CN104196981B - A kind of design method of biradical cone spiral bevel gear form of gear tooth - Google Patents

Abstract

The invention relates to a design method of a double-base bevel spiral bevel gear tooth profile, which relates to gears. The double-base bevel spiral bevel gear has 5 cone surfaces, 4 cone angles, and 2 base cones; the normal tooth profile of a single tooth is composed of addendum circle, dedendum circle, drive side tooth profile curve, non-drive side tooth profile The 5 cone surfaces are face cone, root cone, pitch cone, back cone and front cone respectively, and the 4 cone angles are face angle, root angle, pitch angle and back angle respectively. For the gear transmission mechanism with one-way transmission mainly in vehicles, the performance of the spiral bevel gear is improved, and a large tooth angle is used on the driving surface, and a small tooth angle is used on the non-driving surface, which not only increases the gear load in the driving direction It is a double-base bevel helical bevel gear tooth shape design method that can effectively improve the load-carrying capacity, fatigue life and output torque of the spiral bevel gear, and reduce the transmission vibration and noise.

Description

A Design Method of Double Base Conical Helical Bevel Gear Tooth Profile

technical field

The invention relates to gears, in particular to a method for designing the tooth profile of a double base bevel spiral bevel gear.

Background technique

Compared with straight and helical bevel gears, spiral bevel gears have the advantages of large coincidence, large relative curvature radius of the tooth surface at the contact point, easy control of the surface contact area, and less sensitivity to errors. In the transmission, it has been widely used. With the development trend of high speed and heavy load in gear transmission, the traditional spiral bevel gear has been difficult to meet the requirements of the equipment. Especially in vehicle transmission components, the requirements for the output torque of spiral bevel gears are getting higher and higher.

When the tooth profile angle increases, the bending load capacity and contact load capacity of the gear are significantly enhanced, but as the tooth profile angle increases, the tooth top of the gear gradually becomes sharper. When the tooth top thickness is less than 1.5m (m is the gear mold number), the addendum is prone to breakage under load, resulting in gear transmission failure (Li Huamin, Quality Index Analysis of Large Tooth Angle Involute Gear Transmission, Journal of Harbin Institute of Technology, 1987, Z1:86-98).

Contents of the invention

The purpose of the present invention is to aim at the gear transmission mechanism with one-way transmission as the mainstay, in order to improve the performance of the spiral bevel gear, it is provided to adopt a large tooth profile angle on the driving surface and a small tooth profile angle on the non-driving surface, which not only increases A double-base bevel helical bevel gear tooth shape design method that improves the load capacity of the driving direction gear and avoids the sharpening of the tooth top, which can effectively improve the load capacity, fatigue life and output torque of the spiral bevel gear, and reduce transmission vibration and noise .

The present invention comprises the following steps:

1) The double-base bevel spiral bevel gear has 5 cone surfaces, 4 cone angles, and 2 base cones; the normal tooth profile of a single tooth consists of a top circle, a dedendum circle, a driving side tooth profile curve, and a non-driving side tooth profile. Composed of tooth profile curve and tooth direction line; the five cone surfaces are face cone, root cone, pitch cone, back cone and front cone, and the four cone angles are face angle, root angle, pitch angle and back angle ;

2) The tooth surface Ω ₁ on the driving side of the double base cone spiral bevel gear is tangent to the base cone with the base cone angle δ _bd at OP ₁ . When Ω ₁ performs pure rolling along the base cone, the circle with O as the center of rotation Arcs such as M ₁ N ₁ and M ₂ N ₂ will form the driving side and non-driving side tooth surfaces of the spiral bevel gear in space. Since the tooth profile angles on both sides of the spiral bevel gear are different, the base cones at the beginning of the tooth surfaces on both sides are different. ;

3) Through coordinate transformation, the spherical involute of the large end tooth profile of the double-base conical spiral bevel gear is transformed into a spherical coordinate system. After derivation, in the spherical coordinate system, the large-end tooth profile of the drive side tooth surface of the double-base bevel spiral bevel gear is at The equation in the spherical coordinate system is

