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

A kind of design method of biradical cone spiral bevel gear form of gear tooth Download PDF

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CN104196981B
CN104196981B CN201410512635.XA CN201410512635A CN104196981B CN 104196981 B CN104196981 B CN 104196981B CN 201410512635 A CN201410512635 A CN 201410512635A CN 104196981 B CN104196981 B CN 104196981B
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angle
tooth
cone
bevel gear
base
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CN104196981A (en
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肖望强
潘天龙
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Xiamen Zhenwei Technology Co ltd
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Xiamen University
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    • FMECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16HGEARING
    • F16H55/00Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02Toothed members; Worms
    • F16H55/17Toothed wheels
    • FMECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16HGEARING
    • F16H55/00Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02Toothed members; Worms
    • F16H55/08Profiling
    • F16H55/088Profiling with corrections on tip or foot of the teeth, e.g. addendum relief for better approach contact
    • FMECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16HGEARING
    • F16H55/00Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02Toothed members; Worms
    • F16H55/08Profiling
    • F16H2055/086Silent gear profiles
    • FMECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16HGEARING
    • F16H55/00Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02Toothed members; Worms
    • F16H55/08Profiling
    • F16H2055/0866Profiles for improving radial engagement of gears, e.g. chamfers on the tips of the teeth

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  • Engineering & Computer Science (AREA)
  • General Engineering & Computer Science (AREA)
  • Mechanical Engineering (AREA)
  • Gears, Cams (AREA)

Abstract

一种双基锥螺旋伞齿轮齿形的设计方法,涉及齿轮。双基锥螺旋伞齿轮设有5个锥面、4个锥角、2个基锥;单个轮齿法向齿廓由齿顶圆、齿根圆、驱动侧齿形曲线、非驱动侧齿形曲线、齿向线组成;所述5个锥面分别为面锥、根锥、节锥、背锥和前锥,所述4个锥角分别为面角、根角、节角、背角。针对车辆等以单向传动为主的齿轮传动机构,改善了螺旋伞齿轮的性能,提供在驱动面采用大齿形角,在非驱动面采用小齿形角,既增大了驱动方向齿轮承载能力,又避免了齿顶变尖,能有效提高螺旋伞齿轮承载能力、疲劳寿命和输出扭矩,并降低传动振动和噪声的一种双基锥螺旋伞齿轮齿形的设计方法。

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

一种双基锥螺旋伞齿轮齿形的设计方法A Design Method of Double Base Conical Helical Bevel Gear Tooth Profile

技术领域technical field

本发明涉及齿轮,特别是涉及一种双基锥螺旋伞齿轮齿形的设计方法。The invention relates to gears, in particular to a method for designing the tooth profile of a double base bevel spiral bevel gear.

背景技术Background technique

螺旋伞齿轮与直齿和斜齿伞齿轮相比较,具有重合度大、接触点处齿面的相对曲率半径大、曲面接触区域易于控制、对误差不太敏感等优点,在相交轴线和交错轴线的传动中,得到了广泛的应用。随着齿轮传动中高速重载的发展趋势,传统螺旋伞齿轮已经难以满足设备的要求。尤其在车辆传动部件中,对螺旋伞齿轮输出扭矩的要求越来越高。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.

当齿形角增大时,齿轮的弯曲承载能力和接触承载能力都显著增强,但是随着齿形角的增大,齿轮齿顶逐渐变尖,当齿顶厚度小于1.5m(m为齿轮模数)时,齿顶容易在受载时发生折断,导致齿轮传动失效(李华敏,大齿形角渐开线齿轮传动的质量指标分析,哈尔滨工业大学学报,1987,Z1:86-98)。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).

