CN112555152A - Twisted-blade Roots rotor and design method thereof, compressor and expander - Google Patents

Twisted-blade Roots rotor and design method thereof, compressor and expander Download PDF

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CN112555152A
CN112555152A CN202011192721.9A CN202011192721A CN112555152A CN 112555152 A CN112555152 A CN 112555152A CN 202011192721 A CN202011192721 A CN 202011192721A CN 112555152 A CN112555152 A CN 112555152A
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rotor
segment
line
meshing
female
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CN112555152B (en
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韦炜
何志龙
李丹童
孙崇洲
王嘉辰
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Xian Jiaotong University
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Xian Jiaotong University
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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
    • 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
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F01—MACHINES OR ENGINES IN GENERAL; ENGINE PLANTS IN GENERAL; STEAM ENGINES
    • F01C—ROTARY-PISTON OR OSCILLATING-PISTON MACHINES OR ENGINES
    • F01C1/00—Rotary-piston machines or engines
    • F01C1/08—Rotary-piston machines or engines of intermeshing engagement type, i.e. with engagement of co- operating members similar to that of toothed gearing
    • F01C1/082—Details specially related to intermeshing engagement type machines or engines
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F01—MACHINES OR ENGINES IN GENERAL; ENGINE PLANTS IN GENERAL; STEAM ENGINES
    • F01C—ROTARY-PISTON OR OSCILLATING-PISTON MACHINES OR ENGINES
    • F01C1/00—Rotary-piston machines or engines
    • F01C1/08—Rotary-piston machines or engines of intermeshing engagement type, i.e. with engagement of co- operating members similar to that of toothed gearing
    • F01C1/12—Rotary-piston machines or engines of intermeshing engagement type, i.e. with engagement of co- operating members similar to that of toothed gearing of other than internal-axis type
    • F01C1/126—Rotary-piston machines or engines 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 elements extending radially from the rotor body not necessarily cooperating with corresponding recesses in the other rotor, e.g. lobes, Roots type
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F01—MACHINES OR ENGINES IN GENERAL; ENGINE PLANTS IN GENERAL; STEAM ENGINES
    • F01C—ROTARY-PISTON OR OSCILLATING-PISTON MACHINES OR ENGINES
    • F01C21/00—Component parts, details or accessories not provided for in groups F01C1/00 - F01C20/00
    • F01C21/08—Rotary pistons
    • 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
    • F04C18/12—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 of other than internal-axis type
    • F04C18/126—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 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
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/28—Design optimisation, verification or simulation using fluid dynamics, e.g. using Navier-Stokes equations or computational fluid dynamics [CFD]
    • 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
    • F04C2250/20—Geometry of the rotor

Abstract

A twisted-blade Roots rotor, its design method, compressor and expander are composed of female and male rotors with same structure, and single-tooth profile A of end face profile of female rotor1B1C1D1E1F1G1H1I1By successive connected circular arc sections A1B1Composite curve segment B1C1Composite curve segment C1D1Circular arc segment D1E1Circular arc segment E1F1Composite curve segment F1G1Composite curve segment G1H1Circular arc section H1I1Composition is carried out; rotor profile origin O wound by single-tooth profile for complete profile1Rotate
Figure DDA0002753213760000011
Then the molded lines are connected end to end, and the molding is repeated for n times, wherein n represents the tooth number of the molded line; the two rotors can realize correct meshing in the rotating meshing process. The invention also provides a design method of the twisted-lobe Roots rotor, and a compressor and an expander using the twisted-lobe Roots rotor. The invention overcomes the defect of insufficient design parameter space of the original twisted lobe Roots type line, breaks through the value limitation of the original type line radial distance ratio and improves the area utilization coefficient.

Description

Twisted-blade Roots rotor and design method thereof, compressor and expander
Technical Field
The invention belongs to the field of rotor design, and particularly relates to a twisted-blade Roots rotor, a design method thereof, a compressor and an expander.
Background
The twisted blade Roots compressor is one rotary compressor with two meshed rotors as the core part and completes the sucking, compressing and exhausting process via the synchronous rotation of the two rotors. The compressor has the advantages of high rotating speed, low noise, reliable operation, simple processing and the like, is mainly used in the medium and low pressure gas transportation field, and particularly has wide application in the fields of petrochemical industry, pharmacy, electronics and aerospace.
