CN106844869A - One kind is based on family of surfaces envelope（Face）The high-order curved surface parametric modeling method of principle - Google Patents

Abstract

The step of present invention provides a kind of high-order curved surface parametric modeling method based on family of surfaces envelope principle, its high-order curved surface parametric modeling method be：S1, analysis simple surface and its mathematical notation method；S2. Single parametric surface family method for expressing under one-parameter motion state is determined；The necessary and sufficient condition that S3, the envelope (face) of analysis Single parametric surface family are present；S4, high-order curved surface mathematical modeling；S41. establishment of coordinate system；S42. grinding wheel profile equation；S43. emery wheel, when emery wheel uses straight edge line wheel grinding, sets up high-order curved surface Mathematical Modeling relative to the family of surfaces equation that thrust bearing rotate counterclockwise is formed.The present invention represents the conjugate surface parameter of the high-order curved surface of high-order curved surface watt thrust bearing using the Differential Geometry method in analytic method, and the space diagram of high-order curved surface is given into displaying.

Description

High-order curved surface parametric modeling method of the one kind based on family of surfaces envelope (face) principle

Technical field

The present invention relates to technical field of bearings, a kind of high order based on family of surfaces envelope (face) principle is related in particular to bent Face parametric modeling method.

Background technology

High-order curved surface watt thrust bearing (thrust ring) tiling uses high-order curved surface, and curved surface (has startup load with planar section Structure design) intersection not by thrust ring axis (center), intersection eccentric throw W=D3/2, oil groove is in diametric(al) not insertion Bearing, wall construction of the oil groove at bearing top circle plays prevention lubricant medium leakage.High-order curved surface watt thrust bearing sets Meter mean camber designs eccentric form with plane intersection line, and rational plane and curved surface ratio are for keeping high-mechanic characteristic to weigh very much Will, and appropriateness increase bearing bore diameter size, very little is influenceed on bearing capacity, high-order curved surface watt thrust bearing bearing capacity is to knot in addition Structure parameter is insensitive, is particularly suited for the working environment that armament-related work rotation speed change is big, axial load is big.

High-order curved surface watt thrust bearing suitable at a high speed, heave-load device, such as express pump.High-order curved surface watt thrust bearing Main structure parameters：n：Tile fragment number；D1：Thrust ring external diameter (mm)；D2：Thrust ring internal diameter (mm)；W：Eccentric throw W=D3/2 (mm)；α：Plane opens up angle (°)；L：Oil groove width (mm)；h：Oil groove depth (mm).High-order curved surface watt thrust bearing (thrust ring) watt Face is conjugate surface, and the conjugate surface refers to two curved surfaces with the presence of contact line.Because space diagram is difficult to represent, institute in the hope of The conjugate surface in space often uses analytic method.

The content of the invention

For the defect of above-mentioned prior art, the present invention provides a kind of high-order curved surface based on family of surfaces envelope (face) principle Parametric modeling method, the high-order curved surface of high-order curved surface watt thrust bearing is represented using the Differential Geometry method in analytic method Conjugate surface parameter.

To achieve the above object, the technical solution used in the present invention is：

High-order curved surface parametric modeling method of the one kind based on family of surfaces envelope (face) principle, its high-order curved surface parametrization is built The step of mould method is：

S1, analysis simple surface and its mathematical notation method

In rectangular coordinate system in space, for the curved surface ∑ for giving, regard as moving point M according to certain rule ∑ The formed track of motion.Generally " certain rule " is represented with the equation that the coordinate (x, y, z) of M points is met, its form Have：

I, parameter type

Formula (1) represents that u and v is referred to as the parameter of ∑, and its vector equation is for the parameter type of curved surface ∑, or parameter：

R=r (u, v)={ x (u, v), y (u, v), z (u, v) } (2)

It is II, explicit

If the coordinate (x, y, z) of moving point M meets equation：

Z=f (x, y) or z=z (x, y) (3)

Deserve to be called the explicit representation that formula is curved surface ∑.As long as coordinate (x, y, z) meets (3), then the set that M (x, y, z) puts is just It is curved surface ∑.