发明内容Contents of the invention

本发明的目的是针对车辆等以单向传动为主的齿轮传动机构,为了改善螺旋伞齿轮的性能,提供在驱动面采用大齿形角,在非驱动面采用小齿形角,既增大了驱动方向齿轮承载能力,又避免了齿顶变尖,能有效提高螺旋伞齿轮承载能力、疲劳寿命和输出扭矩,并降低传动振动和噪声的一种双基锥螺旋伞齿轮齿形的设计方法。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 .

本发明包括以下步骤:The present invention comprises the following steps:

1)双基锥螺旋伞齿轮设有5个锥面、4个锥角、2个基锥;单个轮齿法向齿廓由齿顶圆、齿根圆、驱动侧齿形曲线、非驱动侧齿形曲线、齿向线组成;所述5个锥面分别为面锥、根锥、节锥、背锥和前锥,所述4个锥角分别为面角、根角、节角、背角;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)双基锥螺旋伞齿轮驱动侧齿面Ω1与基圆锥角为δbd的基锥相切于OP1,当Ω1沿基锥做纯滚动时,平面上以O为回转中心的圆弧线如M1N1与M2N2将在空间形成螺旋伞齿轮驱动侧与非驱动侧齿面,由于螺旋伞齿轮两侧齿形角不同,因此两侧齿面开始处的基锥不同;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)通过坐标变换,将双基锥螺旋伞齿轮大端齿廓球面渐开线转换到球面坐标系,经推导,在球坐标系下双基锥螺旋伞齿轮驱动侧齿面大端齿廓在球坐标系下方程为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)驱动齿侧分度圆偏角和驱动侧基圆锥角δbd由下式求得4) Driving tooth side indexing circular deflection angle and the driving side base cone angle δ bd are obtained by the following formula

δbd=δ′-arctan[(1-αd)tanδ′] (3)δ bd =δ′-arctan[(1-α d )tanδ′] (3)

5)双基锥螺旋伞齿轮非驱动侧齿面大端齿廓在球坐标系下方程为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)非驱动齿侧分度圆偏角和非驱动齿侧基圆锥角δbc由下式求得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

δbc=δ′-arctan{[1-arccos(kcosαd)]tanδ′} (6)δ bc = δ′-arctan{[1-arccos(kcosα d )]tanδ′} (6)

双基锥螺旋伞齿轮的齿向线由轮坯和刀盘的相对位置形成;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)根据设计要求的螺旋角β及加工时所选用的刀盘半径r0,确定铣刀盘中心位置与轮冠轴心的位置关系,然后通过坐标变换,将齿向线方程转换到球面坐标系,在球坐标系下,其齿向线方程为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

式(7)中,为齿向线偏角,两个夹角S和j由下式求得In formula (7), is the tooth line deflection angle, and the two included angles S and j are obtained by the following formula

8)当刀盘由大端走到小端时,j角对应一个夹角Q,Q=j/sinδ′,因此,最外点及最内点所对应的夹角分别为Q0=j0/sinδ′和Q1=j1/sinδ′,两角之差Q1-Q0即为从大端到小端相应的双基锥齿形曲线在球坐标系中转过的角度,根据Q1-Q0并将式(1)、(4)中rho的值变为R-B,即可求得双基锥螺旋伞齿轮轮齿小端驱动侧与非驱动侧齿廓在球坐标系下的方程;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 ;

双基锥螺旋伞齿轮齿顶圆直径为:The diameter of the addendum circle of the double base bevel spiral bevel gear is:

da=mtz+2hadcosδd (10)d a =m t z+2h ad cosδ d (10)

双基锥螺旋伞齿轮驱动侧和非驱动侧顶锥角为:The top cone angles of the driving side and non-driving side of the double-base bevel spiral bevel gear are:

δad=δdfd (11)δ ad = δ d + θ fd (11)

δac=δcfc (12)δ ac = δ c + θ fc (12)

双基锥螺旋伞齿轮齿根圆直径为:The diameter of the dedendum circle of the double base bevel spiral bevel gear is:

df=mtz-hfdcosδd (13)d f = m t zh fd cosδ d (13)

双基锥螺旋伞齿轮驱动侧和非驱动侧根锥角为:The root cone angles of the driving side and the non-driving side of the double base bevel spiral bevel gear are:

δfd=δdfd (14)δ fd = δ d - θ fd (14)

δfc=δcfc (15)δ fc = δ c - θ fc (15)

按上述步骤建立双基锥螺旋伞齿轮的齿廓曲面,并建立螺旋伞齿轮单齿实体,按照齿数z进行等分阵列,则可建立双基锥螺旋伞齿轮整体模型,完成双基锥螺旋伞齿轮齿形的设计。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.

