WO2020134519A1 - 一种确定罗茨泵转子型线合理设计区域的方法及其应用 - Google Patents

一种确定罗茨泵转子型线合理设计区域的方法及其应用 Download PDF

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WO2020134519A1
WO2020134519A1 PCT/CN2019/114012 CN2019114012W WO2020134519A1 WO 2020134519 A1 WO2020134519 A1 WO 2020134519A1 CN 2019114012 W CN2019114012 W CN 2019114012W WO 2020134519 A1 WO2020134519 A1 WO 2020134519A1
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rotor
roots pump
curve
point
peak curve
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PCT/CN2019/114012
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English (en)
French (fr)
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何雪明
黄海楠
刘振超
于嘉川
刘平
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江南大学
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Publication of WO2020134519A1 publication Critical patent/WO2020134519A1/zh
Priority to US17/137,512 priority Critical patent/US11168682B2/en

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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F04—POSITIVE - DISPLACEMENT MACHINES FOR LIQUIDS; PUMPS FOR LIQUIDS OR ELASTIC FLUIDS
    • F04C—ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT MACHINES FOR LIQUIDS; ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT PUMPS
    • F04C2/00—Rotary-piston machines or pumps
    • F04C2/08—Rotary-piston machines or pumps of intermeshing-engagement type, i.e. with engagement of co-operating members similar to that of toothed gearing
    • F04C2/12—Rotary-piston machines or pumps of intermeshing-engagement type, i.e. with engagement of co-operating members similar to that of toothed gearing of other than internal-axis type
    • F04C2/126—Rotary-piston machines or pumps of intermeshing-engagement type, i.e. with engagement of co-operating members similar to that of toothed gearing of other than internal-axis type with radially from the rotor body extending elements, not necessarily co-operating with corresponding recesses in the other rotor, e.g. lobes, Roots type
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F04—POSITIVE - DISPLACEMENT MACHINES FOR LIQUIDS; PUMPS FOR LIQUIDS OR ELASTIC FLUIDS
    • F04C—ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT MACHINES FOR LIQUIDS; ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT PUMPS
    • F04C2/00—Rotary-piston machines or pumps
    • F04C2/08—Rotary-piston machines or pumps of intermeshing-engagement type, i.e. with engagement of co-operating members similar to that of toothed gearing
    • F04C2/082—Details specially related to intermeshing engagement type machines or pumps
    • F04C2/084—Toothed wheels
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F04—POSITIVE - DISPLACEMENT MACHINES FOR LIQUIDS; PUMPS FOR LIQUIDS OR ELASTIC FLUIDS
    • F04C—ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT MACHINES FOR LIQUIDS; ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT PUMPS
    • F04C15/00—Component parts, details or accessories of machines, pumps or pumping installations, not provided for in groups F04C2/00 - F04C14/00
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F04—POSITIVE - DISPLACEMENT MACHINES FOR LIQUIDS; PUMPS FOR LIQUIDS OR ELASTIC FLUIDS
    • F04C—ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT MACHINES FOR LIQUIDS; ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT PUMPS
    • F04C18/00—Rotary-piston pumps specially adapted for elastic fluids
    • F04C18/08—Rotary-piston pumps specially adapted for elastic fluids of intermeshing-engagement type, i.e. with engagement of co-operating members similar to that of toothed gearing
    • F04C18/082—Details specially related to intermeshing engagement type pumps
    • F04C18/084—Toothed wheels
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F04—POSITIVE - DISPLACEMENT MACHINES FOR LIQUIDS; PUMPS FOR LIQUIDS OR ELASTIC FLUIDS
    • F04C—ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT MACHINES FOR LIQUIDS; ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT PUMPS
    • F04C2240/00—Components
    • F04C2240/20—Rotors
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F04—POSITIVE - DISPLACEMENT MACHINES FOR LIQUIDS; PUMPS FOR LIQUIDS OR ELASTIC FLUIDS
    • F04C—ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT MACHINES FOR LIQUIDS; ROTARY-PISTON, OR OSCILLATING-PISTON, POSITIVE-DISPLACEMENT PUMPS
    • F04C2250/00—Geometry
    • F04C2250/30—Geometry of the stator
    • F04C2250/301—Geometry of the stator compression chamber profile defined by a mathematical expression or by parameters

Definitions

  • the invention relates to a method for determining a reasonable design area of a Roots pump rotor profile and its application, and belongs to the field of mechanical design.
  • Roots pump refers to a variable volume vacuum pump equipped with two vane-shaped rotors rotating synchronously in opposite directions with a small gap between the rotors and the inner wall of the pump casing without contacting each other; as a general machine, it The characteristics of its own structure and performance are widely used in chemical, paper, power generation, food and other industries.
  • Roots pump As a key part of the Roots pump, a pair of rotors of the Roots pump have a great influence on the performance of the Roots pump; after three typical rotor profiles commonly used in the market (ie, arc, involute, pendulum) (Line curve) Research on the performance parameters of Roots pumps found that the higher the utilization factor of the rotor area, the greater the air volume and sobbing rate of the Roots pump, but the rotor profile design also needs to consider whether there will be a closed volume and undercut Problem, you need to frame the reasonable design area of the rotor profile to avoid possible closed volume and undercutting problems.
  • the rotor In order to solve the problem that the rotor profile of the Roots pump currently exists due to improper design parameter selection, the rotor is found to have a closed volume and undercut after the design is completed.