It is III, implicit

If moving point M (x, y, z) meets equation：

F (x, y, z)=0 (4)

And F_z(x, y, z) ≠ 0, then deserve to be called the implicit representation that (4) formula is curved surface ∑, and ∑ is the set of moving point M.

Three of the above expression-form under certain condition, with equivalence.If curved surface ∑ table in the form of formula (1) Show, and function x (u, v), y (u, v), z (u, v) have continuous first-order partial derivative to independent variable u and v, while matrix：

Order rank (J)=2, then curved surface ∑ is called simple surface, the point on ∑ be normal point.In other words, by normal point The curved surface of composition is referred to as simple surface.The normal vector of every bit is non-vanishing vector, i.e. N=r in simple surface_u×r_v≠ 0, therefore Simple surface can use parameter type, explicit or implicit represent.But simultaneously it is to be noted that the singular point on curved surface is not complete Complete determining by curved surface itself, it is often relevant with the selection of coordinate system and the expression way of curved surface.

S2. Single parametric surface family method for expressing under one-parameter motion state is determined

Space curved surface moves (or change) with parameter a, will form family's curved surface.Corresponding certain a value, just has determination Curved surface is corresponded to therewith, then this race's curved surface is called Single parametric surface family.The representation of space curved surface race equally also has three kinds：

The parameter type and vector equation of I, family of surfaces

The parameter type of family of surfaces：

Wherein：(u, v) ∈ U, a ∈ D, U and D are real number sets.

Vector equation：

R=r (u, v, a)=x (u, v, a), y (u, v, a), z (u, v, a) }

The explicit representation of II, family of surfaces：

Z=f (x, y, a) or z=z (x, y, a)

The implicit representation of III, family of surfaces：

F (x, y, z, a)=0

The necessary and sufficient condition that S3, the envelope (face) of analysis Single parametric surface family are present

For the Single parametric surface family { s for giving_a, if space has a curved surface ∑, for arbitrary point p_a∈ ∑s, There is race's mean camber tangent with ∑ in the point；For arbitrary α ∈ D, there must be point p_a∈s_aSo that ∑ is in the point and s_aIt is tangent.Then claim ∑ is Single parametric surface family { s_aEnvelope, p_aReferred to as point of contact.

Therefore can simply be expressed as：

s_aWith ∑ in point p_aIt is tangent；∑ and s_aIn point p_aIt is tangent,

Then ∑ is called Single parametric surface family { s_aEnvelope.

The adequate condition that I, Single parametric surface family envelope are present

(u, v, the adequate condition that envelope a) is present is Single parametric surface family r=r：

Φ=(r_u, r_v, r_a)=0 and Φ_a≠0

The necessary condition that II, Single parametric surface family envelope are present

(u, v, the necessary condition that envelope a) is present is Single parametric surface family r=r：

