CN105179600A - Double large negative shifted involute gear transmission device - Google Patents

Abstract

The invention discloses a double large negative shifted involute gear transmission device. The transmission device comprises one pair of mutually meshed gears, and is characterized in that working profiles of teeth of two gears comprise outer convex first involute parts and outer convex second involute parts which are arranged on the outer parts of the base circles of the gears and inner concave third involute parts which are arranged in the base circles of the gears, wherein the first involute parts, the second involute parts and the third involute parts are sequentially smoothly connected from the tops to the roots of the gears; the first involute part of one gear of the two gears is in concave-convex arc meshing with the third involute part of the third involute part of the other one gear of the two gears, and the second involute parts of the two gears are mutually in concave-convex arc meshing. The double large negative shifted involute gear transmission device has advantages of relatively long service life, relatively high transmission reliability, low maintenance cost, and easy installation process.

Description

Two large negative addendum modification Involutes Gears Transmission device

Technical field

The present invention relates to mechanical transmission middle gear drive technology field, relate to a kind of two large negative addendum modification Involutes Gears Transmission device especially.

Background technique

Mechanical transmission refers to the transmission utilizing machinery type transferring power and motion, conventional machinery transmission system type have gear transmission, Worm Wheel System, V belt translation, Chain conveyer, train etc.Wherein, gear transmission is a kind of type of belt drive most widely used in mechanical transmission.Its velocity ratio is comparatively accurate, and efficiency is high, compact structure, reliable operation, and the life-span is long.During gear transmission, two-wheeled flank profil must meet following condition: " no matter two-wheeled flank profil contacts in any position, the common normal line of point of contact must cross fixed point C---the node on line of centers excessively." the flank profil engagement fundamental law of Here it is circular gear.The curve that can meet this law has a lot, and in fact also will consider the requirement of the aspects such as manufacture, installation and bearing capacity, generally only adopt several curves such as involute, cycloid and circular arc to make the work flank profil of gear, wherein major part is involute profile.

Involute gear has separability, interchangeability, adaptability, the feature such as easily manufactured, is widely applied, occupies the dominant position of gear drive gradually.But because involute gear is convexo-convex conjugate curve, be convexo-convex profile contact in engagement driving, contact stress is large, touch strength of gear is low, the large normal shift flank profil of many employings when solving big speed ratio problem, pressure angle increases, and flexural stress and contact stress increase thereupon.For improving contact strength, resistance to flexure, there is higher requirement to gear manufacture material and accuracy of manufacturing, increasing manufacture cost, not solving root problem simultaneously.For heavy-load high-power, large load transmission, involute gear pitting corrosion and broken tooth failure, the life-span is extremely short, and reliability is low, and maintenance cost is large, is the critical defect of involute gear.

And circular tooth gear is two sections of circular arc convex-concave profile contact in the same way, area of contact is large, stress dispersion, and contact stress is little, and contact strength is high, not easily pitting corrosion failure effect working life.But the centre distance required precision of two gears is high, does not have separability, adaptability, and manufacture difficulty.

How to find a kind of can by the tooth curve of involute profile together with the advantages of circular arc profile, make both to have had during its transmission separability, interchangeability, adaptability, the feature such as easily manufactured, there is again convex-concave profile contact, area of contact is large, stress dispersion, contact stress is little, and contact strength is high, not easily the advantage such as pitting corrosion inefficacy, becomes problem demanding prompt solution.

Summary of the invention

For above-mentioned the deficiencies in the prior art, technical problem to be solved by this invention is: how to provide a kind of working life longer, reliable transmission is higher, and maintenance cost is low, the gear drive easily installed.

In order to solve the problems of the technologies described above, present invention employs following technological scheme:

A kind of two large negative addendum modification Involutes Gears Transmission device, comprise a pair pitch wheel, it is characterized in that, the work flank profil of described two gear tooths includes the first involute portion of the male type being positioned at rolling circle outside and the second involute portion of male type, also comprise the 3rd involute portion of the inner concave shape being positioned at rolling circle, described first involute portion, the second involute portion and the 3rd involute portion by tooth crest to tooth root portion successively smooth connection; In described two gears, the first involute portion of any one gear is all that convex-concave arc engages with the 3rd involute portion of another gear, the second involute portion convexo-convex arc engagement each other mutually of described two gears.

During transmission, because the first involute portion of the tooth crest of small gear is that convex-concave arc engages with the 3rd involute portion in described gearwheel tooth root portion, add the area of profile contact in transmission process, contact stress is little, and not easily spot corrosion was lost efficacy, and improved working life.Meanwhile, adopt involute profile, make two large negative addendum modification Involutes Gears Transmission device have the separability of involute gear, namely the velocity ratio of gear pair and Base radius are inversely proportional to, and it doesn't matter with the operating center distance of two-wheeled.Like this, make the strong adaptability of two large negative addendum modification Involutes Gears Transmission device, interchangeability is good, reduces the difficulty of manufacturing.

Further, the polar coordinates function expression in described 3rd involute portion is:

$r_k=r_{s2}\sqrt{1+{\tan }^2{\mathrm{\α}}_k{\cos }^2{\mathrm{\β}}_k}$

invα _k＝tanα _k-α _k

Wherein: r _kbe in the 3rd involute portion arbitrfary point to the distance of gear axis, r _s2be the Base radius in the 3rd involute portion, α _kfor the pressure angle of flank profil corresponding points, β _kfor flank profil arbitrfary point helix angle;

In described polar coordinates function expression, also comprise following formula:

r _s2＝r _f+ρ _f(1-sinα _t)

Wherein: r _froot radius, ρ _ffor hobboing cutter corner radius, α _tfor gear compound graduation circle transverse pressure angle.

Further, the work flank profil of described two gear tooths also comprises the transition curve being positioned at tooth root, the upper end of described transition curve and the lower end smooth connection in the 3rd involute portion, and lower end and the root circle of transition curve are tangent; On described transition curve, arbitrfary point (x, y) meets following curvilinear equation:

In described calculating formula, for the vertical line of hobboing cutter movement direction and the angle between hobboing cutter tool arc central point and the line of Gear center.

The advantages such as in sum, it is longer that the present invention has working life, and reliable transmission is higher, and maintenance cost is low, easy installation.

Accompanying drawing explanation

Fig. 1 is involute forming process schematic diagram in Double Involute Gear basic circle of the present invention (in figure, level arrow is to the right hobboing cutter moving direction).

Fig. 2 is involute profile starting point schematic diagram in Double Involute Gear basic circle of the present invention.

Fig. 3 is the structural representation of Double Involute Gear engagement driving.

Fig. 4 is the hob-cutter structure schematic diagram for processing Double Involute Gear.

Fig. 5 is the structural representation of the initial circle of gearwheel root engagement contact in Fig. 3.

Fig. 6 is reverse involute profile transverse tooth thickness Computing Principle schematic diagram.

Fig. 7 is the principle schematic that gear root does not occur is cut.

Fig. 8 is the structural representation of Double Involute Gear root involute starting point.

Fig. 9 is the structural representation of small gear root and joggle(d) joint contact, gearwheel top.

Figure 10 is single negative addendum modification meshed transmission gear contact ratio Computing Principle schematic diagram.

Figure 11 is the structural representation of two negative addendum modification involute gear engagement driving.

Embodiment

Below in conjunction with relevant drawings, the present invention is described in further detail.

During concrete enforcement: as depicted in figs. 1 and 2, add man-hour, according to displacement principle, hobboing cutter is extended in basic circle.Gear rotates with angular velocity omega, with space rate v on hobboing cutter _kmove forward, hobboing cutter tool arc center is moved along straight line all the time, v _k=ω r _bs, be equivalent to gear and pure rolling made by hobboing cutter, the track of tool arc central point is exactly an involute, and its Base radius equals root radius r _fwith hobboing cutter corner radius ρ _fsum, i.e. r _bs=r _f+ ρ _f, the straight line of tool arc central point movement is exactly the generation line of the second involute in rolling circle, and the second involute basic circle is tangent.When hob cutting gear, taking rolling circle as separatrix, is r respectively with radius _bsbasic circle and radius be r _bbasic circle do rightabout involute generating motion, obtain inside and outside two contrary involutes, be namely positioned at the first involute outside rolling circle and be positioned at the second involute of rolling circle.

The cross section of hobboing cutter point of a knife is a radius is ρ _fcircular arc, the chip point of hobboing cutter is positioned on tool arc.Cross hobboing cutter tool arc central point O and make the vertical line of cutter tooth cutting edge PZ on hobboing cutter and cutter tooth cutting edge intersects at P point, due to cutter tooth cutting edge PZ and hobboing cutter tool arc tangent, then P point is the points of tangency of the two, and P point is the peak of hobboing cutter cutter tooth cutting edge simultaneously.When adopting generating method to carry out gear hobbing process, hobboing cutter point of a knife always moves to away from Gear center direction, but P point contacts with flank profil in rolling circle all the time, this is also the cutting point of hobboing cutter in the actual cut of rolling circle inner curve, therefore, the tooth curve in rolling circle is the track formed in cutting movement by P point.Because hobboing cutter just moves in parallel, cutter tooth cutting edge peak P can not change, and therefore, the track of P point is also involute.Now, the vertical line OP of hobboing cutter cutter tooth cutting edge and the angle occurred between line equal hobboing cutter transverse-profile angle α _t, because actual involute is the track of P point in cutting movement, then Gear center point O ₂perpendicular distance to P spot moving direction is the Base radius of the second involute in rolling circle; Be called interior Base radius, use r _srepresent.Then

r _s＝r _f+ρ _f(1-sinα _t)(1)

Obtain because the second involute in rolling circle and the first involute outside rolling circle are cut by each continuous print cutting point on hobboing cutter cutter tooth cutting edge, simultaneously the initial point of contact of hobboing cutter tool arc and hobboing cutter cutter tooth cutting edge tangent, therefore the first involute and the second involute are also just in time the contrary involute curves be smoothly connected in direction.

Because hobboing cutter point of a knife must be one section of circular arc, so involute also has starting point in basic circle, again because hobboing cutter cutter tooth has certain thickness, therefore root circle still has transition curve to involute the initial segment.

As shown in Figure 1, in figure, CC curve is exactly transition curve, can be the elliptic arc curve relevant with hobboing cutter tool arc according to Analytic Proof transition curve.

