CN101710350A - Methods for designing and manufacturing double-lead linear contact bias worm drive - Google Patents

Abstract

The invention provides methods for designing and manufacturing double-lead linear contact bias worm drive, belonging to the fields of machinery principle and machinery manufacture. The double-lead linear contact bias worm drive belongs to the spiroid drive mode in the worm gear drive. The designing method in the invention has the following steps: 1. determining basic parameters according to the power, the transmission ratio and the rotating speed required to be transmitted; 2. performing parameter designing by taking a specified reference circle as a standard on the basis of preliminary calculation of the outer diameter of a worm gear; and 3. calculating each parameter in terms of the formula deduced from the above parameter designing. The manufacturing method in the invention comprises a mitting method, a fly-cutter hobbing method and a numerical control turning method. The invention has a simple and precious designing method, which can simplify the structure of the worm gear and machining and manufacturing; the current machine tool and cutter can be used to perform machining on tooth surfaces without adding other devices or using special machine tools and cutters except for adopting a special machine tool for efficient and high precision machining.

Description

Worm geared design of double-lead linear contact bias and manufacture method

Technical field

The invention belongs to mechanical principle and mechanical manufacturing field, relate to gearing mesh theory and worm drive.

Background technology

From eighties of last century the fifties, Procedure for Spiroid Gearing has appearred, this type of belt drive has improved worm geared efficient and load-bearing capacity, increased ratio of gear, and can realize the worm gear material with copper take place of steel, but about the research of this transmission with to use be not a lot, its main cause is a theory of engagement complexity, the mathematical relation that is used for describing its principle and the flank of tooth is also very complicated, thereby causes the difficulty of this worm geared design and manufacturing.The secondary worm screw of theoretic Procedure for Spiroid Gearing is to become the helical pitch helicoid, and the worm gear nodal section is the hyperboloid of one sheet.Up to now, the design of Procedure for Spiroid Gearing and manufacturing still are based upon on the engineering approximation basis.Though can adopt advanced designing and calculating means, but still very complicated, and must adopt dedicated tool and process manufacturing, also having, instantaneous transmission ratio is non-constant, causes this kind of drive to be difficult to popularize.Therefore the present invention proposes a kind of novel Procedure for Spiroid Gearing form---two helical pitch biasing worm drive.

Summary of the invention

The objective of the invention is to be design that solves Procedure for Spiroid Gearing and the problem of making difficulty, a kind of design and calculation method simply and has accurately been proposed, make the designs simplification of worm and gear, can use existing machine tool and the Tool in Cutting processing flank of tooth, and needn't add equipment or use special purpose machine tool and dedicated tool.

The present invention proposes a kind of pair of helical pitch biasing worm drive form, it is based on space crossed axis helical gear drive principle, its worm screw workplace is that the involute helicoid by two constant leads constitutes, and respectively with the both sides workplace engagement of worm-gear toothing, realizes the bi-directional of motion and power.The both sides workplace of worm gear also is an involute helicoid, and they are that straight line contacts with the instantaneous contact condition of worm screw workplace engagement, and the both sides workplace is equivalent to the external toothing and the interior engaged transmission of involute cylindrical gear respectively.Principle has been simplified the design and calculation method of existing spiroid gear, worm screw in view of the above, thereby realizes utilizing common universal machine tools and universal cutter to carry out the cut of the flank of tooth.

One, the worm geared method for designing of double-lead linear contact bias provided by the invention may further comprise the steps:

(1) according to power, ratio of gear and the rotating speed of required transmission, after parameters such as definite centre distance, the number of teeth, worm screw mean radius, while total number of teeth in engagement, determines each base radius, profile angle.

(2) calculate worm gear external diameter R just _aThe basis on, with the regulation reference circle be that benchmark carries out parameter designing, parameter comprises: worm gear external diameter R _a, the reference circle of wormwheel radius R _m, the involute urve starting point angle that interlaces on the reference circle cross section

Modulus m, worm gear internal diameter R _i, worm gear tooth depth h, spiroid gear taper angle theta ₁, worm screw tooth depth h ₂, the spiroid taper angle theta ₂, worm thread length L, worm screw outside diameter d _d, worm screw end diameter d _s, worm and gear installs mesh tooth face helical pitch p ' in offset E, worm screw involute helicoid relative position parameter S, worm and gear setting height(from bottom) a, worm screw external toothing flank of tooth helical pitch p, the worm screw, carries out while total number of teeth in engagement checking computations, cylindrical worm and worm gear calculation of parameter, the calculating of worm geared self-locking critical angle and worm geared efficiency estimation again.

(3) formula of being derived in the parameter designing according to step 2 calculates each parameter.

A. the first calculation worm gear external diameter R described in the step (2) _aBe parameter, comprise worm screw external diameter, while total number of teeth in engagement etc., calculate as follows, and get the rounding value according to step (1):

$R_a=\sqrt{R_{b1}^{\&prime;2}+{(\mathrm{Rctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+2(n+0.5){{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;})}^2}$

Wherein: R ' _H2Be interior engagement side worm screw involute helicoid base radius, R ' _B1Be interior engagement side worm gear involute helicoid base radius, n is the while total number of teeth in engagement, and R is the worm screw radius

B. the reference circle of wormwheel radius R described in the step (2) _mReference circle of wormwheel be to be the circle in the center of circle with the worm-wheel shaft kernel of section, this circle on transverse tooth thickness equate with inter-tooth slots, the mistake this circle the cross section be defined as the reference circle cross section.The reference circle of wormwheel radius R _mCalculating by following equation solution: $\frac{2\mathrm{\&pi;}}{z}-[\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})$

$+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))]=0$

Wherein: r=R _a, R _aBe worm gear external diameter, R _B1Be external toothing side involute helicoid base radius, R ' _B1Be interior engagement side involute helicoid base radius, R _mBe reference radius, z is the worm gear number of teeth.

C. the involute urve starting point angle that interlaces on the reference circle cross section described in the step (2)

Be according to defined reference radius, calculate that its expression formula is in conjunction with involute equation:

D. the modulus (m) described in the step (2) is to calculate according to defined reference radius, and its expression formula is:

$m=\frac{{2R}_m}{z}$

E. the worm gear internal diameter R described in the step (2) _iBe to calculate, press following equation solution according to reference radius and theoretical heel height:

$\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2,$

$-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})=0$

Wherein: r=R _i

F. the worm gear tooth depth h described in the step (2) calculates according to reference radius and theoretical heel height, and its expression formula is:

Ad. h _a=m

Height of teeth root h _f=m+C ^*

h＝h _a+h _f

C ^*Be tip clearance coefficient, get C ^*=(0.1～0.2) m, m is a modulus.

G. the spiroid gear taper angle theta described in the step (2) ₁Calculate according to following formula:

${\mathrm{\&theta;}}_1=\frac{\mathrm{\&pi;}}{2}-{\mathrm{tg}}^{-1}\left(\frac{t_{\max }^{\&prime;}}{R_a-R_i}\right)$

Wherein: $t_{\max }^{\&prime;}=t^{\&prime;}{|}_{r=R_i},$ And

$t^{\&prime;}==\frac{2r\sin \frac{{\mathrm{\&phi;}}^{\&prime;}}{2}}{({\mathrm{tg\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}$

${\mathrm{\&phi;}}^{\&prime;}=\frac{2\mathrm{\&pi;}}{z}-[\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}$

$+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))]$

H. the worm screw tooth depth h described in the step (2) ₂Be to calculate according to the worm gear tooth depth, its expression formula is:

$h_2=h_{2a}+h_{2f}=(m+m+C^{*})/\cos \left({\mathrm{tg}}^{-1}\frac{R_{b2}^{\&prime;}}{r_2}\right)$

Wherein: ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{R_{b2}}^{\&prime;2}}$

I. the spiroid taper angle theta described in the step (2) ₂Be to calculate according to worm gear exradius and eccentric throw, its expression formula is:

${\mathrm{\&theta;}}_2={\mathrm{tg}}^{-1}\left(\frac{(R_a-R_e)\mathrm{tg}(\frac{\mathrm{\&pi;}}{2}-{\mathrm{\&theta;}}_1)}{R_a-E}\right)$

Wherein $R_e=\sqrt{{R_{b1}}^{\&prime;2}+{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">E</figure-callout>}}^2}$

