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.     

DYNAMIC RESPONSE OF A ROTATING FLEXIBLE ARM CARRYING A MOVING MASS

A clamped-free flexible arm rotating in a horizontal plane and carrying a moving mass is studied in this paper. The arm is modelled by the Euler-Bernoulli beam theory in which rotatory inertia and shear deformation effects are ignored. The assumed mode method in conjunction with Hamilton's principle is used to derive the equation of motion of the system which takes into account the effect of centrifugal stiffening due to the rotation of the beam. The eigenfunctions of a cantilever beam which satisfy the prescribed geometric boundary conditions are used as basis functions in the assumed mode method. The equation of motion is expressed in non-dimensional matrix form. Pre-designed transformed cosine profiles are used as trajectory inputs for the hub angle and the moving mass. The equation of motion is solved numerically using the fourth order Runge-Kutta method. Graphical results are presented to show the influence of centrifugal stiffening effect, moving mass values, mass travelling time, hub angle and mass trajectory profile on the deflection of the beam.     

VIBRATION AND STABILITY OF ROTATING POLAR ORTHOTROPIC ANNULAR DISKS SUBJECTED TO A STATIONARY CONCENTRATED TRANSVERSE LOAD

In the present paper, natural frequencies and stability of a spinning polar orthotropic disk subjected to a stationary concentrated transverse load are investigated. The analysis of the free vibration of a spinning disk is performed first to find natural frequencies and corresponding vibration modes. The resulting eigenfunctions obtained from the free vibration are used as deflection functions of the forced vibration of a disk where the load is modelled as a mass-spring-dashpot system fixed in space. By using the Galerkin approximation method, eigenvalues of the whole system are determined. Results show that disks with higher values of modulus ratios or the Poisson ratios have higher natural frequencies, and the stability of the whole system can be improved by raising the value of the modulus ratio or lowering that of the Poisson ratio.     

DYNAMIC MODELLING AND VIBRATION ANALYSIS OF A FLEXIBLE CABLE-STAYED BEAM STRUCTURE

In this paper, the non-linear vibration of a cable-stayed beam with time-varying length and tension in the cable is investigated. A set of non-linear, time-varying differential equations describing this coupling system is derived by Hamilton's principle and the finite element method. According to the results of numerical simulation, the tension of the cable is related to the cable length, which in turn is a function of the longitudinal and transverse displacements of the cable. Furthermore, it is shown that the tension and length of the cable can be considerably different by using linear and non-linear models.     

QUASI-PERIODIC VIBRATION OF CRACKED ROTOR ON FLEXIBLE BEARINGS

A cracked rotor on flexible bearings is studied in this paper. The vibration of such a system has many complexities because of the crack and bearing flexibility. However, if the properties of the bearings are known, the system can be simplified by supposing that, the vibration due to weight is dominant. Equations of motion are derived, and a linear system in which the crack has been considered as an external disturbance described by a series of trigonometric functions is obtained. Consequently, the quasi-periodic vibrations of the rotor and bearings are established by harmonic balance method and approximate values of the vibration determined by truncating the higher order terms. It is believed that the simulated results will be useful for crack detection in the case of weight-dominant rotors.     

Design and analysis of vibration isolation systems for hard disk drives

Hard disk drives (HDD) are increasingly being used in mobile applications and consumer electronic devices in potentially harsh dynamic environments. In mission-critical applications such as in computers on board trains and military vehicles, the vibration experienced by the operating HDD can be very severe. An effective suspension system is necessary to isolate the HDD from the strong vibration transmitted from the ground, vehicle engine, etc. This paper gives details on how to design, test and evaluate vibration isolation systems for operating HDD subjected to very severe random vibrations based on military specifications MIL-STD-810E. One design is presented and evaluated based on well-defined criteria. Design principles on how to achieve the required degree of vibration isolation is described. A computer simulation model of an example design is formulated and correlated with experiments. It is shown that a low-cost, vibration isolation system can indeed be made that will allow an HDD to operate well when subjected to strong input vibration.     

