封闭差动人字齿轮传动系统均载及动力学特性分析研究
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摘要
载荷均衡分配对于提高行星齿轮传动系统寿命、增大可靠性和减小振动尤为重要，强非线性的行星齿轮传动系统需要应用非线性动力学理论研究其振动特性，鉴于此，行星齿轮传动系统均载及非线性动力学特性已成当前研究的热点和难点问题。本文以封闭差动人字齿轮传动（大型舰船主减速器）为研究对象，通过研究其均载及非线性动态特性，为该传动系统的设计提供理论和技术支持。
在传动系统的啮合与切向刚度研究中，确定了传动系统两级啮合齿轮的各种相位关系，引入基于齿轮副瞬时总接触线长度推导出的斜齿轮时变啮合刚度公式按刚度并联方式计算人字齿时变啮合刚度，分析了人字齿轮啮合刚度波动小的原因。应用材料力学原理计算由单位载荷引起的轮齿位移即柔度，进而确定中间浮动构件人字齿轮接触切向刚度。
在传动系统的齿轮误差等效位移研究中，采用简谐函数推导了偏心误差、齿频误差转化到齿轮副啮合线上的等效位移公式；提出了一种齿轮传动几何误差转化为啮合线上的等效位移计算方法，按此方法推导出了传动系统两级各齿轮安装误差转化到啮合线等效位移的计算公式，进而建立了传动系统两级各齿轮误差转化到啮合线上等效位移的完整公式体系。
在传动系统的静力学均载特性研究中，建立了包含中间浮动构件的封闭差动人字齿轮传动系统静力学计算模型；确定了传动系统静力学均载系数计算公式，计算了传动系统的静力学均载系数；分析了传动系统主要参数对传动系统静力学载荷分配的影响，获得主要参数对传动系统静力学均载特性的影响规律。
在传动系统的动力学均载特性研究中，考虑了齿轮重量、时变啮合刚度、各种误差的影响，建立了包含中间浮动构件的封闭差动人字齿轮传动系统动力学计算模型；把动力学方程线性化，采用傅立叶级数法求解；确定了传动系统动力学均载系数计算公式，计算了传动系统的动力学均载系数；分析了传动系统主要参数对传动系统动力学载荷分配的影响，获得封闭差动人字齿轮传动系统动力学均载特性的变化规律。
在传动系统的动力学浮动特性研究中，建立了封闭差动人字齿轮传动系统动态浮动量的计算方法，计算了传动系统两级各齿轮动力学浮动量，分析了传动系统的各种参数对传动系统动力学浮动量的影响。获得封闭差动人字齿轮传动系统动力学浮动特性的变化规律。
在传动系统的非线性动力学特性研究中，建立了多齿侧间隙、时变啮合刚度的封闭差动人字齿轮传动系统的多自由度扭转非线性动力学方程；应用Newmark数值法求解非线性动力学微分方程组，得到了传动系统的非线性动态响应结果；综合运用位移响应时间历程图、啮合力响应时间历程图、相图、庞加莱截面，分析了齿侧间隙、时变啮合刚度、阻尼、综合误差对封闭差动人字齿轮传动系统非线性动态特性的影响；获得了齿侧间隙、时变啮合刚度、阻尼、综合误差对啮合轮齿的受力、运动状态的影响规律。
The load sharing is particularly important for improving the life of planetary gear train, increasingreliability and reducing vibration. The theory of nonlinear dynamics is required to study the vibrationmechanism of the strong nonlinear planetary gear train. So the load sharing and nonlinear dynamicscharacteristics of the planetary gear train have become the current hotspots and difficult issues. Thepaper mainly studies the load sharing and nonlinear dynamics characteristics of encased differentialherringbone train (main reducer of large ships) to provide theoretical and technical support for thedesign of this train.
