Multi-tooth roots rotor based on beizer curve design profile and design method thereof

Abstract

A multi-tooth Roots rotor based on a Bezier curve design profile and a design method thereof, the multi-tooth Roots rotor comprising two rotors with the same Roots profile, the two rotors being different in rotation direction and meshing with each other; a single-tooth profile of the Roots profile is formed by splicing a symmetrical upper half curve segment ABCDE and a lower half curve segment AB1C1D1E1; the upper half curve segment ABCDE is composed 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 circular arc segment AB and a Bezier curve segment BC, the tooth bottom curve segment is jointly composed of a curve segment CD and a dedendum circular arc segment DE, and the curve segment CD is an envelope curve segment of the Bezier curve segment BC; the single-tooth profile of the Roots profile is rotated and spliced to form a complete Roots profile. The multi-tooth Roots rotor has a large adjustable space, so that the performance optimization space based on the existing profile structure is large, and the optimal geometric shape regulation and control under different requirements can be obtained.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of Roots machinery, and particularly relates to a multi-tooth Roots rotor based on a Bezier curve designed profile and a design method thereof. BACKGROUND

[0002] Roots machinery is a positive displacement rotary machinery with forced suction and exhaust functions, and is widely applied as a blower or vacuum pump in modern industry. Compared with other positive displacement machinery, Roots machinery has the characteristics of reliable operation, low cost, no gas valve and the like, compatibility with liquid, and high operation efficiency. The core component of Roots machinery is a pair of meshing rotors, and a key design element of the rotors is a rotor profile, which directly determines the thermal performance of the whole machine.

[0003] In order to ensure the meshing relationship between the two rotors, the existing rotor profile is mostly in the form of a circular arc and a circular arc envelope, an involute, a cycloid and the like. However, the design space of the above curve forms is small, and the shape of the rotor has limited adjustment space, which further makes the performance optimization space based on the existing profile structure small, and the optimal geometric shape adjustment and control cannot be obtained to meet different requirements. SUMMARY

[0004] The present application aims to solve the problems in the prior art, and provides a multi-tooth Roots rotor based on a Bezier curve designed profile and a design method thereof, so as to realize high flexibility adjustment of the Roots rotor profile by adjusting the parameters of the Bezier curve.

[0005] In order to achieve the above-mentioned purpose, the present application has at least the following beneficial effects:

[0006] A multi-tooth Roots rotor based on a Bezier curve designed profile 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 profile is composed of a symmetrical upper half curve segment and a lower half curve segment. ABCDE The upper half curve segment is composed of a tooth crest curve segment and a tooth bottom curve segment, the tooth crest curve segment is a combined curve segment of a tooth crest circular arc segment and a Bezier curve segment, and the tooth bottom curve segment is composed of a curve segment and a tooth root circular arc segment. AB 1 C 1 D 1 E 1The curve segment is an envelope curve segment of a Bezier curve segment. The single-tooth profile of the Roots profile is rotated and spliced to form a complete Roots profile. ABCDE AB BC CD DE CD BC

[0007] ​​​​​​​As a preferred solution, the Bezier curve segment BC The position vector of the upper point is calculated as follows:

[0008]

[0009] where r BC denotes the Bezier curve segment BC The position vector of the upper point, r B denotes the position vector of the point C denotes the position vector of the point B denotes the position vector of the point C denotes the position vector of the point P1 denotes the position vector of the point P2 denotes the position vector of the point P 1 and the position vector of the point P 2, respectively.

[0010] The points P 1 and P 2 are taken from the line segment BP 0 and the line segment CP 0, respectively, and the position vector expressions are as follows:

[0011]

[0012] wherein i 1 and i 2 are the position parameters of the points P 1 and P 2, respectively, and P r P0 is the position vector of the point P 0, the point BP 0 being the intersection of the straight line CP 0 and the straight line BP 0, the straight line B 0 passing through the point AB , the slope being determined by the tangent vector at the point B , the straight line CP 0 passing through the point C , the slope being determined by the direction vector at the point C , the direction vector at the point C being determined by the designable angle β , the designable angle β being the angle between the tangent at the point C and the axis OC , and O the point being the center of the rotor.