Implication of conservative and gyroscopic forces on vibration and stability of an elastically tailored rotating shaft modeled as a composite thin-walled beam

Problems related with the implications of conservative and gyroscopic forces on vibration and the stability of a circular cylindrical shaft modeled as a thin-walled composite beam and spinning with constant angular speed about its longitudinal axis are addressed. Taking into account the directionality property of fiber reinforced composite materials, it is shown that for a shaft featuring flapwise-chordwise-bending coupling, a dramatic enhancement of both the vibrational and stability behavior can be reached. In addition, the effects played in the same context by transverse shear, rotatory inertias as well as by the various boundary conditions are discussed and pertinent conclusions are outlined.     

Suppression of random vibration in flexible structures using a hybrid vibration absorber

A design method is proposed to suppress stationary random vibration in flexible structures using a hybrid vibration absorber (HVA). While the traditional vibration absorber can damp down the vibration mainly at the pre-tuned mode of the primary structure, active damping is generated by the proposed HVA to damp down all resonant modes of interest of the vibrating structure and the spatial average mean square motion of the vibrating structure can be minimized. Only one absorber and one feedback signal are required to achieve global vibration suppression of a flexible structure under stationary random excitation. A special pole-placement controller is designed such that all vibration modes of the flexible structures become critically damped. It is proved analytically that the proposed HVA damps the vibration of the entire structure instead of just the attachment point of the absorber. The proposed optimized HVA is tested on a beam structure and it shows a superior performance on global suppression of broadband vibration in comparison to other published designs of passive and hybrid vibration absorbers.     

Vibrational response of a moving suspension-slider loading system exciting a rotating flexible disk

To investigate the vibrational response of the magnetic read/write head in hard disk drives this paper models a rotating flexible disk excited by a moving suspension-slider system which is considered to be a mass-dashpot-spring loading system, with the initial unstressed transverse runout integrated into the rotating disk dynamic model. The slider motion on the disk surface is driven by the suspension rotating at a constant speed. By subtracting the steady-state deflection component from the instantaneous deflection response of the rotating disk system, the relative vibration transverse deflection of the slider caused by the motion of the suspension-slider loading system is obtained. The effects of the slider initial and final positions, speed of movement, the disk rotational speed, and the disk mode of the initial transverse runout on the maximum amplitude of the relative vibration deflection are analyzed.     

Bending vibration of axially loaded Timoshenko beams with locally distributed Kelvin-Voigt damping

Utilizing the Timoshenko beam theory and applying Hamilton's principle, the bending vibration equations of an axially loaded beam with locally distributed internal damping of the Kelvin-Voigt type are established. The partial differential equations of motion are then discretized into linear second-order ordinary differential equations based on a finite element method. A quadratic eigenvalue problem of a damped system is formed to determine the eigenfrequencies of the damped beams. The effects of the internal damping, sizes and locations of damped segment, axial load and restraint types on the damping and oscillating parts of the damped natural frequency are investigated. It is believed that the present study is valuable for better understanding the influence of various parameters of the damped beam on its vibration characteristics.     

A HYBRID FINITE SEGMENT/FINITE ELEMENT MODELLING AND EXPERIMENT ON A FLEXIBLE DEPLOYMENT SYSTEM

This paper focuses on the dynamic responses of a flexible deployment system that has a central rigid body and four articulated flexible beams and undergoes locking impact. A hybrid finite segment/finite element model and an experiment are presented for the deploy-ment system. The flexible beam components in the system are modelled with the finite segments connected by massless beam elements, wherein the finite segments describe the inertia of the large rotation flexible beam and the massless elastic elements describe the elas-ticity of the flexible beam by taking the advantage of small deformation in the relative co-ordinate system. To model the internal impacts in the articulate joints due to clearances, a continuous contact force model of locking joint is also proposed. The governing differential-algebraic equations of the system are established by the Newton-Euler method with Lagrange multipliers and are solved with the method of generalized co-ordinate partitioning. To accelerate the numerical integration, a “longitudinal constraint” is suggested to alleviate the stiff problem of the dynamic equations. In addition, a physical model of the deployment system is constructed. The deployment is released by the compressed springs in the joints. A position measuring system of linear CCD cameras is used to measure the large displacement of the system. Correlations between the mathematical model and the experiments are also presented. Reasonable results are obtained.     

