extended variational formulations and f e models for nonlinear beams under nonconservative loading abstract for nonconservative mechanical systems classical variational principles do not hold true hence the principle of incremental virtual work is usually taken as a convenient foundation for finite element solution methods 2 8 9 recent results on the so called inverse problem of calculus of variations however lead to extended variational formulations which can be provided even for nonconservative systems in the present paper this approach is used for the particular case of a nonlinear beam subjected to a co planar follower static loading suitable finite element models are deduced from the extended variational principle and their numerical performance is assessed by means of some meaningful examples

stochastic evolution of 2d crystals we study a crystalline version of the modified stefan problem in the plane the feature of our approach is that we consider a model with stochastic perturbations and assume the interfacial curve to be a polygon the existence of solution to our stochastic system is established galerkin s method is one of the main tools used in the proof of our assertion

non linear analysis of elastic plane frames with crooked bars in this investigation the postbuckling response of plane frames consisting of a large number of crooked bars is achieved by means of an efficient method based on non linear kinematic stability analysis the entire formulation of the non linear equilibrium equations presented in matrix form is implemented by the computer program nlsrf non linear stability of rod frames critical loads and critical displacements are obtained through a direct computation by using the additional stability condition resulting from the vanishing of the stability determinant the efficacy of the method proposed herein is demonstrated by a three storey single span frame

divergence of a rotating shaft with an intermediate support and conservative axial loads abstract the dynamic behavior of a rotating shaft with an intermediate support and conservative axial loads is analyzed using euler beam theory and the assumed mode method the equations of motion are then transformed to the standard form of eigenvalue problems for determining the critical dimensionless rotational speeds and the critical axial loads corresponding to the divergence type instability of the shaft results of numerical simulations are presented for various combinations of support locations axial loads and rotational speeds

stability of a timoshenko beam resting on a winkler elastic foundation the influences of a winkler elastic foundation modulus slenderness ratio and elastically restrained boundary conditions on the critical load of a timoshenko beam subjected to an end follower force are investigated the characteristic equation for elastic stability is derived it is found that the critical flutter load for the cantilever timoshenko beam will first decrease as the elastic foundation modulus is increased and when the elastic foundation modulus is greater than the corresponding critical value which corresponds to the lowest critical load it will increase instead in particular if the elastic foundation modulus is large enough the critical flutter load for the cantilever timoshenko beam can be greater than that of the bernoulli euler beam for a clamped translational or clamped rotational elastic spring supported beam resting on an elastic foundation there exists a critical value of the spring constant for each beam at this critical point the critical load jumps and the type of instability mechanism changes the jump mechanisms for beams resting on elastic foundations of different modulus values are different

dynamic stability of a tapered cantilever beam on an elastic foundation subjected to a follower force the equation of motion in matrix form of a tapered cantilever euler beam subjected to a follower force at the free end is formulated based on the lagrangian approach and the assumed mode method the beam is resting on a winkler type elastic foundation the effects of the presence of viscous damping in the foundation which makes the foundation viscoelastic and the presence of internal damping in the beam on the critical flutter loads are examined separately to evaluate their relative importance as expected internal damping of the beam tends to drastically reduce the critical flutter loads for a beam of uniform cross section for tapered beams the effects are however dependent on the taper ratio of the beam as well as the modulus of the elastic foundation the critical flutter loads of both tapered beams and beams of uniform cross section are found to be unaffected by the presence of viscous damping in the elastic foundation the effect of varying the modulus of the elastic foundation on the critical flutter loads is also discussed in the paper

optimum shape design against flutter of a cantilevered column with an end mass of finite size subjected to a non conservative load abstract optimum design for dynamic stability of slender cantilevered columns subjected to a follower force due to a rocket thrust is investigated the aim is to determine the tapering of the column which maximizes the critical value of the rocket thrust at which flutter is initiated under the constraint of constant length and volume of the column the rocket thrust is assumed to be produced by a solid rocket motor mounted at the tip end of the column the rocket motor is simplified as a massive ball with the same material density as the column based on experimental evidence 1 2 it is argued that a mathematical model without damping gives the practical stability limit if internal and external damping is small and the rocket thrust acts only in a short interval of time optimum columns are determined for various sizes of the end ball rocket motor for small sizes the critical thrust can be significantly increased by optimization about eight times by practical experimental realizable values of the mass ratio mass of end ball mass of column the critical thrust can only be increased 1 3 1 4 times which is similar to the case of a pure conservative dead end load also it is found that the great sensitivity to small changes in design parameters which significantly complicates optimization of the pure beck s column is not present for practical values of it is argued then that the pure beck s column should be considered as a theoretical limit case of vanishing end mass

