CN103016678A - Exact modeling method of double circular arc gear - Google Patents

Abstract

The invention relates to an exact modeling method of a double circular arc gear. The method comprises the following steps of: S1, determining basic parameters of the double circular arc gear; S2, drawing a pitch circle, an addendum circle and a root circle of the gear; S3, drawing a circular arc curve according to an end surface tooth profile equation, and establishing a basic tooth profile; S4, duplicating and creating a plurality of gear teeth cross sections; S5, drawing a scanning curve according to a helix equation, and finishing single-tooth modeling through mixed scanning; and S6, creating a fitting model through an array order, and adding hubs to finish the gear modeling. The method has the advantages of high modeling precision and high efficiency; and the method is beneficial for improving the design and manufacturing precision of the double circular arc gear.

Description

A kind of Novikov gears with double circular arc tooth profiles Precise modeling

Technical field

The present invention relates to gear modeling field, more particularly, relate to a kind of Novikov gears with double circular arc tooth profiles Precise modeling.

Background technique

Novikov gears with double circular arc tooth profiles is widely used in the industries such as metallurgy, mine, boats and ships, electric power and handling machinery owing to advantages such as bearing capacity are high, running-in characteristic is good, lubricating condition is superior, cost is low, the life-span is long.The important prerequisite that to set up accurate Novikov gears with double circular arc tooth profiles model be its accurate Design and manufacture.Yet, because double circular arc gear structure is complicated, also do not have at present the accurately reliable modeling method of a cover.There is following defective in the existing modeling method:

At first, calculate a plurality of points on the gear by programming software, import matched curve in the three-dimensional software again, namely complexity is inaccurate again.

Secondly, adopt the method for projection on standard pitch circle to generate helix, in software, can produce very large error during projection.

At last, take helix as guide line, modeling is carried out in two cross section scannings to head and the tail by the two-dimentional generating three-dimensional flank of tooth time, and this scanning results and actual profile of tooth differ greatly.

In sum: present Novikov gears with double circular arc tooth profiles modeling method low precision, be difficult to obtain high-precision gear-profile, can not satisfy its Design and manufacture required precision.

Summary of the invention

The technical problem to be solved in the present invention is, a kind of Novikov gears with double circular arc tooth profiles Precise modeling is provided.

The technical solution adopted for the present invention to solve the technical problems is: construct a kind of Novikov gears with double circular arc tooth profiles Precise modeling, may further comprise the steps:

S1 determines the basic parameter of Novikov gears with double circular arc tooth profiles;

S2, pitch circle, top circle and the root circle of drafting gear;

S3 draws circular curve according to end surface tooth form equation, sets up basic rack tooth profile;

S4 copies creating a plurality of gear teeth transverse section;

S5, curve is scanned in drafting according to helix equation, and finishes the monodentate moulding by mixed sweep;

S6 uses array commands to create whole tooth model, and adds wheel hub and finish the gear moulding.

In Novikov gears with double circular arc tooth profiles Precise modeling of the present invention, when circular curve was double wedge flank profil circular arc, described end surface tooth form equation was:

$\left\{\begin{array}{c}{\mathrm{\φ}}_k=-(-\mathrm{\ρ}\cos \mathrm{\α}\sin \mathrm{\β}+e\cot \mathrm{\α}\cos \mathrm{\β}\cot \mathrm{\β}+N/\sin \mathrm{\β})\tan \mathrm{\β}/r_k\\ x_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\cos {\mathrm{\φ}}_k-(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\sin {\mathrm{\φ}}_k\\ y_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\sin {\mathrm{\φ}}_k+(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\cos {\mathrm{\φ}}_k\end{array}\right.$

When circular curve was recessed tooth flank profil circular arc, described end surface tooth form equation was:

$\left\{\begin{array}{c}{\mathrm{\φ}}_k=-(-\mathrm{\ρ}\cos \mathrm{\α}\sin \mathrm{\β}+e\cot \mathrm{\α}\cos \mathrm{\β}\cot \mathrm{\β}+N/\sin \mathrm{\β})\tan \mathrm{\β}/r_k\\ x_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\cos {\mathrm{\φ}}_k-(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\sin {\mathrm{\φ}}_k\\ y_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\sin {\mathrm{\φ}}_k+(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\cos {\mathrm{\φ}}_k\end{array}\right.$

In the formula, Ф _kBe the corner of gear, ρ is the Relative radius of curvature of flank profil, and α is the flank profil pressure angle, and β is gear helical angle, and e is the side-play amount of the relatively selected coordinate axes x axle in the flank profil center of circle, and N is the side-play amount of the relatively selected coordinate axes y axle in the flank profil center of circle, r _kIt is the Pitch radius of gear.

In Novikov gears with double circular arc tooth profiles Precise modeling of the present invention, described step S4 finishes according to following equation:

L＝b/n

θ'＝Ltanβ/(r _k)

In the formula, L is the level interval between each transverse section, and b is the facewidth, and n is that number is copied in gear teeth transverse section, and θ ' is angle of swing.

In Novikov gears with double circular arc tooth profiles Precise modeling of the present invention, the helix equation among the described step S5 is:

$\left\{\begin{array}{c}x=r_k\sin \left(\frac{180z\tan \mathrm{\β}}{r_k\mathrm{\π}}\right)\\ y=r_k\cos \left(\frac{180z\tan \mathrm{\β}}{r_k\mathrm{\π}}\right)\\ z=\mathrm{bt}\end{array}\right.$

In the formula, z is the number of teeth, and t is variable, and 0＜t＜1.

Implement Novikov gears with double circular arc tooth profiles Precise modeling of the present invention, have following beneficial effect:

(1), draw circular curve according to end surface tooth form equation, set up two circular arc basic rack tooth profiles, saved loaded down with trivial details programming and calculated, and the curve precision is high, has avoided coarse curve.

(2), by setting up helix equation, the error of having avoided software in projection is calculated, to cause, and efficient is higher than projecting method.

(3), for the transition curve of different gears, the basic parameter that only needs to revise transition curve just can obtain corresponding transition curve.

(4), set up a plurality of gear teeth transverse section by copying, and when scanning, pass through a plurality of transverse section, improved the accuracy of gear teeth modelings.

Therefore, a kind of Novikov gears with double circular arc tooth profiles Precise modeling provided by the invention has modeling accuracy height, efficient advantages of higher, is conducive to improve Novikov gears with double circular arc tooth profiles Design and manufacture precision.

Description of drawings

The invention will be further described below in conjunction with drawings and Examples, in the accompanying drawing:

Fig. 1 is the primary circle schematic representation of Novikov gears with double circular arc tooth profiles;

Fig. 2 is Novikov gears with double circular arc tooth profiles basic rack tooth profile parameter schematic representation;

Fig. 3 is the basic rack tooth profile system of coordinates;

Fig. 4 is Novikov gears with double circular arc tooth profiles flank profil end view;

Fig. 5 is the gear teeth schematic cross-sectional view that copies establishment;

Fig. 6 is the helix schematic representation;

Fig. 7 is the monodentate model schematic representation of Novikov gears with double circular arc tooth profiles;

Fig. 8 is complete Novikov gears with double circular arc tooth profiles model schematic representation.

Embodiment

Understand for technical characteristics of the present invention, purpose and effect being had more clearly, now contrast accompanying drawing and describe the specific embodiment of the present invention in detail.

Novikov gears with double circular arc tooth profiles Precise modeling of the present invention may further comprise the steps:

S1 arranges in 3 d modeling software and the definition relevant parameter, and desired parameters comprises:

According to the fundamental formular of Novikov gears with double circular arc tooth profiles, can calculate tip diameter d _a, pitch diameter d, root diameter d _f

S2 according to the value that calculates, draws top circle 1, pitch circle 2 and the root circle 3 of gear, as shown in Figure 1.

