CN1312406C - Whirlpool compressor - Google Patents

Abstract

The present invention relates to a novel scroll compressor. A molded line at the root of a scroll disc is base on a correcting molded line extending on the basis of the theory of a three basic circle. The molded line is composed of an external main involute section J2E2, a straight line connecting section E2E1, an internal main involute section E1J1, an internal involute correcting section J1K1, a big circle arc correcting section K1N, a small circular arc correcting section NK2 and an external involute correcting section K2J2. Circular arc correcting centers O1 and O2 are both not positioned on a tangent line passing through a main scroll line basic circle. The present invention solves the problems that the strength and the compression ratio of the scroll root section (namely a high pressure area position) of a PMP-shaped line are mutually limited, and the compression ratio of the scroll compressor can not be greatly improved. Thereby, the compression ratio of the scroll compressor is improved, and simultaneously, the strength of the scroll root is also improved. The scroll compressor of the present invention is suitable for aspects of air conditioners, refrigeration, vacuum pumps, gas compression, etc.

Description

Scroll compressor having a plurality of scroll members

Technical Field

The present invention relates to a scroll compressor. The scroll compressor is used for air conditioning, refrigeration, vacuum pump, gas compression and the like.

Background

The scroll compressor is a brand new compressor which appears internationally for nearly twenty years and has the advantages of simple mechanism, high efficiency, low noise and the like. In order to obtain high-efficiency compression ratio, a great deal of work is done in various countries on the correction of the molded line of the root part of the vortex of the scroll compressor. At present, the modified molded line of the scroll plate vortex root (namely the high pressure region part) of the scroll compressor widely adopted by various countries is a fully meshed Perfect MeshIndustrial molded line (PMP molded line for short) developed by Mitsubishi heavy industry in Japan. The high-pressure area is formed by correcting two sections of arcs, and the centers of the corrected arcs are on the tangent line passing through the base circle of the main vortex line.

Although the PMP line can exhaust residual gas and avoid repeated compression, the correction principle is analyzed to know that: to obtain a high compression ratio, the discharge angle is increased, but the main vortex line is reduced by the modified section, the wall thickness of the vortex line is increased only in the vicinity of the discharge hole, and the wall thickness of the vortex line is not changed in the remaining portion, so that sufficient strength cannot be ensured in the root section, whereas the strength of the root section is increased only at the cost of reducing the compression ratio. Therefore, the compression ratio of the scroll compressor cannot be greatly improved, and the power is limited, so that most scroll compressors are mostly used under the working condition of an air conditioner.

Disclosure of Invention

The invention aims to solve the defects that the strength and the compression ratio of the PMP type line vortex root section (namely a high-pressure area part) are mutually limited, and the compression ratio of a vortex compressor cannot be greatly improved. The invention relates to a scroll compressor and provides an extension correction molded line based on a Three Base Circle (TBC) theory.

The invention relates to a scroll compressor, wherein the molded line of the root section (namely the high-pressure area part) of the scroll is an extension correction molded line based on the correction theory of a molded line (TBC molded line for short) of a three-base circle, and the center of the molded line is positioned at the outer main involute section J of a main base circle O of an origin₂E₂Connecting straight line segment E₂E₁Inner main involute section E of main base circle O₁J₁The center of the circle is x_o1′、y_o1′(ii) a Auxiliary base circle of₁' inner correction run-off segment J₁K₁Correction of the great circle segment K₁N, correction of the minor arc section NK₂The center of the circle is x_o2′、y_o2′Auxiliary base circle of₂' outer correction run-off segment K₂J₂Composition is carried out; the radius of each of the three base circles is alpha, the distance from the center of the main base circle to the centers of the two auxiliary base circles is D, the centers of the three base circles are located on the same straight line, and the included angle between the straight line and the x-axis is psi; wherein,

(1) outer main involute section J of main base circle O₂E₂The shape curve satisfies the equation:

$\left\{\begin{array}{c}x=x_0+a[\cos (\mathrm{\θ}+\mathrm{\α})+\mathrm{\θ}\sin (\mathrm{\θ}+\mathrm{\α})]\\ y=y_0+a[\sin (\mathrm{\θ}+\mathrm{\α})-\mathrm{\θ}\cos (\mathrm{\θ}+\mathrm{\α})]\end{array}\right.$

wherein x is₀＝0，y₀＝0，α＝α₀，α₀To form the starting angle of the outer main run-off segment, θ is chosen from $\mathrm{\θ}={\mathrm{\θ}}_{\min }=\frac{\mathrm{\π}}{2}+\mathrm{\ψ}+{\mathrm{\α}}_0$ Starting;

(2) connecting straight line segment E₂E₁The shape curve satisfies the equation:

straight line segment E₂E₁End point E of₂、E₁Are respectively connected with an involute J₂E₂、E₁J₁E of₂And E₁The points are the same point, so the coordinate values are known, E₂E₁The parameterized equation of (a) is:

$y=\mathrm{LN}(x_{E_1},y_{E_1},x_{E_2},x_{E_2},x)$

(3) inner main involute section E of main base circle O₁J₁The shape curve satisfies the equation:

$\left\{\begin{array}{c}x=x_0+a[\cos (\mathrm{\θ}+\mathrm{\α})+\mathrm{\θ}\sin (\mathrm{\θ}+\mathrm{\α})]\\ y=y_0+a[\sin (\mathrm{\θ}+\mathrm{\α})-\mathrm{\θ}\cos (\mathrm{\θ}+\mathrm{\α})]\end{array}\right.$

wherein x is₀＝0，y₀＝0，α＝-α₀，-α₀To form the starting angle of the inner main run-off segment, θ is taken to be $\mathrm{\θ}={\mathrm{\θ}}_{\min }=\frac{3\mathrm{\π}}{2}+\mathrm{\ψ}-{\mathrm{\α}}_0$ Starting;