[0013] As a preferred solution, the position vector at the point C is as follows:

[0014]

[0015] In the formula, r p Indicates the radius of the pitch circle. A This represents the center distance, which is a design parameter. α Solve using the following formula:

[0016]

[0017] In the formula, m Indicates the number of teeth, which is a design parameter;

[0018] point B The position vector at that location is:

[0019]

[0020] In the formula, R 2 indicates the tooth tip arc segment AB The radius of the arc, γ Indicates the tooth tip arc segment AB The center turning angle experienced is a design parameter.

[0021] As a preferred option, the Roots profiles of the two rotors are offset by an angle. ε Then, they achieve rotation in different directions and mesh with each other;

[0022] Offset angle ε It can be obtained from the following expression:

[0023] .

[0024] As a preferred embodiment, the curve segment CD In a rectangular coordinate system O xy The calculation expression in is:

[0025]

[0026] In the formula, r CD Indicates curve segment CD The position vector of the upper point; ϕ The intermediate corner variable parameter is obtained from the following relationship:

[0027]

[0028] In the formula, τ BC Represents a segment of a Bézier curve BC The position vector of the tangential vector in the stationary coordinate system; r p Indicates the radius of the pitch circle; , ) represents a segment of a Bézier curve. BCThe position vector in the rotating coordinate system is obtained by the following relationship:

[0029]

[0030] .

[0031] A design method for a multi-tooth Roots rotor based on a Bézier curve design profile includes the following steps:

[0032] The rotor center distance is determined based on the exhaust volume and sealing requirements. A Tooth tip arc segment AB radius of the arc R 2 and number of teeth m And determine the radius of the pitch circle. r p ,point A Location, point E Location and point C Location;

[0033] The tooth tip arc segment is determined based on strength and sealing requirements. AB The central turning point experienced γ Determine the tooth tip arc segment AB to determine the point B Location and point D Location;

[0034] Based on design requirements, the optimal choice is either maximum area utilization or a combination of both of the shortest contact line length. C tangent at the point and OC Angle between axes β , O Point is the rotor center, point P 1 and point P 2 Position Parameters i 1 and i 2;

[0035] Determine the Bézier curve segment based on the above parameters. BC And the curve segment is obtained by using the meshing theorem. CD ;

[0036] By combining the tooth tip arc segment AB Bézier curve segment BC Curved segment CD With the tooth root arc segment DE The upper curved segment that makes up the single tooth profile ABCDE Then, the lower half of the curve segment is obtained by symmetrical arrangement of the upper and lower parts. AB 1 C 1 D 1 E1. Splice together to form a complete single tooth profile, and then spin and splice the complete single tooth profile to form a complete Roots profile;

[0037] Two rotors with the same Roots profile are biased by an angle. ε Then, the two rotors are made to rotate in different directions and mesh with each other.

[0038] As a preferred solution, by adjusting the point C Tangent at the point and OC Angle between axes β ,point P 1 and point P 2 Position Parameters i 1 and i 2, and the tooth tip arc segment AB The central turning point experienced γ Any one or more combinations thereof can be used to adjust the rotor profile shape.

[0039] As a preferred solution, by adjusting the point C Tangent at the point and OC Angle between axes β To achieve adjustment of rotor profile area utilization rate.

