A FINITE ELEMENT FORMULATION FOR COUPLING RIGID AND FLEXIBLE BODY DYNAMICS OF ROTATING BEAMS

The work presented in this paper is based on an existing comprehensive formulation for rotating flexible systems. In the existing formulation the flexible degrees of freedom (d.o.f.) are represented by an analytically computed modal basis and the coupling matrices between the rigid- and the flexible-body d.o.f. are developed based on the analytical modal representation of the flexible d.o.f. In this paper, the existing formulation is generalized for rotating beams by representing the flexible d.o.f. either as physical d.o.f. of a finite element formulation or as a set of retained and internal d.o.f. of a Craig-Bampton formulation. The coupling matrices between the rigid-body rotation and the flexible d.o.f. are developed accordingly. The non-linear effects from the work done by the centrifugal forces are included in the formulation. Finite element shape functions of a beam element in a three-dimensional space and finite element shape functions for solid elements are employed for deriving the coupling terms between the rigid-body d.o.f. and the physical d.o.f. An additional transformation is required and performed when the right-body d.o.f. are coupled with the internal and the retained d.o.f. of a Craig-Bampton formulation. The coupled system of equations is solved in the time domain by combining the Newmark method for time integration and the Newton-Raphson method for solving the non-linear system of equations within each time step. Analyses are performed for a flexible rotating beam in order to validate the development. An analytical solution is compared with the new formulations that represent the rotating beam flexibility with the physical d.o.f. of beam or solid elements. The analytical solution is also compared to the formulation that represents the flexible d.o.f. in terms of retained and internal d.o.f. of a Craig-Bampton formulation. Very good correlation between the analytical and numerical results is observed.     

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 VIBRATION ANALYSIS OF A SPINNING FLEXIBLE DISK-SPINDLE SYSTEM SUPPORTED BY BALL BEARING AND FLEXIBLE SHAFT USING THE FINITE ELEMENT METHOD AND SUBSTRUCTURE SYNTHESIS

Free vibration of a spinning flexible disk-spindle system supported by ball bearing and flexible shaft is analyzed by using Hamilton's principle, FEM and substructure synthesis. The spinning disk is described by using the Kirchhoff plate theory and von Karman non-linear strain. The rotating spindle and stationary shaft are modelled by Rayleigh beam and Euler beam respectively. Using Hamilton's principle and including the rigid body translation and tilting motion, partial differential equations of motion of the spinning flexible disk and spindle are derived consistently to satisfy the geometric compatibility in the internal boundary between substructures. FEM is used to discretize the derived governing equations, and substructure synthesis is introduced to assemble each component of the disk-spindle-bearing-shaft system. The developed method is applied to the spindle system of a computer hard disk drive with three disks, and modal testing is performed to verify the simulation results. The simulation result agrees very well with the experimental one. This research investigates critical design parameters in an HDD spindle system, i.e., the non-linearity of a spinning disk and the flexibility and boundary condition of a stationary shaft, to predict the free vibration characteristics accurately. The proposed method may be effectively applied to predict the vibration characteristics of a spinning flexible disk-spindle system supported by ball bearing and flexible shaft in the various forms of computer storage device, i.e., FDD, CD, HDD and DVD.     

Viscoelastic analysis of multi-body systems using the finite element method

A study of the effect of viscoelastic material damping on the dynamic response of multibody systems, consisting of interconnected rigid, elastic and viscoelastic components, is presented. The motion of each elastic or viscoelastic body is identified by using three sets of modes: rigid body, reference and normal modes. Rigid body modes describe translation and large angular rotation of a body reference. Reference modes are the result of imposing the body-axis conditions. Normal modes define the deformation of the body relative to the body reference. Constraints between different components are formulated by using a set of non-linear algebraic equations that can be introduced to the dynamic formulation by using a Lagrange multiplier technique or can be utilized to eliminate dependent co-ordinates by partitioning the constraint Jacobian matrix. In developing the system equations of motion of the viscoelastic component, an assumption of a linear viscoelastic model is made. A Kelvin-Voigt model is employed, wherein the stress is assumed to be proportional to the strain and its time derivative. The formulation yields a constant damping matrix and the damping forces depend only on the local deformation; thus, no additional coupling between the reference and elastic co-ordinates appears in the formulation when considering the viscoelastic effects. It is demonstrated, by a numerical example, that the viscoelastic material damping can have a significant effect on the dynamic response of multibody systems.     