elastic stability of non uniform columns abstract a simple and efficient method is proposed to investigate the elastic stability of three different tapered columns subjected to uniformly distributed follower forces the influences of the boundary conditions and taper ratio on critical buckling loads are investigated the critical buckling loads of columns of rectangular cross section with constant depth and linearly varied width t 1 constant width and linearly varied depth t 2 and double taper t 3 are investigated among the three different non uniform columns considered taper ratio has the greatest influence on the critical buckling load of column t 3 and the lowest influence on that of column t 1 the types of instability mechanisms for hinged hinged and cantilever non uniform columns are divergence and flutter respectively however for clamped hinged and clamped clamped non uniform columns the type of instability mechanism for column t 1 is divergence while that for columns t 2 and t 3 is divergence only when the taper ratio of the columns is greater than certain critical values and flutter for the rest value of taper ratio when the type of instability mechanism changes from divergence to flutter there is a finite jump for the critical buckling load the influence of taper ratio on the elastic stability of cantilever column t 3 is very sensitive for small values of the taper ratio and there also exist some discontinieties in the critical buckling loads of flutter instability for a hinged hinged non uniform column t 2 or t 3 with a rotational spring at the left end of the column when the taper ratio is less than the critical value the instability mechanism changes from divergence to flutter as the rotational spring constant is increased for a clamped elastically supported non uniform column when the taper ratio is greater than the critical value the instability mechanism changes from flutter to divergence as the translational spring constant is increased

shape sensitivity analysis of flutter response of a laminated wing an approach is presented for calculating the shape sensitivity of a wing flutter response with respect to changes in geometric shape three methods are used to calculate the sensitivity of the flutter eigenvalue

criteria for occurrence of flutter instability before buckling the occurrence of utter instability through a hopf bifurcation before static buckling in regions of divergence in nonconservative nonself adjoint dissipative systems is thoroughly discussed using a qualitative analysis this region where both ziegler s and static criterion may fail to predict the actual critical load is de ned via two values bounds of the nonconservativenessloadingparameter the upperboundcorresponds to 0 5 being invariant with respect to all other parameters whereas the lower bound corresponds to a double critical divergence point beyond which there are no adjacent equilibria the location of the last point lying always between 0 and 0 5 depends on a stiffness parameter it is also found that the region of nonexistence of adjacent equilibria becomes maximum minimum when the double critical point corresponds to 0 5 0 the interaction of vanishing damping with various parameters leads to new phenomena related to point and periodic attractors as well as to a new type of dynamic bifurcation

some general effects of strong high frequency excitation stiffening biasing and smoothening abstract mechanical high frequency hf excitation provides a working principle behind many industrial and natural applications and phenomena this paper concerns three particular effects of hf excitation that may change the apparent characteristics of mechanical systems 1 stiffening by which the apparent linear stiffness associated with an equilibrium is changed along with derived quantities such as stability and natural frequencies 2 biasing by which the system is biased towards a particular state static or dynamic which does not exist or is unstable in the absence of the hf excitation and 3 smoothening referring to a tendency for discontinuities to be effectively smeared out by hf excitation illustrating first these effects for a few specific systems analytical results are provided that quantify them for a quite general class of mechanical systems this class covers systems that can be modelled by a finite number of second order ordinary differential equations generally non linear with periodically oscillating excitation terms of high frequency and small amplitude the results should be useful for understanding the effects in question in a broader perspective than is possible with specific systems for calculating effects for specific systems using well defined formulas and for possibly designing systems that display prescribed characteristics in the presence of hf excitation

parametric instability of stochastic columns columns which have stochastically distributed young s modulus and mass density and are subjected to deterministic periodic axial loadings are considered the general case of a column supported on a winkler elastic foundation of random stiffness and also on discrete elastic supports which are also random is considered material property fluctuations are modeled as independent one dimensional univariate homogeneous real random fields in space in addition to autocorrelation functions or their equivalent power spectral density functions the input random fields are characterized by scale of fluctuations or variance functions for their second order properties the foundation stiffness coefficient and the stiffnesses of discrete elastic supports are treated to constitute independent random variables the system equations of boundary frequencies are obtained using bolotin s method for deterministic systems stochastic fem is used to obtain the discrete system with random as well as periodic coefficients statistical properties of boundary frequencies are derived in terms of input parameter statistics a complete covariance structure is obtained the equations developed are illustrated using a numerical example employing a practical correlation structure

