多面体螺旋曲线啮合齿轮变速器几何学设计理论研究
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摘要
在现代机械中，齿轮变速器被广泛用于各种原动机和工作机之间，起着匹配转速和传递转矩的作用。传统齿轮变速器采用传统曲面啮合齿轮作为传动部件，在结构和大小方面受到一定的限制，其输出数目比较有限。
空间曲线啮合齿轮是一种基于空间曲线啮合原理的新型齿轮。目前，该齿轮已经具备了一定的设计理论和制造基础，其啮合理论、重合度、弹性变形准则、弯曲强度校核准则、制造工艺以及安装精度分析等均取得一定的进展，已开始逐步应用至微小变速器等领域。其中，螺旋曲线啮合齿轮是最常见的空间曲线啮合齿轮，其技术最为成熟。
本文基于螺旋曲线啮合齿轮提出了一种多轴输出的微小变速器——多面体螺旋曲线啮合齿轮变速器，并系统地研究了其几何学设计理论。该变速器的主要特征包括齿轮组的应用、传动轴的多面体三维几何布局、单输入多输出的并联式单级传动和串-并联混合式的多级传动方式等。该变速器具有尺寸小，质量轻，灵活性高等特点，特别适合作为微小机械装置的变速机构。
具体而言，本文的研究内容主要包括以下四个方面：
1．钩杆半径不相等的螺旋曲线啮合齿轮设计理论。根据空间曲线啮合齿轮传动基于钩杆接触线的建模思路，求解了空间曲线啮合齿轮的啮合方程、接触线方程和中心线方程；给出了螺旋曲线啮合齿轮的相关方程，并在设定主、从动钩杆半径不相等的条件下，推导得到了主、从动钩杆的中心线方程，完成了主、从动齿轮的三维实体建模，并通过软件仿真和实物验证了其传动的连续性。
2．多面体螺旋曲线啮合齿轮变速器的布局规范与安装尺寸链计算公式。分析了螺旋曲线啮合齿轮副在斜交轴、正交轴与平行轴三种情况下的安装尺寸链；设计了单主动轮-多从动轮的螺旋曲线啮合齿轮组；研究了齿轮组的初始安装角度；分析了齿轮组的啮合情况，进而提出了完成啮合的必要条件；在齿轮组的基础上，提出了多面体螺旋曲线啮合齿轮变速器的概念与安装规范，并推导得到传动系统和箱体结构尺寸的设计公式。
3．多面体螺旋曲线啮合齿轮变速器几何约束与不干涉条件的计算公式。分别从接触线、齿轮副和齿轮组三个层面对螺旋曲线啮合齿轮传动的几何约束进行了定义和计算，并相应地提出了接触线互不干涉条件、齿轮副互不干涉条件和齿轮组互不干涉条件；以带有螺旋曲线啮合齿轮组的正四面体螺旋曲线啮合齿轮变速器作为例子，运用本文研究的几何约束和不干涉条件进行设计计算，确定了各坐标系位置参数的选择范围。
4．螺旋曲线啮合齿轮基于滑动率的坐标系位置参数选用准则及计算公式。参照传统的空间共轭曲面啮合齿轮，定义并计算了空间共轭曲线、空间曲线啮合齿轮接触线及螺旋曲线啮合齿轮接触线的滑动率；分析了螺旋曲线啮合齿轮中接触线滑动率的单调性与可行域，并确定了其最优啮合条件与坐标系位置参数选用准则及计算公式。
Gear reducers are widely used between prime motors and working mechanisms withinmodern machines to change values and directions of rotation speeds. Commonly usingtraditional space curve meshing gears as transmission components, classical gear reducers arestuck with their structures and sizes as well as their limited output shafts.
The Space Curve Meshing Wheel (SCMW) is an innovative gear, which has possesseddecent design theory and fabrication foundation. Progresses have been achieved in manyaspects like meshing equations, contact ratio, elastic deformation criterion, manufacturingtechnology and assembly accuracy. It has begun to be applied in the fields like micro reducers.The Helix Curve Meshing Wheel (HCMW) is the most common SCMW with the mostmatured technology.
