CN112483405B - Variable cross-section vortex tooth consisting of algebraic spiral and molded line design method thereof - Google Patents

Variable cross-section vortex tooth consisting of algebraic spiral and molded line design method thereof Download PDF

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CN112483405B
CN112483405B CN202011373556.7A CN202011373556A CN112483405B CN 112483405 B CN112483405 B CN 112483405B CN 202011373556 A CN202011373556 A CN 202011373556A CN 112483405 B CN112483405 B CN 112483405B
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algebraic spiral
algebraic
curve
wall
spiral
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CN112483405A (en
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张朋成
彭斌
张宇波
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Lanzhou University of Technology
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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
    • F04C18/02—Rotary-piston pumps specially adapted for elastic fluids of arcuate-engagement type, i.e. with circular translatory movement of co-operating members, each member having the same number of teeth or tooth-equivalents
    • F04C18/0207—Rotary-piston pumps specially adapted for elastic fluids of arcuate-engagement type, i.e. with circular translatory movement of co-operating members, each member having the same number of teeth or tooth-equivalents both members having co-operating elements in spiral form
    • F04C18/0246—Details concerning the involute wraps or their base, e.g. geometry
    • 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
    • F04C29/00—Component parts, details or accessories of pumps or pumping installations, not provided for in groups F04C18/00 - F04C28/00
    • 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
    • F04C2240/00—Components
    • F04C2240/20—Rotors

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  • Mechanical Engineering (AREA)
  • General Engineering & Computer Science (AREA)
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Abstract

The invention belongs to the field of compressors, and particularly relates to a variable cross-section scroll wrap composed of algebraic spirals and a molded line design method thereof. In particular toThe technical scheme is as follows: a variable cross-section scroll wrap comprises a first scroll wrap and a second scroll wrap; the molded lines of the first scroll wrap and the second scroll wrap are the same and are respectively composed of algebraic spirals with two different polar diameters and equidistant curves of the algebraic spirals. The algebraic spiral part in the molded line sequentially comprises a polar diameter of rho1、ρ2、ρ1Part of the curve of the algebraic spiral of (1). The invention provides a novel vortex molded line and a construction method thereof; the advantages of the traditional algebraic spiral gradual-change wall thickness molded line and variable-section vortex molded line are considered, the high compression ratio can be realized by adopting fewer turns, and the retention time of the gas working medium in the working cavity is effectively reduced. In practice, according to different requirements, the method provided by the invention can be used for simply calculating, and then a series of geometric models of algebraic spiral variable cross-section vortex teeth can be quickly established.

Description

Variable cross-section vortex tooth consisting of algebraic spiral and molded line design method thereof
Technical Field
The invention belongs to the field of compressors, and particularly relates to a variable cross-section scroll wrap composed of algebraic spirals and a molded line design method thereof.
Background
The scroll compressor is a novel positive displacement compressor following a reciprocating compressor, a rotor compressor and a screw compressor; the compressor has the characteristics of compact structure, low noise, stable compression and good sealing performance, so the compressor is unique in a plurality of compressors and is widely applied to the fields of food, medical treatment and the like. The molded line of the scroll wrap directly affects the working performance of the scroll compressor, so the molded line design of the scroll wrap is extremely critical.
The existing variable-wall-thickness vortex molded lines can be divided into variable-wall-thickness vortex molded lines and variable-section vortex molded lines. Compared with the vortex model with the equal cross section, the vortex model with the gradually-changed wall thickness has the advantages of high area utilization rate, high internal volume ratio and the like, when the diameters of the vortex disks are the same, the air displacement is improved by 10-15%, the internal volume ratio is increased by 15-20%, and the vortex model has better comprehensive performance. The variable cross-section vortex model can realize high compression ratio/expansion ratio by adopting fewer turns, and the retention time of the gas working medium in the working cavity is reduced. Therefore, establishing a high-efficiency, practical and easy-to-machine variable-wall-thickness vortex model is always a research hotspot.
Disclosure of Invention
The invention aims to provide a variable cross-section scroll wrap consisting of algebraic spirals and a method for designing a molded line of the variable cross-section scroll wrap.
In order to achieve the purpose of the invention, the technical scheme adopted by the invention is as follows: a variable cross-section scroll wrap comprises a first scroll wrap and a second scroll wrap; the molded lines of the first scroll wrap and the second scroll wrap are the same and are respectively composed of algebraic spirals with different polar diameters and equidistant curves of the algebraic spirals.