Φ (u, v, a)=(r_u, r_v, r_a)=0

The expression-form of III, Single parametric surface family envelope

The parametric equation of envelope ∑：

The vector equation of envelope ∑：

S4, high-order curved surface mathematical modeling

S41. establishment of coordinate system

Hydrldynamic pressure lubrication thrust bearing high-order curved surface tiling is manufactured using the method for specific shape of generatrix wheel grinding.It is high Establishment of coordinate system is as follows during secondary curved surface Mathematical Models：Rectangular coordinate system o₁-x₁ y₁ z₁Connected firmly respectively with o-x y z On emery wheel and thrust bearing, wherein grinding wheel spindle and y₁Overlapping of axles, y₁It is W (during the relative bearing of emery wheel is turned round with the distance between z Heart side-play amount), thrust bearing axis is overlapped with z-axis, and thrust bearing planar section is located in o-x y planes, and y-axis is located at thrust axis High-order curved surface and plane intersection line position are held, positioned at y₁Emery wheel section circular curve in=μ+δ planes, P points are t emery wheel sections The contact point (characteristic point) of circular curve and high-order curved surface, initial time x₁Axle position in xoz coordinate surfaces, x₁Distance is s between x-axis (φ) (displacement function of the emery wheel relative to thrust bearing), z-axis and z₁Angle is β (machine tool structure guarantee) between axle.Moving axes It is o₁-x₁ y₁ z₁While (emery wheel) is around z-axis (bearing axis) rotate counterclockwise, and reciprocating linear is made in edge parallel to z-axis direction Motion is (while i.e. emery wheel is around thrust bearing axis rotate counterclockwise and along parallel to thrust bearing axis direction reciprocating linear fortune It is dynamic), t x₁Axle relative initial position corner is φ (t emery wheel corner).The curved surface normal vector of thrust bearing high order tiling It is acute angle (i.e. tiling is upward) with z-axis forward direction angle, revolving body emery wheel bus is non-straight edge line；

Emery wheel displacement function expression formula：

S (φ)=s (ω t)

φ=ω t

Wherein：Angular velocity of rotation (rad/s) of the ω-emery wheel relative to thrust bearing；

z₀- high order face number；

β-z-axis and z₁Angle between axle, the angle between emery wheel axis and thrust bearing axis is 90 ° of-β；

Corner of the φ-t emery wheel axis relative to thrust bearing；

R (δ)-emery wheel bus equation；

S42. grinding wheel profile equation

Vector form：

r₁=r₁(δ, θ)=i [(r+ δ tan α) cos θ]+j (u+ δ)+k [(r+ δ tan α) sin θ] (1)

Parametric form：

S43. emery wheel is relative to the family of surfaces equation that thrust bearing rotate counterclockwise is formed：

R=r₀+A_z(φ)A_x(β)(r_x+r₁)

Formula (2) is substituted into formula (3) to obtain：

Vector form：

Parametric form：

When emery wheel uses straight edge line wheel grinding, high-order curved surface Mathematical Modeling：

Vector form：

Wherein：

I=(r+ δ tan α) tan α cos β+(u+ δ) cos β

H=w sin β-s ' (φ) cos β

J=w tan α cos β+s ' (φ) tan α sin β

Parametric form：

X (δ, θ, φ)=(r+ δ tan α) (cos θ cos φ+sin θ sin β sin φ)-(u+ δ) sin φ cos β+w cos φ

Y (δ, θ, φ)=(r+ δ tan α) (sin φ cos θ-sin θ sin β cos φ)+(u+ δ) cos β cos φ+w sin φ

Z (δ, θ, φ)=(r+ δ tan α) sin θ cos β+(u+ δ) sin β+s (φ)

Wherein：

I=(r+ δ tan α) tan α cos β+(u+ δ) cos β

H=w sin β-s ' (φ) cos β

J=w tan α cos β+s ' (φ) tan α sin β

As the improvement to above-mentioned technical proposal,

Compared with prior art, the present invention has the advantages and positive effects that：

The present invention is the high-order curved surface parametric modeling method based on family of surfaces envelope (face) principle；Using in analytic method Differential Geometry method represents the conjugate surface parameter of the high-order curved surface of high-order curved surface watt thrust bearing.Obtained using this method Mathematical Modeling includes the tool shape and envelope movement parameter information being related in a kind of complicated high-order curved surface manufacturing process, can Design, manufacture integrated modelling are realized, overcomes the complex-curved of traditional modeling method design to introduce principle in the fabrication process The low critical defect of the curved surface accuracy of manufacture caused by error.

Brief description of the drawings

In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing The accompanying drawing to be used needed for having technology description is briefly described, it should be apparent that, drawings in the following description are only this Some embodiments of invention, for those of ordinary skill in the art, without having to pay creative labor, may be used also Other accompanying drawings are obtained with according to these accompanying drawings.

Fig. 1 is emery wheel relief grinding thrust bearing high order watt curved surface schematic diagram；

Fig. 2 is the graph of a relation of emery wheel coordinate system and thrust bearing coordinate system；

Fig. 3 is y₁=μ+δ emery wheels section circular curve figure；

Fig. 4 is grinding wheel shape Parameter Map.