各步骤中的标记为:The labels in each step are:

z——齿数z - the number of teeth

B——齿宽B - tooth width

β——螺旋角β——helix angle

Σ——轴交角Σ——shaft intersection angle

rho——球坐标系极径,图2中的 rho—the polar radius of the spherical coordinate system, in Figure 2

theta——有向线段与z轴正向的夹角theta - directed line segment Angle with positive z axis

phi——从正z轴来看自x轴按逆时针方向转到OS所转过的角,这里S为点P在xOy面上的投影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

mt——端面模数m t ——end modulus

d——分度圆直径d——The diameter of the indexing circle

R——外锥距R——outer cone distance

Rm——中点锥距R m —— Midpoint taper distance

αd——驱动侧齿形角α d —— tooth profile angle of driving side

αc——非驱动侧齿形角α c —— tooth profile angle of non-driving side

k——齿形角系数k——tooth profile angle coefficient

δ1——分锥角δ 1 —— sub-cone angle

——驱动齿侧分度圆偏角 ——The circular deflection angle of the drive tooth side indexing

——非驱动齿侧分度圆偏角 ——Indicating circular angle of non-driving tooth side

δad——驱动侧顶锥角δ ad ——cone angle of driving side

δac——非驱动侧顶锥角δ ac ——cone angle of non-driving side

δfd——驱动侧根锥角δ fd ——drive lateral root taper angle

δfc——非驱动侧根锥角δ fc ——cone angle of non-driving side root

δbd——驱动侧基圆锥角δ bd ——drive side base cone angle

δbc——非驱动侧基圆锥角δ bc ——cone angle of non-driving side base

δ’——节锥角δ’——pitch angle

——齿向线偏角 ——tooth line deflection angle

da——齿轮齿顶圆直径d a ——diameter of gear addendum circle

df——齿轮齿根圆直径d f —— gear dedendum circle diameter

r0——刀盘半径r 0 —— cutter head radius

L1——刀盘中心到锥顶中心的距离L 1 ——the distance from the center of the cutterhead to the center of the cone

S——齿向线的相对于刀盘中心的圆心偏角S——the declination angle of the tooth line relative to the center of the cutter head

j——齿向线的相对于锥顶中心的圆心偏角j——the declination angle of the tooth direction line relative to the center of the cone top

βp——齿根偏角β p —— dedendum deflection angle

θfd——驱动侧齿根角θ fd —— Drive side dedendum angle

θfc——非驱动侧齿根角θ fc —— root angle of non-driving side

had——驱动侧齿顶高h ad ——drive side addendum height

hac——非驱动侧齿顶高h ac ——Height of addendum on non-driving side

hfd——驱动侧齿根高h fd ——drive side dedendum height

hfc——非驱动侧齿根高h fc ——Dedendum height of non-drive side

x——高度变位系数x—coefficient of height variation

xt——切向变位系数。x t —coefficient of tangential displacement.

由于双基锥螺旋伞齿轮的驱动侧和非驱动侧具有不同的基锥、顶锥和根椎等齿轮参数,与传统螺旋伞齿轮的几何特性和设计方法完全不同,因此需要建立一种双基锥螺旋伞齿轮的齿形设计方法,为进行刀具设计、加工制造、齿根弯曲疲劳强度计算和齿面接触疲劳强度计算奠定基础。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.

