超声珩齿变幅器动力学特性研究
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摘要
齿轮相连,世界才会动起来。齿轮作为重要传动部件,不仅是我国装备制造业的基础产业,也是国民经济建设各领域的重要基础。随着国民经济的快速发展,性能良好的齿轮受到了行业认可和青睐,因此,对于齿轮制造技术的研究就显得尤为重要。超声珩齿是将超声振动切削技术应用于齿轮精密加工的一项新技术,是对齿轮进行超声复合加工的新探索。利用超声珩齿加工能够提高珩磨效率并有助于齿轮修形,有效减少珩磨轮堵塞和加工过程中的珩磨力、提高加工效率、加工精度并增加珩磨轮寿命,这样,对于超声珩齿复合加工的研究有很好的理论价值和应用前景。要实现超声珩齿,振动系统的设计是必须解决的关键问题之一,本文正是针对现有超声珩齿非谐振设计理论只能设计薄圆环板类齿轮的问题和不足,在查阅大量技术资料和文献的基础上,提出了基于Mindlin中厚板理论设计超声珩齿振动系统的思路方法,为此,国家自然科学基金(No.50975191)在2009年批准资助了本课题组提出的“非谐振单元变幅器设计理论及其齿轮超声剃珩应用”,做为其子课题“超声珩齿变幅器动力学特性”得到了太原市大学生创新创业专题项目(No.110148050)的资助,对超声珩齿变幅器动力学特性进行了如下的研究：
(1)基于Mindlin理论研究了齿轮横向振动的简化模型
齿轮是典型的圆板类零件,振动特性和圆板振动类似。针对很多普通圆柱齿轮中心带孔,厚径比在中厚板理论范围之内(厚径比大于1／5,小于1／2)的结构特点,将其简化为直径等于分度圆的中厚圆环板,基于Mindlin理论,推导了在自由边界条件下横向振动频率方程,通过MATLAB计算固有频率,并与有限元计算结果和实验测试结果对比分析,三者结果相符,从而验证了简化模型的可行性。该结论对超声珩齿振动系统设计具有一定理论指导意义。
(2)超声珩齿变幅器动力学特性研究
为了扩展非谐振设计理论的应用范围,采用非谐振设计思想和Mindlin理论,根据变幅杆及中厚圆环板的力耦合条件,建立了圆锥形、指数形、悬链线形和阶梯形与中厚圆环板组成的变幅器的数学模型,推导了相应变幅器的频率方程和位移振幅方程,利用MATLAB软件求出了变幅器设计参数和位移振幅的数值解。有限元分析及动力学实验得到的动力学谐振频率和位移振幅与数值解一致。在此基础上,为了验证本文方法对传统非谐振设计方法的兼容性,对含有不同厚径比环板的变幅器进行有限元动力学仿真,发现厚径比不论在经典薄板理论范围之内还是在Mindlin中厚板理论范围之内,变幅器动力学特性与理论设计要求的完全一致,这就使变幅器设计由含薄圆环板类齿轮的情况拓展到含中厚圆环板类齿轮的情况,进而完善了非谐振设计理论。
(3)超声珩齿振动系统设计新方法探索
针对非谐振设计理论齿轮边缘振幅较小的问题。借鉴经典全谐振设计理论和非谐振设计思想,将传统工具头部分改造为由齿轮和圆柱形杆组成的齿轮杆,根据相应边界耦合条件建立频率方程和位移振幅方程,利用MATLAB软件求出圆柱形杆参数和中厚圆环板位移振幅数值解,用有限元法对变幅杆和齿轮杆组成系统进行动力学仿真分析,理论解和有限元法都表明中厚圆环板边缘振幅较非谐振设计理论时变大。旨在对超声珩齿振动系统设计方法探索出一条新的设计思路。
The world will move it up only when the gears are connected. Gears as one of the most inportant transmission, is not only the foundation industry of equipment manufacturing industry in our country, but also the important basis of the national economic construction fields. Along with the rapid economic development, gears with satisfactory performance are recognized and favoured by industry. Hence, the research and application of gears manufacturing technology appears especially important. The technology of ultrasonic gear honing has been a new craft of precision machining for gear on the basis of traditional honing process, and it is a new exploration of an ultrasonic compound machining for gears. Ultrasonic gear honing machining is useful in the modification of gear, decrease the stemming problems of plugged wheel and the force of honing, hence, the machining efficiency, machining accuracy and the life of honing wheels can be improved, so, it is of important academic significance and application value to study the ultrasonic gear honing machining. To realize the ultrasonic gear honing, one of the key problem solved firstly is to design the ultrasonic vibration system. The problems and shortages in non-resonant structure design theory (NRST) which only can be used to design gear including thin annular plate was focused to be solved in this paper. On the basis of consulting a large number of technical data and documents, a completely new design method for ultrasonic gear honing vibration system was brought forward based on Mindlin's theory. Just for this reason, the project "Non-resonant structure design theory of Transformer and application of ultrasonic gear shaving honing" which was proposed by our research team was supported by the National Natural Science Foundation of China (No.50975191). As an important sub subject "Dynamical characteristics of transformer in ultrasonic gear honing" was proposed by the Innovation and Entrepreneurial Projects for College Students of Taiyuan (No.110148050). The following researches about dynamical characteristics of transformer in ultrasonic gear honing have been studied:
(1) Models of transverse vibration for gear based on Mindlin's theory
Gear was simplified to moderately thick annular plate (MTAP) with the outer diameters of reference circle, according to the structural characteristics of a cylindrical gear with a hole in the center and its ratio of the thickness to radius in the scope of the moderately thick plate theory, i.e., the ratio of the thickness to radius is greater than1/5and less than1/2. The frequency equations for transverse vibration of the moderately thick annular plate were derived with the free boundary conditions based on Mindlin's theory and solved computationally with the help of MATLAB software. The solutions are in best agreement with those from the finite element method (FEM) and also with the experimental results. The conclusion has certain theoretical significance to guide to design the vibration system of ultrasonic gear honing.