S3 draws circular curve according to an arc equation, sets up two circular arc basic rack tooth profiles.Novikov gears with double circular arc tooth profiles basic rack tooth profile parameter as shown in Figure 2, the system of coordinates of foundation as shown in Figure 3, ED section circular arc is double wedge flank profil circular arc 4, DC section circular arc is recessed tooth flank profil circular arc 5, CB section circular arc is recessed tooth flank profil circular arc 6, BA section circular arc is recessed tooth flank profil circular arc 7.According to its geometrical relationship, derive the accounting equation of each section of basic rack tooth profile circular arc, can obtain each section circular curve as shown in Figure 4.For each arc section, the unified α that uses ₁Represent the initial pressure angle of circular arc, use α ₂Represent the termination pressure angle of circular arc, α represents the pressure angle variable of circular arc; T is variable, and from 0 to 1 changes.

(1) for double wedge flank profil circular arc 4, its parametric is:

α ₁＝arcsin(h _a-e _a)/ρ _a

In the formula, h _aThe expression addendum, e _aThe side-play amount of circular arc 4 centers of circle relatively selected coordinate x axle, ρ _aIt is the radius of curvature of circular arc 4.

a ₂＝δ ₁

In the formula: δ ₁Be the double wedge process corner.

α＝α ₁+(α ₂-α ₁)*t

ρ＝ρ _a

e＝e _a

N＝l _a

In the formula: l _a Circular arc 4 center of circle side-play amounts.

Tooth profile equation is:

$\left\{\begin{array}{c}{\mathrm{\φ}}_k=-(-\mathrm{\ρ}\cos \mathrm{\α}\sin \mathrm{\β}+e\cot \mathrm{\α}\cos \mathrm{\β}\cot \mathrm{\β}+N/\sin \mathrm{\β})\tan \mathrm{\β}/r_k\\ x_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\cos {\mathrm{\φ}}_k-(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\sin {\mathrm{\φ}}_k\\ y_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\sin {\mathrm{\φ}}_k+(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\cos {\mathrm{\φ}}_k\end{array}\right.$

In the formula, Ф _kBe the corner of gear, ρ is the Relative radius of curvature of flank profil, and α is the flank profil pressure angle, and β is gear helical angle, and e is the side-play amount of the relatively selected coordinate axes x axle in the flank profil center of circle, and N is the side-play amount of the relatively selected coordinate axes y axle in the flank profil center of circle, r _kIt is the Pitch radius of gear.

(2) for recessed tooth flank profil circular arc 5, its parametric is:

a ₁＝-δ ₁

a ₂＝-arcsin(h _ja+r _jsinδ ₁+h _jf)/r _j

α＝α ₁+(α ₂-α ₁)*t

N＝(r _j+ρ _a)cosδ ₁-l _a

e＝(r _j+ρ _a)sinδ ₁+e _a

In the formula, h _JaThe side-play amount of circular arc 5 upper ends relatively selected coordinate x axle, h _JfThe side-play amount of circular arc 5 lower ends relatively selected coordinate x axle, r _jIt is the radius of circular arc 5.

Tooth profile equation is:

$\left\{\begin{array}{c}{\mathrm{\φ}}_k=-(-\mathrm{\ρ}\cos \mathrm{\α}\sin \mathrm{\β}+e\cot \mathrm{\α}\cos \mathrm{\β}\cot \mathrm{\β}+N/\sin \mathrm{\β})\tan \mathrm{\β}/r_k\\ x_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\cos {\mathrm{\φ}}_k-(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\sin {\mathrm{\φ}}_k\\ y_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\sin {\mathrm{\φ}}_k+(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\cos {\mathrm{\φ}}_k\end{array}\right.$

