vibration of rotationally periodic structures abstract the purpose of this paper is to determine eigensolutions of a rotationally periodic structure p of period n in terms of those of the i th substructure s i of p let y i be the generalized displacement vector and f i be the generalized force vector at interface i i where substructures s i and s i 1 are connected transfer function formulation of s i implies that there exists a linear operator g mapping y t and f i at interface i i to y i 1 and f i 1 at interface i i 1 in addition the operator g depends on transfer functions and therefore eigensolutions of s i the periodicity of p then requires that gn be an identity map this results in n self adjoint fredholm integral equations the non trivial solutions of which predict eigensolutions of p as a consequence a periodic structure p with period n will have exactly n eigenvalues lying between two consecutive eigenvalues of the substructure s i if the interface i i contains only one degree of freedom finally three examples are illustrated the first example considers anti plane strain vibration of a linear elastic solid containing four slots of infinitesimal width the second example derives exact eigensolutions of a string mass periodic structure the third example illustrates how eigensolutions of an axisymmetric structure can be recovered under the formulation of rotationally periodic structures

modal structure of centrifugal pendulum vibration absorber systems with multiple cyclically symmetric groups of absorbers this work studies the symmetry breaking effects on the vibration mode structure of centrifugal pendulum vibration absorber cpva systems when multiple groups of absorbers are used an absorber group is a set of equally spaced identical absorbers absorbers within a group are cyclically symmetric while the entire system is asymmetric because the groups have no pre defined relative angular spacing one rotational and two translational degrees of freedom for the rotor and a single arclength degree of freedom for each absorber are considered in the planar model the well defined structure of the vibration modes is obtained by analytical and numerical investigations of the associated eigenvalue problem this vibration mode structure is similar to that for cpva systems with equally spaced identical absorbers thus the disrupted symmetry from multiple absorber groups does not destroy the vibration mode structure resulting from the cyclic symmetry within each group the critical speeds and flutter instability of the system are investigated

dynamic analysis of finite linear elastic solids containing small elastic imperfections theory with application to asymmetric circular plates abstract eigensolutions green s functions and steady state responses of three dimensional homogeneous isotropic finite linear elastic solids containing small linear elastic imperfections are presented in this paper the inhomogeneities are treated as perturbations to otherwise homogeneous solids the spectral eigensolutions of which are known the exact eigensolutions of the imperfect solid formulated through the betti reciprocal theorem satisfy a set of integral equations from which both analytic perturbation formulas and a numerical iteration scheme are developed special emphasis is given to analysis of perfect homogeneous elastic solids that are spectrally degenerate finally the vibration of circular plates with evenly spaced radial inclusions which cause splitting of some of the repeated vibration modes is analyzed both analytically and numerically for the eigensolutions

effects of rotation on the dynamics of a circular cylindrical shell with application to tire vibration abstract for the purpose of understanding the effects of rotation on wave propagation within a tire s treadband the vibration of an inflated circular cylindrical shell rotating about a fixed axis has been considered here the equations of motion of the rotating shell are formulated in a fixed reference frame i e eulerian co ordinates by assuming wave like solutions for the free vibration case the natural frequencies and corresponding wave like basis functions can then be obtained a natural frequency selection procedure is introduced that can be used to associate each of the basis functions with a single natural frequency the basis functions are then superimposed to represent the forced response of the system when driven by a point harmonic force at a fixed location in the reference frame by using the procedure described here the coefficients of the basis functions can be obtained directly by solving an uncoupled ordinary differential equation finally the resulting forced responses are presented in both the spatial and wave number domains and the wave number spectrum of the rotating shell is compared with that of a stationary shell based on the results presented here it is suggested that at typical rotational speeds it may be possible to use a stationary tire analysis to predict the characteristics of a rotating tire after performing a simple kinematic compensation

