global solutions for a semilinear two dimensional klein gordon equation with exponential type nonlinearity we prove the existence and uniqueness of global solutions for a cauchy problem associated to a semilinear klein gordon equation in two space dimensions our result is based on an interpolation estimate with a sharp constant obtained by a standard variational method 2006 wiley periodicals inc

maximal ergodic theorems for some group actions abstract we prove maximal ergodic inequalities for a sequence of operators and for their averages in the noncommutative l p space we also obtain the corresponding individual ergodic theorems applying these results to actions of a free group on a von neumann algebra we get noncommutative analogues of maximal ergodic inequalities and pointwise ergodic theorems of nevo stein

numerical analysis of free vibrations of laminated composite conical and cylindrical shells discrete singular convolution dsc approach a numerical study on the free vibration analysis for laminated conical and cylindrical shell is presented the analysis is carried out using love s first approximation thin shell theory and solved using discrete singular convolution dsc method numerical results in free vibrations of laminated conical and cylindrical shells are presented graphically for different geometric and material parameters free vibrations of isotropic cylindrical shells and annular plates are treated as special cases the effects of circumferential wave number number of layers on frequencies characteristics are also discussed the numerical results show that the present method is quite easy to implement accurate and efficient for the problems considered

numerical presentation of eigenvalue curve veering and mode localization in coupled pendulums to be 0 01 0 01 0 01 0 0 and 0 01 0 the e rst variations and the eigenvalues obtained by the classic method and the improved calculation are given in table 1 the relative errors of the variations are shown in fig 2 the result of the improved calculation is in the very good agreementwith the exact one apartfromthe third mode the e rst variations given by the improved calculation are even better than those with the second variations by the classic method a large modie cation of the truss is considered by multiplying a and b with a factor from 1 through 10 the relative errors of the e rst variations are shown in fig 3 the errors in the e rst mode increase from 0 0827 to 0 8410 whereas those of the classic method change from 0 4159 to 4 1834 the fourth mode shows the biggest difference as the largest error of the improved calculation is 0 0023 but 9 9982 for the classic method conclusions an improved calculation of eigenvalue variations has been presented the eigenvalue approximation from the e rst variation is a rayleigh quotient the variations are different from those in the classic method when there is a variation of b which is the inertiarelated matrix in a structural dynamics although there are more terms in the variations of eigenvalues no extra matrix operation is required thegivenexamplecone rmsthattheaccuracyoftheeigenvalue variations is much better than that obtained with the classic method

scattering for a critical nonlinear wave equation in two space dimensions we show that the solutions to the cauchy problem for a wave equation with critical exponential nonlinearity in 2 space dimensions scatter for arbitrary smooth compactly supported initial data

particularities of newton s method in space frame force determination utilising eigenpair functions abstract redundant space frame dynamics are modelled through the structural eigenproblem which is formulated to account for the geometric stiffness due to self equilibrating membrane forces the resulting mathematical eigenvalues and eigenvectors respectively indicative of natural frequencies and modes of oscillation are then assumed to be continuous functions of a scalar factor upon a normalised set of forces in equilibrium newton s method or sensitivity analysis provides a means for minimising the difference between mathematical and physically observed eigenvalues whereupon the axial forces are inferred from the model issues of convergence and root uniqueness are anticipated eigenvalue coalescence in which eigenvalues transitorily coalesce to permute their arbitrarily numerical ordering is seen to define non smooth eigenvalue functions and hence encumber newton s method mode tracing strategies to overcome this are discussed the existence of various degrees of eigenvalue degeneracy owing to frame cyclic symmetry or otherwise is anticipated in terms of the influence upon newton s method if the degree of frame static redundancy is greater than one and there exist a number of linearly independent force distributions then the required force determination is a multidimensional newton method problems encountered in these manifold dimensions including those arising from frame spatial periodicity and the adversity of newton s method are overcome emphasis is placed upon minimising the dimensionality of newton s method through enforcement of equilibrium constraints illustrative numerical simulations are given and some applications of the method are proposed

free vibration analysis of rotating cantilever plates linearized equations of motion for the free vibration analysis of rotating cantilever plates are derived two inplane stretch variables are introduced and approximated to obtain the ordinary differential equations of motion the use of the two in plane stretch variables enables one to obtain the equations of motion which include proper motion induced stiffness variation terms the equations of motion are transformed into dimensionless forms in which dimensionless parameters are identieed the effects of the dimensionless parameters on the modal characteristics of rotating cantilever plates are investigated through numerical study the eigenvalue loci veering and crossing along with the associated mode shape variations are also presented and discussed

behavior of derivatives of eigenvalues and eigenvectors in curve veering and mode localization and their relation to close eigenvalues the problem to measure the phenomena of eigenvalue curve veering and mode localization is addressed in this paper the second derivative of an eigenvalue and the first derivative of an eigenvector are taken as the measures numerically showing curve veering and mode localization based on the measurement close eigenvalues as a key factor for the occurrence of the phenomena are defined two eigenvalues are considered to be close if their difference is small enough to cause the occurrence of the phenomena the curve veering and mode localization can be noted by comparison of the derivatives with a critical value and hence the associated eigenvalues are close weakly coupled springs are given as an example

buckling of a conical thin shell with variable thickness under a dynamic loading this study considers the buckling of an elastic truncated conical shell having a meridional thickness expressed by an arbitrary function subject to a uniform external pressure which is a power function of time at first the fundamental relations and donnell type dynamic buckling equation of an elastic conical shell with variable thickness have been obtained then employing galerkin s method those equations have been reduced to a time dependent differential equation with variable coefficients finally applying the ritz type variational method the critical static and dynamic loads the corresponding wave numbers dynamic factor and critical stress impulse have been found analytically using the results thus obtained the effects of the thickness variations with a power or an exponential function the variation of the semi vertex angle and the variation of the power of time in the external pressure expression are studied through pertinent computations it is observed that these factors have appreciable effects on the critical parameters of the problem in the heading

