蒙日曲面共轭原理与环面蜗杆珠轮传动技术的研究
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摘要
论文以国家自然科学基金项目“基于Bertrand曲面的传动原理与新型传动技术的研究(No.50275017)”为背景,将目前众多常用的齿轮副廓面提升到蒙日曲面(mouldingsurface)的层面上来认识,在此基础上研究了蒙日曲面共轭的基本原理及其分支——法向圆弧齿轮传动的一般原理,并在解析研究与实验研究相结合的基础上对环面蜗杆珠轮传动进行了初步研究。
蒙日曲面由19世纪法国几何学家蒙日(G.Monge)首先提出,但它的一些优良特性并不为工程界所熟悉。事实上,工程上广泛应用的许多曲面,包括齿廓面,都可以归结为蒙日曲面,这就有必要对蒙日曲面进行再认识和再研究。本论文应用现代微分几何学的曲率带形方法,基于准线及其基本标架,从新的角度来研究了蒙日曲面的形成与几何学性质。指出,准线决定曲面的骨架,母线决定断面形状,并且曲面沿着母线的法线共面。蒙日曲面的形状多样,结构单纯,可以适应各种工程的需要,这就为本论文的研究提供了良好的平台。
蒙日曲面共轭原理是本文重要的基础性工作。依据“面线转化”思路,文中将共轭问题归结到一对准线上来讨论。按照母线的形状特征,蒙日曲面共轭被区分为三种情形,并分别给出各自的共轭条件和微分关系式。在此基础上,应用回转运动群方法进一步研究了共轭条件式的相容性,即共轭的结构条件,并被表示为便于应用的Pythagoras函数形式。作为Oliver理论的继承与发展,文中研究了蒙日曲面的媒介共轭原理与间接展成法,并给出了实现媒介共轭所需的运动条件与几何条件。
法向圆弧齿轮传动的一般原理是蒙日曲面共轭原理的重要分支。从此类传动的特点出发,文中以啮合线为纲、以准面为依托和以准线为骨架,构建出用于研究的理论平台,给出了基于啮合线的共轭条件、限制条件和不干涉条件。在此基础上,系统地研究了平行轴、相交轴和交错轴条件下的三类传动,并构造出双曲型法向圆弧齿轮传动的理论模型,将法向圆弧齿轮传动发展成为可适用于各类轴线配置的一种传动形式,从而形成了较为完整的体系。
作为蒙日曲面共轭原理与齿轮传动技术的交汇点之一,文中提出环面蜗杆珠轮传动。这是一类以钢珠为媒介的法向圆弧齿轮传动的特例,原理上的“点—线‘共轭'”、结构上的“三体两副”以及技术上的点啮合化处理构成这类传动的基本特征。在理论与技术相结合的基础上,对环面蜗杆珠轮传动副的点啮合化与接触域控制、钢珠的冗余运动与受力分析以及误差敏度与适应性等进行了量化研究。在此基础上,针对蜗杆的齿形设计和钢珠的防脱等特殊问题作了讨论,给出了传动副设计方法;讨论了基于数控加工的刀具设计、精度分析与加工方法。原型机的研制与运转接触实验表明,原理正确、技术可行,达到了预期目标。
Using "Research of Transmission Theory and New Transmission Technology based on the Bertrand Surface" supported by National Natural Science Foundation of China as the background,many kinds of tooth flanks of gears are re-understood by promoting them to the concept of moulding surface in the dissertation,and on this basis,the basic principle of moulding-surface conjugation and its branch—the general principle of normal circular-arc gear transmission are investigated,and toroid worm drive with spherical meshing elements are studied by the analytic and experimental method.
Moulding surface was firstly proposed by G.Monge in 19~(th) century,but the engineering society is unfamiliar with some of its surface Characteristics.Actually,many surfaces used widely in the engineering,including tooth flanks,can be boiled down to moulding surface,so it is necessary to re-understand and re-research on this surface.Based on the directrix and basic frame,the generation and geometric properties of moulding surface are studied from a new point of view by the application of Curvature Stripe Method in Modern Differential Geometry. The skeleton and the cross-section of surface are decided respectively by the directrix and the generatrix,and the surface normals along the generatrix are coplanar.The multiformity of shape and the simplicity of structure make moulding surface meet different practical demands, and also provide a good platform for this research.