The rotor end surface profile of the twisted-lobe roots compressor has a significant effect on the performance of the whole machine. The end face molded line commonly used at present is a circular arc-circular arc enveloping line, as shown in fig. 1, and the main parameter indexes are the number of teeth n of the rotor and the pitch circle radius RpRatio of radial to radial
Figure BDA0002753213740000011
(RmTooth tip radius). The main problems with this profile are: single design parameter and low area utilization rate. Due to the number of teeth n and the pitch circle radius RpRatio of radial to radial
Figure BDA0002753213740000012
After the determination, the shape of the mold line is determined, resulting in a lack of flexibility in mold line designThe method is suitable for the optimization design requirements under different working conditions; radial to pitch ratio
Figure BDA0002753213740000013
Is limited in value range when
Figure BDA0002753213740000014
Above a limit value
Figure BDA0002753213740000015
In the meantime, an undercut phenomenon as shown in fig. 2 occurs, which makes it difficult to further improve the area utilization coefficient, thereby limiting the development of the twisted-lobe roots rotor toward miniaturization and high efficiency.
Disclosure of Invention
The invention aims to solve the problems of single design parameter and limited area utilization rate of the end face profile of the circular arc-circular arc envelope linear twisted lobe Roots rotor in the prior art, and provides a twisted lobe Roots rotor, a design method thereof, a compressor and an expander, which can increase the parameter design space of the twisted lobe Roots rotor profile and improve the area utilization coefficient of the twisted lobe Roots rotor profile.
In order to achieve the purpose, the invention has the following technical scheme:
a twisted-blade Roots rotor comprises a female rotor and a male rotor which have the same structure, and a single-tooth profile A of the end face profile of the female rotor1B1C1D1E1F1G1H1I1By successive connected circular arc sections A1B1Composite curve segment B1C1Composite curve segment C1D1Circular arc segment D1E1Circular arc segment E1F1Composite curve segment F1G1Composite curve segment G1H1Circular arc section H1I1Composition is carried out; the end face profile of the female rotor is in a single-tooth form A1B1C1D1E1F1G1H1I1Origin O of wound rotor profile1Rotate
Figure BDA0002753213740000021
Then with tooth form A1B1C1D1E1F1G1H1I1Repeating the steps for n times in an end-to-end mode, wherein n represents the number of teeth of the molded line; the molded line origin of the male rotor molded line is O2The center distance A between the female rotor profile and the male rotor profile is equal to 2 pitch circle radii Rp(ii) a The two rotors can be correctly meshed in the rotating and meshing process, and the arc section A of the female rotor1B1Arc segment D of male rotor2E2Compound curve section B of female rotor engaged with each other1C1Compound curve segment C with male rotor2D2Composite curve section C of female rotor and meshed with each other1D1Compound curve segment B with male rotor2C2Arc segments D of female rotor engaged with each other1E1Arc segment A of male rotor2B2Arc sections E of female rotor engaged with each other1F1And the circular arc section H of the male rotor2I2Compound curve segment F of female rotor engaged with each other1G1Compound curve segment G with male rotor2H2Compound curve segment G of female rotor engaged with each other1H1Compound curve segment F with male rotor2G2Are correspondingly engaged with each other, and the arc sections H of the female rotors1I1Arc section E of male rotor2F2Are correspondingly engaged with each other.
As a preferred scheme of the invention, a meshing line bus is generated, then the meshing line bus is dispersed and transformed to obtain a new complete meshing line, the corner relation corresponding to the new meshing line is solved by utilizing the tooth profile meshing principle, and finally the single-tooth profile with complete molded lines of the end faces of the female rotor and the male rotor is obtained through coordinate transformation.