axisymmetric vibration of thick conical shells determination des frequences propres de coques coniques d epaisseur variable analyse de vibrations libres a symetrie axiale en utilisant une technique d equations integrales etudes en fonction de la deformation de cisaillement transversal et de l inertie de rotation etude parametrique dans le cas d une coque encastree aux deux extremites

axisymmetric vibrations of laminated conical shells of variable thickness abstract axisymmetric free vibrations of laminated conical shells with a linear thickness variation in the meridional direction are studied a rayleigh ritz procedure is adopted for the analysis a general stacking arrangement with orthotropic layers is considered classical thin shell theory is used assumed displacement functions are algebraic polynomials in transformed meridional co ordinate parametric studies are presented to illustrate the effects of geometric material and coupling parameters and of the boundary conditions on the frequencies and mode shapes

modal characteristic of a rotating rectangular cantilever plate modal characteristics of a rotating cantilever plate are investigated in the present work a dynamic modelling method for rectangular plates undergoing prescribed overall motion is employed to derive the equations of motion the general equations are particularized for the modal analysis of a rotating cantilever plate and dimensionless parameters are identified through dimensional analysis the effects of the dimensionless parameters on the modal characteristics of the rotating plate are investigated incidentally eigenvalue loci veering and crossing phenomena along with the corresponding modeshape variations are exhibited and discussed

an analytical numerical and experimental study of the axisymmetric vibrations of a short cylinder this paper presents a study of standing axisymmetric waves in homogeneous isotropic linear and elastic cylinders whose length and diameter are of the same order of magnitude it is analytically demonstrated that only materials with a poisson s ratio equal to zero can vibrate in pure radial and longitudinal axisymmetric modes and hence the universal frequencies independent of poisson s ratio and the corresponding universal aspect ratios can be accurately calculated the ritz method is applied to numerically determine the natural frequencies and the mode shapes of finite cylinders vibration of a cylinder in the neighbourhood of the universal point is analysed both free and forced vibrations are experimentally detected by using a laser speckle interferometer that allows the determination of the natural frequencies and the mode shapes from the numerical calculations it is confirmed that for materials with poisson s ratio not equal to zero no crossings occur between the curves of non dimensional frequency versus aspect ratio neither for antisymmetric modes themselves nor for symmetric modes there is a good agreement between the analytical solutions the numerical results and the experimental measurements

vibration of high speed rotating rings coupled to space fixed stiffnesses abstract this study investigates the vibration of high speed rotating rings coupled to space fixed discrete stiffnesses the ring radial and tangential deformations are defined using space fixed eulerian coordinates where material particles pass through fixed locations in space engineering strain is used in the strain energy expression the derived nonlinear equations from hamilton s principle are linearized about the steady non trivial configuration that results from constant ring rotation comparisons are made to other models in the literature that use different assumptions the governing equations are cast in terms of matrix differential operators that reveal the system s standard gyroscopic system structure the natural frequencies and vibration modes are calculated over a wide range of rotation speeds for axisymmetric free rings and a non axisymmetric ring with a space fixed discrete stiffness element

generalized differential quadrature method for the free vibration of truncated conical panels abstract this paper presents an improved generalized differential quadrature gdq method for the investigation of the effects of boundary conditions on the free vibration characteristics of truncated conical panels the truncated conical panel is an important geometrical shape in the fields of aerospace marine and structural engineering however despite this importance few works in free vibration analysis have dealt with this particular geometry in this work the vibration characteristics of clamped and simply supported truncated conical shells are obtained for various circumferential wave numbers further the effects of the vertex and subtended angles on the frequency parameters are also examined in detail due to limited published results in the open literature results for a range of cases are compared with those generated from the commercial finite element solver mcneal schwendler corporation nastran and excellent agreement is observed

frequency coalescence and mode localization phenomena a geometric theory physical systems the natural frequencies of which depend parametrically on certain quantities may exhibit the phenomenon of frequency coalescence when two or more natural frequencies become equal or the phenomenon of avoided crossings when two or more frequencies approach rapidly and then avoid crossing each other in the process the natural modes undergo significant variations it is shown that both phenomena can be described in a unified manner from the singularities of the mapping from the complex parameter plane into the complex frequency plane in particular the formation of a saddle point in the frequency plane and a branch point in the parameter space is the basis for explaining the properties associated with these phenomena the question of sensitivity of the system response near such singularities is subsequently examined it is shown that the development of a branch point singularity in the complex parameter plane can cause large or even infinite sensitivity of the system about this point depending on the distance of the branch point from the real axis in the parameter space it is further shown that mode localization is also associated with the appearance of a branch point near the real axis of the complex parameter plane which causes the large sensitivity that characterizes the phenomenon the theory is applied to examples of conservative and dissipative systems undergoing free vibrations

