the existing tooth form diamond roller in Example 2;

图6是本申请蜗杆砂轮磨削面齿轮得到的面齿轮的测量结果。FIG. 6 is the measurement result of the face gear obtained by grinding the face gear with the worm grinding wheel of the present application.

【具体实施方式】【Detailed ways】

下面将结合附图及具体实施例对本申请作进一步说明。The present application will be further described below with reference to the accompanying drawings and specific embodiments.

一种齿形式金刚滚轮及蜗杆砂轮的设计方法，包括以下步骤：A method for designing a toothed diamond roller and a worm grinding wheel, comprising the following steps:

步骤一、通过插齿刀齿面方程得出蜗杆砂轮齿面方程Step 1. Obtain the tooth surface equation of the worm grinding wheel through the tooth surface equation of the gear shaper

请参看图1，插齿刀、蜗杆砂轮以及面齿轮之间存在两两啮合的关系，建立插齿刀包络蜗杆砂轮的坐标系，作为插齿刀齿面作为插齿刀齿面rs(us,uz)的包络面，蜗杆砂轮齿面

Figure BDA0002531244090000061
可以通过坐标变换和啮合原理求得，如式(1)所示，Referring to Figure 1, there is a pairwise meshing relationship between the gear shaper, the worm grinding wheel and the face gear, and the coordinate system of the gear shaper enveloping the worm grinding wheel is established, as the gear shaper tooth surface rs ( u s , u z ) envelope surface, worm wheel tooth surface
Figure BDA0002531244090000061
It can be obtained by coordinate transformation and meshing principle, as shown in formula (1),

Figure BDA0002531244090000071
Figure BDA0002531244090000071

其中，

Figure BDA0002531244090000072
表示从插齿刀固联坐标SS到蜗杆砂轮固联坐标系SW的齐次坐标变换矩阵，SW坐标系表示与蜗杆砂轮固联的坐标系，坐标轴如图1所示，画出两个坐标轴yw和Zw,，然后再根据右手定则确定剩下的一根坐标轴即可确定SW坐标系。in,
Figure BDA0002531244090000072
Represents the homogeneous coordinate transformation matrix from the fixed coordinate S S of the gear shaping cutter to the fixed coordinate system SW of the worm grinding wheel. The SW coordinate system represents the coordinate system fixed to the worm grinding wheel. The coordinate axis is shown in Figure 1. Two coordinate axes y w and Z w, , and then determine the remaining one coordinate axis according to the right-hand rule to determine the SW coordinate system.

Figure BDA0002531244090000073
表示插齿刀齿面在蜗杆砂轮固联坐标系Sw中形成的曲面族；
Figure BDA0002531244090000073
Represents the surface family formed by the tooth surface of the gear shaper in the fixed coordinate system S w of the worm grinding wheel;

rs(us，uz)表示插齿刀齿面方程；r s (u s , u z ) represents the tooth surface equation of the gear shaper;

Figure BDA0002531244090000074
表示插齿刀和蜗杆砂轮在图1所示Ss坐标系下的啮合方程，Ss坐标系表示与插齿刀固联的坐标系，坐标轴如图1所示，画出两个坐标轴ys和Zs,，然后再根据右手定则确定剩下的一根坐标轴即可确定Ss坐标系；
Figure BDA0002531244090000074
Represents the meshing equation of the gear shaper and the worm grinding wheel in the S s coordinate system shown in Figure 1. The S s coordinate system represents the coordinate system fixed to the gear shaper. The coordinate axes are shown in Figure 1, and two coordinate axes are drawn. y s and Z s, , and then determine the remaining one coordinate axis according to the right-hand rule to determine the S s coordinate system;

ns(us)表示插齿刀在图1所示Ss坐标系下的单位法向量；n s (u s ) represents the unit normal vector of the gear shaper in the S s coordinate system shown in Figure 1;

Figure BDA0002531244090000075
表示插齿刀和蜗杆砂轮在图1所示Ss坐标系下的相对速度。
Figure BDA0002531244090000075
Indicates the relative speed of the gear shaper and the worm grinding wheel in the S s coordinate system shown in Figure 1.