J. the worm thread length L described in the step (2) is to calculate according to interior engages worm base radius and worm spiral lift angle, and its expression formula is:

L＝2(n+1)R _b2′πtgλ′

K. worm screw outside diameter (d _d), worm screw end diameter (d _s)

Be to calculate according to worm screw mean radius and worm screw cone angle, its expression formula is:

d _d＝d+Ltgθ ₂ d _s＝d-Ltgθ ₂

Wherein: d _dBe outside diameter, d _sBe end diameter

1. the worm and gear described in the step (2) is installed offset E

Be the distance of worm screw tip-to-face distance worm gear center line, be according to the worm and gear flank of tooth not interference condition calculate, its expression formula is:

E＝r ₂ctgβ′ _b1+mtgβ′ _b1

M. the worm and gear setting height(from bottom) a described in the step (2) is the distance of worm axis apart from the reference circle cross section, and its expression formula is:

a＝r ₂-h _a

Wherein ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}^2}$

N. worm screw involute helicoid relative position parameter (S) is used for determining the relative position of two involute helicoid starting points of worm screw, computing formula:

E wherein _Fmin=r ₂Ctg β ' _B1+ mtg β ' _B1

O. the worm screw external toothing flank of tooth helical pitch (p) described in the step (2) is the lead of helix on the external toothing lateral tooth flank, and its expression formula is:: p=2R _B2π tg λ

λ---external toothing lead angle; λ=β ' _B1

P. mesh tooth face helical pitch p ' is for being the lead of helix on the external toothing lateral tooth flank in the worm screw described in the step (2), and its expression formula is:

p′＝2R _b2′πtgλ′

λ '---external toothing lead angle; λ=β _B1

Q. total number of teeth in engagement n checking computations simultaneously are:

$n\&le;\frac{\sqrt{R_a^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}$

Wherein: ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}^2}$

R. the cylindrical worm and worm gear calculation of parameter is:

1. calculate cylindrical worm gear external diameter R _Ac

The tooth depth of getting worm gear is identical with taper worm gear tooth depth, R _AcFind the solution by following equation

$h_a=\frac{2r\sin \frac{{\mathrm{\&phi;}}^{\&prime;}}{2}}{({\mathrm{tg\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{|}_{r=R_{\mathrm{ac}}}$

Wherein:

${\mathrm{\&phi;}}^{\&prime;}=\frac{2\mathrm{\&pi;}}{z}-[\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}$

$+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))]$

2. calculate cylindrical worm gear internal diameter R _Ic

Find the solution by following equation

$h_f=\frac{2r\sin \frac{\mathrm{\&phi;}}{2}}{({\mathrm{tg\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{|}_{r=R_{\mathrm{ic}}}$

Wherein:

$\mathrm{\&phi;}=\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}$

$+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2$

$-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))$

$-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})$

Wherein: R _Ic〉=R ' _B1

3. other geometric parameters of cylindrical worm gear are identical with the computing method of above-mentioned conical worm gear

4. cylindrical worn tooth depth

Be calculated as follows

${\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">h</figure-callout>}}_2=h_{2a}+h_{2f}=(m+m+C^{*})/\cos \left({\mathrm{tg}}^{-1}\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}{r_2}\right)$

5. check total number of teeth in engagement n simultaneously

$n\&le;\frac{\sqrt{R_{\mathrm{ac}}^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}$

6. calculate worm thread length

L＝2(n+1)R _b2′πtgλ′

S. worm geared self-locking critical angle is calculated as:

Critical value λ with worm screw basic circle lead angle _oThe expression, when worm screw basic circle lead angle λ less than critical angle λ _oThe time worm and gear transmission self-locking.

λ _oBe calculated as follows

Get 0.9 ~ 1, the big more value of ratio of gear is big more.Work as r=R _B2The time try to achieve external toothing transmission self-locking critical angle; As r=R ' _B2The time try to achieve in engaged transmission self-locking critical angle;

T. worm geared efficiency estimation

Calculate the av eff of flank of tooth midpoint with following formula

$R_1=\sqrt{{(E+L/2)}^2+{R_{b1}}^2},$ R ₂＝R-h _a

$\mathrm{\&eta;}=\frac{D}{\mathrm{iC}}$

Wherein i is a ratio of gear

Reduce value in 0.15 ~ 0.3 with the ratio of gear increase;

Increase value in 0.9 ~ 1 with ratio of gear;

One, the used design proposal of the present invention is:

With the tooth both sides workplace of worm and gear, respectively by space crossed axis involute cylindrical gear external toothing transmission and interior engaged transmission principle design.Worm and gear comes down to a pair of special cylindrical screw gear, and the gear shape that the number of teeth seldom (is generally 1) is defined as worm screw like screw rod; The gear that the number of teeth is a lot (the general number of teeth is no less than 10), tooth is distributed on the end face of wheel, is defined as worm gear.The gabarit of worm gear and worm screw is generally taper, also can be for cylindricality.According to the rotation direction difference of involute helicoid, the rotation direction of worm gear and worm screw is divided into left-handed and dextrorotation.Worm screw nibbles out from the end face of worm gear is engaging-in during worm gear work.One lateral tooth flank of worm gear tooth and the lateral tooth flank of worm screw constitute external toothing space crossed axis gear-driven form, realize the motion of a direction and the transmission of power; Engagement space crossed axis gear-driven form was realized the motion of another direction and the transmission of power in the opposite side flank of tooth of worm gear tooth and the opposite side flank of tooth of worm screw constituted; The worm screw workplace is the screw rod that the involute helicoid by two constant leads constitutes, thus be defined as two helical pitch worm screws, the two helical pitches biasing of worm drive called after of the present invention in view of the above worm drive, worm screw places end face one side of worm gear during work.The phase alternate angle is 90 ° generally speaking, sees Fig. 1.Design and production method of the present invention is that 90 ° situation is as the criterion with the phase alternate angle all.

The worm gear and the flank of tooth of worm screw external toothing side with interior engagement side---involute helicoid forms by different separately base cylinders and Base spiral angle respectively.

1. basic geometric parameters and definition thereof

Two helical pitches are setovered worm geared ultimate principle and basic geometric parameters as shown in Figure 2.Two involute helicoids that constitute the external toothing transmission are respectively ∑ ₁, ∑ ₂, two base cylinders have public tangent plane Q; In constituting two involute helicoids of engaged transmission be respectively ∑ ' ₁, ∑ ' ₂, two base cylinders have public tangent plane Q ', wherein:

R _B1Be external toothing side worm gear flank of tooth base cylinder radius;

R ' _B1Be interior engagement side worm gear flank of tooth base cylinder radius (the external toothing lateral tooth flank base cylinder of worm gear is coaxial with interior engagement side flank of tooth base cylinder);

R _B2Be external toothing side worm tooth-surface base cylinder radius;

R ' _B2Be interior engagement side worm tooth-surface base cylinder radius (the external toothing lateral tooth flank base cylinder of worm screw is coaxial with interior engagement side flank of tooth base cylinder);

β _B1Be external toothing side worm gear flank of tooth Base spiral angle, in the promptly public tangent plane (Q plane), the angle of external toothing flank of tooth straight edge line and worm gear axis direction.When the phase alternate angle is 90 °, equal external toothing side worm tooth-surface basic circle lead angle λ on the numerical value;

α is an external toothing side profile angle, α=β _B1

β ' _B1Be interior engagement side worm gear flank of tooth Base spiral angle, in the promptly public tangent plane (Q ' plane), the angle of interior mesh tooth face straight edge line and worm axis direction.When the phase alternate angle is 90 °, engagement side worm tooth-surface basic circle lead angle λ ' in equaling on the numerical value;

α ' is interior engagement side profile angle, α '=β ' _B1

β _B1And β ' _B1The two direction is symmetrical with respect to the worm gear axis, as foundation β _B1When the involute helicoid that forms was dextrorotation, worm gear was left-handed, and worm screw is dextrorotation, otherwise is dextrorotation worm gear, left-hand worm.Above-mentioned parameter meets following relation

A＝R _b1+R _b2＝R′ _b1+R′ _b2

2. basic transmission parameter

1. power P of Chuan Diing and torque T;

2. ratio of gear i ₂₁

3. basic structure design

1) determines elementary structure parameter

(1) centre distance A

Power and moment of torsion according to required transmission are definite, and by getting big value rounding after the following formula estimation

$A=28.34{\left(\frac{1.36P}{K_m{K}_v{K}_i}\right)}^{0.373}\left(\mathrm{mm}\right)$

In the formula: P is the power that spiroid transmitted, and unit is kW;

K _mBe material section coefficient; When spiroid gear, worm screw are all made with steel, use the extreme boundary lubrication oil lubrication, getting it is 0.002

K _vBe velocity coefficient, determine by following formula: for low speed transmission, K _v=n ₁ ^0.546-7, n ₁Be the worm screw rotating speed;

K _iBe the ratio of gear coefficient, determine by following formula: $K_i=\frac{36}{i^{0.64}}-1.$

(2) the worm gear number of teeth and number of threads

Require to determine according to ratio of gear,

$i_{21}=\frac{\mathrm{<figure-callout\; id="2"\; label="z\; z"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">z</figure-callout>}}{z_{}}=z,$ Wherein z---the worm gear number of teeth, z ₂---number of threads is generally 1.