vibration of pretwisted cantilever shallow conical shells abstract this paper presents a mathematical model to investigate the effects of initial twist on the vibratory characteristics of cantilever shallow conical shells the energy functional is minimized according to the ritz procedure to arrive at the governing eigenvalue equation a set of orthogonally generated two dimensional polynomials associated with a basic function which accounts for the boundary expressions and constraints is introduced to approximate the in plane and transverse displacement amplitude functions the complete procedure has been automated to compute the vibration frequencies and mode shapes for exemplary problems in the numerical experiments the convergence of eigenvalues is confirmed by increasing the degrees of polynomials employed in the admissible shape functions to enhance the existing literature a set of first known frequency parameters is presented the paper highlights the important effects of angle of twist on the vibration frequencies and mode shapes of conical shells the fundamental physical frequency decreases monotonically for a longer conical shell the result shows that an increase in the angle of twist does not ensure higher torsional stiffness for a conical shell which is in contradiction with previous observation for a pretwisted beam or plate the symmetry of modes is absent when the angle of twist is non zero

on nonlinear frequency veering and mode localization of a beam with geometric imperfection resting on elastic foundation this work presents an investigation on the effect of an initial geometric imperfection wavelength amplitude and degree of localization on the in plane nonlinear natural frequencies veering and mode localization of an elastic euler bernoulli beam resting on a winkler elastic foundation the beam is assumed to be pinned pinned with a linear torsional spring at one end the effect of the axial force induced by mid plane stretching is accounted for in the derivation of the mathematical model due to its known importance and significant effect on the nonlinear dynamic behavior of the beam as it was proved and presented in earlier investigations the governing partial differential equation is discretized using the assumed mode method and the resulting nonlinear temporal equation was solved using the harmonic balance method to obtain results for the nonlinear natural frequencies and mode shapes the results are presented in the form of characteristic curves which show the variations of the nonlinear natural frequencies of the first three modes of vibration for a selected range of physical parameters like torsional spring constant elastic foundation stiffness and amplitude and wavelength of a localized and non localized initial slack

numerical study of vibration localization in disordered cyclic structures a study of mode localization in disordered cyclic structures that have multi degree of freedom substructures in presented particular emphasis is placed on the transition of the natural modes from being extended throughout the structure to being localized on a single substructure a localization length scale is proposed as a measure of the spatial extent of a mode for the purpose of discussing the results of numerical experiments and comparing the localization of different modes different modes in the same mode group were found to have significantly different degrees of localization and similar modes in different mode groups were found to be related to each other via a scaling parameter results also indicate that the particular set of random disorder introduced into the finite structure can have a significant effect on the degree of localization

wave propagation in two dimensional periodic lattices plane wave propagation in infinite two dimensional periodic lattices is investigated using floquet bloch principles frequency bandgaps and spatial filtering phenomena are examined in four representative planar lattice topologies hexagonal honeycomb kagome lattice triangular honeycomb and the square honeycomb these topologies exhibit dramatic differences in their long wavelength deformation properties long wavelength asymptotes to the dispersion curves based on homogenization theory are in good agreement with the numerical results for each of the four lattices the slenderness ratio of the constituent beams of the lattice or relative density has a significant influence on the band structure the techniques developed in this work can be used to design lattices with a desired band structure the observed spatial filtering effects due to anisotropy at high frequencies short wavelengths of wave propagation are consistent with the lattice symmetries

the natural vibration of a conical shell with an annular end plate in this paper natural frequencies and mode shapes for a conical shell with an annular end plate or a round end plate are investigated in detail by combining the vibration theory with the transfer matrix method the governing equations of vibration for this structure are expressed in terms of a matrix differential equation and a novel recurrence formula method that can close in on analytical solution of transfer matrix is presented once the transfer matrix of single component has been determined and the product of each component matrix and the joining matrix can obtain entire structure matrix the frequency equations and mode shape functions are represented in terms of the elements of the structural transfer matrices the 3d finite element numerical simulations have verified the present formulas of natural frequencies and mode shapes the conclusions show that the transfer matrix method can accurately obtain natural vibration characteristics of the conical shell with an annular end plate

effects of initial twist and thickness variation on the vibration behaviour of shallow conical shells abstract a thorough computational investigation is presented into the effects of initial twist and thickness variation on the vibratory characteristics of cantilevered pretwisted thin shallow conical shells with generally varying thickness an energy approach with the ritz minimization procedure is employed to arrive at a governing eigenvalue equation the admissiblepb 2shape functions which comprise sets of mathematically complete two dimensional orthogonal polynomials and a basic function are introduced to account for the boundary constraints and to approximate the three dimensional displacements of the conical shell the effects of angle of twist thickness variation and geometric parameters on the vibration response are presented in a comprehensive parametric study which provides useful new design data selected vibration mode shapes are also included to enhance the existing literature it is found that an increase in the angle of twist does not necessarily increase the torsional stiffness on the contrary the fundamental frequencies is reduced in the present analysis it is also observed that non zero angle of twist and non uniform chordwise thickness variation cause the absent of vibration mode symmetry

free vibration analyses of thin sector plates by the new version of differential quadrature method abstract the new version of differential quadrature dq method is extended to analyze the free vibration of thin sector plates with various sector angles and six combinations of boundary conditions detailed formulations are given various numerical examples are studied comparisons are made with existing analytical and or numerical data numerical results indicate that convergence can be achieved with increasing in number of grid points and accurate results could be obtained with 9 9 gird or even higher grid by using the extended dq method for the cases considered

scattering for the two dimensional energy critical wave equation we investigate existence and asymptotic completeness of the wave operators for the nonlinear klein gordon equation with a defocusing exponential nonlinearity in two space dimensions a certain threshold is defined based on the value of the conserved hamiltonian below which the exponential potential energy is dominated by the kinetic energy via a trudinger moser type inequality we prove that if the energy is below or equal to the critical value then the solution approaches a free klein gordon solution at the time infinity an interesting feature in the critical case is that the strichartz estimate together with sobolev type inequalities cannot control the nonlinear term uniformly on each time interval it crucially depends on how much the energy is concentrated thus we have to trace concentration of the energy along time in order to set up favorable nonlinear estimates and only after that we can apply bourgain s induction argument or any other similar one

