# Profile Design Method of Twin-Screw Compressor Rotors Based on the Pixel Solution

Authors: Zhihuang Shen
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A new design method based on pixel solution is proposed to achieve an efficient and high-precision design of a twin-screw rotor profile. This method avoids the complex analytic calculations in the traditional envelope principle. The best approximation of the pixels of the rotor conjugate motion sweeping surface in the lattice screen pixels is illuminated using a specific color. The sweeping surface of the screw rotor single-tooth profile is roughly scanned to capture the base point set of the sweeping surface boundary pixels. The chord length and tilt angle of each interval are calculated using the value of the base point set to adjust the position, phase, and magnification of each interval sweeping surface. Each interval sweeping surface is finely scanned to capture the data point set of the subinterval, and then the data point set is converted to the same coordinate system to generate the conjugated rotor profile. Finally, an example is used to verify the feasibility and adaptability of this method. The approach provided can be used to design screw rotor profiles with high precision.


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## Body

## 1. Introduction

Twin-screw compressors have the advantages of reliable operation, long service life, compact structure, and high energy efficiency and are, thus, widely used in automotive, compression, and refrigeration, among other industrial fields. A pair of meshing screw rotors forms the key component of the twin-screw compressor, and the rotor profile is the main factor determining the performance of the compressor. Therefore, the methodology used to design a precision screw rotor profile efficiently is particularly important. With the improvements in the technological level and calculation conditions, many representative rotor profiles and their design methods have been developed in recent years.Litvin and Fuentes [1] derived an analytical calculation process of the forming tool profile for machining gears, which could also be used to design screw rotors. Xing et al. [2, 3] designed a rotor profile based on screw rotor geometry, thermal characteristics, and the contact condition between the male rotor and the female rotor, establishing the rotor and the grinding wheel. Stosic et al. [4, 5] used the conjugate principle to design the N profile. Spitas proposed a method to discretize the gear tooth surface into several involutes to calculate the conjugate tooth profile [6]. For the abovementioned literature, the mathematical model of the analytic gearing envelope method was accurate enough. However, it was identified that they have a computing problem concerning the complex contact line equation. Owing to the complexity of the equation of the contact line, the computing process of the equation led to the formation of a singular point and an uneven distribution of the generated profile data.Many scholars have, thus, been continuously working toward finding an easier and more comprehensive method to substitute the analytic gearing envelope method. The point vector envelope method was proposed to calculate the forming tool profile of the spiral surface by He et al. [7]. Lyashkov and Panchuk [8] used the CAD technology to calculate the tool profile of the screw pump. Yang et al. [9] used MATLAB and Unigraphics motion simulation to build the 3D model of the screw rotor and then used edge detection of the graphics method and the shape algorithm to calculate the tool profile. Wu et al. [10] proposed a radial-ray shooting method to simulate the form grinding process of a threaded cylindrical workpiece, such as a spur, helical gear, and screw. The profile of the workpiece can precisely be determined from the intersecting points on each ray without solving the simultaneous system equations of locus and the equation of meshing. However, Wu omitted the details of the calculation of the intersecting points.To generate the conjugate rotor profile, a new method of designing a screw rotor profile, based on pixel solution, has been proposed in this study. This method requires multiple adjustments of the pose and the magnification of the conjugate motion sweeping surface by the known rotor. The critical pixels in the screen pixel matrix of the subinterval sweeping surface were captured to rebuild the unknown rotor profile. The pixel solution method generates a grinding profile without establishing and computing the contact line equation. The improved pixel solution has an automatic scanning position adjustment function and solves the problem of uneven local data points caused by the single scanning direction in the digital graphic scanning method and pixel solution proposed by the studies mentioned earlier [11, 12]. Finally, an example has been proposed to show the difference between the pixel solution and the analytic gearing envelope method. The results show that the pixel solution avoids the singular point and uneven distribution of profile data seen in the analytic gearing envelope method. The deviation between the two methods was found to be within −0.002 mm to 0.002 mm.