(3) worm screw external toothing base radius R _B2, worm gear external toothing base radius R _B1, engagement base radius R in the worm screw _B2', engagement base radius R in the worm gear _B1'.Computing formula

${\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b=\frac{A}{\mathrm{ztg}{\mathrm{\&beta;}}_{b1}+1};$ R _b1＝A-R _b2； ${\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2=\frac{A}{\mathrm{ztg}{\mathrm{\&beta;}}_{b1}^{\&prime;}-1};$

R′ _b1＝A+R′ _b2。

(4) meshing spiral angle (β in the primary election _B1', i.e. profile angle α '), external toothing helix angle (β _B1, i.e. profile angle α), and meet following relation:

$i_{21}=\frac{R_{b1}}{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}\mathrm{ctg}{\mathrm{\&beta;}}_{b1}=\frac{R_{b1}^{\&prime;}}{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}---\left(1\right)$

(5) rough calculation worm screw outside diameter d, d=kA=2R, $k=\frac{1}{2}~\frac{7}{12},$ d＝54；

R is the worm screw mean diameter

(6) determine total number of teeth in engagement n simultaneously

Generally get n=(10%～12%) z for the taper worm gear, generally get n ≈ (5%) z for the cylindricality worm gear

2) other geometrical parameter design is calculated

The geometric parameter of worm and gear is seen Fig. 3, and its definition and computing method and step are as follows:

(1) rough calculation worm gear external diameter R _a

Formula:

$R_a=\sqrt{R_{b1}^{\&prime;2}+{(\mathrm{Rctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+2(n+0.5){{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;})}^2}$

(2) reference circle of wormwheel radius R _m

With the worm-wheel shaft kernel of section is the circle in the center of circle, and transverse tooth thickness equates (Fig. 3,4) with inter-tooth slots on this circle, and defining this circle is reference circle, and the cross section of crossing this circle is defined as the reference circle cross section, the reference circle of wormwheel radius R _mAccording to following equation solution:

$\frac{2\mathrm{\&pi;}}{z}-[\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-c{\mathrm{os}}^{-1}\frac{R_{b1}^{\&prime;}}{R_m}$

$+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))]=0$

Wherein: r=R _a

(3) the involute urve starting point angle that interlaces on the reference circle cross section

Two spiral involute surfaces of worm gear tooth involute urve reference position on reference circle plane angle that interlaces

Determine the relative position of each tooth both sides involute helicoid when being used for the design gear flank of tooth, determine the relative position condition of cutter blade during also as two flank of tooth of cut.Be according to defined reference radius, calculate that its expression formula is in conjunction with involute equation:

(4) modulus m

Characterize the parameter of worm gear size, be used for the intermediate variable that following several parameters calculates.Calculate according to defined reference radius, its expression formula is:

$m=\frac{{2R}_m}{z}$

(5) worm gear tooth depth h

Be to calculate according to reference radius and theoretical heel height, its expression formula is:

Ad. h _a=m

Height of teeth root h _f=m+C ^*

h＝h _a+h _f

C ^*Be tip clearance coefficient, get C ^*=(0.1～0.2) m, m is a modulus.

(6) worm gear internal diameter R _iCalculating is found the solution according to following equation:

$\mathrm{tg}({\cos }^{-1}-\frac{R_{b1}}{r})-{\cos }^{-1}\frac{R_{b1}}{r}+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2$

$-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})=0$

Work as R _i＜R ' _B1The time get R _i=R ' _B1

Wherein: r=R _i

(7) spiroid gear taper angle theta ₁Calculate according to following formula:

${\mathrm{\&theta;}}_1=\frac{\mathrm{\&pi;}}{2}-{\mathrm{tg}}^{-1}\left(\frac{t_{\max }^{\&prime;}}{R_a-R_i}\right)$

Wherein: $t_{\max }^{\&prime;}=t^{\&prime;}{|}_{r=R_i},$ And

$t^{\&prime;}==\frac{2r\sin \frac{{\mathrm{\&phi;}}^{\&prime;}}{2}}{({\mathrm{tg\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}$

${\mathrm{\&phi;}}^{\&prime;}=\frac{2\mathrm{\&pi;}}{z}-[\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}$

$+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))]$

(8) worm screw tooth depth h ₂

Be to calculate according to the worm gear tooth depth, its expression formula is:

${\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">h</figure-callout>}}_2=h_{2a}+h_{2f}=(m+m+C^{*})/\cos \left({\mathrm{tg}}^{-1}\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}{r_2}\right)$

Wherein: ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;2}}$

(9) spiroid taper angle theta ₂

Be to calculate according to worm gear exradius and eccentric throw, its expression formula is:

${\mathrm{\&theta;}}_2={\mathrm{tg}}^{-1}\left(\frac{(R_a-R_e)\mathrm{tg}(\frac{\mathrm{\&pi;}}{2}-{\mathrm{\&theta;}}_1)}{R_a-E}\right)$

Wherein $R_e=\sqrt{{R_{b1}}^{\&prime;2}+{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">E</figure-callout>}}^2}$

(10) worm thread length L

Be that interior engages worm base radius and worm spiral lift angle calculate, its expression formula is:

L＝2(n+1)R _b2′πtgλ′

(11) worm screw outside diameter d _d, worm screw end diameter d _s

Be to calculate according to worm screw mean radius and worm screw cone angle, its expression formula is:

d _d＝d+Ltgθ ₂ d _s＝d-Ltgθ ₂

Wherein: d _dBe outside diameter, d _sBe end diameter

(12) worm and gear is installed offset distance

Be the distance of worm screw tip-to-face distance worm gear center line, be according to the worm and gear flank of tooth not interference condition calculate, its expression formula is:

E＝r ₂ctgβ′ _b1+mtgβ′ _b1

(13) worm and gear setting height(from bottom) a

Be the distance of worm axis apart from the reference circle cross section, its expression formula is:

a＝r ₂-h _a

Wherein ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}^2}$

(14) worm screw involute helicoid relative position parameter S

Be used for determining the relative position of two involute helicoid starting points of worm screw, computing formula

E wherein _Fmin=r ₂Ctg β ' _B1+ mtg β ' _B1

(15) worm screw external toothing flank of tooth helical pitch:

Be the lead of helix on the external toothing lateral tooth flank, its expression formula is:

p＝2R _b2πtgλ

λ---external toothing lead angle; λ=β ' _B1

(16) mesh tooth face helical pitch in the worm screw:

Be the lead of helix on the interior engagement side flank of tooth, its expression formula is:

p′＝2R _b2′πtgλ′

λ '---interior meshing spiral lift angle; λ=β _B1

(17) checking computations while total number of teeth in engagement n

$n\&le;\frac{\sqrt{R_a^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}$

Wherein: ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}^2}$

(18) cylindrical worm and worm gear calculates

1. calculate cylindrical worm gear external diameter R _Ac

The tooth depth of getting worm gear is identical with taper worm gear tooth depth, R _AcFind the solution by following equation

$h_a=\frac{2r\sin \frac{{\mathrm{\&phi;}}^{\&prime;}}{2}}{({\mathrm{tg\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{|}_{r=R_{\mathrm{ac}}}$

Wherein:

${\mathrm{\&phi;}}^{\&prime;}=\frac{2\mathrm{\&pi;}}{z}-[\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}$

