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.     

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.     

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.     

Experimental investigation of a single-plane automatic balancing mechanism for a rigid rotor

We present an experimental investigation of a single-plane automatic balancer that is fitted to a rigid rotor. Two balls, which are free to travel around a circular race, are used to compensate for the mass imbalance in the plane of the device. The experimental rig possesses both cylindrical and conical rigid body modes and the performance of the automatic balancer is assessed for a variety of different levels of imbalance. A non-planar mathematical model that also includes the observed effect of support anisotropy is developed and numerical simulations are compared with the experimental findings. In the highly supercritical frequency range the balls act to balance the rotor and a good quantitative match is found between the model and the experimental data. However, during the rigid body resonances the dynamics of the ball balancer is highly nonlinear and for this speed range the agreement between theory and experiment is mainly qualitative. Nevertheless, the model is able to successfully reproduce many of the solution types that are found experimentally.     

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.     

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.     

Computational analysis for dynamic response of a rotating shaft on flexible support structure with clearances

A finite element-based formulation for modelling the dynamic behavior of a rotating flexible shaft supported by a flexible support structure is presented. The coupling effect between the rigid-body rotation and the flexible deformation of the shaft is considered and represented by non-linear coupling terms in the mass matrix and forcing vectors in the global system of equations. The rigid-body rotation is treated as one of the degrees of freedom (d.o.f.) of the entire system. The interaction between the rotating shaft and the flexible support is modelled by either linear or non-linear springs distributed around the circumference of the shaft. The coupling between the flexibility of the shaft and the flexibility of the support structure are considered. The flexible d.o.f. of both the shaft and the support structure are represented as a set of retained and internal d.o.f. of a Craig-Bampton formulation. An additional transformation is performed when the rigid-body d.o.f. is coupled with the internal and the retained d.o.f. in a Craig-Bampton basis. The equations of motion are solved in the time domain using a modified Newmark method for time integration, in which the Newton-Raphson method is used for handling the non-linear behavior within each time step. Analyses are performed to validate the new development for different combinations of load condition, spring type, and rigid-body rotation.     

INFLUENCE OF A LEVELNESS DEFECT IN A THRUST BEARING ON THE DYNAMIC BEHAVIOUR OF AN ELASTIC SHAFT

This paper examines the non-linear dynamic behaviour of a flexible shaft. The shaft is mounted on two journal bearings and the axial load is supported by a defective hydrodynamic thrust bearing at one end. The defect is a levelness defect of the rotor. The thrust bearing behaviour must be considered to be non-linear because of the effects of the defect. The shaft is modelled with typical beam finite elements including effects such as the gyroscopic effects. A modal technique is used to reduce the number of degrees of freedom. Results show that the thrust bearing defects introduce supplementary critical speeds. The linear approach is unable to show the supplementary critical speeds which are obtained only by using non-linear analysis.     

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.     

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.     

THE APPLICATION OF BALL-TYPE BALANCERS FOR RADIAL VIBRATION REDUCTION OF HIGH-SPEED OPTIC DISK DRIVES

This study is dedicated to the design of a ball-type balancer system installed on the high-speed disk drive in order to reduce radial vibrations of rotors caused by eccentricities of disk's center of gravity and circular runway of the ball balancer. The ball balancer is a promising candidate due to its low cost and capability to completely eliminate radial vibrations under the conditions that runway eccentricity, damping and friction are not present. A mathematical model was established first for the analysis of the dynamics of a rotor-balancer system. The influence of concerned parameters, e.g., runway eccentricity and rolling resistance, on residual vibrations was then explored through solving the equations for steady state solutions. The results were used to evaluate the performance of balancers in terms of vibration reduction. The design guidelines for minimizing the vibrations by controlling the aforementioned concerned parameters were provided based on the parametric analysis conducted. Finally, experimental study was orchestrated and performed to verify the validity of the mathematical model and demonstrate balancer capability for reduction of radial vibrations.     

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.     

Circular whirling and stability due to unbalanced magnetic pull and eccentric force

This rapid communication is concerned with the circular whirling and stability of a model rotor in a synchronous generator under no load, subjected to unbalanced magnetic pull and mass eccentric force. The analysis is focused on the synchronous whirling of the rotor. Based on the existing analytical expression for unbalanced magnetic pull with any pole-pair number, the nonautonomous system of differential equations of motion with parametrically exciting force is transformed to an autonomous one by introducing a rotating coordinate frame. The circular whirlings of the model rotor are thus converted into equilibrium solutions to the autonomous system, which can be obtained by solving a system of polynomial equations with two unknowns only. Furthermore, stability of these equilibrium solutions is determined by applying the linearized stability criterion. An example is used to illustrate the proposed analytical method.     

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.     

