齿侧间隙非线性函数拟合阶次的选取研究

针对齿侧间隙非线性函数拟合阶次的选取问题,以斜齿轮传动系统为例,对齿轮系统进行了理论建模、仿真分析,并对实际齿轮系统进行了实验研究。首先,采用集中质量法建立了考虑轴向振动的齿轮非线性系统模型,并对数学模型进行了无量纲化,对齿侧间隙函数分别进行了无拟合、4阶拟合和8阶拟合处理;然后,采用Runge-Kutta数值积分法进行了数值仿真分析,得到了齿轮系统x、y、z三个方向的幅值-频率比响应曲线图,研究了拟合阶次对系统x、y、z三个方向动态特性的影响;最后,进行了齿轮系统动态测试,得到了齿轮系统x、y、z三个方向的振动响应图,根据仿真与实验结果得出了结论。研究结果表明:不同处理方式下的齿轮系统振动响应在z向的差异较大,拟合阶次的选取应重点考虑z向;在x、y方向,齿侧间隙非线性4阶拟合函数会提前改变齿面接触状态且共振振幅分别增大1倍和1/3倍;对于多间隙的齿轮非线性系统,对齿侧间隙进行无拟合处理,可能导致计算困难和结果不准确;在固有频率附近,对齿侧间隙进行4阶拟合处理能够最大程度弥补其他非线性因素带来的影响;在其他频率范围,对齿侧间隙进行8阶拟合处理会更接近真实齿轮系统。齿侧间隙非线性函数的拟合处理在提高了计算效率的同时,也提高了模型的准确性。

间隙非线性齿轮系统周期解结构及其稳定性研究

基于含有间隙的单自由度齿轮系统非线性动力学模型,运用伪不动点追踪法探讨了系统状态空间中同时存在的多重周期解,研究了阻尼参数和激励频率变化时系统周期解结构的变化,并进一步由预测—校正算法研究了各周期解的稳定性及分岔情况。研究结果表明,伪不动点追踪法可以有效地用于求解非线性动力系统的多重周期解结构,并可为周期运动的稳定性和分岔规律的研究提供良好的迭代初值,而系统的稳定性分岔规律则有助于更加细致地研究通向混沌的途径。

强非线性单级齿轮系统的周期运动与全局分岔

针对含间隙和时变啮合刚度的强非线性齿轮系统动力学模型,用4阶变步长Runge-Kutta法对系统行为进行数值计算,做出了系统在不同参数值下的全局分岔图,显示了动力学系统通过倍周期分岔走向混沌的道路,并且为实际系统进行参数优化提供理论依据。通过分析揭示了强非线性齿轮系统存在着复杂的分岔结构,并为进一步研究其非线性动力学性能及控制提供理论依据。

齿轮系统倍周期分岔和混沌层次结构的研究

针对考虑间隙和时变啮合刚度的强非线性齿轮系统动力学模型,讨论了混乱带中倍周期分岔现象及混沌的层次结构问题。利用频谱分析法对周期运动和混沌运动进行判断,并对嵌于不同混乱带中的周期轨道进行区分。运用分频采样法求解强非线性齿轮系统主倍周期分岔序列与混沌带合并序列,以及混沌带中周期窗口和混沌窗口共存的层次结构问题。通过分析揭示了强非线性齿轮系统存在着复杂的分岔结构和普适规律,并为深入研究机械系统非线性动力学行为的性态提供参考。

齿轮系统周期运动稳定性研究

针对含间隙的强非线性齿轮系统动力学模型,用数值方法研究了当系统参数和初始条件变化时周期运动的稳定性。基于Floquet分岔理论将预测-校正算法用于讨论参数变化时周期解的稳定性,得到精确的分岔点参数值;通过胞映射法求得周期吸引子的吸引域,引入稳定性品质因子用以定量分析初始条件变化时周期运动的稳定性。该研究结果可为非线性动力学行为的分析和齿轮系统的设计提供参考。

Theoretical and experimental study on non-linear vibration characteristic of gear transmission system

In order to investigate the vibration of gear transmission system with clearance, a vibratory test-bed of the gear transmission system was designed. The non-linear dynamic model of the system was presented, with consideration of the effects of nonlinear dynamic gear mesh excitation, flexible rotors and bearings. Integration method was used to investigate the non-linear dynamic response of the system. The results imply that when the mesh frequency is near the natural frequency of gear pair, it is the first primary resonance, the bifurcation appears, and the vibration becomes to be chaotic motion rapidly. When the speed is close to the natural frequency of the first-order bending vibration, it is the second primary resonance, the periodic motion changes to chaos by period doubling bifurcation. The vibratory measurement of test-bed of the gear transmission system was performed. Accelerometers were employed to measure the high frequency vibration. Experimental results show that the vibration acceleration of the gear transmission system includes mesh frequency and sideband. The numerical calculation results of low speed can be validated by experimental results basically. It means that the presented non-linear dynamic model of the gear transmission system is right.

Forced oscillations with time-varying mesh stiffness and backlash in gear systems

A dynamic model to describe the torsional vibration behaviors of a spur gear system is presented in this paper.Differential equations of nonlinear dynamics for the gear system exhibiting combined nonlinearity influence such as time-varying mesh stiffness,backlash and dynamic transmission error(DTE) were obtained.The method of multiple scales was employed to solve the nonlinear differential equations with parametric excitation in gear systems,by which both the frequency-response curves of the primary resonance caused by internal excitation and the analytical periodic solutions of nonlinear differential equations were obtained.The nonlinear influence of stiffness variation,the damping and the internal excitation on the system response was shown by frequency-response curves.Compared with numerical examples,the approximate analytical solutions are in good agreement with exact solutions,which proves that the method of multiple scales is effective for solving nonlinear problems in gear systems.