  • the present invention provides a method for determining the reasonable design area of the rotor profile of the Roots pump and Its application makes it possible to predict whether the designed Roots pump rotor profile will produce closed volume and undercut problems according to this method when designing the Roots pump rotor profile, so as to make corresponding modifications to make the designed Roots pump
  • the pump rotor profile can avoid closed volume and undercut problems as much as possible, and improve the design pass rate and design efficiency of the Roots pump rotor profile.
  • a method for determining a reasonable design area for the rotor profile of a Roots pump includes:
  • Step 1 Before designing the root peak curve of the Roots pump rotor profile, establish the relationship between the Roots pump rotor profile and the performance parameter; the establishment of the relationship between the Roots pump rotor profile and performance parameter includes:
  • s th is the pumping rate of the Roots pump
  • V is the volume of the medium removed by the Roots pump when one rotor of the Roots pump makes one revolution
  • n is the speed of the rotor
  • ⁇ is the rotor area utilization factor
  • R a is the rotor tooth
  • L is the length of the rotor
  • S r refers to the area of the rotor of a single roots pump.
  • Step 2 Determine the constraints to avoid closed volume
  • Step 3 Determine the constraints to avoid undercutting
  • Step 4 Determine the reasonable design area of the rotor blade peak curve of the Roots pump according to steps 1 to 3.
  • the calculation formula of the area S r of the rotor of the single roots pump is:
  • S b is the area of the rotor half tooth; for calculation convenience, S b is divided into 3 areas for area calculation, and the area of the 3 areas is recorded as S 1 , S 2 , and S 3 in order from the center of the roots pump to the edge ,Calculated as follows:
  • R is the pitch circle radius and the size is half of the center distance H of the two rotors of the roots pump;
  • (x e , y e ) is the starting point coordinate of the blade valley curve,
  • (x c , y c ) is the blade valley curve End point coordinates; because the leaf valley curve is connected to the leaf peak curve, the end point of the leaf valley curve is also the start point of the leaf peak curve,
  • (x a , y a ) is the end point coordinate of the leaf peak curve;
  • g(x) represents the function expression of the leaf valley curve Formula
  • f(x) represents the function expression of the leaf peak curve;
  • ⁇ is the angle between the start point and end point of the blade peak curve or blade valley curve and the connection line of the pitch circle center, the size of ⁇ is related to the number of rotor teeth Z, and the calculation formula is (1):
  • determining the constraint condition for avoiding the closed volume includes:
  • step three determining the constraint condition for avoiding undercutting includes:
  • H is the center distance between the two rotors;
  • O 1 O 2 is the line connecting the center points of the two rotors.
  • the left boundary of the leaf peak curve that avoids undercut satisfies equation (25), where x and y are the horizontal and vertical coordinates of each point within the left boundary of the leaf peak curve:
  • step 4 the reasonable design area of the rotor blade peak curve of the Roots pump is determined according to steps 1 to 3, including:
  • step 1 to step 3 draw the area of the root peak curve of the roots pump.
  • the Roots pump rotor profile includes a circular-arc rotor profile, a cycloidal rotor profile or an involute rotor profile.
  • the method adopts the method described in the claims to design the Roots pump rotor profile.
  • the method for determining the reasonable design area of the Roots pump rotor profile determines the Roots by establishing the relationship between the Roots pump rotor profile and the performance parameters, and then further determining the constraint conditions to avoid the closed volume and the undercut
  • the reasonable design area of the pump rotor blade peak curve give a reference when designing the Roots pump rotor profile, that is, the rotor profile of the blade peak curve not in this reasonable design area will inevitably lead to closed volume or undercut in the application Problems, so that when designing the rotor profile of the Roots pump, it is possible to avoid designing a rotor profile that will result in a closed volume or undercut, without having to wait until the design is completed to find that the rotor profile design is unreasonable, thereby improving the rotor profile Design efficiency.
  • Figure 1 is a composition diagram of the Roots pump rotor profile; (a) is the profile of two rotors, and (b) is the profile of a half tooth.
  • Figure 2 is a schematic diagram of rotor area zoning.
  • Figure 3 is a schematic diagram of Ye Feng curve design.
  • Figure 4 is a schematic diagram of the curve of the leaf peak curve mapped to the pitch arc; (a) is the normal meshing state, (b) is the closed volume state, and (c) is partially non-meshing.
  • Figure 5 is a schematic diagram of the undercut phenomenon in the rotor profile.
  • Figure 6 is a schematic diagram of design constraints to avoid undercutting.
  • FIG. 7 is a schematic diagram of the basic design area of the rotor profile design.
  • Fig. 8 is a schematic diagram of determining the lower boundary of the basic area of the rotor profile design.
  • Figure 9 is a schematic diagram of the design of the leaf peak curve beyond the lower boundary of the qualified area.
  • FIG. 10 is a schematic diagram of determining the left boundary of the basic area of the rotor profile design.
  • Fig. 11 is a schematic diagram of the design of the peak curve of the leaf beyond the left boundary of the qualified area.
  • Figure 12 is a schematic diagram of the relationship between a typical rotor profile and a reasonable design area.
  • Figure 13 is a schematic diagram of the basic volume of the rotor.
  • Fig. 14 is a coordinate relationship diagram of the rotor of the Roots pump.
