CN103198226B

CN103198226B - A kind of cycloid bevel gears Analysis of Vibration Characteristic method considering to rub

Info

Publication number: CN103198226B
Application number: CN201310136180.1A
Authority: CN
Inventors: 刘志峰; 张涛; 罗兵
Original assignee: Beijing University of Technology
Current assignee: Beijing University of Technology
Priority date: 2013-04-18
Filing date: 2013-04-18
Publication date: 2016-04-20
Anticipated expiration: 2033-04-18
Also published as: CN103198226A

Abstract

The invention discloses a method for analyzing vibration characteristics of a cycloidal bevel gear considering friction, belonging to the field of nonlinear vibration analysis of gears. The method includes: (1) Simplifying the cycloidal bevel gear system into a torsional vibration system model of a gear pair; (2) Introduce the friction factor into the torsional vibration system of the cycloidal bevel gear pair, and obtain the torsional vibration balance equations of the driving and driven gears respectively by the Lagrange principle; (3) Dimensionless the torsional vibration balance equation of the gear pair, and obtain Dimensionless equation of the vibration model; (4) According to the dimensionless equation of the vibration model of the cycloid bevel gear pair, research and analyze the law of the friction factor and the vibration characteristics of the cycloid bevel gear pair. The method of the present invention not only provides theoretical support for the vibration and noise reduction of the bevel gear transmission system, but also provides a reference for manufacturing cycloid bevel gears with high precision and high load capacity, and improving the transmission accuracy, service life and reliability of the cycloid bevel gear transmission system .

Description

A Method for Analyzing Vibration Characteristics of Cycloidal Bevel Gears Considering Friction

技术领域technical field

本发明属于齿轮非线性振动分析领域，涉及一种摆线锥齿轮振动特性分析方法，更具体涉及一种考虑摩擦的摆线锥齿轮振动特性分析方法。The invention belongs to the field of nonlinear vibration analysis of gears, relates to a cycloid bevel gear vibration characteristic analysis method, and more particularly relates to a cycloid bevel gear vibration characteristic analysis method considering friction.

背景技术Background technique

摆线锥齿轮作为螺旋锥齿轮的两大齿制之一，具有传动平稳、承载能力高、硬齿面刮削技术等特点，从而特别适用于大功率和大扭矩重载传动领域，是重型高档数控机床、汽车传动系统、航空航天装备等重要领域中的核心传动部件。随着机械传动系统日益朝着高速、精密等方向发展，摆线锥齿轮作为传动系统中的关键传动部件，其振动特性对于传动系统性能的影响将会更显著。因此，研究摆线锥齿轮振动特性对于设计和制造高精度、高耐久性、低噪声等高效传动部件有着重要的实用价值和学术意义。Cycloidal bevel gear, as one of the two major tooth systems of spiral bevel gear, has the characteristics of stable transmission, high load capacity, and hard tooth surface scraping technology, so it is especially suitable for high-power and high-torque heavy-duty transmission fields. It is a heavy-duty high-end CNC Core transmission components in important fields such as machine tools, automotive transmission systems, and aerospace equipment. With the development of mechanical transmission systems towards high speed and precision, cycloidal bevel gears are the key transmission components in the transmission system, and their vibration characteristics will have a more significant impact on the performance of the transmission system. Therefore, studying the vibration characteristics of cycloidal bevel gears has important practical value and academic significance for the design and manufacture of high-precision, high-durability, low-noise and other high-efficiency transmission components.

近年来，国内外许多学者以非线性振动理论为基础，以齿轮啮合过程中的时变刚度和齿侧间隙等非线性因素为核心，对齿轮系统的非线性振动进行了较广泛而深入的研究。但近年来研究表明：啮合齿间摩擦也是齿轮非线性振动影响因素之一。但目前多数研究都是针对直齿圆柱齿轮的，而对摆线锥齿轮的研究较少。如何在摆线锥齿轮副动力学模型中引入摩擦因素并正确分析摩擦因素对摆线锥齿轮振动特性的影响规律，仍然具有很大的研究潜力，研究和探索新的动力学模型和分析方法，仍是该领域的重要内容之一。研究摩擦因素对摆线锥齿轮振动特性的影响，不仅为锥齿轮传动系统的减振降噪提供理论支持，而且为制造高精度、高承载能力的摆线锥齿轮，提升摆线锥齿轮传动系统的传动精度、寿命及可靠性提供参考。In recent years, many scholars at home and abroad have carried out extensive and in-depth research on the nonlinear vibration of the gear system based on the theory of nonlinear vibration and focusing on nonlinear factors such as time-varying stiffness and backlash in the gear meshing process. . However, recent studies have shown that the friction between meshing teeth is also one of the factors affecting the nonlinear vibration of gears. However, most of the current researches are on spur gears, but less on cycloidal bevel gears. How to introduce friction factors into the dynamic model of cycloidal bevel gear pairs and correctly analyze the influence of friction factors on the vibration characteristics of cycloidal bevel gears still has great research potential, research and explore new dynamic models and analysis methods, is still one of the important contents in this field. Studying the influence of friction factors on the vibration characteristics of cycloidal bevel gears will not only provide theoretical support for the vibration and noise reduction of bevel gear transmission systems, but also provide high-precision, high-load-carrying cycloidal bevel gears for the manufacture of cycloidal bevel gear transmission systems. The transmission accuracy, life and reliability provide a reference.