(4) auxiliary base circle O₁' inner correction run-off segment J₁K₁The shape curve satisfies the equation:

$\left\{\begin{array}{c}x=x_{0_1^{\′}}+a[\cos (\mathrm{\θ}+\mathrm{\α})+\mathrm{\θ}\sin (\mathrm{\θ}+\mathrm{\α})]\\ y=y_{0_1^{\′}}+a[\sin (\mathrm{\θ}+\mathrm{\α})-\mathrm{\θ}\sin (\mathrm{\θ}+\mathrm{\α})]\end{array}\right.$

wherein, $x_{O_1^{\′}}=-D\cos \mathrm{\ψ},$ $y_{O_1^{\′}}=-D\sin \mathrm{\ψ},$ $\mathrm{\α}={\mathrm{\α}}_{01},$ ${\mathrm{\α}}_{01}=\frac{3\mathrm{\π}}{2}+\mathrm{\ψ}-\frac{L_1}{a},$ α₀₁to form an inner correction run-off segment J₁K₁The starting angle of (a) of (b), $L_1=a(\frac{3\mathrm{\π}}{2}+\mathrm{\ψ}-{\mathrm{\α}}_0)-D,$ θ＝α_K1，α_K1＝π-α₀₁+ψ+δ′， ${\mathrm{\δ}}^{\′}=\frac{\mathrm{\π}}{2}-\mathrm{\δ},$ delta is an inner correction run-off segment J₁K₁An unfolding angle;

(5) correcting large arc section K₁N shapeThe curve satisfies the equation:

correcting large arc section K₁Endpoint K of N₁Correcting the involute segment J for the inner side₁K₁Upper point, center of circle and x₀₁、y₀₁The point coordinates are:

$\left\{\begin{array}{c}{x}_{01}=-\stackrel{\&OverBar;}{{\mathrm{OO}}_1}\cos (\mathrm{\&gamma;}+\mathrm{\&psi;})\\ {\mathrm{<figure-callout\; id="01"\; label="y"\; filenames="C20041002252600181.png"\; state="\{\{state\}\}">y</figure-callout>}}_{01}=-\stackrel{\&OverBar;}{{\mathrm{OO}}_1}\sin (\mathrm{\&gamma;}+\mathrm{\&psi;})\end{array}\right.$

gamma is a corrected great arc section K₁N and corrected minor arc section NK₂The connecting line of the circle centers forms an included angle with the straight line of the circle centers of the three base circles,correcting the great arc section K for the distance between the center of the central base circle and the center of the corrected great arc₁N and corrected minor arc section NK₂Tangent at point N;

$x_N=\frac{(R_1-R_2)\cos {\mathrm{\&gamma;}}_P}{2},$ $y_N=\frac{(R_1-R_2)\sin {\mathrm{\&gamma;}}_P}{2}$

R₁for correcting the large arc section K₁Radius of N, R₂For correcting small arc sections NK₂The radius of (a);

γ_pδ' + ψ _, wherein:

S＝2a(π+ψ-δ)+2D(sinδ′-1)

(6) modified small arc section NK₂The shape curve satisfies the equation:

modified small arc section NK₂Endpoint K of₂Correcting the involute segment K for the outer side₂J₂Upper point, center x₀₂、y₀₂The point coordinates are:

$\left\{\begin{array}{c}{x}_{02}=-x_{01}\\ {\mathrm{<figure-callout\; id="01"\; label="y"\; filenames="C20041002252600181.png"\; state="\{\{state\}\}">y</figure-callout>}}_{01}=-y_{02}\end{array}\right.$

correcting large arc section K₁N and corrected minor arc section NK₂Tangent at point N;

(7) auxiliary base circle O₂' outer correction run-off segment K₂J₂The shape curve satisfies the equation:

$\left\{\begin{array}{c}x=x_{O_2^{\&prime;}}+a[\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})+\mathrm{\&theta;}\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})]\\ y=y_{O_2^{\&prime;}}+a[\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})-\mathrm{\&theta;}\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})]\end{array}\right.$

wherein, $x_{O_2^{\&prime;}}=D\cos \mathrm{\&psi;},$ $y_{O_2^{\&prime;}}=D\sin \mathrm{\&psi;},$ α＝α₀₂， ${\mathrm{\&alpha;}}_{02}=\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-\frac{L_2}{a},$ α₀₂correction of the run-off segment K for the formation of the outer side₂J₂The starting angle of (a) of (b), $L_2=a(\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}+{\mathrm{\&alpha;}}_0)-D,$ θ＝α_K2，

α_K2＝δ′+ψ-α₀₂， ${\mathrm{\&delta;}}^{\&prime;}=\frac{\mathrm{\&pi;}}{2}-\mathrm{\&delta;},$ delta is an inner correction run-off segment J₁K₁The flare angle.

According to the novel scroll compressor, the molded line of the scroll root section (namely the high-pressure area part) is completely formed by an extension correction molded line based on a three-base-circle molded line (TBC molded line for short) correction theory. Correcting the center O of the arc₁And O₂All are not passing through the main vortex lineOn the tangent line of the base circle, the defect of PMP type line is overcome, and the following advantages are brought:

<1>when the value of beta ↓, $R_1=\stackrel{\&OverBar;}{K_1{O}_1}\&DownArrow;$ time involute J₁K₁The wall thickness of the section vortex line can still extend inwards to increase the wall thickness, so that the strength of the root part of the vortex molded line is improved.

Due to the fact that the wall thickness of the section near the root of the molded line is increased, the volume of a pair of closed cavities formed by meshing a pair of vortex cavities at the moment is reduced, and therefore the compression ratio can be improved.

