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.     

New electromagnetic ring balancer for active imbalance compensation of rotating machinery

New electromagnetic ring balancer for active mass imbalance compensation of rotating machinery is proposed in this paper. This type of active balancer maintains the balanced condition of machine by permanent magnets rather than clutch and external energy. Firstly, the imbalance compensation principle is presented, and then the balancer design is completed. For the proposed balancer, the actuation principle and operational process are presented in detail. Based on the finite element stable-state and transient calculations, the reasonable design parameters are obtained to direct the development of balancer prototype machine. The relative experimental tests are completed to validate the balancer design and fabrication. The proposed balancer has a simple structure, a wide range of balance capability, a stable and smooth operational process and a desirable vibration response. This study shows that the machine tool spindle integrated with active balancer indeed has an expected mass imbalance compensation function, and the mass–radius product of 6.03–38.55 g cm can eliminate the vibration of 0.0021–0.1076 gravitational acceleration.     

Three-dimensional modelling and dynamic analysis of an automatic ball balancer in an optical disk drive

Dynamic behaviours and stability of an automatic ball balancer (ABB) in an optical disk drive are analyzed based on the proposed three-dimensional dynamic model. For dynamic analysis, the feeding deck with the ball balancer and a spindle motor is modelled as a rigid body with six degrees of freedom. The nonlinear equations of motion are derived using Lagrange's equation in order to describe the translational and rotational motions of the system. From the derived nonlinear equations, the linearized equations of motion in the neighbourhood of a balanced equilibrium position are obtained by the perturbation method. These equations are coupled, linear, differential equations with time-dependent periodic coefficients, from which the stability of the system is analyzed by using the Floquet theory. Finally, the time responses are computed to verify the results of the stability analysis, and to investigate the balancing performance of the ABB.     

Finite element modelling of a rotating piezoelectric ultrasonic motor

The evaluation of the performance of ultrasonic motors as a function of input parameters such as the driving frequency, voltage input and pre-load on the rotor is of key importance to their development and is here addressed by means of a finite element three-dimensional model. First the stator is simulated as a fully deformable elastic body and the travelling wave dynamics is accurately reproduced; secondly the interaction through contact between the stator and the rotor is accounted for by assuming that the rotor behaves as a rigid surface. Numerical results for the whole motor are finally compared to available experimental data.     

Performance evaluation of traveling wave ultrasonic motor based on a model with visco-elastic friction layer on stator

A new visco-elastic contact model of traveling wave ultrasonic motor (TWUSM) is proposed. In this model, the rotor is assumed to be rigid body and the friction material on stator teeth surface to be visco-elastic body. Both load characteristics of TWUSM, such as rotation speed, torque and efficiency, and effects of interface parameters between stator and rotor on output characteristic of TWUSM can be calculated and simulated numerically by using MATLAB method based on this model. This model is compared with that one of compliant slider and rigid stator. The results show that this model can obtain bigger stall torque. The simulated results are compared with test results, and found that their load characteristics have good agreement.     

DYNAMIC ANALYSIS OF AN AUTOMATIC WASHING MACHINE WITH A HYDRAULIC BALANCER

A mathematical model of a hydraulic balancer in steady state condition was derived from a whirling model of a vertical axis washing machine, with the aim of implementing a dynamic analysis of an automatic washing machine during spin drying mode. The centrifugal force acting on the hydraulic balancer depends on the centroidal distance of the fluid in the hydraulic balancer, and the centroidal distance is a function of an eccentricity of the geometric center of the hydraulic balancer from the axis of rotation. A mathematical model of the hydraulic balancer in steady state is validated by the experimental result of the centrifugal force. Experiments were performed on a washing machine during spin drying mode, and results were compared with the simulation result. The parameters affecting the vibration of the washing machine were investigated by the parameter study.     

Aeroelastic stability of complete rotors with application to a teetering rotor in forward flight

The derivation of a set of non-linear coupled flap-lag-torsion equations of motion for moderately large deflections of an elastic, two-bladed teetering helicopter rotor in forward flight is concisely outlined. The following degrees of freedom are included in the mathematical model: rigid body flapping, rigid body lead-lag, elastic bending in flap and lead-lag, blade root torsion, and shaft torsion. Quasi-steady aerodynamic loads are considered and the effects of reversed flow are included. The aeroelastic stability of the complete rotor is investigated by using a linearized system of equations of motion. The equilibrium position about which the equations are linearized is obtained by considering the trim state of the helicopter, in true or simulated forward flight conditions. The sensitivity of the aeroelastic stability boundaries to interblade structural and mechanical coupling is illustrated by comparing the complete rotor stability boundaries with those obtained from a single blade analysis for a number of hover and forward flight cases.     