Preferably, the first and second scroll wraps have the same profile and are formed by algebraic spirals of two different polar diameters and equidistant curves thereof.
Preferably, in the molded line, the algebraic spiral part sequentially has a polar diameter of rho1、ρ2、ρ1Part of the curve of the algebraic spiral of (1).
Preferably, the first scroll wrap is defined by a first outer wall curve A0A3And a second inner wall curve B0B＇4Forming; said second wrap being defined by a second outer wall curve A0A＇3And a first inner wall curve B0B4Forming; the centers of base circles of the first and second scroll wraps are spaced from each other by a distance RorAnd (5) carrying out installation.
Preferably, the first outer wall curve A0A3From A0A1、A1A2、A2A3Sequentially connected from inside to outside; a near the center of the base circle0A1Is a polar diameter of rho1A part of an algebraic spiral of, A1A2Is a polar diameter of rho2A part of an algebraic spiral of, A2A3Is a polar diameter of rho1A portion of the algebraic spiral of (a);
the first outer wall curve A0A3The drawing method is as follows:
A0A1the drawing formula of (1) is as follows:
Figure BDA0002807529070000021
A1A2the drawing formula of (1) is as follows:
Figure BDA0002807529070000022
A2A3the drawing formula of (1) is as follows:
Figure BDA0002807529070000023
where ρ is1、ρ2The diameter of the electrode is the same as the diameter of the electrode,
Figure BDA0002807529070000024
in order to be the polar angle, the angle of the pole,
Figure BDA0002807529070000025
is the terminal polar angle, k is the helical coefficient, n1、n2Is a positive integer and is a non-zero integer,
Figure BDA0002807529070000026
is an arbitrary value, phi1、φ2Is an included angle; the following formula is satisfied between the parameters:
Figure BDA0002807529070000027
Figure BDA0002807529070000028
Figure BDA0002807529070000031
Figure BDA0002807529070000032
Figure BDA0002807529070000033
preferably, theFirst inner wall curve B0B4From inside to outside in turn from B0B1、B1B2、B2B4Sequentially connected to form; b is0B1Is a polar diameter of rho1A part of the equidistant curve of the algebraic spiral of (A), B1B2Is a polar diameter of rho2A part of the equidistant curve of the algebraic spiral of (A), B2B4Is a polar diameter of rho1Part of an equidistant curve of an algebraic spiral of (a).
Preferably, the first inner wall curve B0B4The drawing method is as follows:
B0B1the drawing formula of (1) is as follows:
Figure BDA0002807529070000034
wherein R isorThe distance between the centers of the base circles of the first scroll wrap and the second scroll wrap is;
B1B2the drawing formula of (1) is as follows:
Figure BDA0002807529070000035
B2B4the drawing formula of (1) is as follows:
Figure BDA0002807529070000036
where ρ is1、ρ2The diameter of the electrode is the same as the diameter of the electrode,
Figure BDA0002807529070000041
in order to be the polar angle, the angle of the pole,
Figure BDA0002807529070000042
is the terminal polar angle, k is the helical coefficient, n1、n2Is a positive integer and is a non-zero integer,
Figure BDA0002807529070000043
is an arbitrary value, phi1、φ2Is an included angle; the following formula is satisfied between the parameters:
Figure BDA0002807529070000044
Figure BDA0002807529070000045
Figure BDA0002807529070000046
Figure BDA0002807529070000047
Figure BDA0002807529070000048
preferably, said second inner wall curve B ″, is0B＇4The drawing method is as follows:
(1) curve A of the first outer wall0A3Translating inwards and inwards at equal intervals to obtain a first initial inner wall curve B of the second vortex tooth0B3；
(2) The first initial inner wall curve B0B3By rotation of pi about the centre of the base circle, the second initial wall curve B' of the first wrap0B＇3；
(3) Deleting the second initial wall curve B0B＇3At an outermost 1/2 turns, giving a second inner wall curve B0B＇4。
Preferably, said second outer wall curve A0A＇3The drawing method is as follows:
curve A of the first outer wall0A3Rotate pi around the center of the base circle to obtain the secondOuter wall curve A0A＇3。
Correspondingly, the design method of the variable cross-section scroll wrap is characterized in that: the method comprises the following steps:
(1) drawing the polar diameter as rho1、ρ2、ρ1Algebraic spiral S1、S2、S3The equation is as follows:
S1：
Figure BDA0002807529070000051
S2：
Figure BDA0002807529070000052
S3：
Figure BDA0002807529070000053
ρ1、ρ2the diameter of the electrode is the same as the diameter of the electrode,
Figure BDA0002807529070000054
in order to be the polar angle, the angle of the pole,
Figure BDA0002807529070000055
is the terminal polar angle, k is the helical coefficient, n1、n2Is a positive integer and is a non-zero integer,
Figure BDA0002807529070000056
is an arbitrary value, phi1、φ2Is an included angle;
Figure BDA0002807529070000057
algebraic spiral S1Has an endpoint of A0、A1Algebraic spiral S3The endpoints of (a) and (b).
