陀螺仪转子系统非线性动力特性及稳定性分析
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摘要
陀螺仪是构成惯性导航系统的基础核心器件,其性能很大程度上取决于陀螺仪转子系统的性能。为了提高卫星陀螺仪的精度、灵敏度、寿命和可靠性,开展对陀螺仪转子系统非线性动力特性及稳定性的分析具有重要的学术意义和应用价值。本文建立了滚动轴承-转子系统非线性动力学模型,分析了陀螺仪转子系统的非线性动力特性和稳定性。
本文的主要内容和结论如下:
1、陀螺仪转子系统非线性动力特性分析
建立了考虑变柔度、Hertzian接触力等多种非线性因素的滚动轴承-转子系统动力学模型;提出了一种计算滚动轴承刚度的新方法,该方法可以得到滚动轴承时变的刚度矩阵;研究了系统转速、滚动体数目、等效黏滞阻尼、轴承径向力等结构和工作参数对陀螺仪转子系统的非线性动力特性的影响。结果表明转子系统具有丰富的周期和非周期(拟周期或混沌)响应形式,转子系统进入混沌的主要途径是倍周期分岔,合理的选择转子系统的结构和工作参数,可降低系统的不稳定性。
2、考虑表面缺陷的陀螺仪转子系统非线性动力特性分析
建立了考虑滚动轴承表面波纹度、表面局部缺陷的转子系统非线性动力学模型;采用数值方法、Poincaré映射方法、频谱分析法等,分析了陀螺仪转子系统的分岔、混沌等特性;分析了波纹度波数、幅值,表面缺陷动荷系数等参数与转子系统非线性动力特性间的关系,发现外圈波纹度波数与滚动轴承滚动体数目相等时,转子系统会产生强烈振动;内圈波纹度引起的振动频率与波数有明确的函数关系。提出了依据振动时域参数对表面局部缺陷进行识别的方法。
3、陀螺仪转子系统非线性动力稳定性分析
根据Floquet理论,判断、分析了滚动轴承-转子系统的周期解的稳定性;采用不动点法、直接数值积分法、Poincaré映射、功率谱分析等方法,分析了不同阻尼情况下转子系统周期解的稳定性以及分岔、混沌等特性;研究了阻尼对陀螺仪转子系统非线性稳定性的影响;发现随着阻尼的增大,系统的拓扑结构逐渐变得简单,系统的失稳区间的数目减少,相应的失稳区间变窄,相应的失稳临界转速提高;找到了三种导致转子系统周期解失稳的方式:倍周期分岔失稳、拟周期分岔失稳和阵发性分岔进入混沌失稳。
4、陀螺仪转子系统非线性参数、强迫联合振动
对工程实际中的参数、强迫联合激励的滚动轴承-转子系统,建立了考虑非线性轴承力、径向游隙、变柔度等非线性因素和不平衡力的滚动轴承-转子系统动力学微分方程;利用分岔图、Poincaré映射图和频谱图,分析了参数、强迫联合激励的陀螺仪转子系统的响应、分岔和混沌等非线性动力特性;发现参数、强迫联合激励的陀螺仪转子系统有多种周期和混沌响应形式,其振动频率不仅有参数振动频率成分和强迫振动频率成分,而且有二者的倍频成分和组合频率成分;不平衡力较小时,系统中参数振动占主导地位,增大不平衡力有利于抑制转子系统的不稳定振动。另一方面,随不平衡力的增大,强迫振动逐渐增强,大的不平衡力会诱发系统产生混沌振动。
5、预紧力对陀螺仪转子系统非线性动力特性及稳定性的影响
建立了三自由度的滚动轴承-转子系统动力学模型;研究了轴向预紧力对陀螺仪转子系统的非线性响应、分岔和稳定性等动力学行为的影响;进行了预紧力与系统固有频率、预紧力与轴承保持架转速关系的两项试验,验证本文建立的滚动轴承-转子系统动力学模型的合理性;随着轴向预紧力的增大,系统变柔度振动的幅值减小,固有频率增大,陀螺仪转子系统的稳定性提高。
As a key device of inertial navigation system, gyroscope is widely used in satellite and other space craft. The performance of gyroscope largely depends on the performance of gyroscope rotor system. It is necessary to research nonlinear dynamic characteristics of gyroscope rotor system to improve the performance of satellite gyroscope. In this paper, a nonlinear dynamic model of rolling element bearing rotor system is presented and nonlinear dynamic characteristics and stability of gyroscope rotor system is investigated.