通过求解

Figure BDA0002531244090000076
可以将uz表示为us和
Figure BDA0002531244090000077
的函数，再将其带入
Figure BDA0002531244090000078
中，消去uz从而得到插齿刀包络的蜗杆砂轮的齿面方程
Figure BDA0002531244090000079
by solving
Figure BDA0002531244090000076
u z can be expressed as u s and
Figure BDA0002531244090000077
function, and then bring it into
Figure BDA0002531244090000078
, eliminate u z to obtain the tooth surface equation of the worm grinding wheel enveloped by the gear shaper cutter
Figure BDA0002531244090000079

步骤二、由步骤一的蜗杆砂轮齿面方程得出蜗杆砂轮轴截面齿廓并以蜗杆砂轮轴截面齿廓作为金刚滚轮齿廓；In step 2, the tooth profile of the worm grinding wheel shaft section is obtained from the tooth surface equation of the worm grinding wheel in step 1, and the tooth profile of the worm grinding wheel shaft section is used as the tooth profile of the diamond roller;

蜗杆砂轮的轴截面是指通过蜗杆砂轮轴线的截面，请参看图3所示zwowyw平面，该截面与蜗杆砂轮齿面的交线被称为蜗杆砂轮的轴截面齿廓，不同的轴截面齿廓通过将zwowyw平面绕zw轴旋转某个角度

Figure BDA0002531244090000081
求旋转后的平面与蜗杆砂轮齿面交线的方法来求取，其满足The axial section of the worm grinding wheel refers to the section passing through the axis of the worm grinding wheel. Please refer to the z w o w y w plane shown in Figure 3. The intersection of this section and the tooth surface of the worm grinding wheel is called the axial section tooth profile of the worm grinding wheel. The shaft section of the tooth profile is obtained by rotating the z w o w y w plane around the z w axis by some angle
Figure BDA0002531244090000081
The method of finding the intersection of the rotated plane and the tooth surface of the worm grinding wheel can be obtained, which satisfies

Figure BDA0002531244090000082
Figure BDA0002531244090000082

Figure BDA0002531244090000083
Figure BDA0002531244090000083

几何特征分析Geometric feature analysis

将对应

Figure BDA0002531244090000084
的轴截面齿廓称为参考齿廓c1，将对应于
Figure BDA0002531244090000085
的轴截面齿廓c2用于与c1进行比较分析，以证明蜗杆砂轮各轴截面齿廓一致，且蜗杆砂轮齿面可被简化为一个根据其轴截面齿廓扫掠得到的齿面。will correspond to
Figure BDA0002531244090000084
The shaft section tooth profile of is called the reference tooth profile c 1 and will correspond to
Figure BDA0002531244090000085
The tooth profile of the shaft section c 2 is used for comparative analysis with c 1 to prove that the tooth profile of each shaft section of the worm grinding wheel is consistent, and the tooth surface of the worm grinding wheel can be simplified as a tooth surface obtained by sweeping the tooth profile of its shaft section.

先将先将c1、c2的起点重合，再将某个齿廓向另一个齿廓的方向旋转一个合适的角度θ，以表1为例，请参看附图3(b)(c)(d)，First, the starting points of c 1 and c 2 are coincident, and then a certain tooth profile is rotated in the direction of the other tooth profile by an appropriate angle θ. Taking Table 1 as an example, please refer to Figure 3(b)(c) (d),

Figure BDA0002531244090000086
Figure BDA0002531244090000086

Figure BDA0002531244090000091
Figure BDA0002531244090000091

表1两组蜗杆砂轮设计参数Table 1 Design parameters of two groups of worm grinding wheels

计算如表1所示的两个不同参数的蜗杆砂轮轴截面齿廓，对应Calculate the tooth profile of the worm grinding wheel shaft section with two different parameters as shown in Table 1, corresponding to