$+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))]$

2. calculate cylindrical worm gear internal diameter R _Ic

Find the solution by following equation

$h_f=\frac{2r\sin \frac{\mathrm{\&phi;}}{2}}{({\mathrm{tg\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{|}_{r=R_{\mathrm{ic}}}$

Wherein:

$\mathrm{\&phi;}=\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}$

$+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2$

$-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))$

$-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})$

Wherein: R _Ic〉=R ' _B1

3. other geometric parameters of cylindrical worm gear are identical with the computing method of above-mentioned conical worm gear

4. cylindrical worn tooth depth

Be calculated as follows

${\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">h</figure-callout>}}_2=h_{2a}+h_{2f}=(m+m+C^{*})/\cos \left({\mathrm{tg}}^{-1}\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}{r_2}\right)$

5. check total number of teeth in engagement n simultaneously

$n\&le;\frac{\sqrt{R_{\mathrm{ac}}^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}$

6. calculate worm thread length

L＝L′+p′＝2(n+1)R _b2′πtgλ′

(19) worm geared condition of self-locking

Critical value λ with worm screw basic circle lead angle _oThe expression, when worm screw basic circle lead angle λ less than critical angle λ _oThe time worm and gear transmission self-locking.

λ _oBe calculated as follows

Get 0.9 ~ 1, the big more value of ratio of gear is big more.Work as r=R _B2The time try to achieve external toothing transmission self-locking critical angle; As r=R ' _B2The time try to achieve in engaged transmission self-locking critical angle;

(20) worm geared efficiency estimation

Estimate av eff with following formula

$R_1=\sqrt{{(E+L/2)}^2+{R_{b1}}^2},$ R ₂＝R-h _a

$\mathrm{\&eta;}=\frac{D}{\mathrm{iC}}$

Wherein i is a ratio of gear

Reduce value in 0.15 ~ 0.3 with the ratio of gear increase;

Increase value in 0.9 ~ 1 with ratio of gear;

Two, the worm geared job operation of double-lead linear contact bias provided by the invention comprises: milling method, fly cutter gear hobbing job operation and numerical control turning job operation.

(1) processing scheme

The formation of the workplace according to worm gear and worm screw---involute helicoid be by the straight line in the plane when the plane with the principle that generates when making pure rolling of tangent base cylinder, if promptly can cut as blade and form worm gear and worm tooth-surface so that line to take place.Realize the needed motion of this flank of tooth cut can be decomposed into straight line moving planar (tool motion, comprise straight line parallel move and straight line along the sliding motion of self direction) and the rotatablely moving of workpiece.Possess this kinematic relation, and the flank of tooth cut that the lathe that can realize the rectilinear edge cutter all can be finished of the present invention pair of helical pitch biasing worm transmission pair can be installed.Wherein the flank of tooth cut of worm screw is suitable for and utilizes especially numerically controlled lathe cut of horizontal lathe, and the flank of tooth processing of worm gear can utilize multiple lathe to realize.

1. milling method

Utilize the milling method cut worm gear of common vertical milling machine or universal mill or CNC milling machine or numerical control machining center.Its basic demand is that the vertical milling unit head can the anglec of rotation, and dividing head or circular turntable are installed on the platen.Workpiece is installed on the dividing head (perhaps circular turntable), and the gyration of dividing head and the in-movement of platen are got in touch transmission in realizing; Select suitable column or tapered mill for use.

As Fig. 4, base cylinder section (Q face) and the parallel adjustment lathe of the horizontal moving direction of platen (YOZ plane) with worm gear make finger cutter cylinder and Q face tangent, constitute the line segment L (rectilinear edge of cutting usefulness, do not draw among the figure), the milling cutter axis becomes β with the z axle _B1The angle is to form the helix angle that requires.Make worm gear axis and milling cutter axis turn over the γ angle around the x axle simultaneously, when dividing head drive workpiece turned round with angular velocity omega, the worktable movement therewith realized aggregate velocity υ.Select the suitable amount of feeding for use along the axis direction of milling cutter, so can cut the flank of tooth, can cut repeatedly up to cutting to complete dark.

H＝(R _a-R _i)tgθ ₁

The engagement side flank of tooth in the cutting

$\mathrm{\&gamma;}={\mathrm{tg}}^{-1}\frac{{\mathrm{\&upsi;}}_2}{{\mathrm{\&upsi;}}_1}$

Wherein:

$\mathrm{\&upsi;}=\sqrt{{\mathrm{\&upsi;}}_1^2+{\mathrm{\&upsi;}}_2^2}$

υ ₁＝ωR′ _b1

$R_a=\frac{R_{b1}^{\&prime;}}{\cos {\mathrm{\&alpha;}}_a^{\&prime;}}$

$R_i=\frac{R_{b1}^{\&prime;}}{\cos {\mathrm{\&alpha;}}_i^{\&prime;}}$

Cutting external toothing lateral tooth flank

$\mathrm{\&gamma;}={\mathrm{tg}}^{-1}\frac{{\mathrm{\&upsi;}}_2}{{\mathrm{\&upsi;}}_1}$

Wherein:

$\mathrm{\&upsi;}=\sqrt{{\mathrm{\&upsi;}}_1^2+{\mathrm{\&upsi;}}_2^2}$

υ ₁＝ωR _b1

$R_a=\frac{R_{b1}}{\cos {\mathrm{\&alpha;}}_a}$

$R_i=\frac{R_{b1}}{\cos {\mathrm{\&alpha;}}_i^{\&prime;}}$

Z is the number of teeth of processed worm gear;

H is the total travel of the working angles of processed worm gear in the tooth depth direction.

Divide tooth by dividing head after a flank of tooth cutting is finished, realize monodentate calibration Milling Process thereby cut next tooth.Figure 6 shows that the left-handed worm gear external toothing flank of tooth of cutting.In the cutting opposite side during mesh tooth face except the flank of tooth itself requires, must meet the designing requirement of transverse tooth thickness, this must be by correct tool setting realization.Same principle can be cut the dextrorotation worm gear.

Use the finger cutter milling can't choose suitable milling cutter because of the milling cutter diameter that requires is undersized for modulus or the less worm gear of size, can select tapered mill this moment for use.The installation of tapered mill and shape are seen Fig. 5,6, and the cone angle of milling cutter and dimensional requirement be formula as follows.

For interior engagement side flank of tooth cutting, the cone angle τ of milling cutter should satisfy

τ＝β′ _b1-γ

The radius R x of milling cutter is a condition not interfere with the interior engagement side flank of tooth, should satisfy

$\mathrm{Rx}\&le;(\sqrt{R_i^2-R_{b1}^{\&prime;2}}-\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;}$

The thickness of milling cutter is being criterion less than worm gear inter-tooth slots minimum value.

The milling of external toothing lateral tooth flank can be used the transverse plane milling flank of tooth of milling cutter, and the installation of milling cutter is adopted finger cutter with reference to Fig. 4.Difference is that cutting blade and cutter shaft are rectangular.For satisfying the cutting needs, cutter spindle and z axle clamp angle are 90 °-γ-β _B1

2. fly cutter gear hobbing job operation

Utilize the fly cutter gear hobbing job operation cut worm gear of general hobbing machine or chain digital control gear hobbing machine.Its basic demand is that hobbing machine has the tangential motion function, and for example general hobbing machine is equipped with tangential hobhead, and knife rest also should be able to be made axial feed motion, preferably auto-feed campaign simultaneously.Workpiece is installed on the workpiece spindle, and fly cutter is installed on cutter shaft.In realizing, the tangential and axially-movable of the gyration of workpiece spindle and the gyration of cutter and knife rest gets in touch transmission.