The basic principle of moulding-surface conjugation is an important theoretical foundation in this dissertation.According to the idea of surface-curve transformation,the conjugate problems are boiled down to the relationships between a pair of directrixes. Because of the difference of generatrix shape,moulding-surface conjugation is divided into three cases,and respective conjugate conditions and differential relational equations are presented.On this basis,Motion Group Method is applied to solve the consistency of conjugate conditions,i.e.the structure condition of conjugation,and it is arranged as a Pythagoras Equation for the convenience.As the succession and development of Oliver's Theorem,the principle of intermediate conjugation and indirect generating method for moulding surface are studied,and the motion condition and the geometric condition are presented to realize this conjugation.
As an important branch of moulding-surface conjugate principle,the general principle of normal circular-arc gear transmission is studied.According to the driving characteristics,a theoretical platform for the research is constituted by taking the meshing line as the program, datum surface as the support and the directrix as the skeleton,and based on the meshing line, the conjugate condition,the limit condition and the non-interference condition are presented, and three kinds of gear drives with parallel axes,intersecting axes and non-parallel non-intersecting axes are systematically studied,a theoretical model of hyperbolic-type normal circular-arc gear drive is proposed.As a result,normal circular-arc gear transmission is developed into one drive suitable for every possibility of axial arrangement,and so an integrated theoretical system is set up.
As one of the combinations of moulding-surface conjugate principle and gear transmission technology,toroid worm drive with spherical meshing elements is proposed.It is a special case of normal circular-arc gear with the steel ball mediums,and has some basic characteristics,such as the point-curve conjugation,the three-body two-pair and the mismatch treatment in the technology,etc.By combining theory with technology,the quantitative researches on toroid worm drive with spherical meshing elements are executed,such as the mismatch treatment and the contact area control,the redundant motion and the force analysis of steel ball,the sensitivity and adaptability to error,etc.On this basis,some special problems, such as the profile design of worm and the anti-dropping of steel-bail,are discussed,and the design method of worm pair is introduced.Aiming at NC machining,the design of tool,the accuracy analysis and the machining method are also discussed.By the development and the running test of prototype machine,the results show that the principle is correct and the techniques are realizable,and the anticipative purposes are achieved.
蒙日曲面由19世纪法国几何学家蒙日(G.Monge)首先提出,但它的一些优良特性并不为工程界所熟悉。事实上,工程上广泛应用的许多曲面,包括齿廓面,都可以归结为蒙日曲面,这就有必要对蒙日曲面进行再认识和再研究。本论文应用现代微分几何学的曲率带形方法,基于准线及其基本标架,从新的角度来研究了蒙日曲面的形成与几何学性质。指出,准线决定曲面的骨架,母线决定断面形状,并且曲面沿着母线的法线共面。蒙日曲面的形状多样,结构单纯,可以适应各种工程的需要,这就为本论文的研究提供了良好的平台。
蒙日曲面共轭原理是本文重要的基础性工作。依据“面线转化”思路,文中将共轭问题归结到一对准线上来讨论。按照母线的形状特征,蒙日曲面共轭被区分为三种情形,并分别给出各自的共轭条件和微分关系式。在此基础上,应用回转运动群方法进一步研究了共轭条件式的相容性,即共轭的结构条件,并被表示为便于应用的Pythagoras函数形式。作为Oliver理论的继承与发展,文中研究了蒙日曲面的媒介共轭原理与间接展成法,并给出了实现媒介共轭所需的运动条件与几何条件。
法向圆弧齿轮传动的一般原理是蒙日曲面共轭原理的重要分支。从此类传动的特点出发,文中以啮合线为纲、以准面为依托和以准线为骨架,构建出用于研究的理论平台,给出了基于啮合线的共轭条件、限制条件和不干涉条件。在此基础上,系统地研究了平行轴、相交轴和交错轴条件下的三类传动,并构造出双曲型法向圆弧齿轮传动的理论模型,将法向圆弧齿轮传动发展成为可适用于各类轴线配置的一种传动形式,从而形成了较为完整的体系。
作为蒙日曲面共轭原理与齿轮传动技术的交汇点之一,文中提出环面蜗杆珠轮传动。这是一类以钢珠为媒介的法向圆弧齿轮传动的特例,原理上的“点—线‘共轭'”、结构上的“三体两副”以及技术上的点啮合化处理构成这类传动的基本特征。在理论与技术相结合的基础上,对环面蜗杆珠轮传动副的点啮合化与接触域控制、钢珠的冗余运动与受力分析以及误差敏度与适应性等进行了量化研究。在此基础上,针对蜗杆的齿形设计和钢珠的防脱等特殊问题作了讨论,给出了传动副设计方法;讨论了基于数控加工的刀具设计、精度分析与加工方法。原型机的研制与运转接触实验表明,原理正确、技术可行,达到了预期目标。
Using "Research of Transmission Theory and New Transmission Technology based on the Bertrand Surface" supported by National Natural Science Foundation of China as the background,many kinds of tooth flanks of gears are re-understood by promoting them to the concept of moulding surface in the dissertation,and on this basis,the basic principle of moulding-surface conjugation and its branch—the general principle of normal circular-arc gear transmission are investigated,and toroid worm drive with spherical meshing elements are studied by the analytic and experimental method.
