车辆复合行星传动系统动力学特性研究
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摘要
随着对车辆性能要求的提高，复合行星传动系统已经在车辆中得到应用。与简单行星齿轮传动系统相比，它可以实现更多的档位，承受更大的载荷与传动比。但复合行星传动系统的结构更加复杂，影响其动力学特性的因素很多，具有更为严重的振动与噪声问题。国内外对于复合行星传动系统的研究尚处于起步阶段，目前缺乏完整的设计准则。大多数工程师在设计复合行星传动系统时仍沿用传统的静态设计方法，而对于系统的动力学特性和参数对其的影响往往通过经验和估计来获得。这种设计方法不仅效率低，而且会对实际应用造成影响。随着对齿轮传动系统的性能要求的提高，实现动态设计已经成为当今齿轮传动系统设计的发展趋势,而了解系统的动态特性是实现动态设计的基础。因此，建立复合行星传动系统的动力学模型，深入研究其动力学特性具有重要的意义。
本文在国家自然科学基金项目“车辆复合行星传动系统动态设计理论研究（项目号：50875189）”与中国北方车辆研究所国防项目“XXX特种传动系统扭转振动特性研究（项目号：9140C340101101）”的资助下，在建立了复合行星传动系统动力学模型的基础上，立足于非线性动力学和混沌理论，结合实验测试和信号分析，对车辆复合行星传动系统动力学特性进行了较为系统和深入的研究，主要研究内容与结论如下：
1.论文以Ravigneaux式复合行星传动系统为研究对象，在相关合理假设的前提下，应用集中质量法建立了系统的“纯扭转”和“平移-扭转”耦合动力学模型。考虑了齿侧间隙、时变啮合刚度与综合啮合误差等非线性因素，并对其进行了相应的数学描述。在推导了构件相对位移的基础上，运用Lagrange方程分别建立了与两种动力学模型对应的运动微分方程。此两组运动微分方程均适用于反复迭代求解的场合，并对任意的行星轮组数、间距和相对位置均适用。
2.基于系统的“纯扭转”动力学模型，运用谐波平衡法(Harmonic Balance Method,HBM)研究了复合行星传动系统的非线性频响特性。首先，为消除系统的刚体位移，同时便于求解，对模型进行了坐标变换和无量纲化处理。进而通过实际算例详细研究了利用谐波平衡法求解系统动态响应的具体过程。该方法将系统的稳态响应展开为平均分量与基频分量叠加的形式，间隙非线性函数利用描述函数(Describing Function)展开为相同的形式，从而将系统运动微分方程转化为非线性代数方程组。为进一步求解此方程组，采用逆Broyden秩1拟Newton法进行迭代求解，获得了系统的基频稳态响应。研究发现，齿侧间隙的存在使得系统的频响曲线出现了跳跃与多值解等典型非线性特征。这种特征受到齿侧间隙、时变啮合刚度和综合啮合误差等参数的影响，文中通过改变以上参数进行了详细计算和说明。
3.基于系统的“平移-扭转”耦合动力学模型，运用数值法研究了系统分别在不同激励频率、啮合阻尼比和齿侧间隙作用下的各种非线性动态响应，运用定性分析方法对系统的不同运动形式进行了研究。首先利用线性变换将Ravigneaux式复合行星传动系统“平移-扭转”耦合运动微分方程写成了矩阵形式，解决了原始方程中复杂的多元非线性函数与线性变量共存而难以求解的困难。在分析了变步长Gill积分法求解原理的基础上，利用该方法，通过改变系统的激励频率、齿侧间隙和阻尼比，获得了系统动态响应的全局分岔图，得到了系统在上述参数影响下所呈现的不同运动状态。综合运用时间历程曲线、相空间轨线、Poincáre截面和功率谱分析了系统的周期运动、拟周期运动和混沌运动。结果表明，由于齿侧间隙、时变啮合刚度和综合啮合误差等非线性因素的存在使复合行星传动系统表现出明显的非线性动力学特性。
4.推导了动态响应灵敏度分析的直接法并给出了计算流程图。以Ravigneaux式复合行星传动系统的“平移-扭转”动力学模型为基础，利用直接法结合Runge-Kutta数值积分法计算了系统的动态响应对构件质量、中心件支承刚度、啮合刚度与啮合阻尼的灵敏度。研究结果表明，复合行星传动系统的动态响应对上述参数的灵敏度均不为零。不仅同一参数对不同啮合副的响应贡献程度有所不同，而且不同参数对同一啮合副的的影响程度也有所不同。在四类参数中，中心件的支承刚度对动态响应的影响相对较小，而啮合阻尼对动态响应的影响相对较大。
5.建造了复合行星传动系统的实验平台，对系统的动态特性进行了实验测试，验证了理论分析的正确性与合理性。通过实验模态分析获得了系统的频响函数。系统的第1阶扭振固有频率得到验证，从而证明了利用集中质量法建立的系统“纯扭转”动力学模型较为合理。随后测试了复合行星传动系统在三种不同输入转速下的动态响应。通过频谱分析可知，在远离系统共振频率的低频区域中，在响应信号中包含基频及其倍频成分，并以基频为主，系统没有出现超谐波或次谐波等非线性振动。通过实验证明了本文理论研究所进行的相关方法与结论是比较合理的。
通过本文的研究揭示了复合行星传动系统的非线性动态特性，掌握了设计参数对系统动态特性的影响，为实现复合行星传动系统的动态设计提供了理论依据。
With the promotion of the vehicles performance requirements, compound planetarygear train sets (CPGT) have been used in vehicle. Compared to simple planet gear train sets,more range shift, larger loads and transmission ratios can be achieved by CPGT. However,there are many factors affecting its dynamic characteristics because of its complexstructures. Thus, more serious problems of vibration and noise are avaliable. The researchesat home and abroad on CPGT are in initial states with the lack of complete design criteria.Most engineers only continue to use conventional static method when they design theCPGT. The dynamic characteristics and the impact of parameters are usually obtained byexperience and estimation. It will exert harmful effect in practice as well as its inefficiency.With higher performance requirements of gear transmission system, dynamic design hasbecome a development trend of transmission system design, which knowledge of thedynamic characteristics of the system is the foundation of dynamic design. Therefore, Ithave great significance on building the dynamic model and researching the dynamiccharacteristics of CPGT.