The main research works can be described as follows:
1. Analysis of nonlinear dynamic characteristics of gyroscope rotor system. With multiple nonlinear factors such as varying compliance and Hertzian elastic contact force considered, a dynamic model of rolling element bearing rotor system is presented. A new method to calculate stiffness of rolling element bearing is proposed, by which the time-varying stiffness matrix can be obtained. The effect of structure and running parameters on nonlinear dynamic characteristics and stability of gyroscope rotor system is studied. Simulation results show that abundant kinds of periodic and non-periodic (quasi-periodic and chaotic) responses exist in the system. The main route to chaos is doubling bifurcation. Instability of the system can be reduced by adopting reasonable structure parameters and running parameters in the system.
2. Study on nonlinear dynamic characteristics of gyroscope rotor system due to surface defect.
With the sources of nonlinearity such as inner and outer race surface waviness, Hertzian elastic contact force and localized defect considered, an analytical model of rolling element bearing rotor system is presented. With the aid of numerical algorithm, Poincarémaps, frequency spectrum diagrams, the bifurcation and chaos behaviors of gyroscope rotor system are analyzed. The effect of number and amplitude of waves, impact factor of localized defect on nonlinear response of gyroscope rotor system is studied. Simulation results show that the severe vibration occurs when the number of balls is equal to the number of waves. There is a definitely functional relationship between the number of waves and frequencies of vibration produced by inner race waviness. A method to spot defect by vibration time domain parameters is proposed.
3. Analysis of nonlinear dynamic stability of gyroscope rotor system.
The stability of rolling element bearing rotor system is analyzed by means of Floquet theory. With the aid of fixed-point method, numerical algorithm, Poincarémaps and frequency spectrum diagrams, the bifurcation, chaos and stability behaviors of gyroscope rotor system are analyzed. The effect of damping on stability of gyroscope rotor system is investigated. It is found that damping plays a significant role in stability of system. Three routes to unstable periodic solution are found: doubling bifurcation, quasi-periodic bifurcation and intermittent bifurcation.
4. Analysis of nonlinear vibration of gyroscope rotor system subject to parametrical and external excitations.
The rolling element bearing rotor system in practice is essentially a nonlinear system under parametrical and external excitations. With unbalance force and the sources of nonlinearity such as Hertzian elastic contact force, internal radial clearance and varying compliance considered, the governing differential equations of motion of a rolling element bearing rotor system are derived first and then solved by numerical algorithm. Meanwhile the nonlinear dynamic behaviors of gyroscope rotor system are illustrated by means of bifurcation diagrams, Poincarémaps and frequency spectrum diagrams. Numerical results show that various periodic responses with frequencies of the external forcing one, the parametrical forcing one, or the linear combinations of them, and even chaotic responses may exist. When the unbalance is weak and the parametrical vibration is the dominating one, proper increase of the unbalance force may relieve the risk of parametrical vibration instability. On the other hand, increase of unbalance force makes forced vibration stronger, and an improper increase of unbalance force may induce chaotic response.
5. The effect of axial preload on nonlinear dynamic characteristics and stability of gyroscope rotor system.
A 3-dimention freedom dynamic model of rolling element bearing rotor system is presented and the effect of axial preload on nonlinear response, bifurcation and stability of gyroscope rotor system is analyzed. Experiments have been conducted to study relationship between axial preload and natural frequency of system, and relationship between axial preload and rotating speed of bearing cage. Numerical results are in agreement with experimental ones. It is found that axial preload is an important parameter. It is also shown that increase of axial preload may decrease the amplitude value of varying compliance vibration, increase natural frequency of system, and enhance stability of the system.