  • This embodiment provides a method for determining a reasonable design area of the rotor profile of a Roots pump. Referring to FIG. 1, the method includes:
  • Step 1 Establish the relationship between the rotor profile of the Roots pump and the performance parameters; use the pumping rate of the Roots pump as the index to measure the performance of the Roots pump.
  • the Roots pump has two rotors, each rotor has Z teeth, and Z is an integer greater than or equal to 2.
  • the shape of the two rotors of the Roots pump is the same. Furthermore, the tooth profile of the rotor on each tooth is the same, and the tooth profile on each tooth is axisymmetric of. As shown in Fig. 1(b), the tooth profile on half of the teeth is composed of the leaf peak curve and the leaf valley curve, with the pitch circle as the boundary, the part outside the pitch circle is called the leaf peak curve, and the part inside the pitch circle is called the leaf valley curve.
  • R a is the radius of the rotor tip circle
  • R is the pitch circle radius
  • the size of R is half of the center distance H of the two rotors
  • ⁇ is the starting point and ending point of the blade peak curve (or blade valley curve) and the pitch, respectively
  • the angle between the lines of the center of the circle, the size of ⁇ is related to the number of rotor teeth Z, the calculation formula is (1):
  • R a is rotor tip radius
  • L is the length of the rotor
  • the Roots pump has two rotors. According to formula (2), the pumping rate of the Roots pump is shown in formula (3):
  • n is the rotation speed of the rotor
  • the rotor area utilization factor ⁇ is the ratio of the cross-sectional area of the basic volume of the rotor to the area of the addendum circle.
  • the calculation formula is formula (4).
  • S b refers to the area of the half tooth of the rotor. Please refer to Figure 2.
  • the calculation process is as follows: (Formula 6-Formula 8 x e , y e , x c , and x a are marked in Figure 2 respectively. )
  • the formula (6)(7)(8)(9) obtains the rotor area utilization factor ⁇ .
  • Step 2 Determine the constraints to avoid the closed volume: if there is no closed volume, the rotor needs to be fully engaged during the rotation process;
  • mapping relation is applied to the entire curve Feng Ye
  • Feng Ye curve can be obtained on the shape of a circular arc section A '1 C 1 same curve; in particular in that: the mapping curve may be obtained exactly a circular arc section a '1 C 1, FIG.
  • the closed volume is not required, it is required to be fully engaged during the rotation of the rotor.
  • the following derivation of the rotor profile does not appear the constraint of the closed volume:
  • Equation normal B 1 B '1 can be expressed as:
  • Equation (13) must have a solution
  • Step 3 Determine the constraints to avoid undercutting
  • intersection point W of the normal line at any point T on the engagement line and O 1 O 2 must be between the rotation centers O 1 O 2 of the two rotors. Therefore, in order to obtain the conditions for avoiding undercutting when designing the rotor profile of the Roots pump, the coordinates of any point T on the engagement line mapped to the point W on O 1 O 2 must be obtained and detected.
  • Step 4 Obtain a reasonable design area of the leaf peak curve to further improve the design efficiency of the rotor profile.
  • the rotor peak curve is designed on the first quadrant and the coordinate axis, and the starting point of the rotor peak curve is designed on the O 1 x 1 axis.
  • the blade peak curve must be on the right side of the pitch arc A'C, the left side of the addendum arc O 1 AF, above the line segment A'A, and below the line segment CF . It is easy to derive the basic design area of the Yefeng curve:
  • the leaf peak curve is on the left side of O 1 AF:
  • mapping relationship between the rotor blade peak curve and the corresponding pitch circle segment.
  • the complete pitch circle segment is obtained. If the curves obtained by the mapping overlap, it will be generated. Closed volume, in order to avoid this kind of situation and push the situation to the limit, the curve obtained by mapping the leaf peak curve is only one point, and considering the continuity requirements of the leaf peak curve at point A, the point obtained by the mapping can only be A 'Point, the leaf peak curve that satisfies the situation is the pin tooth arc A'AG, which is the lower boundary of the leaf peak curve design, as shown in Figure 8.
  • the peak curve of the blade peak is completely adopted, the end point of the peak curve of the blade peak cannot reach the point C of the design end point, so if the arc of the pin tooth is used, the peak curve of the blade peak is composed of at least two curves, and the closedness of the rotor profile And the continuity also has constraints on the curve design outside the first segment of the leaf peak curve, so the left boundary of the design area can be further reduced on the basis of the lower boundary of the reasonable design area of the leaf peak curve defined by equation (24).
  • the closedness of the rotor limits the peak-to-peak curve and the blade-to-valley curve of the rotor to be end-to-end, so the peak-to-peak curve must pass through point C; the continuity of the rotor defines that when the peak-to-peak curve is multistage, the curves must Ensure tangent continuity. Under the constraints of these two conditions, to explore the problem of the left boundary of the reasonable design area will be transformed into exploring the case where the second segment of the blade peak curve can be designed reasonably, when the pin tooth arc is used as the first curve of the blade peak curve segment The problem of using length limit.