发明内容Contents of the invention

本发明的目的是提供一种考虑摩擦的摆线锥齿轮振动特性分析方法，探索摩擦因素对摆线锥齿轮振动特性的影响规律，从而为锥齿轮传动系统的减振降噪提供理论支持，而且为制造高精度、高承载能力的摆线锥齿轮，提升摆线锥齿轮传动系统的传动精度、寿命及可靠性提供参考。The purpose of the present invention is to provide a cycloidal bevel gear vibration characteristic analysis method considering friction, to explore the influence of friction factors on the cycloidal bevel gear vibration characteristics, thereby providing theoretical support for the vibration and noise reduction of the bevel gear transmission system, and It provides a reference for manufacturing cycloidal bevel gears with high precision and high load capacity, and improving the transmission accuracy, life and reliability of cycloidal bevel gear transmission system.

本发明是采用以下技术手段实现的：The present invention is realized by adopting the following technical means:

1、将摆线锥齿轮系统简化处理成为齿轮副的扭转振动系统模型；1. Simplify the cycloidal bevel gear system into a torsional vibration system model of the gear pair;

2、在摆线锥齿轮副的扭转振动系统模型中引入摩擦因素，由Lagrange原理分别得到主、从动齿轮的扭转振动平衡方程。平衡方程如下：2. Introduce the friction factor into the torsional vibration system model of the cycloidal bevel gear pair, and obtain the torsional vibration balance equations of the driving and driven gears respectively by the Lagrange principle. The balance equation is as follows:

${I I}_{p p} {\overset{\overset{\cdot \cdot \cdot &Center Dot;}{~ ~}}{θ θ}}_{p p} + + {λ λ}_{p p} \overset{~ ~}{C C} ((\overset{~ ~}{t t})) [[{λ λ}_{p p} {\overset{\overset{\cdot \cdot}{~ ~}}{θ θ}}_{p p} - - {λ λ}_{g g} {\overset{\overset{\cdot \cdot}{~ ~}}{θ θ}}_{g g} - - \overset{\overset{\cdot \cdot}{~ ~}}{e e} ((\overset{~ ~}{t t}))]] + + {λ λ}_{p p} \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} (({λ λ}_{p p} {\overset{~ ~}{θ θ}}_{p p} - - {λ λ}_{g g} {\overset{~ ~}{θ θ}}_{g g} - - \overset{~ ~}{e e} ((\overset{~ ~}{t t})))) = = {\overset{~ ~}{T T}}_{p p} - - {\overset{~ ~}{T T}}_{f f,, p p} ((\overset{~ ~}{t t}))$

${I I}_{g g} {\overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{θ θ}}_{g g} - - {λ λ}_{g g} \overset{~ ~}{C C} ((\overset{~ ~}{t t})) [[{λ λ}_{p p} {\overset{\overset{\cdot &Center Dot;}{~ ~}}{θ θ}}_{p p} - - {λ λ}_{g g} {\overset{\overset{\cdot &Center Dot;}{~ ~}}{θ θ}}_{g g} - - \overset{\overset{\cdot &Center Dot;}{~ ~}}{e e} ((\overset{~ ~}{t t}))]] - - {λ λ}_{p p} \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} (({λ λ}_{p p} {\overset{~ ~}{θ θ}}_{p p} - - {λ λ}_{g g} {\overset{~ ~}{θ θ}}_{g g} - - \overset{~ ~}{e e} ((\overset{~ ~}{t t})))) = = - - {\overset{~ ~}{T T}}_{g g} + + {\overset{~ ~}{T T}}_{f f,, g g} ((\overset{~ ~}{t t}))$

