CN107345567A - A kind of coplanar axis gear mechanism that active line tooth is constructed with conical spiral - Google Patents

A kind of coplanar axis gear mechanism that active line tooth is constructed with conical spiral Download PDF

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CN107345567A
CN107345567A CN201710768458.5A CN201710768458A CN107345567A CN 107345567 A CN107345567 A CN 107345567A CN 201710768458 A CN201710768458 A CN 201710768458A CN 107345567 A CN107345567 A CN 107345567A
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CN107345567B (en
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陈扬枝
李政
谢雄敦
吕月玲
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South China University of Technology SCUT
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/17—Toothed wheels
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation

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  • Pure & Applied Mathematics (AREA)
  • Mechanical Engineering (AREA)
  • Gears, Cams (AREA)
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Abstract

The invention discloses a kind of coplanar axis gear mechanism that active line tooth is constructed with conical spiral, including driving wheel and driven pulley, the driving wheel and driven pulley meet line gear space curve mesh theory and use and interaxial angle, the adaptable conical spiral of gearratio carries out active exposure line construction, the present invention active line tooth and include two contact lines respectively from the flank of tooth of moving-wire tooth, two active exposure lines and two driven contact lines are two pairs of intermeshing conjugation inter_curves, respectively positioned at active line tooth and from the both sides of moving-wire tooth, it can be achieved without sideshake rotating Bidirectional driving.The present invention has the advantages that the bulk of line gear is small, gearratio is big；The shape of driving wheel can use the conical spiral of any drift angle, have the characteristics of space availability ratio is high, sliding ratio is small；It can be achieved without sideshake rotating Bidirectional driving, and easy to process, microminiature machinery, micromachine and the application of conventional mechanical field especially suitable for limited space.

Description

A kind of coplanar axis gear mechanism that active line tooth is constructed with conical spiral
Technical field
It is specially a kind of that the same flat of active line tooth is constructed with conical spiral the invention belongs to gear-driven technical field Face axis gear mechanism.
Background technology
Gear drive is that a kind of most widely used transmission and transmission technology, line gear transmission can realize axis with any Angle of intersection or transmission staggeredly and big speed ratio.Line gear before this is also known as space curve engagement wheel, joins in transmission process It is to grip space curve, that is, a pair driving wheel contact lines and driven pulley contact line gripped altogether altogether a pair with what is engaged.Existing line tooth Mechanism is taken turns unanimously using circular helix construction active exposure line, as Chinese patent (number of patent application 2010101059023) is public " a kind of angular bevel gear mechanism " opened, in the case of driving of intersecting axes, the constant value of circular helix engagement radius can not and phase The distance between axles of quadrature axis change is adapted, and causes low space utilization, and sliding ratio is big.Therefore, optimize line gear active exposure line Shape is necessary.In many use occasions, it is desirable to which gear mechanism has the function of bi-directional power, but existing line The line tooth of gear mechanism is moved along a contact line by flank profil and formed, such as Chinese patent (number of patent application 2015105709269) Disclosed " a kind of bump arc gear mechanism for the driving of parallel axes ", each only have one to participate in engagement on the line tooth Contact line, therefore backlash be present in the line gear of this structure, and single-direction transmission can only be realized.
The content of the invention
In view of this, the present invention is directed to propose a kind of coplanar axis gear mechanism that active line tooth is constructed with conical spiral Structure, the mechanism is with space availability ratio is high, sliding ratio is small, can realize rotating Bidirectional driving, with higher contact strength With bending strength, be easy to the features such as digital control processing, microminiature machinery, micromachine and conventional machine especially suitable for limited space Apply in tool field.