Specific embodiment

Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is carried out clear, complete Site preparation is described, it is clear that described embodiment is only a part of embodiment of the invention, rather than whole embodiments.It is based on Embodiment in the present invention, it is every other that those of ordinary skill in the art are obtained under the premise of creative work is not made Embodiment, any modification, equivalent substitution and improvements made etc., should be included within the scope of the present invention.

It is as shown in Fig. 1,2,3 and 4, the step of its high-order curved surface parametric modeling method of the invention：

S1, analysis simple surface and its mathematical notation method

In rectangular coordinate system in space, for the curved surface ∑ for giving, regard as moving point M according to certain rule ∑ The formed track of motion.Generally " certain rule " is represented with the equation that the coordinate (x, y, z) of M points is met, its form Have：

I, parameter type

Formula (1) represents that u and v is referred to as the parameter of ∑, and its vector equation is for the parameter type of curved surface ∑, or parameter：

R=r (u, v)={ x (u, v), y (u, v), z (u, v) } (2)

It is II, explicit

If the coordinate (x, y, z) of moving point M meets equation：

Z=f (x, y) or z=z (x, y) (3)

Deserve to be called the explicit representation that formula is curved surface ∑.As long as coordinate (x, y, z) meets (3), then the set that M (x, y, z) puts is just It is curved surface ∑.

It is III, implicit

If moving point M (x, y, z) meets equation：

F (x, y, z)=0 (4)

And F_z(x, y, z) ≠ 0, then deserve to be called the implicit representation that (4) formula is curved surface ∑, and ∑ is the set of moving point M.

Three of the above expression-form under certain condition, with equivalence.If curved surface ∑ table in the form of formula (1) Show, and function x (u, v), y (u, v), z (u, v) have continuous first-order partial derivative to independent variable u and v, while matrix：

Order rank (J)=2, then curved surface ∑ is called simple surface, the point on ∑ be normal point.In other words, by normal point The curved surface of composition is referred to as simple surface.The normal vector of every bit is non-vanishing vector, i.e. N=r in simple surface_u×r_v≠ 0, therefore Simple surface can use parameter type, explicit or implicit represent.But simultaneously it is to be noted that the singular point on curved surface is not complete Complete determining by curved surface itself, it is often relevant with the selection of coordinate system and the expression way of curved surface.

S2. Single parametric surface family method for expressing under one-parameter motion state is determined

Space curved surface moves (or change) with parameter a, will form family's curved surface.Corresponding certain a value, just has determination Curved surface is corresponded to therewith, then this race's curved surface is called Single parametric surface family.The representation of space curved surface race equally also has three kinds：

The parameter type and vector equation of I, family of surfaces

The parameter type of family of surfaces：

Wherein：(u, v) ∈ U, a ∈ D, U and D are real number sets.

Vector equation：

R=r (u, v, a)=x (u, v, a), y (u, v, a), z (u, v, a) }

The explicit representation of II, family of surfaces：

Z=f (x, y, a) or z=z (x, y, a)

The implicit representation of III, family of surfaces：

F (x, y, z, a)=0

The necessary and sufficient condition that S3, the envelope (face) of analysis Single parametric surface family are present

For the Single parametric surface family { s for giving_a, if space has a curved surface ∑, for arbitrary point p_a∈ ∑s, There is race's mean camber tangent with ∑ in the point；For arbitrary α ∈ D, there must be point p_a∈s_aSo that ∑ is in the point and s_aIt is tangent.Then claim ∑ is Single parametric surface family { s_aEnvelope, p_aReferred to as point of contact.

Therefore can simply be expressed as：

s_aWith ∑ in point p_aIt is tangent；∑ and s_aIn point p_aIt is tangent,

Then ∑ is called Single parametric surface family { s_aEnvelope.