(2) Will algebraic spiral S2Rotate counterclockwise around the center of the base circle
Figure BDA0002807529070000058
Or rotate clockwise
Figure BDA0002807529070000059
To obtain an algebraic spiral S4Algebraic spiral S4Has an endpoint of A1、a；
Algebraic spiral S4The equation of (1) is:
Figure BDA00028075290700000510
(3) will algebraic spiral S3Rotate clockwise around point a by phi1To obtain an algebraic spiral S5Algebraic spiral S5The end points of (a) and (c);
algebraic spiral S5The equation of (1) is:
Figure BDA00028075290700000511
wherein phi is1Is an algebraic spiral S3And S4In A1Tangent slope of point, k2Is an algebraic spiral S4In A1The tangent slope of the point is calculated by the following formula:
Figure BDA0002807529070000061
Figure BDA0002807529070000062
(4) as shown in fig. 4, the algebraic spiral S4Around A1Point counterclockwise rotation phi2To obtain an algebraic spiral S6；
Will algebraic spiral S5Around A1Point counterclockwise rotation phi2To obtain an algebraic spiral S7(ii) a Algebraic spiral S6Has an endpoint of A1、A2Algebraic spiral S7Has an endpoint of A2、A3(ii) a Algebraic spiral S6Algebraic spiral S7The equations of (a) are:
S6：
Figure BDA0002807529070000063
S7：
Figure BDA0002807529070000064
wherein phi is2Is an algebraic spiral S1And S4In A1The included angle of the point tangent line is calculated by the formula:
Figure BDA0002807529070000065
wherein k is3Is an algebraic spiral S1In A1Tangent slope of point, k4Is an algebraic spiral S4In A1The tangent slope of the point is calculated by the following formula:
Figure BDA0002807529070000066
Figure BDA0002807529070000067
(5) algebraic spiral S1、S6And S7Form a first outer wall curve A0A3Curve A of the first outer wall0A3Translating inwards and inwards at equal intervals to obtain a first initial inner wall curve B of the second vortex tooth0B3，RorThe distance between the centers of the base circles of the first scroll wrap and the second scroll wrap;
(6) respectively curve the first outer wall0A3First initial inner wall curve B0B3Rotate pi around the center of the base circle to respectively obtainSecond outer wall curve A0A＇3And a second initial inner wall curve B' of the first wrap0B＇3；
(7) Respectively deleting the first initial inner wall curves B0B3And a second initial inner wall curve B0B＇3To obtain a first inner wall curve B0B4And a second inner wall curve B0B＇4；
(8) Curve A of the first outer wall0A3With a second inner wall curve B0B＇4Connected to form a first scroll wrap; curve B of the first inner wall0B4Curve A' with the second outer wall0A＇3Joined to form a second wrap.
The invention has the following beneficial effects: the invention provides a novel vortex molded line and a construction method thereof; the advantages of the traditional algebraic spiral gradual-change wall thickness molded line and variable-section vortex molded line are considered, the high compression ratio can be realized by adopting fewer turns, and the retention time of the gas working medium in the working cavity is effectively reduced. In practice, according to different requirements, the method provided by the invention can be used for simply calculating, and then a series of geometric models of the involute variable-section scroll wrap can be quickly established. The scroll wrap profile provided by the invention is simple in composition and easy to process.