Above-mentioned technical purpose is realized, the present invention adopts the following technical scheme that：
A kind of coplanar axis gear mechanism that active line tooth is constructed with conical spiral, including driving wheel and driven pulley, The driving wheel and driven pulley meet line gear space curve mesh theory, i.e. the active exposure line of spatial conjugation and driven contact Line realizes engaged transmission with a contact form, and the space curve mesh theory is related to following space coordinates, coordinate system o0- x0y0z0With coordinate system o1-x1y1z1The respectively fixed coordinate system of driving wheel and with moving coordinate system, coordinate system op-xpypzpAnd coordinate It is o2-x2y2z2The respectively fixed coordinate system of driven pulley and with moving coordinate system；Driving wheel axis of rotation and z0(z1) overlapping of axles, from Driving wheel axis of rotation and z2(zp) overlapping of axles, angle is θ between driving wheel and driven pulley axis of rotation0, angle can be between the axis Intersected according to the design needs with any angle, any angle refers to the unspecified angle more than or equal to 0 ° and less than 180 °；It is described Driving wheel and driven pulley are respectively with angular speedWithAround axle z1And z2Axle rotates, in certain time, driving wheel and driven pulley difference Turn over angleWithIt is described using conical spiral construction active line tooth coplanar axis gear mechanism mesh equation as：
Wherein, the active exposure line is in coordinate system o1-x1y1z1Parametric equation be：
Wherein, t1For parameter, t1s、t1eFor parameter t1Span, i.e., on active exposure line the starting point of meshing and Engage terminating point position；
The driven contact line is in coordinate system o1-x1y1z1Parametric equation be：
Wherein,I is gearratio.
Further, the active exposure line of the mechanism is conical spiral, and driven contact line requires according to axis angle Circular helix, conical spiral or plane Archimedes spiral, active exposure line and driven contact line parametric equation can be used For：
Wherein, t1And t2For parameter, θ1And θ2The respectively semicircle cone-apex angle of active exposure line and driven contact line, m1、 n1And m2、n2Respectively active exposure line and the helix parameter of driven contact line, k1And k2For rotation direction parameter, when rotation direction parameter is 1 When, conical spiral is left-handed, and when rotation direction parameter is -1, conical spiral is dextrorotation.
Further, there are two contact lines on each line tooth of the driving wheel and driven pulley, and then rotating can be realized Bidirectional driving, on driving wheel, two contact lines include the first active exposure line and the second active exposure line；In driven pulley On, two contact lines include the first driven contact line and the second driven contact line；The second active exposure line is by described First active exposure line z1Axle rotatesObtain, the parametric equation difference of the first active exposure line and the second active exposure line For：
Similarly, the described second driven contact line is by the described first driven contact line z2Axle rotatesObtain, described first The parametric equation of driven contact line and the second driven contact line is respectively：
Further, the line transverse tooth thickness degree of the coplanar axis gear mechanism and space width are equal, the line transverse tooth thickness degree and Space width defines in the shaft section of line gear, i.e. the circular cone on the active exposure line and driven contact line where any point Straight edge line direction defines the line transverse tooth thickness degree and space width of driving wheel and driven pulley respectively.
Further, a lateral tooth flank of the active line tooth of the driving wheel, it is to be led by one section of circular arc profile along described first Dynamic contact line and a driving wheel transverse tooth thickness boost line are moved and formed, another lateral tooth flank of the active line tooth, are by one section of circle Curved tooth exterior feature is moved and formed along the second active exposure line and another driving wheel transverse tooth thickness boost line；The driven pulley it is driven One lateral tooth flank of line tooth, it is to be moved by one section of circular arc profile along the described first driven contact line and a driven pulley transverse tooth thickness boost line And formed, another lateral tooth flank from moving-wire tooth, it is along the described second driven contact line and another by one section of circular arc profile Driven pulley transverse tooth thickness boost line is moved and formed；The circular arc profile of the circular arc profile of the driving wheel and the driven pulley is located at respectively On active exposure line and the normal plane of driven contact line；It is auxiliary that two driving wheel transverse tooth thickness boost line includes the first driving wheel transverse tooth thickness Index contour and the second driving wheel transverse tooth thickness boost line, be respectively the first active exposure line and the second active exposure line in the normal direction Equidistant curve；Two driven pulley transverse tooth thickness boost line includes the first driven pulley transverse tooth thickness boost line and the second driving wheel transverse tooth thickness aids in Line, it is the equidistant curve of the first driven contact line and the second driven contact line in the normal direction respectively.It can be achieved without sideshake Rotating Bidirectional driving.
