External and internal coupling effects of rotor's bending and torsional vibrations under unbalances

External and internal bending–torsion coupling effects of a rotor system with comprehensive unbalances are studied by analytical analysis and numerical simulations. Based on Lagrangian approach, a full-degree-of-freedom dynamic model of a Jeffcott rotor is developed. The harmonic balance method and the Floquet theory are combined to analyze the stability of the system equations. Numerical simulations are conducted to observe the bending–torsion coupling effects. In the formulation of rotordynamic model, two bending–torsion coupling patterns, external coupling and internal coupling, are suggested. By analytical analysis, it is concluded that the periodic solution of the system is asymptotically stable. From numerical simulations, three bending–torsion coupling effects are observed in three cases. Under static unbalance, synchronous torsional response is observed, which is the result of external coupling under unbalanced force. Under dynamic unbalance, two-time synchronous frequency torsional response is observed, which is the result of internal coupling under unbalanced moment. Under comprehensive unbalance, synchronous and two-time synchronous frequency torsional components are observed, which are the results of both external and internal couplings under unbalanced force and moment. These observations agree with the analytical analysis. It is believed that these observed phenomena should make sense in the dynamical design and fault diagnostics of a rotor system.     

Calculation and analysis of unbalanced magnetic pulls of different stator winding setups in static eccentric induction motor

In large-scale electric machines, unbalanced magnetic pull(UMP) caused by eccentricity usually results in stator-rotor rub, so it is necessary to investigate the amplitude and the influencing factors. This paper takes the squirrel-cage induction motor as an example. A magnetic loop model of an induction motor is established by an analytical method. The impact of stator winding setup(parallel branch and pole pairs) on each magnetomotive force(MMF) and unbalanced magnetic pull is analyzed. Using the finite element simulation method, the spatial and time distribution of flux density of the rotor outer circle under static eccentricity is obtained, and the unbalanced magnetic pull calculation caused by static eccentricity is completed. The conclusion of the influence of stator winding on the size of unbalanced magnetic pull provides reliable gist for motor noise and vibration analysis, and especially provides an important reference for large induction motor design.     

THE UNBALANCED MAGNETIC PULL AND ITS EFFECTS ON VIBRATION IN A THREE-PHASE GENERATOR WITH ECCENTRIC ROTOR

The unbalanced magnetic pull (UMP) in a three-phase generator under no-load, caused by dynamic and static eccentricity, is calculated theoretically. The air-gap permeance is expressed as a Fourier series. Analytical expressions of the UMP for any pole-pair number are obtained. Effects of relative eccentricity and pole number on the magnitude of the UMP are obtained. The vibration of a model rotor in a three-phase generator under the action of the UMP and the eccentric force is analyzed by the numerical method and the harmonic analysis.     

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.     

Observation of breathers in Josephson ladders

We report on the observation of spatially localized excitations in a ladder of small Josephson junctions. The excitations are whirling states which persist under a spatially homogeneous force due to the bias current. These states of the ladder are visualized using a low temperature scanning laser microscopy. We also compute breather solutions with high accuracy in corresponding model equations. The stability analysis of these solutions is used to interpret the measured patterns in the I-V characteristics.     

DYNAMIC RESPONSE AND STABILITY ANALYSIS OF AN AUTOMATIC BALL BALANCER FOR A FLEXIBLE ROTOR

Dynamic stability and time responses are studied for an automatic ball balancer of a rotor with a flexible shaft. The Stodola-Green rotor model, of which the shaft is flexible, is selected for analysis. This rotor model is able to include the influence of rigid-body rotations due to the shaft flexibility on dynamic responses. Applying Lagrange's equation to the rotor with the ball balancer, the non-linear equations of motion are derived. Based on the linearized equations, the stability of the ball balancer around the balanced equilibrium position is analyzed. On the other hand, the time responses computed from the non-linear equations are investigated. This study shows that the automatic ball balancer can achieve the balancing of a rotor with a flexible shaft if the system parameters of the balancer satisfy the stability conditions for the balanced equilibrium position.     

Free transverse vibration analysis of an elastically connected annular and circular double-membrane compound system

In this paper, free transversal vibrations of a systems of two annular and circular membranes connected by a Winkler elastic layer are studied using analytical methods and numerical simulation. At first the motion of each system is described by two homogeneous partial differential equations. The general solutions of the free vibrations are derived by the Bernoulli-Fourier method and the boundary problems are solved. The natural frequencies and natural mode shapes of vibrations of systems under consideration are determined. The investigation of free vibrations prove that the double-membrane systems execute two kinds of vibrations, synchronous and asynchronous. Then for each system two models formulated by using finite element representations are prepared. The FE models are manually tuned to reduce the difference between the natural frequencies of the analytical solutions and the natural frequencies of the FE model calculations, respectively.     

