斜齿轮多间隙非线性耦合系统动力学研究
摘要
齿轮传动系统是机械传动系统的重要组成部分。随着科学技术的进步和机械行业的发展,对齿轮系统提出了更高要求,高转速、低噪声成为发展方向。加强对齿轮系统动力学的研究,是提高传动系统性能的重要途径。深入研究齿轮系统的动力学参数,对全面分析齿轮传动系统的动态特性,提高机械传动系统性能有重要意义。
本文以斜齿圆柱齿轮传动系统为研究对象,以齿轮系统动力学理论为基础,在全面考虑时变啮合刚度、传递误差、齿侧间隙和轴承游隙等非线性因素下,建立了更符合实际情况的齿轮系统弯-扭-轴-摆多间隙耦合非线性模型。利用Runge-Kutta数值积分算法对系统动力学方程求解,得到系统动态响应结果,对结果进行分析比较,并讨论了多个动力学参数对系统动态特性的影响。
首先,阐述了刚度激励和误差激励的产生机理,对啮合时变刚度进行了Fourier级数数值模拟,并分析了齿轮重合度以及转速与时变刚度的关系；对两种误差类型进行了分析,并作了数值模拟的近似处理。最终将两种激励进行合成,作为齿轮系统的输入。
其次,分别研究了几种不同的建模方法和模型类型。采用集中质量法,建立了弯-扭-轴-摆的非线性动力学模型,并依据模型建立了动力学微分方程组。由于数值计算的需要,进行了间隙非线性描述函数的拟合和对微分方程组的无量纲化。
第三,以固有特性数值计算的理论为基础,求解研究对象对应的无阻尼自由振动微分方程,得出固有频率的数值结果；与有限元方法的分析结果进行比较,说明了理论方法的可行性。
第四,分析比较了多种动力学方程的求解方法,选用四阶定步长Runge-Kutta数值方法,对齿轮系统模型进行求解,得出系统振动响应的时域图、相图、Poincare图以及FFT图。分析了响应结果的特点,并分别从振动幅值和振动周期性两方面,研究讨论了齿轮侧隙、激励频率以及轴承游隙对系统响应结果的影响方式和影响程度,对参数数值的选择提出合理化的建议。
最后,利用MATLAB的GUI开发平台,将以上关于齿轮系统动力学相关分析的研究方法,编写程序开发成能与用户交互的分析工具软件。
本文针对斜齿轮传动系统展开动力学特性的研究,通过大量数值计算得出动态特性的相应结果,为进一步的理论研究和高性能齿轮系统的设计提供基础。
Gear transmission system is an important part of mechanical transmission system. With the scientific and technological progress and the development of machinery industry, the gear system put a higher requirement, high speed, low noise as the direction of development. Strengthen the dynamic study of gear system is an important way to improve the performance of transmission system. Depth study of gear system dynamic parameters, is important to comprehensive analyze the dynamic characteristics of gear transmission system and improve performance of mechanical transmission system.
In this paper, helical gear transmission system was studied with gear system dynamics theory, and established a more realistic bend-torsion-axes-swing multi-gap nonlinear coupled gear system model, in full consideration of the nonlinear factors such as time-varying mesh stiffness, transmission error, backlash and bearing clearance. The dynamic equations of system was solved by use of Runge-Kutta numerical integration algorithm, to obtain the results of dynamic response, which were analyzed and compared, and discussed the affect of several dynamic parameters on the dynamic characteristics of system.
Firstly, the mechanism of stiffness excitation and error excitation were elaborated, and the time-varying stiffness of meshing was numerical simulated by Fourier series method, to analyze the relationship between time-varying stiffness and gear contact ratio & speed; two types of errors were analyzed and made a numerical simulation. Ultimately two excitations were synthesized as the input of gear system.
Secondly, each of several different types of modeling methods and models were analyzed. The bend-torsion-axes-swing nonlinear dynamic model was established by use of lumped parameter method, and the dynamic differential equations based on the model. As the need for numerical calculation, the clearance nonlinear describing function was fitted and the differential equations were dimensionless.
Thirdly, based on the numerical theory of the inherent characteristics, the undamped free vibration equation was solved to obtain the numerical results of the natural frequency; and compare with the results of the finite element method to illustrate the theoretical feasibility of the method.
Fourth, by comparing a variety of methods for solving the dynamic equations, the model of the gear system was solved by use of the fourth order fixed step Runge-Kutta numerical method to obtain the vibration response of the system, such as time-domain diagram, phase diagram, Poincare map and the FFT graph. Analyzing the characteristics of the response results, and discussing the affect way and level of the parameters, such as gear backlash, driving frequency and the bearing clearance from each two aspects of vibration amplitude and vibration periodicity, to propose reasonable suggestions for the choice of parameter values.
Finally, by use of MATLAB's GUI development platform, the analysis tools software was developed including the above analysis methods on the gear system dynamics, which can be interacted with the user.