(3) for recessed tooth flank profil circular arc 6, its parametric is:

a ₁＝-δ ₂

In the formula, δ ₂Be recessed tooth process corner.

a ₂＝-acsin((h _f-r _gf+e _f)/(ρ _f-r _gf))

α＝α ₁+(α ₂-α ₁)*t

ρ＝ρ _f

N＝πm _n/2+l _f

e＝e _f

In the formula: h _fDedendum of the tooth, r _GfThe radius of curvature of circular arc 7, e _fThe side-play amount of circular arc 6 centers of circle relatively selected coordinate x axle, ρ _fThe radius of curvature of circular arc 6, l _fCircular arc 6 center of circle side-play amounts, m _nIt is normal module.

Tooth profile equation is:

$\left\{\begin{array}{c}{\mathrm{\φ}}_k=-(-\mathrm{\ρ}\cos \mathrm{\α}\sin \mathrm{\β}+e\cot \mathrm{\α}\cos \mathrm{\β}\cot \mathrm{\β}+N/\sin \mathrm{\β})\tan \mathrm{\β}/r_k\\ x_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\cos {\mathrm{\φ}}_k-(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\sin {\mathrm{\φ}}_k\\ y_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\sin {\mathrm{\φ}}_k+(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\cos {\mathrm{\φ}}_k\end{array}\right.$

(4) for recessed tooth flank profil circular arc 7, its parametric is:

a ₁＝-arcsin((h _f-r _gf+e _f)/(ρ _f-r _gf))

a ₂＝-π/2

α＝α ₁+(α ₂-α ₁)*t

ρ＝r _gf

N＝πm _n/2

e＝-(h _f-r _gf)

Tooth profile equation is:

$\left\{\begin{array}{c}{\mathrm{\φ}}_k=-(-\mathrm{\ρ}\cos \mathrm{\α}\sin \mathrm{\β}+e\cot \mathrm{\α}\cos \mathrm{\β}\cot \mathrm{\β}+N/\sin \mathrm{\β})\tan \mathrm{\β}/r_k\\ x_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\cos {\mathrm{\φ}}_k-(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\sin {\mathrm{\φ}}_k\\ y_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\sin {\mathrm{\φ}}_k+(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\cos {\mathrm{\φ}}_k\end{array}\right.$

Behind above-mentioned equation drafting four arcs curve, again by the symmetric replication function, can obtain the tooth curve of symmetrical position.The circular curve that obtains and top circle, root circle are boolean shear computing, just can obtain monodentate flank profil as shown in Figure 4.

S4 in order to guarantee the highi degree of accuracy of profile of tooth, copies creating a plurality of transverse section, in the present embodiment is 4.Finish according to following equation:

L＝b/n

θ'＝Ltanβ/(r _k)

In the formula, L is the level interval between each transverse section, and b is the facewidth, and n is that number is copied in gear teeth transverse section, and θ ' is angle of swing.

S5, drafting is scanned curve 8 as shown in Figure 6 according to helix equation, can obtain the monodentate model by the mixed sweep order subsequently, as shown in Figure 7.Helix equation is:

$\left\{\begin{array}{c}x=r_k\sin \left(\frac{180z\tan \mathrm{\β}}{r_k\mathrm{\π}}\right)\\ y=r_k\cos \left(\frac{180z\tan \mathrm{\β}}{r_k\mathrm{\π}}\right)\\ z=\mathrm{bt}\end{array}\right.$

In the formula, z is the number of teeth, and t is variable, and 0＜t＜1.

S6 uses array commands to create whole tooth model, and adds wheel hub and finish the gear moulding, as shown in Figure 8.

Coordinate in the present embodiment is cartesian coordinate system, but is understandable that, it is preferred embodiment a kind of, and system of coordinates can be other system of coordinates such as spherical coordinate system, cylindrical-coordinate system also, and when system of coordinates changed, corresponding equation also can change.