on the discretization of an elastic rod with distributed sliding friction a one dimensional elastic system with distributed contact under fixed boundary conditions is investigated in order to study dynamic behavior under sliding friction a partial differential equation of motion is established and its exact solution is presented due to the friction the eigenvalue problem is non self adjoint mathematical methods for handling the non self adjoint system such as the non self adjoint eigenvalue problem and the eigenvalue problem with a proper inner product are reviewed and applied the exact solution showed that the undamped elastic system under fixed boundary conditions is neutrally stable when the coefficient of friction is a constant the assumed mode approximation and the lumped parameter discretization method are evaluated and their solutions are compared with the exact solution as a cautionary example the assumed modes approximation leads to false conclusions about stability the lumped parameter discretization algorithm generates reliable results

a crystalline motion uniqueness and geometric properties the author studies a model of crystalline motion in the plane existence and uniqueness of local in time solutions are shown geometric properties of the flow are studied assuming that the wulff shape is a regular n sided polygon the author shows that a small convex grain of ice immersed in the cold melt shrinks provided that the flow does not overly deform the initial interface the author is able to verify the last condition only if the initial interface is a scaled wulff shape in this case the free boundary will remain a scaled wulff shape at later times this is shown by using an isoperimetric inequality

a crystalline motion of spiral shaped curves with symmetry we study the evolution of spiral shaped polygonal curves by crystalline curvature crystalline curvature is known to extend the notion of ordinary curvature to the special class of nonsmooth curves with the assumption of symmetry we show that the motion of our spirals can be analyzed up to the rest state possibly beyond singularities

sensitivity analysis for problems of dynamic stability abstract in mechanics as well as in physics the most general and important thing is to study the dependence of the characteristics of a physical process on problem parameters problems of dynamic stability for non conservative systems involve determination of eigenvalues and eigenvectors for these problems it is shown in general how the different sensitivity analyses can be performed without any new eigenvalue analyses the main question relates to the change in flutter load as a function of change in stiffness mass boundary conditions or load distribution discretized as well as non discretized examples are presented in details

nonlinear chains and painleve equations abstract ode systems which define a periodically closed sequence of backlund transformations bt and which are equivalent to the painleve equations p2 p6 are presented transformation properties of painleve equations can be easily derived from the discrete symmetries of these systems

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

the influence of shear deformations on the stability of thin walled beams under non conservative loading abstract this paper analyses the influence of shear deformations on the stability of thin walled beams of open section subjected to follower type loads the warping of the cross sections is assumed to depend on the shear forces as well as on both primary and secondary torsion the internal stresses are consistently deduced from the kinematic field an algorithm is presented which works in the framework of the least squares method making use of a vector iteration technique an extensive comparison is performed with the results offered by the classical models

nonlinear finite element analysis of elastic systems under nonconservative loading natural formulation part ii dynamic problems abstract the paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces attention is focused on the stability behaviour of such systems this leads mathematically to non self adjoint boundary value problems which are of great theoretical and practical interest in particular in connection with alternative modes of instability like divergence of flutter only quasistatic effects are however considered in the present part the methodology of our theory is general but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement dependent nonconservative follower forces in a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives in general rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix as a result the total tangent stiffness matrix becomes unsymmetric an indication of the non self adjoint character of the problem our theory is based on the natural mode technique 1 2 3 and permits i a a simple but elegant derivation of the load correction matrix the application of the general theory as evolved in this paper is demonstrated on the beam element in space a number of numerical examples are considered including divergence and flutter types of instability for which exact analytic solutions are known the problems demonstrate the efficiency of the present finite element formulation the paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in 2 3

evolution of 3 d crystals from supersaturated vapor with modified stefan condition galerkin method approach abstract we examine the evolution of crystals in three dimensions we assume that the wulff shape is a prism with a hexagonal base we include the gibbs thomson law on the crystal surface and the so called stefan condition we show local in time existence of solutions assuming that the initial crystal has admissible shape

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

integral equation formulation and analysis of the dynamic stability of damped beams subjected to subtangential follower forces abstract this paper presents a mathematical model based on integral equations for numerical investigations of stability analyses of damped beams subjected to subtangential follower forces a mathematical formulation based on euler bernoulli beam theory is presented for beams with variable cross sections on a viscoelastic foundation and subjected to lateral excitation conservative and non conservative axial loads using the boundary element method and radial basis functions the equation of motion is reduced to an algebro differential system related to internal and boundary unknowns generalized formulations for the deflection the slope the moment and the shear force are presented the free vibration of loaded beams is formulated in a compact matrix form and all necessary matrices are explicitly given the load frequency dependence is extensively investigated for various parameters of non conservative loads of internal and viscous dampings and for various positions of the concentrated foundation for an undamped beam a dynamic stability analysis is illustrated numerically based on the coalescence criterion the flutter load and instability regions with respect to various parameters are identified the effects of internal and viscous dampings on the critical flutter load are examined separately and relative effects are evaluated the dynamic responses before near and after the flutter are investigated a simple and quite general methodological approach is presented comprehensive numerical tests for flutter analysis are reported and discussed