其中，~为量纲符号；I_i(i＝p,g)为主、被动齿轮的转动惯量；λ_i(i＝p,g)为主、被动齿轮的齿轮方向旋转半径；θ_i(i＝p,g)为主、被动齿轮的角位移；T_i(i＝p,g)为主、被动齿轮上的扭矩；T_f,i(i＝p,g)为主、被动齿轮上的摩擦力矩；C(t)为齿轮副啮合阻尼；K(t)为齿轮副啮合刚度；f(·)为间隙函数；e(t)为齿轮副静态传递误差函数。Among them, ~ is the dimension symbol; I _i (i=p, g) is the moment of inertia of the main and driven gear; λ _i (i=p, g) is the rotation radius of the main and driven gear in the gear direction; θ _i (i =p,g) the angular displacement of the main and driven gears; T _i (i=p,g) the torque on the main and driven gears; T _f,i (i=p,g) the torque on the main and driven gears Friction torque; C(t) is the meshing damping of the gear pair; K(t) is the meshing stiffness of the gear pair; f(·) is the clearance function; e(t) is the static transmission error function of the gear pair.

3、将齿轮副的扭转振动平衡方程无量纲化，得到振动模型的无量纲化形式；3. The torsional vibration balance equation of the gear pair is dimensionless, and the dimensionless form of the vibration model is obtained;

3.1.引入新变量 ${\tilde{x}}_{i} = λ_{i} {\tilde{θ}}_{i} (i = p, g),$ $m_{i} = \frac{I_{i}}{λ_{i}^{2}} (i = p, g),$ $\tilde{F} = \frac{{\tilde{T}}_{i}}{λ_{i}} (i = p, g),$ 代入步骤2中的平衡方程中得：3.1. Introducing new variables ${\tilde{x}}_{i} = λ_{i} {\tilde{θ}}_{i} (i = p, g),$ $m_{i} = \frac{I_{i}}{λ_{i}^{2}} (i = p, g),$ $\tilde{f} = \frac{{\tilde{T}}_{i}}{λ_{i}} (i = p, g),$ Substitute into the balance equation in step 2 to get:

${m m}_{p p} {\overset{\overset{\cdot \cdot \cdot \cdot}{~ ~}}{x x}}_{p p} + + \overset{~ ~}{C C} ((\overset{~ ~}{t t})) [[{\overset{\overset{\cdot \cdot}{~ ~}}{x x}}_{p p} - - {\overset{\overset{\cdot \cdot}{~ ~}}{x x}}_{g g} - - \overset{\overset{\cdot \cdot}{~ ~}}{e e} ((\overset{~ ~}{t t}))]] + + \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} (({\overset{~ ~}{x x}}_{p p} - - {\overset{~ ~}{x x}}_{g g} - - \overset{~ ~}{e e} ((\overset{~ ~}{t t})))) = = \overset{~ ~}{F f} - - {\overset{~ ~}{F f}}_{f f} ((\overset{~ ~}{t t}))$

${m m}_{g g} {\overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{x x}}_{g g} - - \overset{~ ~}{C C} ((\overset{~ ~}{t t})) [[{\overset{\overset{\cdot &Center Dot;}{~ ~}}{x x}}_{p p} - - {\overset{\overset{\cdot &Center Dot;}{~ ~}}{x x}}_{g g} - - \overset{\overset{\cdot &Center Dot;}{~ ~}}{e e} ((\overset{~ ~}{t t}))]] - - \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} (({\overset{~ ~}{x x}}_{p p} - - {\overset{~ ~}{x x}}_{g g} - - \overset{~ ~}{e e} ((\overset{~ ~}{t t})))) = = \overset{~ ~}{- - F f} + + {\overset{~ ~}{F f}}_{f f} ((\overset{~ ~}{t t}))$

其中，x_i(i＝p,g)为主、被动齿轮轮齿动态传递误差；m_i(i＝p,g)为主、被动齿轮的质量；F为外载荷；F_f为平均摩擦力；Among them, x _i (i=p,g) is the main force and the dynamic transmission error of the driven gear teeth; m _i (i=p,g) is the main force and the mass of the driven gear; F is the external load; F _f is the average friction force ;

3.2.引入新变量将步骤3.1中的两平衡方程式相减并合并得到：3.2. Introducing new variables Subtract and combine the two balance equations in step 3.1 to get:

$M m \overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{x x} + + \overset{~ ~}{C C} ((\overset{~ ~}{t t})) \overset{\overset{\cdot &Center Dot;}{~ ~}}{x x} + + \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} ((\overset{~ ~}{x x})) = = ((\overset{~ ~}{F f} - - {\overset{~ ~}{F f}}_{f f} ((\overset{~ ~}{t t})))) - - M m \overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{e e} ((\overset{~ ~}{t t}))$