Moulding surface was firstly proposed by G.Monge in 19~(th) century,but the engineering society is unfamiliar with some of its surface Characteristics.Actually,many surfaces used widely in the engineering,including tooth flanks,can be boiled down to moulding surface,so it is necessary to re-understand and re-research on this surface.Based on the directrix and basic frame,the generation and geometric properties of moulding surface are studied from a new point of view by the application of Curvature Stripe Method in Modern Differential Geometry. The skeleton and the cross-section of surface are decided respectively by the directrix and the generatrix,and the surface normals along the generatrix are coplanar.The multiformity of shape and the simplicity of structure make moulding surface meet different practical demands, and also provide a good platform for this research.
The basic principle of moulding-surface conjugation is an important theoretical foundation in this dissertation.According to the idea of surface-curve transformation,the conjugate problems are boiled down to the relationships between a pair of directrixes. Because of the difference of generatrix shape,moulding-surface conjugation is divided into three cases,and respective conjugate conditions and differential relational equations are presented.On this basis,Motion Group Method is applied to solve the consistency of conjugate conditions,i.e.the structure condition of conjugation,and it is arranged as a Pythagoras Equation for the convenience.As the succession and development of Oliver's Theorem,the principle of intermediate conjugation and indirect generating method for moulding surface are studied,and the motion condition and the geometric condition are presented to realize this conjugation.
As an important branch of moulding-surface conjugate principle,the general principle of normal circular-arc gear transmission is studied.According to the driving characteristics,a theoretical platform for the research is constituted by taking the meshing line as the program, datum surface as the support and the directrix as the skeleton,and based on the meshing line, the conjugate condition,the limit condition and the non-interference condition are presented, and three kinds of gear drives with parallel axes,intersecting axes and non-parallel non-intersecting axes are systematically studied,a theoretical model of hyperbolic-type normal circular-arc gear drive is proposed.As a result,normal circular-arc gear transmission is developed into one drive suitable for every possibility of axial arrangement,and so an integrated theoretical system is set up.
As one of the combinations of moulding-surface conjugate principle and gear transmission technology,toroid worm drive with spherical meshing elements is proposed.It is a special case of normal circular-arc gear with the steel ball mediums,and has some basic characteristics,such as the point-curve conjugation,the three-body two-pair and the mismatch treatment in the technology,etc.By combining theory with technology,the quantitative researches on toroid worm drive with spherical meshing elements are executed,such as the mismatch treatment and the contact area control,the redundant motion and the force analysis of steel ball,the sensitivity and adaptability to error,etc.On this basis,some special problems, such as the profile design of worm and the anti-dropping of steel-bail,are discussed,and the design method of worm pair is introduced.Aiming at NC machining,the design of tool,the accuracy analysis and the machining method are also discussed.By the development and the running test of prototype machine,the results show that the principle is correct and the techniques are realizable,and the anticipative purposes are achieved.
引文
[1]注:在研究之初,作者所在的学科组曾将此类共轭形式称为白川德共轭[21].
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