CN102979731B - Rotor profile of double-screw vacuum pump, and designing method of rotor profile - Google Patents

Abstract

The invention discloses a rotor profile of a twin-screw vacuum pump and a design method thereof. The tooth profile of the twin-screw vacuum pump produced by the design consists of point meshing cycloid, point meshing epicycloid, arc-arc envelope, class Composed of Archimedes spiral and its conjugate envelope, on the basis of ordinary trapezoidal teeth, the range and range of the Archimedes-like spiral and its conjugate envelope are defined by the upper and lower bottom positions of the axial trapezoid. Cycloidal range, determine the Archimedes-like spiral continuously by the first derivative with the arc, and add the conjugate envelope of the Archimedes-like spiral, and finally use the closed meshing line to determine the point meshing cycloid and Point meshing epicycloid equation, thereby completing the profile design, the tooth profile of the twin-screw vacuum pump of the present invention fully satisfies the meshing theorem, has closed meshing lines and continuous contact lines, small through-hole area, and high volumetric efficiency. The designed two-screw rotors have the same profile, which can reduce the rotor processing cost better.

Description

A twin-screw vacuum pump rotor profile and its design method

technical field

The invention belongs to the field of mechanical engineering design, in particular to a twin-screw vacuum pump rotor profile and a design method thereof.

Background technique

Dry vacuum pumps are widely used in modern industry because there is no liquid working medium and sealing medium in the working process. Examples include vacuum extraction without backflow in semiconductor manufacturing, extraction of condensable vapors or corrosive gases in the petrochemical industry. It inherits the characteristics of screw mechanical forced gas transmission, compact structure, high reliability, long life, good power balance performance, multi-phase mixed transmission, etc., and has low and stable power consumption and no pollution to cooling water. Hotspots in research and manufacture of vacuum pumps.

The core component of a twin-screw vacuum pump is a pair of meshing rotors with opposite rotations. The design of the rotor profile directly affects the performance of the pump. The research on the rotor profile is not only the basis for the study of the thermodynamic performance of the twin-screw vacuum pump, but also the optimization of the profile. The key to design and improve the performance of the whole machine. As a commercial secret, the actually usable screw profiles are rarely reported in foreign literature.

There are two types of traditional screw vacuum pumps, which generally require the same profile of the two rotors to facilitate processing:

1. Rectangular teeth and trapezoidal teeth

The remarkable feature of this twin-screw vacuum pump is that the axial section is rectangular or trapezoidal. This kind of vacuum pump does not meet the meshing conditions, and has large through-holes in the radial direction, and its airtightness is poor.

2. Cycloidal-Archimedes spiral tooth

This twin-screw vacuum pump consists of a point meshing cycloid, a dedendum arc, an Archimedes spiral (or an involute) and an addendum arc. Its air tightness has been greatly improved compared with rectangular teeth and trapezoidal teeth. By adjusting the parameters of the Archimedes spiral section, the Archimedes spirals of the male and female rotors can contact each other, but the meshing theorem is not satisfied, and there are also through holes in the radial direction of the rotor.

The design of the new line hopes to meet: 1) The two rotors of the male and female completely satisfy the meshing theorem, the meshing line is closed, and the contact line is continuous; 2) The composition of the two rotors is the same to reduce the processing cost; 3) The area of the through hole is small.

Contents of the invention

The object of the present invention is to provide a twin-screw vacuum pump rotor profile and its design method. The twin-screw vacuum pump rotor profile obtained by the design method is completely conjugate, and the two rotor profile lines have the same composition, and the through-hole area is small. This molded line has good airtightness, and the formed molded surface is easy to process, and the processing cost is low.

The rotor profile of the twin-screw vacuum pump is composed of point meshing cycloid (ab), point meshing epicycloid (bc), dedendum arc (cd), archimedes-like spiral (de), class A Composed of the conjugate envelope (ef) of the Kimedes spiral and the addendum arc (fa), the profile of the female rotor is the same as that of the male rotor. The curve connecting the addendum arc (fa) and the dedendum arc (cd) is two sections, which are respectively the Archimedes-like spiral and the conjugate envelope of this segment of the Archimedes-like spiral. Through this design, the profiles of the two rotors of the twin-screw vacuum pump can be made the same, effectively reducing the processing cost.