其中， ${\tilde{F}}_{f} (\tilde{t}) = μ (\tilde{K} (\tilde{t}) \tilde{f} (\tilde{x}) + \tilde{C} (\tilde{t}) \overset{\overset{\cdot}{~}}{x})$ 带入上式中得：in, ${\tilde{f}}_{f} (\tilde{t}) = μ (\tilde{K} (\tilde{t}) \tilde{f} (\tilde{x}) + \tilde{C} (\tilde{t}) \overset{\overset{&Center Dot;}{~}}{x})$ Bring it into the above formula to get:

$M m \overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{x x} + + ((11 + + μ μ)) \overset{~ ~}{C C} ((\overset{~ ~}{t t})) \overset{\overset{\cdot &Center Dot;}{~ ~}}{x x} + + ((11 + + μ μ)) \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} ((\overset{~ ~}{x x})) = = \overset{~ ~}{F f} - - M m \overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{e e} ((\overset{~ ~}{t t}))$

其中，x为啮合点位移；M为齿轮相对质量；μ为摩擦因子。Among them, x is the displacement of the meshing point; M is the relative mass of the gear; μ is the friction factor.

3.3.将刚度、阻尼和静态传递误差按傅里叶级数展开，并只考虑主谐波形式有：3.3. Expand the stiffness, damping and static transmission error according to the Fourier series, and only consider the main harmonic form:

且令：最终得到振动模型的无量纲化形式为：And order: The final dimensionless form of the vibration model is:

其中，α为谐波阻尼系数；ρ为谐波刚度系数；γ为传递误差因子；ξ为阻尼因子；为相位角；ω_n为固有频率；ω为激励频率，b为齿轮间隙。Among them, α is the harmonic damping coefficient; ρ is the harmonic stiffness coefficient; γ is the transmission error factor; ξ is the damping factor; is the phase angle; ω _n is the natural frequency; ω is the excitation frequency, and b is the gear gap.

4、根据摆线锥齿轮副振动模型的无量纲化方程式，研究和分析摩擦因子μ与摆线锥齿轮振动特性的规律。4. According to the dimensionless equation of the vibration model of the cycloidal bevel gear pair, research and analyze the law of the friction factor μ and the vibration characteristics of the cycloidal bevel gear pair.

本发明的目的是针对摩擦对摆线锥齿轮振动特性的影响，提出了一种考虑摩擦的摆线锥齿轮振动特性分析方法。特点在于从摆线锥齿轮副的扭转振动模型出发，在其动力学平衡方程中引入摩擦因子得到含摩擦因子的无量纲化方程，最后研究和分析摩擦因子μ与摆线锥齿轮振动特性的规律。发明内容包括三部分。在第一部分中，主要是建立摆线锥齿轮副的扭转振动模型；在第二部分中，主要是推导得到含摩擦因子的摆线锥齿轮副振动模型的无量纲化方程式；在第三部分中，主要是根据摆线锥齿轮副振动模型的无量纲化方程式，研究和分析摩擦因子μ与摆线锥齿轮振动特性的规律。The purpose of the present invention is to propose a cycloidal bevel gear vibration characteristic analysis method considering friction in view of the influence of friction on the vibration characteristics of cycloidal bevel gears. The characteristic is that starting from the torsional vibration model of the cycloidal bevel gear pair, the friction factor is introduced into the dynamic balance equation to obtain a dimensionless equation containing the friction factor, and finally the law of the friction factor μ and the vibration characteristics of the cycloidal bevel gear is studied and analyzed . The content of the invention includes three parts. In the first part, the torsional vibration model of the cycloidal bevel gear pair is mainly established; in the second part, the dimensionless equation of the vibration model of the cycloidal bevel gear pair with friction factor is derived; in the third part , mainly according to the dimensionless equation of the vibration model of the cycloidal bevel gear pair, to study and analyze the law of the friction factor μ and the vibration characteristics of the cycloidal bevel gear pair.