vibration of rings on a general elastic foundation the free vibration eigensolutions of a thin ring on a general elastic foundation are obtained by perturbation and galerkin analyses natural frequencies and vibration modes are determined as closed form expressions for a ring having a circumferentially varying foundation of very general description the elastic foundation consists of two orthogonal distributed springs oriented at an arbitrary inclination angle the foundation stiffnesses vary circumferentially the simple eigensolution expressions explicitly show the parameter dependencies lead to natural frequency splitting rules for degenerate unperturbed eigenvalues at both first and second orders of perturbation and identify which nodal diameter fourier components contaminate a given n nodal diameter base mode of the free ring discrete spring supports are treated as a special case where the natural frequencies are determined by five parameters nondimensional spring stiffness stiffness angle support angle number of springs and location of the springs the predicted effects of these parameters on the natural frequencies are verified numerically as an application and as the motivating problem for the study the natural frequencies and vibration modes of a ring gear used in helicopter planetary gears with unequally spaced planets are investigated

modeling and finite element formulation for acoustic problems including rotating domains a finite element formulation for the efficient numerical simulation of sound in computational domains including rotating regions is presented the mathematical description is based on an arbitrary lagrangian eulerian framework and results in a convective wave equation for the scalar acoustic potential numerically the capability of nonconforming grids is explored by applying a nitsche type mortaring between stationary and rotating regions the formulation can be applied to classical acoustics stagnant fluid as well as moving fluids in the case of aeroacoustics the validation with the analytical solution of a rotating point source in three dimensions demonstrates the accuracy and robustness of the developed numerical scheme additionally the convergence study shows that the intersection mesh operations needed in the nonconforming setting do not deteriorate the accuracy of the numerical solution

on the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models free vibration analysis is a major part of any dynamic analysis natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems although the eigensolutions of large scale structures require large computational efforts these solutions may be achieved simply for symmetric structures we present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry having nodes at the axis of symmetry into eigensubproblems with significantly smaller dimensions this decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant using the concept of the kronecker product and similarity transformations the proposed method is applied to three finite element models discretized by triangular and four node quadrilateral plate and shell elements and its efficiency accuracy and simplicity are evaluated copyright 2009 john wiley sons ltd

vibration mode structure and simplified modelling of cyclically symmetric or rotationally periodic systems this paper proves the modal vibration properties of general rotating cyclically symmetric or rotationally periodic systems including those with central components to which cyclically symmetric

semi implicit extension of a godunov type scheme based on low mach number asymptotics i one dimensional flow a single scale multiple space scale asymptotic analysis provides detailed insight into the low mach number limit behavior of solutions of the compressible euler equations we use the asymptotics as a guideline for developing a low mach number extension of an explicit higher order shock capturing scheme this semi implicit scheme involves multiple pressure variables large scale differencing and averaging procedures that are discretized versions of standard operations in multiple scales asymptotic analysis advection and acoustic wave propagation are discretized explicitly and upwind and only one scalar elliptic equation is to be solved implicitly at each time step this equation is a pressure correction equation for incompressible flows when the mach number is zero in the low mach number limit the time step is restricted by a courant number based essentially on the maximum flow velocity for moderate and large mach numbers the scheme reduces to the underlying explicit higher order shock capturing algorithm

linearized acoustic perturbation equations for low mach number flow with variable density and temperature when the mach number tends to zero the compressible navier stokes equations converge to the incompressible navier stokes equations under the restrictions of constant density constant temperature and no compression from the boundary this is a singular limit in which the pressure of the compressible equations converges at leading order to a constant thermodynamic background pressure while a hydrodynamic pressure term appears in the incompressible equations as a lagrangian multiplier to establish the divergence free condition for the velocity in this paper we consider the more general case in which variable density variable temperature and heat transfer are present while the mach number is small we discuss first the limit equations for this case when the mach number tends to zero the introduction of a pressure splitting into a thermodynamic and a hydrodynamic part allows the extension of numerical methods to the zero mach number equations in these non standard situations the solution of these equations is then used as the state of expansion extending the expansion about incompressible flow proposed by hardin and pope j c hardin d s pope an acoustic viscous splitting technique for computational aeroacoustics theor comput fluid dyn 6 1995 323 340 the resulting linearized equations state a mathematical model for the generation and propagation of acoustic waves in this more general low mach number regime and may be used within a hybrid aeroacoustic approach