Rotordynamical simulations of a fibre refiner during production

A key component in panel board production is the refiner, whose task is to break raw cellulose wood chips into slender fibres, done by a relative angular motion between stator and rotor. The main problem in predicting the dynamics of these machines is to model the complicated fibre breaking process, since the refining process leads to a three-phase flow (solid wood, water and steam) between the stator and rotor. By modelling the rotor as a rigid body, the process can only contribute by a resulting force and a resulting moment. Through this approach and axial force measurements, it has been shown that the refining process can be treated as a time dependent stiffness matrix and external load. The objective for this paper is to predict and explain dynamical characteristics of fibre refiners during production in such a way that the results can be used already at early stages of the product development process. Two different pressure distribution cases are studied, i.e. one axisymmetric with only radial variations and one non-axisymmetric. It is found that the axisymmetric case only excited forward modes, while the non-axisymmetric case excited all modes. The time dependent stiffness matrix resulted in unstable domains, but also in stable domains with intermittent high amplitudes.     

New predictions of a classic model

The predictions of the rigid rotor model concerning electromagnetic transverse form factors of rotational nuclei are explored. Results for inelastic scattering at 180° to the first 2⁺ state in ¹⁶⁶Er are presented and compared to those obtained in the projected Hartree-Fock (PHF) approach. Large discrepancies between both predictions have been found in the low-q region. The rigid rotor model predicts a cross section 50 to 100 times lower than that predicted by the PHF approach in the region0.25 fm^?1 ? q ? 0.4 fm^?1, where the first maximum appears.     

Theoretical and experimental analysis of the electromechanical behavior of a compact spherical loudspeaker array for directivity control

Sound directivity control is made possible by a compact array of independent loudspeakers operating at the same frequency range. The drivers are usually distributed over a sphere-like frame according to a Platonic solid geometry to obtain a highly symmetrical configuration. The radiation pattern of spherical loudspeaker arrays has been predicted from the surface velocity pattern by approximating the drivers membranes as rigid vibrating spherical caps, although a rigorous assessment of this model has not been provided so far. Many aspects concerning compact array electromechanics remain unclear, such as the effects on the acoustical performance of the drivers interaction inside the array cavity, or the fact that voltages rather than velocities are controlled in practice. This work presents a detailed investigation of the electromechanical behavior of spherical loudspeaker arrays. Simulation results are shown to agree with laser vibrometer measurements and experimental sound power data obtained for a 12-driver spherical array prototype at low frequencies, whereas the non-rigid body motion and the first cavity eigenfrequency yield a discrepancy between theoretical and experimental results at high frequencies. Finally, although the internal acoustic coupling affects the drivers vibration in the low-frequency range, it does not play an important role on the radiated sound power.     

Alternating Parity Band in Octupole-Soft ~(140)Xe with Axial Vibrational-Rotational Model and Triaxial Rigid Rotor Model

Energies of the yrast positive-and negative-parity excited states in140 Xe are reproduced by two different models considering quadrupole-octupole deformations, namely the axial vibrational-rotational model and the triaxial rigid rotor model, and compared with the stable octupole-deformed222 Th. The origin of the energy difference between the opposite parity sequences is considered from two different mechanisms, the vibration in axial deformed energy minima and the rotation considering the effective triaxial deformation. The success of reproducing the data in both the models implies that these two mechanisms are equivalent on some level for the octupole-soft nuclei. By investigating the probability distributions for projection of total angular momentum in the triaxial rigid rotor model, it is found that such an energy difference is associated with the difference of orientation of the rotational axis.     

Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.     

Dynamic analysis of horizontal axis wind turbine by thin-walled beam theory

A mixed flexible-rigid multi-body mathematical model is applied to predict the dynamic performance of a wind turbine system. Since the tower and rotor are both flexible thin-walled structures, a consistent expression for their deformations is applied, which employs a successive series of transformations to locate any point on the blade and tower relative to an inertial coordinate system. The kinetic and potential energy terms of each flexible body and rigid body are derived for use in the Lagrange approach to formulate the wind turbine system's governing equation. The mode shapes are then obtained from the free vibration solution, while the distributions of dynamic stress and displacement of the tower and rotor are computed from the forced vibration response analysis. Using this dynamic model, the influence of the tower's stiffness on the blade tip deformation is studied. From the analysis, it is evident that the proposed model not only inherits the simplicity of the traditional 1-D beam element, but also able to provide detailed information about the tower and rotor response due to the incorporation of the flexible thin-walled beam theory.     

The structure of the odd-A nuclei withA≈130 and the problem ofγ-softness

The negative parity states in the^{123, 125, 131}Cs and the¹³¹La nuclei are described in the framework of the particle-core coupling model. In order to study the problem of gamma softness, the following two core models are used:(1) the rigid triaxial rotor model, and(2) theγ-unstable model withγ-dependent inertial functions. The properties of the odd-A nuclei with rigid and soft cores are compared with the experimental data. The results do not allow to draw a definite conclusion about the softness. When seeking properties which could help to distinguish between soft and rigid nuclei it has been found that some spectroscopic factors for the proton stripping reaction are sensitive to the gamma softness.     