Based on the HCMW, a micro reducer with multiple output shafts named PolyhedralHelix Curve Meshing Wheel Reducer (Polyhedral HCMWR) was proposed in thisdissertation, and its design theory was studied from geometrical perspectives. The features ofthis reducer include the application of the transmission train, the polyhedral3D geometriclayout of the input and output shafts, and single input and multiple outputs in either singlestage reducer of parallel transmissions or multiple stage reducer of mixed transmissions. Witha small size, light weight and high flexibility, this reducer is very suitable to serve in a smalldevice.
Specifically speaking, the followings were concerned in the dissertation:
(1) Design of the HCMW with unequal tine radii. The design fundamental based on thecontact curves were expounded, the meshing equation of the SCMW was solved in the spacecurve meshing coordinates as well as its contact curves and center curves. As an illustration,equations of the HCMW were derived, the driving and driven wheels were simulated with3Dmodeling software, and its transmission continuity was testified by virtual simulation andpractical experiment.
(2) Component layout specification and installation dimension chains formulation for thePolyhedral HCMWR. The installation chains of the HCMW were analyzed according to threedifferent cases (intersecting axis, vertical axis, or parallel axis), respectively; the HCMW trainwith a single driving wheel and multiple driven wheels was designed; the initial installationangle of the HCMW train was studied; the mesh requirement was proposed after the analysisof the mesh status in the HCMW train; and based on the HCMW train, the concept of the Polyhedral HCMWR and its component layout specification were defined, and the designformulas of the transmission system and the box structure were derived.
(3) Geometrical constrain demarcations and interference-proof condition deductions inthe Polyhedral HCMWR. The geometric constraints were defined and calculated in the threeaspects: contact curves, HCMW pair and HCMW train; the interference-proof conditionswere proposed in the same three aspects. As a practical example, the parameter selection of atetrahedron HCMWR with HCMW trains was illustrated to show the process of determiningthe feasibility range of the position parameters.
(4) Selection criterion and calculation formulas of the position parameters based on thesliding rates. With the slide rates of classical space conjugate surfaces referred, the slide ratesof two conjugate curves in the space were defined and calculated, and subsequently those ofthe contact curves in the SCMW and the HCMW were derived; and optimal meshingcondition and position-parameter selection criterion of the HCMW were obtained based onthe analysis of the monotonicity and feasible region of its sliding rates.
空间曲线啮合齿轮是一种基于空间曲线啮合原理的新型齿轮。目前，该齿轮已经具备了一定的设计理论和制造基础，其啮合理论、重合度、弹性变形准则、弯曲强度校核准则、制造工艺以及安装精度分析等均取得一定的进展，已开始逐步应用至微小变速器等领域。其中，螺旋曲线啮合齿轮是最常见的空间曲线啮合齿轮，其技术最为成熟。
本文基于螺旋曲线啮合齿轮提出了一种多轴输出的微小变速器——多面体螺旋曲线啮合齿轮变速器，并系统地研究了其几何学设计理论。该变速器的主要特征包括齿轮组的应用、传动轴的多面体三维几何布局、单输入多输出的并联式单级传动和串-并联混合式的多级传动方式等。该变速器具有尺寸小，质量轻，灵活性高等特点，特别适合作为微小机械装置的变速机构。
具体而言，本文的研究内容主要包括以下四个方面：
1．钩杆半径不相等的螺旋曲线啮合齿轮设计理论。根据空间曲线啮合齿轮传动基于钩杆接触线的建模思路，求解了空间曲线啮合齿轮的啮合方程、接触线方程和中心线方程；给出了螺旋曲线啮合齿轮的相关方程，并在设定主、从动钩杆半径不相等的条件下，推导得到了主、从动钩杆的中心线方程，完成了主、从动齿轮的三维实体建模，并通过软件仿真和实物验证了其传动的连续性。
2．多面体螺旋曲线啮合齿轮变速器的布局规范与安装尺寸链计算公式。分析了螺旋曲线啮合齿轮副在斜交轴、正交轴与平行轴三种情况下的安装尺寸链；设计了单主动轮-多从动轮的螺旋曲线啮合齿轮组；研究了齿轮组的初始安装角度；分析了齿轮组的啮合情况，进而提出了完成啮合的必要条件；在齿轮组的基础上，提出了多面体螺旋曲线啮合齿轮变速器的概念与安装规范，并推导得到传动系统和箱体结构尺寸的设计公式。
3．多面体螺旋曲线啮合齿轮变速器几何约束与不干涉条件的计算公式。分别从接触线、齿轮副和齿轮组三个层面对螺旋曲线啮合齿轮传动的几何约束进行了定义和计算，并相应地提出了接触线互不干涉条件、齿轮副互不干涉条件和齿轮组互不干涉条件；以带有螺旋曲线啮合齿轮组的正四面体螺旋曲线啮合齿轮变速器作为例子，运用本文研究的几何约束和不干涉条件进行设计计算，确定了各坐标系位置参数的选择范围。