Drawings

FIG. 1 is a schematic view of the scroll profile root "extension deformation modification";

FIG. 2 is a schematic view of the scroll profile root "extension deformation modification" concept with additional straight line modification;

fig. 3 is a schematic diagram of a japanese patent profile PMP;

FIG. 4 is a comparison diagram of the Japanese molded lines after "extension deformation correction" of the scroll molded line root;

FIG. 5 is the exhaust cavity area when PMP type movable and stationary plates are engaged;

FIG. 6 is the area of the exhaust cavity when the dynamic and static scrolls are engaged with each other by the extension deformation correction molded line at the root of the scroll molded line;

Detailed Description

The basic idea of the "extended deformation correction" theory is as follows:

as shown in FIG. 1, a main vortex line base circle O is centered at a point O, and a left and a right secondary base circles O with a radius a and a distance D from the point O are introduced₁' and O₂' the centers of the three main and auxiliary base circles are on a straight line, and the included angle between the straight line of the three centers and the X axis is psi. The inner and outer main vortex lines generated by the main base circle are involute J₁E₁And J₂E₂Respectively having an initial angle of alpha₀And-alpha₀The radius of the main base circle is a, and the starting point J of the inner and outer vortex lines is now set₁And J₂Not as a function of the angle beta but as a fixed point in fig. 1. To figure 1

J₁The starting angles of the points are: ${\mathrm{\&alpha;}}_1=\frac{3\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-{\mathrm{\&alpha;}}_0---\left(1\right)$

whereby the outward expansion generates a main vortex line J₁E₁Segment of

J₂The starting angles of the points are: ${\mathrm{\&alpha;}}_2=\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}+{\mathrm{\&alpha;}}_0---\left(2\right)$

thereby extending outwardGenerating main vortex line J₂E₂Segment of

From J₁And J₂Two involute curves J extending inwards from the base circle₁K₁Segment and J₂K₂Segment, line J in FIG. 1₁C₁Around the slave base circle O₁' rotating through angle delta tangent from base circle O with pure rolling₁In Q₁Point, line segment J₂C₂Rotated through the same angle and cut from the base circle O₂In Q₂And (4) point. Due to the introduction of the extended deformation involute J₁K₁And J₂K₂Therefore, the circle center O of the circular arc is corrected in the abstract drawing₁And O₂Are all not on the tangent line of the base circle of the main vortex line and are formed by₁And O₂And then, arc correction is carried out, so that the limitation that the circle center of the vortex root molded line correction arc specified in Japanese patent can only be on the tangent line of the base circle of the main vortex line is broken through.

The invention relates to an extension correction molded line based on a three-base-circle molded line (TBC molded line for short) correction theory, which is specifically derived as follows:

theoretical calculation and derivation of the modified profile:

1. correction of straight line segment without additional introduced parameters D and psi

As can be seen from fig. 1: delta-pi-beta is formed by the following steps, $\mathrm{\&delta;}+{\mathrm{\&delta;}}^{\&prime;}=\frac{\mathrm{\&pi;}}{2}$ therefore, it is ${\mathrm{\&delta;}}^{\&prime;}=\mathrm{\&beta;}-\frac{\mathrm{\&pi;}}{2};$

$\left\{\begin{array}{c}{L}_1=\stackrel{\&OverBar;}{J_1{C}_1}=a(\frac{3\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-{\mathrm{\&alpha;}}_0)-D\\ L_2=J_2{C}_2=a(\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}+{\mathrm{\&alpha;}}_0)-D\end{array}\right.---\left(3\right)$

According to the correction idea and the involute property, the following characteristics are obtained: curve segment J₁K₁Segment and J₂K₂The initial angles of the segments are respectively:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_{01}=\frac{3\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-\frac{L_1}{a}\\ {\mathrm{\&alpha;}}_{02}=\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-\frac{L_2}{a}\end{array}\right.---\left(4\right)$

its starting angle (corresponding to K)₁And K₂Point) and end angle (corresponding to J)₁And J₂Points) are respectively:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_{K1}=\mathrm{\&pi;}-{\mathrm{\&alpha;}}_{01}+\mathrm{\&psi;}+{\mathrm{\&delta;}}^{\&prime;}\\ {\mathrm{\&alpha;}}_{J1}=\frac{3\mathrm{\&pi;}}{2}+\mathrm{\&Psi;}-\left|{\mathrm{\&alpha;}}_{01}\right|\end{array}\right.---\left(5\right)$ $\left\{\begin{array}{c}{\mathrm{\&alpha;}}_{K2}={\mathrm{\&delta;}}^{\&prime;}+\mathrm{\&psi;}-{\mathrm{\&alpha;}}_{02}\\ {\mathrm{\&alpha;}}_{J2}=\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-{\mathrm{\&alpha;}}_{02}\end{array}\right.---\left(6\right)$

for proper engagement, there should be:

$R_1+R_2=\stackrel{\&OverBar;}{O_1{O}_2}=2\stackrel{\&OverBar;}{O_{1}O}=2\stackrel{\&OverBar;}{O_{2}O}$

OO is composed of a right triangle in FIG. 1₁B₁Comprises the following steps:

${\left(\frac{R_1+R_2}{2}\right)}^2={\stackrel{\&OverBar;}{O_1{B}_1}}^2+{\stackrel{\&OverBar;}{O{B}_1}}^2---\left(7\right)$

and Δ OO₁B₁≌ΔOO₂B₂，B₁、B₂Respectively is the point of crossing O

Extension line and

the intersection point of the perpendicular lines of the extension lines.