Further, the parametric equation of the first driving wheel transverse tooth thickness boost line and the second driving wheel transverse tooth thickness boost line is distinguished For：
Wherein, r is the radius of the flank profil circular arc, and φ is the driving wheel transverse tooth thickness boost line and driven pulley transverse tooth thickness boost line Directioin parameter, N1It is the number of teeth of driving wheel；
The parametric equation of the first driven pulley transverse tooth thickness boost line and the second driven pulley transverse tooth thickness boost line is respectively：
Wherein, r is the radius of the flank profil circular arc, and φ is the driving wheel transverse tooth thickness boost line and driven pulley transverse tooth thickness boost line Directioin parameter, N2It is the number of teeth of driven pulley.
Further, described driving wheel wheel body is the drift angle cone equal with the first active exposure line drift angle； Described driven pulley wheel body is the drift angle cone equal with the described first driven contact line drift angle, the active line tooth and driven Line tooth is distributed on the wheel body of driving wheel and driven pulley.
The present invention has the following advantages compared with prior art:
(1) it is compact-sized：Driving wheel and driven pulley form a pair of transmissions, should compared with traditional microminiature gear Powertrain arrangement very simple；In the case of driving of intersecting axes, the engagement radius of conical spiral is easy to change with concurrent aces Distance between axles be adapted, space availability ratio is high, can greatly save installing space, and sliding ratio is low, and vibration and noise are small, fatigue Long lifespan.
(2) gearratio is big：With the transmission mechanisms such as existing spur gear, helical gear be driven compared with, it is possible to achieve single-stage it is big Gearratio, high contact ratio transmission.
(3) without backlash：Compared with existing line gear mechanism, active line tooth and the flank of tooth both sides from moving-wire tooth are distinguished Comprising two contact lines, the two-way no sideshake even running of rotating can be achieved；The line tooth footpath to depending on wheel body, have compared with Good bearing capacity and operation stability.
Brief description of the drawings
Fig. 1 is the engagement coordinate schematic diagram in embodiment.
Fig. 2 is shaft section default adopted line transverse tooth thickness degree and space width schematic diagram in embodiment.
Fig. 3 is contact line and transverse tooth thickness boost line schematic diagram in embodiment.
Fig. 4 is driving wheel schematic diagram in embodiment.
Fig. 5 is driven pulley schematic diagram in embodiment.
Fig. 6 is driven wheel schematic diagram in embodiment.
Embodiment
With reference to specific embodiment, the present invention is described in detail.Following examples will be helpful to the technology of this area Personnel further understand the present invention, but the invention is not limited in any way.It should be pointed out that the ordinary skill to this area For personnel, without departing from the inventive concept of the premise, various modifications and improvements can be made.These belong to the present invention The protection domain of patent.
A kind of coplanar axis gear mechanism that active line tooth is constructed with conical spiral, including driving wheel and driven pulley, The driving wheel and driven pulley meet line gear space curve mesh theory, i.e. the active exposure line of spatial conjugation and driven contact Line realizes engaged transmission with a contact form.
The space curve mesh theory is related to following space coordinates, coordinate system o0-x0y0z0With coordinate system o1-x1y1z1 The respectively fixed coordinate system of driving wheel and with moving coordinate system, coordinate system op-xpypzpWith coordinate system o2-x2y2z2It is respectively driven The fixed coordinate system of wheel and with moving coordinate system.Driving wheel axis of rotation and z0(z1) overlapping of axles, driven pulley axis of rotation and z2(zp) Overlapping of axles, angle is θ between driving wheel and driven pulley axis of rotation0, angle can be according to the design needs with any between the axis Angular cross, any angle refer to the unspecified angle more than or equal to 0 ° and less than 180 °, as shown in Figure 1.Driving wheel and driven Wheel is respectively with angular speedWithAround axle z1And z2Axle rotates, and in certain time, driving wheel and driven pulley turn over angle respectivelyWithIt is described using conical spiral construction active line tooth coplanar axis gear mechanism mesh equation as：
Wherein, active exposure line is in coordinate system o1-x1y1z1Parametric equation be：
t1For parameter, t1s、t1eFor parameter t1Span, i.e., on active exposure line the starting point of meshing and engagement eventually Stop position.