Drawings
FIG. 1 shows the radial dimension of the pole as ρ1、ρ2、ρ1Algebraic spiral S1、S2、S3A schematic diagram;
FIG. 2 shows an algebraic spiral S4A schematic diagram;
FIG. 3 shows an algebraic spiral S5A schematic diagram;
FIG. 4 shows an algebraic spiral S6、S7A schematic diagram;
FIG. 5 is a first outer wall curve A0A3First initial inner wall curve B0B3A schematic diagram;
FIG. 6 is a second outside wall curve A ″0A＇3Second initial inner wall curve B0B＇3A schematic diagram;
FIG. 7 is a first inner wall curve B0B4Second inner wall curve B0B＇4A schematic diagram;
FIG. 8 shows the distance R between the centers of the first and second scroll tooth base circlesorAnd (5) installation schematic diagram.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Unless otherwise specified, the technical means used in the examples are conventional means well known to those skilled in the art.
In the description of the present invention, it is to be understood that the terms "longitudinal", "lateral", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", and the like, indicate orientations or positional relationships based on those shown in the drawings, are merely for convenience of description of the present invention, and do not indicate or imply that the referenced devices or elements must have a particular orientation, be constructed and operated in a particular orientation, and thus, are not to be construed as limiting the present invention.
The invention provides a variable cross-section scroll wrap, which consists of a first scroll wrap and a second scroll wrap; the molded lines of the first scroll wrap and the second scroll wrap are the same and are respectively composed of 2 algebraic spirals with different polar diameters and equidistant curves of the algebraic spirals.
The invention provides a profile of a variable cross-section scroll compressor, which consists of algebraic spirals and equidistant curves thereof. After the inventors performed a number of mapping analyses, they found, by chance, that: under a certain condition, the pole diameter is rho1Two points on the algebraic spiral can always have a polar diameter of rho2Find two points on the algebraic spiral of (1) to make them equally distant. Based on the above, the mutual substitution rule is deduced reversely, and the invention is obtained. After the baseline is obtained, a novel variable truncated vortex model is constructed by using an equidistance method. The derivation procedure is as follows:
Figure BDA0002807529070000081
the scroll tooth profile design is described in detail below:
referring to FIGS. 1-8, the first wrap is defined by a first outer wall curve A0A3And a second inner wall curve B0B＇4Forming; said second wrap being defined by a second outer wall curve A0A＇3And a first inner wall curve B0B4Forming; the centers of base circles of the first and second scroll wraps are spaced from each other by a distance RorAnd (5) carrying out installation.
1. As shown in fig. 5, the first outer wall curve a is centered on the base circle0A3Rho is respectively formed by the pole diameters from inside to outside in sequence1、ρ2、ρ1A part of the algebraic spiral of (a) is formed in succession. Wherein, the first section of curve A closest to the center of the base circle0A1Is a polar diameter of rho1A part of the algebraic spiral of (1), a second curve A1A2Is a polar diameter of rho2A part of the algebraic spiral of (1), a third section curve A2A3Is a polar diameter of rho1Is part of the algebraic spiral of (a). Rho1、ρ2、ρ1Respectively corresponding algebraic spiral S1、S2、S3As shown in fig. 1. In addition, S is1、S2、S3Are all only part of the corresponding algebraic spiral, S1And S3In fact different curve segments on an algebraic spiral.
First section curve A0A1The drawing formula of (1) is as follows:
Figure BDA0002807529070000091
second curve A1A2The drawing formula of (1) is as follows:
Figure BDA0002807529070000092
third section curve A2A3The drawing formula of (1) is as follows:
Figure BDA0002807529070000093
where ρ is1、ρ2The diameter of the electrode is the same as the diameter of the electrode,
Figure BDA0002807529070000094
in order to be the polar angle, the angle of the pole,
Figure BDA0002807529070000095
is the terminal polar angle, k is the helical coefficient, n1、n2Is a positive integer and is a non-zero integer,
Figure BDA0002807529070000096
is an arbitrary value, phi1、φ2Is an included angle. The following formula is satisfied between the parameters:
Figure BDA0002807529070000097
Figure BDA0002807529070000098
Figure BDA0002807529070000099
Figure BDA00028075290700000910
Figure BDA0002807529070000101
where the above parameters are referred to throughout, the same shall apply.
