齿轮刚度计算及其有限元分析
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摘要
齿轮传动作为机械传动最常用的形式之一，在机械、电子、纺织、冶金、采矿、汽车、航空、航天及船舶设备(装置)制造中得到了广泛应用。随着机械科学和制造技术的迅速发展，对机械传动系统的运行精度和质量提出了新的更高的要求，齿轮刚度计算及其动态特性研究倍受国内外学者们的高度关注。
本文在对国内外有关齿轮刚度研究进展进行概述和分析的基础上，系统开展了直齿圆柱齿轮和直齿圆锥齿轮刚度计算与分析研究，并利用有限元法对直齿锥齿轮的刚度进行了深入分析，提出了一种适用于小齿数、小直径直齿锥齿轮刚度计算的等效转换方法，进一步丰富和发展了齿轮刚度的计算理论，其主要研究工作进展及其创新点为：
1．针对直齿圆柱齿轮刚度理论计算时采用的当量齿形，本研究采用一种更接近理论齿形的当量齿形，推导出一种新的直齿圆柱齿形刚度计算公式，通过对新的齿轮刚度计算公式与石川公式计算结果的比较，给出了石川公式计算的偏差。
2．通过对直齿圆柱齿轮刚度的系统分析，揭示出影响直齿圆柱齿轮刚度的主要因素及其变化规律，通过对齿轮刚度计算结果的比较，发现了直齿圆柱齿轮刚度随着齿数的增加而增加，而与齿轮的模数无关，并且通过新的直齿圆柱齿轮刚度计算结果证实了本研究结论的正确性。
3．运用ANSYS软件对直齿锥齿轮的刚度进行有限元分析，同时将其有限元计算值与其理论计算值进行比较，结果表明：用新的齿轮刚度计算公式计算的刚度值比用石川公式计算的刚度值有明显的提高，且新的齿轮刚度计算公式的计算值更接近趋向于有限元计算值；而新的齿轮刚度计算公式的计算值与有限元计算值基本上趋于一致，并且在分锥角较小时两者之间误差小，随着分锥角增大两者之间误差会有所增大。
4．在分析了常用等效转换方法的缺陷和直齿锥齿轮刚度的理论与数值计算结果的基础上，提出一种新的适用于小齿数、小直径直齿锥齿轮刚度计算的等效转换新方法，为系统全面研究和解析齿轮刚度提供了新的方法。
本研究是国家自然科学基金项目《锥齿轮局部互换性研究》的内容之一。有关齿轮刚度计算和运用ANSYS软件研究直齿锥齿轮刚度的工作及取得的成果，对进一步丰富和发展了齿轮刚度理论具有重要的理论意义，其应用将产生良好的经济和社会效益。
Gear drive is one of the most common mechanical transmission modes. It has abroadbeen used in many industry fields, such as machine and electron and textile andmetallurgy and mining and automobile and aerospace and ship equipment manufacturing,etc. With the rapid development of mechanical science and manufacturing technology, thenew advanced requirement on the transmission accuracy and quality of mechanicaltransmission system is put forward. Research on the stiffness calculation and dynamiccharacteristic of gears is highly concerned by engineering experts and scholars both hereand abroad.
Based on the summarization and analysis of the progress of gears stiffness both hereand abroad, this paper systematically expands the research of stiffness calculation andanalysis of straight spur and bevel gears and deep analyzes mesh stiffness of straightbevel gears using finite element methods and presents a new equivalent conversionmethod for mesh stiffness calculation of straight bevel gears of the small teeth number anddiameter and further enriches and develops the theoretical calculation of gears stiffness.The major innovative achievements and work progress are listed as follows :
1．Equivalent weight tooth shape is adopted when mesh stiffness of gears iscalculated , so this research adopts a kind of new equivalent weight tooth shape which iscloser to the theoretical tooth shape than old equivalent weight tooth shape and educes thenew formulae of spur gears mesh stiffness. Then ,it is shown that the results calculated byusing Shi Chuan gears stiffness formulae have a certain deviation by means of comparisonbetween the results calculated by using the new gears stiffness formulae and the resultscalculated by using Shi Chuan gears stiffness formulae .
2．By the systematical analysis of spur gears stiffness, the main influencing factorsand laws of spur gears mesh stiffness are found out. It is presented that spur gears stiffnessmagnitude increases with the increase of the teeth number of spur gears but it isindependent of spur gears modulus by means of comparison among the calculated resultsof spur gears stiffness, and the results calculated by the new spur gears stiffness formulaehave further proved its correctness.
3．By using FEM and FEA software ANSYS , the finite element analyses of straightbevel gears mesh stiffness are made .Meanwhile ,all the numerical calculation results havebeen compared with their theoretical calculation results .From the compared results ,theconclusions can be obtained as follow: The value calculated by the new gears stiffness formulae is more enhanced obviously and closer to the numerical calculation value thanthe value calculated by Shi Chuan gears stiffness formulae ;The value calculated by thenew gears stiffness formulae and the finite element methods is parallel ,and error betweenthem is small when reference cone angle is less but when reference cone angleincreases ,error between them increases.
4．Based on analyzing the defects of the usual equivalent conversion methods and theresults calculated by the theoretical formulae and the finite element methods , this paperpresents a new equivalent conversion method for mesh stiffness calculation of straightbevel gears of the small teeth number and diameter . It offers a new method for wholeresearching systems and analyzing gears stiffness.