divergence and flutter of a cantilever rod with an intermediate spring support abstract the equation of motion in matrix form of a cantilever euler beam subject to a tipconcentrated follower force at the free end is formulated based on the lagrangian approach and the assumed mode method the non conservative nature of the system is identified by the non symmetric matrix in the equation of motion the beam is assumed to rest on an intermediate spring support jump phenomenon for the critical follower load is found to occur for both variations in the support location as well as the stiffness of the spring support detailed history of the changes in the load frequency diagram due to variation in the support location and the stiffness of the spring support is presented to explain the occurrence of the jump phenomenon an interesting finding is that the rod is found to be extremely unstable for a certain combination of spring stiffness and support location for a cantilever rod

stability analysis of stochastically parametered nonconservative columns nonconservatively loaded columns which have stochastically distributed material property values and stochastic loadings in space are considered young s modulus and mass density are treated to constitute random fields the support stiffness coefficient and tip follower load are considered to be random variables the fluctuations of external and distributed loadings are considered to constitute a random field the variational formulation is adopted to get the differential equation and boundary conditions the non self adjoint operators are used at the boundary of the regularity domain the statistics of vibration frequencies and modes are obtained using the standard perturbation method by treating the fluctuations to be stochastic perturbations linear dependence of vibration and stability parameters over property value fluctuations and loading fluctuations are assumed bounds for the statistics of vibration frequencies are obtained the critical load is first evaluated for the averaged problem and the corresponding eigenvalue statistics are sought then the frequency equation is employed to transform the eigenvalue statistics to critical load statistics specialization of the general procedure to beck leipholz and pfluger columns is carried out for pfluger column nonlinear transformations are avoided by directly expressing the critical load statistics in terms of input variable statistics

rational solutions of the painleve vi equation in this paper we classify all values of the parameters and of the painleve vi equation such that there are rational solutions we give a formula for them up to the birational canonical transformations and the symmetries of the painleve vi equation

multi parameter linear periodic systems sensitivity analysis and applications for stability analysis of linear periodic systems with more than one degree of freedom the floquet method is a general and valuable practical method in multi parameter periodic systems repeated numerical integration to obtain the floquet matrix may be a limiting factor and effective sensitivity analysis of stability characteristics is therefore needed analytical first and second order sensitivities of the floquet matrix and its eigenvalues multipliers are presented in this paper some numerical applications are given these include effective stabilization by proper change of parameters and optimal design with constraints on stability requirements

collapse of the keldysh chains and stability of continuous nonconservative systems in the present paper eigenvalue problems for non self adjoint linear differential operators smoothly dependent on a vector of real parameters are considered bifurcation of eigenvalues along smooth curves in the parameter space is studied the case of a multiple eigenvalue with the keldysh chain of arbitrary length is investigated explicit expressions describing bifurcation of eigenvalues are found the obtained formulas use eigenfunctions and associated functions of the adjoint eigenvalue problems as well as the derivatives of the differential operator taken at the initial point of the parameter space these results are important for the stability problems and sensitivity analysis of nonconservative systems as a mechanical application the extended beck problem of stability of an elastic column subjected to a partially tangential follower force is considered and discussed in detail

sensitivity analysis and optimization of aeroelastic stability abstract the present paper deals with problems concerning sensitivity analysis and optimization of aeroelastic stability of distributed systems in sections 1 3 the optimization problem of aeroelastic stability of a slender wing in incompressible flow is formulated and solved the optimal control function h y to be determined represents the mass and stiffness distribution along a wing span sensitivity analysis of flutter systems is developed first the gradients of flutter and divergence critical speeds are derived and necessary optimality conditions are obtained then the solution technique is described and numerical results are presented in section 4 the problem of determining the optimal distribution of nonstructural mass along the wing span is considered the optimality conditions are established and the bang bang optimal distributions are obtained

affine weyl groups discrete dynamical systems and painleve equations a new class of representations of affine weyl groups on rational functions are constructed in order to formulate discrete dynamical systems associated with affine root systems as an application some examples of difference and differential systems of painleve type are discussed

