少齿差行星减速器动态特性分析及非线性振动研究
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摘要
课题来源于国防科工委“十一五”民用航天预研项目—“空间环境下的高性能摩擦副与高效传动机构技术”（C4220061319）、国家教育部“长江学者和创新团队发展计划”项目—“高性能机电传动系统的创新设计理论、方法与技术”（IRT0763）、国家自然科学基金重点项目—“新型高性能传动件及系统的可靠性设计理论与方法”（50735008）。
减速器系统的工作状态极其复杂，不仅载荷工况和动力装置多样，且对于齿轮传动系统，由于时变啮合刚度、传动误差、齿侧间隙等因素的影响，引起轮齿接触–脱离–接触周期性、强非线性耦合振动，对传动系统的平稳性、可靠性产生严重影响。因此对其进行动态特性、间隙非线性振动行为及影响因素的研究，为高性能齿轮系统的设计、分析、制造提供了一定的理论依据与实验参考。
论文以NN型少齿差行星减速器为对象，先分析其结构及传动原理，用有限元法分析其固有频率及模态振型，用集中质量法建立减速器系统非线性振动模型与方程，通过数值求解分析减速器非线性振动特性以及参数对它的影响。最后对其模态特性和振动响应进行实验研究。
论文主要研究内容如下：
（1）分析该减速器的结构、传动原理，计算某常见工况下各传动件的理论转速。求得两级传动的理论啮合频率、啮合阻尼。详细分析各滚动轴承的变形、刚度、阻尼等动特性参数。
（2）推导内齿副单齿刚度计算式，考虑理论重合度，计算多对齿啮合刚度。将两级传动多对齿时变啮合刚度拟合为8阶Fourier级数的形式。分别以各级传动啮合角频率为角速度的正弦函数模拟各级齿轮误差。
（3）采用ABAQUS建立该减速器有限元自由模态分析模型，其中轮齿啮合部位采用绑定约束，用弹簧单元模拟轴承。采用Lanczos特征值求解器对该减速器进行自由模态求解，获得该减速器前20阶固有频率及模态振型。
（4）综合考虑齿轮啮合刚度、传动误差、齿侧间隙及支撑刚度和阻尼，用集中质量法建立多自由度、多间隙、变参数、弯–扭耦合的两级齿轮系统非线性振动模型，用Lagrange方程推导齿轮系统的非线性振动微分方程组。用四阶五级的RKF法对非线性微分方程组求解，系统地分析该减速器各齿轮振动位移、速度响应，以及振动位移–速度相图、Poincaré截面。进一步计算得齿轮弹粘啮合力、轴承动载荷、振动加速度响应。最后分析各参数对减速器非线性振动特性的影响。
（5）用LMS Test. Lab对该减速器进行锤击法自由模态实验，验证理论分析结果的正确性。用三向加速度传感器采集减速器壳体振动信号，由FFT变换得到相应的振动频率，经1/3倍频处理分析振动加速度级结构噪声，经积分处理得到振动速度及位移响应。
This research subject stems from the projects “Commission of Science,Technology and Industry for National Defense "Eleventh Five-Year" civilian spacepre-research project-the high performance friction pairs and efficient transmissionmechanism technology under space environment (No. C4220061319)" and “Programfor Changjiang Scholars and Innovative Research Team of Chinese Ministry ofEducation-the innovation design theory, method and technology of high performanceelectric drive system (No. IRT0763)" and “National Natural Science Foundation KeyProject of China-the reliability design theory and methods of new high performancetransmission parts and system (No.50735008)".
Working state of reducer system is extremely complex, also the load cases andpower plants are diverse, and for the gear transmission, the factors of the time-varyingmesh stiffness, transmission errors, backlash, and so on, may cause tooth periodical andstrong nonlinear coupled vibration of contact-detachment-contact, which will seriouslyinfluence the smooth and reliability of transmission system. So the researches on itsdynamic characteristics, the backlash nonlinear vibration behaviors and theirinfluencing factors will provide a theoretical basis and experimental reference for thedesign, analysis, and manufacture of high-performance gear system.