  • This straight line can be used as the left boundary of the reasonable design area of the rotor profile, which can be expressed as:
  • the reasonable design range of the blade peak curve is obtained, as shown by the area marked by the thick dotted line in FIG. 12, and the arc-shaped rotor profile, cycloidal rotor profile, and The relationship between the peak curve of the open-line rotor profile and this reasonable design range. It can be seen from the figure that the blade peak curves of several typical rotor profiles are all designed within a reasonable design area. It is worth mentioning that the reasonable design range still has the possibility of further narrowing, especially the right and upper boundaries of the reasonable design area.
  • Some steps in the embodiments of the present invention may be implemented by software, and corresponding software programs may be stored in a readable storage medium, such as an optical disk or a hard disk.

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Abstract

一种确定罗茨泵转子型线合理设计区域的方法及其应用。所述方法在建立罗茨泵转子型线和性能参数的关系式之后，进一步确定避免封闭容积及根切的约束条件，从而确定罗茨泵转子叶峰曲线的合理设计区域，在设计罗茨泵转子型线时给出参考，即叶峰曲线不在此合理设计区域内的转子型线必然会导致在应用中出现封闭容积或者根切的问题，从而使得罗茨泵转子型线在设计时即可避免设计出会导致封闭容积或者根切的转子型线，而无需等到设计完成之后才发现该转子型线设计不合理，从而提高转子型线的设计效率。

Description

一种确定罗茨泵转子型线合理设计区域的方法及其应用 技术领域
本发明涉及一种确定罗茨泵转子型线合理设计区域的方法及其应用，属于机械设计领域。
背景技术
罗茨泵是指泵内装有两个相反方向同步旋转的叶形转子，转子间、转子与泵壳内壁间有细小间隙而互不接触的一种变容真空泵；其作为一种通用机械，由于自身结构和性能的特点，广泛应用在化工、造纸、发电、食品等行业。
罗茨泵的一对转子作为罗茨泵的关键零部件，对罗茨泵的性能具有很大的影响；经过对市场上常用的三种典型转子型线(即圆弧、渐开线、摆线曲线)对罗茨泵性能参数的研究，发现转子面积的利用系数越高，罗茨泵的风量和抽泣速率就越大，但是转子型线设计时还需要考虑是否会出现封闭容积以及根切的问题，那么就需要对转子型线的合理设计区域进行框定，从而避免可能出现的封闭容积以及根切等问题。
通常在设计转子型线完成后，可以根据设计出的转子型线的参数数据检测所设计的转子型线是否满足无封闭容积、根切等情况。但该检测行为依赖于设计出的转子型线的参数数据，在设计过程前无法对叶峰曲线进行限定，这就可能导致叶峰曲线数据一开始就设计的不合理，但直到该段曲线设计完才发现，降低了转子型线的设计效率。
发明内容
为了解决目前存在的罗茨泵转子型线由于设计参数选择不当导致在设计完成后发现转子出现封闭容积、根切的问题，本发明提供一种确定罗茨泵转子型线合理设计区域的方法及其应用，使得在设计罗茨泵转子型线时能够根据该方法预判出设计出的罗茨泵转子型线是否会产生封闭容积、根切问题，从而进行相应修改，使得设计出的罗茨泵转子型线能够尽可能的避免出现封闭容积、根切问题，提高罗茨泵转子型线的设计合格率以及设计效率。
一种确定罗茨泵转子型线合理设计区域的方法，所述方法包括：
步骤一：在设计罗茨泵转子型线叶峰曲线之前，建立罗茨泵转子型线和性能参数的关系式；所述建立罗茨泵转子型线和性能参数的关系式，包括：
选定罗茨泵的抽气速率作为衡量罗茨泵性能的指标，建立罗茨泵的抽气速率和性能参数的关系式；所述建立罗茨泵的抽气速率和性能参数的关系式包括：
Figure PCTCN2019114012-appb-000001
Figure PCTCN2019114012-appb-000002
其中，s th为罗茨泵的抽气速率，V为罗茨泵一个转子旋转一周时罗茨泵排除的介质的体积，n为转子的转速，λ为转子面积利用系数，R a为转子齿顶圆半径，L为转子长度，S r指单个罗茨泵转子的面积。
步骤二：确定避免封闭容积的约束条件；
步骤三：确定避免根切的约束条件；