附图说明Description of drawings

图1考虑摩擦的摆线锥齿轮振动特性分析方法流程图Fig.1 Flowchart of analysis method of cycloidal bevel gear vibration characteristics considering friction

图2本发明实施例摆线锥齿轮副动力学模型图Fig. 2 cycloidal bevel gear pair dynamics model figure of the embodiment of the present invention

图3本发明实施例间隙函数模型图Fig. 3 gap function model diagram of the embodiment of the present invention

图4本发明实施例摩擦因子影响啮合点振动曲线图Fig. 4 is the curve diagram of friction factor affecting the vibration of meshing point in the embodiment of the present invention

具体实施方式detailed description

本发明实施例的一种考虑摩擦的摆线锥齿轮振动特性分析方法流程图如图1所示，下面结合流程图对本发明的步骤作详细说明。具体实施步骤如下：A flow chart of a vibration characteristic analysis method of a cycloidal bevel gear considering friction in an embodiment of the present invention is shown in FIG. 1 , and the steps of the present invention will be described in detail below in conjunction with the flow chart. The specific implementation steps are as follows:

第一步：将摆线锥齿轮系统简化处理成为齿轮副的扭转振动系统模型；Step 1: Simplify the cycloidal bevel gear system into a torsional vibration system model of the gear pair;

本实施例以航空用摆线锥齿轮副为研究对象，其具体参数见表1。考虑齿面间摩擦的摆线锥齿轮副动力学模型如图2所示。在该模型中，假设两齿轮的支撑刚度较大，且不考虑传动轴、支承轴承和箱体等的弹性变形对摆线锥齿轮系统的影响，最终将摆线锥齿轮系统简化处理成为齿轮副的扭转振动系统模型。In this embodiment, the cycloidal bevel gear pair for aviation is taken as the research object, and its specific parameters are shown in Table 1. The dynamic model of the cycloidal bevel gear pair considering the friction between the tooth surfaces is shown in Fig. 2. In this model, it is assumed that the supporting rigidity of the two gears is relatively large, and the influence of the elastic deformation of the transmission shaft, support bearing and box on the cycloidal bevel gear system is not considered, and finally the cycloidal bevel gear system is simplified as a gear pair The model of the torsional vibration system.

表1摆线锥齿轮系统参数Table 1 Cycloidal bevel gear system parameters

第二步：在摆线锥齿轮副的扭转振动系统模型中引入摩擦因素，由Lagrange原理分别得到主、从动齿轮的扭转振动平衡方程。平衡方程如下：The second step: Introduce the friction factor into the torsional vibration system model of the cycloidal bevel gear pair, and obtain the torsional vibration balance equations of the driving and driven gears respectively by the Lagrange principle. The balance equation is as follows:

${I I}_{p p} {\overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{θ θ}}_{p p} + + {λ λ}_{p p} \overset{~ ~}{C C} ((\overset{~ ~}{t t})) [[{λ λ}_{p p} {\overset{\overset{\cdot &Center Dot;}{~ ~}}{θ θ}}_{p p} - - {λ λ}_{g g} {\overset{\overset{\cdot &Center Dot;}{~ ~}}{θ θ}}_{g g} - - \overset{\overset{\cdot &Center Dot;}{~ ~}}{e e} ((\overset{~ ~}{t t}))]] + + {λ λ}_{p p} \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} (({λ λ}_{p p} {\overset{~ ~}{θ θ}}_{p p} - - {λ λ}_{g g} {\overset{~ ~}{θ θ}}_{g g} - - \overset{~ ~}{e e} ((\overset{~ ~}{t t})))) = = {\overset{~ ~}{T T}}_{p p} - - {\overset{~ ~}{T T}}_{f f,, p p} ((\overset{~ ~}{t t}))$

${I I}_{g g} {\overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{θ θ}}_{g g} - - {λ λ}_{g g} \overset{~ ~}{C C} ((\overset{~ ~}{t t})) [[{λ λ}_{p p} {\overset{\overset{\cdot &Center Dot;}{~ ~}}{θ θ}}_{p p} - - {λ λ}_{g g} {\overset{\overset{\cdot &Center Dot;}{~ ~}}{θ θ}}_{g g} - - \overset{\overset{\cdot &Center Dot;}{~ ~}}{e e} ((\overset{~ ~}{t t}))]] - - {λ λ}_{g g} \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} (({λ λ}_{p p} {\overset{~ ~}{θ θ}}_{p p} - - {λ λ}_{g g} {\overset{~ ~}{θ θ}}_{g g} - - \overset{~ ~}{e e} ((\overset{~ ~}{t t})))) = = {- - \overset{~ ~}{T T}}_{g g} + + {\overset{~ ~}{T T}}_{f f,, g g} ((\overset{~ ~}{t t}))$

第三步：将齿轮副的扭转振动平衡方程无量纲化，得到振动模型的无量纲化形式；Step 3: Dimensionless the torsional vibration balance equation of the gear pair to obtain the dimensionless form of the vibration model;