Figure BDA0002531244090000092
的参考齿廓c1根据上述方法被用来与
Figure BDA0002531244090000093
等于10°，20°,40°,80°,160°,240°,320°的齿廓进行了比较。比较结果显示，任意c2与c1的最大偏差都小于100nm，对于该数量级的偏差在本研究目前的工业应用领域完全可以被忽略。通过改变蜗杆砂轮参数继续做多组实验，也能得到基本相同的结论，蜗杆砂轮的各个轴截面齿廓是一致的，且蜗杆砂轮齿面可被简化为一个根据其轴截面齿廓扫掠得到的扫掠面，进一步地，考虑用廓形与蜗杆砂轮轴截面齿廓相同的金刚滚轮加工蜗杆砂轮。
Figure BDA0002531244090000092
The reference tooth profile c1 according to the above method is used with
Figure BDA0002531244090000093
Tooth profiles equal to 10°, 20°, 40°, 80°, 160°, 240°, 320° were compared. The comparison results show that the maximum deviation of any c 2 and c 1 is less than 100 nm, and the deviation of this order of magnitude can be completely ignored in the current industrial application field of this study. By changing the parameters of the worm grinding wheel and continuing to do multiple sets of experiments, we can also get basically the same conclusion. The tooth profile of each axial section of the worm grinding wheel is consistent, and the tooth surface of the worm grinding wheel can be simplified as a sweep according to the tooth profile of its axial section. The swept surface of , further, consider machining the worm grinding wheel with a diamond roller with the same profile as the tooth profile of the worm grinding wheel shaft.

步骤三、建立金刚滚轮包络蜗杆砂轮的坐标系，得出金刚滚轮包络出的蜗杆砂轮Step 3: Establish the coordinate system of the diamond roller enveloping the worm grinding wheel, and obtain the worm grinding wheel enveloped by the diamond roller

请参看图2，基于上述几何特征分析，齿形式金刚滚轮加工蜗杆砂轮的过程如图所示，齿形式金刚滚轮的齿廓就是待加工蜗杆砂轮的轴截面齿廓。当蜗杆砂轮绕其轴线zw旋转时，金刚滚轮在自转的同时绕着zd轴摆动。蜗杆砂轮的转角

Figure BDA0002531244090000094
与金刚滚轮的摆角
Figure BDA0002531244090000095
满足关系
Figure BDA0002531244090000096
其中，Ns和Nw分别对应表1中的插齿刀齿数和蜗杆砂轮头数。根据该过程，蜗杆砂轮齿面作为金刚滚轮齿面rg(us，θs)的包络在Sw中如式(2)所示，Referring to Figure 2, based on the above geometrical analysis, the process of machining a worm grinding wheel with a toothed diamond roller is shown in the figure. When the worm grinding wheel rotates around its axis zw , the diamond roller swings around the zd axis while rotating. Rotation angle of worm wheel
Figure BDA0002531244090000094
Swing angle with diamond roller
Figure BDA0002531244090000095
Satisfaction relationship
Figure BDA0002531244090000096
Among them, N s and N w correspond to the number of gear shaper teeth and the number of worm grinding wheel heads in Table 1, respectively. According to this process, the worm grinding wheel tooth surface as the envelope of the diamond roller tooth surface r g (u s , θ s ) is shown in Eq. (2) in S w ,

Figure BDA0002531244090000101
Figure BDA0002531244090000101

其中，

Figure BDA0002531244090000102
表示从金刚滚轮固联坐标系Sg到蜗杆砂轮固联坐标系Sw的变换矩阵，Sg坐标系表示与金刚滚轮固联的坐标系，坐标轴如图2所示，画出两个坐标轴zg和xg,，然后再根据右手定则确定剩下的一根坐标轴即可确定Sg坐标系。in,
Figure BDA0002531244090000102
Represents the transformation matrix from the diamond roller fixed coordinate system S g to the worm grinding wheel fixed coordinate system S w . The S g coordinate system represents the coordinate system fixed to the diamond roller. The coordinate axis is shown in Figure 2, and two coordinates are drawn. axis z g and x g, , and then determine the remaining one coordinate axis according to the right-hand rule to determine the S g coordinate system.