This dissertation is supported by the National Natural Science Foundation of China(Project name：Research on Dynamic Design Theory of Vehicle Compound Planetary GearTransmission System.Grant No.：50875189) and national defense project of China NorthVehicle Research Institute (Project name：Research on torsional vibration characteristics ofXXX special type of transmission system. Grant No.：9140C340101101). Based onbuilding the dynamic model of CPGT, by means of nonlinear dynamic and chaos theory, aswell as experimental observations and signal analysis, the dynamic characteristics ofvehicle CPGT are studied in this dissertation systematically and penetratingly. The maincontents and conclusions are as follows：
1. A Ravigneaux CPGT in practical vehicle automatic gearbox is used for the researchsubject in this dissertation. In case of some reasonable assumptions, the system purelytosional and translational-torsional coupled dynamic models are established by lumpedmass method. Considering the gear backlashes, time-varying meshing stiffness andsynthetical meshing errors, the mathematical description is applied. Based on deducing therelative displacements between the components, the corresponding kinematic differentialequations of the two kinds of models are established by means of Lagrange equation. The differential equations are adequate for iterative solving instance, any group number, spacelength and relative position of the planet gears.
2. Based on the pure torsional model of CPGT, Harmonic Balance Method (HBM) isapplied for studying the nonlinear frequency response characteristics. In the first place,coordinate transformation and nondimensional treatment are operated on the model toeliminate rigid body displacement and solve conveniently. The pocess of solving CPGT’sdynamic response by Harmonic Balance Method (HBM) is studied in detail. The steadystate response of the system is expanded as a form of average component together withfundamental frequency component. The backlash nonlinear function is expanded as thesame form by using Describing Function, which convert the differential equations intononlinear algebraic equations. To solve the algebraic equations, single rank inverseBroyden quasi Newton method is applied iteratively for obtaining the fundamentalfrequency steady state response of the CPGT. It shows from the study that the typicalnonlinear feature of jumping and multivalue in the frequency response curves has arisen forthe existence of gear backlashes. This feature is affected by gear backlashes, time-varyingmeshing stiffness and synthetical mesh errors, which the detail calculation and explanationare processed by changing the above parameters in this dissertation.