Driven contact line is in coordinate system o1-x1y1z1Parametric equation be：
Wherein,I is gearratio.
The active exposure line of the mechanism is conical spiral, and driven contact line can use cylinder according to axis angle requirement Helix, conical spiral or plane Archimedes spiral, active exposure line and driven contact line parametric equation are：
Wherein, t1And t2For parameter, θ1And θ2The respectively semicircle cone-apex angle of active exposure line and driven contact line, m1、 n1And m2、n2Respectively active exposure line and the helix parameter of driven contact line, k1And k2For rotation direction parameter, when rotation direction parameter is 1 When, conical spiral is left-handed, and when rotation direction parameter is -1, conical spiral is dextrorotation.
There are two contact lines on each line tooth of driving wheel and driven pulley, and then rotating Bidirectional driving can be realized. On driving wheel, two contact lines include the first active exposure line and the second active exposure line；On the driven wheel, described two Contact line includes the first driven contact line and the second driven contact line.The second active exposure line is by first active exposure Line z1Axle rotatesObtain.The parametric equation of the first active exposure line and the second active exposure line is respectively：
Similarly, the described second driven contact line is by the described first driven contact line z2Axle rotatesObtain.Described first The parametric equation of driven contact line and the second driven contact line is respectively：
The line transverse tooth thickness degree 1 and space width 2 of the mechanism are equal.The shaft section of the line transverse tooth thickness degree and space width in line gear Interior definition, i.e., the circular cone straight edge line direction on the active exposure line and driven contact line where any point define actively respectively The line transverse tooth thickness degree and space width of wheel and driven pulley, as shown in Figure 2.
Specifically, the mechanism can realize the rotating Bidirectional driving of no sideshake.The active line tooth of the driving wheel One lateral tooth flank 3, be moved by one section of circular arc profile along the first active exposure line 4 and the first driving wheel transverse tooth thickness boost line 5 and Formed, another lateral tooth flank of the active line tooth, be by one section of circular arc profile along the second active exposure line 6 and second actively Wheel transverse tooth thickness boost line 7 is moved and formed, as shown in Figure 3.The lateral tooth flank from moving-wire tooth of the driven pulley, it is by one section of circular arc Flank profil is moved along the described first driven contact line and the first driven pulley transverse tooth thickness boost line and formed, the opposite side from moving-wire tooth The flank of tooth, it is to be moved by one section of circular arc profile and formed along the described second driven contact line and Article 2 driven pulley transverse tooth thickness boost line. The circular arc profile of the circular arc profile of the driving wheel and the driven pulley is respectively positioned at active exposure line and the method for driven contact line In plane.The first driving wheel transverse tooth thickness boost line and the second driving wheel transverse tooth thickness boost line, be respectively the first active exposure line and The equidistant curve of second active exposure line in the normal direction；The first driven pulley transverse tooth thickness boost line and the second driving wheel transverse tooth thickness Boost line, it is the equidistant curve of the first driven contact line and the second driven contact line in the normal direction respectively.
The parametric equation of the first driving wheel transverse tooth thickness boost line and the second driving wheel transverse tooth thickness boost line is respectively：
R is the radius of the flank profil circular arc, and φ is the side of the driving wheel transverse tooth thickness boost line and driven pulley transverse tooth thickness boost line To parameter, N1It is the number of teeth of driving wheel.
The parametric equation of the first driven pulley transverse tooth thickness boost line and the second driven pulley transverse tooth thickness boost line is respectively：
R is the radius of the flank profil circular arc, and φ is the side of the driving wheel transverse tooth thickness boost line and driven pulley transverse tooth thickness boost line To parameter, N2It is the number of teeth of driven pulley.
The wheel body of the driving wheel is the drift angle cone equal with the first active exposure line drift angle；The driven pulley Wheel body be the drift angle cone equal with the described first driven contact line drift angle.The active line tooth and it is distributed in from moving-wire tooth On the wheel body of driving wheel and driven pulley.
If primary quantity is m1=2.5mm, n1=4mm, θ1=7 °, m2=12mm, i=8, θ0=90 °, N1=2, N2=16.