$R_1=\stackrel{\&OverBar;}{Q_1{K}_1}+\stackrel{\&OverBar;}{Q_1{B}_1}-\stackrel{\&OverBar;}{O_1{B}_1}---\left(8\right)$

$R_2=\stackrel{\&OverBar;}{Q_2{K}_2}+\stackrel{\&OverBar;}{Q_2{B}_2}-\stackrel{\&OverBar;}{O_2{B}_2}---\left(9\right)$

Wherein: $\stackrel{\&OverBar;}{O_1{B}_1}=\stackrel{\&OverBar;}{O_2{B}_2},\stackrel{\&OverBar;}{Q_1{B}_1}=\stackrel{\&OverBar;}{Q_2{B}_2}---\left(10\right)$ formula (8) + formula (9):

$R_1+R_2=\stackrel{\&OverBar;}{Q_1{K}_1}+\stackrel{\&OverBar;}{Q_1{K}_2}+2\stackrel{\&OverBar;}{Q_1{B}_1}-2\stackrel{\&OverBar;}{O_1{B}_1}---\left(11\right)$

order to $S=\stackrel{\&OverBar;}{Q_1{K}_1}+\stackrel{\&OverBar;}{Q_2{K}_2}+2\stackrel{\&OverBar;}{Q_1{B}_1}---\left(12\right)$ Equation (11) can then be expressed as:

$R_1+R_2=S-2\stackrel{\&OverBar;}{O_1{B}_1}---\left(13\right)$

substituting the formula (7) to obtain:

$O_1{B}_1=\frac{S}{4}-\frac{{\stackrel{\&OverBar;}{O{B}_1}}^2}{S}---\left(14\right)$

from fig. 1 and the nature of the involute:

$\left\{\begin{array}{c}\stackrel{\&OverBar;}{Q_1{K}_1}=L_1-\mathrm{a\&delta;}=L_1-a(\mathrm{\&pi;}-\mathrm{\&beta;})\\ \stackrel{\&OverBar;}{Q_2{K}_2}=L_2-\mathrm{a\&delta;}=L_2-a(\mathrm{\&pi;}-\mathrm{\&beta;})\end{array}\right.---\left(15\right)$

from the graph Q in FIG. 1₁O₁O₁′OB₁Shown in the figure:

$\stackrel{\&OverBar;}{O_1^{\&prime;}O}=D,$ $\stackrel{\&OverBar;}{O_1^{\&prime;}{Q}_1}=a$

for Δ O₁′Q₁O, from the chord length measure:

$\stackrel{\&OverBar;}{O{Q}_1}=\sqrt{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}}$

and is formed by a chord length measure:

$\sin \mathrm{\&sigma;}=\frac{D}{\stackrel{\&OverBar;}{{\mathrm{OQ}}_1}}\sin {\mathrm{\&delta;}}^{\&prime;}$

therefore, the method comprises the following steps: $\stackrel{\&OverBar;}{Q_1{B}_1}=\stackrel{\&OverBar;}{{\mathrm{OQ}}_1}\sin \mathrm{\&sigma;}=D\sin {\mathrm{\&delta;}}^{\&prime;}---\left(16\right)$

$\stackrel{\&OverBar;}{{\mathrm{OB}}_1}=\sqrt{{\stackrel{\&OverBar;}{{\mathrm{OB}}_1}}^2-{\stackrel{\&OverBar;}{Q_1{B}_1}}^2}---\left(17\right)$

thereby: r can be obtained from the formulas (8) and (9)₁、R₂；

As can be seen from fig. 1:

$\mathrm{\&gamma;}={\mathrm{\&delta;}}^{\&prime;}++\mathrm{\&psi;}-{\mathrm{tg}}^{-1}\left(\frac{\stackrel{\&OverBar;}{O_1{B}_1}}{\stackrel{\&OverBar;}{{\mathrm{OB}}_1}}\right)=\mathrm{\&beta;}-\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-{\mathrm{tg}}^{-1}\left(\frac{\stackrel{\&OverBar;}{O_1{B}_1}}{\stackrel{\&OverBar;}{{\mathrm{OB}}_1}}\right)---\left(18\right)$

therefore, the disengaging angle is: theta 2 pi-gamma (19)

2. Introduced with additional straight-line segment correction of parameters D, psi

Let O be determined from FIG. 1₁、O₂Corresponding to O in FIG. 2_1P、O_2PR in FIG. 1₁、R₂Correspond to $R_1^{\&prime;}=\stackrel{\&OverBar;}{K_1{O}_{1P}}.$ $R_2^{\&prime;}=\stackrel{\&OverBar;}{K_2{O}_{2P}}$ Now will

The line segment rotates clockwise around the point O by a certain angle to ensure that O_1PPoint is turned to O₁Point, O_2pPoint to O₂Point, O_1pPoint to O₁The distance between points is ε, then:

$\left\{\begin{array}{c}{R}_1=R_1^{\&prime;}-\mathrm{\&epsiv;}\\ R_2=R_2^{\&prime;}-\mathrm{\&epsiv;}\end{array}\right.---\left(20\right)$

let in FIG. 1In FIG. 2 areThen in fig. 2:

$\stackrel{\&OverBar;}{O_1{B}_1}=\stackrel{\&OverBar;}{O_{1P}{B}_1}+\mathrm{\&epsiv;}---\left(21\right)$

in fig. 2:

$\stackrel{\&OverBar;}{O_1{O}^{\&prime;}}=\frac{(R_1+R_2)}{2}---\left(22\right)$

$\stackrel{\&OverBar;}{{\mathrm{OO}}_1}=\sqrt{{\stackrel{\&OverBar;}{{\mathrm{OB}}_1}}^2+{\stackrel{\&OverBar;}{O_1{B}_1}}^2}---\left(23\right)$

$\mathrm{\&eta;}=\sin \left(\frac{\stackrel{\&OverBar;}{O_1{O}^{\&prime;}}}{\stackrel{\&OverBar;}{{\mathrm{OO}}_1}}\right)---\left(24\right)$

from the non-linear correction formula (18): gamma → gamma_p

Then, the following steps are obtained: $\mathrm{\&gamma;}=\frac{\mathrm{\&pi;}}{2}-\mathrm{\&eta;}+{\mathrm{\&gamma;}}_P+\mathrm{\&psi;}---\left(25\right)$

obtaining a disengaging angle: theta deg. -2 pi-gamma (26)

3. Constraints of parameters D and psi

The parameters D and psi have certain limit conditions with the basic parameters of the main vortex line design. If the values of D and psi are too large, the constraint condition is not satisfied, and the two correction arcs cannot be tangent correctly.