1）、令 ${\tilde{x}}_{i} = λ_{i} {\tilde{θ}}_{i} (i = p, g),$ $m_{i} = \frac{I_{i}}{λ_{i}^{2}} (i = p, g),$ $\tilde{F} = \frac{{\tilde{T}}_{i}}{λ_{i}} (i = p, g),$ ${\tilde{F}}_{f} (\tilde{t}) = \frac{{\tilde{T}}_{f, i} (\tilde{t})}{λ_{i}} (i = p, g)$ 分别代入第二步中的平衡方程中得到：1), order ${\tilde{x}}_{i} = λ_{i} {\tilde{θ}}_{i} (i = p, g),$ $m_{i} = \frac{I_{i}}{λ_{i}^{2}} (i = p, g),$ $\tilde{f} = \frac{{\tilde{T}}_{i}}{λ_{i}} (i = p, g),$ ${\tilde{f}}_{f} (\tilde{t}) = \frac{{\tilde{T}}_{f, i} (\tilde{t})}{λ_{i}} (i = p, g)$ Substitute into the balance equation in the second step to get:

${m m}_{p p} {\overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{x x}}_{p p} + + \overset{~ ~}{C C} ((\overset{~ ~}{t t})) [[{\overset{\overset{\cdot &Center Dot;}{~ ~}}{x x}}_{p p} - - {\overset{\overset{\cdot &Center Dot;}{~ ~}}{x x}}_{g g} - - \overset{\overset{\cdot &Center Dot;}{~ ~}}{e e} ((\overset{~ ~}{t t}))]] + + \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} (({\overset{~ ~}{x x}}_{p p} - - {\overset{~ ~}{x x}}_{g g} - - \overset{~ ~}{e e} ((\overset{~ ~}{t t})))) = = \overset{~ ~}{F f} - - {\overset{~ ~}{F f}}_{f f} ((\overset{~ ~}{t t}))$

${m m}_{g g} {\overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{x x}}_{g g} - - \overset{~ ~}{C C} ((\overset{~ ~}{t t})) [[{\overset{\overset{\cdot &Center Dot;}{~ ~}}{x x}}_{p p} - - {\overset{\overset{\cdot \cdot}{~ ~}}{x x}}_{g g} - - \overset{\overset{\cdot \cdot}{~ ~}}{e e} ((\overset{~ ~}{t t}))]] - - \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} (({\overset{~ ~}{x x}}_{p p} - - {\overset{~ ~}{x x}}_{g g} - - \overset{~ ~}{e e} ((\overset{~ ~}{t t})))) = = \overset{~ ~}{- - F f} + + {\overset{~ ~}{F f}}_{f f} ((\overset{~ ~}{t t}))$

2）、将1）中两式分别除m_p、m_g并相减得到：2) Divide m _p and m _g from the two formulas in 1) and subtract them to get:

${\overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{x x}}_{p p} - - {\overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{x x}}_{g g} + + \frac{{m m}_{p p} + + {m m}_{g g}}{{m m}_{p p} {m m}_{g g}} \overset{~ ~}{C C} ((\overset{~ ~}{t t})) [[{\overset{\overset{\cdot &Center Dot;}{~ ~}}{x x}}_{p p} - - {\overset{\overset{\cdot &Center Dot;}{~ ~}}{x x}}_{g g} - - \overset{\cdot &Center Dot;}{\overset{~ ~}{e e}} ((\overset{~ ~}{t t}))]] + + \frac{{m m}_{p p} + + {m m}_{g g}}{{m m}_{p p} {m m}_{g g}} \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} (({\overset{~ ~}{x x}}_{p p} - - {\overset{~ ~}{x x}}_{g g} - - \overset{~ ~}{e e} ((\overset{~ ~}{t t})))) = = \frac{{m m}_{p p} + + {m m}_{g g}}{{m m}_{p p} {m m}_{g g}} ((\overset{~ ~}{F f} - - {\overset{~ ~}{F f}}_{f f} ((\overset{~ ~}{t t}))))$

令 $\tilde{x} = {\tilde{x}}_{p} - {\tilde{x}}_{g} - \tilde{e} (\tilde{t}),$ $M = \frac{m_{p} m_{g}}{m_{p} + m_{g}}$ 代入上式中得到：make $\tilde{x} = {\tilde{x}}_{p} - {\tilde{x}}_{g} - \tilde{e} (\tilde{t}),$ $m = \frac{m_{p} m_{g}}{m_{p} + m_{g}}$ Substitute into the above formula to get:

$M m \overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{x x} + + \overset{~ ~}{C C} ((\overset{~ ~}{t t})) \overset{\overset{\cdot \cdot}{~ ~}}{x x} + + \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} ((\overset{~ ~}{x x})) = = ((\overset{~ ~}{F f} - - {\overset{~ ~}{F f}}_{f f} ((\overset{~ ~}{t t})))) - - M m \overset{\overset{\cdot \cdot \cdot \cdot}{~ ~}}{e e} ((\overset{~ ~}{t t}))$