As shown in Figure 2, the P point in Fig. 2 is the starting point of the second involute in rolling circle, and normal and the Y-axis of crossing P point work second involute intersect at P _jpoint, i.e. node.Simultaneously tangent with N point with interior basic circle, then P _jn=r _ssin α _t, wherein α _tfor hob profile angle.α in figure _dit is the pressure angle of starting point P.

As seen from the figure:

${\mathrm{tan\α}}_d=\frac{PN}{r_s}$

PN＝P _fN-PP _J

And:

${\mathrm{PP}}_J=\frac{({h_a}^{*}-x_{sn})m_n}{{\mathrm{sin\α}}_t}$

Obtain:

${\mathrm{tan\α}}_d={\mathrm{sin\α}}_t-\frac{({h_a}^{*}-x_{sn})m_n}{r_s{\mathrm{sin\α}}_t}---\left(2\right)$

Wherein, h _a ^*for addendum coefficient, x _snfor the second involute profile modification coefficient in rolling circle, modification coefficient in being called for short, r _sit is interior Base radius.

As shown in Figure 1, be accurate Calculation, with analytic method basic circle inner curve.Set up with gear center of circle O ₂for the rectangular coordinate system of initial point, P is hobboing cutter cutting point, and O is hobboing cutter tool arc central point, and Q is the portable cord of O point and the intersection point of Y-axis.Then straight line O ₂q=r _f+ ρ _f=r _bs, Δ POO ₂with Δ OO ₂q is right-angled triangle.So have

Obtain

In formula, r _skit is the Base radius of cutting point P (x, y).

r _sk＝r _f+ρ _f(1-sinα _k)

R _skwith pressure angle α _kchange has minor variations, and the helix angle for helical gear cutting point P (x, y) is β _k.Then

Formula (4) is substituted into above formula and converts pressure angle α to _kfunctional expression:

${\mathrm{tan\β}}_k=\frac{r_{sk}tan\mathrm{\β}}{{\mathrm{rcos\α}}_k}---\left(5\right)$

The right angled coordinates function of cutting point P (x, y) is:

X＝O ₂Psinα _kcosβ _k

Y＝O ₂Pcosα _k

Formula (3) is substituted in above formula, obtains:

Because hobboing cutter moves horizontally, hobboing cutter cutter tooth has certain thickness, in the initial segment, be a constant constant, above formula abbreviation then had:

This is an oval right angled coordinates function, and therefore the transition curve of root portions is still elliptic curve.

Formula (4) is substituted into formula (6) (7), obtains the right angled coordinates function expression of cutting point P geometric locus:

X＝r _sktanα _kcosβ _k

Y＝r _sk

During due to Gear Processing, hobboing cutter moves horizontally, and the track of the highest cutting point on hobboing cutter cutting limit determines gear-profile.Therefore, r now _skequal interior Base radius r _sbe a constant numerical value, the right angled coordinates function expression of involute changed into the polar coordinates functional equation of involute function:

$r_k=r_s\sqrt{1+{\tan }^2{\mathrm{\α}}_k{\cos }^2{\mathrm{\β}}_k}$

invα _k＝tanα _k-α _k

Work as helixangleβ _kwhen=0, be then the involute polar coordinates function expression of standard:

$r_k=\frac{r_s}{{\mathrm{cos\α}}_k}$

invα _k＝tanα _k-α _k

Can draw thus, the transition curve in the tooth root portion of this gear is elliptic curve, and the curve in rolling circle is also involute from initial lighting.

Double Involute Gear transmission is in two kinds of situation: a kind of is the gearwheel of large negative addendum modification and the small gear engagement driving of normal shift, and the engagement driving of this situation is referred to as single large negative meshed transmission gear; Another kind is both the gear transmission of large negative addendum modification, and the engagement driving of this situation is referred to as two large negative meshed transmission gear.

One, single large negative meshed transmission gear:

As shown in Figure 3 and Figure 5, single large negative meshed transmission gear, wherein, gearwheel is large negative addendum modification gear, in basic circle, have flank profil.Small gear is gear with positive correction, and flank profil is outside basic circle.P _jpoint is the intersection point of two pitch circles, i.e. the node of two gears, straight line NP _jnode P _jthe tangent line of the gearwheel basic circle done, also tangent with the basic circle of small gear.In gearwheel basic circle, the second involute curve will engage with pinion gear teeth top involute, and according to " flank profil engagement fundamental law ", its common normal line also must cross node P _j, cross P _jpoint makes the tangent line of basic circle in gearwheel, also must be tangent with small gear top flank profil involute basic circle.MP in figure _jwith NP _jit is the tangent line of two basic circles, also be flank profil engagement process, the separation that pinion gear teeth top is engaged with gearwheel tooth root portion is exactly the basic circle of gearwheel, and two sections of involutes engage in different phase, and section has convexo-convex flank profil and convex-concave flank profil two kinds of mesh form at one time.The second involute in gearwheel basic circle is that hobboing cutter tool arc is formed in basic circle, second pressure angle of involute and hob profile angle do not have direct relation, therefore pressure angle of involute in not controlling by hob profile angle, ensure pinion gear teeth top portion and the transmission of gearwheel root portions correct engagement, only have and the pressure angle of graduated circle of pinion gear teeth top portion involute is revised, and determine the separation of two sections of involutes.

The pinion gear teeth top pressure angle of involute on standard pitch circle is called at " standard pitch circle internal pressure angle ", uses α _snrepresent; Pinion gear teeth top and tooth root portion involute separation claim top to demarcate circle radius, use r _jarepresent; Boundary circle claims to the height of top circle addendum of demarcating, and uses h _jarepresent, tooth top involute profile modification coefficient claims interior modification coefficient, uses x _snrepresent.Gearwheel tooth root portion claims gearwheel root to engage initial circle with small gear tip circle point of contact, radius r _fcrepresent; It is round that two sections, gearwheel tooth root portion involute separation claims root to demarcate, radius r _jbrepresent, root boundary circle claims basic circle addendum to the height of top circle, uses h _barepresent.

Double Involute Gear is two sections of flank profils, contains the flank profil height, the bottom clearance height that may be used for theoretical engagement and removes wedge angle burr chamfering tooth depth three part flank profil height.The flank profil height component that may be used for theoretical engagement is called for short Working depth, uses h _wrepresent, only comprise Working depth and ensure that the flank profil height of tooth top gap is called for short effective tooth depth, representing with h, contain the flank profil, the bottom clearance that may be used for theoretical engagement and go the whole flank profil height of wedge angle burr chamfering three part to be called for short whole depth, using h _grepresent.Double Involute Gear two sections of flank profils have a circle of demarcating, therefore boundary circle is adopted to be convenient to designing and calculating and cutter manufacture with boundary addendum, involute gear take standard pitch circle as the addendum that is divided into of boundary and dedendum of the tooth Double Involute Gear parameter with calculate in just nonsensical, so just no longer adopt.Addendum coefficient h _a ^*, tip clearance coefficient c ^*involute gear standard can be adopted, remove wedge angle burr chamfering c _a× 45 °, be calculated as follows

c _a＝c _a ^*m _n(8)

In formula, c _a ^*remove wedge angle burr chamfering coefficient, be called for short tooth top chamfering coefficient, c _a ^*=0.1.

As shown in Figure 3, O ₂m=r _s2be Base radius in gearwheel, calculate according to formula (1), O ₂p _j=r _w2gearwheel Pitch radius, O ₁p _j=r _w1it is small gear Pitch radius.Because of Δ P _jmO ₂with Δ P _jqO ₁right-angled triangle, and Δ P _jmO ₂∽ Δ P _jqO ₁, so pinion gear teeth top portion involute Base radius r _s1can calculate according to the character of similar triangles:

$r_{s1}=\frac{r_{w1}}{r_{w2}}{r}_{s2}$

Because of velocity ratio:

$i=\frac{r_{w2}}{r_{w1}}=\frac{r_{b2}}{r_{b1}}$

In small gear, Base radius can calculate with Base radius in velocity ratio and gearwheel:

$r_{s1}=\frac{r_{s2}}{i}---\left(9\right)$

According to Base radius formula r _b=rcos α _t, pinion gear teeth top flank profil involute end face standard pitch circle internal pressure angle α can be tried to achieve _st:

${\mathrm{cos\α}}_{st}=\frac{r_{s2}}{r_2}$ Obtain ${\mathrm{\α}}_{st}=\arccos \left(\frac{r_{s2}}{r_2}\right)---\left(10\right)$

According to normal direction pressure angle of graduated circle and end face pressure angle of graduated circle formula, involute standard pitch circle normal direction internal pressure angle, pinion gear teeth top is:

α _sn＝arctan(tanα _stcosβ)(11)

The working pressure angle of small gear top flank profil and gearwheel root flank profil claims pitch circle internal messing angle, according to transverse pressure angle and end face working pressure angle formula, and interior circle involute flank profil end face pitch circle internal messing angle α _swt:

${\mathrm{\α}}_{swt}=\arccos \left(\frac{a^{\′}}{a}{\mathrm{cos\α}}_{st}\right)---\left(12\right)$

In formula, a ' is theoretical center distance, and a is that mounting center is apart from (or claiming operating center distance).

According to involute character: without involute in basic circle, therefore large negative addendum modification gear two sections of flank profil involute separations should on basic circle; Corresponding small gear boundary circular diameter can calculate according to this condition.Outer teeth profile involute boundary circular diameter d in gearwheel root basic circle _jbrepresent, radius r _jbwith representing, the corresponding boundary circular diameter d at small gear top _jarepresent, radius r _jarepresent.

As shown in Figure 3, P point is the separation of gearwheel two sections of flank profils, is also the separation of the two sections of flank profils in small gear top, therefore, and O ₁p is exactly that pinion gear teeth top and tooth root portion involute are demarcated circle radius, uses r _ja1represent, r _ja1=O ₁p.At Δ O ₁o ₂in P, ∠ O ₁o ₂p=α _swt-α _sbt2, a=O ₁o ₂, r _jb2=O ₂p, obtains according to triangle edges angular dependence:

$r_{Ja1}=\sqrt{a^2+{r_{Jb2}}^2-{\mathrm{ar}}_{Jb2}\cos ({\mathrm{\α}}_{swt}-{\mathrm{\α}}_{sbt2})}---\left(13\right)$

In formula, α _swtbe interior circle involute flank profil end face working pressure angle, calculated by formula (12), α _sbt2be gearwheel root involute at basic circle engagement terminating point transverse pressure angle, be called for short root termination pressure angle, can by following formulae discovery:

${\mathrm{\α}}_{sbt2}=arccos\frac{r_{s2}}{r_{Jb2}}---\left(14\right)$

Small gear with demarcate circle for separation, root is different from top involute, top involute tooth depth claim demarcate addendum be:

h _Ja＝r _a-r _Ja(15)

Small gear has two pressure angles on standard pitch circle, and one calculates for root portions involute profile, is the pressure angle α that design is selected _n; One calculates for tip portion involute profile, is to ensure that small gear top and gearwheel root can correct engagement, according to the pressure angle α calculated by the involute curve of hobboing cutter tool arc self-assembling formation in gearwheel basic circle _sn, be addendum h _jathe hob profile angle of part flank profil, therefore, when making small gear hobboing cutter, with gear boundary circle for boundary is by two tooth-shape angle α _n, α _snmanufacture and design hobboing cutter, as shown in Figure 4.Gearwheel hobboing cutter is then by selected tooth-shape angle α _nmake, only have a tooth-shape angle.