The adequate condition that I, Single parametric surface family envelope are present

(u, v, the adequate condition that envelope a) is present is Single parametric surface family r=r：

Φ=(r_u, r_v, r_a)=0 and Φ_a≠0

The necessary condition that II, Single parametric surface family envelope are present

(u, v, the necessary condition that envelope a) is present is Single parametric surface family r=r：

Φ (u, v, a)=(r_u, r_v, r_a)=0

The expression-form of III, Single parametric surface family envelope

The parametric equation of envelope ∑：

The vector equation of envelope ∑：

S4, high-order curved surface mathematical modeling

S41. establishment of coordinate system

Hydrldynamic pressure lubrication thrust bearing high-order curved surface tiling is manufactured using the method for specific shape of generatrix wheel grinding.It is high Establishment of coordinate system is as follows during secondary curved surface Mathematical Models：Rectangular coordinate system o₁-x₁ y₁ z₁Connected firmly respectively with o-x y z On emery wheel and thrust bearing, wherein grinding wheel spindle and y₁Overlapping of axles, y₁It is W (during the relative bearing of emery wheel is turned round with the distance between z Heart side-play amount), thrust bearing axis is overlapped with z-axis, and thrust bearing planar section is located in o-x y planes, and y-axis is located at thrust axis High-order curved surface and plane intersection line position are held, positioned at y₁Emery wheel section circular curve in=μ+δ planes, P points are t emery wheel sections The contact point (characteristic point) of circular curve and high-order curved surface, initial time x₁Axle position in xoz coordinate surfaces, x₁Distance is s between x-axis (φ) (displacement function of the emery wheel relative to thrust bearing), z-axis and z₁Angle is β (machine tool structure guarantee) between axle.Moving axes It is o₁-x₁ y₁ z₁While (emery wheel) is around z-axis (bearing axis) rotate counterclockwise, and reciprocating linear is made in edge parallel to z-axis direction Motion is (while i.e. emery wheel is around thrust bearing axis rotate counterclockwise and along parallel to thrust bearing axis direction reciprocating linear fortune It is dynamic), t x₁Axle relative initial position corner is φ (t emery wheel corner).The curved surface normal vector of thrust bearing high order tiling It is acute angle (i.e. tiling is upward) with z-axis forward direction angle, revolving body emery wheel bus is non-straight edge line；

Emery wheel displacement function expression formula：

S (φ)=s (ω t)

φ=ω t

Wherein：Angular velocity of rotation (rad/s) of the ω-emery wheel relative to thrust bearing；

z₀- high order face number；

β-z-axis and z₁Angle between axle, the angle between emery wheel axis and thrust bearing axis is 90 ° of-β；

Corner of the φ-t emery wheel axis relative to thrust bearing；

R (δ)-emery wheel bus equation；

S42. grinding wheel profile equation

Vector form：

r₁=r₁(δ, θ)=i [(r+ δ tan α) cos θ]+j (u+ δ)+k [(r+ δ tan α) sin θ] (1)

Parametric form：

S43. emery wheel is relative to the family of surfaces equation that thrust bearing rotate counterclockwise is formed：

R=r₀+A_z(φ)A_x(β)(r_x+r₁)

Formula (2) is substituted into formula (3) to obtain：

Vector form：

Parametric form：

When emery wheel uses straight edge line wheel grinding, high-order curved surface Mathematical Modeling：

Vector form：

Wherein：

I=(r+ δ tan α) tan α cos β+(u+ δ) cos β

H=w sin β-s ' (φ) cos β

J=w tan α cos β+s ' (φ) tan α sin β

Parametric form：

X (δ, θ, φ)=(r+ δ tan α) (cos θ cos φ+sin θ sin β sin φ)-(u+ δ) sin φ cos β+w cos φ

Y (δ, θ, φ)=(r+ δ tan α) (sin φ cos θ-sin θ sin β cos φ)+(u+ δ) cos β cos φ+w sin φ

Z (δ, θ, φ)=(r+ δ tan α) sin θ cos β+(u+ δ) sin β+s (φ)

Wherein：

I=(r+ δ tan α) tan α cos β+(u+ δ) cos β

H=w sin β-s ' (φ) cos β

J=w tan α cos β+s ' (φ) tan α sin β

General principle of the invention, principal character and advantages of the present invention has been shown and described above.The technology of the industry Personnel it should be appreciated that the present invention is not limited to the above embodiments, simply explanation described in above-described embodiment and specification this The principle of invention, various changes and modifications of the present invention are possible without departing from the spirit and scope of the present invention, these changes Change and improvement all fall within the protetion scope of the claimed invention.The claimed scope of the invention by appending claims and its Equivalent is defined.