## 2. Calculation of the Sweeping Surface

### 2.1. Coordinate Transformation between Male and Female Rotor

The coordinate system is established to describe the conjugate envelope process of the screw rotors through mathematical equations, as shown in Figure1. The coordinate systems O1x1y1 and O2x2y2 are attached to the male and female rotors. a is defined as the distance between the male and female rotors. Parameters r1 and r2, φ1 and φ2, and w1 and w2 are the pitch circle radii, phase angle, and angular velocity of the male and female rotors, respectively. According to the spatial geometric relationship, the coordinate transformation formula between the male and the female rotors is given by(1)M1=−cosφ1+φ2−sinφ1+φ2acosφ1−sinφ1+φ2cosφ1+φ2asinφ1001,M2=−cosφ1+φ2−sinφ1+φ2acosφ2−sinφ1+φ2cosφ1+φ2asinφ2001.Figure 1
Coordinate relationship of male and female rotors.


### 2.2. Conjugate Envelope Motion of Rotors

As shown in Figure2, assuming the female rotor profile coordinates x2,y2 are known, we calculate the profile coordinates x1,y1 of the conjugate male rotor. Both rotors are given a rotational angular velocity of −w1 that enables the male rotor to orbit first, after which the female rotor will take planetary motion, while the male rotor stands still. The rotation and revolution angular velocity of the female rotor is w2 and w1. The planetary motion of the female rotor envelops the profile of the male rotor. The conjugate envelope process is shown in Figure 3.Figure 2
Planetary motion of the female rotor.Figure 3
Male rotor enveloped by the female rotor.


## 2.1. Coordinate Transformation between Male and Female Rotor

The coordinate system is established to describe the conjugate envelope process of the screw rotors through mathematical equations, as shown in Figure1. The coordinate systems O1x1y1 and O2x2y2 are attached to the male and female rotors. a is defined as the distance between the male and female rotors. Parameters r1 and r2, φ1 and φ2, and w1 and w2 are the pitch circle radii, phase angle, and angular velocity of the male and female rotors, respectively. According to the spatial geometric relationship, the coordinate transformation formula between the male and the female rotors is given by(1)M1=−cosφ1+φ2−sinφ1+φ2acosφ1−sinφ1+φ2cosφ1+φ2asinφ1001,M2=−cosφ1+φ2−sinφ1+φ2acosφ2−sinφ1+φ2cosφ1+φ2asinφ2001.Figure 1
Coordinate relationship of male and female rotors.


## 2.2. Conjugate Envelope Motion of Rotors

As shown in Figure2, assuming the female rotor profile coordinates x2,y2 are known, we calculate the profile coordinates x1,y1 of the conjugate male rotor. Both rotors are given a rotational angular velocity of −w1 that enables the male rotor to orbit first, after which the female rotor will take planetary motion, while the male rotor stands still. The rotation and revolution angular velocity of the female rotor is w2 and w1. The planetary motion of the female rotor envelops the profile of the male rotor. The conjugate envelope process is shown in Figure 3.Figure 2
Planetary motion of the female rotor.Figure 3
Male rotor enveloped by the female rotor.