Involute profile point of contact in pinion gear teeth tip circle and gearwheel root basic circle, be exactly the gearwheel root starting point of meshing, place circle radius is exactly gearwheel root starting point of meshing radius, uses r _fc2represent.

As shown in Figure 5, r _fc2=O ₂p, at Δ O ₁o ₂in P, ∠ O ₁o ₂p=α _sat1-α _swt, a=O ₁o ₂, r _a1=O ₁p, obtains according to triangle edges angular dependence:

$r_{fc2}=\sqrt{a^2+{r_{a1}}^2-2{\mathrm{ar}}_{a1}cos({\mathrm{\α}}_{sat1}-{\mathrm{\α}}_{swt})}---\left(16\right)$

In formula, α _swtbe pitch circle interior edge face working pressure angle, calculated by formula (12), α _sat1be small gear top flank profil involute terminating point transverse pressure angle, be called for short small gear " top end angle ", can by following formulae discovery:

$i=\frac{r_{w2}}{r_{w1}}=\frac{r_{b2}}{r_{b1}}---\left(17\right)$

Gearwheel top circle and small gear root involute profile point of contact are exactly the small gear root starting point of meshing, and place circle radius is exactly small gear root starting point of meshing radius, uses r _fc1represent.In like manner obtain:

$r_{fc1}=\sqrt{a^2+{r_{a2}}^2-2{\mathrm{ar}}_{a2}cos({\mathrm{\α}}_{at2}-{\mathrm{\α}}_{wt})}---\left(18\right)$

α in formula _at2be gearwheel top circle pressure angle, formula is:

${\mathrm{\α}}_{at2}=arccos\frac{r_{b2}}{r_{a2}}---\left(19\right)$

Interior modification coefficient x _sn: modification coefficient and transverse tooth thickness closely related, interior modification coefficient x can be calculated according to boundary knuckle-tooth thick the same terms _sn.Transverse tooth thickness is the significant dimensions of involute gear engagement driving, and the transverse tooth thickness of gear-profile each several part is different, and the standard pitch circle transverse tooth thickness of two gears is not identical yet, represents standard pitch circle transverse tooth thickness, use s with s _srepresent involute transverse tooth thickness in standard pitch circle basic circle, be called for short transverse tooth thickness in standard pitch circle, small gear is two sections of involute profiles in the same way, and the formula of standard pitch circle transverse tooth thickness and interior transverse tooth thickness is:

$s=\frac{m_n}{cos\mathrm{\β}}(\frac{\mathrm{\π}}{2}+2x_n{\mathrm{tan\α}}_n)---\left(20\right)$

$s_s=\frac{m_n}{cos\mathrm{\β}}(\frac{\mathrm{\π}}{2}+2x_{sn}{\mathrm{tan\α}}_{sn})---\left(21\right)$

Use s _krepresent that any knuckle-tooth is thick, r _kbe any circle radius, r is reference radius, α _kthe pressure angle of circle arbitrarily, α _tbe standard pitch circle transverse pressure angle, the formula that any knuckle-tooth of small gear root is thick is:

$s_k=\frac{r_k}{r}s-2r_k({\mathrm{inv\α}}_k-{\mathrm{inv\α}}_t)---\left(22\right)$

The formula that any knuckle-tooth in small gear top is thick is:

$s_k=\frac{r_k}{r}{s}_s-2r_k({\mathrm{inv\α}}_k-{\mathrm{inv\α}}_{st})---\left(23\right)$

Namely gearwheel flank profil in basic circle is spill flank profil is reverse involute profile, and the upper transverse tooth thickness of circle calculates different from forward involute arbitrarily.Shown in Fig. 6, s _kthe circular thickness on any circle, s _sthe circular thickness on standard pitch circle, according to involute character, the circular thickness arbitrarily on circle:

$s_k=r_k\mathrm{\φ}=r_k\[\frac{s}{r}+2({\mathrm{\θ}}_k-{\mathrm{\θ}}_t)\]$

Because of involute exhibition angle θ _k=inv α _kand θ _t=inv α _st, obtain s _kformula:

$s_k=\frac{r_k}{r}{s}_s+2r_k({\mathrm{inv\α}}_k-{\mathrm{inv\α}}_{st})---\left(24\right)$

In formula, the standard pitch circle transverse tooth thickness s of anti-involute profile _sfor:

$s_s=\frac{{\mathrm{zm}}_n}{\cos \mathrm{\β}}(\frac{\mathrm{\π}}{2}-2x_m{\mathrm{tan\α}}_{sn})---\left(25\right)$

Forward involute profile outside basic circle, the same small gear of formula.

Double Involute Gear modulus, helix angle, standard pitch circle are identical, and top is different from root pressure angle of graduated circle, and therefore modification coefficient is not identical.Can calculate the modification coefficient of tooth crest involute profile according to the thick identical condition of boundary knuckle-tooth, modification coefficient in being called for short, uses x _snrepresent.

(1), modification coefficient x in small gear _sn1: small gear boundary circular thickness s _jrepresent, because two sections of flank profils are forward involutes, obtain according to formula (20), (21), (22), (23):

$s_1=\frac{m_n}{cos\mathrm{\β}}(\frac{\mathrm{\π}}{2}+2x_{n1}{\mathrm{tan\α}}_n)$

$s_{s1}=\frac{m_n}{cos\mathrm{\β}}(\frac{\mathrm{\π}}{2}+2x_{sn1}{\mathrm{tan\α}}_{sn})$

$s_J=\frac{r_{Ja1}}{r_1}{s}_1-2r_{Ja1}({\mathrm{inv\α}}_{Jat1}-{\mathrm{inv\α}}_t)$

$s_J=\frac{r_{Ja1}}{r_1}{s}_{s1}-2r_{Ja1}({\mathrm{inv\α}}_{sJt1}-{\mathrm{inv\α}}_{st})$

In formula, α _jat1be small gear root flank profil involute termination endface pressure angle, be called for short small gear " root termination pressure angle ", α _sJt1be the initial transverse pressure angle of small gear top flank profil involute, be called for short small gear " initial pressure angle, top ", calculated by following formula respectively:

${\mathrm{\α}}_{Jat1}=arccos\left(\frac{r_{b1}}{r_{Ja1}}\right)---\left(26\right)$

${\mathrm{\α}}_{sJt1}=arccos\left(\frac{r_{s1}}{r_{Ja1}}\right)---\left(27\right)$

Try to achieve according to above formula simultaneous:

$x_{sn1}=\frac{{\mathrm{tan\α}}_n}{{\mathrm{tan\α}}_{sn}}{x}_{n1}+\frac{z_1({\mathrm{inv\α}}_t+{\mathrm{inv\α}}_{sJt1}-{\mathrm{inv\α}}_{Jat1}-{\mathrm{inv\α}}_{st})}{2{\mathrm{tan\α}}_{sn}}---\left(28\right)$

(2), modification coefficient x in gearwheel _sn2: according to involute character: without involute in basic circle, therefore gearwheel two sections of flank profil involute separations should near basic circle; Due to two involute curve smooth transition, therefore separation transverse tooth thickness is identical.Use s _brepresent basic circle separation circular thickness, in basic circle, flank profil is reverse involute, obtains according to formula (20), (22), (24), (25):

$s_2=\frac{m_n}{cos\mathrm{\β}}(\frac{\mathrm{\π}}{2}+2x_{n2}{\mathrm{tan\α}}_n)$

$s_{s2}=\frac{m_n}{cos\mathrm{\β}}(\frac{\mathrm{\π}}{2}-2x_{sn2}{\mathrm{tan\α}}_{sn})$

$s_b=\frac{r_{Jb2}}{r_2}{s}_2-2r_{Jb2}({\mathrm{inv\α}}_{Jbt2}-{\mathrm{inv\α}}_t)$

$s_b=\frac{r_{Jb2}}{r_2}{s}_{s2}+2r_{Jb2}({\mathrm{inv\α}}_{sbt2}-{\mathrm{inv\α}}_{st})$

In formula, α _jbt2be gearwheel top (two large negative gear is middle part) involute starting point transverse pressure angle, be called for short at " top (middle part) initial pressure angle "; α _sbt2be gearwheel root terminating point transverse pressure angle, be called for short at " root termination pressure angle ".

${\mathrm{\α}}_{Jbt2}=arccos\left(\frac{r_{b2}}{r_{Jb2}}\right)---\left(29\right)$

${\mathrm{\α}}_{sbt2}=arccos\left(\frac{r_{s2}}{r_{Jb2}}\right)---\left(30\right)$

Try to achieve: $x_{sn2}=\frac{z_2({\mathrm{inv\α}}_{sbt2}+{\mathrm{inv\α}}_{Jbt2}-{\mathrm{inv\α}}_t-{\mathrm{inv\α}}_{st})}{2{\mathrm{tan\α}}_{sn}}-\frac{{\mathrm{tan\α}}_n}{{\mathrm{tan\α}}_{sn}}{x}_{n2}---\left(31\right)$

Double Involute Gear engagement driving is also Involutes Modified Gears transmission, and therefore Involutes Modified Gears open top container ship equation is applicable to Double Involute Gear engagement driving.Gearwheel top and small gear root are reverse involute profile engagement driving, are also convexo-convex flank profil engagement driving, the same with involute gear engagement driving, claim main engagement driving; Gearwheel root and small gear top are the auxiliary engagement driving of involute in the same way, are also convex-concave flank profil engagement driving, different from Involutes Gears Transmission, claim auxiliary engagement driving, calculate respectively.