Theoretical model of ball joint squeak

The theoretical model for the ball joint squeak problem is proposed. The ball joint is modeled as a sphere attached to a rotating flexible beam and it is in contact with a semi-spherical rigid socket. The relative rotation and vibration between the ball and socket produces friction stresses causing the dynamic instability of the system. The squeak propensity of the ball joint system is numerically investigated through the eigenvalue sensitivity analysis at the sliding equilibrium. Stability analysis shows that the bending modes of the beam can produce the dynamic instability under the negative friction-velocity slope. It is also highlighted that the squeak propensity under the poor friction characteristics can be controlled by the system parameters such as the sliding speed, the contact stiffness, the radius of the ball, the axial load and the tilting angle.     

Vibration characteristics of a rotating flexible arm with ACLD treatment

In this paper, the vibration behavior and control of a clamped–free rotating flexible cantilever arm with fully covered active constrained layer damping (ACLD) treatment are investigated. The arm is rotating in a horizontal plane in which the gravitational effect and rotary inertia are neglected. The stress–strain relationship for the viscoelastic material (VEM) is described by a complex shear modulus while the shear deformations in the two piezoelectric layers are neglected. Hamilton's principle in conjunction with finite element method (FEM) is used to derive the non-linear coupled differential equations of motion and the associated boundary conditions that describe the rigid hub angle rotation, the arm transverse displacement and the axial deformations of the three-layer composite. This refined model takes into account the effects of centrifugal stiffening due to the rotation of the beam and the potential energies of the VEM due to extension and bending. Active controllers are designed with PD for the piezosensor and actuator. The vibration frequencies and damping factors of the closed-loop beam/ACLD system are obtained after solving the characteristic complex eigenvalue problem numerically. The effects of different rotating speed, thickness ratio and loss factor of the VEM as well as different controller gain on the damped frequency and damping ratio are presented. The results of this study will be useful in the design of adaptive and smart structures for vibration suppression and control in rotating structures such as rotorcraft blades or robotic arms.     

A CONCENTRATED MASS ON THE SPINNING UNCONSTRAINED BEAM SUBJECTED TO A THRUST

The dynamic stability of a spinning unconstrained beam subjected to a pulsating follower forceP₀ +P₁cos Ωt is analyzed. A concentrated mass is located at an arbitrary location on the beam, and the stability of the beam is studied with the mass at various locations. The beam is analyzed using the Timoshenko-type shear deformation theory with the rotary inertia. Hamilton's principle is used to derive the equations of motion, and the spinning speed of the beam with various non-dimensional parameters subjected to a pulsating follower force is investigated. The finite element method is applied to analyze the spinning beam model, and the method of multiple scales is used to investigate the dynamic stability characteristics. A pulsating follower force is applied, and then the stability regions are changed with the transitions of the stability area in many regions. The results show that the concentrated mass increases the dynamic stability of the spinning unconstrained beam subjected to a thrust. As the spinning speed of the beam is increased, the instability regions are reduced, but various slight instability regions are additionally developed.     