3. Based on the translational-torsonal coupled model of CPGT, numerical method isused for studying the nonlinear dynamic responses of the system under different excitationfrequency, meshing damping ratio and backlashes, which the qualitative methods areapplied to discuss the different movment forms. The translational-torsonal coupleddifferential equation of Ravigneaux CPGT, which the complex nonlinear multi-variablefunction and linear variable are coextensive, is written as matrix form through lineartransform to overcome the difficulty of solving inconvenience. Based on analyzing theprinciple of steplength-varying Gill integral method, the global bifurcation diagrams of theCPGT’s dynamic response are obtained by changing the excitation frequency, gearbacklashes and damping ratio, which the different motion states impacted by aboveparameters are achieved. The CPGT’s periodic, quasi-periodic and chaotic motions areanalyzed through using time history curve, phase-plane diagram, Poincáre section andpower spectrum synthetically. The analysis shows that rich and varied dynamic behavioursare included in CPGT because of the existence of nonlinear factors as gear backlashes,time-varying meshing stiffness and synthetical mesh errors.
4. Direct method of dynamic response sensitivity analysis is derived and the calculation flow graph is given. Based on the Ravigneaux CPGT’s translational-torsionaldynamic model, the dynamic response sensitivities on mass, supporting stiffness of centercomponents, meshing stiffness and meshing damping are calculated by direct methodtogether with Runge-Kutta numerical methods of integration. It shows that the dynamicresponse sensitivities on the parameters above are non-zero totally. The same parameter hasdifferent contribution on different engagement pairs and different parameters have differentcontribution on the same engagement pairs. Among the four kinds of parameters, thesupporting stiffness of center component has minimal influence on the CPGT’s dynamicresponse and the meshing damping has the maximum.
5. Experimental platform of CPGT is built to test the dynamic characteristic and therationality of the theoretical analysis is affirmed. Frequency response function is achievedby means of hammer blow. The primary natural frequency of torsional vibration is proved,which certify the rationality of purely torsional model built by mass lumped method. Thenthe CPGT’s dynamic responses under three different input rotationl speeds are measured. Itshows from spectrum analysis that the response signals include fundamental frequency andits octave parts in the section of low frequency away from CPGT’s resonance frequency.The fundamental frequency is predominant in the signals and nonlinear vibrations likesubharmonic or ultraharmonic vibration are not appeared. The rationality of the method andconlusion made by the theoretical study in the dissertation is proved by physical test.