a projection hybrid finite volume element method for low mach number flows the purpose of this article is to introduce a projection hybrid finite volume element method for low mach number flows of viscous or inviscid fluids starting with a 3d tetrahedral finite element mesh of the computational domain the equation of the transport diffusion stage is discretized by a finite volume method associated with a dual mesh where the nodes of the volumes are the barycenters of the faces of the initial tetrahedra the transport diffusion stage is explicit upwinding of convective terms is done by classical riemann solvers as the q scheme of van leer or the rusanov scheme concerning the projection stage the pressure correction is computed by a piecewise linear finite element method associated with the initial tetrahedral mesh passing the information from one stage to the other is carefully made in order to get a stable global scheme numerical results for several test examples aiming at evaluating the convergence properties of the method are shown

generalized solution of time dependent traveling load problem via moving finite element scheme abstract a finite element methodology is described which can handle the transient response of moving structures which are themselves subject to time dependent traveling load fields the generality of the approach is such that it can handle the overall transient response due to moving loads whose magnitude zone of application and trajectory are time dependent in addition to developing the overall transient solution methodology attention is also given to establishing the overall eigenvalue properties of the linear problem as well as developing a solution for the steady vibrating case to benchmark the overall procedure the results of several numerical experiments are included in the presentation

computation of flow noise using source terms in linearized euler s equations an acoustic analogy using linearized euler s equations lee forced with aerodynamic source terms is investigated to computetheacousticfare eld thishybridmethod isappliedto threemodelproblemssimulatedby solving navier stokes equations in this way its validity is estimated by comparing the predicted acoustic e eld with the reference solution given directly by the navier stokes equations the noise radiated by two corotating vortices is studied e rst in a medium at rest and second in a mean sheared e ow with no convection velocity then the sound e eld generated by vortex pairings in a subsonic mixing layer is investigated in this case a simplie ed formulation of lee is proposed to prevent the exponential growth of instability waves the acoustic e elds obtained by solving lee are in good agreement with the reference solution this study shows that the source terms introduced into the lee are appropriate for free sheared e ows and that acoustic mean e ow interactions are properly taken into account in the wave operator nomenclature b half width of the monopolar source c sound velocity e f h vectors in linearized euler s equations lee f frequency f0 fundamental frequency of the mixing layer k complex wave number kr ciki m mach number p pressure re reynolds number rc vortex core radius r0 initial half distance between the two vortices s sound source vector in lee si source terms in the momentum equations t period tij lighthill s tensor t time u unknown vector in lee u1 slow stream velocity of the mixing layer u2 rapid stream velocity of the mixing layer u velocity vector u1 u2 v initial tangential velocity of vortices

transverse vibrations of elastically connected double string complex system part i free vibrations this paper analyzes the forced transverse vibrations of an elastically connected double string complex continuous system the general solutions of forced vibrations of strings subjected to arbitrarily distributed continuous loads are found by using the method of expansion in a series of the mode shape functions different cases of exciting loadings are analyzed the action of stationary harmonic loads and moving concentrated forces is considered vibrations caused by the harmonic exciting forces are discussed and conditions of resonance and dynamic vibration absorption are formulated thus the string type dynamic absorber can be used to suppress the excessive vibrations of corresponding string systems the dynamic string type damper is a new concept of a continuous dynamic vibration absorber cdva it is shown that a corresponding two degree of freedom discrete system is an analogue of an elastically connected double string complex system theoretical analysis is illustrated by a numerical example

on the in plane vibrations of rotating rings the general problem of the in plane vibrations of a rotating ring is investigated the potential and kinetic energy functionals including the effects of extensional and shear deformation and of rotatory inertia respectively are derived the governing differential equations for the inextensional case are derived by using hamilton s principal and shown to have additional terms not present in previous works numerical studies which show the importance of these additional terms are presented

measurement and analysis of modulated doublet mode response in mock bladed disks the steady state response of spatially modulated doublet modes that occur in low count flexible bladed disks is investigated for the case in which the structure is driven by a harmonic travelling wave excitation source finite element simulation and modal testing of prototypical bladed disk structures demonstrate the presence of particular wavenumbers beyond the base number of nodal diameters which contaminate and distort the appearance of certain doublet modes the manner in which the natural frequency and wavenumber content of such modes shift and split as functions of the number of blades and their span angle is discussed in the light of a companion perturbation analysis for rotationally periodic structures resonance conditions are established and verified through simultaneous measurements made with a spin test stand using sensors that are placed in the rotating structure and stationary excitation frames of reference the travelling wave response components of a repeated frequency doublet mode are shown to propagate either in the same or opposite direction as the excitation source depending on whether certain algebraic relationships between the excitation order the base number of nodal diameters and the contamination wavenumbers are satisfied to the extent that such components can travel at different phase speeds and directions relative to one another the placement of sensors on the structure can be optimized to best measure the response amplitude conversely other placements can result in submaximal measurement of peak vibration amplitude over the structure

