Investigation on steady state deformation and free vibration of a rotating inclined Euler beam

The steady state deformation and infinitesimal free vibration around the steady state deformation of a rotating inclined Euler beam at constant angular velocity are investigated by the corotational finite element method combined with floating frame method. The element nodal forces are derived using the consistent second order linearization of the nonlinear beam theory, the d'Alembert principle and the virtual work principle in a current inertia element coordinates, which is coincident with a rotating element coordinate system constructed at the current configuration of the beam element. The rotating element coordinates rotate about the hub axis at the angular speed of the hub. The equations of motion of the system are defined in terms of an inertia global coordinate system, which is coincident with a rotating global coordinate system rigidly tied to the rotating hub. Numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method and to investigate the steady state deformation and natural frequency of the rotating inclined beam.     

Flapwise bending vibration analysis of rotating composite cantilever beams

A modeling method for the modal analysis of a rotating composite cantilever beam is presented in this paper. Linear differential equations of motion are derived using the assumed mode method. For the modeling, hybrid deformation variables are employed and approximated to derive the equations of motion. Symmetrical laminated composite beams are considered to obtain the numerical results. The effects of the dimensionless angular velocity, the hub radius and the fiber orientation angle on the variations of modal characteristics are investigated.     

Vibration and reliability of a rotating beam with random properties under random excitation

In this paper, vibration and reliability of a rotating beam with random properties under random excitations are studied. The rotating beam is under a stochastic load modeled as a stationary white noise. The cross-sectional area, elasticity modulus, moment of inertia, shear modulus, damping coefficient, mass density and rotational speed are modeled as random variables. To develop the equations of motion, the finite element method and space state analysis are applied. In order to consider the randomness of properties, a second order perturbation method is used. The effects of rotational speed, setting angle, hub radius, variances of random properties, correlation of random variables and damping matrix forms on the vibration and reliability of rotating beams, are studied completely.     

Modeling and free vibration behavior of rotating composite thin-walled closed-section beams with SMA fibers

Smart structure with active materials embedded in a rotating composite thin-walled beam is a class of typical structure which is using in study of vibration control of helicopter blades and wind turbine blades. The dynamic behavior investigation of these structures has significance in theory and practice. However, so far dynamic study on the above-mentioned structures is limited only the rotating composite beams with piezoelectric actuation. The free vibration of the rotating composite thin-walled beams with shape memory alloy(SMA) fiber actuation is studied. SMA fiber actuators are embedded into the walls of the composite beam. The equations of motion are derived based on Hamilton’s principle and the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin’s method. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. Numerical results of natural frequency are obtained for two configuration composite beams. It is shown that natural frequencies of the composite thin-walled beam decrease as SMA fiber volume and initial strain increase and the decrease in natural frequency becomes more significant as SMA fiber volume increases. The actuation performance of SMA fibers is found to be closely related to the rotational speeds and ply-angle. In addition, the effect of the pitch angle appears to be more significant for the lower-bending mode ones. Finally, in all cases, the precone angle appears to have marginal effect on free vibration frequencies. The developed model can be capable of describing natural vibration behaviors of rotating composite thin-walled beam with active SMA fiber actuation. The present work extends the previous analysis done for modeling passive rotating composite thin-walled beam.     

Aeroelastic behavior of cantilevered rotating rectangular plates

Aeroelastic behavior of a supersonic rotating rectangular plate in the air medium is studied. For simulating the plate structure, the Mindlin first-order shear deformation plate theory along with Von Karman nonlinear terms is employed. Air dynamic pressure is modeled using first-order piston theory. The plate is placed inside a rigid baffle to remove shock waves generated by plate rotation and provide a uniform flow passing over the plate. Nonlinear dimensionless generalized equations of motion are presented based on the Kane dynamic method. After linearization of the nonlinear equations of motion, effect of different parameters including plate aspect ratio, thickness ratio, hub radius ratio and dimensionless rotation speed on aeroelastic behavior of the system are investigated. Frequency locking and high frequency flutter phenomenon are observed in the numerical results.     

Buckling and dynamic stability of spinning pre-twisted beams under compressive axial loads

The equations of motion of a spinning pre-twisted beam under compressive axial loads are formulated using Euler beam theory and the assumed mode method. The equations of motion are first transformed to the standard form of an eigenvalue problem for determining the critical buckling loads for various combinations of prescribed spinning speeds, aspect ratio of the cross-section and pre-twist angle of the beam. The equations of motion are then transformed to the form of another eigenvalue problem for determining the critical spinning speeds. Both the critical spinning speeds and critical buckling loads are found to exhibit similar trends of variation and curve veering phenomenon with respect to changes in the aspect ratio of the cross-section. For a spinning pre-twisted beam under axial compressive loads, the critical spinning speeds corresponding to divergent behaviours are found to be no longer the dividing points for separating the speed zones into stable and unstable regions, contrary to the stability behaviour of a nonpre-twisted beam which have distinct regions of stable and unstable spinning speed zones separated by critical spinning speeds.     