4) Driving tooth side indexing circular deflection angle and the driving side base cone angle δ _bd are obtained by the following formula

δ _bd ＝δ′-arctan[(1-α _d )tanδ′] (3)

5) In the spherical coordinate system, the equation of the tooth profile at the large end of the non-driving side tooth surface of the double base bevel spiral bevel gear is

6) Indexing circular angle of non-driving tooth side and the base cone angle δ _bc of the non-driving tooth side are obtained by the following formula

δ _bc = δ′-arctan{[1-arccos(kcosα _d )]tanδ′} (6)

The tooth direction line of the double-base bevel spiral bevel gear is formed by the relative position of the wheel blank and the cutter head;

7) According to the helix angle β required by the design and the radius r ₀ of the cutterhead selected during processing, determine the positional relationship between the center position of the milling cutterhead and the axis center of the crown, and then transform the tooth direction line equation into spherical coordinates through coordinate transformation system, in the spherical coordinate system, the tooth line equation is

In formula (7), is the tooth line deflection angle, and the two included angles S and j are obtained by the following formula

8) When the cutter head goes from the big end to the small end, the j angle corresponds to an included angle Q, Q=j/sinδ′, therefore, the included angles corresponding to the outermost point and the innermost point are Q ₀ =j ₀ /sinδ' and Q ₁ = j ₁ /sinδ', the difference between the two angles Q ₁ -Q ₀ is the angle that the corresponding double-base bevel tooth curve from the big end to the small end turns in the spherical coordinate system, according to Q ₁ -Q ₀ and the value of rho in formulas (1) and (4) is changed to RB, then the equation of the tooth profile of the small end drive side and non-drive side of the double-base bevel spiral bevel gear tooth in the spherical coordinate system can be obtained ;

The diameter of the addendum circle of the double base bevel spiral bevel gear is:

d _a =m _t z+2h _ad cosδ _d (10)

The top cone angles of the driving side and non-driving side of the double-base bevel spiral bevel gear are:

δ _ad = δ _d + θ _fd (11)

δ _ac = δ _c + θ _fc (12)

The diameter of the dedendum circle of the double base bevel spiral bevel gear is:

d _f = m _t zh _fd cosδ _d (13)

The root cone angles of the driving side and the non-driving side of the double base bevel spiral bevel gear are:

δ _fd = δ _d - θ _fd (14)

δ _fc = δ _c - θ _fc (15)

According to the above steps, the tooth profile surface of the double base bevel spiral bevel gear is established, and the single tooth entity of the spiral bevel gear is established, and the array is equally divided according to the number of teeth z, then the overall model of the double base bevel spiral bevel gear can be established to complete the double base bevel spiral bevel Gear tooth design.

The labels in each step are:

z - the number of teeth

B - tooth width

β——helix angle

Σ——shaft intersection angle

rho—the polar radius of the spherical coordinate system, in Figure 2

theta - directed line segment Angle with positive z axis

phi——From the perspective of the positive z-axis, the angle turned from the x-axis to OS in a counterclockwise direction, where S is the projection of point P on the xOy plane

m _t ——end modulus

d——The diameter of the indexing circle

R——outer cone distance

R _m —— Midpoint taper distance

α _d —— tooth profile angle of driving side

α _c —— tooth profile angle of non-driving side

k——tooth profile angle coefficient

δ ₁ —— sub-cone angle

——The circular deflection angle of the drive tooth side indexing

——Indicating circular angle of non-driving tooth side

δ _ad ——cone angle of driving side

δ _ac ——cone angle of non-driving side

δ _fd ——drive lateral root taper angle

δ _fc ——cone angle of non-driving side root

δ _{bd ——drive} side base cone angle

δ _{bc ——cone} angle of non-driving side base

δ’——pitch angle

——tooth line deflection angle

d _a ——diameter of gear addendum circle

d _f —— gear dedendum circle diameter

r ₀ —— cutter head radius

L ₁ ——the distance from the center of the cutterhead to the center of the cone

S——the declination angle of the tooth line relative to the center of the cutter head

j——the declination angle of the tooth direction line relative to the center of the cone top