Figure BDA0002531244090000103
表示金刚滚轮齿面在蜗杆砂轮固联坐标系Sw中形成的曲面族；
Figure BDA0002531244090000103
Represents the surface family formed by the tooth surface of the diamond roller in the fixed coordinate system S w of the worm grinding wheel;

rg(us，θs)表示金刚滚轮齿面方程；r g (u s , θ s ) represents the tooth surface equation of the diamond roller;

Figure BDA0002531244090000104
表示金刚滚轮和蜗杆砂轮在图2所示Sp坐标系下的啮合方程，Sp坐标系表示方便计算的辅助坐标系，坐标轴如图2所示，画出两个坐标轴yp和Zp,，然后再根据右手定则确定剩下的一根坐标轴即可确定Sp坐标系；
Figure BDA0002531244090000104
Represents the meshing equation of the diamond roller and the worm grinding wheel in the Sp coordinate system shown in Figure 2. The Sp coordinate system represents an auxiliary coordinate system that is convenient for calculation. The coordinate axes are shown in Figure 2, and two coordinate axes y p and Z are drawn . p , and then determine the remaining one coordinate axis according to the right-hand rule to determine the Sp coordinate system;

np表示金刚滚轮在图2所示Sp坐标系下的单位法向量； n p represents the unit normal vector of the diamond roller in the Sp coordinate system shown in Figure 2;

vp wg表示金刚滚轮和蜗杆砂轮在图2所示Sp坐标系下的相对速度。v p wg represents the relative speed of the diamond roller and the worm grinding wheel in the Sp coordinate system shown in Figure 2.

通过求解

Figure BDA0002531244090000105
可以将θs表示为us和
Figure BDA0002531244090000106
的函数，再将其带入
Figure BDA0002531244090000107
中，消去θs从而得到金刚滚轮包络的蜗杆砂轮的齿面方程
Figure BDA0002531244090000108
by solving
Figure BDA0002531244090000105
θ s can be expressed as u s and
Figure BDA0002531244090000106
function, and then bring it into
Figure BDA0002531244090000107
, eliminate θ s to obtain the tooth surface equation of the worm grinding wheel enveloped by the diamond wheel
Figure BDA0002531244090000108

步骤四、通过理论计算以及实验验证本申请方法的可行性Step 4. Verify the feasibility of the proposed method through theoretical calculations and experiments

根据表1所示两组蜗杆砂轮参数，分别将根据现有齿形式金刚滚轮加工方法和本申请方法定义的金刚滚轮齿廓包络计算得到的蜗杆砂轮齿面与理论蜗杆砂轮齿面(插齿刀包络得到的蜗杆砂轮齿面，如式(1)所示)进行对比，比较结果请参看图4和图5，可见采用本申请方法定义的金刚滚轮齿廓计算出的蜗杆砂轮精度较目前方法的精度有较大的改善。According to the two groups of worm grinding wheel parameters shown in Table 1, the worm grinding wheel tooth surface and the theoretical worm grinding wheel tooth surface (tooth shape) calculated according to the diamond wheel tooth profile envelope of the existing tooth form diamond wheel processing method and the method of the present application are calculated respectively. The tooth surface of the worm grinding wheel obtained by the knife enveloping, as shown in formula (1), is compared. Please refer to Figure 4 and Figure 5 for the comparison results. It can be seen that the accuracy of the worm grinding wheel calculated by the diamond roller tooth profile defined by the method of the present application is higher than that of the current The accuracy of the method has been greatly improved.

采用本申请方法定义齿廓的金刚滚轮用来加工蜗杆砂轮，能显著的减小蜗杆砂轮的齿面误差，加工出的蜗杆砂轮齿面精度更高，然后用加工得到的蜗杆砂轮磨削面齿轮，测量结果如图6所示，加工得到的面齿轮具有很高的精度。The diamond roller with the tooth profile defined by the method of the present application is used to process the worm grinding wheel, which can significantly reduce the tooth surface error of the worm grinding wheel, and the processed worm grinding wheel has higher tooth surface accuracy. , the measurement results are shown in Figure 6, the surface gear obtained by processing has high precision.

