CN115822963A - Multi-tooth Roots rotor based on Bezier curve design line and design method thereof - Google Patents

Multi-tooth Roots rotor based on Bezier curve design line and design method thereof Download PDF

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CN115822963A
CN115822963A CN202211528930.5A CN202211528930A CN115822963A CN 115822963 A CN115822963 A CN 115822963A CN 202211528930 A CN202211528930 A CN 202211528930A CN 115822963 A CN115822963 A CN 115822963A
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point
tooth
curve segment
segment
profile
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李丹童
何志龙
马凯
齐天宇
谌小倩
高立刚
王闯
邢子文
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Xian Jiaotong University
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Abstract

A multi-tooth Roots rotor based on a Bezier curve design profile and a design method thereof are provided, the multi-tooth Roots rotor comprises two rotors with the same Roots profile, and the two rotors have different rotation directions and are meshed with each other; the single tooth profile of the Roots type line consists of a symmetrical upper half curve segment ABCDE and a symmetrical lower half curve segment AB 1 C 1 D 1 E 1 Splicing to obtain the finished product; the upper half part curve segment ABCDE consists of a tooth top curve segment and a tooth bottom curve segment, the tooth top curve segment is a combined curve segment of a tooth top arc segment AB and a Bezier curve segment BC, the tooth bottom curve segment consists of a curve segment CD and a tooth root arc segment DE, and the curve segment CD is an enveloping curve segment of the Bezier curve segment BC; and rotating and splicing the single-tooth profile of the Roots type line to form a complete Roots type line. The multi-tooth Roots rotor has larger adjustable space, so that the performance optimization space based on the conventional molded line structure is larger, and the multi-tooth Roots rotor can meet different requirementsOptimal geometry regulation of (a).

Description

Multi-tooth Roots rotor based on Bezier curve design line and design method thereof
Technical Field
The invention belongs to the field of Roots machinery, and particularly relates to a multi-tooth Roots rotor with a designed profile based on a Bezier curve and a design method thereof.
Background
The roots machine is a positive displacement rotary machine with forced suction and exhaust functions, and is widely applied to modern industries as a blower or a vacuum pump. Compared with other positive displacement machines, the positive displacement machine has the characteristics of reliable operation, low cost, no wearing parts such as an air valve, compatibility with liquid, high operation efficiency and the like. The core parts of the Roots machine are a pair of rotors meshed with each other, the key design factor of the rotors is the rotor profile, and the design of the profile directly determines the thermal performance of the whole machine.
In order to ensure that the two rotors are in a meshing relationship, the conventional rotor profile mostly adopts a circular arc, a circular arc envelope curve, an involute curve, a cycloid curve and other curve forms, but the design space of the above curve forms is small, and the adjustable space of the rotor shape is limited, so that the performance optimization space based on the conventional profile structure is small, and the optimal geometric shape regulation and control for meeting different requirements cannot be obtained.
Disclosure of Invention
The invention aims to solve the problems in the prior art and provides a multi-tooth Roots rotor based on a Bezier curve design profile and a design method thereof, wherein the high-flexibility adjustment of the Roots rotor profile is realized by adjusting parameters of the Bezier curve.
In order to achieve the above object, the present invention has at least the following advantageous effects:
a multi-tooth Roots rotor based on a Bezier curve design profile comprises two rotors with the same Roots profile, wherein the two rotors have different rotation directions and are meshed with each other; the single tooth profile of the Roots profile is pairedThe upper half curve segment ABCDE and the lower half curve segment AB of the scale 1 C 1 D 1 E 1 Splicing to obtain the finished product; the upper half part curve segment ABCDE consists of a tooth top curve segment and a tooth bottom curve segment, the tooth top curve segment is a combined curve segment of a tooth top arc segment AB and a Bezier curve segment BC, the tooth bottom curve segment consists of a curve segment CD and a tooth root arc segment DE, and the curve segment CD is an envelope curve segment of the Bezier curve segment BC; and rotating and splicing the single-tooth profile of the Roots profile to form the complete Roots profile.