又， ${\tilde{F}}_{f} (\tilde{t}) = μ (\tilde{K} (\tilde{t}) \tilde{f} (\tilde{x}) + \tilde{C} (\tilde{t}) \overset{\cdot}{\tilde{x}})$ 代入上式中得到：again, ${\tilde{f}}_{f} (\tilde{t}) = μ (\tilde{K} (\tilde{t}) \tilde{f} (\tilde{x}) + \tilde{C} (\tilde{t}) \overset{&Center Dot;}{\tilde{x}})$ Substitute into the above formula to get:

$M m \overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{x x} + + ((11 + + μ μ)) \overset{~ ~}{C C} ((\overset{~ ~}{t t})) \overset{\overset{\cdot \cdot}{~ ~}}{x x} + + ((11 + + μ μ)) \overset{~ ~}{K K} ((\overset{~ ~}{t t})) \overset{~ ~}{f f} ((\overset{~ ~}{x x})) = = \overset{~ ~}{F f} - - M m \overset{\overset{\cdot &Center Dot; \cdot &Center Dot;}{~ ~}}{e e} ((\overset{~ ~}{t t}))$

3）、将刚度、阻尼和静态传递误差按傅里叶级数展开，并只考虑主谐波形式，即代入上式得到：3) Expand the stiffness, damping and static transmission errors according to the Fourier series, and only consider the main harmonic form, that is Substitute into the above formula to get:

令 $x = \frac{\tilde{x}}{b},$ $t = ω_{n} \tilde{t},$ $ω_{n} = \sqrt{\frac{k_{m}}{M}},$ $γ = \frac{e_{1}}{b},$ $ξ = \frac{c_{m}}{M ω_{n}},$ $α = \frac{c_{1}}{M ω_{n}},$ $ρ = \frac{k_{1}}{M ω_{n}^{2}},$ $F = \frac{\tilde{F}}{bM ω_{n}^{2}},$ $ω = \frac{\tilde{ω}}{ω_{n}},$ $f (x) = \frac{\tilde{f} (\tilde{x})}{b},$ 代入上式最终得到振动模型的无量纲化形式为：make $x = \frac{\tilde{x}}{b},$ $t = ω_{no} \tilde{t},$ $ω_{no} = \sqrt{\frac{k_{m}}{m}},$ $γ = \frac{e_{1}}{b},$ $ξ = \frac{c_{m}}{m ω_{no}},$ $α = \frac{c_{1}}{m ω_{no}},$ $ρ = \frac{k_{1}}{m ω_{no}^{2}},$ $f = \frac{\tilde{f}}{b ω_{no}^{2}},$ $ω = \frac{\tilde{ω}}{ω_{no}},$ $f (x) = \frac{\tilde{f} (\tilde{x})}{b},$ Substituting the above formula, the final dimensionless form of the vibration model is:

其中， $f (x) = \{\begin{matrix} x - 1 & x &GreaterEqual; 1 \\ 0 & - 1 < x < 1, \\ x + 1 & x \leq - 1 \end{matrix}$ 其模型如图3所示。in, $f (x) = \{\begin{matrix} x - 1 & x &Greater Equal; 1 \\ 0 & - 1 < x < 1, \\ x + 1 & x \leq - 1 \end{matrix}$ Its model is shown in Figure 3.

第四步：根据摆线锥齿轮副振动模型的无量纲化方程式，研究和分析摩擦因子与摆线锥齿轮振动特性的规律。The fourth step: According to the dimensionless equation of the vibration model of the cycloidal bevel gear pair, research and analyze the laws of the friction factor and the vibration characteristics of the cycloidal bevel gear pair.

选定参数ξ＝0.1，α＝0.01，ρ＝0.1，γ＝0.2，F＝2，，和，探究摩擦因子对摆线锥齿轮振动特性的影响。其中当μ分别等于0、0.1、0.2、0.3时，摩擦因子影响齿轮啮合点振动曲线图如图4所示。Selected parameters ξ=0.1, α=0.01, ρ=0.1, γ=0.2, F=2, , and , to investigate the effect of friction factor on the vibration characteristics of cycloidal bevel gears. When μ is equal to 0, 0.1, 0.2, and 0.3 respectively, the friction factor affects the vibration curve of the gear meshing point as shown in Figure 4.