Claims (1)

1. one kind is based on the high-order curved surface parametric modeling method of family of surfaces envelope (face) principle, it is characterised in that：Its high order is bent The step of face parametric modeling method is：

S1, analysis simple surface and its mathematical notation method

In rectangular coordinate system in space, for the curved surface ∑ for giving, ∑ is regarded as moving point M is moved according to certain rule The track for being formed, the equation met with the coordinate (x, y, z) of M points represents that its form has：

I, parameter type

$\mathrm{\Σ}:\left(\begin{array}{c}x=x(u,v)\\ y=y(u,v)\\ z=z(u,v)\end{array}\right.,(u,v)\∈U---\left(1\right)$

Formula (1) represents that u and v is referred to as the parameter of ∑, and its vector equation is for the parameter type of curved surface ∑, or parameter：

R=r (u, v)={ x (u, v), y (u, v), z (u, v) } (2)

It is II, explicit

If the coordinate (x, y, z) of moving point M meets equation：

Z=f (x, y) or z=z (x, y) (3)

The explicit representation that formula is curved surface ∑ is deserved to be called, as long as coordinate (x, y, z) meets (3), then the set of M (x, y, z) points is exactly bent Face ∑；

It is III, implicit

If moving point M (x, y, z) meets equation：

F (x, y, z)=0 (4)

And F_z(x, y, z) ≠ 0, then deserve to be called the implicit representation that (4) formula is curved surface ∑, and ∑ is the set of moving point M；

Three of the above expression-form under certain condition, with equivalence；If curved surface ∑ is represented in the form of formula (1), And function x (u, v), y (u, v), z (u, v) have continuous first-order partial derivative to independent variable u and v, while matrix：

$J=\left(\begin{array}{ccc}{x}_u& y_u& z_u\\ x_v& y_v& z_v\end{array}\right)$

Order rank (J)=2, then curved surface ∑ is called simple surface, the point on ∑ be normal point；In other words, it is made up of normal point Curved surface be referred to as simple surface；The normal vector of every bit is non-vanishing vector, i.e. N=r in simple surface_u×r_v≠ 0, therefore simply Curved surface can use parameter type, explicit or implicit represent；

S2. Single parametric surface family method for expressing under one-parameter motion state is determined

Space curved surface moves (or change) with parameter a, will form family's curved surface, corresponds to certain a value, just has the curved surface of determination Correspond to therewith, then this race's curved surface is called Single parametric surface family；The representation of space curved surface race equally also has three kinds：

The parameter type and vector equation of I, family of surfaces

The parameter type of family of surfaces：

$\left\{s_a\right\}:\left(\begin{array}{c}x=x(u,v,a)\\ y=y(u,v,a)\\ z=z(u,v,a)\end{array}\right.$

Wherein：(u, v) ∈ U, a ∈ D, U and D are real number sets；

Vector equation：

R=r (u, v, a)=x (u, v, a), y (u, v, a), z (u, v, a) }

The explicit representation of II, family of surfaces：

Z=f (x, y, a) or z=z (x, y, a)

The implicit representation of III, family of surfaces：

F (x, y, z, a)=0

The necessary and sufficient condition that S3, the envelope (face) of analysis Single parametric surface family are present

For the Single parametric surface family { s for giving_a, if space has a curved surface ∑, for arbitrary point p_a∈ ∑s, there is race Mean camber is tangent with ∑ in the point；For arbitrary α ∈ D, there must be point p_a∈s_aSo that ∑ is in the point and s_aIt is tangent；Then claiming ∑ is Single parametric surface family { s_aEnvelope, p_aReferred to as point of contact；