(1), main engagement driving is gearwheel top and small gear root flank profil engagement driving, and open top container ship equation is also involute gear open top container ship equation, saves derivation here, and its equation is:

$x_{\mathrm{\Σ}n}=\frac{(z_1+z_2)({\mathrm{inv\α}}_{wt}-{\mathrm{inv\α}}_t)}{2{\mathrm{tan\α}}_n}---\left(32\right)$

x _Σn＝x _n1+x _n2(33)

In formula, z ₁the small gear number of teeth, z ₂the gearwheel number of teeth, α _wtend face working pressure angle, α _tstandard pitch circle transverse pressure angle, α _nstandard pitch circle Normal pressure angle, x _{Σ n}total normal direction modification coefficient, x _n1small gear normal direction modification coefficient, x _n2c is gearwheel normal direction modification coefficient.α _t, α _wt, α _nfollowing relation is had with spiral angle of graduated circle β:

${\mathrm{\α}}_t=arctan\left(\frac{{\mathrm{tan\α}}_n}{cos\mathrm{\β}}\right)$

${\mathrm{\α}}_{wt}=arccos\left(\frac{a^{\′}}{a}{\mathrm{cos\α}}_t\right)$

In formula, a is the mounting center distance of two gear reality, and a ' is the theoretical center distance of two gears.

(2), auxiliary engagement driving is gearwheel root and small gear top engagement driving, this is involute profile engagement driving in the same way, also open top container ship equation to be met, according to transverse tooth thickness formula (24), (25), the net slip coefficient of small gear top and gearwheel root flank profil meets formula:

$x_{s\mathrm{\Σ}n}=\frac{(z_1-z_2)({\mathrm{inv\α}}_{swt}-{\mathrm{inv\α}}_{st})}{2{\mathrm{tan\α}}_{sn}}---\left(34\right)$

x _s∑ _n＝x _sn1-x _sn2(35)

In formula, x _{s Σ n}be total normal direction modification coefficient that in gearwheel root basic circle, involute profile and small gear top involute profile match, be called for short modification coefficient or total interior modification coefficient in total normal direction, α _swtbe the standard pitch circle end face working pressure angle of involute profile and small gear top involute profile in gearwheel root basic circle, be called for short end face internal messing angle, α _stbe the standard pitch circle transverse pressure angle of involute profile and small gear top involute profile in gearwheel root basic circle, be called for short end face internal pressure angle, α _snbe interior involute standard pitch circle Normal pressure angle, be called for short normal direction internal pressure angle, x _sn1be that small gear top flank profil matches the normal direction modification coefficient of involute in gearwheel root basic circle, be called for short modification coefficient in small gear normal direction, x _sn2be involute profile normal direction modification coefficient in gearwheel root basic circle, be called for short modification coefficient in gearwheel normal direction.

Confirmable about modification coefficient in total, modification coefficient x in small gear _sn1with modification coefficient x in gearwheel _sn2and the boundary circle radius r of small gear two sections of flank profils _ja1to demarcate circle radius r with gearwheel two sections of flank profils _jb2four unknown numbers according to formula (13), (28), (31), (34), (35) simultaneous solution, first can will obtain boundary circle radius r _ja1and r _jb2, formula (28), (31), (34) are substituted in formula (35) and obtain equation:

z ₁(invα _Jat1-invα _sJt1)+z ₂(invα _Jbt2+invα _sbt2)＝A(36)

In formula, A is constant.

A＝2x _Σntanα _n+(z ₁+z ₂)invα _t-(z ₁-z ₂)invα _swt(37)

Change formula (13) into equation representation:

r _Ja1 ²＝a ²+r _Jb2 ²-ar _Jb2cos(α _swt-α _sbt2)(38)

Formula (36), (38) constitute binary quadratic equation with (26), (27), (29), (30), and unknown number is r _ja1and r _jb2.As long as calculate r _ja1and r _jb2, just can calculate x _sn1and x _sn2.But (36), (38) set of equation is not only binary quadratic equation group, but also is transcendental equation, and it is very difficult for solving, and conventional method cannot solve.Can prove that the separation of the inside and outside involute of gearwheel basic circle is not on basic circle according to (36), (38) set of equation, but be greater than basic circle, near basic circle, therefore can solve by Base radius close approximation method.Namely with Base radius r _b2equal boundary circle radius r _jb2, calculate boundary circle radius r _ja1, then use r _ja1substitute into formula inverse r _jb2, obtain more accurate numerical value can double counting several times again, until obtain the precision of wishing.

Also can solving by simple averaging method, finding, as long as know t by analyzing _jb2just r can be solved _ja1and x _sn1and x _sn2.Namely first r is established _jb2=r _b2calculate other three unknown number x _sn1, x _sn2, r _j1, wherein x _sn1, x _sn2directly can calculate according to formula (28), (31), then the x that will calculate _sn1and x _sn2substitute into formula (35) respectively and calculate x _sn2and x _sn1, by two numerical value calculating mean value again obtained, it should be noted that when calculating boundary knuckle-tooth and being thick also with the numerical value calculating mean value again that top and root calculate respectively.

Bottom clearance and addendum shorten coefficient: ensure a pair gear correct engagement, leave certain interval between gear root and top, this gap is called tooth top gap, are called for short bottom clearance.Modified gear is according to ensureing that mounting center distance is selected in the requirement of open top container ship, then bottom clearance can not be ensured, according to ensureing that mounting center distance is selected in the requirement of bottom clearance, then open top container ship can not be ensured, therefore, modified gear will ensure that tooth top then shortens by bottom clearance and open top container ship, claims addendum to shorten coefficient, use δ with the ratio of modulus _yrepresent.

δ _y＝x _Σn-y _n(39)

Y in formula _nbe center Separating factor, formula is:

$y_n=\frac{a-a^{\′}}{m_n}=\frac{z_1+z_2}{2}(\frac{cos\mathrm{\α}t}{{\mathrm{cos\α}}_{wt}}-1)---\left(40\right)$

By the δ of formula (32), (40) substitution (39) _yformula obtains:

${\mathrm{\δ}}_y=\frac{z_1+z_2}{2}(1+\frac{{\mathrm{inv\α}}_{wt}-{\mathrm{inv\α}}_t}{{\mathrm{tan\α}}_n}-\frac{{\mathrm{cos\α}}_t}{{\mathrm{cos\α}}_{wt}})---\left(41\right)$

Double involute flank profil each section of high computational formula is:

h _w＝(2h _a ^*-δ _y)m _n(42)

h＝(2h _a ^*-δ _y+c ^*)m _n(43)

h _g＝(2h _a ^*-δ _y+c _*+c _a ^*)m _n(44)

Double involute flank profil formation condition is: r-h _w< r _b, substitute into correlation computations formula and obtain:

${\mathrm{\&delta;}}_y<2{h_a}^{*}-\frac{z(1-{\mathrm{cos\&alpha;}}_t)}{2\cos \mathrm{\&beta;}}---\left(45\right)$

The limit case of a pair involute gear engagement driving is exactly working pressure angle is 0, obtains according to formula (41):

${\mathrm{\&delta;}}_y<\frac{z_1+z_2}{2}(1-\frac{{\mathrm{inv\&alpha;}}_t}{{\mathrm{tan\&alpha;}}_n}-{\mathrm{cos\&alpha;}}_t)---\left(46\right)$

Formula (45) and (46) are all the conditions that Double Involute Gear should be checked.

The number of teeth and modification coefficient and helix angle: according to modified gear principle, modified gear is that hobboing cutter radially moves x by normal place _nm _ndistance institute's cutting gear out.Wherein x _nnormal direction modification coefficient, m _nit is normal module.

As shown in Figure 7, the condition that root does not cut interference is N ₁q > h _a-x _nm _nor x _nm _n> h _a-N ₁q; H in formula _areference addendum, α _tbe gear face pressure angle of graduated circle, r is gear compound graduation circular diameter, respectively by formulae discovery below:

h _a＝h _a ^*m _n

N ₁Q＝PN ₁sinα _t

PN ₁＝rsinα _t

$r=\frac{{\mathrm{zm}}_n}{\cos \mathrm{\&beta;}}$

The calculation formulas obtaining modification coefficient according to mathematical operation is:

$x_n\&GreaterEqual;{h_a}^{*}-\frac{{\mathrm{zsin}}^2{\mathrm{\&alpha;}}_t}{2cos\mathrm{\&beta;}}---\left(47\right)$

Or check the number of teeth according to modification coefficient, helix angle, pressure angle:

$z\&GreaterEqual;\frac{2({h_a}^{*}-x_n)cos\mathrm{\&beta;}}{{\sin }^2{\mathrm{\&alpha;}}_t}---\left(48\right)$

Or check helix angle according to modification coefficient, the number of teeth, pressure angle:

$cos\mathrm{\&beta;}\&le;\frac{{\mathrm{zcos}}^2{\mathrm{\&alpha;}}_t}{2({h_a}^{*}-x_n)}---\left(49\right)$

Formula (47), (48), (49) be check modification coefficient, the number of teeth, helix angle three satisfy condition mutually.

In engagement, transition curve is interfered: make transmission that transition curve does not occur and interfere, it is all involutes that the starting point of meshing of flank profil work must be made to start, i.e. the pressure angle α of flank profil active section starting point _cthe pressure angle α of flank profil involute starting point must be more than or equal to _d.Flank profil involute starting point is determined by during hob cutting, is processing starting point, pressure angle α _drepresent, the flank profil active section starting point of meshing is that two gear transmission engagements and mounting point are determined, pressure angle α _drepresent, calculate two pressure angles respectively.In gearwheel basic circle, involute processing starting point is calculated by formula (2), is calculated as follows for small gear involute processing starting point.