步骤四：根据步骤一至步骤三确定罗茨泵转子叶峰曲线的合理设计区域。
可选的，所述单个罗茨泵转子的面积S r的计算公式为：
S r＝6S b     (5)
其中S b为转子半个齿的面积；为计算方便，将S b划分为3个区域进行面积计算，3个区域的面积由罗茨泵中心至边缘依次记为S 1、S 2、S 3，计算公式如下：
Figure PCTCN2019114012-appb-000003
Figure PCTCN2019114012-appb-000004
Figure PCTCN2019114012-appb-000005
故：
Figure PCTCN2019114012-appb-000006
式(9)中，R为节圆半径、大小为罗茨泵两个转子的中心距H的一半；(x e，y e)为叶谷曲线起点坐标，(x c，y c)为叶谷曲线终点坐标；由于叶谷曲线与叶峰曲线相连，所以叶谷曲线的终点也即叶峰曲线的起点，(x a，y a)为叶峰曲线的终点坐标；g(x)表示叶谷曲线的函数表达式，f(x)表示叶峰曲线的函数表达式；
γ为叶峰曲线或叶谷曲线的起点、终点分别与节圆圆心连线之间的夹角，γ的大小与转子齿数Z相关，计算公式为式(1)：
Figure PCTCN2019114012-appb-000007
可选的，所述步骤二中，确定避免封闭容积的约束条件，包括：
取罗茨泵转子叶峰曲线上任意一点做叶峰曲线在该点的法线，设该法线的斜率为k 1、该法线与节圆弧的交点的坐标为(x 1，y 1)，
Figure PCTCN2019114012-appb-000008
在叶峰曲线上B 1点做法线B 1B’ 1，B 1点坐标为(x b，y b)，则避免封闭容积的约束条件为：
Figure PCTCN2019114012-appb-000009
可选的，所述步骤三中，确定避免根切的约束条件，包括：
在啮合线上任意点做啮合线在该点的法线，设该法线斜率为k 2、该法线与O 1O 2的交点坐标为(X 1，Y 1)，则避免根切的约束条件为：
Figure PCTCN2019114012-appb-000010
其中，H为两个转子的中心距；O 1O 2为两个转子的中心点的连线。
可选的，所述避免根切的叶峰曲线的左边界满足式(25)，其中x、y分别为叶峰曲线的左边界内各点的横坐标和纵坐标：
Figure PCTCN2019114012-appb-000011
式中：
Figure PCTCN2019114012-appb-000012
Figure PCTCN2019114012-appb-000013
可选的，所述步骤四中，根据步骤一至步骤三确定罗茨泵转子叶峰曲线的合理设计区域，包括：
根据步骤一至步骤三确定的约束条件画出罗茨泵叶峰曲线所处的区域范围。
可选的，所述罗茨泵转子型线包括圆弧型转子型线、摆线型转子型线或渐开线型转子型线。
可选的，所述方法采用权利要求上述方法设计罗茨泵转子型线。
本发明有益效果是：
本发明提供的确定罗茨泵转子型线合理设计区域的方法通过在建立罗茨泵转子型线和性 能参数的关系式之后，进一步确定确定避免封闭容积及根切的约束条件，从而确定罗茨泵转子叶峰曲线的合理设计区域，在设计罗茨泵转子型线时给出参考，即叶峰曲线不在此合理设计区域内的转子型线必然会导致在应用中出现封闭容积或者根切的问题，从而使得罗茨泵转子型线设计时即可避免设计出会导致封闭容积或者根切的转子型线，而无需等到设计完成之后才发现该转子型线设计不合理，从而提高转子型线的设计效率。
附图说明
为了更清楚地说明本发明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发明的一些实施例，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。
图1为罗茨泵转子型线的组成图；(a)是两个转子的型线组成，(b)是半个齿的型线组成。
图2为转子面积分区示意图。
图3为叶峰曲线设计示意图。
图4为叶峰曲线映射到节圆弧上的曲线示意图；(a)是正常啮合状态，(b)是封闭容积状态，(c)是局部不啮合。
图5为转子型线中的根切现象示意图。
图6为避免根切的设计约束示意图。
图7为转子型线设计的基本设计区域示意图。
图8为转子型线设计的基本区域下边界的确定示意图。
图9为设计超出合格区域下边界的叶峰曲线设计示意图。
图10为转子型线设计的基本区域左边界的确定示意图。
图11为设超出合格区域左边界的叶峰曲线设计示意图。
图12为典型转子型线与合理设计区域的关系示意图。
图13为转子基本容积示意图。
图14为罗茨泵转子的坐标关系图。
具体实施方式
为使本发明的目的、技术方案和优点更加清楚，下面将结合附图对本发明实施方式作进一步地详细描述。
实施例一：
本实施例提供一种确定罗茨泵转子型线合理设计区域的方法，参见图1，所述方法包括：
步骤一：建立罗茨泵转子型线和性能参数的关系式；采用罗茨泵的抽气速率作为衡量罗茨泵性能的指标。罗茨泵具有两个转子，每个转子具有Z个齿，Z为大于等于2的整数。本发明实施例以Z＝3为例进行说明。
如图1(a)所示，罗茨泵两个转子的形状是一样的，进一步的，转子在每个齿上的齿廓都是相同的，而每个齿上的齿廓又是轴对称的。如图1(b)所示，半个齿上的齿廓由叶峰曲线和叶谷曲线组成，以节圆为界，节圆以外的部分称为叶峰曲线，节圆以内的部分称为叶谷曲线。
图1中：R a为转子齿顶圆半径；R为节圆半径，R的大小是两个转子的中心距H的一半；γ为叶峰曲线(或叶谷曲线)的起点、终点分别与节圆圆心连线之间的夹角，γ的大小与转子齿数Z相关，计算公式为式(1)：