As a preferred embodiment of the present invention, the point a on the meshing line2Arc segment A corresponding to female rotor1B1、I2H2With the circular arc segment of the male rotorD2E2、E2F2At position XpOpYpMeshing line segment S of first quadrant of coordinate system2Compound curve segment B corresponding to female rotor1C1Compound curve segment C with male rotor2D2At position XpOpYpMeshing line segment S of the third quadrant of the coordinate system3Compound curve segment C corresponding to female rotor1D1Compound curve segment B with male rotor2C2Point d on the meshing line2Arc segment D corresponding to female rotor1E1、E1F1Arc segment A of male rotor2B2、I2H2At position XpOpYpMeshing line segment S of the second quadrant of the coordinate system4Composite curve segment F corresponding to female rotor1G1Compound curve segment H with male rotor2G2At position XpOpYpMeshing line segment S of the third quadrant of the coordinate system5Corresponding to the compound curve segment G of the female rotor1H1Compound curve segment G with male rotor2F2。
The invention also provides a design method of the twisted-lobe Roots rotor, which comprises the following steps:
1) determining pitch circle radius RpTarget radial to pitch ratio
Figure BDA0002753213740000031
And a number of teeth n;
2) generating meshing line bus S according to the parameters determined in the step 1)1；
3) To mesh the line bus S1Performing transformation to obtain a meshing line S2And making the radial-to-radial ratio take on the value
Figure BDA0002753213740000032
Increase to
Figure BDA0002753213740000033
4) Meshing using cubic spline curve pairsWire S2Fitting is performed to construct
Figure BDA0002753213740000034
About
Figure BDA0002753213740000035
Of a multi-segment fitting function
Figure BDA0002753213740000036
5) According to the symmetry principle, a complete meshing line S is formedj,j＝2,3,4,5；
6) And generating end face molded lines corresponding to the female rotor and the male rotor according to the meshing line and the meshing angle.
Preferably, the specific steps of step 2) are as follows:
the pitch circle tangent point of the male and female rotors is set as OpIn the coordinate system XpOpYpMiddle, meshing line bus
Figure BDA0002753213740000037
The parameter equation of (1) is as follows:
Figure BDA0002753213740000038
wherein:
Figure BDA0002753213740000039
Figure BDA00027532137400000310
Rr、θLMis an intermediate variable formed by the pitch circle radius RpIs a center distance A ═ O1O2Half of l, get:
Figure BDA0002753213740000041
preferably, the specific steps of step 3) are as follows:
let S1Right end point a1Highest point b1Left end point and OpAre superposed and respectively paired with a1b1、b1OpAfter the segments are dispersed, normalization processing is carried out, and t is set to be the element [0,1 ∈]For discrete points numbered i
Figure BDA0002753213740000042
Comprises the following steps:
Figure BDA0002753213740000043
each discrete point is replaced by the following method to obtain a new meshing line S2Each point on
Figure BDA0002753213740000044
Figure BDA0002753213740000045
Wherein:
Figure BDA0002753213740000046
wherein u is1、v1As a scaling factor, u2、u3、v2Is a position parameter;
Figure BDA0002753213740000051
preferably, the specific steps of step 4) are as follows:
Figure BDA0002753213740000052
about
Figure BDA0002753213740000053
Of a multi-segment fitting function
Figure BDA0002753213740000054
The expression is as follows:
Figure BDA0002753213740000055
wherein, c1,k,c2,k,c3,k,c4,kPolynomial coefficients of the k-th section;
obtaining a fitting curve
Figure BDA0002753213740000056
Then, for S2Solving a fitting curve S between the left endpoint and the right endpoint2Corresponding angle of engagement
Figure BDA0002753213740000057
In that
Figure BDA0002753213740000058
The complete expression above is as follows:
Figure BDA0002753213740000059
Figure BDA00027532137400000510
namely the arc section A of the female rotor1B1Corresponding to the central angle.
Preferably, the specific steps of step 5) are as follows:
meshing line S3And the line of mesh S2With respect to OpCentral symmetry, S3The coordinates of the meshing line and the corresponding meshing angle have the following relations:
Figure BDA00027532137400000511
meshing line S4And S3With respect to XpThe axis is symmetrical up and down, and comprises:
Figure BDA00027532137400000512
meshing line S5And S2With respect to XpThe axis is symmetrical up and down, and comprises:
Figure BDA0002753213740000061
preferably, the specific steps of step 6) are as follows: for the
Figure BDA0002753213740000062
Corresponding angle of engagement therewith
Figure BDA0002753213740000063
Curve of tooth
Figure BDA0002753213740000064
For the composite curve, it is given by:
Figure BDA0002753213740000065
the invention provides a compressor, which uses the twisted-blade roots rotor.
The invention provides an expander using the twisted-lobe roots rotor.
Compared with the prior art, the invention has the following beneficial effects: the rotor meshing line can be flexibly adjusted according to actual working conditions to improve the working performance of the rotor, and then corresponding rotor molded lines are generated according to the improved meshing line. The invention can greatly increase the parameter design space of the existing twisted-blade Roots rotor profile by improving the rotor profile designed by the meshing line, and effectively improve the thermal dynamic performance of the existing rotor. The invention breaks through the application range of the traditional arc-arc envelope line radius ratio, under the condition of a certain center distance, the improved twisted-blade roots rotor profile has a larger tooth crest arc radius, and the area utilization coefficient is obviously improved, thereby becoming a favorable technical means for solving the difficult problems of miniaturization and high efficiency of the twisted-blade roots compressor.