As a preferred solution, the position vector of the point on the bezier curve segment BC is calculated according to the following formula:
Figure BDA0003973879750000021
in the formula, r BC Representing a position vector, r, of a point on the Bezier curve segment BC B And r C Respectively representing the position vectors of point B and point C, r P1 And r P2 Respectively represent points P 1 And point P 2 A position vector of (a);
point P 1 And point P 2 Are respectively taken from the line segments BP 0 And the line segment CP 0 At point above, the direction vector expression is as follows:
Figure BDA0003973879750000022
in the formula i 1 And i 2 Are respectively a point P 1 And point P 2 The position parameter of (2) is a design parameter; r is P0 Is a point P 0 Position vector of (1), point P 0 Is a straight line BP 0 And straight line CP 0 Cross point of, straight line BP 0 Passing through point B, the slope is determined by the tangential vector of addendum arc segment AB at point B, straight line CP 0 Through point C, the slope is determined by the direction vector at point C, which is determined by the programmable angle β, which is the angle between the tangent at point C and the x-axis.
As a preferred solution, the position vector at the point C is:
Figure BDA0003973879750000023
in the formula, r p Representing pitch circle radius, a represents center distance, and alpha is solved according to the following formula as a design parameter:
Figure BDA0003973879750000024
in the formula, m represents the number of teeth and is a design parameter;
the position vector at point B is:
r B ＝[R 2 cosγ R 2 sinγ]
in the formula, R 2 The radius of the addendum arc segment AB is represented, and γ represents the central rotation angle experienced by the addendum arc segment AB, which is a design parameter.
As a preferred scheme, after the Roots-type lines of the two rotors pass through an offset angle epsilon, the rotors rotate in different rotation directions and are meshed with each other;
the offset angle ε is determined by the following expression:
Figure BDA0003973879750000031
preferably, the curve segment CD is in a rectangular coordinate system O xy The calculation expression in (1) is:
Figure BDA0003973879750000032
in the formula, r CD A position vector representing a point on the curve segment CD; phi is an intermediate turning angle variable parameter and is obtained by the following relational expression:
τ BC (θ,φ)·[x s,BC (θ,φ)-r p y s,BC (θ,φ)] T ＝0
in the formula, τ BC A position vector in the stationary coordinate system representing a tangential vector of the bezier curve segment BC; r is a radical of hydrogen p Represents the pitch circle radius; (x) s,BC ，y s,BC ) A position vector representing the bezier curve segment BC in the rotating coordinate system is obtained from the following relation:
Figure BDA0003973879750000033
Figure BDA0003973879750000034
a design method of the multi-tooth Roots rotor based on the Bezier curve design line comprises the following steps:
determining the center distance A of the rotor and the arc radius R of the addendum arc section AB according to the air displacement and the sealing requirement 2 With the number m of teeth, and determining the pitch radius r p The position of point a, the position of point E and the position of point C;
determining a central rotation angle gamma experienced by the addendum circular-arc section AB according to the strength and the sealing requirement, determining the addendum circular-arc section AB, and further determining the position of the point B and the position of the point D;
according to design requirements, with the optimization goal of any one or two of the maximum area utilization rate and the shortest contact line length, selecting an included angle beta between a tangent line at a point C and an x axis and a point P 1 And point P 2 Position parameter i 1 And i 2 ；
Determining a Bezier curve segment BC according to the parameters, and solving a curve segment CD through a meshing theorem;
the curve section AB of the lower half part is obtained by the up-down symmetry of the curve section ABCDE of the upper half part of the single-tooth profile formed by combining the addendum arc section AB, the Bessel curve section BC, the curve section CD and the dedendum arc section DE 1 C 1 D 1 E 1 Splicing to form a complete single-tooth profile, and then rotationally splicing the complete single-tooth profile to form a complete Roots profile;
after two rotors with the same Roots profile pass through the offset angle epsilon, the two rotors rotate along different rotation directions and are meshed with each other.
As a preferable scheme, the included angle beta between the tangent line at the point C and the x axis and the point P are adjusted 1 And point P 2 Position parameter i 1 And i 2 And any one or more of the central rotation angles γ experienced by the addendum arc segment AB to achieve the adjustment of the rotor profile shape.
As a preferable scheme, the area utilization rate of the rotor profile is adjusted by adjusting the included angle beta between the tangent line at the point C and the x axis.