## 3. Principle of Bresenham Algorithm

The line, circle, and ellipse are the basic elements in graphic design. Pixel function is necessary to build the graphics according to the display principle of the monitor and realistic graphic technology. The existing monitor represents a line, approximately, by capturing the nearest pixel point set of the region boundary, rather than by drawing the line from one pixel to another directly. The graph scan conversion illuminates the pixels that are closer to the ideal graph compared with all other pixels in the matrix of the screen in a specified color. For instance, in Figure4, the black line can only be approximated by lightening the blue pixels which are nearest to the blue line. The Bresenham algorithm is adopted for the line scan conversion due to its high efficiency in drawing lines. In the Bresenham algorithm, the active pixel moves one unit with every frequency change along the principal direction. The secondary directional movement is determined by the midpoint deviation discrimination [13, 14].Figure 4
Line scan conversion.The starting point of the ideal line is defined asP0x0,y0, and the ending point is defined as Pnxn,yn. The implicit function equation of the line is described as(2)fx,y=y−ux−v=0,where u is the linear slope and v is the intercept in the y direction. As shown in Figure 5, assuming that the active pixel coordinate point is Pixi,yi and by moving one unit along the x direction, the next pixel can be chosen between Paxi+1,yi+1 and Pbxi+1,yi. The midpoint of Pa and Pb is Mxi+1,yi+0.5. Pa is chosen when the ideal line is above the midpoint M, indicating that Pa is nearer to the line compared to Pb, as if otherwise, Pb would have been chosen.Figure 5
Midpoint Bresenham algorithm.Midpoint deviation discrimination is integrant to acquire the next pixel after the first step. The midpointM, in the implicit function equation, is substituted to build the midpoint deviation discrimination di and is given by(3)di=fxi+1,yi+0.5=yi+0.5−uxi+1−v.Pa is chosen when di<0 and the ideal line is above the midpoint M, which means that the active pixel moves one unit along the y direction. Pb is chosen when di>0 and the ideal line is below the midpoint M, which means that the active pixel has no movement. Either Pa or Pb can be chosen when di=0, indicating that M is located on the line. The deviation discrimination can also be presented as [15](4)yi+1=yi+1,di<0,yi,di≥0.Recurrence formulation is a prerequisite when the midpoint deviation discriminationdi is acquired to judge every point on the line continuously. As shown in Figure 6, there are two possibilities when choosing a midpoint to substitute the midpoint deviation discrimination and when choosing the pixel after the active pixel has already moved along the principle direction.Figure 6
Recurrence formulation of midpoint deviation discrimination. (a)di<0 and (b) di≥0.
(a)(b)The midpoint coordinates of the next step areM′xi+2,yi+1.5. When di<0, the midpoint deviation discrimination can be prescribed as(5)di+1=fxi+2,yi+1.5=yi+1.5−uxi+2−v=yi+0.5−uxi+1−v+1−u=di+1−u.The midpoint coordinate of the next step isM′xi+2,yi+0.5. When di≥0, the midpoint deviation discrimination can be prescribed as(6)di+1=fxi+2,yi+0.5=yi+0.5−uxi+2−v=yi+0.5−uxi+1−v−u=di−u.