Dynamic stability of two rigid rotors connected by a flexible coupling with angular misalignment

The effect of misalignment on the stability of two rotors connected by a flexible mechanical coupling subjected to angular misalignment is examined. The study performed is to understand the effect of angular misalignment on the stability of rotating machinery. The dimensionless stability criteria of the non-linear system of differential equations of two misaligned rigid rotors are derived using Liapunov's direct method. A rigid disk is attached at the middle of each rotor, where the rotor-disk assembly is mounted on two hydrodynamic bearings with four stiffness and four damping coefficients. Sets of dimensionless conditions for sufficient whirl stability of the two misaligned rotors are derived. The stability conditions are presented in graphical form for deeper understanding of the effect of the flexible mechanical coupling stiffness and angular misalignment on rotating machinery stability. The results show that an increase in angular misalignment or mechanical coupling stiffness terms leads to an increase of the model stability region.     

Effects of isolators internal resonances on force transmissibility and radiated noise

Vibration isolators have been extensively used to reduce the vibration and noise transmitted between the components of mechanical systems. Although some previous studies on vibration isolation considered the inertia of isolators, they only examined its effects on the vibration of single degree-of-freedom (d.o.f.) systems. These studies did not emphasize the importance of the isolators’ inertia, especially from the perspective of noise reduction. This paper shows that the internal dynamics of the isolator, which are also known as internal resonances (IRs) or wave effects, can significantly affect the isolator performance at high frequencies. To study the IR problem, a model of a primary mass connected to a flexible foundation through three isolators is used. In this model, the isolator is represented as a one-dimensional continuous rod that accounts for its internal dynamics. The primary mass is modelled as a rigid body with three d.o.f.'s. The effects of the IRs on the force transmissibility and the radiated sound power from the foundation are examined. It is shown that the IRs significantly increase the force transmissibility and the noise radiation level at some frequencies. These effects cannot be predicted using a traditional model that neglects the inertia of the isolator. The influence of the foundation flexibility on the IRs is also investigated. It is shown that the foundation flexibility greatly affects the noise radiation level but it affects only slightly the force transmissibility, especially at high frequencies where the IRs occur.     

Acceleration of unbalanced flexible rotors through the critical speeds

The bending vibrational behaviour of a flexible rotor with a continuous mass distribution passing its critical speeds under a driving torque is considered. It is shown that the (non-linear) equations of motion for an actual shaft can be formally traced back to those of a Laval rotor. In this way, the results for a Laval rotor, which, in an earlier publication by the authors [1], have been presented generally for constant load torque can be applied to actual rotors. The system parameters of the Laval rotor merely have to be replaced by the generalized parameters of the respective bending modes. A special study shows that the effect of the torsional flexibility of the shaft on the bending vibrational behaviour is negligible.     

Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cables

This paper presents a model formulation capable of analyzing large-amplitude free vibrations of a suspended cable in three dimensions. The virtual work-energy functional is used to obtain the non-linear equations of three-dimensional motion. The formulation is not restricted to cables having small sag-to-span ratios, and is conveniently applied for the case of a specified end tension. The axial extensibility effect is also included in order to obtain accurate results. Based on a multi-degree-of-freedom model, numerical procedures are implemented to solve both spatial and temporal problems. Various numerical examples of arbitrarily sagged cables with large-amplitude initial conditions are carried out to highlight some outstanding features of cable non-linear dynamics by accounting also for internal resonance phenomena. Non-linear coupling between three- and two-dimensional motions, and non-linear cable tension responses are analyzed. For specific cables, modal transition phenomena taking place during in-plane vibrations and ensuing from occurrence of a dominant internal resonance are observed. When only a single mode is initiated, a higher or lower mode can be accommodated into the responses, making cable spatial shapes hybrid in some time intervals.     

