CN102151911A - Machining method for dual-lead linear contact offset worm drive - Google Patents

Abstract

The invention discloses a machining method for dual-lead linear contact offset worm drive, which belongs to the technical field of manufacture. The dual-lead linear contact offset worm drive belongs to a spiroid drive form in worm wheel and worm drive. The machining method disclosed by the invention comprises a milling machining method, a fly-cutter hobbing machining method and a numerical control lathe machining method. By using the invention, accurate and structure-simplified worm wheel and worm are obtained aiming at a dual-lead linear contact offset worm drive designing method without efficient and high-precision machining of a special lathe, and a tooth surface is accurately machined by using the traditional lathe and the traditional tool without addition of other equipment or use of a special lathe and a special tool.

Description

The worm geared processing method of double-lead linear contact bias

Divide an application

The present invention be name be called " worm geared design of double-lead linear contact bias and manufacture method " divide an application original application day: on May 27th, 2009, application number: 200910067019.7.

Technical field

The invention belongs to theory of mechanics and mechanical manufacturing field, relate to gears engaged 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 bearing capacity, increased gearratio, 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 relationship 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 pitches worm geared processing method of setovering.

Summary of the invention

The objective of the invention is to problem for the processing difficulties that solves Procedure for Spiroid Gearing, on a kind of design and calculation method basis simply and accurately, propose a kind ofly can use existing machine tool and Tool in Cutting machining worm wheel worm tooth-surface, and needn't add equipment or use the processing method of 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 working face is that the involute helicoid by two constant leads constitutes, and respectively with the both sides working face engagement of worm-gear toothing, realizes the bi-directional of motion and power.The both sides working face of worm gear also is an involute helicoid, and they are that straight line contacts with the instantaneous contact condition of worm screw working face engagement, and the both sides working face 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 machining 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, gearratio and the rotating speed of required transmission, after parameters such as definite centre-to-centre spacing, 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 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):

$R_a=\sqrt{R_{b1}^{\&prime;2}+{(\mathrm{Rctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+2(n+0.5){R^{\&prime;}}_{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">b</figure-callout>}2}{\mathrm{\&pi;tg\&lambda;}}^{\&prime;})}^2}$

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 λ '---interior engages worm basic circle lead angle, 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 of wormwheel cross section.Reference circle of wormwheel radius (R _m) calculating by following equation solution:

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

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

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

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

Be according to defined reference circle of wormwheel 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 circle of wormwheel 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 circle of wormwheel 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: γ=R _i

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

Height of teeth top 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 described in the step (2) (θ 1) calculates according to following formula:

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

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

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

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

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

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: ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{R^{\&prime;}}_{b2}}^2}$

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 $R_e=\sqrt{{R_{b1}}^{\&prime;2}+{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">E</figure-callout>}}^2}$

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′ _b2πtgλ′

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

a＝r ₂-h _a

Wherein ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}^2}$

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 _b2πtgλ

λ---external toothing worm screw basic circle lead angle; R _B2---external toothing side worm screw involute helicoid base radius; 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′ _b2πtgλ′

λ '---interior engages worm basic circle lead angle; R ' _B2---interior engagement side worm screw involute helicoid base radius; λ '=β ' _B1

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

$n\&le;\frac{\sqrt{R_a^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{R^{\&prime;}}_{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">b</figure-callout>}2}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}$

Wherein: ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}^2}$

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

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

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. the reference circle of wormwheel radius (R of cylindrical worm gear _m), the involute starting point angle that interlaces on the reference circle cross section

Modulus (m), worm gear internal diameter (R _i), worm gear tooth depth (h) is identical with the computational methods of above-mentioned conical worm gear;

4. cylindrical worn tooth depth

Be calculated as follows

${\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">h</figure-callout>}}_2=h_{2a}+h_{2f}=(m+m+C^{*})/\cos \left({\mathrm{tg}}^{-1}\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}{r_2}\right)$

5. check total number of teeth in engagement n simultaneously

$n\&le;\frac{\sqrt{R_{\mathrm{ac}}^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{R^{\&prime;}}_{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">b</figure-callout>}2}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}$

6. calculate worm thread length

L＝2(n+1)R′ _b2πtgλ′

S. the worm geared self-locking critical angle described in the step (2) is calculated as: with the critical value λ of external toothing worm screw basic circle lead angle _oThe expression, when external toothing 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 gearratio is big more.As γ=R ' _B2The time try to achieve in engaged transmission self-locking critical angle.R ' wherein _B2--interior engagement side worm screw involute helicoid base radius.