Drawings
FIG. 1 original arc-arc envelope profile and parameter Rm1、RpSchematic diagram of the geometrical meaning of (1);
FIG. 2 is a schematic view of an original rotor profile tooth curve undercut;
FIG. 3 rotor profile and parameter R of the present inventionm2Schematic diagram of the geometrical meaning of (1);
FIG. 4 meshing line bus S of the present invention1Mesh line S2And parameter u2,u3,v2Schematic diagram of the geometrical meaning of (1);
FIG. 5 is a schematic diagram of the generation process of the complete meshing line of the present invention: (a) s2The generation process of (1); (b) s3The generation process of (1); (c) s4The generation process of (1); (d) s5The generation process of (1);
FIG. 6 is a schematic view of rotor profile engagement according to the present invention;
FIG. 7 is a schematic view of the meshing state of the rotors of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
Referring to fig. 1, among original arc-arc envelope profiles, an arc envelope segment J1K1Circular arc section K1L1Circular arc segment L1M1Circular arc envelope line segment M1N1Formed tooth curve segment J1K1L1M1N1A single-tooth profile forming the profile of the end face of the female rotor, the single-tooth profile J1K1L1M1N1Origin O of wound rotor profile1Rotate
Figure BDA0002753213740000071
Then is combined with the tooth form J1K1L1M1N1This is repeated n times end to form the original rotor profile, where n represents the number of teeth of the profile and is taken to be 4 in fig. 1.
Arc section K of original molded line1L1And arc section L1M1With respect to O1L1Symmetrical, circular arc enveloping line segment J1K1And the arc envelope line segment M1N1With respect to O1L1Circular arc enveloping line segment J of symmetric female rotor1K1Arc section K of male rotor2L2Arc segment K of meshed female rotor1L1Segment J of circular arc envelope with male rotor2K2Arc segment L of meshed female rotor1M1Circular arc enveloping line segment M with male rotor2N2Arc envelope line segment M of meshed female rotors1N1Arc segment L of male rotor2M2Are engaged with each other. The only parameter determining the shape of the original rotor profile curve segment is the pitch radius RpRadius to pitch ratio
Figure BDA0002753213740000072
And the number of teeth n of the rotor, the number of design parameters of the rotor profile is less, and the area utilization coefficient of the original profile is the radial distance ratio
Figure BDA0002753213740000073
The value is limited and is difficult to increase further. With following
Figure BDA0002753213740000074
Increase to the limit value
Figure BDA0002753213740000075
The original profile will exhibit an undercut phenomenon as shown in fig. 2.
In order to solve the problems of small quantity of designable parameters and limited area utilization rate of the original rotor profile, the invention provides a twisted lobe Roots rotor profile design method based on improved meshing line design. Single tooth profile A of female rotor end surface molded line1B1C1D1E1F1G1H1I1By successive connected circular arc sections A1B1Composite curve segment B1C1Composite curve segment C1D1Circular arc segment D1E1Circular arc segment E1F1Composite curve segment F1G1Composite curve segment G1H1Circular arc section H1I1Composition, as shown in fig. 3.
The complete female rotor profile is in a single tooth form A1B1C1D1E1F1G1H1I1Origin O of wound rotor profile1Rotate
Figure BDA0002753213740000076
Then with tooth form A1B1C1D1E1F1G1H1I1And repeating the steps for n times in an end-to-end manner, wherein n represents the number of teeth of the molded line.
Winding the complete female rotor profile around the rotor profile origin O1Rotate
Figure BDA0002753213740000077
Then, go along O1O2Is directionally translated by a center distance A (A is equal to 2 times of pitch circle radius R)p) The complete male rotor profile can be obtained.
The two rotors can be engaged correctly during the rotation engagement process, as shown in fig. 3, wherein the circular arc section A of the female rotor1B1Arc segment D of male rotor2E2Compound curve section B of female rotor engaged with each other1C1Compound curve segment C with male rotor2D2Arc segments D of female rotor engaged with each other1E1Arc segment A of male rotor2B2Arc sections E of female rotor engaged with each other1F1And the circular arc section H of the male rotor2I2Compound curves of female and male rotors engaged with each otherLine segment F1G1Compound curve segment G with male rotor2H2Compound curve segment G of female rotor engaged with each other1H1Compound curve segment F with male rotor2G2Are correspondingly engaged with each other, and the arc sections H of the female rotors1I1Arc section E of male rotor2F2Are correspondingly engaged with each other.
