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考虑温度效应的斜齿轮时变啮合刚度解析算法

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   摘   要:以斜齿轮副为研究对象,基于切片法和积分思想,计入齿面接触温度变化引起的齿廓形变,结合轮齿接触、弯曲、剪切、轴向压缩及基体弹性变形,提出了考虑温度效应的斜齿轮啮合刚度解析算法,并通过有限元法验证了算法的准确性. 分析了不同摩擦因数、输入转矩、输入转速等工况参数对斜齿轮啮合刚度的影响规律. 结果表明,考虑齿轮温升影响后,轮齿从啮入到啮出整个过程的啮合刚度均有所增大;随着摩擦因数、输入转矩和输入转速的增大,斜齿轮本体温度及啮合齿面瞬时闪温升高,单齿啮合刚度和综合啮合刚度均值呈增大趋势. 研究结果可为高速重载齿轮系统准确高效的动力学分析提供理论依据.
   关键词:斜齿轮;温度效应;时变啮合刚度;势能法
   中图分类号:TH132.41                  文献标志码:A
   Abstract:Taking a helical gear pair as the research object,the analytic algorithm of time-varying mesh stiffness of helical gears with temperature was proposed based on slicing method and integral thought. In this algorithm,the tooth profile deformation caused by change of tooth contact temperature,elastic deformation of tooth contact,bending,shear,axial compression and the wheel were taken into account. The accuracy of the algorithm was verified by using finite element method. Then the influence of working parameters,such as friction coefficient,input torque and input speed,on the mesh stiffness of gears was analyzed. The results show that the mesh stiffness of teeth increases in the whole meshing process after considering the influence of temperature. Besides,the body temperature and instantaneous flash temperature of helical gears increase with the increase of friction coefficient,input torque and input speed,so the single mesh stiffness and the mean of total mesh stiffness increase. The research results can provide a theoretical basis for accurate and efficient dynamic analysis of the high-speed and heavy-duty gear system.
   Key words:helical gears;temperature effect;time-varing mesh stiffness;potential energy method
   齒轮系统作为机械装置中最为广泛的动力和运动传递形式,正朝着大功率、高转速、低噪声和轻量化方向发展. 在高速重载工况下,由于啮合齿面相对滑动速度大,瞬时温升高,直接影响齿轮系统内部的温度分布,引起结构热应力及热变形,进而对齿轮啮合刚度产生较大影响,而准确高效的啮合刚度计算方法又是齿轮系统动力学分析的关键. 因此,综合考虑温度影响,开展斜齿轮时变啮合刚度解析算法研究,对高速重载齿轮系统动力学设计有着重要的工程意义.
   近年来,对于齿轮系统时变啮合刚度的研究非常活跃. 在解析算法方面,Cui等[1]、Chaari等[2]和
  Liang等[3]基于材料力学理论,运用势能法计算了直齿轮的啮合刚度;在直齿轮啮合刚度算法的基础上,Wan等[4]提出了一种累积积分势能法计算斜齿轮的啮合刚度,并研究了齿轮参数与齿根裂纹对啮合刚度的影响;万志国等[5]、刘文等[6]考虑了基圆与齿根圆不重合的问题,运用势能法分别提出了求解直齿轮及斜齿轮啮合刚度的改进算法. 在有限元法方面,Cooley等[7]、Liang等[8]提出了多种基于有限元法的直齿轮时变啮合刚度计算方法,并评估了各种方法的应用条件及优缺点;Fernandez等[9-10]和Ma等[11]综合考虑加工误差、齿顶修形或齿轮摩擦等非线性因素,采用有限元法与弹性接触理论相结合的方式,计算了直齿轮的时变啮合刚度. 在考虑温度效应方面,苟向锋等[12]建立了由齿面接触温度变化引起直齿轮齿廓形变的数学表征,而后基于Hertz接触理论研究了接触温度对直齿轮啮合刚度的影响;罗彪等[13]基于石川模型,将轮齿齿廓简化为由梯形和矩形组成的当量齿形,综合考虑温度对直齿轮刚度的影响,引入了热刚度的概念,并提出了一种直齿轮热刚度的解析算法,计算结果与有限元法基本吻合. 目前有关考虑温度效应的齿轮啮合刚度研究已取得一定的成果,但有限元法计算规模较大,解析法仅针对直齿轮开展了相关研究,关于考虑温度效应的斜齿轮时变啮合刚度解析算法鲜有报道.    在上述研究成果的基础上,本文以斜齿轮副为研究对象,提出一种考虑温度效应的斜齿轮啮合刚度解析算法. 将轮齿简化为齿根圆上的变截面悬臂梁,基于切片法和积分思想,在考虑基圆与齿根圆不重合因素的同时,计入齿面接触温度变化引起的轮齿齿廓形变,以确保啮合刚度计算结果准确性;而后分析摩擦因数、输入转矩、输入转速等工况参数对斜齿轮啮合刚度的影响规律.
  1   考虑热变形的斜齿轮端面齿廓方程
  1.1   斜齿轮基体热变形
   斜齿轮副达到热平衡状态后,本体温度场基本稳定,但各处温度非均一. 对于齿轮基体,尽管本体温度场稳定,但与轮齿相固联的基体部分温度不同,因此将斜齿轮基体温度场处理为无内热源稳态非均匀温度场,其在柱面坐标系下的导热微分方程为[14]:
  5   结   论
   1)将轮齿简化为齿根圆上的变截面悬臂梁,计
  入齿面接触温度变化引起的轮齿齿廓形变,基于势能法提出了一种考虑温度效应的斜齿轮啮合刚度解析算法,通过与有限元法计算结果对比分析,验证了解析算法的准确性,提升了斜齿轮啮合刚度的计算效率.
   2)考虑斜齿轮温升影响后,轮齿从啮入到啮出整个过程的啮合刚度均有所增大. 对于单齿啮合刚度,在啮入和啮出端增大量较小,在节点附近增大量较大.
   3)通过不同摩擦因数、输入转矩、输入转速等工况参数对斜齿轮啮合剛度的影响分析,得出考虑温度效应后单齿啮合刚度及综合啮合刚度均值均随上述工况参数的增大而增大,其中输入转矩对啮合刚度的影响最大.
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