In this paper, dynamic characteristics of helical gear transmission system was studied, and obtained the dynamic characteristics of the corresponding results by a large number of values calculated to provide the basis for further theoretical research and design of high-performance gear system.
本文以斜齿圆柱齿轮传动系统为研究对象,以齿轮系统动力学理论为基础,在全面考虑时变啮合刚度、传递误差、齿侧间隙和轴承游隙等非线性因素下,建立了更符合实际情况的齿轮系统弯-扭-轴-摆多间隙耦合非线性模型。利用Runge-Kutta数值积分算法对系统动力学方程求解,得到系统动态响应结果,对结果进行分析比较,并讨论了多个动力学参数对系统动态特性的影响。
首先,阐述了刚度激励和误差激励的产生机理,对啮合时变刚度进行了Fourier级数数值模拟,并分析了齿轮重合度以及转速与时变刚度的关系；对两种误差类型进行了分析,并作了数值模拟的近似处理。最终将两种激励进行合成,作为齿轮系统的输入。
其次,分别研究了几种不同的建模方法和模型类型。采用集中质量法,建立了弯-扭-轴-摆的非线性动力学模型,并依据模型建立了动力学微分方程组。由于数值计算的需要,进行了间隙非线性描述函数的拟合和对微分方程组的无量纲化。
第三,以固有特性数值计算的理论为基础,求解研究对象对应的无阻尼自由振动微分方程,得出固有频率的数值结果；与有限元方法的分析结果进行比较,说明了理论方法的可行性。
第四,分析比较了多种动力学方程的求解方法,选用四阶定步长Runge-Kutta数值方法,对齿轮系统模型进行求解,得出系统振动响应的时域图、相图、Poincare图以及FFT图。分析了响应结果的特点,并分别从振动幅值和振动周期性两方面,研究讨论了齿轮侧隙、激励频率以及轴承游隙对系统响应结果的影响方式和影响程度,对参数数值的选择提出合理化的建议。
最后,利用MATLAB的GUI开发平台,将以上关于齿轮系统动力学相关分析的研究方法,编写程序开发成能与用户交互的分析工具软件。
本文针对斜齿轮传动系统展开动力学特性的研究,通过大量数值计算得出动态特性的相应结果,为进一步的理论研究和高性能齿轮系统的设计提供基础。
Gear transmission system is an important part of mechanical transmission system. With the scientific and technological progress and the development of machinery industry, the gear system put a higher requirement, high speed, low noise as the direction of development. Strengthen the dynamic study of gear system is an important way to improve the performance of transmission system. Depth study of gear system dynamic parameters, is important to comprehensive analyze the dynamic characteristics of gear transmission system and improve performance of mechanical transmission system.
In this paper, helical gear transmission system was studied with gear system dynamics theory, and established a more realistic bend-torsion-axes-swing multi-gap nonlinear coupled gear system model, in full consideration of the nonlinear factors such as time-varying mesh stiffness, transmission error, backlash and bearing clearance. The dynamic equations of system was solved by use of Runge-Kutta numerical integration algorithm, to obtain the results of dynamic response, which were analyzed and compared, and discussed the affect of several dynamic parameters on the dynamic characteristics of system.
Firstly, the mechanism of stiffness excitation and error excitation were elaborated, and the time-varying stiffness of meshing was numerical simulated by Fourier series method, to analyze the relationship between time-varying stiffness and gear contact ratio & speed; two types of errors were analyzed and made a numerical simulation. Ultimately two excitations were synthesized as the input of gear system.
Secondly, each of several different types of modeling methods and models were analyzed. The bend-torsion-axes-swing nonlinear dynamic model was established by use of lumped parameter method, and the dynamic differential equations based on the model. As the need for numerical calculation, the clearance nonlinear describing function was fitted and the differential equations were dimensionless.
Thirdly, based on the numerical theory of the inherent characteristics, the undamped free vibration equation was solved to obtain the numerical results of the natural frequency; and compare with the results of the finite element method to illustrate the theoretical feasibility of the method.
Fourth, by comparing a variety of methods for solving the dynamic equations, the model of the gear system was solved by use of the fourth order fixed step Runge-Kutta numerical method to obtain the vibration response of the system, such as time-domain diagram, phase diagram, Poincare map and the FFT graph. Analyzing the characteristics of the response results, and discussing the affect way and level of the parameters, such as gear backlash, driving frequency and the bearing clearance from each two aspects of vibration amplitude and vibration periodicity, to propose reasonable suggestions for the choice of parameter values.
Finally, by use of MATLAB's GUI development platform, the analysis tools software was developed including the above analysis methods on the gear system dynamics, which can be interacted with the user.
In this paper, dynamic characteristics of helical gear transmission system was studied, and obtained the dynamic characteristics of the corresponding results by a large number of values calculated to provide the basis for further theoretical research and design of high-performance gear system.
引文
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