Large displacement analysis of elastically constrained rotating disks with rigid body degrees of freedom

A central aspect of the linear vibration theory of rotating disks involves the concept of critical speeds. At such rotation speeds an axisymmetric disk can support a standing wave as recorded by a stationary observer. In such situations an applied space fixed constant force can give rise to a resonance in the disk. Such a response is of concern in industrial applications as diverse as circular saw blades and computer floppy disks. In such situations the magnitude of response may exceed the limits of linear theory. The present paper is concerned with the effects of large displacements upon the disk response in the neighborhood of such critical speeds. The effects of geometric nonlinearities and the influence of rigid body tilting and translation (caused by the boundary conditions) are considered. The equations of motion are based on Von Karman plate theory. The eigenfunctions of two self-adjoint eigenvalue problems, corresponding to the stress function and the transverse displacement, are determined and used as approximation functions in a numerically efficient Galerkin formulation. The coupled nonlinear ordinary differential equations of motion are solved using the Runge–Kutta method. Numerical results are presented for disks that are free to translate and rotate at their inner boundary and are constrained from lateral motion by space fixed linear springs. The effects of vibration magnitude on system response in the sub and super-critical speed regions are computed and the effects of large displacements on critical speed behavior and forced response are investigated. Experiments are conducted to verify the accuracy of the numerical results obtained in this paper.     

旋转变截面扭梁的变形应力分析

本文研究了在剪切变形和转动的影响下,在轴向变形、弯曲变形和剪切变形的作用下的旋转变截面扭梁的单元刚度矩阵。假设扭转角、宽度和厚度沿着梁的长度方向都是线性变化的。给出了转速和轮毂半径对于梁的变形和应力的影响：转速增大时能增加轴线方向的变形和应力,但是却减小弯曲变形、剪切变形以及由它们产生的应力；而轮毂半径主要是影响梁的轴线方向的变形和应力,对弯曲变形和剪切变形的影响比较小。     

In-plane vibration analysis of cantilevered circular arc beams undergoing rotational motion

Equations of motion of cantilevered circular arc beams undergoing rotational motion are derived based on a dynamic modeling method developed in this paper. Kane’s method is employed to derive the equations of motion. Different from the classical linear modeling method which employs two cylindrical deformation variables, the present modeling method employs a non-cylindrical variable along with a cylindrical variable to describe the elastic deformation. The derived equations (governing the stretching and the bending motions) are coupled but linear, so they can be directly used for vibration analysis. The coupling effect between the stretching and the bending motions, which could not be considered in the conventional modeling method, is considered in this modeling method. The effects of rotational speed, arc angle, and hub radius ratio on the natural frequencies of the rotating circular arc beam are investigated through numerical analysis.     

Dynamic modeling, input-shaped maneuvering and vibration suppression of flexible body using quasi-coordinates and euler parameters

In dealing with the dynamics of a flexible body, the rigid-body motions and elastic vibrations are analyzed separately. However, rigidbody motions cause vibrations, and elastic vibrations affect rigid-body motions, indicating the inherent coupling between rigid-body motions and elastic vibrations. The coupled equations of motion for a flexible body can be derived by means of Lagrange’s equations in terms of quasi-coordinates. The resulting equations of motion are hybrid and nonlinear. This paper proposes a unified approach for the equations of motion for a flexible body based on the perturbation method and the Lagrange’s equations of motion in terms of quasicoordinates and Euler parameters to analyze a more general case maneuvering. The resulting equations consist of zero-order nonlinear equations of motion which depict rigid-body motions and first-order time-varying linear equations of motion which depict perturbed rigid-body motions and elastic vibration. Hence, the input-shaped maneuvering can be applied to the zero-order equation considering the induced vibrations. Since the input-shaped maneuvering alone cannot achieve vibration suppression, the vibration suppression controller combined with the input-shaped maneuvering is proposed in this study. As a numerical example, a hub with elastic appendages is considered. Numerical results show that the unified modeling approach proposed in this paper is effective in numerical simulation and control design.     