本发明针对车辆等以单向传动为主的齿轮传动机构,改善了螺旋伞齿轮的性能,提供在驱动面采用大齿形角,在非驱动面采用小齿形角,既增大了驱动方向齿轮承载能力,又避免了齿顶变尖,能有效提高螺旋伞齿轮承载能力、疲劳寿命和输出扭矩,并降低传动振动和噪声的一种双基锥螺旋伞齿轮齿形的设计方法。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.

附图说明Description of drawings

图1为双基锥螺旋伞齿轮锥面和锥角。Figure 1 shows the cone surface and cone angle of a double base bevel spiral bevel gear.

图2为双基锥螺旋伞齿轮驱动侧、非驱动侧球面渐开线形成。Figure 2 shows the formation of spherical involutes on the driving side and non-driving side of the double base bevel spiral bevel gear.

图3为双基锥螺旋伞齿轮刀盘与轮坯相对位置示意图。Fig. 3 is a schematic diagram of the relative positions of the double-base conical spiral bevel gear cutter head and the wheel blank.

图4为双基锥螺旋伞齿轮的建模过程。Figure 4 shows the modeling process of the double base bevel spiral bevel gear.

图5为双基锥螺旋伞齿轮副啮合模型。Figure 5 is the meshing model of the double-base conical spiral bevel gear pair.

在图中,各标记为:1——双基锥螺旋伞齿轮驱动侧齿形,2——双基锥螺旋伞齿轮非驱动侧齿形,3——双基锥螺旋伞齿轮齿顶,4——双基锥螺旋伞齿轮齿根,5——双基锥螺旋伞齿轮齿向线。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.

具体实施方式detailed description

参见图1~5,双基锥螺旋伞齿轮设有5个锥面、4个锥角、2个基锥;单个轮齿法向齿廓由齿顶圆、齿根圆、驱动侧齿形曲线、非驱动侧齿形曲线、齿向线组成,所述5个锥面分别为面锥、根锥、节锥、背锥和前锥,所述4个锥角分别为面角、根角、节角、背角;如图1所示。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.

双基锥螺旋伞齿轮驱动侧齿面Ω1与基圆锥角为δbd的第1基锥(在图2中表示为基锥11)相切于OP1,当Ω1沿基锥1做纯滚动时,平面上以O为回转中心的圆弧线如M1N1与M2N2将在空间形成螺旋伞齿轮驱动侧与非驱动侧齿面。由于螺旋伞齿轮两侧齿形角不同,所以两侧齿面开始处的基锥不同,如图2所示。在图2中还标出第2基锥(在图2中表示为基锥21)。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 ).

通过坐标变换,将双基锥螺旋伞齿轮大端齿廓球面渐开线转换到球面坐标系,经推导,在球坐标系下双基锥螺旋伞齿轮驱动侧齿面大端齿廓在球坐标系下方程为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

驱动齿侧分度圆偏角和驱动侧基圆锥角δbd可由下式求得Driving gear flank indexing circular deflection angle and the driving side base cone angle δ bd can be obtained by the following formula

δbd=δ′-arctan[(1-αd)tanδ′] (3)δ bd =δ′-arctan[(1-α d )tanδ′] (3)

双基锥螺旋伞齿轮非驱动侧齿面大端齿廓在球坐标系下方程为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

非驱动齿侧分度圆偏角和非驱动齿侧基圆锥角δbc可由下式求得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

δbc=δ′-arctan{[1-arccos(kcosαd)]tanδ′} (6)δ bc = δ′-arctan{[1-arccos(kcosα d )]tanδ′} (6)

双基锥螺旋伞齿轮的齿向线是由轮坯和刀盘的相对位置形成的,如图3所示。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.