The dynamic characteristics of compound planetary gear train sets are came out andthe influences of design parameters on dynamic characteristics are possessed, whichprovide theoretical foundation on dynamic design of compound planetary gear train sets.
本文在国家自然科学基金项目“车辆复合行星传动系统动态设计理论研究（项目号：50875189）”与中国北方车辆研究所国防项目“XXX特种传动系统扭转振动特性研究（项目号：9140C340101101）”的资助下，在建立了复合行星传动系统动力学模型的基础上，立足于非线性动力学和混沌理论，结合实验测试和信号分析，对车辆复合行星传动系统动力学特性进行了较为系统和深入的研究，主要研究内容与结论如下：
1.论文以Ravigneaux式复合行星传动系统为研究对象，在相关合理假设的前提下，应用集中质量法建立了系统的“纯扭转”和“平移-扭转”耦合动力学模型。考虑了齿侧间隙、时变啮合刚度与综合啮合误差等非线性因素，并对其进行了相应的数学描述。在推导了构件相对位移的基础上，运用Lagrange方程分别建立了与两种动力学模型对应的运动微分方程。此两组运动微分方程均适用于反复迭代求解的场合，并对任意的行星轮组数、间距和相对位置均适用。
2.基于系统的“纯扭转”动力学模型，运用谐波平衡法(Harmonic Balance Method,HBM)研究了复合行星传动系统的非线性频响特性。首先，为消除系统的刚体位移，同时便于求解，对模型进行了坐标变换和无量纲化处理。进而通过实际算例详细研究了利用谐波平衡法求解系统动态响应的具体过程。该方法将系统的稳态响应展开为平均分量与基频分量叠加的形式，间隙非线性函数利用描述函数(Describing Function)展开为相同的形式，从而将系统运动微分方程转化为非线性代数方程组。为进一步求解此方程组，采用逆Broyden秩1拟Newton法进行迭代求解，获得了系统的基频稳态响应。研究发现，齿侧间隙的存在使得系统的频响曲线出现了跳跃与多值解等典型非线性特征。这种特征受到齿侧间隙、时变啮合刚度和综合啮合误差等参数的影响，文中通过改变以上参数进行了详细计算和说明。
3.基于系统的“平移-扭转”耦合动力学模型，运用数值法研究了系统分别在不同激励频率、啮合阻尼比和齿侧间隙作用下的各种非线性动态响应，运用定性分析方法对系统的不同运动形式进行了研究。首先利用线性变换将Ravigneaux式复合行星传动系统“平移-扭转”耦合运动微分方程写成了矩阵形式，解决了原始方程中复杂的多元非线性函数与线性变量共存而难以求解的困难。在分析了变步长Gill积分法求解原理的基础上，利用该方法，通过改变系统的激励频率、齿侧间隙和阻尼比，获得了系统动态响应的全局分岔图，得到了系统在上述参数影响下所呈现的不同运动状态。综合运用时间历程曲线、相空间轨线、Poincáre截面和功率谱分析了系统的周期运动、拟周期运动和混沌运动。结果表明，由于齿侧间隙、时变啮合刚度和综合啮合误差等非线性因素的存在使复合行星传动系统表现出明显的非线性动力学特性。
4.推导了动态响应灵敏度分析的直接法并给出了计算流程图。以Ravigneaux式复合行星传动系统的“平移-扭转”动力学模型为基础，利用直接法结合Runge-Kutta数值积分法计算了系统的动态响应对构件质量、中心件支承刚度、啮合刚度与啮合阻尼的灵敏度。研究结果表明，复合行星传动系统的动态响应对上述参数的灵敏度均不为零。不仅同一参数对不同啮合副的响应贡献程度有所不同，而且不同参数对同一啮合副的的影响程度也有所不同。在四类参数中，中心件的支承刚度对动态响应的影响相对较小，而啮合阻尼对动态响应的影响相对较大。
5.建造了复合行星传动系统的实验平台，对系统的动态特性进行了实验测试，验证了理论分析的正确性与合理性。通过实验模态分析获得了系统的频响函数。系统的第1阶扭振固有频率得到验证，从而证明了利用集中质量法建立的系统“纯扭转”动力学模型较为合理。随后测试了复合行星传动系统在三种不同输入转速下的动态响应。通过频谱分析可知，在远离系统共振频率的低频区域中，在响应信号中包含基频及其倍频成分，并以基频为主，系统没有出现超谐波或次谐波等非线性振动。通过实验证明了本文理论研究所进行的相关方法与结论是比较合理的。
通过本文的研究揭示了复合行星传动系统的非线性动态特性，掌握了设计参数对系统动态特性的影响，为实现复合行星传动系统的动态设计提供了理论依据。
With the promotion of the vehicles performance requirements, compound planetarygear train sets (CPGT) have been used in vehicle. Compared to simple planet gear train sets,more range shift, larger loads and transmission ratios can be achieved by CPGT. However,there are many factors affecting its dynamic characteristics because of its complexstructures. Thus, more serious problems of vibration and noise are avaliable. The researchesat home and abroad on CPGT are in initial states with the lack of complete design criteria.Most engineers only continue to use conventional static method when they design theCPGT. The dynamic characteristics and the impact of parameters are usually obtained byexperience and estimation. It will exert harmful effect in practice as well as its inefficiency.With higher performance requirements of gear transmission system, dynamic design hasbecome a development trend of transmission system design, which knowledge of thedynamic characteristics of the system is the foundation of dynamic design. Therefore, Ithave great significance on building the dynamic model and researching the dynamiccharacteristics of CPGT.