response of rotating rings to harmonic and periodic loading and comparison with the inverted problem abstract the harmonic and periodic forced vibrations of rotating rings are derived and investigated the modal expansion technique yields the forced solution which is characterized by four generalized co ordinates associated with each n circumferential wave number the inextensional assumption is presumed when flexural vibration is the only important component to reduce the order of the system the closed form solutions to the harmonic load cases once concentrated once distributed are demonstrated and interpreted the approach is then extended to periodic loads where fourier sine and cosine series is applied examples depict the numerical responses to all the cases being derived the solutions of a stationary ring subjected to traveling loads are also solved for comparison their difference is investigated and interpreted from various viewpoints

constrained spanning trees and the traveling salesman problem abstract minimum weight 1 trees provide a well known lower bound for the symmetric traveling salesman problem we propose to strengthen this bound by imposing degree constraints upon the 1 trees the vertices for the constraints are chosen to form a stable set s we propose an o m log log n s n s m m n algorithm for this problem and report on its use in a lagrangian approach to the traveling salesman problem

flexural vibration of a thin rotating ring abstract flexural vibration of a thin rotating ring has been investigated experimentally and from some theoretical points of view the frequencies of forward and backward flexural traveling waves were measured by using strain gages the range of rotational speeds of the ring was such as to include frequencies comparable with that of the fundamental flexural mode of a non rotating ring and the stability of flexural vibration was examined with consideration of the effects of the centrifugal and coriolis forces it was ascertained that for all the flexural modes of the rotating ring no instability phenomenon exists in the sense that the natural frequencies of the rotating ring never become zero over the whole range of rotational speed in the light of the experimental results previous theories for a rotating ring or a circular cylindrical shell and the author s theory for a ring are examined and some differences among them discussed

excitation of rotating circumferentially periodic structures abstract a circumferentially periodic structure consists of a finite number of identical substructures coupled together in identical ways to form a closed ring type structure the vibrational behaviour of such structures when rotating about the axis of structural symmetry is considered especially the forced response due to forces fixed in space it is known that for a circumferentially symmetric structure rotating about its axis of structural symmetry and excited by a force fixed in space resonance for the n nodal diameters mode is obtained when the corresponding natural frequency is n n where is the angular velocity of rotation the deformation at these resonances consists of waves travelling backwards in relation to the direction of the structure rotation this resonance condition also holds for a circumferentially periodic structure but then additional resonance possibilities exist given by n kn n where n is the number of sub structures and k 0 1 2 these results are known to hold true for a structure fixed in space and acted upon by a rotating force in this paper these results are shown to hold true also for a rotating structure acted upon by a force distribution fixed in space both backward and forward travelling waves can be excited in those cases when the rotating structure exhibits coriolis accelerations the natural frequencies for backward 1n and forward 2n rotating mode shapes are different this well known property has a pronounced effect on the forced response due to forces fixed in space in the absence of coriolis effects the magnification factor is essentially 1 n 2 2 where is the forcing frequency due to rotation but in the presence of coriolis effects the magnification factor has the characteristics 1 1n 2n

a unified method for computing incompressible and compressible flows in boundary fitted coordinates a unified method for computing incompressible and compressible flows with mach uniform accuracy and efficiency is described the method is equally applicable to stationary and nonstationary flows a pressure based discretisation on a staggered grid in general boundary fitted coordinates is used for the euler equations extension to navier stokes is straightforward dimensionless variables that remain finite for all mach numbers are used mach number independent accuracy and efficiency is shown by numerical experiments

the response of a non uniformly tensioned circular string to a moving load abstract a non uniformly tensioned circular string on a damped elastic foundation is acted upon by a moving concentrated load this configuration can be viewed as a very simplified representation of a spinning rectangularly orthotropic disk acted upon by a spatially fixed force or of a stationary orthotropic disk subjected to a circumferentially moving load the natural frequencies and normal modes of vibration of the string on elastic foundation are determined for a range of values of the tension inhomogeneity the response of the string to a moving load is then determined the results indicate that there are many speeds below the critical speed of the uniformly tensioned string for which the forced response is large it is anticipated that the response of this simple model to a moving load can provide insight into the behavior of the mathematically more complicated rectangularly orthotropic circular disk