β _p —— dedendum deflection angle

θ _fd —— Drive side dedendum angle

θ _fc —— root angle of non-driving side

h _{ad ——drive} side addendum height

h _{ac ——Height} of addendum on non-driving side

h _{fd ——drive} side dedendum height

h _{fc ——Dedendum} height of non-drive side

x—coefficient of height variation

x _t —coefficient of tangential displacement.

Since the driving side and non-driving side of the double base bevel spiral bevel gear have different gear parameters such as base cone, top cone and root vertebra, which are completely different from the geometric characteristics and design methods of the traditional spiral bevel gear, it is necessary to establish a double base The tooth profile design method of bevel spiral bevel gear lays the foundation for tool design, manufacturing, root bending fatigue strength calculation and tooth surface contact fatigue strength calculation.

The invention improves the performance of the spiral bevel gear for vehicles and other gear transmission mechanisms that mainly use one-way transmission, and adopts a large tooth shape angle on the driving surface and a small tooth shape angle on the non-driving surface, which not only increases the driving direction It is a double-base bevel spiral bevel gear design method that can effectively improve the load capacity, fatigue life and output torque of the spiral bevel gear, and reduce the transmission vibration and noise.

Description of drawings

Figure 1 shows the cone surface and cone angle of a double base bevel spiral bevel gear.

Figure 2 shows the formation of spherical involutes on the driving side and non-driving side of the double base bevel spiral bevel gear.

Fig. 3 is a schematic diagram of the relative positions of the double-base conical spiral bevel gear cutter head and the wheel blank.

Figure 4 shows the modeling process of the double base bevel spiral bevel gear.

Figure 5 is the meshing model of the double-base conical spiral bevel gear pair.

In the figure, each mark is: 1——Double base bevel spiral bevel gear driving side tooth shape, 2——Double base bevel spiral bevel gear non-driving side tooth shape, 3——Double base bevel spiral bevel gear tooth top, 4 —Dendum root of double base bevel spiral bevel gear, 5—tooth direction line of double base bevel spiral bevel gear.

detailed description

Referring to Figures 1 to 5, the double-base bevel spiral bevel gear has 5 cone surfaces, 4 cone angles, and 2 base cones; the normal tooth profile of a single tooth consists of a top circle, a dedendum circle, and a tooth profile curve on the driving side. , non-drive side tooth profile curve and tooth direction line, the five cone surfaces are face cone, root cone, pitch cone, back cone and front cone, and the four cone angles are face angle, root angle, pitch angle Angle, dorsal angle; as shown in Figure 1.

The drive side tooth surface Ω _{1 of} the double base cone spiral bevel gear is tangent to OP ₁ with the base cone angle δ _bd of the first base cone (represented as base cone 11 in Fig. 2). When rolling, the arc lines on the plane with O as the center of rotation, such as M ₁ N ₁ and M ₂ N ₂ , will form the drive side and non-drive side tooth surfaces of the spiral bevel gear in space. Since the tooth profile angles on both sides of the spiral bevel gear are different, the base cones at the beginning of the tooth surfaces on both sides are different, as shown in Figure 2. Also marked in FIG. 2 is a second base cone (shown as base cone 21 in FIG. 2 ).