Compared with the prior art, the invention has the following beneficial effects:
the tooth crest part curve of the tooth form of the Roots rotor is formed by adopting the Bezier curve, the tooth root part curve is solved according to the meshing relation, and the high-flexibility adjustment of the molded line of the Roots rotor can be realized by adjusting the parameters of the Bezier curve, wherein the high-flexibility adjustment comprises the angle beta formed by the tangent line at the adjusting point C and the x axis and the point P 1 And point P 2 Position parameter i 1 And i 2 And any one or more of the central rotation angles γ experienced by the addendum arc segment AB may realize the adjustment of the rotor profile shape. The flexible adjustment of the tooth number of the rotor profile can be realized by adjusting the tooth number m. The adjustment of the area utilization rate of the rotor profile can be realized by adjusting the included angle beta between the tangent line at the point C and the x axis. The multi-tooth Roots rotor has larger adjustable space, so that the performance optimization space based on the conventional molded line structure is larger, and the optimal geometric shape regulation and control under different requirements can be obtained.
Drawings
FIG. 1 (a) is a schematic representation of a multi-tooth Roots rotor tooth profile based on a Bezier curve design profile of an embodiment of the present invention;
FIG. 1 (b) is a schematic view of a multi-tooth Roots rotor profile solving for a Bezier curve-based design profile according to an embodiment of the present invention;
FIG. 2 is a schematic view of a multi-tooth Roots rotor profile engagement process based on a Bezier curve design profile according to an embodiment of the present invention;
FIG. 3 (a) is a schematic view of the rotor profile of the present invention adjusted by adjusting the angle β between the tangent at point C and the x-axis;
FIG. 3 (b) by adjusting the point P 1 And point P 2 Position parameter i 1 And i 2 Adjusting the profile shape schematic diagram of the rotor of the invention;
FIG. 3 (c) is a schematic view illustrating the shape of the rotor profile according to the present invention by adjusting the center rotation angle γ experienced by the addendum arc segment AB;
FIG. 4 (a) is a schematic view of a rotor profile of the present invention with 3 teeth;
FIG. 4 (b) is a schematic view of a rotor profile of the present invention with 4 teeth;
FIG. 5 is a comparison graph of the area utilization of the conventional arc-shaped line and the line at different beta angles.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
Referring to fig. 1 (a) and fig. 1 (b), the embodiment of the invention provides a multi-tooth roots rotor based on a bezier curve design profile, wherein the high-flexibility adjustment of the roots rotor profile can be realized by adjusting parameters of the bezier curve, the multi-tooth roots rotor specifically comprises two rotors with the same roots profile, and the two rotors have different rotation directions and are meshed with each other; the single tooth profile of the Roots type line consists of a symmetrical upper half curve segment ABCDE and a symmetrical lower half curve segment AB 1 C 1 D 1 E 1 And (4) splicing to obtain the finished product. The upper half part curve segment ABCDE consists of a tooth top curve segment and a tooth bottom curve segment, the tooth top curve segment is a combined curve segment of a tooth top arc segment AB and a Bezier curve segment BC, the tooth bottom curve segment consists of a curve segment CD and a tooth root arc segment DE together, and the curve segment CD is an envelope curve segment of the Bezier curve segment BC; the single-tooth profiles of the Roots profile are rotated and spliced to form a complete and smooth-transition multi-tooth Roots rotor profile structure, as shown in FIG. 2, two identical Roots rotor profiles can complete a correct meshing relationship.
In this embodiment, the position vector of a point on the bezier curve segment BC is calculated as follows:
Figure BDA0003973879750000051
in the formula, r BC Representing a position vector, r, of a point on the Bezier curve segment BC B And r C Respectively representing the position vectors, r, of points B and C P1 And r P2 Respectively represent point P 1 And point P 2 A position vector of (a);
point P 1 And point P 2 Taken from line segments BP respectively 0 And the line segment CP 0 At point above, the direction vector expression is as follows:
Figure BDA0003973879750000052
in the formula i 1 And i 2 Are respectively a point P 1 And point P 2 The position parameter of (2) is a design parameter; r is a radical of hydrogen P0 Is a point P 0 Position vector of (1), point P 0 Is a straight line BP 0 And straight line CP 0 Cross point of, straight line BP 0 Passing through point B, the slope is determined by the tangential vector of addendum arc segment AB at point B, straight line CP 0 Through point C, the slope is determined by the direction vector at point C, which is determined by the programmable angle β, which is the angle between the tangent at point C and the x-axis.