## 4. Principle of Pixel Solution

The pixel solution first illuminates the best approximation pixels of the rotor conjugate swept surface in the screen pixel lattice with the specified color and then adjusts the pose and magnification of the sweeping surface. The conjugate rotor profile is calculated by capturing the coordinates of the best approximation pixel. The best approximation pixel is the boundary pixel that distinguishes between the specified color and the background color of the screen. The calculation procedure is divided into the following steps:(1)
Assuming the female rotor profile is known, the trajectory of its planetary motion is obtained by the envelope principle, as shown in Figure7.(2)
The sweeping surface of the female rotor profile is adjusted and enlarged (magnification isK) to fill the monitor screen (display resolution of the monitor is xs×ys). According to the Bresenham algorithm, blue color RGB=0,0,255 was used to lighten the best approximated pixels of the enveloping surface in the pixel matrix of the screen and the other region was lightened by yellow color RGB=255,255,0, as shown in Figure 8.(3)
The sweeping surface is rough, scanning to get the basic point sets of the conjugate rotor profile. The basic point sets areP0,P1,P2,…,Pi,…,Pm, where m is the number of the segmented intervals of the sweeping surface (in the actual calculation, m=200).(4)
The sweeping surface is divided into several basic intervals, andA is the i-th interval. As shown in Figure 9, the chord length si and the chord length tilt angle αi of the sweeping surface in the interval A can be calculated according to the coordinate of the points Pi and Pi+1.(5)
The chord lengthsi and chord length tilt angle αi are used to adjust the position, phase, and magnification of the sweep surface between the two points Pi and Pi+1. The purpose of the readjustment is to ensure the A interval fills the entire screen smoothly, as shown in Figure 10. The coordinate system changed from S0 to Si during the readjustment; the rotation angle and the new magnification are αi and Ki=xs/siK, respectively. The critical pixels of the interval A are captured, and the critical pixel set is Pi,0,Pi,1,…,Pi,ti,…,Pi,ni. The critical pixel sets of the other intervals can be obtained using the abovementioned method, as shown in Table 1.(6)
After capturing the critical pixel sets of each interval, they need to be transformed into the unified coordinate system. As shown in Figure11, OXY is the unified coordinate system, xs×ys is the screen resolution, and Oixiyi is the coordinate of each interval. The formula for transforming critical pixel sets of each interval to the unified coordinate system is given by(7)Xi,tiYi,ti=1Kicosαi−sinαisinαicosαixi,ti+Xi,0yi,ti+Yi,0,where Xi,0,Yi,0 is the coordinate of Pi,0.(7)
As shown in Figure12, through the above method, tens of thousands of sweeping surface profile points can be captured. The profile points can be reorganized by setting an offset to meet the accuracy requirement of the male rotor, as shown in Figure 13.Table 1
Critical pixel sets of each interval.
Interval01…t…n0P0,0P0,1…P0,t0…P0,n01P1,0P1,1…P1,t1…P1,n1…………………iPi,0Pi,1…Pi,ti…Pi,ni…………………mPm,0Pm,1…Pm,tm…Pm,nmFigure 7
Planetary trajectory of the female rotor profile.Figure 8
Sweeping surface posture adjustment and basic point set scanning.Figure 9
Local enlargement diagram of intervalA.Figure 10
Regeneration of sweeping surface of intervalA.Figure 11
Coordinate transformation of each interval.Figure 12
Profile data of sweeping surface. (a) Profile of sweeping surface. (b) Enlarged profile of area I. (c) Enlarged profile of area II.
(a)(b)(c)Figure 13
Restructured male rotor profile.


## 5. Numerical Examples

An example of the rotors from a screw compressor manufacturer was investigated to verify the correctness and the feasibility of the method proposed; the parameters of the rotors are as given in Table2. Assuming that the female rotor profile is known, the profile of the male rotor needs to be calculated. Figure 14 shows the cross-sectional contour of the female rotor. The selected screen resolution was 1440×900.Table 2
Parameters of the rotors.
TypeMaleFemaleRevolving directionRight handLeft handNumber of teeth54Outer diameter (mm)47.62159.795Root diameter (mm)24.20936.380Lead (mm)145116Helical angle (deg)45.316Center distance (mm)42Figure 14
Profile of female rotors.The analytic gearing envelope method and the pixel solution are adopted to calculate the male rotor profile. When using the pixel solution, the chosen computer screen has a resolution of1440×900 and the magnification of the male rotor was K1=50. After performing the above steps, 38,152 points of the male rotor were collected. The male rotor profile developed using different generating methods was compared. The result is shown in Figures 15 and 16.Figure 15
Comparison of the male rotor profile.Figure 16
Normal deviation of the male rotor profile.It can be seen from Figure15 that the performance of the pixel solution was clearly better than the analytic gearing envelope method and that there are no singular points or appearance of uneven distribution phenomena. Figure 16 showed that the deviation between different male rotor profiles, generated using the pixel solution and the analytic gearing envelope methods, was within −0.002 mm to 0.002 mm.


## 6. Conclusions

This study presents a pixel solution method, based on computer scanning graphics to generate a conjugate rotor profile, that avoids the difficulties that appear in the analytical gearing envelope method, by using the complex nonlinear equations of the contact line. Conjugate rotor profiles were collected by scanning the pixel matrix of the screen and capturing the coordinates of the indicated color of the best possible pixels, guaranteeing the accuracy of the generated profile.As shown by the numerical examples, the deviation between the different conjugate rotor profiles, generated by the pixel solution and the analytic gearing envelope method, was within −0.002 mm to 0.002 mm. The results showed that pixel solution is a powerful tool to generate the conjugate rotor profile of the rotors and that it is also suitable for threaded cylindrical workpieces, such as helical gears, worms, spurs, and millings, among others.



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