Through coordinate transformation, the spherical involute of the large end tooth profile of the double-base conical spiral bevel gear is transformed into a spherical coordinate system. After derivation, the large-end tooth profile of the driving side tooth surface of the double-base conical spiral bevel gear is in the spherical coordinate system The following equation is

Driving gear flank indexing circular deflection angle and the driving side base cone angle δ _bd can be obtained by the following formula

δ _bd ＝δ′-arctan[(1-α _d )tanδ′] (3)

The equation of the tooth profile at the large end of the non-drive side tooth surface of the double base bevel spiral bevel gear in the spherical coordinate system is

Non-drive tooth side indexing circular deflection angle and the base cone angle δ _bc of the non-driving tooth side can be obtained by the following formula

δ _bc = δ′-arctan{[1-arccos(kcosα _d )]tanδ′} (6)

The tooth direction line of the double-base bevel spiral bevel gear is formed by the relative position of the wheel blank and the cutter head, as shown in Figure 3.

According to the helix angle β required by the design and the radius r ₀ of the cutterhead used during machining, the relationship between the center position of the milling cutterhead and the center of the crown axis is determined, and then the equation of the tooth direction line is transformed into a spherical coordinate system through coordinate transformation. In the spherical coordinate system, the tooth line equation is:

In the formula, is the tooth line deflection angle, and the two included angles S and j can be obtained by the following formula

From the position of the cutter head shown in Figure 3, it can be seen that when the cutter head moves from the big end to the small end, the angle j corresponds to an included angle Q, Q=j/sinδ′, therefore, the outermost point and the innermost point correspond to The included angles are Q ₀ =j ₀ /sinδ' and Q ₁ =j ₁ /sinδ' respectively. The difference between the two angles Q ₁ -Q ₀ is the angle that the corresponding double-base bevel tooth curve from the big end to the small end turns in the spherical coordinate system. According to Q ₁ -Q ₀ and formulas (1), (4) The value of rho in the middle is changed to RB, and the equation of the tooth profile of the driving side and the non-driving side of the small end of the double base bevel helical bevel gear tooth in the spherical coordinate system can be obtained.

The diameter of the addendum circle of the double base bevel spiral bevel gear is:

d _a =m _t z+2h _ad cosδ _d (10)

The top cone angles of the driving side and non-driving side of the double-base bevel spiral bevel gear are:

δ _ad = δ _d + θ _fd (11)

δ _ac = δ _c + θ _fc (12)

The diameter of the dedendum circle of the double base bevel spiral bevel gear is:

d _f = m _t zh _fd cosδ _d (13)

The root cone angles of the driving side and the non-driving side of the double base bevel spiral bevel gear are:

δ _fd = δ _d - θ _fd (14)

δ _fc = δ _c - θ _fc (15)

According to the above steps, the tooth profile surface of the double-base bevel spiral bevel gear can be established, and the single-tooth entity of the spiral bevel gear can be established, and the array can be equally divided according to the number of teeth z, and the overall model of the double-base bevel spiral bevel gear can be established.

According to the design requirements of the double-base bevel spiral bevel gear, select the appropriate module, normal surface drive side, non-drive side tooth profile angle, addendum height coefficient, displacement coefficient, radial clearance coefficient, number of teeth, helix angle, and direction of rotation and other basic design parameters, and then calculate each parameter according to the conversion relationship.

a) According to the parameters such as end face modulus m _t , number of teeth z, tooth width B, helix angle β, shaft intersection angle Σ, driving side tooth profile angle α _d , non-driving side tooth profile angle α _c , and displacement coefficient x, it can be obtained Driving gear flank indexing circular deflection angle Drive tooth side base cone angle δ _bd , non-drive tooth side indexing circular deflection angle and non-driving tooth side base cone angle δ _bc and other parameters, according to the tooth profile equations (1) to (6), the tooth profile of the driving side of the big end of the double-base bevel spiral bevel gear, the tooth profile of the non-driving side of the big end, and the tooth profile of the small end can be established Drive side toothing and small end non-drive side toothing.

b) According to the above parameters, according to equations (10)~(15), the large-end dedendum arc, large-end dedendum arc, small-end dedendum arc and small-end dedendum circle of the double base bevel spiral bevel gear can be established arc.

c) According to the above parameters, according to equations (7) to (9), the tooth direction line of the drive side and the tooth direction line of the non-drive side of the double-base bevel helical bevel gear can be established.

d) Create the first entity of a single tooth of a double base bevel spiral bevel gear.

e) By dividing the array equally according to the number of teeth z, the full-tooth model of the double-base bevel helical bevel gear can be established.