The research is a part of the project <> of National Natural Science Foundation of China. The work about calculation ofgears mesh stiffness and researching mesh stiffness of straight bevel gears using softwareANSYS and the obtained production have important theoretical significance for furtherenriching and developing the theories of gears stiffness, and its application will bring goodeconomical and societal benefit.
本文在对国内外有关齿轮刚度研究进展进行概述和分析的基础上，系统开展了直齿圆柱齿轮和直齿圆锥齿轮刚度计算与分析研究，并利用有限元法对直齿锥齿轮的刚度进行了深入分析，提出了一种适用于小齿数、小直径直齿锥齿轮刚度计算的等效转换方法，进一步丰富和发展了齿轮刚度的计算理论，其主要研究工作进展及其创新点为：
1．针对直齿圆柱齿轮刚度理论计算时采用的当量齿形，本研究采用一种更接近理论齿形的当量齿形，推导出一种新的直齿圆柱齿形刚度计算公式，通过对新的齿轮刚度计算公式与石川公式计算结果的比较，给出了石川公式计算的偏差。
2．通过对直齿圆柱齿轮刚度的系统分析，揭示出影响直齿圆柱齿轮刚度的主要因素及其变化规律，通过对齿轮刚度计算结果的比较，发现了直齿圆柱齿轮刚度随着齿数的增加而增加，而与齿轮的模数无关，并且通过新的直齿圆柱齿轮刚度计算结果证实了本研究结论的正确性。
3．运用ANSYS软件对直齿锥齿轮的刚度进行有限元分析，同时将其有限元计算值与其理论计算值进行比较，结果表明：用新的齿轮刚度计算公式计算的刚度值比用石川公式计算的刚度值有明显的提高，且新的齿轮刚度计算公式的计算值更接近趋向于有限元计算值；而新的齿轮刚度计算公式的计算值与有限元计算值基本上趋于一致，并且在分锥角较小时两者之间误差小，随着分锥角增大两者之间误差会有所增大。
4．在分析了常用等效转换方法的缺陷和直齿锥齿轮刚度的理论与数值计算结果的基础上，提出一种新的适用于小齿数、小直径直齿锥齿轮刚度计算的等效转换新方法，为系统全面研究和解析齿轮刚度提供了新的方法。
本研究是国家自然科学基金项目《锥齿轮局部互换性研究》的内容之一。有关齿轮刚度计算和运用ANSYS软件研究直齿锥齿轮刚度的工作及取得的成果，对进一步丰富和发展了齿轮刚度理论具有重要的理论意义，其应用将产生良好的经济和社会效益。
Gear drive is one of the most common mechanical transmission modes. It has abroadbeen used in many industry fields, such as machine and electron and textile andmetallurgy and mining and automobile and aerospace and ship equipment manufacturing,etc. With the rapid development of mechanical science and manufacturing technology, thenew advanced requirement on the transmission accuracy and quality of mechanicaltransmission system is put forward. Research on the stiffness calculation and dynamiccharacteristic of gears is highly concerned by engineering experts and scholars both hereand abroad.
Based on the summarization and analysis of the progress of gears stiffness both hereand abroad, this paper systematically expands the research of stiffness calculation andanalysis of straight spur and bevel gears and deep analyzes mesh stiffness of straightbevel gears using finite element methods and presents a new equivalent conversionmethod for mesh stiffness calculation of straight bevel gears of the small teeth number anddiameter and further enriches and develops the theoretical calculation of gears stiffness.The major innovative achievements and work progress are listed as follows :
1．Equivalent weight tooth shape is adopted when mesh stiffness of gears iscalculated , so this research adopts a kind of new equivalent weight tooth shape which iscloser to the theoretical tooth shape than old equivalent weight tooth shape and educes thenew formulae of spur gears mesh stiffness. Then ,it is shown that the results calculated byusing Shi Chuan gears stiffness formulae have a certain deviation by means of comparisonbetween the results calculated by using the new gears stiffness formulae and the resultscalculated by using Shi Chuan gears stiffness formulae .
2．By the systematical analysis of spur gears stiffness, the main influencing factorsand laws of spur gears mesh stiffness are found out. It is presented that spur gears stiffnessmagnitude increases with the increase of the teeth number of spur gears but it isindependent of spur gears modulus by means of comparison among the calculated resultsof spur gears stiffness, and the results calculated by the new spur gears stiffness formulaehave further proved its correctness.
3．By using FEM and FEA software ANSYS , the finite element analyses of straightbevel gears mesh stiffness are made .Meanwhile ,all the numerical calculation results havebeen compared with their theoretical calculation results .From the compared results ,theconclusions can be obtained as follow: The value calculated by the new gears stiffness formulae is more enhanced obviously and closer to the numerical calculation value thanthe value calculated by Shi Chuan gears stiffness formulae ;The value calculated by thenew gears stiffness formulae and the finite element methods is parallel ,and error betweenthem is small when reference cone angle is less but when reference cone angleincreases ,error between them increases.
4．Based on analyzing the defects of the usual equivalent conversion methods and theresults calculated by the theoretical formulae and the finite element methods , this paperpresents a new equivalent conversion method for mesh stiffness calculation of straightbevel gears of the small teeth number and diameter . It offers a new method for wholeresearching systems and analyzing gears stiffness.
The research is a part of the project <
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