As shown in Figure 7, involute urve AP can be regarded as radius R _bBasic circle form movement velocity with the rotation of angular velocity omega and straight line along the rectilinear motion of NP direction

υ _t＝ωR _b。

As Fig. 8, on the gear hobbing lathe, tangential hobhead is installed, perhaps make knife rest can do tangential motion, fly cutter is installed on the knife bar, workpiece is installed on the workpiece spindle.The installation of knife bar, fly cutter and workpiece meets worm and gear transmission mounting condition, i.e. centre distance A and highly a, and when the fly cutter rectilinear edge was positioned at the Q plane, rectilinear edge and worm gear axis angle were the helixangle of worm gear _B1

Adjust lathe according to above-mentioned principle and promptly can process the flank of tooth.Ratio of gear is at first satisfied in the rotation of worm gear and worm screw $i=\frac{{\mathrm{\&omega;}}_2}{{\mathrm{\&omega;}}_1}$ Requirement, thus realize dividing the tooth motion.To cut feeding and form involute helicoid in order to realize simultaneously, knife bar must add tangential motion.Tangential motion should meet the involute urve generating principle, establishes additional rotation angle Δ θ, then tangential displacement

S＝ΔθR _b

Tangential motion speed meets formula (11).The more little flank of tooth of tangential admission speed should be smooth more.

In order to satisfy the processing needs of taper worm gear, should there be axially-movable to realize conical insert simultaneously.Axially-movable should meet following relation: tangential admission worm thread total length L ', axial feeding is

ΔH＝(R _a-R _b)tgθ ₁

So axial feed velocity

${\mathrm{\&upsi;}}_z={\mathrm{\&upsi;}}_t\frac{L^{\&prime;}}{\mathrm{\&Delta;H}}$

This processing can continuous division, can process the less gear teeth.Can not process the taper worm gear for the lathe that does not have the axial feed function.Figure 8 shows that the left-handed worm gear external toothing flank of tooth of cutting.Mesh tooth face and dextrorotation worm gear in same principle can be cut.The same transverse tooth thickness designing requirement that needs correct tool setting with the assurance gear with other job operation.

3. numerical control turning job operation

Lathe can machining screw, and numerically controlled lathe has also has the C s function.If worm gear is regarded as special multi-step thread, then utilize the multiple-threaded function of lathe grinding can the cut worm gear flank of tooth.

As shown in Figure 7, involute urve AP can be regarded as the rotation and some P rectilinear motion formation along straight line OP direction of the basic circle of radius R b with angular velocity omega.

Involute urve polar equation formula according to circle

Basic circle is with uniform rotation, and some P moves along the OP direction, and the relation derivation of the two is as follows, establishes the rotational angle θ of basic circle, sets up coordinate system to see Fig. 9, and the basic circle axis overlaps with Y-axis, and OP overlaps with X-axis, then has

$\frac{\mathrm{\&rho;}}{R_b}=\frac{1}{\cos \mathrm{\&alpha;}},$

$\mathrm{\&theta;}=\frac{\sqrt{{\mathrm{\&rho;}}^2-{R_b}^2}}{R_b}$

Perhaps

$\mathrm{\&rho;}=R_b\sqrt{1+{\mathrm{\&theta;}}^2}$

$\mathrm{\&omega;}=\frac{\mathrm{d\&theta;}}{\mathrm{dt}},$

${\mathrm{\&upsi;}}_x=\frac{\mathrm{d\&rho;}}{\mathrm{dt}}$

${\mathrm{\&upsi;}}_x=\frac{R_b\sqrt{{\mathrm{\&rho;}}^2-{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^{}}}{\mathrm{\&rho;}}\mathrm{\&omega;}$

Meet above-mentioned kinematic relation and can form plane involute.υ _xCan be considered as cutting speed, because actual worm gear internal diameter is all greater than base circle diameter (BCD), also promptly cutting only can be from ρ＞R _bBeginning is so have cutting speed forever.

According to above-mentioned analysis, desire forms the motion of also need spinning of involute helicoid plane involute.If the helix angle of the worm gear flank of tooth is β _b, helical motion should meet following relation, i.e. feeding S distance vertically, and basic circle turns over angle

${\mathrm{\&theta;}}_2-{\mathrm{\&theta;}}_1=\mathrm{\&Delta;\&theta;}=\frac{\mathrm{Stg}{\mathrm{\&beta;}}_b}{R_b},$

See Fig. 9.

If

${\mathrm{\&theta;}}_1=\frac{\sqrt{{{\mathrm{\&rho;}}_1}^2-{R_b}^2}}{R_b}$

Then

${\mathrm{\&theta;}}_2={\mathrm{\&theta;}}_1+\mathrm{\&Delta;\&theta;}={\mathrm{\&theta;}}_1+\frac{\mathrm{Stg}{\mathrm{\&beta;}}_b}{R_b}=\frac{\sqrt{{{\mathrm{\&rho;}}_2}^2-{R_b}^2}}{R_b}$

${\mathrm{\&rho;}}_2=\sqrt{{({\mathrm{\&theta;}}_1+\mathrm{Stg}{\mathrm{\&beta;}}_b)}^2+{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^2}$

If starting point θ ₁=0, then ${\mathrm{\&theta;}}_2=\frac{\mathrm{Stg}{\mathrm{\&beta;}}_b}{R_b}$

If to establish the P point is point of a knife, then point of a knife in shaft section according to $\mathrm{\&theta;}=\frac{\sqrt{{\mathrm{\&rho;}}^2-{R_b}^2}}{R_b}$ Regular movement then can form plane involute, and this involute urve is β along helix angle _bHelix for the helical movement, meet formula ${\mathrm{\&rho;}}_2=\sqrt{{({\mathrm{\&theta;}}_1+\mathrm{Stg}{\mathrm{\&beta;}}_b)}^2+{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^2},$ Form involute helicoid.The numerically controlled lathe that utilizes turning multi-step thread function or have a C s function can be realized the branch tooth of worm gear.

Above-mentioned kinematic relation can only cutting circle column type worm gear.

(2) implementation step of each processing scheme

1. milling method may further comprise the steps:

A. common vertical or can install on the milling machine of vertical milling head dividing head is installed, make worktable and dividing head get in touch transmission in realizing, select suitable column or tapered mill for use; Go up installation circular turntable or dividing head in CNC milling machine (perhaps machining center), make worktable and dividing head (perhaps circular turntable) realize interior contact transmission, select suitable column or tapered mill for use.

B. workpiece is installed on (perhaps on the circular turntable) on the dividing head, with base cylinder section (Q face) and the parallel adjustment lathe of the horizontal moving direction of platen, make finger cutter or taper edge of milling cutter be positioned at the base cylinder section, constitute the linear interpolation blade, rectilinear edge becomes β with the z axle _B1Perhaps β ' _B1The angle is to form the helix angle that requires.

C. make worktable with respect to workpiece moving linearly in the Q face of workpiece base cylinder section, this motion must meet workpiece and PURE ROLLING is done on the Q plane, forms the worm gear cone angle simultaneously.

D. divide tooth by dividing head (perhaps circular turntable) after a flank of tooth cutting is finished, cut next tooth.

2. fly cutter gear hobbing job operation may further comprise the steps:

A. on the lathe that knife rest not only can be done tangential motion but also can axially move, for example cut spiroid gear on the chain digital control gear hobbing machine is installed fly cutter on the lathe knife bar, and workpiece is installed on the workpiece spindle.

B. the installation of knife bar, fly cutter and workpiece meets worm and gear transmission mounting condition, promptly install according to centre distance A and height a, and when the fly cutter rectilinear edge was positioned at the Q plane, rectilinear edge and worm gear axis angle is the helixangle of the flank of tooth of worm gear _B1Perhaps β ' _B1

C. adjust lathe, the processing flank of tooth.

3. the numerical control turning job operation may further comprise the steps:

A. workpiece is installed on the workpiece spindle.

B. for the C s function is arranged, and numerically controlled lathe or the machining center that unit head can Milling Process is installed, can adopts step identical and method cut with above-mentioned method for milling.

Process according to the following steps when C. being used for the cylindrical biasing worm gear of turning

1. basic circle is with the uniform rotation of ω rotating speed, and cutter is with speed υ _xRadial feed.

2. cutter shaft is to every displacement S, and workpiece must turn over Δ θ angle.

Wherein: ${\mathrm{\&upsi;}}_x=\frac{R_b\sqrt{{\mathrm{\&rho;}}^2-{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^{}}}{\mathrm{\&rho;}}\mathrm{\&omega;},$ υ _xBe tool feeding speed radially, ω is a workpiece rotational frequency, $\mathrm{\&Delta;\&theta;}=\frac{\mathrm{Stg}{\mathrm{\&beta;}}_b}{R_b},$ Δ θ is the angle that cutter shaft turns over to feeding distance S workpiece, R _bBe base radius, ρ is the distance of point of a knife to axis of workpiece, β _bBe Base spiral angle.