Vibration analysis of a rotating system due to the effect of ball bearing waviness

This research presents an analytical model to investigate vibration due to ball bearing waviness in a rotating system supported by two or more ball bearings, taking account of the centrifugal force and gyroscopic moment of the ball. The waviness of rolling elements is modelled by the sinusoidal function, and it is incorporated into the position vectors of the race curvature center. The Hertzian contact theory is applied to calculate the elastic deflection and non-linear contact force, while the rotor has translational and angular motions. Both the centrifugal force and gyroscopic moment of the ball and the waviness of the rolling elements are included in the kinematic constraints and force equilibrium equations of a ball to derive the non-linear governing equations of the rotor, which are solved by using the Runge–Kutta–Fehlberg algorithm to determine the new position of the rotor. The proposed model is validated by the comparison of the results of the prior researchers. This research shows that the centrifugal force and gyroscopic moment of the ball plays the important role in determining the bearing frequencies, i.e., the principal frequencies, their harmonics and the sideband frequencies resulting from the waviness of the rolling elements of ball bearing. It also shows that the bearing vibration frequencies are generated by the waviness interaction not only between the rolling elements of one ball bearing, but also between those of two or more ball bearings constrained by the rotor.     

Vibration interaction in a multiple flywheel system

This paper investigates vibration interaction in a multiple flywheel system. Flywheels can be used for kinetic energy storage in a satellite Integrated Power and Attitude Control System (IPACS). One hitherto unstudied problem with IPACS is vibration interaction between multiple unbalanced wheels. This paper uses a linear state-space dynamics model to study the impact of vibration interaction. Specifically, imbalance-induced vibration inputs in one flywheel rotor are used to cause a resonant whirling vibration in another rotor. Extra-synchronous resonant vibrations are shown to exist, but with damping modeled the effect is minimal. Vibration is most severe when both rotors are spinning in the same direction.     

Nonlinear traveling wave vibration of a circular cylindrical shell subjected to a moving concentrated harmonic force

This is a study of nonlinear traveling wave response of a cantilever circular cylindrical shell subjected to a concentrated harmonic force moving in a concentric circular path at a constant velocity. Donnell's shallow-shell theory is used, so that moderately large vibrations are analyzed. The problem is reduced to a system of ordinary differential equations by means of the Galerkin method. Frequency-responses for six different mode expansions are studied and compared with that for single mode to find the more contracted and accurate mode expansion investigating traveling wave vibration. The method of harmonic balance is applied to study the nonlinear dynamic response in forced oscillations of this system. Results obtained with analytical method are compared with numerical simulation, and the agreement between them bespeaks the validity of the method developed in this paper. The stability of the period solutions is also examined in detail.     

自治广义Birkhoff系统的平衡稳定性

研究广义Birkhoff系统的平衡稳定性问题.建立了自治广义Birkhoff系统的平衡方程;给出了自治广义Birkhoff系统的一次近似方程,利用Lyapunov一次近似理论,建立了系统平衡状态稳定性的判据;构建了Lyapunov函数,利用Lyapunov直接法,建立了自治广义Birkhoff系统平衡状态稳定性的判据.给出了若干算例以说明结果的应用.     

圆截面弹性螺旋杆的稳定性与振动

基于Kirchhoff理论讨论圆截面弹性螺旋杆的动力学问题.以杆中心线的Frenet坐标系为参考系，建立用欧拉角描述的弹性杆动力学方程.讨论其在端部轴向力和扭矩作用下保持的无扭转螺旋线平衡状态.在静力学和动力学领域内讨论其平衡稳定性问题.还讨论了弹性杆平衡的Lyapunov稳定性和欧拉稳定性两种不同稳定性概念之间的区别和联系.在一次近似意义下证明了螺旋杆在空间域内的欧拉稳定性条件是时域内Lyapunov稳定性的必要条件.导出了解析形式螺旋杆三维弯曲振动的固有频率，为螺旋线倾角和受扰挠性线波数的函数. 关键词： 弹性螺旋杆 Kirchhoff动力学比拟 Lyapunov稳定性 欧拉稳定性     

Self-organized synchronization and voltage stability in networks of synchronous machines

The integration of renewable energy sources in the course of the energy transition is accompanied by grid decentralization and fluctuating power feed-in characteristics. This development raises novel challenges for power system stability and design. We investigate power system stability from the viewpoint of self-organized synchronization aspects. In this approach, the power grid is represented by a network of synchronous machines. We supplement the classical Kuramoto-like network model, which assumes constant voltages, with dynamical voltage equations, and thus obtain an extended model, that incorporates the coupled categories voltage stability and rotor angle synchronization. We compare disturbance scenarios in small systems simulated on the basis of both classical and extended model and we discuss resultant implications and possible applications to complex modern power grids.     

Stability of limit cycles in frictionally damped and aerodynamically unstable rotor stages

This paper deals with the stability of limit cycles (Steady-State Oscillations) associated with the multi-degree-of-freedom model of a frictionally damped and aerodynamically unstable rotor stage. By using the first order averaging technique, a generalized criterion has been established to sort out those unstable limit cycles which govern the maximum transient amplitude beyond which the rotor stage becomes unstable. The stability of the remaining steady-state solutions is analyzed by linearizing the averaged system of differential equations. Numerical results are discussed for three-, four- and five-bladed disks.     