[0040] Compared with the prior art, the present invention has at least the following beneficial effects:

[0041] The tooth tip curve of the Roots rotor tooth profile is formed using Bézier curves, and the tooth root curve is then solved based on the meshing relationship. High-flexibility adjustment of the Roots rotor profile can be achieved by adjusting the parameters of the Bézier curves, including by adjusting the adjustment points. C Tangent at the point and OC Angle between axes β ,point P 1 and point P 2 Position Parameters i 1 and i 2, and the tooth tip arc segment AB The central turning point experienced γ Any one or more combinations of these can achieve adjustment of the rotor profile shape. This is achieved by adjusting the number of teeth. m It allows for flexible adjustment of the rotor profile tooth count. This can be achieved through the adjustment point. C Tangent at the point and OC Angle between axes β The rotor profile area utilization rate can be adjusted. The multi-tooth Roots rotor of this invention has a large adjustable range, which in turn allows for a large space for performance optimization based on existing profile structures, thereby enabling the acquisition of optimal geometric shape control to meet different requirements. Attached Figure Description

[0042] Figure 1(a) Schematic diagram of the tooth profile of the multi-tooth Roots rotor based on the Bézier curve design profile in an embodiment of the present invention;

[0043] Figure 1(b) Schematic diagram of solving the profile of a multi-tooth Roots rotor based on the design profile of the Bezier curve in an embodiment of the present invention;

[0044] Figure 2 A schematic diagram of the meshing process of a multi-tooth Roots rotor profile designed based on Bézier curves in an embodiment of the present invention;

[0045] Figure 3(a) via adjustment point C tangent at the point and OC Angle between axes β Schematic diagram of adjusting the rotor profile shape of the present invention;

[0046] Figure 3(b) shows the adjustment point. P 1 and point P 2 Position Parameters i 1 and i 2. Schematic diagram of adjusting the rotor profile shape of the present invention;

[0047] Figure 3(c) shows the adjustment of the tooth tip arc segment. AB The central turning point experienced γ Schematic diagram of adjusting the rotor profile shape of the present invention;

[0048] Figure 4(a) Schematic diagram of the rotor profile of the present invention with 3 teeth;

[0049] Figure 4(b) is a schematic diagram of the rotor profile of the present invention with 4 teeth;

[0050] Figure 5 A comparison chart of the area utilization rate of traditional circular arc profiles and the profiles of this invention under different β angles. Detailed Implementation

[0051] The present invention will now be described in further detail with reference to the accompanying drawings.

[0052] Referring to Figures 1(a) and 1(b), this embodiment of the invention proposes a multi-tooth Roots rotor with a Bézier curve design profile. High flexibility in adjusting the Roots rotor profile can be achieved by adjusting the parameters of the Bézier curve. Specifically, it includes two rotors with the same Roots profile, but with different rotational directions and meshing with each other. The single tooth profile of the Roots profile consists of symmetrical upper curve segments. ABCDE With the lower half of the curve segment AB 1 C 1 D 1 E It is composed of 1 piece. The upper curved section is one of the pieces. ABCDE It consists of a tooth tip curve segment and a tooth root curve segment, with the tooth tip curve segment being a tooth tip circular arc segment. ABThe combination curve segment of the Bezier curve segment BC The tooth bottom curve segment is composed of the curve segment CD and the dedendum circular arc segment DE The curve segment CD is the envelope curve segment of the Bezier curve segment BC The single-tooth profile of the Roots type line is rotated and spliced to form a complete and smooth multi-tooth Roots rotor profile structure, as shown in Figure 2 Two identical Roots rotor profiles can complete the correct meshing relationship.

[0053] In the embodiment, the position vector of the point on the Bezier curve segment BC is calculated as follows:

[0054]

[0055] In the formula, r BC represents the position vector of the point on the Bezier curve segment BC r B and r C represent the position vectors of the point B and the point C r P1 and r P2 represent the position vectors of the point P 1 and the point P 2 respectively.

[0056] The point P 1 and the point P 2 are taken from a point on the line segment BP 0 and the line segment CP 0 respectively, and the position vector expressions are as follows:

[0057]

[0058] In the formula, s i 1 and s i 2 are the position parameters of the point P 1 and the point P 2 respectively, and are design parameters; r P0 is the position vector of the point P 0, the point P 0 is the intersection of the straight line BP 0 and the straight line CP 0, the straight line BP 0 passes through the point B , the slope is determined by the tangent vector of the addendum circular arc segment AB at the point B , the straight line CP 0 passes through the point C , the slope is determined by the direction vector at the point C , and the pointC The direction vector at that location is determined by the designable angle. β Decision, design angle β For point C Tangent at the point and OC Angle between the axes O The point is the rotor center.