Then the parametric equation of the first active exposure line and the second active exposure line is respectively：
The parametric equation of first driven contact line and the second driven contact line is respectively：
The parametric equation of first driving wheel transverse tooth thickness boost line and the second driving wheel transverse tooth thickness boost line is respectively：
The parametric equation of first driven pulley transverse tooth thickness boost line and the second driven pulley transverse tooth thickness boost line is respectively：
The driving wheel that is constructed as shown in figure 4, the driven pulley that is constructed as shown in figure 5, driving wheel 8 and driven pulley 9 are nibbled Conjunction situation is as shown in Figure 6.
The above embodiment of the present invention is only intended to clearly illustrate example of the present invention, and is not to the present invention Embodiment restriction.For those of ordinary skill in the field, can also make on the basis of the above description Other various forms of changes or variation.There is no necessity and possibility to exhaust all the enbodiments.It is all the present invention All any modification, equivalent and improvement made within spirit and principle etc., should be included in the protection of the claims in the present invention Within the scope of.

Claims (7)

1. a kind of coplanar axis gear mechanism that active line tooth is constructed with conical spiral, including driving wheel and driven pulley, its It is characterised by：The driving wheel and driven pulley meet the active exposure line of line gear space curve mesh theory, i.e. spatial conjugation Engaged transmission is realized with a contact form with driven contact line, the space curve mesh theory is related to following space coordinates, Coordinate system o0-x0y0z0With coordinate system o1-x1y1z1The respectively fixed coordinate system of driving wheel and with moving coordinate system, coordinate system op- xpypzpWith coordinate system o2-x2y2z2The respectively fixed coordinate system of driven pulley and with moving coordinate system；Driving wheel axis of rotation and z0 (z1) overlapping of axles, driven pulley axis of rotation and z2(zp) overlapping of axles, angle is θ between driving wheel and driven pulley axis of rotation0, it is described Angle can be intersected with any angle according to the design needs between axis, and any angle refers to more than or equal to 0 ° and less than 180 ° Unspecified angle；The driving wheel and driven pulley are respectively with angular speedWithAround axle z1And z2Axle rotates, in certain time, actively Wheel and driven pulley turn over angle respectivelyWithThe coplanar axis gear mechanism that active line tooth is constructed with conical spiral Mesh equation is：
Wherein, the active exposure line is in coordinate system o1-x1y1z1Parametric equation be：
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Wherein, t1For parameter, t1s、t1eFor parameter t1Span, i.e., the starting point of meshing and engagement on active exposure line Terminating point position；
The driven contact line is in coordinate system o1-x1y1z1Parametric equation be：
Wherein,I is gearratio.
2. the coplanar axis gear mechanism according to claim 1 that active line tooth is constructed with conical spiral, its feature It is：The active exposure line of the mechanism is conical spiral, and driven contact line can use cylinder spiral shell according to axis angle requirement Spin line, conical spiral or plane Archimedes spiral, active exposure line and driven contact line parametric equation are：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msubsup> <mi>x</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>(</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <msub> <mi>t</mi> <mn>1</mn> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>)</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>y</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <msub> <mi>t</mi> <mn>1</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>z</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <msub> <mi>t</mi> <mn>1</mn> </msub> <msub> <mi>cos&amp;theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mn>2</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <msub> <mi>t</mi> <mn>2</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>y</mi> <mn>2</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <msub> <mi>t</mi> <mn>2</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>z</mi> <mn>2</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <msub> <mi>t</mi> <mn>2</mn> </msub> <msub> <mi>cos&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>it</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>a</mi> <mo>-</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> <msub> <mi>sec&amp;theta;</mi> <mn>0</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>in</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>&amp;theta;</mi> <mn>0</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, t1And t2For parameter, θ1And θ2The respectively semicircle cone-apex angle of active exposure line and driven contact line, m1、n1With m2、n2Respectively active exposure line and the helix parameter of driven contact line, k1And k2For rotation direction parameter, when rotation direction parameter is 1, Conical spiral is left-handed, and when rotation direction parameter is -1, conical spiral is dextrorotation.