本申请的金刚滚轮加工蜗杆砂时，蜗杆砂轮绕其轴线zw旋转时，金刚滚轮在自转的同时绕着zd轴摆动，蜗杆砂轮的转角

Figure BDA0002531244090000111
与金刚滚轮的摆角
Figure BDA0002531244090000112
满足关系
Figure BDA0002531244090000113
其中，Ns和Nw分别对应插齿刀齿数和蜗杆砂轮头数。根据该过程，蜗杆砂轮齿面作为金刚滚轮齿面rg(us，θs)的包络在Sw中如式(3)所示，When the diamond wheel of the present application processes worm sand, when the worm wheel rotates around its axis z w , the diamond wheel swings around the z d axis while rotating, and the rotation angle of the worm wheel
Figure BDA0002531244090000111
Swing angle with diamond roller
Figure BDA0002531244090000112
Satisfaction relationship
Figure BDA0002531244090000113
Among them, N s and N w correspond to the number of gear shaper teeth and the number of worm grinding wheel heads, respectively. According to this process, the worm grinding wheel tooth surface as the envelope of the diamond roller tooth surface r g (u s , θ s ) is shown in Eq. (3) in S w ,

Figure BDA0002531244090000114
Figure BDA0002531244090000114

其中，

Figure BDA0002531244090000115
表示从金刚滚轮固联坐标系Sg到蜗杆砂轮固联坐标系Sw的变换矩阵，
Figure BDA0002531244090000116
该式表示蜗杆砂轮的转角
Figure BDA0002531244090000117
与金刚滚轮的摆角
Figure BDA0002531244090000118
满足的关系。啮合方程式(3)中的Mwg本应该表示为
Figure BDA0002531244090000121
根据本式子将一个未知数用另一个未知数表示，消去一个未知数，从而表示为
Figure BDA0002531244090000122
任意一个曲面都可以用两个参数表示，金刚滚轮包络的蜗杆砂轮齿面由两个参数
Figure BDA0002531244090000123
表示，现有三个参数
Figure BDA0002531244090000124
通过式(3)，将θs表示为
Figure BDA0002531244090000125
的函数，从而消除该参数，用
Figure BDA0002531244090000126
表示蜗杆砂轮齿面，这就是式(3)(啮合方程)的含义。in,
Figure BDA0002531244090000115
represents the transformation matrix from the fixed coordinate system S g of the diamond roller to the fixed coordinate system S w of the worm grinding wheel,
Figure BDA0002531244090000116
This formula represents the angle of rotation of the worm wheel
Figure BDA0002531244090000117
Swing angle with diamond roller
Figure BDA0002531244090000118
satisfying relationship. M wg in meshing equation (3) should have been expressed as
Figure BDA0002531244090000121
According to this formula, one unknown is represented by another unknown, and one unknown is eliminated, so that it is expressed as
Figure BDA0002531244090000122
Any surface can be represented by two parameters. The tooth surface of the worm grinding wheel enveloped by the diamond wheel is represented by two parameters.
Figure BDA0002531244090000123
means that there are three parameters
Figure BDA0002531244090000124
By formula (3), θ s is expressed as
Figure BDA0002531244090000125
function, thereby eliminating the parameter, using
Figure BDA0002531244090000126
Represents the tooth surface of the worm grinding wheel, which is the meaning of formula (3) (meshing equation).

综上，本申请定义的齿形式金刚滚轮加工蜗杆砂轮方法有如下优点：To sum up, the method for machining a worm grinding wheel with a toothed diamond roller defined in this application has the following advantages:

1、本申请采用新的齿形式金刚滚轮齿廓来加工蜗杆砂轮，加工效率高，相对于锥面金刚滚轮加工蜗杆砂轮的方法，采用齿形式金刚滚轮加工蜗杆砂轮需要规划的刀具路径大大减少。1. This application adopts the new tooth profile of the diamond roller to process the worm grinding wheel, and the processing efficiency is high. Compared with the method of processing the worm grinding wheel with the tapered diamond roller, the tool path that needs to be planned for processing the worm grinding wheel by the tooth-shaped diamond roller is greatly reduced.

2、本申请采用新的齿形式金刚滚轮齿廓来加工蜗杆砂轮，不存在因为蜗杆砂轮齿面奇异性而导致无法规划刀具路径的问题。2. The present application adopts the new tooth profile of diamond roller to process the worm grinding wheel, and there is no problem that the tool path cannot be planned due to the singularity of the tooth surface of the worm grinding wheel.

3、本申请采用新的齿形式金刚滚轮齿廓来加工蜗杆砂轮，大大降低了对机床轴行程以及控制精度的要求。3. The application adopts the new tooth profile of diamond roller to process the worm grinding wheel, which greatly reduces the requirements on the travel of the machine tool axis and the control accuracy.