4．螺旋曲线啮合齿轮基于滑动率的坐标系位置参数选用准则及计算公式。参照传统的空间共轭曲面啮合齿轮，定义并计算了空间共轭曲线、空间曲线啮合齿轮接触线及螺旋曲线啮合齿轮接触线的滑动率；分析了螺旋曲线啮合齿轮中接触线滑动率的单调性与可行域，并确定了其最优啮合条件与坐标系位置参数选用准则及计算公式。
Gear reducers are widely used between prime motors and working mechanisms withinmodern machines to change values and directions of rotation speeds. Commonly usingtraditional space curve meshing gears as transmission components, classical gear reducers arestuck with their structures and sizes as well as their limited output shafts.
The Space Curve Meshing Wheel (SCMW) is an innovative gear, which has possesseddecent design theory and fabrication foundation. Progresses have been achieved in manyaspects like meshing equations, contact ratio, elastic deformation criterion, manufacturingtechnology and assembly accuracy. It has begun to be applied in the fields like micro reducers.The Helix Curve Meshing Wheel (HCMW) is the most common SCMW with the mostmatured technology.
Based on the HCMW, a micro reducer with multiple output shafts named PolyhedralHelix Curve Meshing Wheel Reducer (Polyhedral HCMWR) was proposed in thisdissertation, and its design theory was studied from geometrical perspectives. The features ofthis reducer include the application of the transmission train, the polyhedral3D geometriclayout of the input and output shafts, and single input and multiple outputs in either singlestage reducer of parallel transmissions or multiple stage reducer of mixed transmissions. Witha small size, light weight and high flexibility, this reducer is very suitable to serve in a smalldevice.
Specifically speaking, the followings were concerned in the dissertation:
(1) Design of the HCMW with unequal tine radii. The design fundamental based on thecontact curves were expounded, the meshing equation of the SCMW was solved in the spacecurve meshing coordinates as well as its contact curves and center curves. As an illustration,equations of the HCMW were derived, the driving and driven wheels were simulated with3Dmodeling software, and its transmission continuity was testified by virtual simulation andpractical experiment.
(2) Component layout specification and installation dimension chains formulation for thePolyhedral HCMWR. The installation chains of the HCMW were analyzed according to threedifferent cases (intersecting axis, vertical axis, or parallel axis), respectively; the HCMW trainwith a single driving wheel and multiple driven wheels was designed; the initial installationangle of the HCMW train was studied; the mesh requirement was proposed after the analysisof the mesh status in the HCMW train; and based on the HCMW train, the concept of the Polyhedral HCMWR and its component layout specification were defined, and the designformulas of the transmission system and the box structure were derived.
(3) Geometrical constrain demarcations and interference-proof condition deductions inthe Polyhedral HCMWR. The geometric constraints were defined and calculated in the threeaspects: contact curves, HCMW pair and HCMW train; the interference-proof conditionswere proposed in the same three aspects. As a practical example, the parameter selection of atetrahedron HCMWR with HCMW trains was illustrated to show the process of determiningthe feasibility range of the position parameters.
(4) Selection criterion and calculation formulas of the position parameters based on thesliding rates. With the slide rates of classical space conjugate surfaces referred, the slide ratesof two conjugate curves in the space were defined and calculated, and subsequently those ofthe contact curves in the SCMW and the HCMW were derived; and optimal meshingcondition and position-parameter selection criterion of the HCMW were obtained based onthe analysis of the monotonicity and feasible region of its sliding rates.
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