The Cartesian coordinate parametric equation of the described class Archimedes spiral (de) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; de</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; de</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\end{array}\right.$

In the equation, x _de represents the X coordinate of the Archimedes-like spiral, y _de represents the Y coordinate of the Archimedes-like spiral, f(t) represents the function that replaces the angle parameter in the Archimedes spiral, and t represents Coordinate parameters, the equation ρ(t)=α×f(t)+β satisfies t _d , t _f represent the start and end angles of the Archimedes-like spiral and its conjugate envelope (def) respectively, and ρ(t _d )=d/2, ρ(t _f )=D/ 2. d represents the diameter of the dedendum circle, and D represents the diameter of the addendum circle. From this, the conditions to be satisfied by the constants α, β and f(t) can be obtained. The conjugate envelope of the Archimedes-like spiral is defined by the class The conjugate curve of the Archimedes spiral is obtained by translation and rotation.

The conjugate envelope (ef) of the class Archimedes spiral is determined by the following rectangular coordinate parametric equation:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; ef</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; ef</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\end{array}\right.$

θ is the polar angle difference between the end point of the Archimedes-like spiral and the starting point of the conjugate curve of the Archimedes-like spiral transformed into polar coordinates;

Wherein k=i+1, i represents transmission ratio, is taken as 1, Indicates the position parameter of the rotor profile (obtained from the above equation), A indicates the center distance between the male and female rotors, It is determined by the rotation direction of the yin and yang rotors when they are working. When the yang rotor rotates clockwise, it is taken as –, otherwise it is +.

The lengths of the point meshing cycloids and point meshing epicycloids are determined by the meshing lines and contact lines of the profiled lines, and it is necessary to ensure that the meshing lines of the profiled lines are closed and the contact lines are continuous.

The Cartesian coordinate parametric equation of the point meshing cycloid (ab) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; ab</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; ab</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\end{array}\right.$

The Cartesian coordinate parametric equation of the point meshing epicycloid (bc) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; bc</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; bc</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\end{array}\right.$

In the equation, x _ab represents the X coordinate of the point meshing cycloid, y _ab represents the Y coordinate of the point meshing cycloid, x _bc represents the X coordinate of the point meshing epicycloid, y _bc represents the Y coordinate of the point meshing epicycloid, and A represents The center distance between the male and female rotors, d represents the diameter of the root circle, D represents the diameter of the addendum circle, t ₁ represents the starting angle of the point meshing cycloid, t ₂ represents the starting angle of the point meshing epicycloid, and t represents the coordinate parameter .

The connection between the Archimedes-like spiral and the dedendum arc and the connection between the conjugate envelope of the Archimedes-like spiral and the addendum arc have continuous first-order derivatives and are smooth connections. Its conjugate rotor (female rotor) is also smoothly connected.

The Cartesian coordinate parametric equation of the dedendum arc (cd) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; cd</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; cd</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right.$

The Cartesian coordinate parameter equation of the addendum arc (fa) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; fa</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; fa</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right.$

In the equation, x _cd indicates the X coordinate of the dedendum arc, y _cd indicates the Y coordinate of the dedendum arc, x _fa indicates the X coordinate of the addendum arc, y _fa indicates the Y coordinate of the addendum arc, and D indicates the addendum circle diameter, d represents the diameter of the dedendum circle, and t represents the coordinate parameter.