Aimed at a NN-type planetary reducer with small tooth number difference reducer,firstly, the structure and transmission principle are analyzed, secondly, the naturalfrequencies and mode shapes are analyzed by means of the FE method, then, thenonlinear vibration model and equations of reducer system are established by thelumped mass method, and the equations are solved by numerical method to analyze thenonlinear vibration characteristics and influences of parameters on them, finally, themodal characteristics and vibration response testing is carried out, respectively.
The main contents of this paper are as follows:
(1) The structure and transmission principle of this reducer are analyzed, thetheoretical speed of each transmission part is calculated under a common operatingcondition, also the theoretical meshing frequency and meshing damping are calculated,the dynamic characteristics parameters such as deformation, stiffness, damping of eachrolling bearing are analyzed in detail.
(2) The stiffness calculation formula of single tooth in internal gear pair is deduced. Then, the mesh stiffness of multi-pair of teeth is calculated considering the theoreticalcontact ratio, the time-varying mesh stiffness of two stage transmissions are fitted to8-order Fourier series form, and a sine function, whose angular velocity is meshingangle frequency, is simulated as gear error.
(3) The FE modal analysis model of this reducer is established using the ABAQUS,in which the tooth engaging portions are constrained as binding, the bearings aresimulated by spring units. The Lanczos eigenvalue solver is adopted for the free modalsolving of the reducer, the first20orders natural frequencies and mode shapes of thereducer are obtained.
(4) Considering the gear meshing stiffness, transmission error, backlash andsupport stiffness and damping, the multi-degree-of-freedom, multi-backlash, variableparameters, bending-torsion coupled nonlinear vibration model of the two-stage gearsystem is established by the lumped mass method, the nonlinear vibration differentialequations of the gear system is deduced by the Lagrange equation. The fourth-orderfifth-grade RKF method is adopted to solve the nonlinear differential equations, the gearvibration displacements, velocity responses, and the vibration displacement-velocityphase portraits, Poincaré sections are analyzed systematically. The elasticity viscousmeshing forces of gear pair, bearing dynamic loads, and vibration accelerationresponses are further calculated, finally, the influences of various parameters on thenonlinear vibration characteristics of the reducer are analyzed.
(5) The Hammer free modal experiment on the reducer is carried out by utilizingthe LMS Test. Lab to verify the theoretical analysis results. The three-directionacceleration sensors are adopted to capture the vibration signal of the reducer shell, thevibration frequency can be obtained by the FFT transform, the vibration accelerationstructural noise is analyzed through1/3octave processing, the vibration velocity anddisplacement response can be obtained by integral processing.
减速器系统的工作状态极其复杂，不仅载荷工况和动力装置多样，且对于齿轮传动系统，由于时变啮合刚度、传动误差、齿侧间隙等因素的影响，引起轮齿接触–脱离–接触周期性、强非线性耦合振动，对传动系统的平稳性、可靠性产生严重影响。因此对其进行动态特性、间隙非线性振动行为及影响因素的研究，为高性能齿轮系统的设计、分析、制造提供了一定的理论依据与实验参考。
论文以NN型少齿差行星减速器为对象，先分析其结构及传动原理，用有限元法分析其固有频率及模态振型，用集中质量法建立减速器系统非线性振动模型与方程，通过数值求解分析减速器非线性振动特性以及参数对它的影响。最后对其模态特性和振动响应进行实验研究。
论文主要研究内容如下：
（1）分析该减速器的结构、传动原理，计算某常见工况下各传动件的理论转速。求得两级传动的理论啮合频率、啮合阻尼。详细分析各滚动轴承的变形、刚度、阻尼等动特性参数。
（2）推导内齿副单齿刚度计算式，考虑理论重合度，计算多对齿啮合刚度。将两级传动多对齿时变啮合刚度拟合为8阶Fourier级数的形式。分别以各级传动啮合角频率为角速度的正弦函数模拟各级齿轮误差。
（3）采用ABAQUS建立该减速器有限元自由模态分析模型，其中轮齿啮合部位采用绑定约束，用弹簧单元模拟轴承。采用Lanczos特征值求解器对该减速器进行自由模态求解，获得该减速器前20阶固有频率及模态振型。
（4）综合考虑齿轮啮合刚度、传动误差、齿侧间隙及支撑刚度和阻尼，用集中质量法建立多自由度、多间隙、变参数、弯–扭耦合的两级齿轮系统非线性振动模型，用Lagrange方程推导齿轮系统的非线性振动微分方程组。用四阶五级的RKF法对非线性微分方程组求解，系统地分析该减速器各齿轮振动位移、速度响应，以及振动位移–速度相图、Poincaré截面。进一步计算得齿轮弹粘啮合力、轴承动载荷、振动加速度响应。最后分析各参数对减速器非线性振动特性的影响。
（5）用LMS Test. Lab对该减速器进行锤击法自由模态实验，验证理论分析结果的正确性。用三向加速度传感器采集减速器壳体振动信号，由FFT变换得到相应的振动频率，经1/3倍频处理分析振动加速度级结构噪声，经积分处理得到振动速度及位移响应。
This research subject stems from the projects “Commission of Science,Technology and Industry for National Defense "Eleventh Five-Year" civilian spacepre-research project-the high performance friction pairs and efficient transmissionmechanism technology under space environment (No. C4220061319)" and “Programfor Changjiang Scholars and Innovative Research Team of Chinese Ministry ofEducation-the innovation design theory, method and technology of high performanceelectric drive system (No. IRT0763)" and “National Natural Science Foundation KeyProject of China-the reliability design theory and methods of new high performancetransmission parts and system (No.50735008)".