The constraint method formula is derived as follows:

when in use $y_{Q_1}\&GreaterEqual;y_{K_2}$ And is $y_{O_2}\&GreaterEqual;y_{K_2}$ The molded lines are smoothly connected.

To pair $y_{O_2}\&GreaterEqual;y_{K_2}$ I.e. by $\stackrel{\&OverBar;}{{\mathrm{OO}}_2}\cos (\mathrm{\&gamma;}+\mathrm{\&psi;})\&GreaterEqual;\stackrel{\&OverBar;}{{\mathrm{OO}}_2}\cos (\mathrm{\&gamma;}+\mathrm{\&psi;})+R_2\sin (\mathrm{\&delta;}-\mathrm{\&psi;})---\left(27\right)$

The formula is simplified to obtain: r₂sin(δ-ψ)≤0 (28)

Analytical formula (28), R₂Can only be positive, so

sin (delta-psi) < 0, it is possible to form a profile because R₂And δ, ψ.

General formula (12) $R_1+R_2=S-2\stackrel{\&OverBar;}{O_1{B}_1}---\left(29\right)$

Represented by the formulae (8), (9) R₁-R₂＝a(π-2α) (30)

The vertical type (29) and (30) are combined: $R_2=\frac{1}{2}[S-2\stackrel{\&OverBar;}{O_1{B}_1}-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})]---\left(31\right)$

in formula (31):

$S=\stackrel{\&OverBar;}{Q_1{K}_1}+\stackrel{\&OverBar;}{Q_2{K}_2}+2\stackrel{\&OverBar;}{Q_1{B}_1}$

$=L_1-\mathrm{a\&delta;}+L_2-\mathrm{a\&delta;}+2D\sin {\mathrm{\&delta;}}^{\&prime;}$

$=a(\frac{3\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-\mathrm{\&alpha;})-D-\mathrm{a\&delta;}+a(\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}+\mathrm{\&alpha;})-D-\mathrm{a\&delta;}+2D\sin {\mathrm{\&delta;}}^{\&prime;}$

$=2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin \mathrm{\&delta;}-1)---\left(32\right)$

$O_1{B}_1=\frac{S}{4}-\frac{{\stackrel{\&OverBar;}{{\mathrm{OB}}_1}}^2}{S}$

$=\frac{S}{4}-\frac{{\stackrel{\&OverBar;}{{\mathrm{OQ}}_1}}^2-{\stackrel{\&OverBar;}{Q_1{B}_1}}^2}{S}$

$=\frac{S}{4}-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{S}$

$=\frac{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{4}-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}$

$=\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}---\left(33\right)$

formulae (32) and (33) are substituted for formula (31):

$R_2=\frac{1}{2}[S-2\stackrel{\&OverBar;}{O_1{B}_1}-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})]$

$=\frac{1}{2}\{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)-2[\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}---\left(34\right)$

$-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}]-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})\}$

therefore, it is to $y_{O_2}\&GreaterEqual;y_{K_2}$

R₂When the sum is more than 0 and sin (delta-psi) < 0, a smooth molded line can be formed;

to pair $y_{Q_2}\&GreaterEqual;y_{K_2},$ Namely, it is

$D\sin \mathrm{\&psi;}+a\sin ({\mathrm{\&delta;}}^{\&prime;}+\mathrm{\&psi;})\&GreaterEqual;\stackrel{\&OverBar;}{{\mathrm{OO}}_2}\cos (\mathrm{\&gamma;}+\mathrm{\&psi;})+R_2\sin (\mathrm{\&delta;}-\mathrm{\&psi;})---\left(35\right)$

In the formula (35), the reaction mixture is,

${\stackrel{\&OverBar;}{{\mathrm{OO}}_2}}^2={\stackrel{\&OverBar;}{{\mathrm{OB}}_1}}^2+{\stackrel{\&OverBar;}{O_1{B}_1}}^2$

$={\stackrel{\&OverBar;}{{\mathrm{OQ}}_1}}^2-{\stackrel{\&OverBar;}{Q_1{B}_1}}^2+{\stackrel{\&OverBar;}{O_1{B}_1}}^2$

$=a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}$

$+\&lsqb;\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{\&rsqb;}^2---\left(36\right)$

formula (36) is substituted for formula (35) to give:

$D\sin \mathrm{\&psi;}+a\sin ({\mathrm{\&delta;}}^{\&prime;}+\mathrm{\&psi;})\&GreaterEqual;$

$\{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}$

$+\&lsqb;\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{\&rsqb;}^2{\}}^{\frac{1}{2}}\cos (\mathrm{\&gamma;}+\mathrm{\&psi;})$

$+\frac{1}{2}\{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)-2\&lsqb;\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}$

$-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}\&rsqb;-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})\}\sin (\mathrm{\&delta;}-\mathrm{\&psi;})---\left(37\right)$

the γ angle in the formula (37) can be obtained from the formula (18). Namely, it is

$\mathrm{\&gamma;}={\mathrm{\&delta;}}^{\text{'}}+\mathrm{\&psi;}-{\mathrm{tg}}^{-1}\left(\frac{\stackrel{\&OverBar;}{{\mathrm{\&Omicron;}}_1{\mathrm{\&Bgr;}}_1}}{\stackrel{\&OverBar;}{{\mathrm{\&Omicron;\&Bgr;}}_1}}\right)=\mathrm{\&beta;}-\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-{\mathrm{tg}}^{-1}\left(\frac{\stackrel{\&OverBar;}{{\mathrm{\&Omicron;}}_1{\mathrm{\&Bgr;}}_1}}{\stackrel{\&OverBar;}{{\mathrm{\&Omicron;\&Bgr;}}_1}}\right)$

Therefore, the obtained formula (37) is a judgment formula. When the parameters a, δ, δ', ψ are fixed, the range of D can be determined, and the resulting range can form a constraint of a smooth closed curve. Whether the molded line can be smoothly closed can be judged.