a system level model reduction technique for the efficient simulation of flexible multibody systems flexible multibody systems are governed by sets of non linear differential algebraic equations dae the number of degrees of freedom dofs required for accurate body flexibility modeling as well as the presence of the algebraic constraint equations limit the simulation speed body level model reduction and general purpose system level model reduction techniques do not result in optimal model dimension reduction and do not necessarily transform the dae set into a cheaper to solve set of ordinary differential equations as opposed to model reduction techniques such as global modal parameterization gmp which are based on describing the system motion by the contribution of its dominant configuration dependent system level modal patterns their use as a simulation tool has not been thoroughly investigated this paper investigates the influence of the modal content of the configuration dependent mode set on the simulation accuracy through numerical experiments by including the dynamically excited eigenmodes and static deformation patterns for dofs in which the system is externally loaded the system can be accurately represented by a very limited number of dofs phenomena such as mode veering and mode crossing in which the modal content of eigenmodes rapidly varies can be expected to be problematic for such a description which is confirmed with numerical experiments copyright 2010 john wiley sons ltd

the vibration and stability of a friction guided translating string abstract eyelets capstans and cylindrical surfaces are often used in thread fiber and paper handling machinery to guide the axially moving element in addition to providing positional control these guides introduce dry friction forces that alter the vibration and stability characteristics of the system this paper examines the lateral response of a string that slides through an elastically supported dry friction guide exact expressions are derived for the linear response under free and forced conditions solutions for the eigenvalue spectrum exhibit unusual features including multiple divergence instabilities regions of flutter instability and regions of curve veering associated with mode localization a second order perturbation solution is derived to examine the behavior of the eigenvalue spectrum in regions of flutter instability and curve veering the analysis highlights the common features of these two phenomena and suggests ways to minimize vibration by adjusting various design variables the analysis also demonstrates that the eigenvalue loci in regions of flutter instability and curve veering are naturally described by a local hyperbolic approximation

effects of subtended and vertex angles on the free vibration of open conical shell panels a conical co ordinate approach this paper presents a computational investigation on the effects of subtended and vertex angles on the free vibration characteristics of open conical shell panels unlike the conventional approximation using a rectangular or cylindrical co ordinate system this analysis adopts a natural conical co ordinate system so that any approximation in geometry is eliminated the strain and curvature components formulated in this orthogonal conical co ordinate system have been employed to derive the strain and kinetic energy integrals the energy functional is minimized in accordance with the ritz procedure to arrive at a governing eigenvalue equation admissible shape functions compromising sets of two dimensional orthogonal polynomials and a basic function are employed to account for the boundary constraints and to approximate the three dimensional displacements of the conical shell comparison of natural frequency shows excellent agreement with the solution of finite element finite strip and integral equation methods the effects of subtended and vertex angles and other geometric parameters on vibration are investigated in a comprehensive parametric study selected vibration mode shapes are illustrated to enhance the physical understanding of vibration of such open conical shell panels

ergodic theorems for free group actions on von neumann algebras abstract we extend a recent ergodic theorem of a nevo and e stein to the non commutative case let be a faithful normal state on the von neumann algebra a let a i r i 1 generate f r the free group on r generators and let i r i 1 be automorphisms of a which leave invariant define to be the group homomorphism from f r to the automorphisms of a defined on base elements by a i i define w n as the set of all reduced words in f r of length n the identity is a neutral element and w n as the number of elements of w n let n 1 w n a w n a and s n 1 n n 1 k 0 k we then show that if x is in a then s n x converges almost uniformly to an element a to prove the above theorem we prove an ergodic theorem involving completely positive maps of which the free group situation is a special case roughly if p 1 p 2 0 p 1 p 2 1 n positive maps such that 1 n p 1 n 1 p 2 n 1 n 1 with 0 x x and 1 1 1 then with a few technical assumptions we show a convergence result for lim n n x and show that lim n n 1 n 1 k 0 k x converges almost uniformly in the case p 2 0 1 a automorphism our theorems correspond to the non commutative pointwise ergodic theorem of e c lance the results partially generalize a result of kummerer our theorems also include results concerning normal operators on a hilbert space which generalizes work of guivarc h

multi parameter perturbation methods for the eigensolution sensitivity analysis of nearly resonant non defective multi degree of freedom systems the dynamic behavior of structural systems may be strongly characterized by the occurrence of multiple internal resonances for particular combinations of the mechanical parameters the linear models governing these resonant or nearly resonant systems tend to exhibit high sensitivity of the eigenvalues and eigenvectors to small parameter modifications this pathological condition is recognized as a source of relevant phenomena such as frequency veering and mode localization or hybridization the paper presents the generalization of uniformly valid perturbation methods to perform eigensolution sensitivity analyses in multi degree of freedom hamiltonian systems with a generic number of close eigenvalues the leading idea is to treat systematically nearly resonant systems as multi parameter perturbations of a perfectly resonant non defective though a priori unknown reference system given a single nearly resonant system a multi parameter perturbation method is presented to achieve a two fold objective first identify a close resonant system suited to serve as a starting point for sensitivity analyses inverse problem second to approximate asymptotically the eigensolution of all the nearly resonant systems which may arise from its generic perturbation direct problem the conditions of existence and uniqueness of the inverse problem solution are discussed the direct problem solution is analyzed with a focus on the eigensolution sensitivity to parameter perturbations with different physical meanings such as a slight geometric disorder or weak elastic coupling in periodic structures finally the procedure is verified on a prototypical structural system describing the section dynamics of a suspended bridge