The position vector at point C is:
Figure BDA0003973879750000061
in the formula, r p Representing pitch circle radius, a represents center distance, and alpha is solved according to the following formula as a design parameter:
Figure BDA0003973879750000062
in the formula, m represents the number of teeth and is a design parameter;
the position vector at point B is:
r B ＝[R 2 cosγ R 2 sinγ]
in the formula, R 2 The radius of the addendum arc segment AB is represented, and γ represents the central rotation angle experienced by the addendum arc segment AB, which is a design parameter.
After the Roots-type lines of the two rotors pass through an offset angle epsilon, the two rotors rotate along different rotation directions to complete a correct meshing relationship, and the offset angle epsilon is calculated by the following expression:
Figure BDA0003973879750000063
the curve segment CD is obtained by the meshing theorem and the curve segment BC is in the rectangular coordinate system O xy The calculation expression in (1) is:
Figure BDA0003973879750000064
in the formula, r CD A position vector representing a point on the curve segment CD; phi is an intermediate rotation angle variable parameter and is obtained by the following relational expression:
τ BC (θ,φ)[x s,BC (θ,φ)-r p y s,BC (θ,φ)] T ＝0
in the formula, τ BC A position vector in the stationary coordinate system representing a tangential vector of the bezier curve segment BC; r is p Represents the pitch circle radius; (x) s,BC ，y s,BC ) A position vector representing the bezier curve segment BC in the rotating coordinate system is obtained from the following relation:
Figure BDA0003973879750000071
Figure BDA0003973879750000072
the invention can realize the screwIn this embodiment, the shape of the rotor profile can be flexibly adjusted as shown in fig. 3 (a) by adjusting the angle β between the tangent line at the point C and the x-axis, and by adjusting the point P 1 And point P 2 Position parameter i 1 And i 2 The flexible adjustment of the shape of the rotor profile shown in fig. 3 (b) can be achieved, and the flexible adjustment of the shape of the rotor profile shown in fig. 3 (c) can be achieved by adjusting the central rotation angle γ experienced by the addendum arc segment AB.
The flexible adjustment of the number of teeth of the rotor profile can be realized by adjusting the number of teeth m, as shown in fig. 4 (a) and 4 (b). The flexible adjustment of the area utilization rate of the rotor profile shown in fig. 5 can be realized by adjusting the included angle beta between the tangent line at the point C and the x axis.
Another embodiment of the present invention further provides a method for designing a multi-tooth roots rotor based on a bezier curve design profile, including the following steps:
s1, determining a rotor center distance A and an arc radius R of an addendum arc section AB according to the requirements of air displacement and sealing performance 2 With the number m of teeth, and determining the pitch radius r p The position of point a, the position of point E and the position of point C.
And S2, determining a central rotation angle gamma experienced by the addendum circular-arc section AB according to the strength and the sealing requirement, determining the addendum circular-arc section AB, and further determining the position of the point B and the position of the point D.
S3, according to specific design requirements, selecting an included angle beta between a tangent line at a point C and an x axis and a point P by taking any one or two of the maximum area utilization rate and the shortest contact line length as an optimal target 1 And point P 2 Position parameter i 1 And i 2 。
S4, designing according to the parameters, wherein the Bezier curve segment BC is as follows:
Figure BDA0003973879750000073
in the formula, r BC Representing a position vector, r, of a point on the Bezier curve segment BC B And r C Representing points B and C, respectivelyPosition vector r P1 And r P2 Respectively represent point P 1 And point P 2 The position vector of (2).
Point P 1 And point P 2 Are respectively selected as line segments BP 0 And the line segment CP 0 The direction vector of the point (c) is expressed as:
Figure BDA0003973879750000081
in the formula i 1 And i 2 Are respectively a point P 1 And point P 2 The position parameter of (2) is a design parameter; r is P0 Is a point P 0 Position vector of (1), point P 0 Is a straight line BP 0 And straight line CP 0 Cross point of, straight line BP 0 Passing through point B, the slope is determined by the tangential vector of addendum arc segment AB at point B, straight line CP 0 Through point C, the slope is determined by the direction vector at point C, which is determined by the programmable angle β, which is the angle between the tangent at point C and the x-axis.