global stabilty analysis of a classical non conservative system under a follower load abstract the bifurcations critical conditions and global stability of the classical two degree of freedom ziegler s dissipative model under a partial follower load are thoroughly discussed with the aid of a complete non linear dynamic analysis attention is mainly focused on the range of values of the non conservativeness loading parameter for which adjacent equilibria do not exist various characteristic findings are established by studying the nature of the jacobian matrix eigenvalues associated with the above non linear autonomous dissipative system it is found that the system exhibits stable hopf bifurcations global stability for the entire region of non adjacent equilibria while the classical linear stability analysis leads to flutter instability local instability moreover in an explicit general form the significant effect of damping on the critical dynamic bifurcation load is established on the basis of the latter the critical loads for the case of vanishing damping are found to be completely different from those of the classical local stability analysis all findings of this analysis are checked against numerical solution of the equations of motion

the existence of regions of divergence instability for nonconservative systems under follower forces abstract in this investigation the existence of regions of divergence instability for an elastically restrained column under a follower compressive force at its end is discussed necessary and sufficient conditions for the existence of regions of divergence instability are established the boundary between flutter and divergence instability passes always through a double critical point where the first and second static buckling eigenmodes coincide the eigenvalues buckling loads of nonself adjoint problems of this type are positive and distinct for the entire region of divergence instability except at its boundary where the first and second eigenvalues coincide at this boundary where the buckling mechanism changes from divergence to flutter and vice versa a sudden increase in the critical load occurs with the flutter load always being higher than the corresponding divergence load

the role of prestressing in establishing regions of instability for a compound column under conservative or non conservative load the influence of prestressing of a non linear two member column on its natural vibration and stability is studied the perturbation method is used for solving the problem regions of divergence and flutter instabilities for a column has been determined on the basis of courses of eigencurves in relation to prestress rate discontinuities of the critical force have been observed for the border value of the prestress leading to the instability of the column without the external load although each prestress lowers the critical force it can be used for passive vibration control

the influence of an intermediate support on the stability behaviour of cantilever beams subjected to follower forces abstract a non uniform undamped beam which is simply supported at one end resting on an elastically flexible support at an intermediate abscissa q is examined the beam is subjected to a generic set of follower forces along the axis so that beck leipholz and hauger problems can be easily recovered as particular cases in all these systems a location for the intermediate support exists which corresponds to passage from flutter to divergence

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

flutter and divergence instability of nonconservative beams and plates for a conservative system the only possible instability is of divergence type for a nonconservative system however instability can be of divergence flutter or both depending on the amount of nonconservativeness in this paper we study the instability of a cantilevered beam and a simply supported plate subjected to a combination of fixed and follower forces a nonconservative parameter is introduced to provide all possible combinations of these forces for the beam instability changes from divergence to flutter at a critical value of this parameter for values of the parameter above the critical value the flutter instability remains as the only instability pattern in contrast to implications in the literature for the plate the instability is governed by flutter for a certain range of the nonconservative parameter even though divergence instability still exists the range depends on the geometry aspect ratio and material property poisson s ratio of the plate

derivatives of eigenvalues and eigenvectors of a general complex matrix a survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non hermitian matrices is presented in addition a modification of one method based on a better normalizing condition is proposed methods are classified as direct or adjoint and are evaluated for efficiency operation counts are presented in terms of matrix size number of design variables and number of eigenvalues and eigenvectors of interest the effect of the sparsity of the matrix and its derivatives is also considered and typical solution times are given general guidelines are established for the selection of the most efficient method

rational solutions of the noumi and yamada system of type a4 1 we completely classify the rational solutions of the noumi and yamada system of type a4 1 which is a generalization of the fourth painleve equation the rational solutions are classified to three by the backlund transformation group

the crystalline dynamics of spiral shaped curves we study the motion of spiral shaped polygonal curves by crystalline curvature we describe this dynamics by the corresponding infinitely dimensional system of ordinary differential equations and show that the considered model is uniquely solvable banach s contraction mapping theorem and the bellman gronwall inequality are the main tools applied in our proof

series solution of painleve ii in electrodiffusion conjectured convergence a perturbation series solution is constructed with the use of airy functions for a nonlinear two point boundary value problem arising in an established model of steady electrodiffusion in one dimension with two ionic species carrying equal and opposite charges the solution includes a formal determination of the associated electric field which is known to satisfy a form of the painleve ii differential equation comparisons with the numerical solution of the boundary value problem show excellent agreement following termination of the series after a sufficient number of terms for a much wider range of values of the parameters in the model than suggested by previously presented analysis or admitted by previously presented approximation schemes these surprising results suggest that for a wide variety of cases a convergent series expansion for the painleve transcendent describing the electric field has been obtained a suitable weighting of error measures for the approximations to the field and its first derivative provides a monotonically decreasing overall measure of the error in a subset of these cases it is conjectured that the series does converge for this subset