on ergodic theorems for free group actions on noncommutative spaces we extend in a noncommutative setting the individual ergodic theorem of nevo and stein concerning measure preserving actions of free groups and averages on spheres s 2 n of even radius here we study state preserving actions of free groups on a von neumann algebra a and the behaviour of s 2 n x for x in noncommutative spaces l p a for the cesaro means this problem was solved by walker our approach is based on ideas of bufetov we prove a noncommutative version of rota alternierende verfahren theorem to this end we introduce specific dilations of the powers of some noncommutative markov operators

free vibration of layered truncated conical shell frusta of differently varying thickness by the method of collocation with cubic and quintic splines abstract free vibrations of layered conical shell frusta of differently varying thickness are studied using the spline function approximation technique the equations of motion for layered conical shells in the longitudinal circumferential and transverse displacement components are derived using extension of love s first approximation theory assuming the displacement components in a separable form a system of coupled equations on three displacement functions are obtained since no closed form solutions are generally possible a numerical solution procedure is adopted in which the displacement functions are approximated by cubic and quintic splines a generalized eigenvalue problem is obtained which is solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients the vibrations of two layered conical shells made up of several types of layer materials and supported differently at the ends are considered linear sinusoidal and exponential variations in thickness of layers are assumed parametric studies are made on the variation of frequency parameter with respect to the relative layer thickness cone angle length ratio type of thickness variation and thickness variation parameter the effect of neglecting the coupling between bending and stretching is also analysed

analysis of in plane wave propagation in hexagonal and re entrant lattices plane wave propagation in hexagonal and re entrant lattices is investigated through the application of bloch s theorem two dimensional dispersion relations are estimated and analyzed to illustrate peculiar properties of re entrant configurations and investigate the directional behavior of hexagonal lattices for varying construction angles special attention is devoted to the determination of phase and group wave velocities velocity plots are obtained for varying frequency and direction of wave propagation to analyze the dispersion characteristics of the lattices and further investigate their directional characteristics

nonlinear stability analysis of thin doubly curved orthotropic shallow shells by the differential quadrature method the geometric nonlinear buckling problem of a thin doubly curved shallow shell with all edges hinged is complicated and difficult to obtain an exact analytical solution thus differential equations are solved incrementally by using the differential quadrature method in this paper detailed formulations are worked out convergence study is performed several examples with various material properties curvatures and dimensions are investigated comparisons are made with existing semi analytical data or finite element data it is shown that the critical loads are lower than the data obtained by either so called adjacent equilibrium method or partitioned solution method the possible reasons to cause the difference are discussed

linear vibration characteristics of cable buoy systems abstract a theoretical model for the linear vibration of a cable tensioned by a subsurface buoy is developed the equilibrium of the cable buoy system subject to drag is evaluated using an approximate closed form solution whose range of validity is confirmed through comparison with numerical solutions the three dimensional equations of cable buoy motion are linearized about this equilibrium and then used to assess vibration characteristics the characteristic equations for the natural frequencies of both in plane and out of plane vibration modes are derived the in plane natural frequency spectrum exhibits the curve veering phenomena due to asymmetry of the associated mode shapes parameter studies reveal the dependencies of the in plane and out of plane vibration modes on the cable tension the buoy mass and the current velocity

vibrations of circular cylindrical shells theory and experiments in the present paper a method for analysing linear and nonlinear vibrations of circular cylindrical shells having different boundary conditions is presented the method is based on the sanders koiter theory displacement fields are expanded in a mixed double series based on harmonic functions and chebyshev polynomials simply supported and clamped clamped boundary conditions are analysed as well as connections with rigid bodies in the latter case experiments are carried out comparisons with experiments and finite element analyses show that the technique is computationally efficient and accurate in modelling linear vibrations of shells with different boundary conditions an application to large amplitude of vibration shows that the technique is effective also in the case of nonlinear vibration comparisons with the literature confirm the accuracy of the approach the method proposed is a general framework suitable for analysing vibration of circular cylindrical shells both in the case of linear and nonlinear vibrations r 2007 elsevier ltd all rights reserved

free vibration of a conical shell with variable thickness abstract an analysis is presented for the free vibration of a truncated conical shell with variable thickness by use of the transfer matrix approach the applicability of the classical thin shell theory is assumed and the governing equations of vibration of a conical shell are written as a coupled set of first order differential equations by using the transfer matrix of the shell once the matrix has been determined by quadrature of the equations the natural frequencies and the mode shapes of vibration are calculated numerically in terms of the elements of the matrix under any combination of boundary conditions at the edges the method is applied to truncated conical shells with linearly parabolically or exponentially varying thickness and the effects of the semi vertex angle truncated length and varying thickness on the vibration are studied

strichartz type estimates and the well posedness of an energy critical 2d wave equation in a bounded domain abstract we study the well posedness of the cauchy problem with dirichlet or neumann boundary conditions associated to an h 1 critical semilinear wave equation on a smooth bounded domain r 2 first we prove an appropriate strichartz type estimate using the l q spectral projector estimates of the laplace operator our proof follows burq lebeau and planchon 2008 4 then we show the global well posedness when the energy is below or at the threshold given by the sharp moser trudinger inequality finally in the supercritical case we prove an instability result using the finite speed of propagation and a quantitative study of the associated ode with oscillatory data