本文的主要内容和结论如下:
1、陀螺仪转子系统非线性动力特性分析
建立了考虑变柔度、Hertzian接触力等多种非线性因素的滚动轴承-转子系统动力学模型;提出了一种计算滚动轴承刚度的新方法,该方法可以得到滚动轴承时变的刚度矩阵;研究了系统转速、滚动体数目、等效黏滞阻尼、轴承径向力等结构和工作参数对陀螺仪转子系统的非线性动力特性的影响。结果表明转子系统具有丰富的周期和非周期(拟周期或混沌)响应形式,转子系统进入混沌的主要途径是倍周期分岔,合理的选择转子系统的结构和工作参数,可降低系统的不稳定性。
2、考虑表面缺陷的陀螺仪转子系统非线性动力特性分析
建立了考虑滚动轴承表面波纹度、表面局部缺陷的转子系统非线性动力学模型;采用数值方法、Poincaré映射方法、频谱分析法等,分析了陀螺仪转子系统的分岔、混沌等特性;分析了波纹度波数、幅值,表面缺陷动荷系数等参数与转子系统非线性动力特性间的关系,发现外圈波纹度波数与滚动轴承滚动体数目相等时,转子系统会产生强烈振动;内圈波纹度引起的振动频率与波数有明确的函数关系。提出了依据振动时域参数对表面局部缺陷进行识别的方法。
3、陀螺仪转子系统非线性动力稳定性分析
根据Floquet理论,判断、分析了滚动轴承-转子系统的周期解的稳定性;采用不动点法、直接数值积分法、Poincaré映射、功率谱分析等方法,分析了不同阻尼情况下转子系统周期解的稳定性以及分岔、混沌等特性;研究了阻尼对陀螺仪转子系统非线性稳定性的影响;发现随着阻尼的增大,系统的拓扑结构逐渐变得简单,系统的失稳区间的数目减少,相应的失稳区间变窄,相应的失稳临界转速提高;找到了三种导致转子系统周期解失稳的方式:倍周期分岔失稳、拟周期分岔失稳和阵发性分岔进入混沌失稳。
4、陀螺仪转子系统非线性参数、强迫联合振动
对工程实际中的参数、强迫联合激励的滚动轴承-转子系统,建立了考虑非线性轴承力、径向游隙、变柔度等非线性因素和不平衡力的滚动轴承-转子系统动力学微分方程;利用分岔图、Poincaré映射图和频谱图,分析了参数、强迫联合激励的陀螺仪转子系统的响应、分岔和混沌等非线性动力特性;发现参数、强迫联合激励的陀螺仪转子系统有多种周期和混沌响应形式,其振动频率不仅有参数振动频率成分和强迫振动频率成分,而且有二者的倍频成分和组合频率成分;不平衡力较小时,系统中参数振动占主导地位,增大不平衡力有利于抑制转子系统的不稳定振动。另一方面,随不平衡力的增大,强迫振动逐渐增强,大的不平衡力会诱发系统产生混沌振动。
5、预紧力对陀螺仪转子系统非线性动力特性及稳定性的影响
建立了三自由度的滚动轴承-转子系统动力学模型;研究了轴向预紧力对陀螺仪转子系统的非线性响应、分岔和稳定性等动力学行为的影响;进行了预紧力与系统固有频率、预紧力与轴承保持架转速关系的两项试验,验证本文建立的滚动轴承-转子系统动力学模型的合理性;随着轴向预紧力的增大,系统变柔度振动的幅值减小,固有频率增大,陀螺仪转子系统的稳定性提高。
As a key device of inertial navigation system, gyroscope is widely used in satellite and other space craft. The performance of gyroscope largely depends on the performance of gyroscope rotor system. It is necessary to research nonlinear dynamic characteristics of gyroscope rotor system to improve the performance of satellite gyroscope. In this paper, a nonlinear dynamic model of rolling element bearing rotor system is presented and nonlinear dynamic characteristics and stability of gyroscope rotor system is investigated.