Therefore can simply be expressed as：

s_aWith ∑ in point p_aIt is tangent；∑ and s_aIn point p_aIt is tangent,

Then ∑ is called Single parametric surface family { s_aEnvelope；

The adequate condition that I, Single parametric surface family envelope are present

(u, v, the adequate condition that envelope a) is present is Single parametric surface family r=r：

Φ=(r_u, r_v, r_a)=0 and Φ_a≠0

The necessary condition that II, Single parametric surface family envelope are present

(u, v, the necessary condition that envelope a) is present is Single parametric surface family r=r：

Φ (u, v, a)=(r_u, r_v, r_a)=0

The expression-form of III, Single parametric surface family envelope

The parametric equation of envelope ∑：

$\mathrm{\Σ}:\left(\begin{array}{c}x=x(u,v,a)\\ y=y(u,v,a)\\ z=z(u,v,a)\\ \mathrm{\Φ}=\frac{\∂(x,y,z)}{\∂(u,v,a)}=0\end{array}\right.$

The vector equation of envelope ∑：

$\mathrm{\Σ}:\left(\begin{array}{c}r=r(u,v,a)\\ (r_u,r_v,r_a)=0\end{array}\right.$

S4, high-order curved surface mathematical modeling

S41. establishment of coordinate system

Establishment of coordinate system is as follows during high-order curved surface Mathematical Models：Rectangular coordinate system o₁-x₁y₁z₁Connected firmly respectively with o-xyz On emery wheel and thrust bearing, wherein grinding wheel spindle and y₁Overlapping of axles, y₁It is W (during the relative bearing of emery wheel is turned round with the distance between z Heart side-play amount), thrust bearing axis is overlapped with z-axis, and thrust bearing planar section is located in o-xy planes, and y-axis is located at thrust axis High-order curved surface and plane intersection line position are held, positioned at y₁Emery wheel section circular curve in=μ+δ planes, P points are t emery wheel sections The contact point (characteristic point) of circular curve and high-order curved surface, initial time x₁Axle position in xoz coordinate surfaces, x₁Distance is s between x-axis (φ) (displacement function of the emery wheel relative to thrust bearing), z-axis and z₁Angle is β (machine tool structure guarantee) between axle；Moving axes It is o₁-x₁y₁z₁While (emery wheel) is around z-axis (bearing axis) rotate counterclockwise, and make reciprocating linear fortune along parallel to z-axis direction It is dynamic (to be transported while i.e. emery wheel is around thrust bearing axis rotate counterclockwise and along parallel to thrust bearing axis direction reciprocating linear It is dynamic), t x₁Axle relative initial position corner is φ (t emery wheel corner)；The curved surface normal vector of thrust bearing high order tiling It is acute angle (i.e. tiling is upward) with z-axis forward direction angle, revolving body emery wheel bus is non-straight edge line；

Emery wheel displacement function expression formula：

S (φ)=s (ω t)

φ=ω t

Wherein：Angular velocity of rotation (rad/s) of the ω-emery wheel relative to thrust bearing；

z₀- high order face number；

β-z-axis and z₁Angle between axle, the angle between emery wheel axis and thrust bearing axis is 90 ° of-β；

Corner of the φ-t emery wheel axis relative to thrust bearing；

R (δ)-emery wheel bus equation；

S42. grinding wheel profile equation

Vector form：

r₁=r₁(δ, θ)=i [(r+ δ tan α) cos θ]+j (u+ δ)+k [(r+ δ tan α) sin θ] (1)

Parametric form：

$\left(\begin{array}{c}{x}_1(\mathrm{\δ},\mathrm{\θ})=(r+\mathrm{\δ}tan\mathrm{\α})cos\mathrm{\θ}\\ y_1(\mathrm{\δ},\mathrm{\θ})=u+\mathrm{\δ}\\ z_1(\mathrm{\δ},\mathrm{\θ})=(r+\mathrm{\δ}tan\mathrm{\α})sin\mathrm{\θ}\end{array}\right.---\left(2\right)$