[0059] point C The position vector at that location is:

[0060]

[0061] In the formula, r p Indicates the radius of the pitch circle. A This represents the center distance, which is a design parameter. α Solve using the following formula:

[0062]

[0063] In the formula, m Indicates the number of teeth, which is a design parameter;

[0064] point B The position vector at that location is:

[0065]

[0066] In the formula, R 2 indicates the tooth tip arc segment AB The radius of the arc, γ Indicates the tooth tip arc segment AB The center turning angle experienced is a design parameter.

[0067] The Roots profiles of the two rotors are adjusted by an offset angle. ε Then, the two rotors can achieve the correct meshing relationship by rotating in different directions, with an offset angle. ε It can be obtained from the following expression:

[0068]

[0069] Curve segment CD Bézier curve segment BC The curve segment is obtained by applying the meshing theorem. CD In a rectangular coordinate system O xy The calculation expression in is:

[0070]

[0071] In the formula, r CD Indicates curve segment CD The position vector of the upper point; ϕThe intermediate corner variable parameter is obtained from the following relationship:

[0072]

[0073] In the formula, τ BC Represents a segment of a Bézier curve BC The position vector of the tangential vector in the stationary coordinate system; r p Indicates the radius of the pitch circle; , ) represents a segment of a Bézier curve. BC The position vector in the rotating coordinate system is obtained by the following relationship:

[0074]

[0075]

[0076] This invention enables highly flexible design of the Roots rotor profile. In this embodiment, the adjustment point... C Tangent at the point and OC Angle between axes β It can achieve flexible adjustment of the rotor profile shape as shown in Figure 3(a), by adjusting the point P 1 and point P 2 Position Parameters i 1 and i 2. The rotor profile shape shown in Figure 3(b) can be flexibly adjusted by adjusting the tooth tip arc segment. AB The central turning point experienced γ It can achieve flexible adjustment of the rotor profile shape as shown in Figure 3(c).

[0077] By adjusting the number of teeth m The rotor profile tooth count can be flexibly adjusted, as shown in Figures 4(a) and 4(b). This can be achieved by adjusting the point... C Tangent at the point and OC Angle between axes β It can achieve such as Figure 5 The rotor profile area utilization rate is flexibly adjusted as shown.

[0078] Another embodiment of the present invention also proposes a design method for a multi-tooth Roots rotor based on the Bézier curve design profile, comprising the following steps:

[0079] S1. Determine the rotor center distance based on exhaust volume and sealing requirements. A Tooth tip arc segment AB radius of the arc R 2 and number of teeth m And determine the radius of the pitch circle. r p, the position of point A , the position of point E , and the position of point C .

[0080] S2, determining the center angle of the tooth tip arc segment AB experienced by the tooth tip arc segment γ , and further determining the position of point AB and the position of point B . D

[0081] S3, according to the specific design requirements, taking either or both of the maximum area utilization and the shortest contact line length as the preferred target, selecting the angle C between the tangent at point OC and the axis β , O with the rotor center at point P 1 and the position parameters P 1 and i 2 of point i 1 and point 2.

[0082] BC S4, according to the above parameters, the Bezier curve segment is:

[0083]

[0084] In the formula, r BC represents the position vector of a point on the Bezier curve segment BC , r B and r C represent the position vectors of point B and point C , respectively, and r P1 and r P2 represent the position vectors of point P 1 and point P 2, respectively.

[0085] Point P 1 and point P 2 are selected as a point on line segment BP 0 and line segment CP 0, respectively, and the position vector expressions are:

[0086]

[0087] In the formula, i 1 and i 2 are the position parameters of point P 1 and point P 2, respectively, and are design parameters; r P0 is point PPosition vector of point 0 P 0 is a straight line BP 0 is a straight line CP 0 is a straight line BP 0 passes through point B The slope is determined by the tooth tip arc segment AB The tangent vector at point B 0 passes through point CP The slope is determined by the direction vector at point C The direction vector at point C The direction vector at point C is determined by the designable angle β The designable angle β is the angle between the tangent at point C and the axis OC .