According to the parameters of the double-base bevel helical bevel gear given in Table 1, the process of establishing the model is shown in Figure 4.

Table 1 Parameters of double base bevel spiral bevel gear

According to the above steps, the tooth profile surface of the double-base bevel spiral bevel gear can be established, and the single-tooth entity of the double-base bevel spiral bevel gear can be established, and the array can be equally divided according to the number of teeth z, and then the driving wheel and the passive gear of the double-base bevel spiral bevel gear can be established. The wheel meshing model is shown in Figure 5.

Claims (1)

1. a method for designing a double-base conical spiral bevel gear tooth profile, is characterized in that comprising the following steps: 1) The double-base bevel spiral bevel gear has 5 cone surfaces, 4 cone angles, and 2 base cones; the normal tooth profile of a single tooth consists of a top circle, a dedendum circle, a drive tooth side tooth profile curve, and a non-drive tooth profile. Tooth flank tooth profile curve and tooth direction line; the five cone surfaces are face cone, root cone, pitch cone, back cone and front cone, and the four cone angles are face angle, root angle, pitch angle, dorsal angle; 2) The tooth surface Ω _{1 of} the driving tooth side of the double base bevel spiral bevel gear is tangent to the base cone with the base cone angle δ _bd at OP ₁ . The arc lines M ₁ N ₁ and M ₂ N ₂ will form the driving tooth side and the non-driving tooth side tooth surface of the spiral bevel gear in space. Since the tooth profile angles on both sides of the spiral bevel gear are different, the base at the beginning of the tooth surface on both sides different cones; 3) Through coordinate transformation, the spherical involute of the large end tooth profile of the double base bevel spiral bevel gear is transformed into the spherical coordinate system. After derivation, the large end tooth profile of the drive tooth side of the double base bevel spiral bevel gear is in the spherical coordinate system The equation is 4) Driving tooth side indexing circular deflection angle and the drive tooth side base cone angle δ _bd are obtained by the following formula δ _bd ＝δ′-arctan[(1-α _d )tanδ′] (3) 5) In the spherical coordinate system, the equation of the tooth profile at the large end of the non-driving tooth side of the double-base bevel helical bevel gear is 6) Indexing circular angle of non-driving tooth side and the base cone angle δ _bc of the non-driving tooth side are obtained by the following formula δ _bc = δ′-arctan{[1-arccos(kcosα _d )]tanδ′} (6) The tooth direction line of the double-base bevel spiral bevel gear is formed by the relative position of the wheel blank and the cutter head; 7) According to the helix angle β required by the design and the radius r ₀ of the cutter head selected during processing, determine the positional relationship between the center position of the milling cutter head and the center of the crown axis, and then transform the tooth direction line equation into spherical coordinates through coordinate transformation system, in the spherical coordinate system, the tooth line equation is In formula (7), is the tooth line deflection angle, and the two included angles S and j are obtained by the following formula <mrow> <mi>S</mi> <mo>=</mo> <mi>&amp;pi;</mi> <mo>-</mo> <mi>arccos</mi> <mfrac> <mrow> <msubsup> <mi>L</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msup> <mi>R</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>r</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>+</mo> <mo>&amp;lsqb;</mo> <mi>arccos</mi> <mfrac> <mrow> <msubsup> <mi>L</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msup> <mi>R</mi> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>r</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>-</mo> <mi>arccos</mi> <mfrac> <mrow> <msubsup> <mi>L</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>r</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mi>R</mi> <mo>-</mo> <mi>B</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mn>2</mn> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>r</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>&amp;rsqb;</mo> <mo>&amp;times;</mo> <mi>t</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mi>j</mi> <mo>=</mo> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mfrac> <mrow> <msub> <mi>r</mi> <mn>0</mn> </msub> <mi>sin</mi> <mi> </mi> <mi>S</mi> </mrow> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>r</mi> <mn>0</mn> </msub> <mi>cos</mi> <mi> </mi> <mi>S</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow> 