4、本申请采用新的齿形式金刚滚轮齿廓来加工蜗杆砂轮，通过对加工蜗杆砂轮的齿形式金刚滚轮齿廓进行合理设计，使得加工出的蜗杆砂轮齿面精度更高，从而保证了蜗杆砂轮磨削出的面齿轮具有较高的精度。4. The application adopts the new tooth profile of diamond roller to process the worm grinding wheel. By rationally designing the tooth profile of the diamond roller with the tooth form for processing the worm grinding wheel, the precision of the tooth surface of the processed worm grinding wheel is higher, so as to ensure the worm grinding wheel. The face gears ground by the grinding wheel have high precision.

以上所述仅为本申请的较佳实施例，并非用来限定本申请实施的范围，其他凡其原理和基本结构与本申请相同或近似的，均在本申请的保护范围之内。The above descriptions are only preferred embodiments of the present application, and are not intended to limit the scope of implementation of the present application. Other principles and basic structures that are the same or similar to those of the present application are within the protection scope of the present application.

Claims (4)

1.金刚滚轮及蜗杆砂轮的设计方法，其特征在于包括：1. the design method of diamond roller and worm grinding wheel, is characterized in that comprising: 步骤一、通过插齿刀齿面方程得出蜗杆砂轮齿面方程；Step 1. Obtain the tooth surface equation of the worm grinding wheel through the tooth surface equation of the gear shaper; 步骤二、由步骤一的蜗杆砂轮齿面方程得出蜗杆砂轮轴截面齿廓并以蜗杆砂轮轴截面齿廓作为金刚滚轮齿廓；In step 2, the tooth profile of the worm grinding wheel shaft section is obtained from the tooth surface equation of the worm grinding wheel in step 1, and the tooth profile of the worm grinding wheel shaft section is used as the tooth profile of the diamond roller; 步骤三、建立金刚滚轮包络蜗杆砂轮的坐标系，得出金刚滚轮包络出的蜗杆砂轮，金刚滚轮与蜗杆砂轮的齿廓包络满足Step 3: Establish the coordinate system of the diamond roller enveloping the worm grinding wheel, and obtain the worm grinding wheel enveloped by the diamond roller. The tooth profile envelope of the diamond roller and the worm grinding wheel satisfies
Figure FDA0002531244080000011
Figure FDA0002531244080000011
 其中，Mwg(φg)表示从金刚滚轮固联坐标系Sg到蜗杆砂轮固联坐标系Sw的齐次坐标变换矩阵；Wherein, M wg (φ g ) represents the homogeneous coordinate transformation matrix from the fixed coordinate system S g of the diamond roller to the fixed coordinate system S w of the worm grinding wheel; rwg(us，φg，θs)表示金刚滚轮齿面在蜗杆砂轮固联坐标系Sw中形成的曲面族；r wg (u s , φ g , θ s ) represents the surface family formed by the tooth surface of the diamond roller in the fixed coordinate system S w of the worm grinding wheel; rg(us，θs)表示金刚滚轮齿面方程；r g (u s , θ s ) represents the tooth surface equation of the diamond roller; fwg(us，φg，θs)表示金刚滚轮和蜗杆砂轮在Sp坐标系下的啮合方程；f wg (u s , φ g , θ s ) represents the meshing equation of the diamond roller and the worm grinding wheel in the Sp coordinate system; np表示金刚滚轮在Sp坐标系下的单位法向量； n p represents the unit normal vector of the diamond roller in the Sp coordinate system; vp wg表示金刚滚轮和蜗杆砂轮在Sp坐标系下的相对速度。v p wg represents the relative speed of the diamond roller and the worm grinding wheel in the Sp coordinate system. 2.根据权利要求1所述的金刚滚轮及蜗杆砂轮的设计方法，其特征在于：步骤一得出的蜗杆砂轮齿面方程满足2. the design method of diamond roller according to claim 1 and worm grinding wheel, it is characterized in that: the worm grinding wheel tooth surface equation that step 1 draws satisfies
Figure FDA0002531244080000012
Figure FDA0002531244080000012