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 AND STABILITY CONTROL OF SMART COMPOSITE ROTATING SHAFT VIA STRUCTURAL TAILORING AND PIEZOELECTRIC STRAIN ACTUATION

A dual approach based on both structural tailoring and piezoelectric strain actuation, aimed at controlling the free vibration and stability of a spinning circular shaft subjected to axial forces is presented. Due to the involvement in these structural systems of gyroscopic forces and, consequently, of the possible occurrence of divergence and flutter instabilities, the dual control methodology shows a high degree of efficiency toward postponement of the occurrence of these instabilities. The structural model of the shaft as considered in this paper is based on an advanced thin-walled beam that includes the effects of transverse shear, anisotropy of constituent materials, rotatory inertias, etc. The displayed results reveal the synergistic implications of the application of this dual technology toward the enhancement of the dynamic response characteristics and expansion of the domain of stability of these systems.     

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.     

Dynamic time responses of a flexible spinning disk misaligned with the axis of rotation

Using the finite element method, this study investigates the dynamic time responses of a flexible spinning disk of which axis of rotation is misaligned with the axis of symmetry. The misalignment between the axes of symmetry and rotation is one of major vibration sources in optical disk drives such as CD-ROM, CD-R, CD-RW and DVD drives. Based upon the Kirchhoff plate theory and the von Karman strain theory, three coupled equations of motion for the misaligned disk are obtained: two of the equations are for the in-plane motion while the other is for the out-of-plane motion. After transforming these equations into two weak forms for the in-plane and out-of-plane motions, the weak forms are discretized by using newly defined annular sector finite elements. Applying the generalized-α time integration method to the discretized equations, the time responses and the displacement distributions are computed and then the effects of misalignment on the responses and the distributions are analyzed. The computation results show that the misalignment has an influence on the magnitudes of the in-plane displacements. It is also found that the misalignment results in the amplitude modulation or the beat phenomenon in the time responses of the out-of-plane displacement.     

Genetic algorithm based active vibration control for a moving flexible smart beam driven by a pneumatic rod cylinder

A rod cylinder based pneumatic driving scheme is proposed to suppress the vibration of a flexible smart beam. Pulse code modulation (PCM) method is employed to control the motion of the cylinder's piston rod for simultaneous positioning and vibration suppression. Firstly, the system dynamics model is derived using Hamilton principle. Its standard state-space representation is obtained for characteristic analysis, controller design, and simulation. Secondly, a genetic algorithm (GA) is applied to optimize and tune the control gain parameters adaptively based on the specific performance index. Numerical simulations are performed on the pneumatic driving elastic beam system, using the established model and controller with tuned gains by GA optimization process. Finally, an experimental setup for the flexible beam driven by a pneumatic rod cylinder is constructed. Experiments for suppressing vibrations of the flexible beam are conducted. Theoretical analysis, numerical simulation and experimental results demonstrate that the proposed pneumatic drive scheme and the adopted control algorithms are feasible. The large amplitude vibration of the first bending mode can be suppressed effectively.     

Magneto-elastic dynamics and bifurcation of rotating annular plate

In this paper, magneto-elastic dynamic behavior, bifurcation, and chaos of a rotating annular thin plate with various boundary conditions are investigated. Based on the thin plate theory and the Maxwell equations, the magneto-elastic dynamic equations of rotating annular plate are derived by means of Hamilton's principle. Bessel function as a mode shape function and the Galerkin method are used to achieve the transverse vibration differential equation of the rotating annular plate with different boundary conditions. By numerical analysis, the bifurcation diagrams with magnetic induction, amplitude and frequency of transverse excitation force as the control parameters are respectively plotted under different boundary conditions such as clamped supported sides, simply supported sides, and clamped-one-side combined with simply-anotherside. Poincare′ maps, time history charts, power spectrum charts, and phase diagrams are obtained under certain conditions,and the influence of the bifurcation parameters on the bifurcation and chaos of the system is discussed. The results show that the motion of the system is a complicated and repeated process from multi-periodic motion to quasi-period motion to chaotic motion, which is accompanied by intermittent chaos, when the bifurcation parameters change. If the amplitude of transverse excitation force is bigger or magnetic induction intensity is smaller or boundary constraints level is lower, the system can be more prone to chaos.