3. the coplanar axis gear mechanism according to claim 2 that active line tooth is constructed with conical spiral, its feature It is：There are two contact lines on each line tooth of driving wheel and driven pulley, and then rotating Bidirectional driving can be realized, in master On driving wheel, two contact lines include the first active exposure line and the second active exposure line；On the driven wheel, described two connect Touching line includes the first driven contact line and the second driven contact line；The second active exposure line is by the first active exposure line z1Axle rotatesObtain, the parametric equation of the first active exposure line and the second active exposure line is respectively：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mn>11</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>y</mi> <mn>11</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>y</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>z</mi> <mn>11</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>z</mi> <mn>1</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo>&amp;le;</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>&amp;le;</mo> <msub> <mi>t</mi> <mrow> <mn>1</mn> <mi>e</mi> </mrow> </msub> </mrow>
Similarly, the described second driven contact line is by the described first driven contact line z2Axle rotatesObtain, described first is driven The parametric equation of contact line and the second driven contact line is respectively：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mn>21</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>x</mi> <mn>2</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>y</mi> <mn>21</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>y</mi> <mn>2</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>z</mi> <mn>21</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>z</mi> <mn>2</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mn>2</mn> <mi>s</mi> </mrow> </msub> <mo>&amp;le;</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>&amp;le;</mo> <msub> <mi>t</mi> <mrow> <mn>2</mn> <mi>e</mi> </mrow> </msub> </mrow>
4. the coplanar axis gear mechanism according to claim 3 that active line tooth is constructed with conical spiral, its feature It is：The line transverse tooth thickness degree and space width of the coplanar axis gear mechanism are equal, the line transverse tooth thickness degree and the online tooth of space width Defined in the shaft section of wheel, i.e., the circular cone straight edge line direction on the active exposure line and driven contact line where any point point The line transverse tooth thickness degree and space width of driving wheel and driven pulley are not defined.
5. the coplanar axis gear mechanism according to claim 4 that active line tooth is constructed with conical spiral, its feature It is：One lateral tooth flank of the active line tooth of the driving wheel, it is along the first active exposure line and one by one section of circular arc profile Bar driving wheel transverse tooth thickness boost line is moved and formed, another lateral tooth flank of the active line tooth, is as described in one section of circular arc profile edge Second active exposure line and another driving wheel transverse tooth thickness boost line are moved and formed；The side tooth from moving-wire tooth of the driven pulley Face, it is to be moved by one section of circular arc profile and formed along the described first driven contact line and a driven pulley transverse tooth thickness boost line, it is described It is auxiliary along the described second driven contact line and another driven pulley transverse tooth thickness by one section of circular arc profile from another lateral tooth flank of moving-wire tooth Index contour is moved and formed；The circular arc profile of the circular arc profile of the driving wheel and the driven pulley respectively positioned at active exposure line and On the normal plane of driven contact line；Two driving wheel transverse tooth thickness boost line includes the first driving wheel transverse tooth thickness boost line and the second master Driving wheel transverse tooth thickness boost line, it is the equidistant curve of the first active exposure line and the second active exposure line in the normal direction respectively；Institute Stating two driven pulley transverse tooth thickness boost lines includes the first driven pulley transverse tooth thickness boost line and the second driving wheel transverse tooth thickness boost line, is respectively The equidistant curve of one driven contact line and the second driven contact line in the normal direction.
