﻿ 基于Frenet标架的变截面涡旋齿齿厚变化规律研究
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (6): 1169-1174  DOI: 10.11990/jheu.201803025 0

### 引用本文

HOU Caisheng, LIU Tao. Law governing the change in the thickness of a variable cross-section scroll tooth based on Frenet frame[J]. Journal of Harbin Engineering University, 2019, 40(6), 1169-1174. DOI: 10.11990/jheu.201803025.

### 文章历史

Law governing the change in the thickness of a variable cross-section scroll tooth based on Frenet frame
HOU Caisheng , LIU Tao
School of Mechanical and Electrical Engineering, Lanzhou University of Technology, Lanzhou 730050, China
Abstract: A variable cross-section scroll tooth formed by circular involutes with different base circle radii was constructed by using the Frenet frame to investigate the trends shown by the change in the tooth thickness of a variable cross-section scroll compressor quantitatively. A mathematical model for calculating the thickness of a variable cross-section scroll tooth was constructed in accordance with the geometric model. The profile parameters (radius of base circle, radius of rotation, and connection points) that influence tooth thickness were analyzed systematically. The relationship between tooth thickness and profile parameters were established precisely. The relationship between the variations in profile parameters and in scroll tooth thickness was summarized by solving the mathematical model. Results show that the proposed mathematical model can accurately describe the law governing the change in the thickness of the variable cross-section scroll tooth. Moreover, scroll teeth can be flexibly designed given that tooth thickness and the initial tooth thickness can be quantitatively adjusted in accordance with design requirements. The mathematical model can also be applied to calculate tooth thickness in other combination profiles.
Keywords: Frenet frame    variable cross-section    scroll tooth    scroll compressor    changing law of tooth thickness    circular involute    profile parameters    mathematical model

1 变截面涡旋齿的建立 1.1 涡旋型线的微分几何关系

 Download: 图 1 型线的微分几何关系示意 Fig. 1 Schematic diagram of differential geometry relation for scroll profile
 ${\mathit{\boldsymbol{R}}_t}(\varphi ) = \frac{{{\rm{d}}{\mathit{\boldsymbol{R}}_{\rm{n}}}(\varphi )}}{{{\rm{d}}\varphi }}$ (1)

 $\mathit{\boldsymbol{r}} = \mathit{\boldsymbol{r}}(s) = (x(s), y(s))$ (2)

 $\mathit{\boldsymbol{\alpha }}(s) = \left( {{x^\prime }(s), {y^\prime }(s)} \right) = (\cos \varphi (s), \sin \varphi (s))$ (3)
 $\mathit{\boldsymbol{\beta }}(s) = \left( { - {y^\prime }(s), {x^\prime }(s)} \right) = ( - \sin \varphi (s), \cos \varphi (s))$ (4)

 ${\mathit{\boldsymbol{\alpha }}^\prime }(s) = {\kappa _r}\mathit{\boldsymbol{\beta }}(s)$ (5)

 ${\mathit{\boldsymbol{\alpha }}^\prime }(s) = {\varphi ^\prime }(s)( - \sin \varphi (s), \cos \varphi (s)) = {\varphi ^\prime }(s)\mathit{\boldsymbol{\beta }}(s)$ (6)

 ${\kappa _r} = {\varphi ^\prime }(s)$ (7)

 $\rho (\varphi ) = \frac{1}{{{\kappa _r}}} = \frac{{{\rm{d}}s}}{{{\rm{d}}\varphi }}$ (8)

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{r}}^\prime }(s) = \mathit{\boldsymbol{\alpha }}(s)}\\ {{\mathit{\boldsymbol{\alpha }}^\prime }(s) = {\mathit{\boldsymbol{\kappa }}_r}\mathit{\boldsymbol{\beta }}(s)}\\ {{\mathit{\boldsymbol{\beta }}^\prime }(s) = - {\mathit{\boldsymbol{\kappa }}_r}\mathit{\boldsymbol{\alpha }}(s)} \end{array}} \right.$ (9)