1 8) When the cutter head goes from the big end to the small end, the j angle corresponds to an included angle Q, Q=j/sinδ′, therefore, the included angles corresponding to the outermost point and the innermost point are Q ₀ =j ₀ /sinδ' and Q ₁ = j ₁ /sinδ', the difference between the two angles Q ₁ -Q ₀ is the angle that the corresponding double-base bevel tooth curve from the big end to the small end turns in the spherical coordinate system, according to Q ₁ -Q ₀ and the value of rho in formulas (1) and (4) is changed to RB, that is, the tooth profile of the small end of the double-base bevel spiral bevel gear and the tooth profile of the non-driving tooth side in the spherical coordinate system equation; The diameter of the addendum circle of the double base bevel spiral bevel gear is: d _a =m _t z+2h _ad cosδ _d (10) The top cone angles of the driving tooth side and the non-driving tooth side of the double base bevel spiral bevel gear are: δ _ad = δ _d + θ _fd (11) δ _ac = δ _c + θ _fc (12) The diameter of the dedendum circle of the double base bevel spiral bevel gear is: d _f = m _t zh _fd cosδ _d (13) The root taper angles of the driving tooth side and the non-driving tooth side of the double base bevel spiral bevel gear are: δ _fd = δ _d - θ _fd (14) δ _fc = δ _c - θ _fc (15) According to the above steps, the tooth profile surface of the double base bevel spiral bevel gear is established, and the single tooth entity of the spiral bevel gear is established, and the array is equally divided according to the number of teeth z, then the overall model of the double base bevel spiral bevel gear is established, and the double base bevel spiral bevel gear is completed Tooth shape design; The labels in each step are: z - the number of teeth B - tooth width β——helix angle Σ——shaft intersection angle rho——polar radius of spherical coordinate system theta——the angle between the polar radius of the spherical coordinate system and the positive direction of the z-axis phi——From the perspective of the positive z-axis, the angle turned from the x-axis to OS in a counterclockwise direction, where S is the projection of point P on the xOy plane m _t ——end modulus d——The diameter of the indexing circle R——outer cone distance R _m —— Midpoint taper distance α _d —— tooth profile angle of drive tooth side α _c —— tooth profile angle of non-driving tooth side k——tooth profile angle coefficient δ ₁ —— sub-cone angle ——The circular deflection angle of the drive tooth side indexing ——Indicating circular angle of non-driving tooth side δ _ad ——cone angle of drive tooth side δ _ac ——Cone angle of non-driving tooth side δ _fd —— root taper angle of drive tooth side δ _fc —— Root taper angle of non-driving tooth side δ _bd —— base cone angle of drive tooth side δ _bc —— base cone angle of non-driving tooth side δ’——pitch angle ——tooth line deflection angle d _a ——diameter of gear addendum circle d _f —— gear dedendum circle diameter r ₀ —— cutter head radius L ₁ ——the distance from the center of the cutterhead to the center of the cone S——the declination angle of the tooth line relative to the center of the cutter head j——the declination angle of the tooth direction line relative to the center of the cone top β _p —— dedendum deflection angle θ _fd —— Root angle of drive tooth side θ _fc —— root angle of non-driving tooth side h _{ad ——drive} tooth side addendum height h _{ac ——Height} of addendum on non-driving tooth side h _fd —— root height of driving tooth side h _fc —— root height of non-driving tooth side x—coefficient of height variation x _t —coefficient of tangential displacement δ _f ——cone angle of dedendum δ _a ——cone angle of addendum δ _d ——Cone sub-angle of drive tooth side δ _c ——Cone sub-angle of non-driving tooth side t - variable.