T. worm geared efficiency estimation

Calculate the average efficiency 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 gearratio

Reduce value in 0.15～0.3 with the gearratio increase;

Increase value in 0.9～1 with gearratio;

One, the used design of the present invention is:

With the tooth both sides working face of worm and gear, respectively by space crossed axis involute cylindrical gear external toothing transmission and the design of interior engaged transmission principle.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 working face 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 basic 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. gearratio i ₂₁

3. basic structure design

1) determines elementary structure parameter

(1) centre-to-centre spacing 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, get it and be 0.002K _vBe velocity coeffficient, 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 gearratio 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 gearratio,

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

${\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b=\frac{A}{\mathrm{ztg}{\mathrm{\&beta;}}_{b1}+1};$ R _b1＝A-R _b2； ${\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2=\frac{A}{\mathrm{ztg}{\mathrm{\&beta;}}_{b1}^{\&prime;}-1};$

R′ _b1＝A+R′ _b2。

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

$i_{21}=\frac{R_{b1}}{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}\mathrm{ctg}{\mathrm{\&beta;}}_{b1}=\frac{R_{b1}^{\&prime;}}{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}---\left(1\right)$

(5) rough calculation worm screw outside diameter d, d=kA=2R,

D=54;

R is the worm screw average 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 geometric parameter designing and calculating

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

(1) rough calculation worm gear external diameter R _a

Formula:

$R_a=\sqrt{R_{b1}^{\&prime;2}+{(\mathrm{Rctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+2(n+0.5){{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;}{\mathrm{tg\&lambda;}}^{\&prime;})}^2}$

(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:

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

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

Wherein: r=R _a

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

Two spiral involute surfaces of worm gear tooth involute original 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 machining.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:

Height of teeth top 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:

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

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

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

Wherein: γ=R _i

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

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

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

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

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

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

(8) worm screw tooth depth h ₂

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

${\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">h</figure-callout>}}_2=h_{2a}+h_{2f}=(m+m+C^{*})/\cos \left({\mathrm{tg}}^{-1}\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}{r_2}\right)$

Wherein: ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;2}}$

(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 $R_e=\sqrt{{R_{b1}}^{\&prime;2}+{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">E</figure-callout>}}^2}$

(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 ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}^2}$

(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

$n\&le;\frac{\sqrt{R_a^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}$

Wherein: ${\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2=\sqrt{R^2-{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}^2}$

(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

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

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

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

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 computational methods of above-mentioned conical worm gear

4. cylindrical worn tooth depth

Be calculated as follows

${\mathrm{<figure-callout\; id="2"\; label="h"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">h</figure-callout>}}_2=h_{2a}+h_{2f}=(m+m+C^{*})/\cos \left({\mathrm{tg}}^{-1}\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^2}{r_2}\right)$

5. check total number of teeth in engagement n simultaneously

$n\&le;\frac{\sqrt{R_{\mathrm{ac}}^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{R^{\&prime;}}_{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">b</figure-callout>}2}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}$

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 gearratio is big more.As γ=R _B2The time try to achieve external toothing transmission self-locking critical angle; As γ=R ' _B2The time try to achieve in engaged transmission self-locking critical angle;

(20) worm geared efficiency estimation

Estimate average efficiency 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 gearratio

Reduce value in 0.15～0.3 with the gearratio increase;

Increase value in 0.9～1 with gearratio;

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

(1) processing scheme

The formation of the working face 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 machining 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 movement relation, and the flank of tooth machining 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 machining of worm screw is suitable for and utilizes especially numerically controlled lathe machining 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 machining worm gear of common vertical knee-type milling machine or universal milling machine 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 taper 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 helical 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 workbench 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

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

Wherein:

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

υ ₁＝ωR′ _b1

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

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

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 taper mill this moment for use.The installation of taper 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 of a 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 processing method

Utilize the fly cutter gear hobbing processing method machining worm gear of general hobbing machine or chain digital control gear hobbing machine.Its basic demand is that gear-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 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-to-centre spacing 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.Gearratio is at first satisfied in the rotation of worm gear and worm screw

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 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 tooth 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 processing method.