The method for designing the rotor profile of the above-mentioned twin-screw vacuum pump includes the following steps:

1) Carry out radial cutting on the trapezoidal rotor pair to obtain the radial profile line of the twin-screw vacuum pump tooth profile. In the profile line of the male rotor, the fa segment is the addendum arc, the ac segment is the Archimedes spiral, and the cd segment is the tooth root arc, segment df is the Archimedes spiral, the profile of the female rotor is the same as that of the male rotor, in the profile of the female rotor, segment f'a' is the addendum arc, and segment a'c' is Archimedes German spiral, c'd' section is the root arc, and d'f' section is the Archimedes spiral;

2) replace the Archimedes spiral of the ac section of the male rotor with the point meshing epicycloid at point a' of the female rotor;

3) Use the Archimedes-like spiral whose equation is ρ=α×f(t)+β under the polar diameter coordinates to replace the Archimedes spiral in the df segment, and f(t) means to replace the original Archimedes spiral The function of the angle parameter, which satisfies the smooth transition with the addendum arc and the dedendum arc at the two endpoints, that is, at the two endpoints $\frac{\mathrm{d\&rho;}}{\mathrm{dt}}=0;$

4) Divide the Archimedes-like spiral into two sections at point e, and keep one section. Assuming that section de is reserved, obtain the conjugate curve e'f' of section de, where the meshing point corresponding to point e is The meshing point corresponding to point e' and d is f', and e'f' is connected to segment de through rotation and translation;

5) Take the intersection point with the pitch circle on the point meshing epicycloid, and calculate the point meshing cycloid corresponding to the intersection point.

The tooth shape of the twin-screw vacuum pump produced by the design of the present invention is composed of point meshing cycloids, point meshing epicycloids, arc-arc envelopes, Archimedes-like spirals and their conjugate envelopes. On the basis of ordinary trapezoidal teeth, the range of the Archimedes-like spiral and its conjugate envelope and the range of the cycloid are defined by the position of the upper and lower bases of the axial trapezoid, and the class is continuously determined by the first-order derivative of the arc. Archimedes spiral, and add the conjugate envelope of Archimedes spiral, and finally use the closed meshing line to determine the point meshing cycloid and the point meshing epicycloid equation, so as to complete the profile design; the present invention The profile line of the twin-screw vacuum pump tooth profile fully satisfies the meshing theorem, and has a closed meshing line and a continuous contact line, so it has better performance; at the same time, the profile line generated by this design method can make the rotor profile of the two screws The lines are the same, which can better reduce the rotor processing cost.

Description of drawings

Fig. 1 is the schematic diagram of twin-screw vacuum pump rotor profile of the present invention;

Figure 2 is the radial profile of a traditional trapezoidal screw vacuum pump;

Fig. 3 is the profile line after engaging the cycloid at the radial profile correction point of the traditional trapezoidal screw vacuum pump;

Fig. 4 is the axial section of the twin-screw vacuum pump rotor of the present invention;

Fig. 5 is a projection view of the rotor profile and its meshing curve on the radial end face of the present invention;

Fig. 6 is the instantaneous contact line of the rotor of the present invention;

Fig. 7 is the molding line shown in Fig. 3 and its meshing line;

Fig. 8 is the profiled line and meshing line after the profiled line correction point meshes with the cycloid shown in Fig. 7 .

Specific implementation

The present invention will be further described below in conjunction with accompanying drawing. The profile of the screw vacuum pump of the present invention is shown in Figure 1, which consists of a point meshing cycloid, an arc-arc envelope, and an Archimedes-like spiral and its conjugate envelope. The specific composition of the male rotor: point meshing cycloid (a ₁ b ₁ ), point meshing epicycloid (b ₁ c ₁ ), dedendum arc (c ₁ d ₁ ), Archimedes-like spiral (d ₁ e ₁ ), the conjugate envelope of the Archimedes-like spiral (e ₁ f ₁ ), and the addendum arc (f ₁ a ₁ ). The profile line of the female rotor is composed of the corresponding conjugate curve of the male rotor, which can be the same as the profile line of the male rotor. The specific composition of the female rotor: point meshing cycloid (a ₂ b ₂ ), point meshing epicycloid (b ₂ c ₂ ), dedendum arc (c ₂ d ₂ ), Archimedes-like spiral (d ₂ e ₂ ), the conjugate envelope of the Archimedes-like spiral (e ₂ f ₂ ), and the addendum arc (f ₂ a ₂ ). The subscript 1 indicates the male rotor, and the subscript 2 indicates the female rotor. By adjusting the parameters, the male and female rotors can be made the same.