In summary, the constraint conditions of D and psi are

When in use

$R_2=\frac{1}{2}[S-2\stackrel{\&OverBar;}{O_1{B}_1}-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})$

$=\frac{1}{2}\{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)-2[\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})=D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}$

$-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}]-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})\}>0$

And sin (delta-psi) < 0 and

$D\sin \mathrm{\&psi;}+a\sin ({\mathrm{\&delta;}}^{\&prime;}+\mathrm{\&psi;})\&GreaterEqual;$

$\{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}$

$+[\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{]}^2{\}}^{\frac{1}{2}}\cos (\mathrm{\&gamma;}+\mathrm{\&psi;})$

$+\frac{1}{2}\{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)-2[\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}$

$-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}]-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})\}\sin (\mathrm{\&delta;}-\mathrm{\&psi;})$

wherein $\mathrm{\&gamma;}={\mathrm{\&delta;}}^{\&prime;}+\mathrm{\&psi;}-{\mathrm{tg}}^{-1}\left(\frac{\stackrel{\&OverBar;}{O_1{B}_1}}{\stackrel{\&OverBar;}{{\mathrm{OB}}_1}}\right)=\mathrm{\&beta;}-\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-{\mathrm{tg}}^{-1}\left(\frac{\stackrel{\&OverBar;}{O_1{B}_1}}{\stackrel{\&OverBar;}{{\mathrm{OB}}_1}}\right)$ In this case, a smooth profile can be formed.

The above is the parameter D and psi constraint method formula derivation.

The following is another formula derivation of the D and ψ constraint method, which can be used as a check formula:

two correction circle arc center distance of $L_{O_1{O}_2}=\sqrt{{(x_{O_1}-x_{O_2})}^2+{(y_{O_1}-y_{O_2})}^2}---\left(38\right)$

When in use $L_{O_1{O}_2}<(R_1+R_2)$ The smooth connectivity of the profile is destroyed, in which case the correction arc is too large.

When in use $L_{O_1{O}_2}=(R_1+R_2)$ And meanwhile, the two correction circular arcs are tangent, and the molded line is corrected only by the circular arc.

When in use $L_{O_1{O}_2}>(R_1+R_2)$ In this case, it must be further corrected by a straight line segment, in which case its tangent point N₁，N₂The coordinates are determined by:

$\left\{\begin{array}{c}{(x_{N_1}-x_{O_1})}^2+{(y_{N_1}-y_{O_1})}^2=R_1^{2-}\\ \frac{y_{N_1}-y_{O_1}}{x_{N_1}-x_{O_1}}=-\frac{x_{N_1}-x_{O_2}}{y_{N_1}-y_{N_2}}\\ {(x_{N_2}-x_{O_2})}^2+{(y_{N_2}-y_{O_2})}^2=R_2^2\\ \frac{y_{N_2}-y_{O_2}}{x_{N_2}-x_{O_2}}=-\frac{x_{N_1}-x_{O_2}}{y_{N_1}-y_{N_2}}\end{array}\right.---\left(39\right)$

4. deriving coordinates of points

The point on the involute is obtained by the following equation:

$\left\{\begin{array}{c}x=x_0+a[\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})+\mathrm{\&theta;}\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})]\\ y=y_0+a[\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})-\mathrm{\&theta;}\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})]\end{array}\right.----\left(40\right)$

J₁point:

x₀＝0，y₀＝0，θ＝α₁[ formula (1)]，α＝α₀(ii) a Substitution (40);

J₂point:

x₀＝0，y₀＝0，θ＝α₂[ formula (2)]，α＝-α₀(ii) a Substitution (40);

K₁point:

x₀＝-Dcosψ，y₀＝-Dsinψ，θ＝α_K1[ formula (5)]，α＝α₀₁[ formula (4)](ii) a Substitution (40);

K₂point:

x₀＝Dcosψ，y₀＝Dsinψ，θ＝α_k2[ formula (6)]，α＝α₀₂[ formula (4)](ii) a Substitution (40);

x₀₁、y₀₁point:

$\left\{\begin{array}{c}{x}_{01}=-\stackrel{\&OverBar;}{{\mathrm{OO}}_1}\cos ({\mathrm{\&gamma;}}_P+\mathrm{\&psi;})\\ {\mathrm{<figure-callout\; id="01"\; label="y"\; filenames="C20041002252600181.png"\; state="\{\{state\}\}">y</figure-callout>}}_{01}=-\stackrel{\&OverBar;}{{\mathrm{OO}}_1}\sin ({\mathrm{\&gamma;}}_P+\mathrm{\&psi;})\end{array}\right.---\left(41\right)$

x₀₂、y₀₂point:

$\left\{\begin{array}{c}{x}_{02}=-x_{02}\\ y_{02}=-y_{02}\end{array}\right.---\left(42\right)$

determined by equation (23);

γ_Pdetermined by formula (8);

x_N1、y_N1point:

$\left\{\begin{array}{c}{x}_{N1}=x_{01}+R_1\cos \mathrm{\&gamma;}\\ x_{N1}=x_{01}+R_1\sin \mathrm{\&gamma;}\end{array}\right.---\left(43\right)$

x_N2、y_N2point:

$\left\{\begin{array}{c}{x}_{N2}=x_{02}-R_2\cos \mathrm{\&gamma;}\\ y_{N2}=y_{02}-R_2\sin \mathrm{\&gamma;}\end{array}\right.---\left(44\right)$

$\stackrel{\&OverBar;}{N_1{N}_2}=\sqrt{{(x_{N2}-x_{N1})}^2+{(y_{N2}-y_{N21})}^2}---\left(45\right)$

when s is 0, x_N1 ＝x_N2 ＝x_N，y_N1＝y_N2 ＝y_N；

To reduce the computational accumulated error, then:

$x_N=\frac{(R_1-R_2)\cos {\mathrm{\&gamma;}}_P}{2},y_N=\frac{(R_1-R_2)\sin {\mathrm{\&gamma;}}_P}{2}---\left(46\right)$

from the above derivation, a new type of vortex profile can be formed.