2. As shown in fig. 7, the first inner wall curve B0B4Sequentially from inside to outside, the polar diameter is rho1、ρ2、ρ1Is formed by connecting a part of equidistant curves of the algebraic spiral. Wherein, the first section of curve B closest to the center of the base circle0B1Is a polar diameter of rho1A part of equidistant curves of the algebraic spiral of (A), a second segment of curve B1B2Is a polar diameter of rho2A part of the equidistant curve of the algebraic spiral of (a), a third section of curve B2B4Is a polar diameter of rho1Part of an equidistant curve of an algebraic spiral of (a).
First section curve B0B1The drawing formula of (1) is as follows:
Figure BDA0002807529070000102
wherein R isorIs the distance between the centers of the base circles of the first and second scroll wraps.
Second curve B1B2The drawing formula of (1) is as follows:
Figure BDA0002807529070000103
third section curve B2B4The drawing formula of (1) is as follows:
Figure BDA0002807529070000104
3. the design method of the variable cross-section scroll wrap comprises the following steps:
(1) as shown in FIG. 1, the respective pole diameters are plotted as ρ1、ρ2、ρ1Algebraic spiral S1、S2、S3The equation is as follows:
S1：
Figure BDA0002807529070000111
S2：
Figure BDA0002807529070000112
S3：
Figure BDA0002807529070000113
ρ1、ρ2the diameter of the electrode is the same as the diameter of the electrode,
Figure BDA0002807529070000114
in order to be the polar angle, the angle of the pole,
Figure BDA0002807529070000115
is the terminal polar angle, k is the helical coefficient, n1、n2Is a positive integer and is a non-zero integer,
Figure BDA0002807529070000116
is an arbitrary value, phi1、φ2Is an included angle.
And satisfies the following conditions:
Figure BDA0002807529070000117
algebraic spiral S1Has an endpoint of A0、A1Algebraic spiral S3The endpoints of (a) and (b).
(2) As shown in fig. 2, the algebraic spiral S2Rotate counterclockwise around the center of the base circle
Figure BDA0002807529070000118
Or rotate clockwise
Figure BDA0002807529070000119
To obtain an algebraic spiral S4Algebraic spiral S4Has an endpoint of A1And a. Algebraic spiral S4The equation of (1) is:
Figure BDA00028075290700001110
(3) as shown in fig. 3, the algebraic spiral S3Rotate clockwise around point a by phi1To obtain an algebraic spiral S5Algebraic spiral S5The endpoints of (a) and (c). Algebraic spiral S5The equation of (1) is:
Figure BDA00028075290700001111
wherein phi is1Is an algebraic spiral S3And S4In A1Tangent slope of point, k2Is an algebraic spiral S4In A1The tangent slope of the point is calculated by the following formula:
Figure BDA0002807529070000121
Figure BDA0002807529070000122
(4) as shown in fig. 4, the algebraic spiral S4Around A1Point counterclockwise rotation phi2To obtain an algebraic spiral S6. Will algebraic spiral S5Around A1Point counterclockwise rotation phi2To obtain an algebraic spiral S7. Algebraic spiral S6Has an endpoint of A1、A2Algebraic spiral S7Has an endpoint of A2、A3. Algebraic spiral S6Algebraic spiral S7The equations of (a) are:
S6：
Figure BDA0002807529070000123
S7：
Figure BDA0002807529070000124
wherein the content of the first and second substances,φ2is an algebraic spiral S1And S4In A1The included angle of the point tangent line is calculated by the formula:
Figure BDA0002807529070000125
wherein k is3Is an algebraic spiral S1In A1Tangent slope of point, k4Is an algebraic spiral S4In A1The tangent slope of the point is calculated by the following formula:
Figure BDA0002807529070000126
Figure BDA0002807529070000127
(5) algebraic spiral S1、S6And S7Form a first outer wall curve A0A3Curve A of the first outer wall0A3Inward normal equidistant translation RorObtaining a first initial inner wall curve B of the second wrap0B3，RorIs the distance between the centers of the base circles of the first and second wraps.
(6) Respectively curve the first outer wall0A3First initial inner wall curve B0B3Rotating pi around the center of the base circle to respectively obtain a second outer wall curve A0A＇3And a second initial inner wall curve B' of the first wrap0B＇3。
(7) Respectively deleting the first initial inner wall curves B0B3And a second initial inner wall curve B0B＇3To obtain a first inner wall curve B0B4And a second inner wall curve B0B＇4；
(8) Curve A of the first outer wall0A3With a second inner wall curve B0B＇4Connecting to form a complete first scroll wrap; curve B of the first inner wall0B4Curve A' with the second outer wall0A＇3And joined to form a complete second wrap.