Working state of reducer system is extremely complex, also the load cases andpower plants are diverse, and for the gear transmission, the factors of the time-varyingmesh stiffness, transmission errors, backlash, and so on, may cause tooth periodical andstrong nonlinear coupled vibration of contact-detachment-contact, which will seriouslyinfluence the smooth and reliability of transmission system. So the researches on itsdynamic characteristics, the backlash nonlinear vibration behaviors and theirinfluencing factors will provide a theoretical basis and experimental reference for thedesign, analysis, and manufacture of high-performance gear system.
Aimed at a NN-type planetary reducer with small tooth number difference reducer,firstly, the structure and transmission principle are analyzed, secondly, the naturalfrequencies and mode shapes are analyzed by means of the FE method, then, thenonlinear vibration model and equations of reducer system are established by thelumped mass method, and the equations are solved by numerical method to analyze thenonlinear vibration characteristics and influences of parameters on them, finally, themodal characteristics and vibration response testing is carried out, respectively.
The main contents of this paper are as follows:
(1) The structure and transmission principle of this reducer are analyzed, thetheoretical speed of each transmission part is calculated under a common operatingcondition, also the theoretical meshing frequency and meshing damping are calculated,the dynamic characteristics parameters such as deformation, stiffness, damping of eachrolling bearing are analyzed in detail.
(2) The stiffness calculation formula of single tooth in internal gear pair is deduced. Then, the mesh stiffness of multi-pair of teeth is calculated considering the theoreticalcontact ratio, the time-varying mesh stiffness of two stage transmissions are fitted to8-order Fourier series form, and a sine function, whose angular velocity is meshingangle frequency, is simulated as gear error.
(3) The FE modal analysis model of this reducer is established using the ABAQUS,in which the tooth engaging portions are constrained as binding, the bearings aresimulated by spring units. The Lanczos eigenvalue solver is adopted for the free modalsolving of the reducer, the first20orders natural frequencies and mode shapes of thereducer are obtained.
(4) Considering the gear meshing stiffness, transmission error, backlash andsupport stiffness and damping, the multi-degree-of-freedom, multi-backlash, variableparameters, bending-torsion coupled nonlinear vibration model of the two-stage gearsystem is established by the lumped mass method, the nonlinear vibration differentialequations of the gear system is deduced by the Lagrange equation. The fourth-orderfifth-grade RKF method is adopted to solve the nonlinear differential equations, the gearvibration displacements, velocity responses, and the vibration displacement-velocityphase portraits, Poincaré sections are analyzed systematically. The elasticity viscousmeshing forces of gear pair, bearing dynamic loads, and vibration accelerationresponses are further calculated, finally, the influences of various parameters on thenonlinear vibration characteristics of the reducer are analyzed.
(5) The Hammer free modal experiment on the reducer is carried out by utilizingthe LMS Test. Lab to verify the theoretical analysis results. The three-directionacceleration sensors are adopted to capture the vibration signal of the reducer shell, thevibration frequency can be obtained by the FFT transform, the vibration accelerationstructural noise is analyzed through1/3octave processing, the vibration velocity anddisplacement response can be obtained by integral processing.
引文
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