Adaptive complete suppression of imbalance vibration in AMB systems using gain phase modifier

Synchronous vibration can be caused by rotor imbalance in high-speed rotors of momentum exchange devices, and the imbalance vibration is the main disturbance for attitude control of spacecrafts. Active magnetic bearing (AMB) is widely used in momentum exchange devices due to its active vibration control ability. To suppress the imbalance vibration completely, an adaptive control approach based on the AMB is proposed. First, dynamics of the AMB rotor with both static imbalance and dynamic imbalance are introduced, and the model of power amplifier is particularly analyzed. Large temperature change range and overpowering cosmic ray will induce considerable errors and variations in parameters of the power amplifier, which has to work in space for about ten years. Therefore, adaptive compensation should be made for these errors and variations. Conditions, on which the imbalance vibration can be completely suppressed, are analyzed, and the results show that these conditions can be satisfied with notch filters and feedforward compensations (FFCs). However, the FFC contains an inverse function of the power amplifier, whose errors and variations can result in gain and phase differences and changes between the output voltage of the controller and the actual output current of the power amplifier. Consequently, the FFC becomes inaccurate, and residual vibration occurs. Finally, a gain phase modifier (GPM) is proposed to form two closed loops to tune the gain and phase of the FFC adaptively and precisely. The effectiveness of the proposed approach has been demonstrated by simulations and experiments. Compared with the existing methods, this method can achieve adaptive complete suppression of the imbalance vibration unaffected by the errors and variations of the power amplifier.     

Stability analyses of a vertical axis automatic washing machine without balancer

This paper analyzes the nonlinear vibration characteristics associated with the spin drying process of a vertical axis automatic washing machine without any balancer. At first, damping properties born with the machine's suspension system are discussed and a mathematical model involving tangential damping forces is built. Based on a rotating coordinate transformation, this model is then converted to an autonomous form for stability analyses. The continuation and bifurcation software AUTO [1] is applied and a Hopf bifurcation phenomenon is observed from a one-parameter bifurcation diagram. Based on several two-parameter bifurcation diagrams, several parameters affecting the Hopf bifurcation are then discussed. At last, bifurcation results are validated by time responses of the autonomous system. For a further view of the spin drying process, simulations of the non-autonomous system are also provided. This paper provides a new insight into the spin drying process of the vertical axis automatic washing machine.     

Scale effects in the bending vibrations of helicopter rotor blades

Scale effects and dynamic similarity in the bending vibrations of helicopter rotor blades are examined by expressing the first three modes of bending vibration of a uniform, conventional rotor blade by a series of Legendre polynomials as suggested by Wilde and others. The natural frequency ratios for these three modes are then determined as functions of a dynamic similarity parameter over the entire range of designs and operating conditions from very flexible, rapidly rotating blades to highly rigid, slowly turning conditions.     

Energy Staggering in the High K Rotational Band

Introducing the staggering index S(I) to describe the energy spectra of the λrigid and λ-soft nuclei, it becomes very clear that there are two kinds of energy staggering for which the S(I) -I plots have opposite zigzag behavior. They can be described using the axially asymmebtric rotor model with vibration-rotation coupling and the interaction boson model O(6) limit with three-body potential or the angular momentum projection deformed Hartree-Fock method (PDHF ), respectively. The theoretical predictions for the characteristics of the staggering in high K rotational band are given. Analyzing the experimental data of high K band spectra of the nuclei in the mass range. 160, it is demoastrated that the energy staggering does exist in the high K band. At the same time, some evidences for the shape transition indicated by zigzag phase change of S(I)-I plots are undoubtedly found.     

Rigid finite element model of a cracked rotor

The article introduces a new mathematical model for the cracked rotating shaft. The model is based on the rigid finite element (RFE) method, which has previously been successfully applied for the dynamic analysis of many complicated, mechanical structures. In this article, the RFE method is extended and adopted for the modeling of rotating machines. An original concept of crack modeling utilizing the RFE method is developed. The crack is presented as a set of spring–damping elements of variable stiffness connecting two sections of the shaft. An alternative approach for approximating the breathing mechanism of the crack is introduced. The approach is simple and allows one to intuitively and systematically prepare and analyze the model of a cracked rotor.The proposed method is illustrated with numerical and experimental results. The experiments conducted for the uncracked free–free rotor as well as the numerical results obtained with other software confirm the accuracy of the RFE model. The numerical analysis conducted for a set of cracked rotors has shown that, depending on the eccentricity and its angular location, the breathing behavior of the crack may take different forms. In spite of this, the frequency spectra for different cracks are almost identical.Due to its simplicity and numerous advantages, the proposed approach may be useful for rotor crack detection, especially if methods utilizing the mathematical model of the rotor are applied.