convergence studies on static and dynamic analyses of plates by using the u transformation and the finite difference method this paper presents the exact static and dynamic analyses of simply supported rectangular plates the analytical solutions for displacements buckling loads natural frequencies and dynamic responses are obtained by using the double u transformation method and the finite difference method after an equivalent system with cyclic periodicity in two directions is established the difference governing equation for such an equivalent system can be uncoupled by applying the double u transformation then the exact analytical finite difference solutions the exact error expressions and the exact convergence rates are derived these results cannot be obtained if other methods are used instead

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

response of modulated doublet modes to travelling wave excitation the forced vibration of a rotationally periodic structure when subjected to travelling wave excitation is discussed with emphasis placed on the steady state response of doublet modes having either repeated or split frequencies such vibration modes have spatially modulated shapes defined by 1 the number of nodal diameters present in the limiting case of axisymmetry and 2 certain additional fourier harmonics which contaminate and distort their appearances the natural frequency and mode structure of a model periodic structure is discussed in the context of an otherwise axisymmetric disk having evenly spaced sector shaped line distributions of stiffness and inertia through perturbation analysis the contamination wavenumbers present in a doublet having repeated frequency are shown to comprise two subsets the members of which have sine and cosine coefficients of the same or of differing signs for each wavenumber in the mode shape s fourier expansion the structure of the wavenumber content is explored further with respect to the response of repeated and split doublets to a harmonic travelling wave excitation the individual fourier components comprising a modulated doublet can respond and propagate in the same direction as the excitation or opposite to it depending on the wavenumber of the excitation and the subset to which the contamination wavenumber belongs the response of the split frequency doublets and the circumstances under which travelling or standing wave responses or a blend of the two can occur in the structure s reference frame are also examined in the context of the model periodic structure the qualitative character of the response the forward or backward propagation direction of each mode s constituent wavenumber components and the phase speeds of those components are discussed in illustrative case studies

sub trans and supercritical variable magnitude travelling loads on cylindrical rings abstract the problem of time dependent variable magnitude travelling loads in a closed symmetric structure is considered this includes the circumferential propagation in viscoelastically supported cylindrical rings which can rotate about their axes the main emphasis of the work is to establish the effects of time dependent load variability on the response characteristics when travelling velocities are in the sub trans and supercritical speed ranges this will enable the doppler like interactions among the propagating bending and in plane waves at the various travelling load speed ranges to be evaluated as an example simulation the model is tuned to define the dynamic response characteristics of tires rolling on bumpy surfaces this enables the sub trans and the supercritical response characteristics to be evaluated

a matroid algorithm and its application to the efficient solution of two optimization problems on graphs we address the problem of finding a minimum weight baseb of a matroid when in addition each element of the matroid is colored with one ofm colors and there are upper and lower bound restrictions on the number of elements ofb with colori fori 1 2 m this problem is a special case of matroid intersection we present an algorithm that exploits the special structure and we apply it to two optimization problems on graphs when applied to the weighted bipartite matching problem our algorithm has complexity o e v v 2log v herev denotes the node set of the underlying bipartite graph ande denotes its edge set the second application is defined on a general connected graphg v e whose edges have a weight and a color one seeks a minimum weight spanning tree with upper and lower bound restrictions on the number of edges with colori in the tree for eachi our algorithm for this problem has complexity o e v m 2 v m v 2 a special case of this constrained spanning tree problem occurs whenv is a set of pairwise nonadjacent nodes ofg one must find a minimum weight spanning tree with upper and lower bound restrictions on the degree of each node ofv then the complexity of our algorithm is o v e v v 2 finally we discuss a new relaxation of the traveling salesman problem

a simple theory for the analysis and correction of frequency splitting in slightly imperfect rings abstract the flexural radial vibration of circular rings with imperfections represented by small attached masses and springs is considered the concept of an equivalent imperfection mass is introduced this is used to quantify the frequency splitting which occurs in slightly imperfect axisymmetric bodies the concept is used to estimate the trimming mass which must be added to or removed from the ring and at which point in order to reduce or eliminate the frequency split in a given pair of modes the method can be applied to any axisymmetric body for which the mode shapes can be measured or computed