根据设计要求的螺旋角β及加工时所选用的刀盘半径r0,确定铣刀盘中心位置与轮冠轴心的位置关系,然后通过坐标变换,将齿向线方程转换到球面坐标系。在球坐标系下,其齿向线方程为: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:

式中,为齿向线偏角,两个夹角S和j可由下式求得In the formula, is the tooth line deflection angle, and the two included angles S and j can be obtained by the following formula

由图3中所示的刀盘位置可知,当刀盘由大端走到小端时,j角对应一个夹角Q,Q=j/sinδ′,因此,最外点及最内点所对应的夹角分别为Q0=j0/sinδ′和Q1=j1/sinδ′。两角之差Q1-Q0即为从大端到小端相应的双基锥齿形曲线在球坐标系中转过的角度,根据Q1-Q0并将式(1)、(4)中rho的值变为R-B,即可求得双基锥螺旋伞齿轮轮齿小端驱动侧与非驱动侧齿廓在球坐标系下的方程。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.

双基锥螺旋伞齿轮齿顶圆直径为:The diameter of the addendum circle of the double base bevel spiral bevel gear is:

da=mtz+2hadcosδd (10)d a =m t z+2h ad cosδ d (10)

双基锥螺旋伞齿轮驱动侧和非驱动侧顶锥角为:The top cone angles of the driving side and non-driving side of the double-base bevel spiral bevel gear are:

δad=δdfd (11)δ ad = δ d + θ fd (11)

δac=δcfc (12)δ ac = δ c + θ fc (12)

双基锥螺旋伞齿轮齿根圆直径为:The diameter of the dedendum circle of the double base bevel spiral bevel gear is:

df=mtz-hfdcosδd (13)d f = m t zh fd cosδ d (13)

双基锥螺旋伞齿轮驱动侧和非驱动侧根锥角为:The root cone angles of the driving side and the non-driving side of the double base bevel spiral bevel gear are:

δfd=δdfd (14)δ fd = δ d - θ fd (14)

δfc=δcfc (15)δ fc = δ c - θ fc (15)

按上述将步骤可以建立双基锥螺旋伞齿轮的齿廓曲面,并建立螺旋伞齿轮单齿实体,按照齿数z进行等分阵列,则可建立双基锥螺旋伞齿轮整体模型。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.

根据双基锥螺旋伞齿轮的设计要求,选择合适的模数、法面驱动侧、非驱动侧齿形角、齿顶高系数、变位系数、径向间隙系数、齿数、螺旋角、旋向等基本设计参数,然后根据换算关系,计算各参数。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)根据端面模数mt、齿数z、齿宽B、螺旋角β、轴交角Σ、驱动侧齿形角αd、非驱动侧齿形角αc、变位系数x等参数可以求得驱动齿侧分度圆偏角驱动齿侧基圆锥角δbd、非驱动齿侧分度圆偏角和非驱动齿侧基圆锥角δbc等参数,根据齿形方程(1)~(6),可以建立双基锥螺旋伞齿轮大端驱动侧齿形、大端非驱动侧齿形、小端驱动侧齿形和小端非驱动侧齿形。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)按照上述参数,根据方程(10)~(15),可以建立双基锥螺旋伞齿轮大端齿根圆弧、大端齿顶圆弧、小端齿根圆弧和小端齿顶圆弧。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)按照上述参数,根据方程(7)~(9),可以建立双基锥螺旋伞齿轮驱动侧齿向线和非驱动侧齿向线。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)生成第一个双基锥螺旋伞齿轮单齿的实体。d) Create the first entity of a single tooth of a double base bevel spiral bevel gear.

e)按照齿数z进行等分阵列,则可建立双基锥螺旋伞齿轮全齿模型。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.

按照表1给定的双基锥螺旋伞齿轮的参数,建立模型过程如图4所示。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.