(2)Research on dynamical characteristics of transformer in ultrasonic gear honing
To extended the range of application the NRST, the mathematical models of the transformer which were consists of cone, exponential, catenary, step horn and MTAP were built respectively by means of NRST, Mindlin's theory and force coupling conditions. Then the frequency equations and displacement amplitude equations of corresponding transformer were derived. The design parameters and displacement amplitude of the transformer were solved by using MATLAB software. The results from FEM and dynamical experiment were consistent with the numerical solutions of the frequency equations and displacement amplitude equations. On this basis, in order to verify the compatibility of the proposed transformer design method for the traditional NRST, FEM simulation for the transformer were calculated under various ratio of the thickness to radius. It can be found that dynamical characteristics of transformer are consistent with the theoretical design demands when the ratio of the thickness to radius that covers the range from the CPT to MTAP. The experimental result demonstrates that the method of this paper can extend the transformer design range from gear including thin annular plate to gear including MTAP, and NRST is further improved and perfected.
(3)Exploratory research on the new design method for vibration system in ultrasonic gear horning
According to problem of displacement amplitude on the edge of MTAP is comparatively small in the NRST generally; the whole resonant design theory (WRDT) and NRST ideas were referenced. The traditional tool head was replaced by gear pole which was consists of MTAP and cylindrical pole. The frequency equations and displacement amplitude equations of the gear pole were derived based on force coupling boundary condition. Then the parameter of the cylindrical pole and the displacement amplitude of the MTAP were solved by using MATLAB software, and the dynamical characteristics simulation of the system which was consists of horn and gear pole were got by FEM. It is proved from theoretical solution and FEM that the displacement amplitude on the edge of MTAP is increased than NRST. This part aims to explore a new design ideas for vibration system in ultrasonic gear horning.
(1)基于Mindlin理论研究了齿轮横向振动的简化模型
齿轮是典型的圆板类零件,振动特性和圆板振动类似。针对很多普通圆柱齿轮中心带孔,厚径比在中厚板理论范围之内(厚径比大于1／5,小于1／2)的结构特点,将其简化为直径等于分度圆的中厚圆环板,基于Mindlin理论,推导了在自由边界条件下横向振动频率方程,通过MATLAB计算固有频率,并与有限元计算结果和实验测试结果对比分析,三者结果相符,从而验证了简化模型的可行性。该结论对超声珩齿振动系统设计具有一定理论指导意义。
(2)超声珩齿变幅器动力学特性研究
为了扩展非谐振设计理论的应用范围,采用非谐振设计思想和Mindlin理论,根据变幅杆及中厚圆环板的力耦合条件,建立了圆锥形、指数形、悬链线形和阶梯形与中厚圆环板组成的变幅器的数学模型,推导了相应变幅器的频率方程和位移振幅方程,利用MATLAB软件求出了变幅器设计参数和位移振幅的数值解。有限元分析及动力学实验得到的动力学谐振频率和位移振幅与数值解一致。在此基础上,为了验证本文方法对传统非谐振设计方法的兼容性,对含有不同厚径比环板的变幅器进行有限元动力学仿真,发现厚径比不论在经典薄板理论范围之内还是在Mindlin中厚板理论范围之内,变幅器动力学特性与理论设计要求的完全一致,这就使变幅器设计由含薄圆环板类齿轮的情况拓展到含中厚圆环板类齿轮的情况,进而完善了非谐振设计理论。
(3)超声珩齿振动系统设计新方法探索
针对非谐振设计理论齿轮边缘振幅较小的问题。借鉴经典全谐振设计理论和非谐振设计思想,将传统工具头部分改造为由齿轮和圆柱形杆组成的齿轮杆,根据相应边界耦合条件建立频率方程和位移振幅方程,利用MATLAB软件求出圆柱形杆参数和中厚圆环板位移振幅数值解,用有限元法对变幅杆和齿轮杆组成系统进行动力学仿真分析,理论解和有限元法都表明中厚圆环板边缘振幅较非谐振设计理论时变大。旨在对超声珩齿振动系统设计方法探索出一条新的设计思路。
The world will move it up only when the gears are connected. Gears as one of the most inportant transmission, is not only the foundation industry of equipment manufacturing industry in our country, but also the important basis of the national economic construction fields. Along with the rapid economic development, gears with satisfactory performance are recognized and favoured by industry. Hence, the research and application of gears manufacturing technology appears especially important. The technology of ultrasonic gear honing has been a new craft of precision machining for gear on the basis of traditional honing process, and it is a new exploration of an ultrasonic compound machining for gears. Ultrasonic gear honing machining is useful in the modification of gear, decrease the stemming problems of plugged wheel and the force of honing, hence, the machining efficiency, machining accuracy and the life of honing wheels can be improved, so, it is of important academic significance and application value to study the ultrasonic gear honing machining. To realize the ultrasonic gear honing, one of the key problem solved firstly