基于ANSYS的大型屏蔽电机泵转子系统建模及动力学分析

基于ANSYS建立了大型屏蔽电机泵转子系统的有限元模型,通过模态分析和谐响应分析研究了陀螺效应对转子系统临界转速和质量不平衡响应的影响。分析结果表明:陀螺效应对临界转速计算结果影响较大,转子动力学分析时应考虑陀螺效应;考虑陀螺效应后,转子系统临界转速大于设计临界转速;各测点质量不平衡响应一次共振峰值小于横向振动设计限制;叶轮中心和上、下飞轮中心振幅远大于其它位置振幅,为易振动部位,实际运行中应重点选择这些位置进行振动监控。     

具有局部表面损伤的滚动球轴承动力学建模

提出了一种滚动球轴承局部表面损伤故障的动力学分析方法。模型中每个轴承元件具有6个运动自由度,并完整地考虑了陀螺效应、离心力及润滑牵引等动力学因素。通过几何特征的变化对局部表面损伤进行了建模。采用四阶变步长Runge-Kutta-Fehlberg积分法对动力学方程进行了数值求解,并在时域和频域中对动力学响应进行了分析。研究结果表明,由于考虑了润滑剂的牵引效果,计算得到的故障特征频率较纯滚动假设下的计算结果有一定的差异。文中综合考虑了轴承元件的三维运动、相对滑动、润滑效应以及局部表面损伤等因素,因此所提出的模型更为全面、有效和实用。     

计入齿圈柔性的直齿行星传动动力学建模

为揭示齿圈柔性对传动系统动态性能的影响,在系杆随动坐标系下建立计入齿圈柔性的直齿行星传动精细化动力学模型,建模中计入时变啮合刚度、支承刚度、陀螺效应和齿圈柔性等诸多影响因素。采用有限段单元的离散化建模方法,将连续体的柔性齿圈离散成由等效虚拟弹簧连接的刚性齿圈段,并推导齿圈段与行星轮的啮合判据。通过分析各构件间的相对位移关系及其受力,最终建立系统的运动微分方程。以三行星轮的NGW型直齿行星传动为例对其进行固有特性分析,得出系统的低阶固有频率分布及相应的振动模式,并进一步分析齿圈柔性和安装方式对系统固有特性的影响。研究结果表明,直齿行星传动的振型可归结为中心构件(太阳轮、系杆)扭转振动和中心构件平移振动两种模式,两种模式下齿圈和行星轮均做复杂平面振动;齿圈的柔性会降低系统的低阶固有频率,且其影响程度与齿圈安装方式有关,齿圈完全固定时其影响最小,径向浮动或周向转动时影响较明显。     

Chatter suppression in micro-milling using shank-mounted Two-DOF tuned mass damper

The tuned mass damper (TMD) has been used in machining processes for reducing forced vibration, suppressing chatter, and improving machined surface quality. In micro-milling process, the tiny size of the cutting tool-tip and the high rotating speed bring challenges in implementing the TMD. Besides, the TMD needs to have two degrees-of-freedom (DOFs) for reducing vibrations of micro-mill in two orthogonal directions. This paper presents the chatter suppression for micro-milling by attaching a two-DOF TMD to the tool shank and rotates with the cutting tool. The frequency response function (FRF) at the tip of the micro-mill clamped by an aerostatic spindle is predicted using receptance coupling analysis. A two-DOF TMD is designed via graphical approach based on the FRF result at the tool-tip. The natural frequencies and damping ratio of the TMD are optimized under different spindle speeds in order to enhance the cutting stability. The chatter stability of micro-milling is predicted considering the gyroscopic and centrifugal effects of the TMD structure. Modal tests and micro-milling experiments are conducted to validate the effect of the TMD on chatter stability. The results show that the TMD is able to improve the critical depth of cut by 13 folds, and satisfy the compact design requirement for micro-milling.     

Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well

In this paper a nonlinear model was developed for a drillstring system in deviated well with axially moving motion and axial loading, using the perturbation techniques.The drillstring was considered as a simply supported axially moving rotor. Governing equations of motion were obtained based on Hamilton's principle and method of multiple scales was employed to solve the nonlinear motion equations in order to obtain the steady state response and stability region of the system.Then the effects of rotating speed, axial compression load, imbalance mass and nonlinear fluid force on the drillstring responses were investigated in detail and nonlinear natural frequencies and their corresponding mode shapes were presented.Analytical and numerical results reveal the rich and interesting nonlinear phenomena such as primary and parametric resonance that is being investigated for the first time in this study on the nonlinear vibration of the drillstring system.Finally in effort to validate the theoretical approach employed in this study, the numerical solutions obtained here were compared with a set of existing experimental data.     

The eigenvalue problem for damped gyroscopic systems

This paper develops the dynamic orthogonal decoupling method in order to decouple damped gyroscopic structural dynamic systems. The Arnoldi method is extended to solve the generalised eigenvalue problem via a real Schur decomposition. Very interesting results are obtained, which show that this method cannot only obtain the eigenvalues of the system, but can also solve dynamic response of damped gyroscopic dynamic systems in a decoupled way. Compared to the conventional complex mode decoupling method, this method virtually uses only real number computation, remains valid when the corresponding eigenvalue problem is defective and does not require solution of the left eigenvector problem. The structure of the decoupled dynamic equation is very satisfying, because it enhances understanding of the combination of damping with gyroscopic effects. For undamped gyroscopic systems, the decoupling method proposed degenerates to that of Meirovitch.     

Experimental determination of time lag due to phase shift on a flexible pipe conveying fluid

A finite element model of a flexible tube conveying fluid is developed in MATLAB^© based on the principle of virtual work, using a three-node isoparametric beam element. Finite element equations are formulated by applying Galerkin technique on the coupled equations of pipe conveying fluid. The present element developed is capable to model three-dimensional flexible tubes by including curved geometry, effects of damping, velocity and gyroscopic effects. The external excitation applied at the middle of the tube in the lateral direction produced a time lag between the lateral responses, which were measured at two equidistant points from the excitation point. This is due to the Coriolis effect, and the same is simulated using the developed code. An experiment, supported with a robust error analysis, is also conducted on a straight polyurethane tube conveying water, subjected to a sinusoidal excitation at the center between the clamped supports. The measured time responses are compared with the numerical values predicted by the code. The time lags for both cases are obtained from the temporal shift along the time axis, between the zero crossing points of the time–response curves. The results obtained agreed well. The code can be used to predict the time lag, which is correlated to the mass flow rate. The proposed method will help to design Coriolis mass flow meters for existing pipelines, without altering the system.     

Analysis of rotating nonuniform pretwisted beams with an elastically restrained root and a tip mass

Using Hamilton's principle derives the governing differential equations for the coupled bending–bending vibration of a rotating pretwisted beam with an elastically restrained root and a tip mass, subjected to the external transverse forces and rotating at a constant angular velocity. Using the mode expansion method derives the closed-form solutions of the dynamic and static systems. The orthogonal condition for the eigenfunctions of the system with elastic boundary conditions is discovered. The self-adjointness of the system is proved. Moreover, the Green functions of the system are obtained. The symmetric properties of the Green functions are revealed. The frequency response on the steady response of the beam is also investigated.     

Loading rate effects of high damping seismic isolation rubber bearing on earthquake responses

This paper investigates the effects of the loading rates of high damping seismic isolation rubber bearing on earthquake responses. For this purpose, a seismically isolated system is formulated by the Runge-Kutta numerical algorithm with nonlinear rate model of high damping rubber bearing to carry out the seismic time history response analysis. Results, of the seismic time history response analyses using both the present parameter equations and Fujita's equations are compared with those of pseudo dynamic tests. From these results it is confirmed that the parameter equations of high damping rubber bearing should be obtained by tests with the loading rate equivalent to the isolation frequency as close as possible.     