Of the present invention pair of helical pitch biasing worm drive is the novel biasing worm drive mode that a kind of theory of engagement is based on the space crossed axis helical tooth column gear transmission of straight line contact, its essential characteristic is that the worm screw workplace is the involute helicoid of two different leads, the worm gear flank of tooth be with the tangent plane of two different base cylinders on oblique straight hair give birth to line and launch the involute helicoid that forms to both direction respectively, be equivalent to the interior field of conjugate action and the outer engagement surface of cylindrical gear.Therefore can the analysis of worm couple and design be simplified greatly fully according to the existing Involutes Gears Transmission principle analysis and this kind of drive of design; Because worm gear is the straight line contact with instantaneous contact of worm mesh transmission, thereby this worm geared load-bearing capacity is big; Because the worm and gear working flank is involute helicoid, thereby simplicity of design not only, can be the existing cutting working method processing of this characteristics utilization of the involute helicoid flank of tooth fully also according to working flank.For worm screw, involute helicoid worm is processed with ripe method and technology, and setovering for two helical pitches uniquely the worm geared worm screw different is, it is two helical pitches, but since helical pitch fix thereby process and do not have difficulty.Can use commonly, existing machine tool and cutter are processed.

Good effect of the present invention is to be design that solves Procedure for Spiroid Gearing and the problem of making difficulty, the two helical pitches that propose are setovered worm geared method for designing simply and accurately, can make the designs simplification of worm and gear, make processing and manufacturing simple simultaneously, special purpose machine tool is efficient except that adopting, the high-precision processing, also can utilize the accurate cut flank of tooth of existing machine tool and cutter, and needn't add miscellaneous equipment or use special purpose machine tool and dedicated tool.

Description of drawings

Fig. 1 is two helical pitch biasing worm transmission structure figure

Fig. 2 is two helical pitch biasing worm drive principle figure

Fig. 3 is the geometric parameter synoptic diagram of worm and gear

Fig. 4 is a worm gear Milling Process synoptic diagram

Fig. 5 is a tapered mill shape synoptic diagram

Fig. 6 is the scheme of installation of mesh tooth face milling cutter in the milling

Fig. 7 forms synoptic diagram for plane involute

Fig. 8 is a fly cutter rolling cut worm gear synoptic diagram

Fig. 9 is a turning worm gear flank of tooth synoptic diagram

Figure 10 forms flank of tooth synoptic diagram for the involute urve helical motion

Figure 11 is cylindricality worm and gear design example geometric model figure

Figure 12 is the scheme of installation of milling cutter

Embodiment

One, taper over-type worm gear design example

A. cylindrical worn transmission design example:

Known centre distance A=100, ratio of gear i12 and number of teeth z, z=55.Rough calculation worm screw outside diameter d, d=kA=2R, $k=\frac{1}{2}~\frac{7}{12},$ D=54; Selected helixangle _B1, β ' _B1, β _B1=20 °, β ' _B1=30 °, calculate the base cylinder radius, R _B2=4.758, R _B1=95.242, R ' _B2=3.251, R ' _B1=103.251; Get n=10%z=5

1) primary Calculation worm gear external diameter R _a, get R _a=152

2) calculate the reference circle of wormwheel radius R _m, R _m≈ 109.1

3) according to R _mCalculate modulus m, $m=\frac{2R_m}{z}=3.967$

4) calculate the worm gear tooth depth

The ad. t ' at reference circle place _mWith height of teeth root t ' _m, $t_m^{\&prime;}=t_m=\frac{2R_m\sin \frac{\mathrm{\&pi;}}{z}}{(\mathrm{tg}{\mathrm{\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}=13.233$

Tooth depth h=h _a+ h _f=m+m+C ^*=m+m+0.2m=2.2m=8.7274

Eligible h _a＜t ' _m, h _f＜t _m

5) the involute urve starting point angle that interlaces on the reference circle cross section

6) calculate worm gear internal diameter R _i

Cause

For on the occasion of, according to condition R _i＞R ' _B1So, get minimum value R _i=R ' _B1=103.251.

7) calculate the spiroid gear taper angle theta ₁

θ ₁＝17°

8) calculate the worm screw tooth depth

h ₂＝8.79

9) worm screw diameter d=2R

10) worm screw is installed offset distance

E＝48.709

11) worm screw is installed end face height (apart from the reference circle plane)

a＝r ₂-h _a＝22.833

12) calculate cylindrical worm gear external diameter, internal diameter

Worm gear external diameter R _Ac≈ 139, worm gear internal diameter R _Ic=R _i≈ 103.5

13) checking computations while total number of teeth in engagement

$n\&le;\frac{\sqrt{R_a^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}=3.76$

14) worm thread length

L＝30

According to aforementioned calculation consequence devised worm and gear, geometric model as shown in Figure 1.

B. Procedure for Spiroid Gearing calculated examples:

Known centre distance A=100, ratio of gear i12 and number of teeth z, z=100.Rough calculation worm screw small end outside diameter d=2R gets k=6.5/12, gets d=54; Selected helixangle _B1=10 °, β ' _B1=20 °, calculate the base cylinder radius and get R _B2=5.367, R _B1=94.633, R ' _B2=2.825, R ' _B1=102.825; Get n=10%z=10.

According to calculating with quadrat method and step with above-mentioned, the result is as follows: calculate and get R _a=173, R _m=144, modulus m=2.88, tooth depth h=6.336, phase alternate angle

R _i=119.5, the spiroid gear taper angle theta ₁=74 °, worm thread length L=71, worm screw tooth depth $h_{}$ ${}_2=h/\cos \left({\mathrm{tg}}^{-1}\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="DEST\_PATH\_HSB00000011275800011.png,H2009100670197E00031.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}{r_2}\right)\&ap;6.37;$ The spiroid taper angle theta ₂=12 °, worm screw outside diameter d _d≈ 69, worm screw end diameter d _s≈ 39; Worm screw is installed offset E ≈ 73.3; Worm and gear setting height(from bottom) (apart from the reference circle plane) a=23.97.Checking computations are total number of teeth in engagement n ≈ 10 simultaneously.

According to aforementioned calculation consequence devised worm and gear, geometric model as shown in Figure 1.

Worm geared condition of self-locking is got μ=0.1,

External toothing transmission self-locking critical angle is 40.9 degree; Interior engaged transmission self-locking critical angle is 20.2 degree.

Worm geared efficient is got

μ=0.03.

$\mathrm{\&eta;}=\frac{D}{\mathrm{iC}}=0.58$

Two, cut embodiment

The taper worm gear Milling Process of above-mentioned design

Select common vertical milling machine for use, the vertical milling unit head can the anglec of rotation, and dividing head is installed on the platen.Workpiece is installed on the dividing head, and the gyration of dividing head and the in-movement of platen are got in touch transmission in realizing; Select tapered mill for use.

As the base cylinder section (Q face) and platen horizontal moving direction (YOZ plane) parallel adjustment lathe of Figure 12 with worm gear, make the tapered mill conical surface and Q face tangent, constitute the rectilinear edge of cutting usefulness, milling cutter axis and z axle meet at right angles.Worm-wheel shaft wire-wound z axle turns over the γ angle, and when dividing head drive workpiece turned round with angular velocity omega, the worktable movement therewith realized aggregate velocity υ.Select the suitable amount of feeding for use along the rectilinear edge direction, the cutting flank of tooth can cut repeatedly up to cutting to complete dark.

The engagement side flank of tooth in the cutting

Z-100

H-15.34

R′ _b1＝103.251

Get workpiece rotational frequency n1=0.2rpm, $\mathrm{\&omega;}=2\mathrm{\&pi;n}=2\mathrm{\&pi;}\frac{0.2}{60}\&ap;0.0209$

$\mathrm{\&upsi;}=\sqrt{{\mathrm{\&upsi;}}_1^2+{\mathrm{\&upsi;}}_2^2}=2.338\mathrm{mm}/s$

υ ₁＝ωR _b＝0.0209×103.251＝2.16mm/s

$R_a=\frac{R_{b1}^{\&prime;}}{\cos {\mathrm{\&alpha;}}_a^{\&prime;}}=173=\frac{103.251}{\cos {\mathrm{\&alpha;}}_a^{\&prime;}}$ α′ _a＝53.357°

$R_i=\frac{R_{b1}^{\&prime;}}{\cos {\mathrm{\&alpha;}}_i^{\&prime;}}=119.5=\frac{103.251}{\cos {\mathrm{\&alpha;}}_i^{\&prime;}}$ α′ _i＝30.228°

The radius R x of milling cutter is a condition not interfere with the interior engagement side flank of tooth, should satisfy

$\mathrm{Rx}\&le;(\sqrt{R_i^2-R_{b1}^{\&prime;2}}-\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;}=33.4$

Get milling cutter radius R x=30

The cone angle τ of milling cutter should satisfy

τ＝β′ _b1-γ＝30-22.55＝7.45°

If select τ=30 ° milling cutter for use,, cutter spindle is rotated counterclockwise 22.55 ° around the x axle for satisfying the cutting needs.