[0088] The position vector of point C is:

[0089]

[0090] In the formula, r p R represents the pitch circle radius, A C represents the center distance, which is a design parameter, α and is solved by the following formula:

[0091]

[0092] In the formula, m N represents the number of teeth, which is a design parameter;

[0093] The position vector of point B is:

[0094]

[0095] The overall structure of the Roots type line is determined by the rotor center distance A , the circular arc radius AB 2 of the tooth tip arc segment R , the number of teeth m , the central rotation angle AB of the tooth tip arc segment γ , the angle C between the tangent at point OC and the axis β , and the position parameters P 1 and P 2 of point i 1 and point i 2.

[0096] The curve segment CD is in a rectangular coordinate system Oxy The calculation expression in is:

[0097]

[0098] In the formula, r CD Indicates curve segment CD The position vector of the upper point; ϕ The intermediate corner variable parameter is obtained from the following relationship:

[0099]

[0100] In the formula, τ BC Represents a segment of a Bézier curve BC The position vector of the tangential vector in the stationary coordinate system; r p Indicates the radius of the pitch circle; , ) represents a segment of a Bézier curve BC The position vector in the rotating coordinate system is obtained by the following relationship:

[0101]

[0102]

[0103] S5, through the combination of tooth tip arc segment AB Bézier curve segment BC Curved segment CD With the tooth root arc segment DE The upper part of the single tooth profile is formed, and then the complete single tooth profile is formed by the symmetrical arrangement of the upper and lower parts. Then, through rotation and transformation, a complete and smoothly transitioning multi-tooth Roots rotor profile structure is formed.

[0104] S6. The two rotors have identical Roots rotor profiles, adjusted by an offset angle. ε Then, the two rotors rotate in different directions and can achieve the correct meshing relationship, with an offset angle. ε It can be calculated using the following formula:

[0105] 1.

[0106] S7. Using the above method, Roots rotor profiles with different numbers of teeth can also be obtained, as shown in Figure 4(a) and Figure 4(b).

[0107] Compared with the prior art, the Roots rotor of the present invention has a larger adjustable space for shape, which in turn makes the performance optimization space based on the existing profile structure larger, so as to obtain the optimal geometric shape control to meet different needs.

[0108] The above-described embodiments are only used to illustrate the technical solutions of the present application, but not limit them; although the present application is described in detail with reference to the foregoing embodiments, those skilled in the art should understand that: it can still modify the technical solutions recorded in the foregoing embodiments, or make equivalent replacement to part of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present application, and should be included in the protection scope of the present application.

Claims

1. A multi-tooth Roots rotor designed based on a Bezier curve profile, characterized in that, The two rotors have the same roots profile, and the two rotors are different in rotation direction and meshed with each other; the single-tooth profile of the roots profile is composed of a symmetrical upper half curve segment ABCDE and a lower half curve segment AB 1 C 1 D 1 E 1 ; the upper half curve segment ABCDE is composed of a tooth top curve segment and a tooth bottom curve segment, the tooth top curve segment is a combination curve segment of a tooth top circular arc segment AB and a Bezier curve segment BC , and the tooth bottom curve segment is composed of a curve segment CD and a dedendum circular arc segment DE , the curve segment CD is an envelope curve segment of a Bezier curve segment BC ; the single-tooth profile of the roots profile is rotated and spliced to form a complete roots profile; The Bezier curve segment BC The position vector of the upper point is calculated as follows: where r BC denotes a Bezier curve segment BC the position vector of the upper point, r B where r C denotes the position vector of the point B 1 and the position vector of the point C 2, r P1 where r P2 denotes the position vector of the point P 1 and the position vector of the point P 2, r point P 1 and point P 2 are taken from the line segment BP 0 and point CP 0 on the line segment CP 0, the position vector expressions are as follows: wherein i 1 and i 2 are the position parameters of points P 1 and P 2, respectively, are design parameters; r P0 is the position vector of point P 0, point P 0 is the intersection of straight line BP 0 and straight line CP 0, straight line BP 0 passes through point B , the slope is determined by the tangent vector at point AB at point B , straight line CP 0 passes through point C , the slope is determined by the direction vector at point C , the direction vector at point C is determined by the designable angle β , the designable angle β is the angle between the tangent at point C and the axis OC , O point is the rotor center.