In the research on the meshing and tangential stiffness, the phase relationships of the meshinggears are determined in the train, and the mesh stiffness formula of the helical gear which is derivedbased on the instantaneous total length of the contact line for the gear pair is introduced to calculatethe time-varying mesh stiffness of the herringbone on the stiffness parallel. So the cause of the smallfluctuation of the herringbone meshing stiffness is analyzed. In the meantime the materials mechanicsprinciple is applied to calculate the tooth displacement which is also called as flexibility caused by theunit load. And then the herringbone tangential stiffness of the intermediate floating component for thistrain is calculated.
The equivalent displacement formulas of the run-out and meshing-frequency errors along themeshing line are deduced by using harmonic function. An equivalent displacement calculation methodof the geometric errors for the gear transmission is proposed and the equivalent displacement formulaof the gear installation error along the meshing line is deduced based on this method. So the completeequivalent displacement formula system of the gear errors along the meshing line is set up in thistrain.
The static model which includes the intermediate floating component of this train is set up. Thestatic formulas of load sharing coefficients for this train are defined. The static load sharingcoefficients are calculated and the impact of the main parameters on the static load sharing isanalyzed.
The impact of the gear weight, the time-varying mesh stiffness, the gear errors on this train hasbeen considered, and the dynamics model which include the intermediate floating component of thistrain is set up. The dynamic equation is linearized and the Fourier series method is used to solve thisequation. The dynamics formulas of load sharing coefficients for this train are defined. The dynamics load sharing coefficients are calculated and the influence of the main parameters on the dynamics loadsharing for this train is analyzed. The dynamics load sharing characteristics for the encaseddifferential herringbone train are obtained.
The calculating method of the dynamics floating displacements for this train is established.Thedynamics floating displacements of the gears for the train are calculated, and the impact of the mainparameters on dynamics floating displacements of the train are analyzed. The dynamics floatingcharacteristics for the encased differential herringbone train are obtained.
A nonlinear torsional dynamic equation with multi-backlash, time-varying mesh stiffness,multi-degree of freedom is set up. The Newmark numerical integration method is used to solve thenonlinear dynamics differential equations from which the result of the nonlinear dynamic response isgot. The influences of the backlash, time-varying mesh stiffness, meshing ratio of damping, integrateerror on nonlinear dynamics characteristics for the encased differential herringbone train are analyzedby using time process diagram of displacement response, time process diagram of meshing forceresponse, phase diagram, Poincaré section. This impact on the gear meshing force, motion state is got.
在传动系统的啮合与切向刚度研究中，确定了传动系统两级啮合齿轮的各种相位关系，引入基于齿轮副瞬时总接触线长度推导出的斜齿轮时变啮合刚度公式按刚度并联方式计算人字齿时变啮合刚度，分析了人字齿轮啮合刚度波动小的原因。应用材料力学原理计算由单位载荷引起的轮齿位移即柔度，进而确定中间浮动构件人字齿轮接触切向刚度。
在传动系统的齿轮误差等效位移研究中，采用简谐函数推导了偏心误差、齿频误差转化到齿轮副啮合线上的等效位移公式；提出了一种齿轮传动几何误差转化为啮合线上的等效位移计算方法，按此方法推导出了传动系统两级各齿轮安装误差转化到啮合线等效位移的计算公式，进而建立了传动系统两级各齿轮误差转化到啮合线上等效位移的完整公式体系。
在传动系统的静力学均载特性研究中，建立了包含中间浮动构件的封闭差动人字齿轮传动系统静力学计算模型；确定了传动系统静力学均载系数计算公式，计算了传动系统的静力学均载系数；分析了传动系统主要参数对传动系统静力学载荷分配的影响，获得主要参数对传动系统静力学均载特性的影响规律。
在传动系统的动力学均载特性研究中，考虑了齿轮重量、时变啮合刚度、各种误差的影响，建立了包含中间浮动构件的封闭差动人字齿轮传动系统动力学计算模型；把动力学方程线性化，采用傅立叶级数法求解；确定了传动系统动力学均载系数计算公式，计算了传动系统的动力学均载系数；分析了传动系统主要参数对传动系统动力学载荷分配的影响，获得封闭差动人字齿轮传动系统动力学均载特性的变化规律。
在传动系统的动力学浮动特性研究中，建立了封闭差动人字齿轮传动系统动态浮动量的计算方法，计算了传动系统两级各齿轮动力学浮动量，分析了传动系统的各种参数对传动系统动力学浮动量的影响。获得封闭差动人字齿轮传动系统动力学浮动特性的变化规律。