vibration characteristics of thin rotating cylindrical shells with various boundary conditions abstract an analysis is presented for the vibration characteristics of thin rotating cylindrical shells with various boundary conditions by use of fourier series expansion method based on sanders shell equations the governing equations of motion which take into account the effects of centrifugal and coriolis forces as well as the initial hoop tension due to rotating are derived the displacement field is expressed as a product of fourier series expressions which represents the axial modal displacements and trigonometric functions which represents the circumferential modal displacements stokes transformation is employed to derive the derivatives of the fourier series expressions then through the process of formula derivation an explicit expression of the exact frequency equation can be obtained for a thin rotating cylinder with classical boundary conditions of any type once the frequency equation has been determined the frequencies are calculated numerically to validate the present analysis comparisons between the results of the present method and previous studies are performed and very good agreement is achieved finally the method is applied to investigate the vibration characteristics of thin rotating cylindrical shells under various boundaries and the results are presented

free vibration analysis of functionally graded conical cylindrical shell and annular plate structures with a four parameter power law distribution based on the first order shear deformation theory fsdt this paper focuses on the dynamic behavior of moderately thick functionally graded conical cylindrical shells and annular plates the last two structures are obtained as special cases of the conical shell formulation the treatment is developed within the theory of linear elasticity when materials are assumed to be isotropic and inhomogeneous through the thickness direction the two constituent functionally graded shell consists of ceramic and metal these constituents are graded through the thickness from one surface of the shell to the other a generalization of the power law distribution presented in literature is proposed two different four parameter power law distributions are considered for the ceramic volume fraction some material profiles through the functionally graded shell thickness are illustrated by varying the four parameters of power law distributions for the first power law distribution the bottom surface of the structure is ceramic rich whereas the top surface can be metal rich ceramic rich or made of a mixture of the two constituents and on the contrary for the second one symmetric and asymmetric volume fraction profiles are presented in this paper the homogeneous isotropic material can be inferred as a special case of functionally graded materials fgm the governing equations of motion are expressed as functions of five kinematic parameters by using the constitutive and kinematic relationships the solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell the discretization of the system equations by means of the generalized differential quadrature gdq method leads to a standard linear eigenvalue problem where two independent variables are involved without using the fourier modal expansion methodology numerical results concerning six types of shell structures illustrate the influence of the power law exponent of the power law distribution and of the choice of the four parameters on the mechanical behaviour of shell structures considered

effects of mode localization on input output directional properties of structures abstract the singular value decomposition svd is employed to study the effects of localization phenomena on input output relationships and power and energy transmission ratios of structures for weakly and strongly coupled systems existence of strong localization of singular vectors and abrupt veering of singular value loci are shown occurrence of strong localization causes abrupt changes in input output directional properties in contrast to eigenvalues singular values do not veer away by introducing a disorder in weakly coupled systems in particular the use of singular values and vectors is computationally advantageous in considering multiple load cases while eigenvalue based analyses give information about the resonance frequencies and vibration modes of a structure singular values of the structure are related to the forced response characteristics and give the dynamic behavior in the frequency domain representative examples in structural dynamics are presented

governing equations of a stiffened laminated inhomogeneous conical shell this study presents the dynamic equations of a stiffened composite laminated conical thin shell under the influence of initial stresses the governing equations of a truncated conical shell are based on the donnell mushtari theory of thin shells including the transverse shear deformation and rotary inertia the extension bending coupling is considered in the derivation the composite laminated conical shell is also reinforced at uniform intervals by elastic rings and or stringers the stiffening elements are relatively closely spaced and therefore the stiffeners are smeared out along the conical shell the inhomogeneity of material properties because of temperature moisture or manufacturing processes is taken into account in the constitutive equations a generalized variational theorem is derived so as to describe the complete set of the fundamental equations of the conical shell next the uniqueness is examined in solutions of the dynamic equations of the conical shell and the boundary and initial conditions are shown to be sufficient for the uniqueness in solutions the equations of the laminated composite conical shell are solved by the use of the finite difference method as an illustrative example the accuracy of results is tested by certain earlier results and a good agreement is found

eigensolutions of joined hermetic shell structures using the state space method abstract a substructure synthesis method based on state space mathematics is proposed for the eigensolution of axisymmetric joined hermetic thin shell structures in the state space method ssm a system of eight coupled first order differential equations is solved for each shell substructure using the pade approximation for matrix exponentiation the substructures are then joined by matching all of the displacement and force boundary variables the pade method allows a space invariant substructure e g uniform cylinder to be treated as a continuum further a space variant substructure e g cone is assumed to be a piecewise continuum which allows one to still use the pade method ssm is validated by applying it to structural elements such as a cylinder a cone a sphere and a toroidal segment and comparing the predicted natural frequencies with well known theoretical solutions the strength of ssm from the substructure synthesis viewpoint is demonstrated by analyzing several shell structures including a hermetic capsule and a refrigeration compressor shell in all examples finite element method fem predictions are used to support ssm results for the compressor shell ssm results compare well with the limited experimental data

free vibration analysis of conical shells via the element free kp ritz method in this paper we consider the free vibration analysis of thin conical shells under different boundary conditions the analysis is carried out using the element free kp ritz method the present study is based on the classical thin shell theory the kernel particle kp functions are employed in hybridized form with harmonic functions to approximate the two dimensional displacement field in order to examine the numerical stability of the present approach convergence studies are performed based on the influences of the support size and the number of nodes to verify the accuracy of this method comparisons of the present results are made with results available in the open literature this study also examines in detail the effects of semi vertex angles and boundary conditions on the frequency characteristics of the conical shells