The main research works can be described as follows:
1. Analysis of nonlinear dynamic characteristics of gyroscope rotor system. With multiple nonlinear factors such as varying compliance and Hertzian elastic contact force considered, a dynamic model of rolling element bearing rotor system is presented. A new method to calculate stiffness of rolling element bearing is proposed, by which the time-varying stiffness matrix can be obtained. The effect of structure and running parameters on nonlinear dynamic characteristics and stability of gyroscope rotor system is studied. Simulation results show that abundant kinds of periodic and non-periodic (quasi-periodic and chaotic) responses exist in the system. The main route to chaos is doubling bifurcation. Instability of the system can be reduced by adopting reasonable structure parameters and running parameters in the system.
2. Study on nonlinear dynamic characteristics of gyroscope rotor system due to surface defect.
With the sources of nonlinearity such as inner and outer race surface waviness, Hertzian elastic contact force and localized defect considered, an analytical model of rolling element bearing rotor system is presented. With the aid of numerical algorithm, Poincarémaps, frequency spectrum diagrams, the bifurcation and chaos behaviors of gyroscope rotor system are analyzed. The effect of number and amplitude of waves, impact factor of localized defect on nonlinear response of gyroscope rotor system is studied. Simulation results show that the severe vibration occurs when the number of balls is equal to the number of waves. There is a definitely functional relationship between the number of waves and frequencies of vibration produced by inner race waviness. A method to spot defect by vibration time domain parameters is proposed.
3. Analysis of nonlinear dynamic stability of gyroscope rotor system.
The stability of rolling element bearing rotor system is analyzed by means of Floquet theory. With the aid of fixed-point method, numerical algorithm, Poincarémaps and frequency spectrum diagrams, the bifurcation, chaos and stability behaviors of gyroscope rotor system are analyzed. The effect of damping on stability of gyroscope rotor system is investigated. It is found that damping plays a significant role in stability of system. Three routes to unstable periodic solution are found: doubling bifurcation, quasi-periodic bifurcation and intermittent bifurcation.
4. Analysis of nonlinear vibration of gyroscope rotor system subject to parametrical and external excitations.
The rolling element bearing rotor system in practice is essentially a nonlinear system under parametrical and external excitations. With unbalance force and the sources of nonlinearity such as Hertzian elastic contact force, internal radial clearance and varying compliance considered, the governing differential equations of motion of a rolling element bearing rotor system are derived first and then solved by numerical algorithm. Meanwhile the nonlinear dynamic behaviors of gyroscope rotor system are illustrated by means of bifurcation diagrams, Poincarémaps and frequency spectrum diagrams. Numerical results show that various periodic responses with frequencies of the external forcing one, the parametrical forcing one, or the linear combinations of them, and even chaotic responses may exist. When the unbalance is weak and the parametrical vibration is the dominating one, proper increase of the unbalance force may relieve the risk of parametrical vibration instability. On the other hand, increase of unbalance force makes forced vibration stronger, and an improper increase of unbalance force may induce chaotic response.
5. The effect of axial preload on nonlinear dynamic characteristics and stability of gyroscope rotor system.
A 3-dimention freedom dynamic model of rolling element bearing rotor system is presented and the effect of axial preload on nonlinear response, bifurcation and stability of gyroscope rotor system is analyzed. Experiments have been conducted to study relationship between axial preload and natural frequency of system, and relationship between axial preload and rotating speed of bearing cage. Numerical results are in agreement with experimental ones. It is found that axial preload is an important parameter. It is also shown that increase of axial preload may decrease the amplitude value of varying compliance vibration, increase natural frequency of system, and enhance stability of the system.
引文
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