(1), involute processing starting point pressure angle α _d

As shown in Figure 8, B point is involute starting point.Involute processing starting point pressure angle α _das shown in Figure 8,

${\mathrm{tan\&alpha;}}_d=\frac{{\mathrm{BN}}_1}{r_b}=\frac{{\mathrm{PN}}_1-PB}{r_b}$

$PB=\frac{H}{{\mathrm{sin\&alpha;}}_t}$

PN ₁＝rsinα _t

r _b＝rcosα _t

In formula, r is small gear reference radius, α _tbe end face pressure angle of graduated circle, H is the actual tooth depth of small gear root.Because of H=(h ^* _a-x _n1) m _n; Therefore can obtain:

${\mathrm{tan\&alpha;}}_d={\mathrm{tan\&alpha;}}_t-\frac{4({h_a}^{*}-x_{n1})cos\mathrm{\&beta;}}{z_1{\sin }^2{\mathrm{\&alpha;}}_t}$

(2), flank profil starting point of meshing pressure angle α _c

As shown in Figure 9, B point is the small gear root flank profil starting point of meshing and gearwheel tip circle job initiation point.α _cflank profil work starting point of meshing pressure angle, then:

${\mathrm{tan\&alpha;}}_c=\frac{{\mathrm{BN}}_1}{r_{b1}}$

BN ₁＝N ₁N ₂-N ₂B

N ₁N ₂＝(r _b1+r _b2)tanα _wt

N ₂B＝r _b2tanα _at2

In formula, r _b1small gear Base radius, r _b2gearwheel Base radius, α _wtworking pressure angle, α _at2be the tooth top pressure angle of large gearwheel, its formula is:

${\mathrm{\&alpha;}}_{at2}=arccos\frac{r_{b2}}{r_{a2}}---\left(50\right)$

Therefore tan α is had _c=tan α _wt-i (tan α _at2-tan α _wt).

According to the condition that transition curve interference does not occur, then there is tan α _c>=tan α _d, namely small gear do not occur transition curve interfere check formula be:

${\mathrm{tan\&alpha;}}_{wt}-i({\mathrm{tan\&alpha;}}_{at2}-{\mathrm{tan\&alpha;}}_{wt})\&GreaterEqual;{\mathrm{tan\&alpha;}}_t-\frac{4({h_a}^{*}-x_{n1})cos\mathrm{\&beta;}}{z_1\sin 2{\mathrm{\&alpha;}}_t}---\left(51\right)$

In like manner can obtain the tan relation of small gear tip circle at gearwheel root engagement initial angle:

${\mathrm{tan\&alpha;}}_c={\mathrm{tan\&alpha;}}_{swt}-\frac{1}{i}({\mathrm{tan\&alpha;}}_{sat1}-{\mathrm{tan\&alpha;}}_{swt})$

Gearwheel involute starting point by formula (2) calculate, in like manner obtain gearwheel root do not occur transition curve interfere check formula be:

${\mathrm{tan\&alpha;}}_{swt}-\frac{1}{i}({\mathrm{tan\&alpha;}}_{sat1}-{\mathrm{tan\&alpha;}}_{swt})\&GreaterEqual;{\mathrm{sin\&alpha;}}_{st}-\frac{({h_a}^{*}-x_{sn2})m_n\cos \mathrm{\&beta;}}{r_{s2}{\mathrm{sin\&alpha;}}_{st}}---\left(52\right)$

In formula, α _swtend face internal messing angle, α _stend face internal pressure angle, α _sat1be involute profile termination pressure angle, small gear top, be called for short termination pressure angle, small gear top.

Gearwheel root and small gear root flank profil engage and contact starting point and calculate, more accurately, so generally use α _fct1, α _fct2check.

Contact ratio ε _γrepresent.As shown in Figure 10, line of contact is by C ₁c, B ₁b two sections composition, therefore, Double Involute Gear engagement driving is by involute contact ratio ε in basic circle _f, the outer involute contact ratio ε of basic circle _α, end face contact ratio ε _βthree part compositions.

As shown in Figure 10, C ₁c is involute engagement line length in basic circle, B ₁b is basic circle outer involute engagement line length.∠ 1=α _fJt1small gear " initial pressure angle, top ", ∠ 2=α _sat1small gear " termination pressure angle, top ", ∠ 3=α _bct1small gear " root termination pressure angle ".

(1), involute contact ratio ε in basic circle _f

As shown in Figure 10, C ₁c is involute engagement line length in basic circle:

C ₁C＝r _s1(tanα _sat1-tanα _sJt1)

In formula, r _s1be Base radius in small gear, calculated by formula (9), initial pressure angle, small gear top α _sJt1calculated by formula (26), small gear " termination pressure angle, top α _sat1by formula (17) calculating below, namely formula is:

${\mathrm{\&alpha;}}_{sat1}=arccos\left(\frac{r_{s1}}{r_{a1}}\right)$

${\mathrm{\&alpha;}}_{sJt1}=arccos\left(\frac{r_{s1}}{r_{Ja1}}\right)$

R in formula _j1calculated by formula (13), gear method joint equals interior basic circle Transverse circular pitch.

$P_{sbt}=\frac{{\mathrm{\&pi;m}}_n{\mathrm{cos\&alpha;}}_{st}}{cos\mathrm{\&beta;}}$

Therefore, involute contact ratio ε in basic circle _f

${\mathrm{\&epsiv;}}_f=\frac{({\mathrm{tan\&alpha;}}_{sat1}-{\mathrm{tan\&alpha;}}_{sJt1})r_{s1}cos\mathrm{\&beta;}}{{\mathrm{\&pi;m}}_n{\mathrm{cos\&alpha;}}_{st}}---\left(53\right)$

(2), the outer involute contact ratio ε of basic circle _α

As shown in Figure 10, B ₁b is basic circle outer involute engagement line length, B ₁b=PB ₁+ PB.

PB ₁＝r _b1(tanα _Jat1-tanα _wt)

PB＝r _b2(tanα _at2-tanα _wt)

In formula, α _at2be gearwheel top circle pressure angle, calculated by formula (50),

${\mathrm{\&alpha;}}_{at2}=arccos\frac{r_{b2}}{r_{a2}}$

α _jatbe small gear terminate partway pressure angle, calculated by formula (24), namely

${\mathrm{\&alpha;}}_{Jat1}=arccos\frac{r_{b1}}{r_{Ja1}}$

In formula, r _ja1calculate according to formula (13).The ratio that B1B and method save is:

${\mathrm{\&epsiv;}}_a=\frac{1}{2\mathrm{\&pi;}}\&lsqb;z_1({\mathrm{tan\&alpha;}}_{Jat1}-{\mathrm{tan\&alpha;}}_{wt})+z_2({\mathrm{tan\&alpha;}}_{at2}-{\mathrm{tan\&alpha;}}_{wt})\&rsqb;---\left(54\right)$

(3) end face contact ratio ε _βfor:

${\mathrm{\&epsiv;}}_{\mathrm{\&beta;}}=\frac{b\sin \mathrm{\&beta;}}{{\mathrm{\&pi;m}}_n}---\left(55\right)$

In formula, b is gear tooth width.

(4), total contact ratio (contact ratio) is:

ε _γ＝ε _α+ε _β+ε _β(56)

Total number of teeth and minimum modification coefficient: Double Involute Gear is negative addendum modification gear, and net slip coefficient is negative value, also has the limit, this limit claims minimum net slip coefficient.According to " flank profil engagement fundamental law ", the pitch circle of a pair involute gear engagement driving must beyond basic circle, and namely two rolling circle radius sums are less than mounting center distance, therefore have:

$\frac{(z_1+z_2)m_n{\mathrm{cos\&alpha;}}_t}{2cos\mathrm{\&beta;}}\&le;a$

The number of teeth sum of gearwheel and small gear is called total number of teeth, if

$z_v=\frac{2a}{m_n}---\left(68\right)$

Be called that the total number of teeth of primary Calculation is called for short and just calculate the number of teeth, obtain

$z_1+z_2\&le;\frac{cos\mathrm{\&beta;}}{{\mathrm{cos\&alpha;}}_t}{z}_v---\left(69\right)$

Above formula (69) is substituted in open top container ship equation (32) and obtains

$X_{\mathrm{\&Sigma;}n}\&GreaterEqual;\frac{z_v({\mathrm{inv\&alpha;}}_{wt}-{\mathrm{inv\&alpha;}}_t)cos\mathrm{\&beta;}}{2{\mathrm{cos\&alpha;}}_t{\mathrm{tan\&alpha;}}_n}---\left(70\right)$

Limit case, α _wt=0, the limiting value obtained is:

$x_{\mathrm{\&Sigma;}n}>-\frac{z_v{\mathrm{inv\&alpha;}}_{t}cos\mathrm{\&beta;}}{2{\mathrm{cos\&alpha;}}_t{\mathrm{tan\&alpha;}}_n}---\left(71\right)$

Double involute formation condition is that flank profil Working depth smallest circle should be less than Base radius, namely

r-(h _a ^*-x _n)m _n＜r _b

Substitution formula obtains

$x_n\&le;{h_a}^{*}-\frac{z(1-{\mathrm{cos\&alpha;}}_t)}{2\cos \mathrm{\&beta;}}---\left(72\right)$

Gearwheel modification coefficient x is obtained according to formula (47) and (72) _nrange of choice be:

${h_a}^{*}-\frac{{\mathrm{zsin}}^2{\mathrm{\&alpha;}}_t}{2cos\mathrm{\&beta;}}\&le;x_n\&le;h_a*-\frac{z(1-{\mathrm{cos\&alpha;}}_t)}{2\cos \mathrm{\&beta;}}---\left(73\right)$

Double Involute Gear two sections of flank profils are uniformly distributed, and work flank profil is symmetrical line with basic circle, then have:

$r_b=\frac{1}{2}\&lsqb;r+({h_a}^{*}+x_n-{\mathrm{\&delta;}}_y)m_n+r-({h_a}^{*}-x_n)m_n\&rsqb;$

R is reference radius, r _bbe Base radius, substitute into formula and obtain:

$x_n=\frac{{\mathrm{\&delta;}}_y}{2}-\frac{z(1-{\mathrm{cos\&alpha;}}_t)}{2\cos \mathrm{\&beta;}}---\left(74\right)$

Meet the best modification coefficient that formula (74) is design Double Involute Gear.