Referring to fig. 1: a main vortex line base circle O with the center at the point O, a left and a right secondary base circles O with the radii a and the distances D from the point O₁' and O₂' the centers of the three main and auxiliary base circles are on a straight line, and the included angle between the straight line of the three centers and the X axis is psi. The inner and outer main vortex lines generated by the main base circle are involute J₁E₁And J₂E₂The initial angles are alpha and-alpha, the radius of the main base circle is a, and the initial point J of the inner and outer vortex lines is set₁And J₂Not as a function of the angle beta but as a fixed point in fig. 3.

From J₁And J₂Two involute curves J extending inwards from the base circle₁K₁Segment and J₂K₂Segment, line segment J₁C₁Around the slave base circle O₁' rotating through angle delta tangent from base circle O with pure rolling₁In Q₁Point, line segment J₂C₂Rotated through the same angle and cut from the base circle O₂In Q₂And (4) point. Due to the introduction of the extended deformation involute J₁K₁And J₂K₂Thus, the center O of the arc is corrected₁And O₂Are all not on the tangent line of the base circle of the main vortex line and are formed by₁And O₂Then, arc correction is performed.

In fig. 1, the centers of the three master and slave base circles are on a straight line, the included angle between the straight line of the three centers and the X-axis is ψ, and the distances D between the centers of the left and right slave base circles and the master base circle are the following conditions:

when in use

$R_2=\frac{1}{2}[S-2\stackrel{\&OverBar;}{O_1{B}_1}-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})$

$=\frac{1}{2}\{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)-2[\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}$

$-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}]-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})\}>0$

And sin (delta-psi) < 0 and

Dsinψ+asin(δ′+ψ)≥

{a²+D²+2aDcosδ′-D²sin²δ′

$+[\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{]}^2{\}}^{\frac{1}{2}}\cos (\mathrm{\&gamma;}+\mathrm{\&psi;})$

$+\frac{1}{2}\{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)-2[\frac{a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}{2}$

$-\frac{a^2+D^2+2\mathrm{aD}\cos {\mathrm{\&delta;}}^{\&prime;}-D^2{\sin }^2{\mathrm{\&delta;}}^{\&prime;}}{2a(\mathrm{\&pi;}+\mathrm{\&psi;}-\mathrm{\&delta;})+2D(\sin {\mathrm{\&delta;}}^{\&prime;}-1)}]-a(\mathrm{\&pi;}-2\mathrm{\&alpha;})\}\sin (\mathrm{\&delta;}-\mathrm{\&psi;})$

wherein $\mathrm{\&gamma;}={\mathrm{\&delta;}}^{\&prime;}+\mathrm{\&psi;}-{\mathrm{tg}}^{-1}\left(\frac{\stackrel{\&OverBar;}{O_1{B}_1}}{\stackrel{\&OverBar;}{{\mathrm{OB}}_1}}\right)=\mathrm{\&beta;}-\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-{\mathrm{tg}}^{-1}\left(\frac{\stackrel{\&OverBar;}{O_1{B}_1}}{{\mathrm{OB}}_1}\right)$ In this case, a smooth profile can be formed.

The molded line of the scroll root section (i.e. the high-pressure area part) of the scroll compressor is composed of an extension correction molded line based on a three-base-circle molded line (TBC molded line for short) correction theory, and is compared with a scroll compressor composed of PMP molded lines with the same proportion size, as shown in FIGS. 5 and 6. It can be known that, under the same air input, the area of the exhaust cavity of the scroll compressor related to the invention is obviously smaller than that of the Japanese PMP type compressor, which means that the compression ratio is improved, and the thickness of the vortex line at the root is also increased, thus completely overcoming the defects of the PMP type compressor.

Claims (1)

1. A vortex compressor is composed of a vortex disk, and features that the root line of the vortex disk is the main involute segment J outside the main base circle O whose center is at original point₂E₂Connecting straight line segment E₂E₁Inner main involute section E of main base circle O₁J₁The center of the circle is x_O1′、y_O1′Auxiliary base circle of₁' inner correction run-off segment J₁K₁Correction of the great circle segment K₁N, correction of the minor arc section NK₂The center of the circle is x_O2′、y_O2′Auxiliary base circle of₂' outer correction run-off segment K₂J₂Composition is carried out; the radius of each of the three base circles is alpha, the distance from the center of the main base circle to the centers of the two auxiliary base circles is D, the centers of the three base circles are located on the same straight line, and the included angle between the straight line and the X axis is psi; wherein:

(1) outer main involute section J of main base circle O₂E₂The shape curve satisfies the equation:

$\left\{\begin{array}{c}x=x_0+\mathrm{\&alpha;}\&lsqb;\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})+\mathrm{\&theta;}\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})\&rsqb;\\ y=y_0+\mathrm{\&alpha;}\&lsqb;\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})-\mathrm{\&theta;}\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})\&rsqb;\end{array}\right.$

wherein x is₀＝0，y₀＝0，α＝α₀，α₀To form the starting angle of the outer main run-off segment, θ is chosen from $\mathrm{\&theta;}={\mathrm{\&theta;}}_{\min }=\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}+{\mathrm{\&alpha;}}_0$ Starting;