Adjust lathe by above-mentioned parameter and can cut the interior engagement side flank of tooth.Divide tooth by dividing head after a flank of tooth cutting is finished, cut next tooth, thereby realize monodentate calibration Milling Process.Figure 12 shows that mesh tooth face in the left-handed worm gear of cutting.

Cutting external toothing lateral tooth flank

Z-100

H-15.34

R _b1＝94.633

Get workpiece rotational frequency n1=0.2rpm, $\mathrm{\&omega;}=2\mathrm{\&pi;n}=2\mathrm{\&pi;}\frac{0.2}{60}\&ap;0.0209$

$\mathrm{\&upsi;}=\sqrt{{\mathrm{\&upsi;}}_1^2+{\mathrm{\&upsi;}}_2^2}=2.12\mathrm{mm}/s$

υ ₁＝ωR _b＝0.0209×94.633＝1.98mm/s

$R_a=\frac{R_{b1}}{\cos {\mathrm{\&alpha;}}_a}=173=\frac{94.633}{\cos {\mathrm{\&alpha;}}_a}$ α _a＝56.837°

$R_i=\frac{R_{b1}}{\cos {\mathrm{\&alpha;}}_i^{\&prime;}}=119.5=\frac{94.633}{\cos {\mathrm{\&alpha;}}_i^{\&prime;}}$ α _i＝37.635°

This moment available milling cutter the in addition side end face milling flank of tooth, the installation of milling cutter is adopted finger cutter with reference to Fig. 3.Difference is that cutting blade and cutter shaft are rectangular.For satisfying the cutting needs, cutter spindle and z axle clamp angle are 90 °-γ-β _B1=59.11 °.

During the cutting external toothing flank of tooth, adjust the lathe except pressing above-mentioned parameter, must meet the designing requirement of transverse tooth thickness, this must realize by correct tool setting.Tool setting is basis then

Value realizes.

Same principle can be cut the dextrorotation worm gear.

Claims (6)

1. worm geared method for designing of double-lead linear contact bias is characterized in that may further comprise the steps:

(1) according to power, ratio of gear and the rotating speed of required transmission, after parameters such as definite centre distance, the number of teeth, worm screw mean radius, while total number of teeth in engagement, determines each base radius, profile angle;

(2) calculate worm gear external diameter (R just _a) the basis on, with the regulation reference circle be that benchmark carries out parameter designing, parameter comprises: worm gear external diameter (R _a, reference circle of wormwheel radius (R _m), the involute urve starting point angle that interlaces on the reference circle cross section

Modulus (m), worm gear internal diameter (R _i), worm gear tooth depth (h), spiroid gear cone angle (θ ₁), worm screw tooth depth (h ₂), spiroid cone angle (θ ₂), worm thread length (L), worm screw outside diameter (d _d), worm screw end diameter (d _s), worm and gear installs mesh tooth face helical pitch (p ') in offset distance (E), worm screw involute helicoid relative position parameter (S), worm and gear setting height(from bottom) (a), worm screw external toothing flank of tooth helical pitch (p), the worm screw, carries out while total number of teeth in engagement checking computations, cylindrical worm and worm gear calculation of parameter, the calculating of worm geared self-locking critical angle and efficiency estimation again.

(3) formula of being derived in the parameter designing according to step 2 calculates each parameter.

2. by the worm geared method for designing of the described double-lead linear contact bias of claim 1, both be applicable to the conical design that also is applicable to cylindrical worm and worm gear, it is characterized in that:

A. the first calculation worm gear external diameter (Ra) described in the step (2) is the parameter according to step (1), comprises worm screw external diameter, while total number of teeth in engagement etc., calculates as follows, and gets the rounding value:

$R_a=\sqrt{R_{b1}^{\&prime;2}+{(\mathrm{Rctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+2(n+0.5)R_{b2}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;})}^2}$

Wherein: R ' _B2Be interior engagement side worm screw involute helicoid base radius, R ' _B1Be interior engagement side worm gear involute helicoid base radius, n is the while total number of teeth in engagement, and R is the worm screw radius

B. the reference circle of wormwheel radius (R described in the step (2) _m) reference circle of wormwheel be to be the circle in the center of circle with the worm-wheel shaft kernel of section, this circle on transverse tooth thickness equate with inter-tooth slots, the mistake this circle the cross section be defined as the reference circle cross section.Reference circle of wormwheel radius (R _m)

$\frac{2\mathrm{\&pi;}}{z}-[\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})$

$+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))]=0$

Calculating by following equation solution:

Wherein: r=R _a, R _aBe worm gear external diameter, R _B1Be external toothing side involute helicoid base radius, R ' _B1Be interior engagement side involute helicoid base radius, R _mBe reference radius, z is the worm gear number of teeth.

C. the involute urve starting point angle that interlaces on the reference circle cross section described in the step (2)

Be according to defined reference radius, calculate that its expression formula is in conjunction with involute equation:

D. the modulus (m) described in the step (2) is to calculate according to defined reference radius, and its expression formula is:

$m=\frac{{2R}_m}{z}$

E. the worm gear internal diameter (R described in the step (2) _i) be to calculate according to reference radius and theoretical heel height, press following equation solution:

$\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2$

$-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m})-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})=0,$

Wherein: r=R _i

F. the worm gear tooth depth (h) described in the step (2) is to calculate according to reference radius and theoretical heel height, and its expression formula is:

Ad. h _a=m

Height of teeth root h _f=m+C ^*

h＝h _a+h _f

C ^*Be tip clearance coefficient, get C ^*=(0.1～0.2) m, m is a modulus.

G. the spiroid gear cone angle (θ described in the step (2) ₁) calculate according to following formula:

${\mathrm{\&theta;}}_1=\frac{\mathrm{\&pi;}}{2}-{\mathrm{tg}}^{-1}\left(\frac{t_{\max }^{\&prime;}}{R_a-R_i}\right)$

Wherein: t ' _Max=t ' | r=R _i, and

$t^{\&prime;}=\frac{2r\sin \frac{{\mathrm{\&phi;}}^{\&prime;}}{2}}{(\mathrm{tg}{\mathrm{\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}$

${\mathrm{\&phi;}}^{\&prime;}=\frac{2\mathrm{\&pi;}}{z}-[\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}$

$+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m})\left)\right]$

H. the worm screw tooth depth (h described in the step (2) ₂) be to calculate according to the worm gear tooth depth, its expression formula is:

$h_2=h_{2a}+h_{2f}=(m+m+C^{*})/\cos \left({\mathrm{tg}}^{-1}\frac{R_{b2}^{\&prime;}}{r_2}\right)$

Wherein: $r_2=\sqrt{R^2-{R_{b2}}^{\&prime;2}}$

I. the spiroid cone angle (θ described in the step (2) ₂) be to calculate according to worm gear exradius and eccentric throw, its expression formula is:

${\mathrm{\&theta;}}_2={\mathrm{tg}}^{-1}\left(\frac{(R_a-R_e)\mathrm{tg}(\frac{\mathrm{\&pi;}}{2}-{\mathrm{\&theta;}}_1)}{R_a-E}\right)$

Wherein $R_e=\sqrt{{R_{b1}}^{\&prime;2}+E^2}$

J. the worm thread length (L) described in the step (2) is to calculate according to interior engages worm base radius and worm spiral lift angle, and its expression formula is:

L＝2(n+1)R _b2′πtgλ′

K. worm screw outside diameter (d _d), worm screw end diameter (d _s) be to calculate according to worm screw mean radius and worm screw cone angle, its expression formula is:

d _d＝d+Ltgθ ₂?d _s＝d-Ltgθ ₂

Wherein: d _dBe outside diameter, d _sBe end diameter

L. offset distance (E) is installed is the distance of worm screw tip-to-face distance worm gear center line to the worm and gear described in the step (2), according to the worm and gear flank of tooth not interference condition calculate, its expression formula is:

E＝r ₂ctgβ′ _b1+mtgβ′ _b1

M. the worm and gear setting height(from bottom) (a) described in the step (2) is the vertical range of worm axis apart from the reference circle cross section, and its expression formula is:

a＝r ₂-h _a

Wherein $r_2=\sqrt{R^2-R_{b2}^{\&prime;2}}$

N. worm screw involute helicoid relative position parameter (S) is used for determining the relative position of two involute helicoid starting points of worm screw, computing formula:

E wherein _Fmin=r ₂Ctg β ' _B1+ mtg β ' _B1

O. the worm screw external toothing flank of tooth helical pitch (p) described in the step (2) is the lead of helix on the external toothing lateral tooth flank, and its expression formula is:

p＝2R _b2πtgλ

λ---external toothing lead angle; λ=β ' _B1

P. the interior mesh tooth face helical pitch (p) of the worm screw described in the step (2) is the lead of helix on the interior engagement side flank of tooth, and its expression formula is:

p′＝2R _b2′πtgλ′

λ '---external toothing lead angle; λ=β _B1

Q. total number of teeth in engagement n checking computations are in the time of described in the step (2):

$n\&le;\frac{\sqrt{R_a^2-R_{b1}^{\&prime;2}}-(r_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{{{2R}_{b2}}^{\&prime;}{\mathrm{\&pi;tg\&lambda;}}^{\&prime;}}$

Wherein: $r_2=\sqrt{R^2-R_{b2}^{\&prime;2}}$

R. can be by following step designing and calculating for cylindrical worm and worm gear:

1. calculate cylindrical worm gear external diameter R _Ac

The tooth depth of getting worm gear is identical with taper worm gear tooth depth, R _AcFind the solution by following equation

$h_a=\frac{2r\sin \frac{{\mathrm{\&phi;}}^{\&prime;}}{2}}{(\mathrm{tg}{\mathrm{\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{|}_{r=R_{\mathrm{ac}}}$

Wherein:

${\mathrm{\&phi;}}^{\&prime;}=\frac{2\mathrm{\&pi;}}{z}-[\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}$

$+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m})\left)\right]$

2. calculate cylindrical worm gear internal diameter R _Ic

Find the solution by following equation

$h_f=\frac{2r\sin \frac{\mathrm{\&phi;}}{2}}{({\mathrm{tg\&beta;}}_{b1}+\mathrm{tg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{|}_{r=R_{\mathrm{ic}}}$

Wherein:

$\mathrm{\&phi;}=\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{r}\right)-{\cos }^{-1}\frac{R_{b1}}{r}$

$+(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{R_m}+\left(\frac{2\mathrm{\&pi;}}{z}\right)/2$

$-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}}{R_m}\right)-{\cos }^{-1}\frac{R_{b1}}{R_m}))$

$-(\mathrm{tg}\left({\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r}\right)-{\cos }^{-1}\frac{R_{b1}^{\&prime;}}{r})$

Wherein: R _Ic〉=R ' _B1

3. other geometric parameters of cylindrical worm gear are identical with the computing method of above-mentioned conical worm gear

4. cylindrical worn tooth depth

Be calculated as follows

$h_2=h_{2a}+h_{2f}=(m+m+C^{*})/\cos \left({\mathrm{tg}}^{-1}\frac{R_{b2}^{\&prime;}}{r_2}\right)$

5. check total number of teeth in engagement n simultaneously

$n\&le;\frac{\sqrt{R_{\mathrm{ac}}^2-R_{b1}^{\&prime;2}}-(r_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{R_{b2}}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}$

6. calculate worm thread length

L＝L′+p′＝2(n+1)R _b2′πtgλ′

S. the worm geared self-locking critical angle described in the step (2) is calculated as: with the critical value λ of worm screw basic circle lead angle _oThe expression, when worm screw basic circle lead angle λ less than critical angle λ _oThe time worm and gear transmission self-locking.

λ _oBe calculated as follows

Get 0.9 ~ 1, the big more value of ratio of gear is big more.Work as r=R _B2The time try to achieve external toothing transmission self-locking critical angle; As r=R ' _B2The time try to achieve in engaged transmission self-locking critical angle;

T. the worm geared efficiency estimation described in the step (2) is estimated av eff with following formula

$R_1=\sqrt{{(E+L/2)}^2+{R_{b1}}^2},$ R ₂＝R-h _a

$\mathrm{\&eta;}=\frac{D}{\mathrm{iC}}$

Wherein i is a ratio of gear

Reduce value in 0.15 ~ 0.3 with the ratio of gear increase;

Increase value in 0.9 ~ 1 with ratio of gear.

3. the worm geared job operation of double-lead linear contact bias is characterized in that job operation comprises: milling method, fly cutter gear hobbing job operation and numerical control turning job operation.

4. by the worm geared job operation of the described double-lead linear contact bias of claim 3, it is characterized in that described milling method may further comprise the steps:

A. common vertical or can install on the milling machine of vertical milling head dividing head is installed, make worktable and dividing head get in touch transmission in realizing, select suitable column or tapered mill for use; Go up installation circular turntable or dividing head in CNC milling machine (perhaps machining center), make worktable and dividing head (perhaps circular turntable) realize interior contact transmission, select suitable column or tapered mill for use.

B. workpiece is installed on (perhaps on the circular turntable) on the dividing head, with base cylinder section (Q face) and the parallel adjustment lathe of the horizontal moving direction of platen, make finger cutter or taper edge of milling cutter be positioned at the base cylinder section, constitute the linear interpolation blade, rectilinear edge becomes β with the z axle _B1Perhaps β ' _B1The angle is to form the helix angle that requires.

C. make worktable with respect to workpiece moving linearly in the Q face of workpiece base cylinder section, this motion must meet workpiece and PURE ROLLING is done on the Q plane, forms the worm gear cone angle simultaneously.

D. divide tooth by dividing head (perhaps circular turntable) after a flank of tooth cutting is finished, cut next tooth.

5. by the worm geared job operation of the described double-lead linear contact bias of claim 3, it is characterized in that described fly cutter gear hobbing job operation may further comprise the steps:

A. on the lathe that knife rest not only can be done tangential motion but also can axially move, for example cut spiroid gear on the chain digital control gear hobbing machine is installed fly cutter on the lathe knife bar, and workpiece is installed on the workpiece spindle.

B. the installation of knife bar, fly cutter and workpiece meets worm and gear transmission mounting condition, promptly install according to centre distance A and height a, and when the fly cutter rectilinear edge was positioned at the Q plane, rectilinear edge and worm gear axis angle is the helixangle of the flank of tooth of worm gear _B1Perhaps β ' _B1

C. adjust lathe, the processing flank of tooth.

6. by the worm geared job operation of the described double-lead linear contact bias of claim 3, it is characterized in that described numerical control turning job operation may further comprise the steps:

A. workpiece is installed on the workpiece spindle.

B. for the C s function is arranged, and numerically controlled lathe or the machining center that unit head can Milling Process is installed, can adopts step identical and method cut with above-mentioned method for milling.

Process according to the following steps when C. being used for the cylindrical biasing worm gear of turning

1. basic circle is with the uniform rotation of ω rotating speed, and cutter is with speed υ _xRadial feed.

2. cutter shaft is to every displacement S, and workpiece must turn over Δ θ angle.

Wherein: ${\mathrm{\&upsi;}}_x=\frac{R_b\sqrt{{\mathrm{\&rho;}}^2-R_b^2}}{\mathrm{\&rho;}}\mathrm{\&omega;},$ υ _xBe tool feeding speed radially, ω is a workpiece rotational frequency, $\mathrm{\&Delta;\&theta;}=\frac{\mathrm{Stg}{\mathrm{\&beta;}}_b}{R_b},$ Δ θ is the angle that cutter shaft turns over to feeding distance S workpiece, R _bBe base radius, ρ is the distance of point of a knife to axis of workpiece, β _bBe Base spiral angle.

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