Modal balancing of flexible rotors without test runs: An experimental investigation

In all methods of balancing flexible rotors currently in use (modal balancing and influence coefficient techniques) test masses and test runs are required for the calculation of correction masses. In the modal balancing procedure suggested by Gasch and Drechsler [1] a simple identification method is used to find the generalized unbalances without test runs. In this paper the Gasch and Drechsler balancing procedure is reported and extended to multibearing rotors with initial shaft bow. Systematic experiments done on two test rotors are described. The results show that flexible shafts mounted in ball bearings can be balanced with very good results. Even rotors in fluid film bearings, exceeding the scope of modal theory, were balanced with good results.     

Design sensitivity analysis of dynamic responses for a BLDC motor with mechanical and electromagnetic interactions

This paper presents a design sensitivity analysis of dynamic responses of a BLDC motor with mechanical and electromagnetic interactions. Based on the equations of motion which consider mechanical and electromagnetic interactions of the motor, the sensitivity equations for the dynamic responses were derived by applying the direct differential method. From the sensitivity equation along with the equations of motion, the time responses for the sensitivity analysis were obtained by using the Newmark time integration method. The sensitivities of the motor performances such as the electromagnetic torque, rotating speed, and vibration level were analyzed for the six design parameters of rotor mass, shaft/bearing stiffness, rotor eccentricity, winding resistance, coil turn number, and residual magnetic flux density. Furthermore, to achieve a higher torque, higher speed, and lower vibration level, a new BLDC motor was designed by applying the multi-objective function method. It was found that all three performances are sensitive to the design parameters in the order of the coil turn number, magnetic flux density, rotor mass, winding resistance, rotor eccentricity, and stiffness. It was also found that the torque and vibration level are more sensitive to the parameters than the rotating speed. Finally, by applying the sensitivity analysis results, a new optimized design of the motor resulted in better performances. The newly designed motor showed an improved torque, rotating speed, and vibration level.     

NON-LINEAR MODELLING OF ROTOR DYNAMIC SYSTEMS WITH SQUEEZE FILM DAMPERS—AN EFFICIENT INTEGRATED APPROACH

Squeeze film dampers used in rotor assemblies such as aero-engines introduce non-linear damping forces into an otherwise linear rotor dynamic system. The steady state periodic response of such rotor dynamic systems to rotating out-of-balance excitation can be efficiently determined by using periodic solution techniques. Such techniques are essentially faster than time marching techniques. However, the computed periodic solutions need to be tested for stability and recourse to time marching is necessary if no periodic attractor exists. Hence, an efficient integrated approach, as presented in this paper, is necessary. Various techniques have been put forward in order to determine the periodic solutions, each with its own advantages and disadvantages. In this paper, a receptance harmonic balance method is proposed for such a purpose. In this method, the receptance functions of the rotating linear part of the system are used in the non-linear analysis of the complete system. The advantages of this method over current periodic solution techniques are two-fold: it results in a compact model, and the receptance formulation gives the designer the widest possible choice of modelling techniques for the linear part. Stability of these periodic solutions is efficiently tested by applying Floquet Theory to the modal equations of the system and time marching carried out on these equations, when necessary. The application of this integrated approach is illustrated with simulations and an experiment on a test rig. Excellent correlation was achieved between the periodic solution approach and time marching. Good correlation was also achieved with the experiment.     

Influence of load torque on stability of rotor driven by flexible shaft

The influence of the load torque on the stability of a symmetric rotor, driven by a flexible shaft, is studied. Both linear and angular displacements of the rotor are considered. The analysis—which is approximate, and with the deflection, the damping and the load torque assumed to be small—shows that the main destabilizing effect of the load torque is due to the transverse moments acting while the rotor is inclined. The reaction pattern at the ends of the shaft, determined by means of the Kirchhoff equations, indicates that the semitangential mode of loading (conservative) is operative.     

Modeling of a rotor speed transient response with radial rubbing

A rotor-stator model of a turbogenerator is introduced in order to investigate speed transients with rotor-to-stator rubbing caused by an accidental blade-off imbalance. In order to assess the angular deceleration of the rotor due to rubbing, the angular position of its cross-section is considered as an unknown of the problem. Displacement fields are discretized through a finite element formulation. The highly nonlinear equations due to contact conditions are solved through an explicit prediction-correction time-marching procedure combined with the Lagrange multiplier approach dealing with a node-to-line contact strategy. The developed numerical tool is suitable for analyzing rotor-stator interactions in turbomachines as the system passes through critical speeds during an accidental shutdown. The sensitivity of the system response to modeling, physical and numerical parameters is investigated. The results highlight the significant role of the friction coefficient together with the diaphragm modeling, from rigid to fully flexible, in the interaction phenomenon. Rigid models have the advantage of simplicity and provide reasonable estimations of the overall response of the turbine. A flexible model, however, may be more computationally intensive but is more appropriate in order to accurately capture quantities of interest such as shaft eccentricity and bearing loads.