 其中，MWS(φs)表示从插齿刀固联坐标SS到蜗杆砂轮固联坐标系SW的齐次坐标变换矩阵；Among them, M WS (φ s ) represents the homogeneous coordinate transformation matrix from the fixed coordinate S S of the gear shaper cutter to the fixed coordinate system SW of the worm grinding wheel; rws(us，uz，φs)表示插齿刀齿面在蜗杆砂轮固联坐标系Sw中形成的曲面族；r ws (u s , u z , φ s ) represents the family of surfaces formed by the tooth surface of the gear shaper in the fixed coordinate system S w of the worm grinding wheel; rs(us，uz)表示插齿刀齿面方程；r s (u s , u z ) represents the tooth surface equation of the gear shaper; fws(us，uz，φs)表示插齿刀和蜗杆砂轮在Ss坐标系下的啮合方程；f ws (u s , u z , φ s ) represents the meshing equation between the gear shaper cutter and the worm grinding wheel in the S s coordinate system; ns(us)表示插齿刀在Ss坐标系下的单位法向量；n s (u s ) represents the unit normal vector of the gear shaper in the S s coordinate system; vs (sw)(us，uz，φs)表示插齿刀和蜗杆砂轮在Ss坐标系下的相对速度。v s (sw) (u s , u z , φ s ) represents the relative speed of the gear shaper and the worm grinding wheel in the S s coordinate system. 3.根据权利要求1所述的金刚滚轮及蜗杆砂轮的设计方法，其特征在于：步骤二的蜗杆砂轮轴截面齿廓通过将zwowyw平面绕zw轴旋转某个角度φ*(0<φ*<360)，求旋转后的平面与蜗杆砂轮齿面交线的方法来求取，其满足3. the design method of diamond roller according to claim 1 and worm grinding wheel, it is characterized in that: the tooth profile of the worm grinding wheel shaft section of step 2 is rotated by a certain angle φ * of the z w o w y w plane around the z w axis (0<φ * <360), find the intersection of the rotated plane and the tooth surface of the worm grinding wheel, which satisfies
Figure FDA0002531244080000021
Figure FDA0002531244080000021
Figure FDA0002531244080000022
Figure FDA0002531244080000022
 其中，pi表示蜗杆砂轮的轴截面齿廓点。Among them, pi represents the tooth profile point of the shaft section of the worm grinding wheel. 4.根据权利要求1所述的金刚滚轮及蜗杆砂轮的设计方法，其特征在于：所述蜗杆砂轮的转角φw与金刚滚轮的摆角φg满足关系φw/φg＝Ns/Nw，其中，Ns和Nw分别对应插齿刀齿数和蜗杆砂轮头数。4. the design method of diamond roller according to claim 1 and worm grinding wheel, it is characterized in that: the rotation angle φ w of described worm grinding wheel and the swing angle φ g of diamond roller satisfy relation φ w /φ g =N s /N w , where N s and N w correspond to the number of gear shaper teeth and the number of worm grinding wheel heads, respectively.
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CN113400197A (en) * 2021-06-23 2021-09-17 重庆大学 Forming and trimming method for drum-shaped worm grinding wheel for grinding face gear
CN114690709A (en) * 2022-04-01 2022-07-01 重庆大学 Face gear grinding tooth face error model creation method considering whole-process machine tool geometric error influence and grinding tooth precision evaluation method

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CN106141326A (en) * 2016-08-05 2016-11-23 西北工业大学 A kind of method for trimming of gear grinding worm abrasion wheel
CN107609231A (en) * 2017-08-24 2018-01-19 中南大学 A kind of worm screw grinding worm surface microscopic topographic emulation mode and system

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CN102423820A (en) * 2011-10-10 2012-04-25 唐进元 Face gear grinding machining method based on worm grinding wheel
CN106141326A (en) * 2016-08-05 2016-11-23 西北工业大学 A kind of method for trimming of gear grinding worm abrasion wheel
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112643143A (en) * 2020-11-13 2021-04-13 重庆大学 Profile design method for drum-shaped worm grinding wheel of grinding face gear
CN112643143B (en) * 2020-11-13 2022-05-06 重庆大学 A method for designing the profile shape of drum-shaped worm sand for grinding face gears
CN113400197A (en) * 2021-06-23 2021-09-17 重庆大学 Forming and trimming method for drum-shaped worm grinding wheel for grinding face gear
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