A generalized modelling technique for linearized motions of mechanisms with flexible parts

Analysis and control of vibrations of agricultural machines to improve machine performance and vibration comfort of the operator is a major concern of manufacturers these days. In this paper, an analytical method to build the linearized equations of motion of an elastic tree structured mechanism, is presented. The method is based on the principle of virtual work resulting in a set of parameterized linear equations that are functions of the mechanical parameters and the geometry and the interconnection structure of different bodies in the mechanism. The rigid-body motions of the mechanical system are represented by Lagrangian generalized co-ordinates while elastic deformations are described by nodal co-ordinates from a finite element formulation. Explicit expressions for external distributed and concentrated forces and internal concentrated forces acting on the mechanism are given.     

Airy Equation for the Topological String Partition Function in a Scaling Limit

We use the polynomial formulation of the holomorphic anomaly equations governing perturbative topological string theory to derive the free energies in a scaling limit to all orders in perturbation theory for any Calabi–Yau threefold. The partition function in this limit satisfies an Airy differential equation in a rescaled topological string coupling. One of the two solutions of this equation gives the perturbative expansion and the other solution provides geometric hints of the non-perturbative structure of topological string theory. Both solutions can be expanded naturally around strong coupling.     

Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries

An adaptive mesh projection method for the time-dependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volume-of-fluid approach. Second-order convergence in space and time is demonstrated on regular, statically and dynamically refined grids. The quad/octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features. The source code of the method implementation is freely available.     

NON-LINEAR DYNAMIC MODEL OF AN INEXTENSIBLE ROTATING FLEXIBLE ARM SUPPORTED ON A FLEXIBLE BASE

A mathematical model for a flexible arm undergoing large planar flexural deformations, continuously rotating under the effect of a hub torque and supported by a flexible base is developed. The position of a typical material point along the span of the arm is described using the inertial reference frame via a transformation matrix from the body co-ordinate system, which is attached to the flexible root of the rotating arm. The condition of inextensibility is employed to relate the axial and transverse deflections of the material point, within the beam body co-ordinate system. The position and velocity vectors obtained, after imposing the inextensibility conditions, are used in the kinetic energy expression while the exact curvature is used in the potential energy. Lagrangian dynamics in conjunction with the assumed modes method is utilized to derive, directly, the non-linear equivalent temporal equations of motion. The resulting non-linear model, which is composed of four coupled non-linear ordinary differential equations, is discussed, simulated and the results of this simulation are presented. The effects of the base flexibility are explored by comparing the resulting simulation results, for various flexibility coefficients, with previously published works of the authors. Moreover, the numerical results show that the base flexibility has a very important effect on the stability of rotating flexible arms that should be accounted for when simulating such systems.     

任意加速带电动态黑洞中Dirac粒子的Hawking辐射

在任意加速带电动态时空中，选取零标架、计算出旋系数，把四个耦合的Dirac方程中化成两个耦合的方程，采用Tortoise坐标变换将其两个耦合的方程变换成Tortoise坐标下的形式，在黑洞视界面附近化成了典型的波动方程，得到在视界面附近Dirac粒子的Hawking辐射温度，成功地导出了Hawking热谱公式. 关键词： Dirac方程 Hawking辐射 黑洞 Tortoise坐标变换     

新型涡旋多相泵原理及其工作过程研究

提出了一种新型涡旋式气液多相混输泵,通过构建变啮合间隙涡旋齿型线,使得所形成的压缩腔存在一条通向排出口的内泄漏卸压通道。构建了一种液相压缩卸荷的新方法,解决了容积式多相泵在混输较高含液率介质时易出现的压缩腔液击问题。介绍了新型涡旋多相泵的工作原理,得到了变啮合间隙涡旋齿型线的生成方法和型线方程,分析了工作腔容积和啮合间隙的变化规律。建立了涡旋多相泵工作过程中气液介质在伴有内泄漏时的增压过程热力模型,求解得到了工作过程中压力的变化规律。     