ANALYSIS OF THE MULTIPLE-SOLUTION RESPONSE OF A FLEXIBLE ROTOR SUPPORTED ON NON-LINEAR SQUEEZE FILM DAMPERS

The multiple-solution response of rotors supported on squeeze film dampers is a typical non-linear phenomenon. The behaviour of the multiple-solution response in a flexible rotor supported on two identical squeeze film dampers with centralizing springs is studied by three methods: synchronous circular centred-orbit motion solution, numerical integration method and slow acceleration method using the assumption of a short bearing and cavitated oil film; the differences of computational results obtained by the three different methods are compared in this paper. It is shown that there are three basic forms for the multiple-solution response in the flexible rotor system supported on the squeeze film dampers, which are the resonant, isolated bifurcation and swallowtail bifurcation multiple solutions. In the multiple-solution speed regions, the rotor motion may be subsynchronous, super-subsynchronous, almost-periodic and even chaotic, besides synchronous circular centred, even if the gravity effect is not considered. The assumption of synchronous circular centred-orbit motion for the journal and rotor around the static deflection line can be used only in some special cases; the steady state numerical integration method is very useful, but time consuming. Using the slow acceleration method, not only can the multiple-solution speed regions be detected, but also the non-synchronous response regions.     

A review on the solution of Grad–Shafranov equation in the cylindrical coordinates based on the Chebyshev collocation technique

Equilibrium reconstruction consists of identifying, from experimental measurements, a distribution of the plasma current density that satisfies the pressure balance constraint. Numerous methods exist to solve the Grad–Shafranov equation, describing the equilibrium of plasma confined by an axisymmetric magnetic field. In this paper, we have proposed a new numerical solution to the Grad–Shafranov equation (an axisymmetric, magnetic field transformed in cylindrical coordinates solved with the Chebyshev collocation method) when the source term (current density function) on the right-hand side is linear. The Chebyshev collocation method is a method for computing highly accurate numerical solutions of differential equations. We describe a circular cross-section of the tokamak and present numerical result of magnetic surfaces on the IR-T1 tokamak and then compare the results with an analytical solution.     

电磁搅拌中旋转磁场及洛伦兹力的数值模拟

由于工程中应用广泛的电磁搅拌问题很难得到解析解,通过有限元软件ANSYS对其进行了数值求解。结果表明：旋转磁场在结晶器搅拌区域内产生电磁力,使钢液在水平方向形成旋转流动。随着频率的增加,磁感应强度逐渐减小、电磁力密度则先增加后减小。在频率一定的情况下,通电线圈电压的增加可导致电磁力密度的增大,通过相关物理理论可对其进行合理解释。     

Hamiltonian formulation of ferromagnetic hydrodynamics

The equations of ideal ferromagnetic hydrodynamics (FMHD) in three dimensions are formulated as a hamiltonian system in terms of a noncanonical Poisson bracket. The conservation laws for this system are determined and used to construct Lyapunov functionals that are potentially useful for dealing with stability properties of FMHD equilibrium solutions. Classes of equilibrium solutions are identified with critical points of these Lyapunov functionals. The FMHD system is also formulated in n dimensions and the relations among magnetic induction, magnetic field intensity, and magnetization density are discussed from a Lie-algebraic viewpoint.     

Lifting the singular nature of a model for peeling of an adhesive tape

We investigate the dynamics of peeling of an adhesive tape subjected to a constant pull speed. Due to the constraint between the pull force, peel angle and the peel force, the equations of motion derived earlier fall into the category of differential-algebraic equations (DAE) requiring an appropriate algorithm for its numerical solution. By including the kinetic energy arising from the stretched part of the tape in the Lagrangian, we derive equations of motion that support stick-slip jumps as a natural consequence of the inherent dynamics itself, thus circumventing the need to use any special algorithm. In the low mass limit, these equations reproduce solutions obtained using a differential-algebraic algorithm introduced for the earlier singular equations. We find that mass has a strong influence on the dynamics of the model rendering periodic solutions to chaotic and vice versa. Apart from the rich dynamics, the model reproduces several qualitative features of the different waveforms of the peel force function as also the decreasing nature of force drop magnitudes.     

Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches

This paper investigates the in-plane linear dynamic behaviour of multi-stepped and multi-damaged circular arches under different boundary conditions. Cracked cross-sections are modelled as massless elastic rotational hinges. In damaged configuration, cracks can be located both at the interface between two adjacent portions as well as inside the portion itself. For each arch portion bounded by two cracks, the differential equations of motion have been established considering axial extension, transverse shear effects and rotatory inertia. The equilibrium equations of arch portions are combined in the coupled fundamental system in terms of radial displacement, tangential displacement and rotation. Analytical and numerical solutions for multi-stepped arches, in undamaged as well as in damaged configurations, are proposed. The analytical solution is based on the Euler characteristic exponent procedure involving the roots of characteristic polynomials, while the numerical method is focused on the Generalized Differential Quadrature (GDQ) method and the Generalized Differential Quadrature Element (GDQE) technique. Numerical results are shown in terms of the first 10 analytical and numerical frequencies of multi-stepped and multi-damaged arches with different boundary conditions. Finally, convergence and stability characteristics of the GDQE procedure are investigated. The convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical procedure is very good.