Figure PCTCN2019114012-appb-000014
一个转子旋转一周时罗茨泵排除的介质的体积计算公式为式(2)：
Figure PCTCN2019114012-appb-000015
式(2)中，R a为转子齿顶圆半径，L为转子长度，λ为转子面积利用系数；
罗茨泵具有两个转子，根据式(2)可得罗茨泵的抽气速率为式(3)所示：
Figure PCTCN2019114012-appb-000016
式(3)中，n为转子的转速；
式(2)和式(3)中转子面积利用系数λ为转子基本容积的横截面面积与齿顶圆面积的比值，其计算公式为式(4)，转子转动的过程中，也即介质转移的过程，与进口连通腔室内的介质随着转子转动逐渐被封闭在一个相对密闭的容积内，此容积即转子基本容积，如图13中阴影所示；
Figure PCTCN2019114012-appb-000017
式(4)中，S r指单个罗茨泵转子的面积，计算公式如式(5)所示：
S r＝6S b      (5)
式(5)中S b指转子半个齿的面积，请参考图2，计算过程如下：(式6-式8中x e、y e、x c、x a请分别在图2中标出。)
Figure PCTCN2019114012-appb-000018
Figure PCTCN2019114012-appb-000019
Figure PCTCN2019114012-appb-000020
Figure PCTCN2019114012-appb-000021
由式(3)、式(4)和式(5)可知，在齿顶圆半径R a不变的情况下，单个罗茨泵转子的面积S r越小，转子的面积利用系数λ就会越大，那么相同转速下罗茨泵的抽气速率越大，即罗茨泵的排放效率越高。
所以在设计罗茨泵转子型线时，在确定转子齿顶圆大小后，应使转子的面积尽可能小。根据图2，公式(6)(7)(8)(9)得到转子的面积利用系数λ。
步骤二：确定避免封闭容积的约束条件：若要不出现封闭容积则需要转子转动过程中全部啮合；
如图3所示，过叶峰曲线上任意点B 1、D 1做叶峰曲线在该点的法线，分别交节圆弧A’ 1C 1于B’ 1、D’ 1，根据啮合原理可知：随着转子的转动，当B’ 1点转动到静坐标系的O 1X 1轴上时，B 1点将成为此时两转子的啮合点；当D’ 1点转动到静坐标系的O 1X 1上时，D 1点将成为此时两转子的啮合点。
如果将叶峰曲线上点B 1、D 1到节圆弧上的B’ 1、D’ 1这种映射关系应用到整个叶峰曲线上，可以得到一条形状与节圆弧A’ 1C 1相同的曲线；特别的在于：映射得到的曲线可能正好是节圆弧曲线A’ 1C 1，如图4(a)，此时转子在转动中啮合良好，不会出现封闭容积；映射得到的曲线也可能是重叠的，如图4(b)所示，叶峰曲线段A 1B 1映射到节圆弧上的曲线段为A’ 1B’ 1，叶峰曲线段B 1D 1映射到节圆弧上的曲线段为B’ 1D’ 1，叶峰曲线段D 1C 1映射的节圆弧D’ 1C’ 1，映射的曲线中B’ 1D’ 1为重叠段，当这段节圆弧上的点转动到两转子旋转中心O 1O 2连线上时，转子会出现封闭容积；也可能叶峰曲线上某段曲线上的点在节圆弧上不能产生这种映射关系，如图4(c)中叶峰曲线段B 1D 1所示，此时转子在转动过程中将会有局部不参与啮合。
而若要求不出现封闭容积，则要求转子转动过程中全部啮合，以下推导转子型线不出现封闭容积的约束条件：
在已知左侧转子的叶峰曲线的方程为：
Figure PCTCN2019114012-appb-000022
请参考图(3)，做叶峰曲线在B 1点的法线B 1B’ 1，设B 1点的坐标为(x b，y b)法线的斜率为k 1，则有：
Figure PCTCN2019114012-appb-000023
所以法线B 1B’ 1的方程可以表达为：
y 1＝k 1(x 1-x b)+y b        (12)
结合节圆弧的方程可以得到叶峰曲线映射的曲线的方程：
Figure PCTCN2019114012-appb-000024
解方程组时，只需要求出x 1值，再联立公式(12)即可求得叶峰曲线映射的曲线的方程。
特别的，此时求解叶峰曲线映射的曲线方程组(13)时，将会产生如下几种情况：
1.k 1＝∞时，叶峰曲线的法线是垂直于x 1轴的，所以x 1＝x b；
2.k 1≠∞时，方程组可能存在无解、一解、两解的情况。以下对这三种情况分别进行解释。
(1)当(k 1 2+1)R 2-(k 1x b-y b) 2＜0时，方程组无解，此时对应的是转子型线局部不参与啮合的情况。
(2)当(k 1 2+1)R 2-(k 1x b-y b) 2＝0时，方程组有一解，此时转子型线在转动过程中，有且仅有一个时刻，B 1点会成为两转子的接触点。
(3)当(k 1 2+1)R 2-(k 1x b-y b) 2＞0时，方程组存在两解，此时转子型线在转动的过程中，可能会有两个时刻，B 1点是满足成为两转子接触点的条件的。但在实际设计型线过程中，根据斜率k 1的正负，只让B 1点在其中一个时刻成为了两转子的啮合点，通常这一点较另一交点更靠近B’点。所以，根据斜率k 1的不同，我们可以得到映射曲线的x 1坐标为：
Figure PCTCN2019114012-appb-000025
根据转子不出现局部不啮合与封闭容积的转子型线设计要求，将其转换成数学概念，则是：
1.方程组(13)必须存在解；
2.方程组的解中x 1必须是关于t的单调不增函数；
3.方程组的解中y 1必须不小于零。
即：