从图4中可以看出，不考虑摩擦，即μ＝0时，位移响应曲线幅值约为3.33，峰值频率在ω=0.9～1.0之间;当μ＝0.1时，位移响应曲线幅值约为3.18，峰值频率在ω＝1.0处；随μ增加到0.2和0.3，其响应幅值分别为3.02和2.84，峰值频率分别为ω＝1.0和ω＝1.1。由此可知，随着摩擦因子的增大，啮合点位移振动幅值随之降低，峰值频率出现漂移，有随之增大的趋势，其它频率点响应值均随摩擦因子增大而减小。摩擦能改变齿轮系统的运动状态，增加系统运动的复杂性。It can be seen from Figure 4 that without considering friction, that is, when μ=0, the amplitude of the displacement response curve is about 3.33, and the peak frequency is between ω=0.9 and 1.0; when μ=0.1, the amplitude of the displacement response curve is about is 3.18, and the peak frequency is at ω=1.0; as μ increases to 0.2 and 0.3, the response amplitudes are 3.02 and 2.84, respectively, and the peak frequency is ω=1.0 and ω=1.1. It can be seen that with the increase of the friction factor, the displacement vibration amplitude of the meshing point decreases, the peak frequency drifts and tends to increase, and the response values of other frequency points decrease with the increase of the friction factor. Friction can change the motion state of the gear system and increase the complexity of the system motion.

通过以上实例分析总结出：本发明方法能够运用于摆线锥齿轮振动特性分析中，并能得出摩擦因子对摆线锥齿轮振动特性的影响规律。本发明方法不仅为锥齿轮传动系统的减振降噪提供理论支持，而且为制造高精度、高承载能力的摆线锥齿轮，提升摆线锥齿轮传动系统的传动精度、寿命及可靠性提供参考。Through the analysis of the above examples, it is concluded that the method of the present invention can be applied to the analysis of the vibration characteristics of cycloidal bevel gears, and the law of the influence of friction factors on the vibration characteristics of cycloidal bevel gears can be obtained. The method of the present invention not only provides theoretical support for the vibration and noise reduction of the bevel gear transmission system, but also provides a reference for manufacturing cycloidal bevel gears with high precision and high load capacity, and improving the transmission accuracy, life and reliability of the cycloidal bevel gear transmission system .

Claims

1. A method for analyzing vibration characteristics of a cycloidal bevel gear in consideration of friction is characterized by comprising the following steps:

1) simplifying and processing the cycloidal bevel gear system into a torsional vibration system model of a gear pair;

2) introducing friction factors into a torsional vibration system model of the cycloid bevel gear pair, and respectively obtaining torsional vibration balance equations of the driving gear and the driven gear according to the Lagrange principle, wherein the balance equations are as follows:

wherein-is a dimension symbol; i is_i(i ═ p, g) is the moment of inertia of the driving and driven gears; lambda [ alpha ]_i(i ═ p, g) represents the gear direction rotation radius of the driving and driven gears; theta_i(i ═ p, g) is the angular displacement of the driving and driven gears; t is_i(i ═ p, g) is the torque on the driving and driven gears; t is_f,i(i ═ p, g) is the friction torque on the driving and driven gears; c (t) is gear pair meshing damping; k (t) is the gear pair meshing stiffness; f (-) is a gap function; e (t) is a gear pair static transfer error function;

3) carrying out dimensionless on a torsional vibration balance equation of the gear pair to obtain a dimensionless form of a torsional vibration system model;

4) researching and analyzing the law of the friction factor mu and the vibration characteristic of the cycloidal bevel gear according to a dimensionless equation of a torsional vibration system model of the cycloidal bevel gear pair;

wherein, the concrete steps of obtaining the dimensionless form of the torsional vibration system model in the step 3) are as follows:

3.1) introduction of New variables Substituting into the balance equation in the step 2 to obtain:

wherein x is_i(i ═ p, g) as driving and driven gearsDynamic transmission error of gear teeth; m is_i(i ═ p, g) represents the mass of the driving and driven gears; f is an external load; f_fIs the average friction;

3.2) introduction of New variablesSubtracting the two equilibrium equations in step 3.1 and combining to obtain:

wherein,taken into the above formula:

wherein x is the displacement of the meshing point; m is the relative mass of the gear; μ is a friction factor;

3.3) expanding the rigidity, the damping and the static transfer error according to Fourier series, and only considering the main harmonic wave form:

and order:the dimensionless form of the vibration model is finally obtained as follows:

wherein α is a harmonic damping coefficient, ρ is a harmonic stiffness coefficient, γ is a transfer error factor, ξ is a damping factor;is a phase angle; omega_nIs the natural frequency; ω is the excitation frequency and b is the gear backlash.