Study on slip displacement and stick-slip characteristics in reciprocating friction drive system

The effects of follower mass, time ratio, driver speed, and normal load on the slip displacement and stick-slip characteristics are experimentally and theoretically investigated in the reciprocating friction drive system under dry contact using a 0.45% carbon steel pair. Results show that the accumulative slip displacement is linearly proportional to the rotating cycle under various operating conditions. The slope for this linear relationship is defined as the slip rate. The slip rate increases with increasing driver speed, the mass of the follower, and |δ−1|, but decreasing normal load. There are three modes of relative motion between the driver ring and the follower in the present study, namely, unstable stick-slip (USS), stable stick-slip (SSS), and sticking (ST) regimes. They are significantly influenced by the driver speed, normal load, time ratio and mass of the follower. The critical operating conditions among unstable stick-slip, stable stick-slip, and sticking regimes were also established. The critical frequency can be theoretically calculated and agrees well with experimental results.     

Dynamic analysis of Coriolis flow meter using Timoshenko beam element

Coriolis mass flow meter (CFM) is used to measure the rate of mass flow through a pipe conveying fluid. In the present work, the Coriolis effect produced in the pipe due to a lateral excitation is modeled using the finite element (FE) method in MATLAB©. The coupled equation of motion for the fluid and pipe is converted to FE equations by applying Galerkin technique. The pipe conveying fluid is excited at its fundamental natural frequency. The time lag observed between symmetrically located measurement points which are equidistant from the point of excitation, is utilized to predict the mass flow rate. The results predicted by the present code is validated using the experimental, and numerical results published in the literature. The main contribution is the development of a FE model, using three node Timoshenko beam element to analyse the dynamics of fluid conveying pipes subjected to external excitation. The direction of the Coriolis force is perpendicular to the plane containing the velocity of flow vector and angular velocity vector of the pipe. Hence a three dimensional FE model is essential. This model can include curved geometry, damping, velocity and gyroscopic effects for three dimensional flexible tubes. The reduced integration used for overcoming shear locking in two node elements, will result in the formation of spurious modes leading to an incorrect prediction of natural frequencies and velocity. These modes will not occur while using three node elements. Influence of spatial as well as temporal discretisation on the time lag and frequency are also discussed. The sensitivity analysis shows that the time lag varies linearly with the mass flow rate.     

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.     

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.     

Dynamic stability of a rotating shaft embedded in an isotropic winkler-type foundation

The dynamic instability of a rotating shaft subjected to axial periodic forces and embedded in an isotropic, Winkler-type elastic foundation is studied by the finite element technique. The equations of motion for such a rotating shaft element are formulated using deformation shape functions developed from the Timoshenko beam theory. The effects of translational and rotatory inertia, gyroscopic moments, bending and shear deformation are included in the mathematical model. The numerical results show that the elastic foundation can shift the regions of dynamic instability away from the dynamic load factor axis and thus reduces the sizes of these regions, whereas the effect of gryoscopic moments is to shift the boundaries of the regions of dynamic instability outwardly and, therefore, increases the sizes of these regions.     

Nonlinear analysis of the buckling and vibration of a rotating elasticum

We investigate the static and dynamic behaviour of an elasticum which is clamped radially to the inside of a rotating ring and carries a mass at the free end. A nonlinear model for buckling and vibration of the structure in the plane of the ring is established and a subsequent Galerkin procedure leads to a system of two first order ODEs, accessible to a bifurcation and phase plane analysis. It will be shown that these results can only claim limited validity. Afterwards, more adequate results are obtained by a Poincaré-Lindstedt perturbation method. This approach is possible, since the linearized model equation can be solved analytically. Regarding the angular velocity of the ring as a bifurcation parameter, the analysis shows the existence of a supercritical bifurcation for any length of the elasticum. Finally, the case of out-of-plane buckling is shortly investigated, with the result that no buckling occurs for a sufficiently long elasticum.     