Figure PCTCN2019114012-appb-000026
由此便得到了罗茨泵转子型线设计中的避免封闭容积与转子局部不啮合的设计约束条件。
步骤三：确定避免根切的约束条件；
在转子型线设计阶段，发生根切的转子型线特征很明显，曲线上将会出现打圈的现象，如图5所示。
如图6，过啮合线上任意点T做啮合线在该点的法线TW，交O 1O 2于点W，根据齿轮啮合原理中齿轮不发生根切的条件可以推断出，若要求罗茨泵转子型线不发生根切，必须满足以下条件：
啮合线上任意点T的法线与O 1O 2的交点W必须处于两转子旋转中心O 1O 2之间。所以要得到罗茨泵转子型线设计时避免根切的条件，必须先求得啮合线上任意点T映射到O 1O 2上的点W的坐标，并对其进行检测。
啮合线是两转子啮合转动时，节点的运动轨迹，因此将式a·b＝0代入式(1)，即可得到左侧转子的叶峰曲线对应的啮合线方程，其中需要注意：两转子齿廓在每个瞬间都是相切接触，用数学表达式可表示为：a·b＝0，式中：
a——转子型线在接触点的法向量；
b——转子型线在接触点相对滑动速度矢量；
设法线WT的斜率为k 2，则有：
Figure PCTCN2019114012-appb-000027
则可得W点的坐标为：
Figure PCTCN2019114012-appb-000028
则不发生根切的条件可转变为不等式：
Figure PCTCN2019114012-appb-000029
根据啮合线表达式可得：
Figure PCTCN2019114012-appb-000030
式中：
Figure PCTCN2019114012-appb-000031
其中
Figure PCTCN2019114012-appb-000032
是罗茨泵右侧转子的转角，根据两个啮合转子间的关系推导出包络条件式。
因此，要计算出法线WT的斜率k 2还必须进一步对
Figure PCTCN2019114012-appb-000033
进行求导。
参考图14，在
Figure PCTCN2019114012-appb-000034
公式包络条件式(26)中，
Figure PCTCN2019114012-appb-000035
令
Figure PCTCN2019114012-appb-000036
则进一步可以得到
Figure PCTCN2019114012-appb-000037
的公式：
Figure PCTCN2019114012-appb-000038
式中：
Figure PCTCN2019114012-appb-000039
Figure PCTCN2019114012-appb-000040
由此可以得到在设计罗茨泵转子型线时的不根切条件。
步骤四：得到叶峰曲线合理设计区域，进一步提高转子型线的设计效率。
规定转子叶峰曲线都设计在第一象限和坐标轴上，转子的叶峰曲线的起点都设计在O 1x 1 轴上。
如图7所示，由于转子叶峰曲线自身位置的布局特点，叶峰曲线必须处于节圆弧A’C右侧、齿顶圆弧O 1AF左侧、线段A’A上方、线段CF下方。很容易得出叶峰曲线的基本设计区域：
1.叶峰曲线处于O 1A’C右侧：
Figure PCTCN2019114012-appb-000041
2.叶峰曲线处于O 1AF左侧：
Figure PCTCN2019114012-appb-000042
3.叶峰曲线处于线段A’A上方：
y 1≥0     (22)
4.叶峰曲线处于线段CF下方：
Figure PCTCN2019114012-appb-000043
转子叶峰曲线和对应的节圆段的存在一种映射关系，理论上叶峰曲线映射到节圆弧段后，刚好得到完整的节圆弧段，若映射得到的曲线出现重叠，则会产生封闭容积，为了避免此类情况，并将情况推至极限，则叶峰曲线映射得到的曲线只为一点，而考虑叶峰曲线在A点的连续性要求，所以映射得到的点只能是A’点，满足情况的叶峰曲线是销齿圆弧A‘AG，这就是叶峰曲线设计的下边界，如图8所示。
若叶峰曲线设计超出销齿圆弧框定的边界，即将叶峰曲线段继续向节圆弧A’AG内压缩，则在该段齿廓的曲线上必然存在与节圆弧段不存在映射关系的点，如图9中的叶峰曲线段A 1B 1所示，此时转子在啮合时将存在封闭容积或者有转子局部不参与啮合的现象。所以这段销齿圆弧是叶峰曲线设计的下边界，可表达为：
(x-R) 2+y 2≥(R a-R) 2，x∈[R,R a]      (24)
叶峰曲线若完全采用销齿圆弧，叶峰曲线的终点不能到达设计终点C点，所以若使用销齿圆弧，则叶峰曲线至少是两段曲线组成的，而转子型线的封闭性及连续性对叶峰曲线第一段外的曲线设计也必然存在约束，所以在式(24)限定的叶峰曲线合理设计区域的下边界的基础上可以进一步缩小设计区域的左边界。
转子的封闭性限定了转子的叶峰曲线和叶谷曲线必然是首尾相接的，所以叶峰曲线必然过C点；转子的连续性则限定了叶峰曲线为多段时，各曲线之间必须至少保证相切连续。 在这两个条件的约束下探究合理设计区域的左边界的问题，将转变为探究在保证第二段叶峰曲线能合理设计的情况下，销齿圆弧作为叶峰曲线段的第一段曲线时的使用长度极限的问题。要在转子型线设计中满足上述的要求，且同时让销齿圆弧使用长度最长，则只有让第一段曲线外的曲线段为直线，如图10所示。若叶峰曲线所设计的曲线段超出设计边界BC，如图11中的叶峰曲线段B 1C 1所示，则造成叶峰曲线段的曲线弯曲方向发生改变，从而使转子的力学性能大大降低。
该直线可作为转子型线的合理设计区域的左边界，用数学表达式可表达为：