The design method of the twisted-blade roots rotor is based on the following principle: for a dual rotor profile, the right end of the meshing line must lie on the tip circle. Moving the right end of the meshing line corresponds to increasing the radius of the tooth top arc. Based on the theory, the design method firstly generates meshing line generatrix, then substitutes, overturns and splices the meshing line generatrix after being dispersed according to a specific mode to obtain a new complete meshing line, mainly aims at moving the right end point of the meshing line, solves the corresponding corner relation of the new meshing line by utilizing the tooth profile meshing principle, and finally obtains the complete single-tooth profile A on the end surfaces of the male and female rotors through coordinate transformation1B1C1D1E1F1G1H1I1. Since the new meshing line is a compound curve, the resulting tooth curve is also a compound curve.
Examples
A design method of a twisted-lobe Roots rotor comprises the following steps:
1) determining pitch circle radius RpTarget radial to pitch ratio
Figure BDA0002753213740000081
And a number of teeth n. In the present case, it is preferred that,
Figure BDA0002753213740000082
take 1.4.
2) Generating meshing line generatrix S1，S1Is an original arc-shaped molded line arc segment L1M1The corresponding meshing lines are shown in fig. 4. S1Radial to pitch ratio of
Figure BDA0002753213740000083
Satisfy the requirement of
Figure BDA0002753213740000084
Figure BDA0002753213740000085
The maximum radial distance ratio when the tooth curve does not generate the undercut phenomenon, when n is 4,
Figure BDA0002753213740000086
therefore, the original arc-shaped line can not reach the target radial-to-radial ratio
Figure BDA0002753213740000087
The requirements of (1). In this case
Figure BDA0002753213740000088
Take 1.32.
The pitch circle tangent point of the male and female rotors is set as OpIn the coordinate system XpOpYpIn, L1M1Segment-corresponding meshing line S1Equation of parameters (2)
Figure BDA0002753213740000089
Comprises the following steps:
Figure BDA0002753213740000091
wherein the content of the first and second substances,
Figure BDA0002753213740000092
is a circular arc section L1M1The equation of the tooth curve of (a),
Figure BDA0002753213740000093
the corresponding engagement angle can be given by the corresponding engagement relationship. Let θ be a variable, having:
Figure BDA0002753213740000094
in the formula
Figure BDA0002753213740000095
Rr、tLMGiven by the following geometric relationship, point M lies in a circular arc segment L1M1Up and from the origin O1A distance of RpWherein the pitch circle radius RpIs a center distance A ═ O1O2Half of l, the concrete solving equation is:
Figure BDA0002753213740000096
the solution is as follows:
Figure BDA0002753213740000097
from the above formula, the arc segment L1M1Corresponding tooth curve and meshing line S1Can be determined by a parameter Rp、
Figure BDA0002753213740000101
n is determined.
3) Will mesh with the line S1By substitution in a particular way, to obtain a meshing line S2And making the radial-to-radial ratio take on the value
Figure BDA0002753213740000102
Increase to
Figure BDA0002753213740000103
As shown in fig. 4. Let S1Right end point a1Highest point b1Left end point and OpAre superposed and respectively paired with a1b1、b1OpAfter the segments are dispersed, normalization processing is carried out, and t is set to be the element [0,1 ∈]For discrete points numbered i
Figure BDA0002753213740000104
Comprises the following steps:
Figure BDA0002753213740000105
each discrete point is replaced by the following method to obtain a new meshing line S2Each point on
Figure BDA0002753213740000106
Figure BDA0002753213740000107
Wherein:
Figure BDA0002753213740000108
wherein u is1、v1As a scaling factor, u2、u3、v2As position parameters:
Figure BDA0002753213740000109
according to the above formula, the parameter u is introduced1,u2,u3,v1,v2Controllable meshing line S1Right endpoint a1Highest point b1And a new meshing line S is obtained2. When a is1When the point moves outward to a2The radius of the tooth crest is from Rm1Increase to Rm2。
4) Using cubic spline curve to mesh line S2Fitting is performed to construct
Figure BDA00027532137400001010
About
Figure BDA00027532137400001011
Of a multi-segment fitting function
Figure BDA00027532137400001012
Figure BDA00027532137400001013
Wherein, c1,k,c2,k,c3,k,c4,kIs the polynomial coefficient of the k-th segment. Obtaining a fitting curve
Figure BDA0002753213740000111
Then, for S2In the portion between the left and right end points, the fitting curve S is solved in the following manner2Corresponding angle of engagement
Figure BDA0002753213740000112
Figure BDA0002753213740000113
In order to satisfy the relationship between the central angle and the number of teeth n corresponding to the complete tooth curve, i.e. in FIG. 3
Figure BDA0002753213740000114
U needs to be adjusted simultaneously1,u2,u3,v1,v2Such that the following holds:
Figure BDA0002753213740000115
in order to prevent the formed tooth curve from generating undercut phenomenon, the undercut discriminant
Figure BDA0002753213740000116
Figure BDA0002753213740000117
In that
Figure BDA0002753213740000118
The sign is not changed.