The invention proposes an improved curve of the Archimedes spiral, which is called the Archimedes-like spiral. If the Archimedes spiral commonly used in the current design of screw vacuum pumps is replaced by the Archimedes-like spiral and its meshing envelope in the present invention, the obtained rotor curves are connected smoothly, and the yin and yang rotors fully comply with the meshing theorem , and the yin and yang rotors are exactly the same, realizing the conjugate of the yin and yang rotors with the same tooth shape.

Figure 4 is the axial section of this design, and the curves of each section correspond to the profiled lines of the end face. It can be seen that the tooth top section is wider than the tooth bottom section, which is beneficial to the design of the rotor machining tool.

Fig. 5 is the projection diagram of the meshing curve of the rotor in this design on the radial end face.

Figure 6 shows the instantaneous contact line of the rotor of this design. It can be seen from the figure that the contact line of this design is completely continuous, and the rotor has good airtightness.

Each section composition curve of the present invention is as follows under Cartesian coordinate system equation:

The parametric equation of the point meshing cycloid (ab) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; ab</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; ab</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\end{array}\right.$

The parametric equation of the points engaging the epicycloid (bc) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; bc</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; bc</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\end{array}\right.$

The parametric equation of the dedendum arc (cd) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; cd</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; cd</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right.$

The parametric equation of the class Archimedes spiral (de) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; de</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; de</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\end{array}\right.$

The conjugate envelope (ef) of the Archimedes-like spiral is obtained by translation and rotation of the conjugate curve of section de:

Calculate the conjugate curve e'f' of de through the following equation, and transform the obtained conjugate curve from the X ₂ O ₂ Y ₂ coordinate system to the X ₁ O ₁ Y ₁ coordinate system:

in It is determined by the rotation direction of the two rotors when they are working. When the male rotor rotates clockwise, it is –, otherwise it is +.

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; ef</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; ef</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\end{array}\right.$

θ is the polar angle difference between e and e' converted to polar coordinates.

The parametric equation of the addendum arc (fa) is:

$\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; fa</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; fa</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right.$

In the formula: t—coordinate parameter, which can be regarded as the polar angle value under polar coordinates

D——rotor addendum circle diameter

d——Rotor root circle diameter

A——The center distance between the male and female rotors

i——transmission ratio, take it as 1

t ₁ ——the starting angle of cycloid ab

t ₂ ——the starting angle of cycloid bc

——Rotor Profile Position Parameters

k—add 1 to the transmission ratio, which is 2

f(t)——a function that replaces the angle parameter in the Archimedes spiral

The profile line of the female rotor is composed of the corresponding conjugate curves of the male rotor, which can be obtained by using the envelope condition relational expression.

Since the profile of this design can be obtained by improving the trapezoidal teeth, the trapezoidal teeth are taken as the basis of this design:

1. Obtain the traditional trapezoidal tooth profile

The radial profile of the tooth profile of the traditional trapezoidal screw vacuum pump shown in Figure 2 is obtained by radially cutting the trapezoidal screw rotor pair.

In Fig. 2, segment fa is the addendum arc, ac is the Archimedes spiral, cd is the root arc, and df is the Archimedes spiral (involute). The female rotor is the same as the male rotor, and the corresponding acdf point can be changed to a'c'd'f'.

2. Correction of point meshing epicycloid

Replace the segment ac of the male rotor in Figure 2 with the meshing epicycloid at point a' of the female rotor, and the formed rotor profile is shown in Figure 3.

3. Archimedes-like spiral

The conjugate curve of the Archimedes spiral in section df intersects and separates from the addendum arc and the dedendum arc. The main reason for this phenomenon is that the first-order derivatives of the Archimedes spiral (involute) ρ=αt+β and the root arc and addendum arc at the connection points d and f are discontinuous, and the rotor profile Points d and f are not smooth, and there are convex and concave points, as shown in Figure 3. Through calculation, it can be found that the traditional Archimedes spiral (involute) is difficult to meet the profile design requirements.