6. the coplanar axis gear mechanism according to claim 5 that active line tooth is constructed with conical spiral, its feature It is：The parametric equation of the first driving wheel transverse tooth thickness boost line and the second driving wheel transverse tooth thickness boost line is respectively：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow> <mn>11</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mi>r</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>y</mi> <mrow> <mn>11</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>-</mo> <mi>r</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;phi;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>-</mo> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>z</mi> <mrow> <mn>11</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <msub> <mi>t</mi> <mn>1</mn> </msub> <mi>cos</mi> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <mi>r</mi> <mi> </mi> <mi>sin</mi> <mi>&amp;phi;</mi> <mi>sin</mi> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <mi>c</mi> <mi>o</mi> <mi>s</mi> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <msub> <mi>t</mi> <mn>1</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mrow> <mn>12</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <mi>cos</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mi>r</mi> <mi> </mi> <msub> <mi>sin&amp;phi;cos&amp;theta;</mi> <mn>1</mn> </msub> <mi>cos</mi> <mi> </mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mi>sin</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mo>)</mo> <mo>+</mo> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mi>cos</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>y</mi> <mrow> <mn>12</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>r</mi> <mi> </mi> <msub> <mi>sin&amp;phi;cos&amp;theta;</mi> <mn>1</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mi>cos</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mo>)</mo> <mo>-</mo> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> <mi>sin</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>1</mn> </msub> </mfrac> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>z</mi> <mrow> <mn>12</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <msub> <mi>t</mi> <mn>1</mn> </msub> <msub> <mi>cos&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <mi>r</mi> <mi> </mi> <msub> <mi>sin&amp;phi;sin&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>cos&amp;theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <msub> <mi>t</mi> <mn>1</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, r is the radius of the flank profil circular arc, and φ is the side of the driving wheel transverse tooth thickness boost line and driven pulley transverse tooth thickness boost line To parameter, N1It is the number of teeth of driving wheel；
The parametric equation of the first driven pulley transverse tooth thickness boost line and the second driven pulley transverse tooth thickness boost line is respectively：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mrow> <mn>21</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>r</mi> <mi> </mi> <msub> <mi>sin&amp;phi;cos&amp;theta;</mi> <mn>2</mn> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>+</mo> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>y</mi> <mrow> <mn>21</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>-</mo> <mi>r</mi> <mi> </mi> <msub> <mi>sin&amp;phi;cos&amp;theta;</mi> <mn>2</mn> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mi>cos</mi> <mi> </mi> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>-</mo> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>z</mi> <mrow> <mn>21</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <msub> <mi>t</mi> <mn>2</mn> </msub> <msub> <mi>cos&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>r</mi> <mi> </mi> <msub> <mi>sin&amp;phi;sin&amp;theta;</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>cos&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <msub> <mi>t</mi> <mn>2</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>x</mi> <mrow> <mn>22</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <mi>cos</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>r</mi> <mi> </mi> <msub> <mi>sin&amp;phi;cos&amp;theta;</mi> <mn>2</mn> </msub> <mi>cos</mi> <mi> </mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mi>sin</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mrow> <mo>)</mo> <mo>+</mo> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> <mi>cos</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>y</mi> <mrow> <mn>22</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>r</mi> <mi> </mi> <msub> <mi>sin&amp;phi;cos&amp;theta;</mi> <mn>2</mn> </msub> <mi>sin</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mi>cos</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mrow> <mo>)</mo> <mo>-</mo> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> <mi>sin</mi> <mo>(</mo> <mrow> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>+</mo> <mfrac> <mi>&amp;pi;</mi> <msub> <mi>N</mi> <mn>2</mn> </msub> </mfrac> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>z</mi> <mrow> <mn>22</mn> <mi>c</mi> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <msub> <mi>t</mi> <mn>2</mn> </msub> <msub> <mi>cos&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <mi>r</mi> <mi> </mi> <msub> <mi>sin&amp;phi;sin&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <mfrac> <mrow> <mi>r</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> <msqrt> <mrow> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mrow> <mo>&amp;prime;</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <msub> <mi>n</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <msub> <mi>cos&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <msub> <mi>t</mi> <mn>2</mn> </msub> <msub> <mi>sin&amp;theta;</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, r is the radius of the flank profil circular arc, and φ is the side of the driving wheel transverse tooth thickness boost line and driven pulley transverse tooth thickness boost line To parameter, N2It is the number of teeth of driven pulley.
7. the coplanar axis gear mechanism according to claim 6 that active line tooth is constructed with conical spiral, its feature It is：Described driving wheel wheel body is the drift angle cone equal with the first active exposure line drift angle；Described driven pulley Wheel body is the drift angle cone equal with the described first driven contact line drift angle, the active line tooth and is distributed in master from moving-wire tooth On the wheel body of driving wheel and driven pulley.
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CN108533685A (en) * 2018-06-12 2018-09-14 中国地质大学（武汉） Male-female engagement pure rolling spiral bevel gear mechanism for intersecting axle transmission
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