 $\left\{ {\begin{array}{*{20}{l}} {x(\varphi ) = x\left( {{\varphi _0}} \right) + \int_{{\varphi _0}}^\varphi \rho (\varphi )\cos \varphi (s){\rm{d}}\varphi }\\ {y(\varphi ) = y\left( {{\varphi _0}} \right) + \int_{{\varphi _0}}^\varphi \rho (\varphi )\sin \varphi (s){\rm{d}}\varphi } \end{array}} \right.$ (10)
1.2 母线方程

 Download: 图 2 组合型线的母线 Fig. 2 Generating line of the combination profile

 $\left\{ {\begin{array}{*{20}{l}} {{x_1} = {x_0}\left( {{\varphi _0}} \right) + \int_0^\varphi {{\rho _1}} (\varphi )\cos \varphi {\rm{d}}\varphi }\\ {{y_1} = {y_0}\left( {{\varphi _0}} \right) + \int_0^\varphi {{\rho _1}} (\varphi )\sin \varphi {\rm{d}}\varphi } \end{array}, \varphi \in \left[ {0, {\varphi _1}} \right)} \right.$ (11)

 $\left\{ {\begin{array}{*{20}{l}} {{x_2} = {x_1}\left( {{\varphi _1}} \right) + \int_{{\varphi _1}}^\varphi {{\rho _2}} (\varphi )\cos \varphi {\rm{d}}\varphi }\\ {{y_2} = {y_1}\left( {{\varphi _1}} \right) + \int_{{\varphi _1}}^\varphi {{\rho _2}} (\varphi )\sin \varphi {\rm{d}}\varphi } \end{array}, \varphi \in \left[ {{\varphi _1}, {\varphi _2}} \right)} \right.$ (12)

 $\left\{ {\begin{array}{*{20}{l}} {{x_3} = {x_2}\left( {{\varphi _2}} \right) + \int_{{\varphi _2}}^\varphi {{\rho _3}} (\varphi )\cos \varphi {\rm{d}}\varphi }\\ {{y_3} = {y_2}\left( {{\varphi _2}} \right) + \int_{{\varphi _2}}^\varphi {{\rho _3}} (\varphi )\sin \varphi {\rm{d}}\varphi } \end{array}, \varphi \in \left[ {{\varphi _2}, 6\pi } \right)} \right.$ (13)

 $\left\{ {\begin{array}{*{20}{l}} {{\rho _1}(\varphi ) = \int_0^\varphi {{a_1}} {\rm{d}}\varphi }\\ {{\rho _2}(\varphi ) = {\rho _1}\left( {{\varphi _1}} \right) + \int_{{\varphi _1}}^\varphi {{a_2}} {\rm{d}}\varphi }\\ {{\rho _3}(\varphi ) = {\rho _2}\left( {{\varphi _2}} \right) + \int_{{\varphi _2}}^\varphi {{a_1}} {\rm{d}}\varphi } \end{array}} \right.$ (14)

1.3 涡旋齿内外壁型线的生成

 $\begin{array}{l} {C_{f, i}} = \left[ {{R_{\rm{n}}}(\varphi ) + \frac{{{R_{{\rm{or}}}}}}{2}} \right] + \exp ({\rm{j}}\varphi ) + \\ \;\;\;\;\;\;\;\;\;\;\;\;{R_{\rm{t}}}(\varphi )\exp \left[ {{\rm{j}}\left( {\varphi \frac{\pi }{2}} \right)} \right] \end{array}$ (15)
 $\begin{array}{l} {C_{f, o}} = \left[ {{R_n}(\varphi ) - \frac{{{R_{{\rm{or}}}}}}{2}} \right]\exp [{\rm{j}}(\varphi + \pi )] + \\ \;\;\;\;\;\;\;\;\;\;\;\;{R_{\rm{t}}}(\varphi )\exp \left[ {{\rm{j}}\left( {\varphi + \frac{\pi }{2} + \pi } \right)} \right] \end{array}$ (16)