The position vector for point C is:
Figure BDA0003973879750000082
in the formula, r p Representing pitch circle radius, a represents center distance, and alpha is solved according to the following formula as a design parameter:
Figure BDA0003973879750000083
in the formula, m represents the number of teeth and is a design parameter;
the position vector for point B is:
r B ＝[R 2 cosγ R 2 sinγ]
the integral structure of the Roots type line consists of a rotor center distance A and an arc radius R of an addendum arc section AB 2 The number of teeth m, the central rotation angle gamma of the addendum arc section AB, the tangent line at the point C, and the x-axisAngle beta, point P 1 And point P 2 Position parameter i 1 And i 2 And (6) determining.
Curve segment CD in rectangular coordinate system O xy The calculation expression in (1) is:
Figure BDA0003973879750000084
in the formula, r CD A position vector representing a point on the curve segment CD; phi is an intermediate turning angle variable parameter and is obtained by the following relational expression:
τ BC (θ,φ)·[x s,BC (θ,φ)-r p y s,BC (θ,φ)] T ＝0
in the formula, τ BC A position vector in the stationary coordinate system representing a tangential vector of the bezier curve segment BC; r is p Represents the pitch circle radius; (x) s,BC ，y s,BC ) A position vector representing the bezier curve segment BC in the rotating coordinate system is obtained by the following relation:
Figure BDA0003973879750000091
Figure BDA0003973879750000092
and S5, combining the addendum arc section AB, the Bessel curve section BC, the curve section CD and the dedendum arc section DE to form the upper half part of a single-tooth profile, forming a complete single-tooth profile by up-down symmetry, and forming a complete and smooth-transition multi-tooth Roots rotor profile structure by rotary transformation.
S6, after the Roots rotor molded lines of the two rotors which are completely the same pass through an offset angle epsilon, the two rotors rotate along different rotation directions and can complete a correct meshing relation, and the offset angle epsilon can be obtained by the following formula:
Figure BDA0003973879750000093
s7, the Roots rotor profiles with different tooth numbers can be obtained by the method, as shown in fig. 4 (a) and 4 (b).
Compared with the prior art, the Roots rotor has larger adjustable space of the shape, so that the performance optimization space based on the conventional molded line structure is larger, and the optimal geometric shape regulation and control under different requirements can be obtained.
The above-mentioned embodiments are only used for illustrating the technical solutions of the present application, and not for limiting the same; although the present application has been described in detail with reference to the foregoing embodiments, it should be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; such modifications and substitutions do not substantially depart from the spirit and scope of the embodiments of the present application and are intended to be included within the scope of the present application.

Claims (8)

1. A multi-tooth Roots rotor based on a Bezier curve design profile is characterized by comprising two rotors with the same Roots profile, wherein the two rotors have different rotation directions and are meshed with each other; the single tooth profile of the Roots type line consists of an upper half curve segment ABCDE and a lower half curve segment AB which are symmetrical 1 C 1 D 1 E 1 Splicing to obtain the finished product; the upper half part curve segment ABCDE consists of a tooth top curve segment and a tooth bottom curve segment, the tooth top curve segment is a combined curve segment of a tooth top arc segment AB and a Bezier curve segment BC, the tooth bottom curve segment consists of a curve segment CD and a tooth root arc segment DE, and the curve segment CD is an envelope curve segment of the Bezier curve segment BC; and rotating and splicing the single-tooth profile of the Roots type line to form a complete Roots type line.
2. The multi-tooth Roots rotor based on a Bezier curve design profile of claim 1, wherein the position vector of a point on the Bezier curve segment BC is calculated as follows:
Figure FDA0003973879740000011
in the formula, r BC Representing a position vector, r, of a point on the Bezier curve segment BC B And r C Respectively representing the position vectors, r, of points B and C P1 And r P2 Respectively represent points P 1 And point P 2 A position vector of (a);
point P 1 And point P 2 Taken from line segments BP respectively 0 And the line segment CP 0 At point above, the direction vector expression is as follows:
Figure FDA0003973879740000012
in the formula i 1 And i 2 Are respectively a point P 1 And point P 2 The position parameter of (2) is a design parameter; r is P0 Is a point P 0 Position vector of (1), point P 0 Is a straight line BP 0 And straight line CP 0 Cross point of, straight line BP 0 Passing through point B, the slope is determined by the tangential vector of addendum arc segment AB at point B, straight line CP 0 Through point C, the slope is determined by the direction vector at point C, which is determined by the programmable angle β, which is the angle between the tangent at point C and the x-axis.
