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首页> 外文学位 >Vibration of a spinning, cyclic symmetric rotor assembled to a flexible stationary housing via multiple bearings.
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Vibration of a spinning, cyclic symmetric rotor assembled to a flexible stationary housing via multiple bearings.

机译：通过多个轴承将旋转的对称循环转子装配到一个固定的固定壳体上的振动。

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摘要

A spinning cyclic symmetric rotor mounted on a stationary housing via multiple bearings is a very common platform used in modern rotary machinery. Representative examples include propellers, wind turbines, bladed turbine disks, and compressors. Nowadays, there are two main industrial trends in designing machines with the cyclic symmetric rotors. The first is to use larger rotors and lighter housing in order to increase efficiency and reduce costs. The second is to employ ground-based measurements.;Motivated by the industrial trends, this research is to develop a reduced-order formulation that accommodates arbitrary geometry of the spinning rotor and the stationary housing. Such a formulation is universal and will be valid for various cyclic symmetric rotors, ranging from wind turbines to bladed turbine disks. Mathematically, the governing equation of motion with reduced order takes the form of a set of ordinary differential equations with periodic coefficients associated with the spin speed. Characteristics drawn from this formulation can then be applied to any cyclic symmetric rotor mounted on a flexible housing. Furthermore, unstable response (e.g., parametric resonances) and stable response (e.g., rotor-based and ground-based response) of the rotor-bearing-housing system will be studied analytically, numerically, and experimentally, as follows.;In the analytical study, the system is first shown to have instabilities in terms of combination resonance of the sum type as a result of the periodic coefficients. All the resonance peaks and corresponding bandwidth within a proper range of spin speeds, where the system remains positive definite, are analytically predicted by the method of multiple scales. As a result of combination resonance of the sum type, the instability occurs at extremely high spin speeds.;Next, the stable response of the stationary and spinning system is studied analytically. The stationary system has two types of modes: rotor-dominant modes and housing-dominant modes. For the spinning system, two types of stable response are studied: the rotor-based response and the ground-based response. Both responses of rotor-dominant modes are similar to the case with rigid housing. The rotor-based response of housing-dominant modes, however, possesses a specific frequency splitting due to dominant vibration of the housing. For a housing-dominant mode with natural frequency o(H) obtained from the stationary system, when the system is spinning at spin speed o 3, the rotor-based response splits into a forward and backward frequency branch equal to o(H)+/-o3. Such frequency splitting is defined as gyroscopic splitting. The gyroscopic splitting is analytically predicted via a perturbation analysis. Subsequently, the ground-based response is theoretically predicted. The theoretical prediction is briefly summarized as follows. The rotor-based response of a housing-dominant mode has frequency components o(H)+/-o 3 due to gyroscopic splitting. Furthermore, if a rotor-based, cylindrical coordinate (r,θ, z) is employed to describe the vibration mode shape of a cyclic symmetric rotor, the mode shape is circumferentially modulated by the exponential function ejk , where k is the harmonic number which follows the identity k = n + M( N). In this identity, n is the phase index governed by the cyclic symmetry of the rotor while M(N) is multiples of numbers of identical substructures N. When the response is viewed from a ground-based observer, the circumferential harmonics kθ gives rise to additional frequency splitting -ko 3. Together with the gyroscopic splitting, the ground-based response splits into multiple forward and backward frequency branches following the rule o(H)-(k+/-1)o 3.;To confirm the results from the analytical study, a benchmark numerical model consisting of a cyclic symmetric rotor, a stationary housing, and two bearings is