The Archimedes spiral is extended, and the Archimedes spiral is replaced by a new curve whose equation is ρ=α×f(t)+β in polar radial coordinates, and it is called an Archimedes-like spiral. The Archimedes-like spiral must be continuous with the addendum and root arcs and the first derivative must be continuous, that is, at the two endpoints This allows you to select a curve that satisfies the condition. For example, ρ=α×sin(t)+β, etc.

At this point, the two rotors that meet the meshing conditions can be designed.

4. Conjugate curves of rotating Archimedes-like spirals

In order to make the yin and yang rotors identical and conjugate, the original Archimedes spiral section is divided into two sections at point e, and one section is kept, assuming that section de is kept, and the conjugate curve e of section de is obtained 'f', where point e corresponds to the meshing point e', and point d corresponds to point f'. Connect e'f' with de by rotation translation. In this way, a profile line consisting of point meshing epicycloid, arc-arc envelope, Archimedes-like spiral and its conjugate envelope is obtained. The yin and yang rotors composed of such profiles satisfy the conjugate condition, and the yin and yang rotors are the same.

If this correction is adopted, the Archimedes-like spiral should be continuous with the addendum and dedendum arcs, and the first-order derivative should satisfy $\frac{\mathrm{d\&rho;}}{\mathrm{dt}}{|}_{t=t_d}=0$ That's it. when $\frac{\mathrm{d\&rho;}}{\mathrm{dt}}{|}_{t=t_e/2}=\frac{\mathrm{\&pi;}}{4},$ That is, the slope at the Archimedes-like point e is , a smooth transition curve can be obtained.

4. Correction of point meshing epicycloids through meshing lines

The meshing line of the above profile is shown in Figure 7, and the meshing line in Figure 7 is not closed, which is not conducive to the airtight type of the vacuum pump. Pick any point on the point meshing epicycloid, and calculate its corresponding point meshing cycloid. After the same operation for the male and female rotors, the meshing line diagram shown in Figure 8 can be obtained.

By changing the position of this correction point and moving it to the dedendum circle on the path of the meshing epicycloid at the origin, the degree of closure of the meshing line can be changed, and finally the closure of the meshing line can be achieved. The resulting meshing line is shown in Figure 4, and the molded line is shown in Figure 1.

Claims (6)