Then
Figure BDA0002753213740000119
In that
Figure BDA00027532137400001110
The complete expression above is as follows:
Figure BDA00027532137400001111
in this example, u1＝-0.3,u2＝0.05A,u3＝0.04A,v1＝-0.45,v2＝1.2A。
5) According to the symmetry principle, a complete meshing line S is formedj，j＝2,3,4,5。
Wherein the meshing line S3And the line of mesh S2With respect to OpCentrosymmetric, as shown in FIG. 5(b), S3The coordinates of the meshing line and its corresponding meshing angle have the following relationships:
Figure BDA00027532137400001112
meshing line S4And S3With respect to XpThe axis is vertically symmetrical, and as shown in FIG. 5(c), there are:
Figure BDA0002753213740000121
meshing line S5And S2With respect to XpThe axis is vertically symmetrical, and as shown in FIG. 5(c), there are:
Figure BDA0002753213740000122
6) and generating a tooth curve corresponding to the left rotor according to the meshing line and the meshing angle.For the line of engagement
Figure BDA0002753213740000123
Corresponding angle of engagement therewith
Figure BDA0002753213740000124
j-2, 3,4,5, tooth curve
Figure BDA0002753213740000125
For a complex curve, it can be derived from the following formula:
Figure BDA0002753213740000126
line of engagement
Figure BDA0002753213740000127
Corresponding tooth curve
Figure BDA0002753213740000128
Is E1G1A segment in which, among other things,
Figure BDA0002753213740000129
when the corresponding tooth curve is the tooth top arc E1F1(ii) a Line of engagement
Figure BDA00027532137400001210
Corresponding tooth curve
Figure BDA00027532137400001211
Is G1I1A segment in which, among other things,
Figure BDA00027532137400001212
when the corresponding tooth curve is a tooth root circular arc H1I1(ii) a In the same way, the line of engagement
Figure BDA00027532137400001213
Corresponding tooth curve
Figure BDA00027532137400001214
Is A1C1A segment in which, among other things,
Figure BDA00027532137400001215
when the corresponding tooth curve is the tooth root circular arc A1B1(ii) a Line of engagement
Figure BDA00027532137400001216
Corresponding tooth curve
Figure BDA00027532137400001217
Is C1E1A segment in which, among other things,
Figure BDA00027532137400001218
when the corresponding tooth curve is the tooth top arc D1E1(ii) a The central angle corresponding to each segment of circular arc is
Figure BDA00027532137400001219
Proper meshing of the complete left and right rotor profiles is achieved as shown in fig. 6. The rotor structure generated by the rotor profile of this example is shown in fig. 7.
The area utilization coefficient of the molded line can be calculated by using relevant mathematical software. Under the condition that the center distances are equal and the number n of teeth is 4, the area utilization coefficient of each molded line is as follows:
Figure BDA0002753213740000131
the molded line overcomes the defect of insufficient design parameter space of the original twisted-blade Roots molded line, breaks through the value limitation of the original molded line radial distance ratio, and can be designed to have larger addendum circle radius under the condition of certain center distance, so that the molded line has a larger area utilization coefficient. In the embodiment, compared with the area utilization coefficient of the original molded line under the limiting radial distance ratio, the area utilization coefficient of the molded line is higher than 0.048, so that the overall performance of the twisted-blade roots compressor is effectively improved, and the actual significance is good.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the technical solution of the present invention, and it should be understood by those skilled in the art that the technical solution can be modified and replaced by a plurality of simple modifications and replacements without departing from the spirit and principle of the present invention, and the modifications and replacements also fall within the protection scope defined by the claims.