a highly accurate discontinuous galerkin method for complex interfaces between solids and moving fluids we have extended a new highly accurate numerical scheme for unstructured 2d and 3d meshes based on the discontinuous galerkin approach to simulate seismic wave propagation in heterogeneous media containing fluid solid interfaces because of the formulation of the wave equations as a unified first order hyperbolic system in velocity stress the fluid can be in movement along the interface the governing equations within the moving fluid are derived from well known first principles in fluid mechanics the discontinuous galerkin approach allows for jumps of the material parameters and the solution across element interfaces which are handled by riemann solvers or numerical fluxes the use of riemann solvers at the element interfaces makesthe treatment of the fluid particularly simple bysetting the shearmodulus in the fluid region to zero no additional compatibility relations such as vanishing shear stress or continuity of normal stresses are necessary to couple the solid and fluid along an interface the

second order implicit explicit total variation diminishing schemes for the euler system in the low mach regime abstract in this work we consider the development of implicit explicit total variation diminishing tvd methods also termed ssp strong stability preserving for the compressible isentropic euler system in the low mach number regime the proposed scheme is asymptotically stable with a cfl condition independent of the mach number in addition it degenerates in the low mach number regime to a consistent discretization of the incompressible system since it has been proved that implicit schemes of order higher than one cannot be tvd ssp 30 we construct a new paradigm of implicit time integrators by coupling first order in time schemes with second order ones in the same spirit as highly accurate shock capturing tvd methods in space for this particular class of schemes the tvd property is first proved on a linear model advection equation and then extended to the isentropic euler case the result is a method which interpolates from the first to the second order both in space and time it preserves the monotonicity of the solution and is highly accurate for all choices of the mach number moreover the time step is only restricted by the non stiff part of the system we finally show thanks to one and two dimensional test cases that the method indeed possesses the claimed properties

vibrations of double string complex system under moving forces closed solutions in this paper the dynamic response of a double string system traversed by a constant or a harmonically oscillating moving force is considered the force is moving with a constant velocity on the top string the strings are identical parallel one upon the other and continuously coupled by a linear winkler elastic element the classical solution of the response of a double string system subjected to a force moving with a constant velocity has a form of an infinite series the main goal of this paper is to show that in the considered case a part of the solution can be presented in a closed analytical form instead of an infinite series the presented method of finding the solution in a closed analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations

parametric instability of a circular ring subjected to moving springs this work investigates parametric instabilities of in plane bending vibrations of a thin elastic ring subject to forces from discrete rotating springs of arbitrary number spacing and orientation several configurations are examined including systems with symmetric and asymmetric circumferential spring spacing and systems with and without fixed springs the method of multiple scales is applied to analytically identify principal and combination instability boundaries as closed form expressions two different numerical approaches are used to verify the analytical results the effects of different system parameters on the instability boundaries are studied analytically the bending stiffness of the ring the number of springs and their stiffness location orientation and rotation speed for several cases well defined properties for the occurrence or suppression of instabilities are obtained as simple relations in the system parameters

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 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

the in plane free vibration of an elastically supported thin ring rotating at high speeds revisited abstract the in plane free vibration of a rotating thin ring is revisited in this paper a new model is proposed which accounts for the elastic foundation and the through thickness variation of the radial stress the emphasis is placed on a proper consideration of the geometrical nonlinearity which is essential for the consistent modelling of the ring stiffening resulting from the radial expansion caused by rotation the in plane stability of a thin ring rotating at relatively high speeds is analysed thoroughly it is shown that the ring can become unstable should the rotational speed exceed a critical value this result is new as in most known to the authors previous studies the stability problem is either not considered or it is stated that the in plane vibration of a rotating ring is stable in the studies which did address the instability the conclusions and the employed models are prone to criticism a parametric study is conducted to illustrate the effects of the ring properties on the in plane stability finally modes which appear as stationary displacement patterns of the ring to an observer in the space fixed reference system are investigated it is shown that the stationary patterns can occur prior to the onset of the instability for certain ring parameters