Transverse tooth thickness calculates: transverse tooth thickness is gear-driven significant dimensions, is also to manufacture hobboing cutter, the important parameter of machining gears, but gear each position transverse tooth thickness is not identical, to double involute, knuckle-tooth of demarcating is thick is very important size, the formula of large and small gear boundary circular thickness

(1), small gear boundary circular thickness s _j1represent, then:

$s_{J1}=\frac{r_{Ja1}}{r_1}{s}_1-2r_{Ja1}({\mathrm{inv\&alpha;}}_{Jat}-{\mathrm{inv\&alpha;}}_t)---\left(80\right)$ Or

$s_{J1}=\frac{r_{J1}}{r_{s1}}{s}_{s1}-2r_{J1}({\mathrm{inv\&alpha;}}_{sJt1}-{\mathrm{inv\&alpha;}}_{st})---\left(81\right)$

If boundary circular diameter do not have accurate Calculation out, these two numerical value are unequal, before said, carry out calculating mean value with two numerical value.Use s respectively _jf1represent that the boundary knuckle-tooth that root calculates is thick, s _ja1represent that the boundary knuckle-tooth that top calculates is thick, then formula is:

$s_{Jf1}=\frac{r_{Ja1}}{r_1}{s}_1-2r_{Ja1}({\mathrm{inv\&alpha;}}_{Jbt}-{\mathrm{inv\&alpha;}}_t)$

$s_{Ja1}=\frac{r_{Ja1}}{r_{s1}}{s}_{s1}-2r_{Ja1}({\mathrm{inv\&alpha;}}_{sJt1}-{\mathrm{inv\&alpha;}}_{st})$

$s_{J1}=\frac{1}{2}(s_{Ja1}+s_{Jf1})---\left(82\right)$

(2), gearwheel boundary circular thickness s _b2represent, then:

$s_{b2}=\frac{r_{Jb2}}{r_2}{s}_2-2r_{Jb2}({\mathrm{inv\&alpha;}}_{Jbt2}-{\mathrm{inv\&alpha;}}_t)---\left(83\right)$ Or

$s_{b2}=\frac{r_{Jb2}}{r_{s2}}{s}_{s2}+2r_{Jb2}({\mathrm{inv\&alpha;}}_{fJt2}-{\mathrm{inv\&alpha;}}_{st})---\left(83\right)$

In like manner, if boundary circular diameter r _jb2not having accurate Calculation out, is boundary circle radius, i.e. r with Base radius _jb2=r _b2, then

$s_{b2}=\frac{r_{b2}}{r_2}{s}_2+2r_{b2}{\mathrm{inv\&alpha;}}_t$

The numerical value that upper formulae discovery numerical value out and formula (83) calculate is unequal, also wants calculating mean value.

(3), pitch circle circular thickness s _wrepresent

Double Involute Gear transmission, the pitch circle of gearwheel and small gear is all greater than circle of demarcating and is less than basic circle, and on positive contacting profile, therefore pitch circle circular thickness graduated arc transverse tooth thickness calculates.

Small gear pitch circle circular thickness s _w1for:

$s_{w1}=\frac{r_{w1}}{r_1}{s}_1-2r_{w1}({\mathrm{inv\&alpha;}}_{wt}-{\mathrm{inv\&alpha;}}_t)---\left(84\right)$

Gearwheel pitch circle circular thickness s _w2for:

$s_{w2}=\frac{r_{w2}}{r_2}{s}_2-2r_{w2}({\mathrm{inv\&alpha;}}_{wt}-{\mathrm{inv\&alpha;}}_t)---\left(85\right)$

Pitch circle circular thickness convenience of calculation is reliable measuring gear and the important parameter making the cutters such as hobboing cutter.

Two, two large negative meshed transmission gear:

The basis of single large negative meshed transmission gear is analyzed two large negative meshed transmission gear, some formula and the calculation formulas of single large negative meshed transmission gear can be utilized.

As shown in figure 11, the Double Involute Gear of two large negative meshed transmission gear is actual is three sections of involute profile engagement driving.According to " flank profil engagement fundamental law ", three sections of flank profils of two engagement driving gears have a common pitch circle and node, therefore, the modulus of gear three sections of involute profiles, the number of teeth, spiral angle of graduated circle, standard pitch diameter, pitch diameter, boundary knuckle-tooth are thick is identical, the difference such as base circle diameter (BCD), pressure angle of graduated circle, standard pitch circle transverse tooth thickness, normal direction modification coefficient, base tangent length of three sections of involute gears.Formula can be derived according to this principle and design of gears and processing are calculated.In order to distinguish, the Base radius be processed to form in gearwheel basic circle, Base radius in being called for short, uses r _s2represent that the Base radius of corresponding small gear top flank profil is called for short Base radius in small gear, uses r _s1represent, the corresponding standard pitch circle Normal pressure angle of interior basic circle flank profil is called for short normal direction internal pressure angle, uses α _snrepresent, end face internal pressure angle, uses α _strepresent.The Base radius of the involute be processed to form in small gear basic circle claims aglucon circle radius, uses r _p1represent, corresponding gearwheel aglucon circle radius r _p2represent, standard pitch circle normal direction joins pressure angle α _pnrepresent.The gear teeth of large small gear all have three sections of involute profiles, and three sections of involutes have r _b, r _sand r _pthree basic circles, when velocity ratio is 1, interior basic circle overlaps with aglucon circle, interior Base radius r _swith aglucon circle radius r _pequal.

1, interior Base radius and aglucon circle radius calculate

(1), Base radius r in gearwheel _s2with small gear aglucon circle radius r _p1calculate according to formula (1), be respectively:

r _s2＝r _f2+ρ _f(1-sinαt)(86)

r _p1＝r _f1+ρ _f(1-sinαt)(87)

(2), in small gear Base radius and gearwheel aglucon circle radius calculate according to velocity ratio formula (9), obtain:

$r_{s1}=\frac{1}{i}{r}_{s2}---\left(88\right)$

r _p2＝ir _p1(89)

2, standard pitch circle normal direction internal pressure angle and normal direction are joined pressure angle and are calculated

(1), standard pitch circle end face internal pressure angle with join pressure angle and calculate according to formula (14), obtain:

${\mathrm{\&alpha;}}_{st}=arccos\frac{r_{s2}}{r_2}---\left(90\right)$

${\mathrm{\&alpha;}}_{pt}=arccos\frac{r_{p1}}{r_1}---\left(91\right)$

(2), standard pitch circle normal direction internal pressure angle with join pressure angle and calculate according to formula (11), obtain:

α _sn＝arctan(tanα _stcosβ)(92)

α _pn＝arctan(tanα _ptcosβ)(93)

3, interior modification coefficient and distribution transforming potential coefficient calculate

(1), interior modification coefficient calculates

Interior modification coefficient is that the engagement driving of small gear top flank profil and gearwheel root flank profil calculates, and calculates according to formula (28) and (31),

$x_{sn1}=\frac{{\mathrm{tan\&alpha;}}_n}{{\mathrm{tan\&alpha;}}_{sn}}{x}_{n1}+\frac{z_1({\mathrm{inv\&alpha;}}_t+{\mathrm{inv\&alpha;}}_{sJt1}-{\mathrm{inv\&alpha;}}_{Jat1}-{\mathrm{inv\&alpha;}}_{st})}{2{\mathrm{tan\&alpha;}}_{sn}}---\left(94\right)$

$x_{sn2}=\frac{z_2({\mathrm{inv\&alpha;}}_{sbt2}+{\mathrm{inv\&alpha;}}_{Jbt2}-{\mathrm{inv\&alpha;}}_t-{\mathrm{inv\&alpha;}}_{st})}{2{\mathrm{tan\&alpha;}}_{sn}}-\frac{{\mathrm{tan\&alpha;}}_n}{{\mathrm{tan\&alpha;}}_{sn}}{x}_{n2}---\left(95\right)$

In total, modification coefficient is calculated by formula (34), obtains:

$x_{\mathrm{\&Sigma;}sn}=\frac{(z1-z2)({\mathrm{inv\&alpha;}}_{swt}-{\mathrm{inv\&alpha;}}_{st})}{2{\mathrm{tan\&alpha;}}_{sn}}---\left(96\right)$

In small gear, in modification coefficient and gearwheel, modification coefficient is calculated by formula (35), obtains:

x _sn1＝x _Σsn+x _sn2(97)

x _sn2＝x _sn1-x _Σsn(98)

Tooth crest boundary circle radius r _ja1with root portions circle circle radius r _jb2according to solving equations, r _ja1with r _jb2set of equation is:

z ₁(invα _Jat1-invα _sJt1)+z ₂(invα _Jbt2+invα _sbt2)＝A(99)

r _Ja1 ²＝a ²+rJ _Jb2 ²-ar _Jb ²cos(α _swt-α _Jbt2)(100)

${\mathrm{\&alpha;}}_{Jat1}=arccos\left(\frac{r_{b1}}{r_{Ja1}}\right)---\left(101\right)$

${\mathrm{\&alpha;}}_{sJt1}=arccos\frac{r_{s1}}{r_{Ja1}}---\left(102\right)$

${\mathrm{\&alpha;}}_{Jbt2}=arccos\left(\frac{r_{b2}}{r_{Jb2}}\right)---\left(103\right)$

${\mathrm{\&alpha;}}_{sbt2}=arccos\left(\frac{r_{s2}}{r_{Jb2}}\right)---\left(104\right)$

In formula, A is constant.

A＝2x _Σntanα _n+(z ₁+z ₂)invα _t-(z ₁-z ₂)invα _swt(105)

(2), distribution transforming potential coefficient calculates

Interior modification coefficient is that the engagement driving of small gear top flank profil and gearwheel root flank profil calculates, and substitutes into corresponding numerical value obtain according to formula (28) and (31):

$x_{pn1}=\frac{z_1({\mathrm{inv\&alpha;}}_{pbt1}+{\mathrm{inv\&alpha;}}_{Jbt1}-{\mathrm{inv\&alpha;}}_t-{\mathrm{inv\&alpha;}}_{pt})}{2{\mathrm{tan\&alpha;}}_{pn}}-\frac{{\mathrm{tan\&alpha;}}_n}{{\mathrm{tan\&alpha;}}_{pn}}{x}_{n1}---\left(106\right)$

$x_{pn2}=\frac{{\mathrm{tan\&alpha;}}_n}{{\mathrm{tan\&alpha;}}_{pn}}{x}_{n2}+\frac{z_2({\mathrm{inv\&alpha;}}_t+{\mathrm{inv\&alpha;}}_{pJt2}-{\mathrm{inv\&alpha;}}_{Jat2}-{\mathrm{inv\&alpha;}}_{st})}{2{\mathrm{tan\&alpha;}}_{pn}}---\left(107\right)$

Total distribution transforming potential coefficient is calculated by formula (34), obtains:

$x_{\mathrm{\&Sigma;}pn}=\frac{(z_2-z_1)({\mathrm{inv\&alpha;}}_{pwt}-{\mathrm{inv\&alpha;}}_{pt})}{2{\mathrm{tan\&alpha;}}_{pn}}---\left(108\right)$

In small gear, in modification coefficient and gearwheel, modification coefficient is calculated by formula (35), obtains:

x _pn1＝x _pn2-x _Σpn(109)

x _pn2x _Σpn+x _pn1(110)

Tooth crest boundary circle radius r _ja2with root portions circle circle radius r _jb1according to solving equations, r _ja2with r _jb1set of equation is:

z ₂(invα _Jat2-invα _pJt2)+z ₁(invα _pbt1+invα _Jbt1)＝B(111)

r _Ja2 ²＝a ²+r _Jb1 ²-ar _Jb1cos(α _swt-α _Jbt1)(112)

${\mathrm{\&alpha;}}_{pbt1}=arccos\left(\frac{r_{p1}}{r_{Jb1}}\right)---\left(113\right)$

${\mathrm{\&alpha;}}_{Jbt1}=arccos\frac{r_{b1}}{r_{Jb1}}---\left(114\right)$

${\mathrm{\&alpha;}}_{pJt2}=arccos\left(\frac{r_{p2}}{r_{Ja2}}\right)---\left(115\right)$

${\mathrm{\&alpha;}}_{Jat2}=arccos\left(\frac{r_{b2}}{r_{Ja2}}\right)---\left(116\right)$

In formula, B is constant.