eigenvalue curve veering in stressed structures an experimental study there have been extensive research works on the veering phenomenon in dynamic systems as eigenvalues change under parametric variation converging loci are commonly seen to veer away suddenly in a small region of the graph such that the modes swap trajectories all of the modal properties are swapped in the process leading to some curious behaviour in the transition zone theoretical studies of this behaviour have been reported for half a century but despite this heritage explicit references to experimental results are scarce in this paper detailed experimental and numerical investigations on veering are reported the parameter varied is the internal pre load of a redundant frame an fe model is presented and stress stiffening approximations are employed to obtain a tangent stiffness for a nonlinear static solution which is then used in a linear dynamic analysis experimental results are given and the behaviour is found to correspond well with the analytical results in particular the mode shape variation is found to be consistent and the implications of this finding with regard to modal correlation and model validation are noted analyses of the mode shapes in terms of eigenvector rotations are presented and are found to form a valuable tool for the interpretation of experimental results

three dimensional vibrations of hollow cones and cylinders with linear thickness variations a three dimensional method of analysis is presented for determining the free vibration frequencies and mode shapes of hollow cones and cylinders with variable thickness displacement components us uz and u in the meridional normal and circumferential directions respectively are taken to be sinusoidal in time periodic in and algebraic polynomials in the s and z directions potential strain and kinetic energies of the cones and cylinders are formulated and upper bound values of the frequencies are obtained by minimizing the frequencies as the degree of the polynomials is increased frequencies converge to the exact values novel numerical results are presented for thick linearly tapered hollow cones and cylinders with completely free boundaries convergence to four digit exactitude is demonstrated for the first five frequencies of the cones and cylinders the method is applicable to thin cones and cylinders as well as thick and very thick ones

2 d differential quadrature solution for vibration analysis of functionally graded conical cylindrical shell and annular plate structures this paper focuses on the dynamic behavior of functionally graded conical cylindrical shells and annular plates the last two structures are obtained as special cases of the conical shell formulation the first order shear deformation theory fsdt is used to analyze the above moderately thick structural elements the treatment is developed within the theory of linear elasticity when materials are assumed to be isotropic and inhomogeneous through the thickness direction the two constituent functionally graded shell consists of ceramic and metal that are graded through the thickness from one surface of the shell to the other two different power law distributions are considered for the ceramic volume fraction the homogeneous isotropic material is inferred as a special case of functionally graded materials fgm the governing equations of motion expressed as functions of five kinematic parameters are discretized by means of the generalized differential quadrature gdq method the discretization of the system leads to a standard linear eigenvalue problem where two independent variables are involved without using the fourier modal expansion methodology for the homogeneous isotropic special case numerical solutions are compared with the ones obtained using commercial programs such as abaqus ansys nastran straus pro mechanica very good agreement is observed furthermore the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good different typologies of non uniform grid point distributions are considered finally for the functionally graded material case numerical results illustrate the influence of the power law exponent and of the power law distribution choice on the mechanical behavior of shell structures

dynamic buckling of a cylindrical shell with variable thickness subject to a time dependent external pressure varying as a power function of time in this study the dynamic buckling of an elastic cylindrical shell with variable thickness subject to a uniform external pressure which is a power function of time has been considered initially the fundamental relations and donnell type dynamic buckling equation of an elastic cylindrical shell with variable thickness have been obtained then employing galerkin s method these equations have been reduced to a time dependent differential equation with variable coefficients finally applying a special ritz type method the critical static and dynamic loads the corresponding wave numbers and dynamic factor have been found analytically using those results the effects of the variation of the thickness with a linear a parabolic or an exponential function in the axial direction and the effect of the variation of the power of time in the external pressure expression are studied using pertinent computations it is observed that these effects change appreciably the critical parameters of the problem the present method has been verified comparing the results of the present work and those of previous works in the literature for a shell with constant thickness subject to a uniform external pressure varying linearly with time

the element free kp ritz method for free vibration analysis of conical shell panels in this paper the free vibration of conical panels is analyzed by the mesh free kp ritz method both 1 d and 2 d versions of the kp ritz approach are formulated for conical panels for conical panels with two simply supported straight edges the 1 d kp ritz version is used where the kernel particle estimation is employed in hybridized form with harmonic functions to approximate the 2 d displacement field for conical panels having arbitrary boundary conditions the displacement field is approximated by the 2 d kp ritz version with 2 d form of kernel particle functions employed the classical thin shell theory based on love s hypothesis is employed in the present analyses and based on the kernel particle concept and ritz technique the eigenequations of the frequencies of the conical panels are obtained to validate the accuracy of stability of the present method convergence studies were carried out based on the influences of the support size and the number of nodes comparisons were also made with existing results available in the open literature this study also examines in detail the effects of variation in the semi vertex angle and boundary conditions on the frequency characteristics of conical panels

general anisotropic doubly curved shell theory a differential quadrature solution for free vibrations of shells and panels of revolution with a free form meridian abstract the generalized differential quadrature gdq method is applied to study the dynamic behavior of anisotropic doubly curved shells and panels of revolution with a free form meridian the first order shear deformation theory fsdt is used to analyze the above mentioned moderately thick structural elements in order to include the effect of the initial curvature in the evaluation of the stress resultants three different approaches specifically qatu approach toorani lakis approach and reissner mindlin approach are considered and compared an improvement of the classical reissner mindlin theory crmt using a different kinematical model is considered by so doing a generalization of the theory of anisotropic doubly curved shells and panels of revolution is proposed four different anisotropic shell theories namely general first order shear deformation theory by qatu gfsdtq general first order shear deformation theory by toorani lakis gfsdttl general first order shear deformation theory by reissner mindlin gfsdtrm and classical reissner mindlin theory crmt are compared in order to show the differences and the accuracy of these theories the solution is given in terms of generalized displacement components of points lying on the middle surface of the shell simple rational bezier curves are used to define the meridian curve of the revolution structures results are obtained taking the meridian and circumferential coordinates into account without using the fourier modal expansion methodology furthermore gdq results are compared with those obtained by using commercial programs such as abaqus ansys nastran straus and pro mechanica very good agreement is observed