Claims (10)

1. A twisted-blade Roots rotor is characterized in that: comprises a female rotor and a male rotor which have the same structure, and a single-tooth profile A of the end surface profile of the female rotor1B1C1D1E1F1G1H1I1By successive connected circular arc sections A1B1Composite curve segment B1C1Composite curve segment C1D1Circular arc segment D1E1Circular arc segment E1F1Composite curve segment F1G1Composite curve segment G1H1Circular arc section H1I1Composition is carried out; the end face profile of the female rotor is in a single-tooth form A1B1C1D1E1F1G1H1I1Origin O of wound rotor profile1Rotate
Figure FDA0002753213730000011
Then with A1B1C1D1E1F1G1H1I1Repeating the steps for n times in an end-to-end mode, wherein n represents the number of teeth of the molded line; the molded line origin of the male rotor molded line is O2The center distance A between the female rotor profile and the male rotor profile is equal to 2 pitch circle radii Rp(ii) a The two rotors can be correctly meshed in the rotating and meshing process, and the arc section A of the female rotor1B1Arc segment D of male rotor2E2Compound curve section B of female rotor engaged with each other1C1Compound curve segment C with male rotor2D2Composite curve section C of female rotor and meshed with each other1D1Compound curve segment B with male rotor2C2Arc segments D of female rotor engaged with each other1E1Arc segment A of male rotor2B2Arc sections E of female rotor engaged with each other1F1And the circular arc section H of the male rotor2I2Compound curve segment F of female rotor engaged with each other1G1Compound curve segment G with male rotor2H2Compound curve segment G of female rotor engaged with each other1H1Compound curve segment F with male rotor2G2Are correspondingly engaged with each other, and the arc sections H of the female rotors1I1Arc section E of male rotor2F2Are correspondingly engaged with each other.
2. The twisted roots rotor of claim 1, wherein: generating meshing line generatrix, then transforming the meshing line generatrix after dispersion to obtain a new complete meshing line, solving a corner relation corresponding to the new meshing line by utilizing a tooth profile meshing principle, and finally obtaining a single tooth profile with complete molded lines of the end faces of the female rotor and the male rotor through coordinate transformation.
3. The twisted roots rotor of claim 2, wherein: point a on the meshing line2Arc segment A corresponding to female rotor1B1、I2H2Arc segment D of male rotor2E2、E2F2At position XpOpYpMeshing line segment S of first quadrant of coordinate system2Compound curve segment B corresponding to female rotor1C1Compound curve segment C with male rotor2D2At position XpOpYpMeshing line segment S of the third quadrant of the coordinate system3Compound curve segment C corresponding to female rotor1D1Compound curve segment B with male rotor2C2Point d on the meshing line2Arc segment D corresponding to female rotor1E1、E1F1Arc segment A of male rotor2B2、I2H2At position XpOpYpMeshing line segment S of the second quadrant of the coordinate system4Composite curve segment F corresponding to female rotor1G1Compound curve segment H with male rotor2G2At position XpOpYpMeshing line segment S of the third quadrant of the coordinate system5Corresponding to the compound curve segment G of the female rotor1H1Compound curve segment G with male rotor2F2。
4. A method of designing a twisted roots rotor as claimed in any one of claims 1 to 3, comprising the steps of:
1) determining pitch circle radius RpTarget radial to pitch ratio
Figure FDA0002753213730000021
And a number of teeth n;
2) generating meshing line bus S according to the parameters determined in the step 1)1；
3) To mesh the line bus S1Performing transformation to obtain a meshing line S2And making the radial-to-radial ratio take on the value
Figure FDA0002753213730000022
Increase to
Figure FDA0002753213730000023
4) Using cubic spline curve to mesh line S2Fitting is performed to construct
Figure FDA0002753213730000024
About
Figure FDA0002753213730000025
Of a multi-segment fitting function
Figure FDA0002753213730000026
5) According to the symmetry principle, a complete meshing line S is formedj,j＝2,3,4,5；
6) And generating end face molded lines corresponding to the female rotor and the male rotor according to the meshing line and the meshing angle.