B＝2x _Σntanα _n+(z ₁+z ₂)invα _t-(z ₂-z ₁)invα _pwt)(117)

(3), the calculating of two large negative gear ratio i=1

Be the two negative transmission transmission of 1 to velocity ratio, the modification coefficient of two gears is equal, and interior modification coefficient is equal with distribution transforming potential coefficient, namely

$x_{n1}=x_{n2}=\frac{1}{2}{x}_{\mathrm{\&Sigma;}n}=x_n$

x _sn1＝x _pn2

x _sn2＝x _pn1

x _Σsn＝x _Σpn＝0

r _Ja1＝r _Ja2＝r _Ja

r _Jb1＝r _Jb2＝r _Jb

Therefore, velocity ratio is the two large negative gear transmission calculating of 1, only need separate a prescription journey, obtain r _jaand r _jb, then calculate x _sn1and x _sn2.

4, flank profil boundary circle and each portion involute tooth depth calculate

(1), flank profil top boundary circle radius according to formula (13) calculate, obtain:

$r_{Ja1}=\sqrt{a^2+{r_{Jb2}}^2-{\mathrm{ar}}_{Jb2}cos({\mathrm{\&alpha;}}_{swt}-{\mathrm{\&alpha;}}_{sbt2})}---\left(118\right)$

$r_{Ja2}=\sqrt{a^2+{r_{Jb1}}^2-{\mathrm{ar}}_{Jb1}cos({\mathrm{\&alpha;}}_{pwt}-{\mathrm{\&alpha;}}_{pbt1})}---\left(119\right)$

(2), boundary addendum according to formula (2-12) calculate, obtain:

h _Ja1＝r _a1-r _Ja1(120)

h _Ja2＝r _a2-r _Ja2(121)

(3), middle part tooth depth calculates:

h _Jz1＝r _Ja1-r _Jb1(122)

h _Jz2＝r _Ja2-r _Jb2(123)

(4), root tooth depth calculates

h _bf1＝h _w1-h _Ja1-h _Jz1(124)

h _bf2＝h _w2-h _Ja2-h _Jz2(125)

(5), working addendum calculates

h _wa1＝r _a1-r _w1(126)

h _wa2＝r _a2-r _w2(127)

5, end face internal messing angle with join working pressure angle and calculate

Calculate according to formula (12), obtain:

${\mathrm{\&alpha;}}_{swt}=arccos\left(\frac{a^{\&prime;}}{a}{\mathrm{cos\&alpha;}}_{st}\right)---\left(128\right)$

${\mathrm{\&alpha;}}_{pwt}=arccos\left(\frac{a^{\&prime;}}{a}{\mathrm{cos\&alpha;}}_{pt}\right)---\left(129\right)$

6, contact ratio calculates

As shown in Figure 7, contact ratio is made up of four parts, small gear top and gearwheel root, middle mate, end face part, and this three part still uses ε respectively _f, ε _α, ε _βrepresent, the gearwheel top of increase and small gear root contact ratio part, use ε _prepresent, formula is also not quite similar.

(1), small gear top and gearwheel root contact ratio ε _fcalculate

${\mathrm{\&epsiv;}}_f=\frac{({\mathrm{tan\&alpha;}}_{sat1}-{\mathrm{tan\&alpha;}}_{sJt1})r_{s1}cos\mathrm{\&beta;}}{{\mathrm{\&pi;m}}_n{\mathrm{cos\&alpha;}}_{st}}---\left(130\right)$

α in formula _sat1calculated by formula (17), α _sJt1calculated by formula (27).

(2), intermediate portion contact ratio ε _αcalculate

${\mathrm{\&epsiv;}}_{\mathrm{\&alpha;}}=\frac{1}{2\mathrm{\&pi;}}(z_1+z_2){\mathrm{tan\&alpha;}}_{wt}---\left(131\right)$

(3), gearwheel top and small gear root contact ratio ε _pcalculate

${\mathrm{\&epsiv;}}_p=\frac{({\mathrm{tan\&alpha;}}_{pat2}-{\mathrm{tan\&alpha;}}_{pJt2})r_{p2}cos\mathrm{\&beta;}}{{\mathrm{\&pi;m}}_n{\mathrm{cos\&alpha;}}_{pt}}---\left(132\right)$

${\mathrm{\&alpha;}}_{pat2}=arccos\frac{r_{p2}}{r_{a2}}---\left(133\right)$

${\mathrm{\&alpha;}}_{pJt2}=arccos\frac{r_{p2}}{r_{Ja2}}---\left(134\right)$

(4), transverse contact ratio ε _βcalculate

${\mathrm{\&epsiv;}}_{\mathrm{\&beta;}}=\frac{bsin\mathrm{\&beta;}}{{\mathrm{\&pi;m}}_n}$

(5), total contact ratio

ε _γ＝ε _f+ε _α+ε _p+ε _β(135)

7, transition curve Interference Check

It is in tooth root portion and tooth crest that transition curve is interfered, and identical with single large negative gear transmission, just symbolic significance is different, obtains two large negative gear transmission transition Interference Check formula according to preceding formula.

${\mathrm{tan\&alpha;}}_{swt}-\frac{1}{i}({\mathrm{tan\&alpha;}}_{sat1}-{\mathrm{tan\&alpha;}}_{swt})\&GreaterEqual;{\mathrm{sin\&alpha;}}_{st}-\frac{({h_a}^{*}-x_{sn2})m_{n}cos\mathrm{\&beta;}}{r_{s2}{\mathrm{sin\&alpha;}}_{st}}$

${\mathrm{tan\&alpha;}}_{pwt}-i({\mathrm{tan\&alpha;}}_{pat2}-{\mathrm{tan\&alpha;}}_{pwt})\&GreaterEqual;{\mathrm{s\&alpha;}}_{pt}-\frac{({h_a}^{*}-x_{pn1})m_{n}cos\mathrm{\&beta;}}{r_{p1}{\mathrm{sin\&alpha;}}_{pt}}$

Because gearwheel root starting point pressure angle calculates, can check with following formula.

${\mathrm{tan\&alpha;}}_{fct2}\&GreaterEqual;{\mathrm{sin\&alpha;}}_{st}-\frac{({h_a}^{*}-x_{sn2})m_{n}cos\mathrm{\&beta;}}{r_{s2}{\mathrm{sin\&alpha;}}_{st}}---\left(142\right)$

In like manner obtain small gear root Interference Check formula

${\mathrm{tan\&alpha;}}_{fct1}\&GreaterEqual;{\mathrm{sin\&alpha;}}_{pt}-\frac{({h_a}^{*}-x_{pn1})m_{n}cos\mathrm{\&beta;}}{r_{p1}{\mathrm{sin\&alpha;}}_{pt}}---\left(143\right)$

It is relatively more accurate that root starting point pressure angle calculates according to reality engagement situation, and therefore root transition interferes general application of formula (142) and (143) to be checked.

8, base tangent length calculates

Two large negative gear transmission, gearwheel z ₁with small gear z ₂are all three sections of flank profils, top and middle part are that double wedge is wide, and can measure with common normal micrometer, root is recessed flank profil, is not easy at present measure.Top spanning measure tooth number small gear k _sawith termination pressure angle, top α _satcalculate, gearwheel k _pawith termination pressure angle, top α _patcalculate, middle part spanning measure tooth number k _szwith terminate partway pressure angle α _jatcalculate, formula is as follows:

$k_{sa}=\frac{{\mathrm{\&alpha;}}_{sat}}{\mathrm{\&pi;}}z+0.5---\left(144\right)$

$k_{pa}=\frac{{\mathrm{\&alpha;}}_{pat}}{\mathrm{\&pi;}}z+0.5---\left(145\right)$

$k_{sz}=\frac{{\mathrm{\&alpha;}}_{Jat}}{\mathrm{\&pi;}}z+0.5---\left(146\right)$

Base tangent length is calculated, with corresponding pressure angle, modification coefficient by formula (78) and (79).Middle part flank profil base tangent length w _szrepresent, small gear top flank profil base tangent length w _sarepresent, gearwheel top flank profil base tangent length w _parepresent.Two large negative other parameters of gear of double involute are identical with single large negative gear with Size calculation.

w _sz＝m _ncosα _n[π(k _sz-0.5)+zinvα _t+2x _ntanα _n](147)

w _sa＝m _ncosα _sn[π(k _sa-0.5)+z ₁invα _st+2x _sn1tanα _sn](148)

w _pa＝m _ncosα _pn[π(k _pa-0.5)+z ₂invα _pt+2x _pn2tanα _pn](149)

9, the thick calculating of boundary knuckle-tooth

Boundary knuckle-tooth is thick is the important parameter designing hobboing cutter with boundary addendum, and two large negative gear transmission will calculate tooth top boundary circular thickness s _jawith boundary addendum h _ja, accurate Calculation wants solving equations to obtain boundary circle radius r _ja, because solving equations difficulty does not have solving equations, average after will calculating respectively according to top and middle part formula.Formula is with single negative driving pinion.

The two large negative gear transmission of Double Involute Gear, the same with single large negative gear transmission, the pitch circle of gearwheel and small gear is all greater than circle of demarcating and is less than basic circle, and on positive contacting profile, therefore pitch circle circular thickness graduated arc transverse tooth thickness calculates.