Rotational Brownian dynamics of semiflexible broken rods

Using the Brownian dynamics simulation technique, we study the rotational dynamics of a semiflexible broken rod. We employ a suitable bead model with stiff springs between beads and strong forces opposing to bending, except at the joint where flexibility is variable. We consider mostly broken rods with equal arms. From the simulated Brownian trajectories we obtain the correlation function for the second order Legendre polynomial of the reorientational angle of the end-to-end vector and of the arm vector. These correlation functions are closely related to fluorescence anisotropy decay and electric birefringence decay, respectively. In the first case, the relaxation time for a completely flexible rod agrees with the Harvey-Wegener theory, and in the second, the longest relaxation time agrees well with that obtained from the rigid-body treatment over the whole range of flexibility. Furthermore, we discuss the relative importance of flexibility in both types of decay. Finally, we present results for a case with unequal arms, confirming the validity of the Harvey-Wegener theory and the rigid-body treatment.     

Vibration of large turbo-rotors in fluid-film bearings on an elastic foundation

In the first sections of this paper the flexible rotating shaft of a turbo-rotor is treated by finite element analysis. Internal and external damping, gyroscopic forces, fluid-film forces, aerodynamic cross-coupling from steam flow and magnetic pull are taken into account. Although some hundred degrees of freedom have to be introduced to describe a realistic turbo-rotor, computational effort can be enormously reduced by making use of the banded structure of the system matrices.In the second part foundation dynamics are introduced into the rotor equations via a receptance formulation. The receptance matrices of the foundation or supporting systems can be obtained either from shaker tests or from the mode analysis of the foundation without shaft. Numerical examples are given.     

Theory of the coupling between conduction electrons and moments of 3d and 4f ions in metals

The theory of the effective coupling between conduction electrons and the magnetic (or electric) moments of 3d and 4f ions in metals is reviewed, with emphasis on a systematic treatment of the generalized coupling forms which usually apply for real ions with orbital degrees of freedom. With the help of the irreducible-tensor method, the coupling is expanded in forms consistent with general symmetry requirements, and the effects of intra-ionic level structure of LS or LSJ type are taken into account. Coupling contributions from the direct and exchange parts of the Coulomb interaction between ionic and conduction electrons, and from the Anderson-Clogston-Schrieffer-Wolff virtual-mixing mechanism, are evaluated. The symmetry analysis is performed according to the full rotation group, meaning that the conduction band is represented as a free-electron band and effects of crystal-field splitting are not explicitly considered.     

非惯性系下考虑剪切变形的柔性梁的动力学建模

研究非惯性坐标系下考虑剪切变形的柔性梁的动力学建模. 首先借鉴Euler-Bernoulli梁的几何非线性变形模式,考虑了Timoshenko梁弯曲以及剪切变形产生的几何非线性效应对纵向、横向变形位移的影响,在考虑两个方向的变形耦合项后,利用有限元法对柔性梁进行了离散,采用Lagrange方程建立了柔性梁的动力学模型,首次建立了包含变形二次耦合量的Timoshenko梁的动力学方程. 关键词： 非惯性坐标系 剪切变形 柔性梁 动力学建模     

Applications of a generalized Galerkin's method to non-linear oscillations of two-degree-of-freedom systems

A generalized Galerkin's method is formulated for multi-degree-of-freedom holonomic systems. Two autonomous non-linear two-d.o.f. oscillators are used as the examples for the applications of the method: (i) free oscillations of a non-linear mass-spring system with two d.o.f., (ii) two weakly non-linearly coupled identical van der Pol oscillators. The accuracy of the approximate solutions is discussed. The effect of different time intervals of integration on the results is also investigated.     

Gyroscopic behaviour of stressed rotating shafts

An analysis is presented of the secondary effects due to deformation as a result of the shearing force, to the kinetic energy of rotation, and to the gyroscopic coupling, on the linearized equation of a straight shaft turning at a constant angular velocity and which is subjected to an axial torque and a compressive force. The local equations of motion of a section of shaft are derived and the magnitudes of the coupling terms are studied, non-dimensional variables being used. Results obtained by different authors in certain particular cases are rediscovered, and, in particular, the case of a shaft on short bearings is studied. Some comparisons of the theoretical results with those from experiments are given.