S43. emery wheel is relative to the family of surfaces equation that thrust bearing rotate counterclockwise is formed：

R=r₀+A_z(φ)A_x(β)(r_x+r₁)

$\left[\begin{array}{c}x(\mathrm{\δ},\mathrm{\θ},\mathrm{\φ})\\ y(\mathrm{\δ},\mathrm{\θ},\mathrm{\φ})\\ z(\mathrm{\δ},\mathrm{\θ},\mathrm{\φ})\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ s\left(\mathrm{\φ}\right)\end{array}\right]+\left[\begin{array}{ccc}\cos \mathrm{\φ}& -\sin \mathrm{\φ}& 0\\ \sin \mathrm{\φ}& \cos \mathrm{\φ}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \mathrm{\β}& -\sin \mathrm{\β}\\ 0& \sin \mathrm{\β}& \cos \mathrm{\β}\end{array}\right]\left[\begin{array}{c}{x}_1(\mathrm{\δ},\mathrm{\θ})+w\\ y_1(\mathrm{\δ},\mathrm{\θ})\\ z_1(\mathrm{\δ},\mathrm{\θ})\end{array}\right]$

$\left[\begin{array}{c}x(\mathrm{\δ},\mathrm{\θ},\mathrm{\φ})\\ y(\mathrm{\δ},\mathrm{\θ},\mathrm{\φ})\\ z(\mathrm{\δ},\mathrm{\θ},\mathrm{\φ})\end{array}\right]=\left[\begin{array}{c}{x}_1(\mathrm{\δ},\mathrm{\θ})cos\mathrm{\φ}+wcos\mathrm{\φ}-y_1(\mathrm{\δ},\mathrm{\θ})cos\mathrm{\β}sin\mathrm{\φ}+z_1(\mathrm{\δ},\mathrm{\θ})sin\mathrm{\β}sin\mathrm{\φ}\\ x_1(\mathrm{\δ},\mathrm{\θ})sin\mathrm{\φ}+w\sin \mathrm{\φ}+y_1(\mathrm{\δ},\mathrm{\θ})cos\mathrm{\β}\cos \mathrm{\φ}-z_1(\mathrm{\δ},\mathrm{\θ})\sin \mathrm{\β}cos\mathrm{\φ}\\ y_1(\mathrm{\δ},\mathrm{\θ})sin\mathrm{\β}+z_1(\mathrm{\δ},\mathrm{\θ})cos\mathrm{\β}+s\left(\mathrm{\φ}\right)\end{array}\right]---\left(3\right)$

Formula (2) is substituted into formula (3) to obtain：

Vector form：

Parametric form：

When emery wheel uses straight edge line wheel grinding, high-order curved surface Mathematical Modeling：

Vector form：

Wherein：

$\mathrm{\θ}=2atan\left(\frac{-H+\sqrt{H^2+I^2-J^2}}{J-I}\right)$

I=(r+ δ tan α) tan α cos β+(u+ δ) cos β

H=wsin β-s ' (φ) cos β

J=wtan α cos β+s ' (φ) tan α sin β

Parametric form：

X (δ, θ, φ)=(r+ δ tan α) (cos θ cos φ+sin θ sin β sin φ)-(u+ δ) sin φ cos β+w cos φ

Y (δ, θ, φ)=(r+ δ tan α) (sin φ cos θ-sin θ sin β cos φ)+(u+ δ) cos β cos φ+w sin φ

Z (δ, θ, φ)=(r+ δ tan α) sin θ cos β+(u+ δ) sin β+s (φ)

Wherein：

$\mathrm{\θ}=2atan\left(\frac{-H+\sqrt{H^2+I^2-J^2}}{J-I}\right)$

I=(r+ δ tan α) tan α cos β+(u+ δ) cos β

H=w sin β-s ' (φ) cos β

J=w tan α cos β+s ' (φ) tan α sin β.