This dissertation is supported by the National Natural Science Foundation of China(Project name：Research on Dynamic Design Theory of Vehicle Compound Planetary GearTransmission System.Grant No.：50875189) and national defense project of China NorthVehicle Research Institute (Project name：Research on torsional vibration characteristics ofXXX special type of transmission system. Grant No.：9140C340101101). Based onbuilding the dynamic model of CPGT, by means of nonlinear dynamic and chaos theory, aswell as experimental observations and signal analysis, the dynamic characteristics ofvehicle CPGT are studied in this dissertation systematically and penetratingly. The maincontents and conclusions are as follows：
1. A Ravigneaux CPGT in practical vehicle automatic gearbox is used for the researchsubject in this dissertation. In case of some reasonable assumptions, the system purelytosional and translational-torsional coupled dynamic models are established by lumpedmass method. Considering the gear backlashes, time-varying meshing stiffness andsynthetical meshing errors, the mathematical description is applied. Based on deducing therelative displacements between the components, the corresponding kinematic differentialequations of the two kinds of models are established by means of Lagrange equation. The differential equations are adequate for iterative solving instance, any group number, spacelength and relative position of the planet gears.
2. Based on the pure torsional model of CPGT, Harmonic Balance Method (HBM) isapplied for studying the nonlinear frequency response characteristics. In the first place,coordinate transformation and nondimensional treatment are operated on the model toeliminate rigid body displacement and solve conveniently. The pocess of solving CPGT’sdynamic response by Harmonic Balance Method (HBM) is studied in detail. The steadystate response of the system is expanded as a form of average component together withfundamental frequency component. The backlash nonlinear function is expanded as thesame form by using Describing Function, which convert the differential equations intononlinear algebraic equations. To solve the algebraic equations, single rank inverseBroyden quasi Newton method is applied iteratively for obtaining the fundamentalfrequency steady state response of the CPGT. It shows from the study that the typicalnonlinear feature of jumping and multivalue in the frequency response curves has arisen forthe existence of gear backlashes. This feature is affected by gear backlashes, time-varyingmeshing stiffness and synthetical mesh errors, which the detail calculation and explanationare processed by changing the above parameters in this dissertation.
3. Based on the translational-torsonal coupled model of CPGT, numerical method isused for studying the nonlinear dynamic responses of the system under different excitationfrequency, meshing damping ratio and backlashes, which the qualitative methods areapplied to discuss the different movment forms. The translational-torsonal coupleddifferential equation of Ravigneaux CPGT, which the complex nonlinear multi-variablefunction and linear variable are coextensive, is written as matrix form through lineartransform to overcome the difficulty of solving inconvenience. Based on analyzing theprinciple of steplength-varying Gill integral method, the global bifurcation diagrams of theCPGT’s dynamic response are obtained by changing the excitation frequency, gearbacklashes and damping ratio, which the different motion states impacted by aboveparameters are achieved. The CPGT’s periodic, quasi-periodic and chaotic motions areanalyzed through using time history curve, phase-plane diagram, Poincáre section andpower spectrum synthetically. The analysis shows that rich and varied dynamic behavioursare included in CPGT because of the existence of nonlinear factors as gear backlashes,time-varying meshing stiffness and synthetical mesh errors.
4. Direct method of dynamic response sensitivity analysis is derived and the calculation flow graph is given. Based on the Ravigneaux CPGT’s translational-torsionaldynamic model, the dynamic response sensitivities on mass, supporting stiffness of centercomponents, meshing stiffness and meshing damping are calculated by direct methodtogether with Runge-Kutta numerical methods of integration. It shows that the dynamicresponse sensitivities on the parameters above are non-zero totally. The same parameter hasdifferent contribution on different engagement pairs and different parameters have differentcontribution on the same engagement pairs. Among the four kinds of parameters, thesupporting stiffness of center component has minimal influence on the CPGT’s dynamicresponse and the meshing damping has the maximum.
5. Experimental platform of CPGT is built to test the dynamic characteristic and therationality of the theoretical analysis is affirmed. Frequency response function is achievedby means of hammer blow. The primary natural frequency of torsional vibration is proved,which certify the rationality of purely torsional model built by mass lumped method. Thenthe CPGT’s dynamic responses under three different input rotationl speeds are measured. Itshows from spectrum analysis that the response signals include fundamental frequency andits octave parts in the section of low frequency away from CPGT’s resonance frequency.The fundamental frequency is predominant in the signals and nonlinear vibrations likesubharmonic or ultraharmonic vibration are not appeared. The rationality of the method andconlusion made by the theoretical study in the dissertation is proved by physical test.
The dynamic characteristics of compound planetary gear train sets are came out andthe influences of design parameters on dynamic characteristics are possessed, whichprovide theoretical foundation on dynamic design of compound planetary gear train sets.
引文
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