Numerical techniques for computing nonlinear dynamic characteristic of rotor-seal system

A transient CFD procedure to compute the nonlinear dynamic characteristic of the coupled rotor-seal system was presented in this study. In each time step, the displacement diffusion model was implemented to govern the mesh deformation, and the URANS (Unsteady Reynolds averaged Navier-Stokes) equations were solved to obtain the transient fluid force on the rotor surface for the free vibrations. With the obtained fluid force from the CFD solver, the nonlinear equations of motion for a simplified rotor-seal system were numerically solved on the basis of external user defined routines. During each transient time step, the computed fluid force from the CFD solver and the rotor motion from the user defined routines were transferred to each other timely. The rotor center trajectories, frequency spectrum and projection of Poincaré section were calculated to investigate the nonlinear dynamic performance of the single disk rotor-seal system. The effects of the rotational speed and pressure ratio on the vibration characteristic of the rotor-seal system were analyzed by the bifurcation theorem. The results show that the coupled rotor-seal system experiences a period-one motion, resonance, periodic-doubling motion, quasi-periodic motion, and finally possible chaotic motion as the rotor speed increases. The pressure ratio has pronounced effect on the frequency response of the first-order critical speed; however, it has little influence on the motion state as well as the frequency response of the rotating speed. Although a constant-clearance annular smooth stator seal was selected as the research object in the current paper, the presented transient CFD method is still available for other complex annular seals, such as labyrinth seal, honeycomb seal, pocket damper seal, etc.     

C0-continuous isoparametric Timoshenko beam element for rotating

This paper presents a C0-continuous isoparametric finite element for free vibration analysis of a rotating, tapered Timoshenko beam mounted on the periphery of a rotating rigid hub, at a setting angle with the plane of rotation. The finite element has three nodes and two degrees of freedom per node and employs modified shape functions for rotational displacement associated with the shear strain energy to avoid shear locking. To obtain a finite element equation of the generalized eigenvalue problem, Hamilton’s principle is applied to kinetic and potential energy expressions of a rotating Timoshenko beam with non-zero setting angle. The numerical solutions for various situations including variations of rotational speed, taper ratio, slenderness ratio, hub radius and setting angle are compared with other numerical results available in the literature whenever possible. The results show that the new 3-noded isoparametric element yields accurate results when compared to other numerical ones.     

非惯性系下旋转叶片的动力学分析

利用Ho-Hoon Lee提出的在满足叶片弯曲机制情形下的新的动力学建模方法对盘叶这种常见的机械系统进行动力学建模。并用拉格郎日方法和假设模态法离散推导出了系统的动力学方程。在建立系统动力学方程的过程中考虑了未被Ho-Hoon Lee[2]所考虑的盘的半径,同时给出了盘叶系统的数值仿真结果。并与文献[3]中简化的一阶近似耦合模型在理论上和数值上进行了比较。     

Dynamics of a rotating Rayleigh beam subject to a repetitively travelling force

The dynamics of a rotating Rayleigh beam subject to a force travelling at a constant speed along the axial direction is studied. The beam is chosen as a simple model of the workpiece in the lathing process. A technique is developed for modeling the repetitive cutting force on the workpiece. The amplitude of the cutting force is chosen to be either constant or dependent on the motion of the beam. The discretized equations of motion of the rotating beam are obtained by Galerkin’s method. The time response of the rotating beam subject to the external force is discussed. The possible resonant conditions resulting in divergent solutions are studied. The stability of the response, due to a travelling motion-dependent force, is determined by the method of multiple scales. The effects of varying the rotating frequency, the travelling speed of the external force, and the movement of the force are also investigated.     

轴向载荷作用下细长柔性旋转梁横向振动响应分析

将水平井柔性钻柱简化为轴向载荷作用下的细长柔性梁,建立了考虑几何大变形的非线性动力学方程,并且利用多重尺度法得到了柔性梁横向振动的平均方程。因钻柱转速和轴向力是影响钻柱振动的两个主要参数,故用改变转速的方法来对平均方程进行非线性动力学响应分析,给出了一阶、二阶模态下钻柱随转速变化的分叉图和不同转速下的相图。分析结果表明,系统响应经历了从周期到混沌再到二倍周期的变化过程,振动幅值有跳变现象出现,在共振条件下,柔性旋转梁振动幅值是有限值,并非无穷大,当轴向力发生变化时,可以通过控制转速变化达到定性、定量地控制柔性钻柱横向振动的目的。     

Theoretical and experimental motion analysis of Self-Compensating Dynamic Balancer

A theoretical and experimental approach was used to investigate the motion and effectiveness of a Self-Compensating Dynamic Balancer (SCDB). This is a device intended to minimize the effects of rotor imbalance and vibratory forces on a rotating system during normal operation. The basic concept of an automatic dynamic balancer has been described in many U.S. patents. The SCDB is composed of a circular disk with a groove containing massive balls and low viscosity damping fluid. The objective of this research is to, determine the motion of the balls and how this ball motion is related to the vibration of the rotating system using both theoretical and experimental methods. The equations of motion of the balls were derived by the Lagrangian method. Static and dynamic solutions were derived from the analytic model. To consider the dynamic stability of the motion, perturbation equations were investigated by two different methods: Floquet theory and direct computer simulation. On the basis of the results of the stability investigation, ball positions which result in a balance system are stable above the critical speed and unstable at critical speed and below critical speed. To determine the actual critical speed of the rotating system used in the experimental work, a modal analysis was conducted. Experimental results confirm the predicted ball positions. Based on the theoretical and experimental results, when the system operates below and near the first critical speed, the balls do not balance the system. However, when the system operates above the first critical speed the balls can balance the system.     