Small gear pitch circle circular thickness s _w1for:

$s_{w1}=\frac{r_{w1}}{r_1}{s}_1-2r_{w1}({\mathrm{inv\&alpha;}}_{wt}-{\mathrm{inv\&alpha;}}_t)---\left(150\right)$

Gearwheel pitch circle circular thickness s _w2for:

$s_{w2}=\frac{r_{w2}}{r_2}{s}_2-2r_{w2}({\mathrm{inv\&alpha;}}_{wt}-{\mathrm{inv\&alpha;}}_t)---\left(151\right)$

Double Involute Gear, no matter no matter is single large negative gear transmission or two large negative gear transmission, be gearwheel or small gear, pitch circle circular thickness is all on positive contacting profile, therefore, pitch circle circular thickness convenience of calculation is reliable measuring gear and the important parameter making the cutters such as hobboing cutter.

Gear Stress calculation and strength check

Mounting center is apart from a and modulus m _nit is the important parameter determining driving mechanism volume size, multiple factors and multiple method is had to determine, wherein calculate to determine it is the most general method according to transmission power and rotating speed and input torque, even if be not determine by the method, also will carry out strength check according to input torque.About Double Involute Gear mounting center apart from a and modulus m _naccording to the method that driving torque Ne calculates.

1, the designing and calculating of as installed centre distance a

Mounting center distance or operating center distance according to input torque and gear ratio calculation, then are chosen according to evaluation, and formula is:

a≥k _a(Nei) ^1/3(152)

In formula, k _abe that rule of thumb choose with test data, Ne is input torque, and unit is N.m, and i is velocity ratio according to coefficient of colligation such as the mechanical property of materials, condition of heat treatment, gear structure, flank profil forms, a calculates mounting center distance or operating center distance, and unit is mm.

Ordinary circumstance, rule of thumb adds up k with test data _aselect between 7.5-8.5, Hardened gear face gets the small value, and the soft flank of tooth takes large values.

2, modulus m _ndesigning and calculating

Modulus calculates according to input torque, then chooses according to evaluation, formula be for:

mn≥k _m(Ne) ^1/3(153)

In formula, k _mbe that rule of thumb choose with test data, Ne is input torque, and unit is N.m according to coefficient of colligation such as the mechanical property of materials, condition of heat treatment, gear structure, flank profil forms, calculating modulus unit is mm.

Ordinary circumstance, rule of thumb adds up k with test data _mselect between 0.35-0.45, Hardened gear face gets the small value, and the soft flank of tooth takes large values.

3, Tooth Number Calculation

The Double Involute Gear number of teeth can primary Calculation be determined according to mounting center distance, modulus, velocity ratio, requires and pressure angle and the helix angle selected adjust again according to integer, and the modification coefficient that meet gearwheel meets the condition of formation double involute.The number of teeth of gearwheel and small gear is calculated respectively according to formula (69) and velocity ratio.

$z_1=\frac{z_{v}cos\mathrm{\&beta;}}{(1+i){\mathrm{cos\&alpha;}}_t}---\left(154\right)$

$z_2=\frac{{\mathrm{iz}}_{v}cos\mathrm{\&beta;}}{(1+i){\mathrm{cos\&alpha;}}_t}---\left(155\right)$

$z_1+z_2\&le;\frac{z_{v}cos\mathrm{\&beta;}}{{\mathrm{cos\&alpha;}}_t}---\left(156\right)$

Calculate according to formula (154) (155) and choose by integer, then calculate velocity ratio, and press formula (156) check.If will reselect too greatly with the ratio error required, until meet velocity ratio and total number of teeth condition (156) simultaneously.

During concrete enforcement, for addendum coefficient h _a ^*, tip clearance coefficient c ^*, general involute gear gets h ^* _a=1, c ^*=0.25, the negative greatly gear transmission of Double Involute Gear list is still determined according to involute gear, and two large negative gear transmission transition curve will occur and interferes, and suitably will increase c ^*numerical value.For hobboing cutter corner radius coefficient ρ ^*, in order to tooth root smooth transition, Double Involute Gear hobboing cutter is still chosen according to hobboing cutter standard, and hobboing cutter corner radius coefficient gets ρ ^*=0.3.In order to protect working flank, tooth top adopts and goes wedge angle burr chamfering, and angle 45 °, chamfer height changes according to modulus, c _a=c ^* _am _n, chamfering coefficient c ^* _a=0.1.Velocity ratio i will change requirement according to transmission speed and determine, modulus m _ndetermine according to transmission power P and rotating speed n or moment of torsion Ne, installing (reality) centre distance a will determine according to driving torque or mechanism's bulk.Therefore, addendum coefficient ha ^*, tip clearance coefficient c ^*, hobboing cutter corner radius coefficient ρ ^*, chamfering coefficient c ^* _a, velocity ratio i, modulus m _n, operating center distance a etc. is all known conditions.The number of teeth draws at first according to velocity ratio i, standard pitch circle Normal pressure angle α _nwith spiral angle of graduated circle β and modification coefficient x _nalso tentatively can choose, then determine according to meshing condition check amendment.

Embodiment 1: two large negative meshed transmission gear (i=1)

Steyr ransaxle cylindrical gears, centre distance a=193, velocity ratio i=1, calculate and determine small gear z ₁=31, gearwheel z ₂=31, modulus m _n=6.5, pressure angle α _n=22.5 °, helixangleβ=14.92833333 ° (14 ° 55 ' 42 ").Concrete gear and hobboing cutter parameter be as shown in Table 1 and Table 2:

Table 1: gear parameter list (two large negative gear transmission i=1)

Table 2: hobboing cutter parameter list

Sequence number	Parameter name	Symbol	Numerical value	Remarks
					1	Modulus	m n	6.5
2	Standard pitch circle Normal pressure angle	α n	22.5°
					3	Effective tooth depth	h	10.675	Not containing tooth top chamfer height
4	Boundary knuckle-tooth is thick	S Jl	9.064
					5	Boundary addendum	h Ja	4.097
6	Thickness on pitch circle	S w	9.779
					7	Working addendum	h wa	6.066
8	Normal pressure angle in standard pitch circle	α sn	29°16′37″
					9	Corner radius	ρ f	1.95±0.025
10	Tooth top chamfer height and angle	c a×45°	0.65×45°
					11	Hobboing cutter number of effective actual teeth	n	7
12	Cutter diameter	D	100
					13	Hobboing cutter length	L	120
14	Hobboing cutter diameter of bore	d	32

Embodiment 2: two large negative meshed transmission gear (i > 1)

Steyr ransaxle cylindrical gears, centre distance a=193, velocity ratio i=1.2, calculate and get small gear z ₁=28, gearwheel z ₂=34, modulus m _n=6.5, pressure angle α _n=22.5 °, helixangleβ=14.92833333 ° (14 ° 55 ' 42 "), wheel facewidth b=36.Concrete gear and hobboing cutter parameter are as shown in table 3 ~ table 5:

Table 3: gear parameter list (two large negative gear transmission i > 1)

Table 4: hobboing cutter parameter list (small gear)

Sequence number	Parameter name	Symbol	Numerical value	Remarks
					1	Modulus	m n	6.5
2	Standard pitch circle Normal pressure angle	α n	22.5°
					3	Effective tooth depth	h	10.675	Not containing tooth top chamfer height
4	Boundary knuckle-tooth is thick	S Jl	8.708
					5	Boundary addendum	h Ja	3.048
	Thickness on pitch circle	s w	9.520
						Working addendum	h wa	3.843
6	Normal pressure angle in standard pitch circle	α sn	26°54′38″
					7	Corner radius	ρ f	1.9±0.02
8	Tooth top chamfer height and angle	c a×45°	0.65×45°
					9	Hobboing cutter number of effective actual teeth	n	7
10	Cutter diameter	D	100
					11	Hobboing cutter length	L	120
12	Hobboing cutter diameter of bore	d	32

Table 5: hobboing cutter parameter list (gearwheel)

Sequence number	Parameter name	Symbol	Numerical value	Remarks
					1	Modulus	m n	6.5
2	Standard pitch circle Normal pressure angle	α n	22.5°
					3	Effective tooth depth	h	10.675	Not containing tooth top chamfer height
4	Boundary knuckle-tooth is thick	S Jl	9.618
					5	Boundary addendum	h Ja	4.267
	Thickness on pitch circle	s w	10.003
						Working addendum	h wa	4.896
6	Normal pressure angle in standard pitch circle	α sn	29°12′57″
					7	Corner radius	ρ f	1.9±0.02
8	Tooth top chamfer height and angle	c a×45°	0.65×45°
					9	Hobboing cutter number of effective actual teeth	n	7
10	Cutter diameter	D	100
					11	Hobboing cutter length	L	120
12	Hobboing cutter diameter of bore	d	32

Claims (3)

1. a two large negative addendum modification Involutes Gears Transmission device, comprise a pair pitch wheel, it is characterized in that, the work flank profil of described two gear tooths includes the first involute portion of the male type being positioned at rolling circle outside and the second involute portion of male type, also comprise the 3rd involute portion of the inner concave shape being positioned at rolling circle, described first involute portion, the second involute portion and the 3rd involute portion by tooth crest to tooth root portion successively smooth connection; In described two gears, the first involute portion of any one gear is all that convex-concave arc engages with the 3rd involute portion of another gear, the second involute portion convexo-convex arc engagement each other mutually of described two gears.

2. two large negative addendum modification Involutes Gears Transmission device as claimed in claim 1, it is characterized in that, the polar coordinates function expression in described 3rd involute portion is:

$r_k=r_{s2}\sqrt{1+{\tan }^2{\mathrm{\&alpha;}}_k{\cos }^2{\mathrm{\&beta;}}_k}$

invα _k＝tanα _k-α _k

Wherein: r _kbe in the 3rd involute portion arbitrfary point to the distance of gear axis, r _s2be the Base radius in the 3rd involute portion, α _kfor the pressure angle of flank profil corresponding points, β _kfor flank profil arbitrfary point helix angle;

In described polar coordinates function expression, also comprise following formula:

r _s2＝r _f+ρ _f(1-sinα _t)

Wherein: r _froot radius, ρ _ffor hobboing cutter corner radius, α _tfor gear compound graduation circle transverse pressure angle.

3. two large negative addendum modification Involutes Gears Transmission device as claimed in claim 2, it is characterized in that, the work flank profil of described two gear tooths also comprises the transition curve being positioned at tooth root, the upper end of described transition curve and the lower end smooth connection in the 3rd involute portion, lower end and the root circle of transition curve are tangent; On described transition curve, arbitrfary point (x, y) meets following curvilinear equation:

In described calculating formula, for the vertical line of hobboing cutter movement direction and the angle between hobboing cutter tool arc central point and the line of Gear center.