在传动系统的非线性动力学特性研究中，建立了多齿侧间隙、时变啮合刚度的封闭差动人字齿轮传动系统的多自由度扭转非线性动力学方程；应用Newmark数值法求解非线性动力学微分方程组，得到了传动系统的非线性动态响应结果；综合运用位移响应时间历程图、啮合力响应时间历程图、相图、庞加莱截面，分析了齿侧间隙、时变啮合刚度、阻尼、综合误差对封闭差动人字齿轮传动系统非线性动态特性的影响；获得了齿侧间隙、时变啮合刚度、阻尼、综合误差对啮合轮齿的受力、运动状态的影响规律。
The load sharing is particularly important for improving the life of planetary gear train, increasingreliability and reducing vibration. The theory of nonlinear dynamics is required to study the vibrationmechanism of the strong nonlinear planetary gear train. So the load sharing and nonlinear dynamicscharacteristics of the planetary gear train have become the current hotspots and difficult issues. Thepaper mainly studies the load sharing and nonlinear dynamics characteristics of encased differentialherringbone train (main reducer of large ships) to provide theoretical and technical support for thedesign of this train.
In the research on the meshing and tangential stiffness, the phase relationships of the meshinggears are determined in the train, and the mesh stiffness formula of the helical gear which is derivedbased on the instantaneous total length of the contact line for the gear pair is introduced to calculatethe time-varying mesh stiffness of the herringbone on the stiffness parallel. So the cause of the smallfluctuation of the herringbone meshing stiffness is analyzed. In the meantime the materials mechanicsprinciple is applied to calculate the tooth displacement which is also called as flexibility caused by theunit load. And then the herringbone tangential stiffness of the intermediate floating component for thistrain is calculated.
The equivalent displacement formulas of the run-out and meshing-frequency errors along themeshing line are deduced by using harmonic function. An equivalent displacement calculation methodof the geometric errors for the gear transmission is proposed and the equivalent displacement formulaof the gear installation error along the meshing line is deduced based on this method. So the completeequivalent displacement formula system of the gear errors along the meshing line is set up in thistrain.
The static model which includes the intermediate floating component of this train is set up. Thestatic formulas of load sharing coefficients for this train are defined. The static load sharingcoefficients are calculated and the impact of the main parameters on the static load sharing isanalyzed.
The impact of the gear weight, the time-varying mesh stiffness, the gear errors on this train hasbeen considered, and the dynamics model which include the intermediate floating component of thistrain is set up. The dynamic equation is linearized and the Fourier series method is used to solve thisequation. The dynamics formulas of load sharing coefficients for this train are defined. The dynamics load sharing coefficients are calculated and the influence of the main parameters on the dynamics loadsharing for this train is analyzed. The dynamics load sharing characteristics for the encaseddifferential herringbone train are obtained.
The calculating method of the dynamics floating displacements for this train is established.Thedynamics floating displacements of the gears for the train are calculated, and the impact of the mainparameters on dynamics floating displacements of the train are analyzed. The dynamics floatingcharacteristics for the encased differential herringbone train are obtained.
A nonlinear torsional dynamic equation with multi-backlash, time-varying mesh stiffness,multi-degree of freedom is set up. The Newmark numerical integration method is used to solve thenonlinear dynamics differential equations from which the result of the nonlinear dynamic response isgot. The influences of the backlash, time-varying mesh stiffness, meshing ratio of damping, integrateerror on nonlinear dynamics characteristics for the encased differential herringbone train are analyzedby using time process diagram of displacement response, time process diagram of meshing forceresponse, phase diagram, Poincaré section. This impact on the gear meshing force, motion state is got.
引文
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