5. The design method according to claim 4, wherein the specific steps of the step 2) are as follows:
the pitch circle tangent point of the male and female rotors is set as OpIn the coordinate system XpOpYpMiddle, meshing line bus
Figure FDA0002753213730000027
The parameter equation of (1) is as follows:
Figure FDA0002753213730000028
wherein:
Figure FDA0002753213730000029
Figure FDA00027532137300000210
Rr、θLMis an intermediate variable formed by the pitch circle radius RpIs a center distance A ═ O1O2Half of l, get:
Figure FDA0002753213730000031
6. the design method according to claim 4, wherein the specific steps of the step 3) are as follows:
let S1Right end point a1Highest point b1Left end point and OpAre superposed and respectively paired with a1b1、b1OpAfter the segments are dispersed, normalization processing is carried out, and t is set to be the element [0,1 ∈]For discrete points numbered i
Figure FDA0002753213730000032
Comprises the following steps:
Figure FDA0002753213730000033
each discrete point is replaced by the following method to obtain a new meshing line S2Each point on
Figure FDA0002753213730000034
Figure FDA0002753213730000035
Wherein:
Figure FDA0002753213730000036
wherein u is1、v1As a scaling factor, u2、u3、v2Is a position parameter;
Figure FDA0002753213730000041
7. the design method according to claim 4, wherein the specific steps of the step 4) are as follows:
Figure FDA0002753213730000042
about
Figure FDA0002753213730000043
Of a multi-segment fitting function
Figure FDA0002753213730000044
The expression is as follows:
Figure FDA0002753213730000045
wherein, c1,k,c2,k,c3,k,c4,kPolynomial coefficients of the k-th section;
obtaining a fitting curve
Figure FDA0002753213730000046
Then, for S2Solving a fitting curve S between the left endpoint and the right endpoint2Corresponding angle of engagement
Figure FDA0002753213730000047
In that
Figure FDA0002753213730000048
The complete expression above is as follows:
Figure FDA0002753213730000049
Figure FDA00027532137300000410
namely the arc section A of the female rotor1B1Corresponding to the central angle.
8. The design method according to claim 4, wherein the specific steps of the step 5) are as follows:
meshing line S3And the line of mesh S2With respect to OpCentral symmetry，S3The coordinates of the meshing line and the corresponding meshing angle have the following relations:
Figure FDA00027532137300000411
meshing line S4And S3With respect to XpThe axis is symmetrical up and down, and comprises:
Figure FDA00027532137300000412
meshing line S5And S2With respect to XpThe axis is symmetrical up and down, and comprises:
Figure FDA0002753213730000051
9. the design method according to claim 4, wherein the specific steps of the step 6) are as follows: for the
Figure FDA0002753213730000052
Corresponding angle of engagement therewith
Figure FDA0002753213730000053
Curve of tooth
Figure FDA0002753213730000054
For the composite curve, it is given by:
Figure FDA0002753213730000055
10. a compressor or expander, characterized by: use of a twisted roots rotor as claimed in any one of claims 1 to 3.
CN202011192721.9A 2020-10-30 2020-10-30 Twisted-blade Roots rotor and design method thereof, compressor and expander Active CN112555152B (en)

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Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
GB282752A (en) * 1926-12-30 1928-05-31 Josef Kozousek Improvements in rotary machines for compressing and conveying liquids
CN203321827U (en) * 2013-05-31 2013-12-04 上海齐耀螺杆机械有限公司 High-efficiency double screw rod compressor rotor line
CN107120273A (en) * 2017-06-19 2017-09-01 中南大学 The molded line and processing technology of a kind of Roots's mechanical supercharger Twisted impeller
CN108757448A (en) * 2018-07-12 2018-11-06 中国石油大学(华东) Three leaf sectional circular camber roots rotors of one kind and its Profile Design method
CN110762004A (en) * 2019-11-01 2020-02-07 西安交通大学 Asymmetric elliptic twisted-blade roots rotor, compressor and expander

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
GB282752A (en) * 1926-12-30 1928-05-31 Josef Kozousek Improvements in rotary machines for compressing and conveying liquids
CN203321827U (en) * 2013-05-31 2013-12-04 上海齐耀螺杆机械有限公司 High-efficiency double screw rod compressor rotor line
CN107120273A (en) * 2017-06-19 2017-09-01 中南大学 The molded line and processing technology of a kind of Roots's mechanical supercharger Twisted impeller
CN108757448A (en) * 2018-07-12 2018-11-06 中国石油大学(华东) Three leaf sectional circular camber roots rotors of one kind and its Profile Design method
CN110762004A (en) * 2019-11-01 2020-02-07 西安交通大学 Asymmetric elliptic twisted-blade roots rotor, compressor and expander

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