1. A rotor profile of a twin-screw vacuum pump, characterized in that: the rotor profile of the twin-screw vacuum pump consists of point meshing cycloids, point meshing epicycloids, dedendum arcs, and Archimedes-like spirals connected sequentially from head to tail , the conjugate envelope of the Archimedes-like spiral and the addendum arc, the profile of the female rotor is the same as that of the male rotor; The Cartesian coordinate parametric equation of the described Archimedes-like spiral is: $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; de</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; de</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\end{array}\right.$ In the equation, x _de represents the X coordinate of the Archimedes-like spiral, y _de represents the Y coordinate of the Archimedes-like spiral, f(t) represents the function that replaces the angle parameter in the Archimedes spiral, and t represents Coordinate parameters, the parametric equation of the polar coordinate system of the Archimedes spiral ρ(t)=α×f(t)+β satisfies t _d , t _f represent the start and end angles of the Archimedes-like spiral and its conjugate envelope respectively, and ρ(t _d )=d/2, ρ(t _f )=D/2,d Indicates the diameter of the dedendum circle, and D indicates the diameter of the addendum circle, from which the constants α, β can be obtained; The conjugate envelope of the Archimedes-like spiral is determined by the following rectangular coordinate parametric equation: $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; ef</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; ef</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}\mathrm{<span\; class="notranslate">\; \&theta;</span>}\end{array}\right.$ θ is the polar angle difference between the end point of the Archimedes-like spiral and the starting point of the conjugate curve of the Archimedes-like spiral transformed into polar coordinates; x _ef represents the conjugate envelope of the Archimedes-like spiral X coordinate, y _ef represents the Y coordinate of the conjugate envelope of the Archimedes spiral; Wherein k=i+1, i represents transmission ratio, is taken as 1, Indicates the position parameter of the rotor profile, A indicates the center distance between the male and female rotors, It is determined by the rotation direction of the yin and yang rotors when they are working. When the yang rotor rotates clockwise, it is taken as –, otherwise it is +. 2. A twin-screw vacuum pump rotor profile according to claim 1, characterized in that: the lengths of the point meshing cycloid and point meshing epicycloid are determined by the meshing line and contact line of the profile, and it is necessary to ensure that the profile line The meshing line is closed and the contact line is continuous. 3. A twin-screw vacuum pump rotor profile according to claim 1, characterized in that: the Cartesian coordinate parameter equation of the point meshing cycloid is: $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; ab</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; ab</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; \&times;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>\end{array}\right.$ The Cartesian coordinate parametric equation of a point meshing with an epicycloid is: $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; bc</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; bc</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; t</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\end{array}\right.$ In the equation, x _ab represents the X coordinate of the point meshing cycloid, y _ab represents the Y coordinate of the point meshing cycloid, x _bc represents the X coordinate of the point meshing epicycloid, y _bc represents the Y coordinate of the point meshing epicycloid, and A represents The center distance between the male and female rotors, d represents the diameter of the root circle, D represents the diameter of the addendum circle, t ₁ represents the starting angle of the point meshing cycloid, t ₂ represents the starting angle of the point meshing epicycloid, and t represents the coordinate parameter . 4. A twin-screw vacuum pump rotor profile according to claim 1, characterized in that: the junction between the Archimedes-like spiral and the dedendum arc and the conjugate envelope of the Archimedes-like spiral The first-order derivative of the connection between the winding line and the addendum arc is continuous, and they are all smooth connections. 5. A twin-screw vacuum pump rotor profile according to claim 1, characterized in that: the Cartesian coordinate parameter equation of the dedendum arc is: $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; cd</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; cd</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right.$ The Cartesian coordinate parametric equation of the addendum arc is: $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; fa</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\\ {\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; fa</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; cos</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}\end{array}\right.$ In the equation, x _cd indicates the X coordinate of the dedendum arc, y _cd indicates the Y coordinate of the dedendum arc, x _fa indicates the X coordinate of the addendum arc, y _fa indicates the Y coordinate of the addendum arc, and D indicates the addendum circle diameter, d represents the diameter of the dedendum circle, and t represents the coordinate parameter. 6. A design method of twin-screw vacuum pump rotor profile as claimed in claim 1, characterized in that: comprising the following steps: 1) Carry out radial cutting on the trapezoidal rotor to obtain the radial profile of the tooth profile of the twin-screw vacuum pump. In the male rotor profile, the fa section is the addendum arc, the ac section is the Archimedes spiral, and the cd section is the tooth root Circular arc, segment df is Archimedes spiral, the profile of the female rotor is the same as that of the male rotor, in the profile of the female rotor, segment f'a' is the addendum arc, segment a'c' is Archimedes Spiral, the c'd' section is the root arc, and the d'f' section is the Archimedes spiral; 2) replace the Archimedes spiral of the ac section of the male rotor with the point meshing epicycloid at point a' of the female rotor; 3) Use the Archimedes-like spiral whose equation is ρ=α×f(t)+β under the polar diameter coordinates to replace the Archimedes spiral in the df segment, and f(t) means to replace the original Archimedes spiral The function of the angle parameter, which satisfies the smooth transition with the addendum arc and the dedendum arc at the two endpoints; 4) Divide the Archimedes-like spiral into two sections at point e, and keep one section. Assuming that section de is reserved, obtain the conjugate curve e'f' of section de, where the meshing point corresponding to point e is The meshing point corresponding to point e' and d is f', and e'f' is connected to segment de through rotation and translation; 5) Take the intersection point with the pitch circle on the point meshing epicycloid, and calculate the point meshing cycloid corresponding to the intersection point.