3. numerical control turning processing method

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

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

Involute 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

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

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

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

${\mathrm{\&upsi;}}_x=\frac{R_b\sqrt{{\mathrm{\&rho;}}^2-{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b^{}}}{\mathrm{\&rho;}}\mathrm{\&omega;}$

Meet above-mentioned movement 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 helical angle of the worm gear flank of tooth is β _b, screw 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

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

${\mathrm{\&rho;}}_2=\sqrt{{({\mathrm{\&theta;}}_1+\mathrm{Stg}{\mathrm{\&beta;}}_b)}^2+{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^2}$

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

Regular movement then can form plane involute, and this involute is β along helical angle _bHelix for the helical movement, meet formula

Form involute helicoid.The numerically controlled lathe that utilizes turning multiple thread function or have a C s function can be realized the branch tooth of worm gear.

Above-mentioned movement 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 workbench and dividing head get in touch transmission in realizing, select suitable column or taper mill for use; Go up installation circular turntable or dividing head at CNC milling machine (perhaps machining center), make workbench and dividing head (perhaps circular turntable) realize interior contact transmission, select suitable column or taper 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 helical angle that requires.

C. make workbench 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 processing method 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 machining 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-to-centre spacing 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 processing method 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 machining 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:

υ _xBe tool feeding speed radially, ω is a workpiece rotational frequency, 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 working face 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 roller 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 engaging transmission, thereby this worm geared 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,, high accuracy efficient except that the employing special purpose machine tool processed, also can utilize the accurate machining 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 schematic diagram of worm and gear

Fig. 4 is a worm gear Milling Process schematic diagram

Fig. 5 is a taper mill shape schematic diagram

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

Fig. 7 forms schematic diagram for plane involute

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

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

Figure 10 forms flank of tooth schematic diagram for the involute screw

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

Figure 12 is the scheme of installation of milling cutter

The specific embodiment

One, taper Worm and Worm Gear Driving design example

A. cylindrical worn transmission design example:

Known centre-to-centre spacing A=100, gearratio i12 and number of teeth z, z=55.Rough calculation worm screw outside diameter d, d=kA=2R,

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

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 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 of a 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

$n\&le;\frac{\sqrt{R_a^2-R_{b1}^{\&prime;2}}-({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">r</figure-callout>}}_2\mathrm{ctg}{\mathrm{\&beta;}}_{b1}^{\&prime;}+\mathrm{mtg}{\mathrm{\&beta;}}_{b1}^{\&prime;})}{2{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000425630900051.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^{\&prime;}\mathrm{\&pi;tg}{\mathrm{\&lambda;}}^{\&prime;}}=3.76$

14) worm thread length

L＝30

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

B. Procedure for Spiroid Gearing calculated examples:

Known centre-to-centre spacing A=100, gearratio 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

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, geometrical 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, machining embodiment

The taper worm gear Milling Process of above-mentioned design

Select common vertical knee-type 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 taper 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 taper 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 workbench 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

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

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 (1)

1. worm geared processing method of double-lead linear contact bias, comprising: milling method, fly cutter gear hobbing processing method and numerical control turning processing method is characterized in that

1) 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 workbench and dividing head get in touch transmission in realizing, select suitable column or taper mill for use; Circular turntable or dividing head are installed on CNC milling machine, are made in the realization of workbench and dividing head and get in touch transmission, select suitable column or taper mill for use;

B. workpiece is installed on the dividing head, with the parallel adjustment lathe of the horizontal moving direction of base cylinder section and platen, makes 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 helical angle that requires;

C. make workbench 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 after a flank of tooth cutting is finished, cut next tooth;

2) described fly cutter gear hobbing processing method 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 machining 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-to-centre spacing 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) described numerical control turning processing method 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 machining 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: υ _xBe tool feeding speed radially, ω is a workpiece rotational frequency,

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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