2. The multi-lobe Roots rotor based on the design profile of a Bezier curve according to claim 1, characterized in that, The position vector of the point C at which the point is located is: wherein r p denotes the pitch radius, A denotes the center distance, which is a design parameter, α is solved by the following equation: In the formula, m N represents the number of teeth, which is a design parameter; The position vector of the point B is: wherein R 2 represents the radius of the circular-arc segment of the addendum circle, AB γ represents the central rotation angle experienced by the circular-arc segment of the addendum circle, AB is a design parameter.​ 3. The multi-lobe Roots rotor based on the design profile of a Bezier curve of claim 2, characterized in that, Roots-type line with offset angle for two rotors ε Afterwards, the rotations of different directions are realized and engaged with each other. Bias angle ε is found from the expression: 。 4. The multi-lobe Roots rotor based on the design profile of a Bezier curve of claim 1, wherein, The curve segment CD In a rectangular coordinate system O xy The calculation expression in the rectangular coordinate system is: where r CD represents a curve segment CD the position vector of the upper point; ϕ is an intermediate corner variable parameter, which is calculated from the following relation: where τ BC denotes the position vector of the tangent vector of the Bezier curve segment BC in the stationary coordinate system; r p denotes the pitch circle radius;( , ) denotes the position vector of the Bezier curve segment BC in the rotating coordinate system, which is determined by the following relation: 。 5. A design method of a multi-lobe Roots rotor based on a Bezier curve design profile as claimed in any one of claims 1 to 4, characterized in that, comprising the steps of: Determination of rotor center distance according to displacement and sealing requirements A , Circular arc segment of addendum circle AB , Circular arc radius of addendum circle R 2 and number of teeth m , and determination of pitch circle radius r p , Position of point A , Position of point E , and position of point C ; determining the addendum arc segment depending on the strength and sealing requirements AB the central angle of rotation experienced γ determining the addendum arc segment AB and thereby determining the position of the point B and the position of the point D ​ According to design requirements, with either or both of maximum area utilization and shortest contact line length as the preferred goal, select the tangent at point C and the angle between the tangent and the axis OC , β , O Point as the rotor center, point P 1 and point P 2 position parameters i 1 and i 2; Determine the Bezier curve segment according to the above parameters BC , and obtain the curve segment by the engagement theorem CD ; By combining the addendum arc segment AB , the Bezier curve segment BC , the curve segment CD and the dedendum arc segment DE , the upper half curve segment of the single-tooth profile is formed ABCDE , the lower half curve segment is obtained by symmetry AB 1 C 1 D 1 E 1, the complete single-tooth profile is spliced, and the complete Roots type line is formed by rotating and splicing the complete single-tooth profile Rotors with the same lobes profile are offset by an angle ε After that, the two rotors are rotated in different directions and mesh with each other.

6. The design method of claim 5, wherein, By adjusting the point C tangent at the point and OC Angle between axes β ,point P 1 and point P 2 Position Parameters i 1 and i 2, and the tooth tip arc segment AB The central turning point experienced γ Any one or more combinations thereof can be used to adjust the rotor profile shape.

7. The method of designing according to claim 5, wherein, The angle of the tangent at the point of intersection of the curve and the axis is adjusted. C The angle of the tangent at the point of intersection of the curve and the axis is adjusted. OC The angle of the tangent at the point of intersection of the curve and the axis is adjusted. β The utilization rate of the rotor profile area is adjusted.

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