(2) connecting straight line segment E₂E₁The shape curve satisfies the equation:

straight line segment E₂E₁End point E of₂、E₁Are respectively connected with an involute J₂E₂、E₁J₁E of₂And E₁The points are the same point, so the coordinate values are known, E₂E₁The parameterized equation of (a) is:

$y=\mathrm{LN}(x_{E_1},y_{E_1},x_{E_2},y_{E_2},x)$

(3) inner main involute section E of main base circle O₁J₁The shape curve satisfies the equation:

$\left\{\begin{array}{c}x=x_0+\mathrm{\&alpha;}\&lsqb;\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})+\mathrm{\&theta;}\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})\&rsqb;\\ y=y_0+\mathrm{\&alpha;}\&lsqb;\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})-\mathrm{\&theta;}\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})\&rsqb;\end{array}\right.$

wherein x is₀＝0，y₀＝0，α＝-α₀，-α₀To form the starting angle of the inner main run-off segment, θ is taken to be $\mathrm{\&theta;}={\mathrm{\&theta;}}_{\min }=\frac{3\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-{\mathrm{\&alpha;}}_0$ Starting;

(4) auxiliary base circle O₁' inner correction run-off segment J₁K₁The shape curve satisfies the equation:

$\left\{\begin{array}{c}x={x_o}_{\stackrel{\&prime;}{1}}+\mathrm{\&alpha;}\&lsqb;\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})+\mathrm{\&theta;}\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})\&rsqb;\\ y={y_o}_{\stackrel{\&prime;}{1}}+\mathrm{\&alpha;}\&lsqb;\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})-\mathrm{\&theta;}\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})\&rsqb;\end{array}\right.$

wherein, $x_{O_1^{\&prime;}}=-D\cos \mathrm{\&psi;},$ $y_{O_1^{\&prime;}}=-D\sin \mathrm{\&psi;},$ ，α＝α₀₁， ${\mathrm{\&alpha;}}_{01}=\frac{3\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-\frac{L_1}{\mathrm{\&alpha;}},$ α₀₁to form an inner correction run-off segment J₁K₁The starting angle of (a) of (b), $L_1=\mathrm{\&alpha;}(\frac{3\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-{\mathrm{\&alpha;}}_0)-D,\mathrm{\&theta;}={\mathrm{\&alpha;}}_{k1},$ α_k1＝π-α₀₁+ψ+δ′， ${\mathrm{\&delta;}}^{\&prime;}=\frac{\mathrm{\&pi;}}{2}-\mathrm{\&delta;},$ delta is an inner correction run-off segment J₁K₁Unfolding angle: (5) correcting large arc section K₁The N-shaped curve satisfies the equation:

correcting large arc section K₁Endpoint K of N₁Correcting the involute segment J for the inner side₁K₁Upper point, center x₀₁、

y₁₀The point coordinates are:

$\left\{\begin{array}{c}{x}_{01}=-\stackrel{-}{{\mathrm{OO}}_1}\cos (\mathrm{\&gamma;}+\mathrm{\&psi;})\\ y_{01}=-\stackrel{-}{{\mathrm{OO}}_1}\sin (\mathrm{\&gamma;}+\mathrm{\&psi;})\end{array}\right.$

gamma is a corrected great arc section K₁N and corrected minor arc section NK₂The connecting line of the circle centers forms an included angle with the straight line of the circle centers of the three base circles,

correcting the great arc section K for the distance between the center of the central base circle and the center of the corrected great arc₁N and corrected minor arc section NK₂Tangent at point N;

$x_N=\frac{(R_1-R_2){\cos \mathrm{\&gamma;}}_P}{2},$ $y_N=\frac{(R_1-R_2){\sin \mathrm{\&gamma;}}_P}{2}$

R_tfor repairing large arc section K₁Radius of N, R₂For correcting small arc sections NK₂The radius of (a);

γ_Pδ' + ψ _, wherein:

(6) modified small arc section NK₂The shape curve satisfies the equation:

modified small arc section NK₂Endpoint K of₂Correcting the involute segment K for the outer side₂J₂Upper point, center of circle X₀₂、y₀₂The point coordinates are: .

$\left\{\begin{array}{c}{x}_{02}=-x_{01}\\ y_{02}=-y_{02}\end{array}\right.$

Correcting large arc section K₁N and corrected minor arc section NK₂Tangent at point N; (7) auxiliary base circle O₂' outer correction run-off segment K₂J₂The shape curve satisfies the equation:

$,\left\{\begin{array}{c}x=x_{O_2^{\&prime;}}+\mathrm{\&alpha;}\&lsqb;\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})+\mathrm{\&theta;}\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})\&rsqb;\\ y=y_{O_2^{\&prime;}}+\mathrm{\&alpha;}\&lsqb;\sin (\mathrm{\&theta;}+\mathrm{\&alpha;})-\mathrm{\&theta;}\cos (\mathrm{\&theta;}+\mathrm{\&alpha;})\&rsqb;\end{array}\right.$

wherein, $x_{O_2^{\&prime;}}=D\cos \mathrm{\&psi;},$ $Y_{O_2^{\&prime;}}=D\sin \mathrm{\&psi;}$ α＝α₀₂， ${\mathrm{\&alpha;}}_{02}=\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}-\frac{L_2}{\mathrm{\&alpha;}},$ α₀₂correction of the run-off segment K for the formation of the outer side₂J₂The starting angle of (a) of (b), $L_2=\mathrm{\&alpha;}(\frac{\mathrm{\&pi;}}{2}+\mathrm{\&psi;}+{\mathrm{\&alpha;}}_0)-D,$ θ=α_K2，

α_K2＝δ′+ψ-α₀₂， ${\mathrm{\&delta;}}^{\&prime;}=\frac{\mathrm{\&pi;}}{2}-\mathrm{\&delta;},$ delta is a side-corrected involute segment J₁K₁The flare angle.

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