The above is described embodiments of the invention by reference to the accompanying drawings; but the present invention is not limited to above-mentioned embodiment; above-mentioned embodiment only is schematic; rather than restrictive; those of ordinary skill in the art is under enlightenment of the present invention; not breaking away from the scope situation that aim of the present invention and claim protect, also can make a lot of forms, these all belong within the protection of the present invention.

Claims (4)

1. a Novikov gears with double circular arc tooth profiles Precise modeling is characterized in that, may further comprise the steps:

S1 determines the basic parameter of Novikov gears with double circular arc tooth profiles;

S2, pitch circle, top circle and the root circle of drafting gear;

S3 draws circular curve according to end surface tooth form equation, sets up basic rack tooth profile;

S4 copies creating a plurality of gear teeth transverse section;

S5, curve is scanned in drafting according to helix equation, and finishes the monodentate moulding by mixed sweep;

S6 uses array commands to create whole tooth model, and adds wheel hub and finish the gear moulding.

2. Novikov gears with double circular arc tooth profiles Precise modeling according to claim 1 is characterized in that, when circular curve was double wedge flank profil circular arc, described end surface tooth form equation was:

$\left\{\begin{array}{c}{\mathrm{\φ}}_k=-(-\mathrm{\ρ}\cos \mathrm{\α}\sin \mathrm{\β}+e\cot \mathrm{\α}\cos \mathrm{\β}\cot \mathrm{\β}+N/\sin \mathrm{\β})\tan \mathrm{\β}/r_k\\ x_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\cos {\mathrm{\φ}}_k-(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\sin {\mathrm{\φ}}_k\\ y_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\sin {\mathrm{\φ}}_k+(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\cos {\mathrm{\φ}}_k\end{array}\right.$

When circular curve was recessed tooth flank profil circular arc, described end surface tooth form equation was:

$\left\{\begin{array}{c}{\mathrm{\φ}}_k=-(-\mathrm{\ρ}\cos \mathrm{\α}\sin \mathrm{\β}+e\cot \mathrm{\α}\cos \mathrm{\β}\cot \mathrm{\β}+N/\sin \mathrm{\β})\tan \mathrm{\β}/r_k\\ x_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\cos {\mathrm{\φ}}_k-(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\sin {\mathrm{\φ}}_k\\ y_k=(\mathrm{\ρ}\sin \mathrm{\α}+e+r_k)\sin {\mathrm{\φ}}_k+(\mathrm{\ρ}\cos \mathrm{\α}+e\cot \mathrm{\α})\cos \mathrm{\β}\cos {\mathrm{\φ}}_k\end{array}\right.$

In the formula, Ф _kBe the corner of gear, ρ is the Relative radius of curvature of flank profil, and α is the flank profil pressure angle, and β is gear helical angle, and e is the side-play amount of the relatively selected coordinate axes x axle in the flank profil center of circle, and N is the side-play amount of the relatively selected coordinate axes y axle in the flank profil center of circle, r _kIt is the Pitch radius of gear.

3. Novikov gears with double circular arc tooth profiles Precise modeling according to claim 1 is characterized in that, described step S4 finishes according to following equation:

L＝b/n

θ'＝Ltanβ/(r _k)

In the formula, L is the level interval between each transverse section, and b is the facewidth, and n is that number is copied in gear teeth transverse section, and θ ' is angle of swing.

4. Novikov gears with double circular arc tooth profiles Precise modeling according to claim 1 is characterized in that, the helix equation among the described step S5 is:

$\left\{\begin{array}{c}x=r_k\sin \left(\frac{180z\tan \mathrm{\β}}{r_k\mathrm{\π}}\right)\\ y=r_k\cos \left(\frac{180z\tan \mathrm{\β}}{r_k\mathrm{\π}}\right)\\ z=\mathrm{bt}\end{array}\right.$

In the formula, z is the number of teeth, and t is variable, and 0＜t＜1.

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