The variable cross-section scroll wrap can be applied to scroll compressors, scroll expanders, scroll vacuum pumps and similar devices requiring the use of molded lines.
The above-described embodiments are merely illustrative of the preferred embodiments of the present invention, and do not limit the scope of the present invention, and various changes, modifications, alterations, and substitutions which may be made by those skilled in the art without departing from the spirit of the present invention shall fall within the protection scope defined by the claims of the present invention.

Claims (1)

1. A design method of a variable cross-section scroll wrap is characterized in that: the method comprises the following steps:
(1) drawing the polar diameter as rho1、ρ2、ρ1Algebraic spiral S1、S2、S3The equation is as follows:
S1：
Figure FDA0003318220440000011
S2：
Figure FDA0003318220440000012
S3：
Figure FDA0003318220440000013
ρ1、ρ2the diameter of the electrode is the same as the diameter of the electrode,
Figure FDA0003318220440000014
in order to be the polar angle, the angle of the pole,
Figure FDA0003318220440000015
is the terminal polar angle, k is the helical coefficient, n1、n2Is a positive integer and is a non-zero integer,
Figure FDA0003318220440000016
is an arbitrary value, phi1、φ2Is an included angle;
Figure FDA0003318220440000017
algebraic spiral S1Has an endpoint of A0、A1Algebraic spiral S3The end points of (A) and (B);
(2) will algebraic spiral S2Rotate counterclockwise around the center of the base circle
Figure FDA0003318220440000018
Or rotate clockwise
Figure FDA0003318220440000019
To obtain an algebraic spiral S4Algebraic spiral S4Has an endpoint of A1、a；
Algebraic spiral S4The equation of (1) is:
Figure FDA00033182204400000110
(3) will algebraic spiral S3Rotate clockwise around point a by phi1To obtain an algebraic spiral S5Algebraic spiral S5The end points of (a) and (c);
algebraic spiral S5The equation of (1) is:
Figure FDA00033182204400000111
wherein k is1Is an algebraic spiral S3And S4In A1Tangent slope of point, k2Is an algebraic screwWire S4In A1The tangent slope of the point is calculated by the following formula:
Figure FDA0003318220440000021
Figure FDA0003318220440000022
(4) will algebraic spiral S4Around A1Point counterclockwise rotation phi2To obtain an algebraic spiral S6；
Will algebraic spiral S5Around A1Point counterclockwise rotation phi2To obtain an algebraic spiral S7(ii) a Algebraic spiral S6Has an endpoint of A1、A2Algebraic spiral S7Has an endpoint of A2、A3(ii) a Algebraic spiral S6Algebraic spiral S7The equations of (a) are:
S6：
Figure FDA0003318220440000023
S7：
Figure FDA0003318220440000024
wherein phi is2Is an algebraic spiral S1And S4In A1The included angle of the point tangent line is calculated by the formula:
Figure FDA0003318220440000025
wherein k is3Is an algebraic spiral S1In A1Tangent slope of point, k4Is an algebraic spiral S4In A1The tangent slope of the point is calculated by the following formula:
Figure FDA0003318220440000026
Figure FDA0003318220440000027
(5) algebraic spiral S1、S6And S7Form a first outer wall curve A0A3Curve A of the first outer wall0A3Inward normal equidistant translation RorObtaining a first initial inner wall curve B of the second wrap0B3，RorThe distance between the centers of the base circles of the first scroll wrap and the second scroll wrap;
(6) respectively curve the first outer wall0A3First initial inner wall curve B0B3Rotating pi around the center of the base circle to respectively obtain a second outer wall curve A0A＇3And a second initial inner wall curve B' of the first wrap0B＇3；
(7) Respectively deleting the first initial inner wall curves B0B3And a second initial inner wall curve B0B＇3To obtain a first inner wall curve B0B4And a second inner wall curve B0B＇4；
(8) Curve A of the first outer wall0A3With a second inner wall curve B0B＇4Connected to form a first scroll wrap; curve B of the first inner wall0B4Curve A' with the second outer wall0A＇3Joined to form a second wrap.
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