developed. The rotor is a circular disk with four evenly spaced radial slots and a rigid hub. The stationary housing is a square plate with a central shaft subjected to fixed boundary conditions on the displacements at four corners. Based on this model, a numerical integration of the equation of motion ad use of the Floquet theory confirms the parametric resonance frequency and the instability bandwidth obtained from the method of multiple scales. Through the benchmark model, the gyroscopic splitting is also numerical confirmed for the rotor-based response. Moreover, ground-based response at various speed in the form of waterfall plots confirms that a housing-dominant mode splits following the rule (k +/- o3).;In order to verify the theoretical prediction of the ground-based response, a series of experiments on a stationary and spinning test rig is carried out. First of all, frequency response functions (FRFs) of the stationary rig are measured. Two FRFs are obtained using two excitation mechanisms. The first is to use an automatic hammer while the second is to use a piezoelectric (PZT) actuator. Two housing-dominant modes are identified by comparing the FRFs. Their mode shapes are characterized by one-nodal diameter and one-nodal line on the rotor and housing, respectively. Next, ground-based response of the spinning rig is measured to obtain waterfall plots. For the waterfall plot obtained form the hammer excitation, both housing-dominant modes reveal forward frequency branches which agree very well with the theoretical prediction. Only one housing-dominant mode presents a backward frequency branch. Nonetheless, the backward branch also agrees well with the theoretical prediction.;Lastly, a closed-form solution of rotor-bearing-housing systems with a special class of cyclic symmetry is derived. Specifically, the equation of motion can be transformed into a set of ordinary differential equations with constant coefficients, when the hub is rigid and the flexible portion of the rotor has only out-of-plane vibration motion. The transformed equation of motion appears as a time invariant gyroscopic system, whose closed-form solution is hence readily available. Both the original and transformed equation of motion are shown to have identical instabilities and rotor-based response through numerical simulations via the benchmark model.
机译：通过多个轴承安装在固定壳体上的旋转对称循环转子是现代旋转机械中非常常见的平台。代表性的例子包括螺旋桨，风力涡轮机，叶片涡轮盘和压缩机。如今，设计带有循环对称转子的机器有两个主要的工业趋势。首先是使用更大的转子和更轻的外壳，以提高效率并降低成本。第二是采用基于地面的测量。受工业趋势的影响，本研究旨在开发一种降序配方，以适应纺纱转杯和固定壳体的任意几何形状。这样的表述是通用的，并且将适用于从风力涡轮机到叶片式涡轮盘的各种循环对称转子。在数学上，降序运动的控制方程采用一组常微分方程的形式，其周期系数与自旋速度相关。从该公式得出的特性然后可以应用于安装在柔性外壳上的任何循环对称转子。此外，将对转子轴承壳体系统的不稳定响应（例如，参数谐振）和稳定响应（例如，基于转子和基于地面的响应）进行分析，数值和实验研究，如下所示：研究表明，由于周期系数，该系统首先表现出在求和型组合共振方面的不稳定性。通过多标度的方法分析预测了在系统保持正定旋转的适当转速范围内的所有共振峰和相应的带宽。由于求和类型的组合共振，在极高的自旋速度下会出现不稳定。接下来，将对固定和旋转系统的稳定响应进行分析研究。固定系统有两种模式：转子主导模式和壳体主导模式。对于纺纱系统，研究了两种类型的稳定响应：基于转子的响应和基于地面的响应。转子主导模式的两种响应都与刚性外壳的情况相似。但是，由于壳体的主导振动，壳体主导模式的基于转子的响应具有特定的频率分割。对于从固定系统获得的固有频率为o（H）的壳体主导模式，当系统以自旋速度o 3旋转时，基于转子的响应分为正向和反向频率分支，等于o（H）+ / -o3。这种频率分割被定义为陀螺分割。通过扰动分析来解析地预测陀螺的分裂。随后，从理论上预测了地面响应。理论预测简要总结如下。由于陀螺分裂，壳体主导模式的基于转子的响应具有频率分量o（H）+/- o 3。此外，如果采用基于转子的圆柱坐标（r，θ，z）来描述循环对称转子的振动模式形状，则该模式形状将通过指数函数ejk进行圆周调制，其中k为谐波数，其中遵循恒等式k = n + M（N）。在这种情况下，n是由转子的循环对称性决定的相位指数，而M（N）是相同子结构N的数目的倍数。当从地面观察者观察响应时，圆周谐波kθ会增加额外的频率分割-ko 3.结合陀螺分割，将地面响应按照规则o（H）-（k +/- 1）o 3.分成多个前向和后向频率分支。通过分析研究，开发了一个由循环对称转子，固定壳体和两个轴承组成的基准数值模型。转子是一个圆盘，具有四个均匀分布的径向槽和一个刚性轮毂。固定壳体是一个方形板，其中心轴在四个角的位移上受到固定的边界条件的影响。在此模型的基础上，通过使用Floquet理论对运动方程进行数值积分，可以确定参数共振频率和从多尺度方法获得的不稳定性带宽。通过基准模型，陀螺劈裂也得到了基于转子响应的数值确认。此外，以瀑布图形式出现的各种速度下的地面响应，都证实了住房主导模式遵循规则（k +/- o3）分裂。为了验证地面响应的理论预测，在固定式和旋转式试验台上进行了一系列实验。首先，测量固定钻机的频率响应函数（FRF）。使用两种激励机制可获得两个FRF。第一种是使用自动锤，第二种是使用压电（PZT）致动器。通过比较FRF可以确定两种住房主导模式。它们的模态形状分别由转子和壳体上的一节点直径和一节点线来表征。接下来，对纺纱机的地面响应进行测量以获得瀑布图。对于通过锤子激励获得的瀑布图，这两种壳体主导模式都揭示了正向频率分支，这与理论预测非常吻合。只有一种以房屋为主的模式会出现向后的频率分支。尽管如此，后向分支也与理论预测相吻合。最后，推导了具有特殊循环对称性的转子轴承壳体系统的闭式解。具体地，当轮毂是刚性的并且转子的柔性部分仅具有平面外的振动运动时，运动方程可以被变换为具有恒定系数的一组常微分方程。变换后的运动方程显示为时不变陀螺仪系统，因此可以轻松获得其闭式解。通过基准模型进行的数值模拟表明，原始运动方程和变换后的运动方程都具有相同的不稳定性和基于转子的响应。

著录项

  • 作者

    Tai, Wei Che.;

  • 作者单位

    University of Washington.;

  • 授予单位 University of Washington.;
  • 学科 Engineering Mechanical.
  • 学位 Ph.D.
  • 年度 2014
  • 页码 119 p.
  • 总页数 119
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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