表1双基锥螺旋伞齿轮参数Table 1 Parameters of double base bevel spiral bevel gear

按上述将步骤可以建立双基锥螺旋伞齿轮的齿廓曲面,并建立双基锥螺旋伞齿轮单齿实体,按照齿数z进行等分阵列,则可建立双基锥螺旋伞齿轮主动轮、被动轮啮合模型,如图5所示。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)

1.一种双基锥螺旋伞齿轮齿形的设计方法,其特征在于包括以下步骤:1. a method for designing a double-base conical spiral bevel gear tooth profile, is characterized in that comprising the following steps: 1)双基锥螺旋伞齿轮设有5个锥面、4个锥角、2个基锥;单个轮齿法向齿廓由齿顶圆、齿根圆、驱动齿侧齿形曲线、非驱动齿侧齿形曲线、齿向线组成;所述5个锥面分别为面锥、根锥、节锥、背锥和前锥,所述4个锥角分别为面角、根角、节角、背角;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)双基锥螺旋伞齿轮驱动齿侧齿面Ω1与基圆锥角为δbd的基锥相切于OP1,当Ω1沿基锥做纯滚动时,平面上以O为回转中心的圆弧线M1N1与M2N2将在空间形成螺旋伞齿轮驱动齿侧与非驱动齿侧齿面,由于螺旋伞齿轮两侧齿形角不同,因此两侧齿面开始处的基锥不同;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)通过坐标变换,将双基锥螺旋伞齿轮大端齿廓球面渐开线转换到球坐标系,经推导,双基锥螺旋伞齿轮驱动齿侧齿面大端齿廓在球坐标系下的方程为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)驱动齿侧分度圆偏角和驱动齿侧基圆锥角δbd由下式求得4) Driving tooth side indexing circular deflection angle and the drive tooth side base cone angle δ bd are obtained by the following formula δbd=δ′-arctan[(1-αd)tanδ′] (3)δ bd =δ′-arctan[(1-α d )tanδ′] (3) 5)双基锥螺旋伞齿轮非驱动齿侧齿面大端齿廓在球坐标系下方程为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)非驱动齿侧分度圆偏角和非驱动齿侧基圆锥角δbc由下式求得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 δbc=δ′-arctan{[1-arccos(kcosαd)]tanδ′} (6)δ bc = δ′-arctan{[1-arccos(kcosα d )]tanδ′} (6) 双基锥螺旋伞齿轮的齿向线由轮坯和刀盘的相对位置形成;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)根据设计要求的螺旋角β及加工时所选用的刀盘半径r0,确定铣刀盘中心位置与轮冠轴心的位置关系,然后通过坐标变换,将齿向线方程转换到球坐标系,在球坐标系下,其齿向线方程为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 式(7)中,为齿向线偏角,两个夹角S和j由下式求得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>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 <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)当刀盘由大端走到小端时,j角对应一个夹角Q,Q=j/sinδ′,因此,最外点及最内点所对应的夹角分别为Q0=j0/sinδ′和Q1=j1/sinδ′,两角之差Q1-Q0即为从大端到小端相应的双基锥齿形曲线在球坐标系中转过的角度,根据Q1-Q0并将式(1)、(4)中rho的值变为R-B,即求得双基锥螺旋伞齿轮轮齿小端驱动齿侧与非驱动齿侧齿廓在球坐标系下的方程;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; 双基锥螺旋伞齿轮齿顶圆直径为:The diameter of the addendum circle of the double base bevel spiral bevel gear is: da=mtz+2hadcosδd (10)d a =m t z+2h ad cosδ d (10) 双基锥螺旋伞齿轮驱动齿侧和非驱动齿侧顶锥角为:The top cone angles of the driving tooth side and the non-driving tooth side of the double base bevel spiral bevel gear are: δad=δdfd (11)δ ad = δ d + θ fd (11) δac=δcfc (12)δ ac = δ c + θ fc (12) 双基锥螺旋伞齿轮齿根圆直径为:The diameter of the dedendum circle of the double base bevel spiral bevel gear is: df=mtz-hfdcosδd (13)d f = m t zh fd cosδ d (13) 双基锥螺旋伞齿轮驱动齿侧和非驱动齿侧根锥角为:The root taper angles of the driving tooth side and the non-driving tooth side of the double base bevel spiral bevel gear are: δfd=δdfd (14)δ fd = δ d - θ fd (14) δfc=δcfc (15)δ fc = δ c - θ fc (15) 按上述步骤建立双基锥螺旋伞齿轮的齿廓曲面,并建立螺旋伞齿轮单齿实体,按照齿数z进行等分阵列,则建立双基锥螺旋伞齿轮整体模型,完成双基锥螺旋伞齿轮齿形的设计;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; 各步骤中的标记为:The labels in each step are: z——齿数z - the number of teeth B——齿宽B - tooth width β——螺旋角β——helix angle Σ——轴交角Σ——shaft intersection angle rho——球坐标系极径rho——polar radius of spherical coordinate system theta——球坐标系极径与z轴正向的夹角theta——the angle between the polar radius of the spherical coordinate system and the positive direction of the z-axis phi——从正z轴来看自x轴按逆时针方向转到OS所转过的角,这里S为点P在xOy面上的投影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 mt——端面模数m t ——end modulus d——分度圆直径d——The diameter of the indexing circle R——外锥距R——outer cone distance Rm——中点锥距R m —— Midpoint taper distance αd——驱动齿侧齿形角α d —— tooth profile angle of drive tooth side αc——非驱动齿侧齿形角α c —— tooth profile angle of non-driving tooth side k——齿形角系数k——tooth profile angle coefficient δ1——分锥角δ 1 —— sub-cone angle ——驱动齿侧分度圆偏角 ——The circular deflection angle of the drive tooth side indexing ——非驱动齿侧分度圆偏角 ——Indicating circular angle of non-driving tooth side δad——驱动齿侧顶锥角δ ad ——cone angle of drive tooth side δac——非驱动齿侧顶锥角δ ac ——Cone angle of non-driving tooth side δfd——驱动齿侧根锥角δ fd —— root taper angle of drive tooth side δfc——非驱动齿侧根锥角δ fc —— Root taper angle of non-driving tooth side δbd——驱动齿侧基圆锥角δ bd —— base cone angle of drive tooth side δbc——非驱动齿侧基圆锥角δ bc —— base cone angle of non-driving tooth side δ’——节锥角δ’——pitch angle ——齿向线偏角 ——tooth line deflection angle da——齿轮齿顶圆直径d a ——diameter of gear addendum circle df——齿轮齿根圆直径d f —— gear dedendum circle diameter r0——刀盘半径r 0 —— cutter head radius L1——刀盘中心到锥顶中心的距离L 1 ——the distance from the center of the cutterhead to the center of the cone S——齿向线的相对于刀盘中心的圆心偏角S——the declination angle of the tooth line relative to the center of the cutter head j——齿向线的相对于锥顶中心的圆心偏角j——the declination angle of the tooth direction line relative to the center of the cone top βp——齿根偏角β p —— dedendum deflection angle θfd——驱动齿侧齿根角θ fd —— Root angle of drive tooth side θfc——非驱动齿侧齿根角θ fc —— root angle of non-driving tooth side had——驱动齿侧齿顶高h ad ——drive tooth side addendum height hac——非驱动齿侧齿顶高h ac ——Height of addendum on non-driving tooth side hfd——驱动齿侧齿根高h fd —— root height of driving tooth side hfc——非驱动齿侧齿根高h fc —— root height of non-driving tooth side x——高度变位系数x—coefficient of height variation xt——切向变位系数x t —coefficient of tangential displacement δf——齿根的圆锥角δ f ——cone angle of dedendum δa——齿顶的圆锥角δ a ——cone angle of addendum δd——驱动齿侧分锥角δ d ——Cone sub-angle of drive tooth side δc——非驱动齿侧分锥角δ c ——Cone sub-angle of non-driving tooth side t——变量。t - variable.
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