flexural vibration of shear deformable circular and annular plates on ring supports this paper presents an analysis of free flexural vibration of circular and annular mindlin plates lying on multiple internal concentric ring supports the rayleigh ritz method with an admissible displacement function expressed in terms of a set of simple polynomials is used to obtain the governing eigenvalue equation the natural frequencies of circular and annular mindlin plates with one or two ring supports having inner and outer peripheries with different combinations of free simply supported and clamped conditions are determined a comprehensive set of vibration results is presented and where possible these have been verified with existing solutions published in the open literature no previous theoretical results for thick annular plates supported with internal rings are known to exist thus the study should be useful and valuable to engineers and researchers

vibratory characteristics of cantilevered rectangular shallow shells of variable thickness in this paper a comprehensive study on the modeling of vibratory response of variable thickness cantilevered shallow cylindrical shells of rectangular planform is carried out through the principle of minimum total energy the equations of stretching and bending strain energies kinetic energy and the associated boundary conditions of the shells are derived the present model is developed based on the ritz formulation with the assumption of sets of mathematicall y complete admissible two dimensional polynomials to approximate the in plane and transverse displacement amplitude functions a basic function is introduced in the approximate method to enforce automatic satisfaction of the kinematic boundary conditions sets of comprehensive reasonably accurate vibration frequencies of the shallow shells are presented for wide ranges of aspect ratio thickness variation ratios and shallowness ratio these results where possible are verified by comparing with other established experimental and theoretical solutions few selected contour plots for the first known mode shapes of these cylindrical shells of variable thickness are included

mode localization and eigenvalue loci veering phenomena in disordered structures an investigation of the effects of disorder on the modes of vibration of nearly periodic structures is presented it is shown that in structures with close eigenvalues small structural irregularities result in both strong localization of the mode shapes and abrupt veering away or mutual repulsion of the loci of the eigenvalues when these are plotted against a parameter representing the disorder in the system perturbation methods for the eigenvalue problem are applied to predict the occurrence of strong localization and eigenvalue loci veering which are shown to be two manifestations of the same phenomenon also perturbation methods that handle the dramatic effects of small disorder are developed to analyze eigenvalue loci veering and strong localization two representative disordered nearly periodic structures are studied a mistuned assembly of coupled oscillators and a multi span beam with irregular spacing of the supports

comments on curve veering in eigenvalue problems abstract the dependence of eigenvalues on a system parameter is frequently illustrated by a family of loci when two loci approach each other they often cross or abruptly diverge the latter case called curve veering has been observed in approximate solutions associated with discretized models the influence of discretization in producing curve veering has raised doubt on the validity of many approximate solutions the existence of curve veering in continuous models is illustrated by presenting the exact solution of an elementary eigenvalue problem veering is then examined in a general eigenvalue problem criteria are established to distinguish veerings from crossings in both continuous and discretized models the application of the criteria is illustrated by examples

vibration of shallow conical shells with shear flexibility a first order theory a formulation of shear deformation theory implemented numerically for the prediction of vibratory characteristics of shallow conical shell panels is presented the derivation of thickness shear is assumed in a linear approximation the lame parameter for the transverse shear strain component which has previously been neglected is considered this consideration accounts for the replacement of a term in transverse strain distribution through the shell thickness which results in linear transverse shear strain distribution in contrast to the constant distribution hitherto known to researchers in this field the energy integral which incorporates the shear deformation and rotary inertia is minimized to derive the governing eigen matrix equation a set of benchmark frequency solutions is presented for two exemplary conical shells the cantilever and the fully clamped shells some selected mode shapes in terms of mid surface contour plots and three dimensional meshes are also illustrated

hermitian matrices depending on three parameters coalescing eigenvalues abstract we consider hermitian matrix valued functions depending on three parameters that vary in a bounded surface of r 3 we study how to detect when such functions have coalescing eigenvalues inside this surface our criterion to locate these singularities is based on a construction suggested by stone 20 for generic coalescings any such singularity is related to a particular accumulation of a certain phase or lack thereof as we cover the surface

semi analytic solutions for the free in plane vibrations of confocal annular elliptic plates with elastically restrained edges abstract a two dimensional analytical model is developed to describe the free extensional vibrations of thin elastic plates of elliptical planform with or without a confocal cutout under general elastically restrained edge conditions based on the navier displacement equation of motion for a state of plane stress the model has been simplified by invoking the helmholtz decomposition theorem and the method of separation of variables in elliptic coordinates is used to solve the resulting uncoupled governing equations in terms of products of even and odd angular and radial mathieu functions extensive numerical results are presented in an orderly fashion for the first three anti symmetric symmetric natural frequencies of elliptical plates of selected geometries under different combinations of classical clamped and free and flexible boundary conditions also the occurrences of frequency veering between various modes of the same symmetry group and interchange of the associated mode shapes in the veering region are noted and discussed moreover selected 2d deformed mode shapes are presented in vivid graphical form the accuracy of solutions is checked through appropriate convergence studies and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature the set of data reported herein is believed to be the first rigorous attempt to obtain the in plane vibration frequencies of solid and annular thin elastic elliptical plates for a wide range of plate eccentricities