Parametric instability of a spinning pretwisted beam under periodic axial force

This paper addresses the parametric instability of a cantilever pretwisted beam rotating around its longitudinal axis under a time-dependent conservative end axial force which contains a steady-state part and a small periodically fluctuating component. This structural element can be used to model fluted cutting tools such as the twist drill bit and the end milling cutter, etc. Using the Euler—Bernoulli beam theory and Hamilton's principle, the present study derives the equation of motion which governs the lateral vibration of a spinning pretwisted beam. Rotary inertia, structural viscous damping and conservative end axial force are included. The Galerkin method is then applied to obtain the associated finite element equations of motion. Due to the existence of the Coriolis force, the resulting finite element equations of motion are transformed into a set of first-order simultaneous differential equations by a special modal analysis procedure. This set of simultaneous differential equations is solved by the method of multiple scales, yielding the system response and expressions for the boundaries of the unstable regions. Numerical results are presented to demonstrate the effects of pretwist angle, spinning speed and steady-state part of the end axial force on the parametric instability regions of the present problem.     

非线性油膜力作用下联轴器耦合转子系统的稳定性和分岔

在非线性油膜力作用下 ,建立了转子 -联轴器系统的运动微分方程。非线性油膜力则采用有限差分法求解 ,对联轴器耦合的刚性转子系统和柔性转子系统分别进行了计算和分析。结果表明对于平衡转子系统在平衡点失稳后 ,在一个较大的转速范围内存在着稳定的涡动轨迹 ,联轴器两侧转子的质量相差越大 ,则发生 Hopf分岔所对应的转速越高。对于不平衡转子系统 ,在同步涡动轨道失稳后 ,系统将产生准周期和倍周期等一系列的分岔现象 ,并且质量偏心的位置对系统的动力学行为有影响。     

机械密封动环变形对端面温度场的影响

在合理的假设条件下,考虑到液膜对动环的热量分配系数,基于液体润滑非接触机械密封稳态传热模型,利用ANSYS软件分别对动环变形前后温度场进行计算,得出动环端面温度分布规律,并分析主轴转速、变形角β、液膜厚度、密封环导热系数等参数对温度场的影响。分析结果表明:变形后温度要明显高于变形前温度;动环的高温区出现在内径处;主轴转速、材料的导热系数对动环端面温度有较大影响。     

Robust rotor dynamics for high-speed air bearing spindles

For ultra-precision machining machine tool components need to operate outside critical frequencies of the machining system to avoid insufficient surface finish caused by vibrations. This particularly applies to tooling spindles as those are generally the component of a machine tool with low stiffness and damping values. Surface finish and shape of a machined part rely directly on the overall accuracy in motion of the tooling spindle over the entire machining parameter and speed range. Thus spindle designs for an operation outside critical frequencies combined with high stiffness and damping values are crucial for ultra-precision machining.For sufficient stiffness properties bearing gaps of gas bearings have to have a size of only a few microns and show a distinct sensitivity on temperature and for journal bearings also on speed. This again means that bearing properties change with temperature and speed. Considering a spindle system comprising a rigid shaft rotating in a radial/axial bearing system with changing stiffness and damping properties leads to a resonance speed map with changing rigid mode resonance speeds.This paper treats the influence of shaft speed and temperature on bearing gaps from which rigid mode resonance speeds for a shaft spinning in a bearing system are derived. The quoted influence of centrifugal load and temperature on bearing stiffness, damping and load capacity can be applied to any kind of gas bearing. Therefore the calculation of bearing stiffness, damping or load capacity is not treated in detail. The reader will be shown that there are simple design rules for air bearing systems and shafts of high-speed tooling spindles to avoid critical speeds through the entire speed range. Finally, methods of how to prove the initial design goals and how to verify dynamics of high-speed spindles in production will be presented to the reader. It will also be shown that there are production high-speed spindles available which do not include any critical speed within their speed range and thus show robust rotor dynamics with extremely low errors in motion.Procedures in design, validation and application treated in this paper shall give the reader not only design guidelines for spindles to avoid critical spindle speeds within its speed range, but also recommendations for machine tool builders and end-users for a machine operation taking machine and rotor dynamics into account. As the knowledge for this paper is predominantly based on the experience and work of the author himself only a few references are used. However presented testing results entirely confirm the approach presented in this paper.