is to design the ultrasonic vibration system. The problems and shortages in non-resonant structure design theory (NRST) which only can be used to design gear including thin annular plate was focused to be solved in this paper. On the basis of consulting a large number of technical data and documents, a completely new design method for ultrasonic gear honing vibration system was brought forward based on Mindlin's theory. Just for this reason, the project "Non-resonant structure design theory of Transformer and application of ultrasonic gear shaving honing" which was proposed by our research team was supported by the National Natural Science Foundation of China (No.50975191). As an important sub subject "Dynamical characteristics of transformer in ultrasonic gear honing" was proposed by the Innovation and Entrepreneurial Projects for College Students of Taiyuan (No.110148050). The following researches about dynamical characteristics of transformer in ultrasonic gear honing have been studied:
(1) Models of transverse vibration for gear based on Mindlin's theory
Gear was simplified to moderately thick annular plate (MTAP) with the outer diameters of reference circle, according to the structural characteristics of a cylindrical gear with a hole in the center and its ratio of the thickness to radius in the scope of the moderately thick plate theory, i.e., the ratio of the thickness to radius is greater than1/5and less than1/2. The frequency equations for transverse vibration of the moderately thick annular plate were derived with the free boundary conditions based on Mindlin's theory and solved computationally with the help of MATLAB software. The solutions are in best agreement with those from the finite element method (FEM) and also with the experimental results. The conclusion has certain theoretical significance to guide to design the vibration system of ultrasonic gear honing.
(2)Research on dynamical characteristics of transformer in ultrasonic gear honing
To extended the range of application the NRST, the mathematical models of the transformer which were consists of cone, exponential, catenary, step horn and MTAP were built respectively by means of NRST, Mindlin's theory and force coupling conditions. Then the frequency equations and displacement amplitude equations of corresponding transformer were derived. The design parameters and displacement amplitude of the transformer were solved by using MATLAB software. The results from FEM and dynamical experiment were consistent with the numerical solutions of the frequency equations and displacement amplitude equations. On this basis, in order to verify the compatibility of the proposed transformer design method for the traditional NRST, FEM simulation for the transformer were calculated under various ratio of the thickness to radius. It can be found that dynamical characteristics of transformer are consistent with the theoretical design demands when the ratio of the thickness to radius that covers the range from the CPT to MTAP. The experimental result demonstrates that the method of this paper can extend the transformer design range from gear including thin annular plate to gear including MTAP, and NRST is further improved and perfected.
(3)Exploratory research on the new design method for vibration system in ultrasonic gear horning
According to problem of displacement amplitude on the edge of MTAP is comparatively small in the NRST generally; the whole resonant design theory (WRDT) and NRST ideas were referenced. The traditional tool head was replaced by gear pole which was consists of MTAP and cylindrical pole. The frequency equations and displacement amplitude equations of the gear pole were derived based on force coupling boundary condition. Then the parameter of the cylindrical pole and the displacement amplitude of the MTAP were solved by using MATLAB software, and the dynamical characteristics simulation of the system which was consists of horn and gear pole were got by FEM. It is proved from theoretical solution and FEM that the displacement amplitude on the edge of MTAP is increased than NRST. This part aims to explore a new design ideas for vibration system in ultrasonic gear horning.
引文
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