机械密封环的传热特性分析

研究机械密封端面摩擦热在动环、静环、端面间液膜和密封介质组成的传热系统中的传递规律。按换热面积守恒的原则将密封环简化为当量圆筒,提出动环和静环获得的摩擦热的计算方法,推导密封环的温度分布方程。结果表明,液膜摩擦热量随角频率的增加和平均膜厚的减小而增加。绝大部分摩擦热通过动环传递到介质,静环端面的温升较小。动环靠近介质侧的温度低于空气侧的温度,端面上的温度较高,且端面径向存在温度梯度。增大动环与介质的接触面积或选用热导率大的材料可降低动环上的最高温度和端面上内外径处的温差,提高机械密封的性能。     

1/2 SUBHARMONIC RESONANCE OF A SHAFT WITH UNSYMMETRICAL STIFFNESS

The 1/2 subharmonic resonance of a shaft with unsymmetrical stiffness is studied. By means of the Hamilton's principle the nonlinear differential equations of motion of the rotating shaft are derived in the rotating rectangular coordinate system. Transforming the equations of motion from rotating coordinate system into stationary coordinate system and introducing a complex variable, the equation of motion in complex variable form is obtained, in which the stiffness coefficient varies periodically with time. It presents a nonlinear oscillation system under parametric excitation. By applying the method of multiple scales (MMS) the averaged equation, the bifurcating response equations and local bifurcating set are obtained. Via the theory of singularity, the stability of constant solutions is analyzed and bifurcating response curves are obtained. This study shows that the rotating shaft has rich bifurcation phenomena.     

非对称刚度转轴的参激共振和分叉分析

研究非对称刚度转轴的参激共振和分叉。用Hamilton原理导出运动微分方程 ,这是刚度系数周期性变化的参激振动方程 ,再用平均法求得平均方程 ,分叉响应方程和定常解。讨论了横截面的不对称性 ,外阻尼和非线性对幅频响应曲线的影响 ,最后用奇异性理论分析定常解的稳定性和分叉。     

考虑螺栓联接结构的轴承-转子系统振动特性分析

针对含螺栓联接结构的轴承-转子系统,建立考虑陀螺力矩及因螺栓预紧力不均匀产生的初始变形量的非线性转子系统动力学模型。采用法求解转子系统运动方程,通过分岔图、时域曲线、频谱及Poincaré映射图研究存在轴承游隙时转子系统的混沌路径,并分析不同初始变形量及轴承游隙对转子系统非线性振动特性的影响,通过试验验证所得结论的准确性。研究表明,当存在轴承游隙时,预紧力不均匀产生的初始变形量增加会抑制低转速下盘的混沌运动,拟周期运动进入混沌运动状态的转速升高,临界转速附近的振动幅值增加,系统混沌路径发生变化;存在初始变形量时,随着轴承径向游隙增大,系统在低转速工作状态下即进入混沌运动运动状态,拟周期运动进入混沌运动状态的转速降低。研究结果可为含螺栓联接结构的轴承-转子系统设计提供理论参考。     

Frequency Effects in Tilting-Pad Journal Bearing Dynamic Coefficients

This paper examines the effects of damped vibrational frequencies on the linear reduced dynamical stiffness and damping coefficients of tilting-pad journal bearings. The frequency ratio (damped frequency/running speed) can be used to judge the accuracy of employing synchronously reduced linear coefficients in rotordynamic stability analyses. The use of these coefficients can result in simpler formulations of the system dynamical equations of motion and solution techniques as well as reduced computational and analysis time. Results presented here indicate that synchronously reduced bearing dynamical coefficients are generally adequate for stability analyses with positively preloaded tilting-pad bearings. Plots of dynamic coefficients are included for a five-pad bearing.