 $\begin{array}{l} {C_{o, i}} = \left[ {{R_{\rm{n}}}(\varphi ) + \frac{{{R_{{\rm{or}}}}}}{2}} \right]\exp [{\rm{j}}(\varphi + \pi )] + \\ \;\;\;\;\;\;\;\;\;\;\;\;{R_{\rm{t}}}(\varphi )\exp \left[ {{\rm{j}}\left( {\varphi + \frac{\pi }{2} + \pi } \right)} \right] \end{array}$ (17)
 $\begin{array}{l} {C_{o, o}} = \left[ {{R_{\rm{n}}}(\varphi ) - \frac{{{R_{{\rm{or}}}}}}{2}} \right] + \exp ({\rm{j}}\varphi ) + \\ \;\;\;\;\;\;\;\;\;\;\;\;{R_{\rm{t}}}(\varphi )\exp \left[ {{\rm{j}}\left( {\varphi \frac{\pi }{2}} \right)} \right] \end{array}$ (18)

2 变截面涡旋齿的齿厚计算模型

 Download: 图 5 涡旋齿厚计算模型 Fig. 5 The calculation model of scroll tooth thickness
 ${\varphi _{12}} = {\varphi _{11}} - \pi + \zeta$ (19)

C点的坐标为C(x0, y0)，则OC之间的距离由下式计算可得到：

 ${L_{oc}} = \sqrt {x_o^2\left( {{\varphi _{11}}} \right) + y_o^2\left( {{\varphi _{11}}} \right)}$ (20)

OB的长度为外壁型线的基圆半径，OA的长度为内壁型线的基圆半径，可分别由Rt(φ11)和Rt(φ12)计算得出。

 $t = {L_{CD}} = {L_{AC}} - {L_{AD}} = {L_{oC}}\cos {\zeta _2} - \left[ {{R_{\rm{n}}}\left( {{\varphi _{12}}} \right) + \frac{{{R_{{\rm{or}}}}}}{2}} \right]$ (21)

 $\left\{ {\begin{array}{*{20}{l}} {\sin {\zeta _1} = \frac{{{L_{OB}}}}{{{L_{OC}}}} = \frac{{{R_{\rm{t}}}\left( {{\varphi _{11}}} \right)}}{{{L_{OC}}}}}\\ {\sin {\zeta _2} = \frac{{{L_{OA}}}}{{{L_{OC}}}} = \frac{{{R_{\rm{t}}}\left( {{\varphi _{12}}} \right)}}{{{L_{OC}}}}}\\ {\zeta = {\zeta _1} - {\zeta _2}} \end{array}} \right.$
3 齿厚影响因素分析 3.1 基圆半径a1的影响

 Download: 图 6 基圆半径a1变化下的齿厚曲线 Fig. 6 Tooth thickness curves under the change of the radius of base circle a1
3.2 基圆半径a2的影响

 Download: 图 7 基圆半径a2变化下的齿厚曲线 Fig. 7 Tooth thickness curves under the change of the radius of base circle a2
3.3 回转半径Ror的影响

 Download: 图 8 回转半径Ror变化下的齿厚曲线 Fig. 8 Tooth thickness curves under the change of the radius of rotation Ror
3.4 连接点φ1φ2的影响

 Download: 图 9 连接点φ1和φ2变化下的齿厚曲线 Fig. 9 Tooth thickness curves under the change of the connection points φ1 and φ2
 Download: 图 10 不同连接点对涡旋齿的影响 Fig. 10 The effect of different connection points on the scroll teeth
4 结论

1) 利用Frenet标架的运动公式，唯一确定了涡旋型线的母线方程。并且在构造组合型线时，选用了不同基圆半径的基圆渐开线，既充分利用了圆渐开线容易加工且具有良好工作性能的优势，同时又克服了使用单一的圆渐开线而使压缩比减小和涡旋盘尺寸过大的缺点。

2) 建立的变截面涡旋齿的齿厚计算模型简洁直观，且具有一定的通用性，该模型能够准确描述涡旋齿的变化规律，可以将其灵活运用到其他类型的组合型线中。

3) 涡旋齿的齿厚受型线参数的影响，它们之间存在着精确、定量的关系，利用这种定量关系，能够很好地帮助工程设计人员设计变截面涡旋型线。

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