3. The multi-tooth Roots rotor based on Bezier curve design profile of claim 2, wherein the position vector at point C is:
Figure FDA0003973879740000013
in the formula, r p Representing pitch circle radius, a represents center distance, and alpha is solved according to the following formula as a design parameter:
Figure FDA0003973879740000014
in the formula, m represents the number of teeth and is a design parameter;
the position vector at point B is:
r B ＝[R 2 cosγ R 2 sinγ]
in the formula, R 2 The radius of the addendum arc segment AB is represented, and γ represents the central rotation angle experienced by the addendum arc segment AB, which is a design parameter.
4. The multi-tooth Roots rotor based on Bezier curve design profile according to claim 3, wherein the Roots profiles of the two rotors pass through the offset angle ε to realize rotation in different rotation directions and mesh with each other;
the offset angle ε is determined by the following expression:
Figure FDA0003973879740000021
5. the multi-tooth Roots rotor based on a Bezier curve design profile of claim 1, wherein the curve segment CD is in a rectangular coordinate system O xy The calculation expression in (1) is:
Figure FDA0003973879740000022
in the formula, r CD A position vector representing a point on the curve segment CD; phi is an intermediate turning angle variable parameter and is obtained by the following relational expression:
τ BC (θ,φ)·[x s,BC (θ,φ)-r p y s,BC (θ,φ)] T ＝0
in the formula, τ BC A position vector in the stationary coordinate system representing a tangential vector of the bezier curve segment BC; r is p Represents the pitch circle radius; (x) s,BC ，y s,BC ) Representing BesseltThe position vector of the line segment BC in the rotating coordinate system is obtained by the following relational expression:
Figure FDA0003973879740000023
Figure FDA0003973879740000024
6. a method of designing a multi-tooth roots rotor based on a bezier curve design profile as set forth in any one of claims 1 to 5, comprising the steps of:
determining the center distance A of the rotor and the arc radius R of the addendum arc section AB according to the air displacement and the sealing requirement 2 With the number m of teeth, and determining the pitch circle radius r p The position of point a, the position of point E and the position of point C;
determining a central rotation angle gamma experienced by the addendum circular-arc section AB according to the strength and the sealing requirement, determining the addendum circular-arc section AB, and further determining the position of the point B and the position of the point D;
according to design requirements, with the optimization goal of any one or two of the maximum area utilization rate and the shortest contact line length, selecting an included angle beta between a tangent line at a point C and an x axis and a point P 1 And point P 2 Position parameter i 1 And i 2 ；
Determining a Bezier curve segment BC according to the parameters, and solving a curve segment CD through a meshing theorem;
the curve section AB of the lower half part is obtained by the up-down symmetry of the curve section ABCDE of the upper half part of the single-tooth profile formed by combining the addendum arc section AB, the Bessel curve section BC, the curve section CD and the dedendum arc section DE 1 C 1 D 1 E 1 Splicing to form a complete single-tooth profile, and then rotationally splicing the complete single-tooth profile to form a complete Roots profile;
after two rotors with the same Roots profile pass through the offset angle epsilon, the two rotors rotate along different rotation directions and are meshed with each other.
7. The design method according to claim 6, wherein the angle β formed by the tangent at the point C and the x-axis and the point P are adjusted 1 And point P 2 Position parameter i 1 And i 2 And any one or more of the central rotation angles γ experienced by the addendum arc segment AB to achieve the adjustment of the rotor profile shape.
8. The design method according to claim 6, wherein the adjustment of the area utilization rate of the rotor profile is realized by adjusting an angle β between a tangent at the point C and the x-axis.
CN202211528930.5A 2022-11-30 2022-11-30 Multi-tooth Roots rotor based on Bezier curve design line and design method thereof Pending CN115822963A (en)

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