Figure PCTCN2019114012-appb-000044
式中：
Figure PCTCN2019114012-appb-000045
Figure PCTCN2019114012-appb-000046
通过以上的约束条件便得到了叶峰曲线的合理设计范围，如图12中粗虚线标出的区域所示，图中同时标出了圆弧型转子型线、摆线型转子型线、渐开线型转子型线的叶峰曲线和该合理设计范围的关系。从图中可以看出几种典型的转子型线的叶峰曲线都是设计在了合理设计区域内。值得一提的是，该合理设计范围仍具备进一步缩小的可能性，尤其是合理设计区域的右边界和上边界。
本发明实施例中的部分步骤，可以利用软件实现，相应的软件程序可以存储在可读取的存储介质中，如光盘或硬盘等。
以上所述仅为本发明的较佳实施例，并不用以限制本发明，凡在本发明的精神和原则之内，所作的任何修改、等同替换、改进等，均应包含在本发明的保护范围之内。

Claims (8)

  1. 一种确定罗茨泵转子型线合理设计区域的方法，其特征在于，所述方法包括：
    步骤一：在设计罗茨泵转子型线叶峰曲线之前，建立罗茨泵转子型线和性能参数的关系式；所述建立罗茨泵转子型线和性能参数的关系式，包括：
    选定罗茨泵的抽气速率作为衡量罗茨泵性能的指标，建立罗茨泵的抽气速率和性能参数的关系式；所述建立罗茨泵的抽气速率和性能参数的关系式包括：
    Figure PCTCN2019114012-appb-100001
    Figure PCTCN2019114012-appb-100002
    其中，s th为罗茨泵的抽气速率，V为罗茨泵一个转子旋转一周时罗茨泵排除的介质的体积，n为转子的转速，λ为转子面积利用系数，R a为转子齿顶圆半径，L为转子长度，S r指单个罗茨泵转子的面积；
    步骤二：确定避免封闭容积的约束条件；
    步骤三：确定避免根切的约束条件；
    步骤四：根据步骤一至步骤三确定罗茨泵转子叶峰曲线的合理设计区域。
  2. 根据权利要求1所述的方法，其特征在于，所述单个罗茨泵转子的面积S r的计算公式为：
    S r＝6S b  (5)
    其中S b为转子半个齿的面积；为计算方便，将S b划分为3个区域进行面积计算，3个区域的面积由罗茨泵中心至边缘依次记为S 1、S 2、S 3，计算公式如下：
    Figure PCTCN2019114012-appb-100003
    Figure PCTCN2019114012-appb-100004
    Figure PCTCN2019114012-appb-100005
    故：
    Figure PCTCN2019114012-appb-100006
    式(9)中，R为节圆半径、大小为罗茨泵两个转子的中心距H的一半；(x e，y e)为叶谷曲线起点坐标，(x c，y c)为叶谷曲线终点坐标；由于叶谷曲线与叶峰曲线相连，所以叶谷曲线的终点也即叶峰曲线的起点，(x a，y a)为叶峰曲线的终点坐标；g(x)表示叶谷曲线的函数表达式，f(x)表示叶峰曲线的函数表达式；
    γ为叶峰曲线或叶谷曲线的起点、终点分别与节圆圆心连线之间的夹角，γ的大小与转 子齿数Z相关，计算公式为式(1)：
    Figure PCTCN2019114012-appb-100007
  3. 根据权利要求2所述的方法，其特征在于，所述步骤二中，确定避免封闭容积的约束条件，包括：
    取罗茨泵转子叶峰曲线上任意一点做叶峰曲线在该点的法线，设该法线的斜率为k 1、该法线与节圆弧的交点的坐标为(x 1，y 1)，
    Figure PCTCN2019114012-appb-100008
    在叶峰曲线上B 1点做法线B 1B’ 1，B 1点坐标为(x b，y b)，则避免封闭容积的约束条件为：
    Figure PCTCN2019114012-appb-100009
  4. 根据权利要求3所述的方法，其特征在于，所述步骤三中，确定避免根切的约束条件，包括：
    在啮合线上任意点做啮合线在该点的法线，设该法线斜率为k 2、该法线与O 1O 2的交点坐标为(X 1，Y 1)，则避免根切的约束条件为：
    Figure PCTCN2019114012-appb-100010
    其中，H为两个转子的中心距；O 1O 2为两个转子的中心点的连线。
  5. 根据权利要求4所述的方法，其特征在于，所述避免根切的叶峰曲线的左边界满足式(25)，其中x、y分别为叶峰曲线的左边界内各点的横坐标和纵坐标：
    Figure PCTCN2019114012-appb-100011
    式中：
    Figure PCTCN2019114012-appb-100012
    Figure PCTCN2019114012-appb-100013
  6. 根据权利要求5所述的方法，其特征在于，所述步骤四中，根据步骤一至步骤三确定罗茨泵转子叶峰曲线的合理设计区域，包括：
    根据步骤一至步骤三确定的约束条件画出罗茨泵叶峰曲线所处的区域范围。
  7. 根据权利要求1-5任一所述的方法，其特征在于，所述罗茨泵转子型线包括圆弧型转子型线、摆线型转子型线或渐开线型转子型线。
  8. 一种罗茨泵转子型线的设计方法，其特征在于，所述方法采用权利要求1-7任一所述的方法设计罗茨泵转子型线。
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