dynamic analysis for axially moving viscoelastic panels abstract in this study stability and dynamic behaviour of axially moving viscoelastic panels are investigated with the help of the classical modal analysis we use the flat panel theory combined with the kelvin voigt viscoelastic constitutive model and we include the material derivative in the viscoelastic relations complex eigenvalues for the moving viscoelastic panel are studied with respect to the panel velocity and the corresponding eigenfunctions are found using central finite differences the governing equation for the transverse displacement of the panel is of fifth order in space and thus five boundary conditions are set for the problem the fifth condition is derived and set at the in flow end for clamped clamped and clamped simply supported panels the numerical results suggest that the moving viscoelastic panel undergoes divergence instability for low values of viscosity they also show that the critical panel velocity increases when viscosity is increased and that the viscoelastic panel does not experience instability with a sufficiently high viscosity coefficient for the cases with low viscosity the modes and velocities corresponding to divergence instability are found numerically we also report that the value of bending rigidity bending stiffness affects the distance between the divergence velocity and the flutter velocity the higher the bending rigidity the larger the distance

free vibration analysis of a uniform multi span beam carrying multiple spring mass systems abstract the literature regarding the free vibration analysis of single span beams carrying a number of spring mass systems is plenty but that of multi span beams carrying multiple spring mass systems is fewer thus this paper aims at determining the exact solutions for the natural frequencies and mode shapes of a uniform multi span beam carrying multiple spring mass systems firstly the coefficient matrices for an intermediate pinned support an intermediate spring mass system left end support and right end support of a uniform beam are derived next the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system finally equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes in this paper the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made thus the last solutions are the exact ones the effects of attached spring mass systems on the free vibration characteristics of the 1 4 span beams are studied

vibrations of axially moving viscoelastic plate with parabolically varying thickness based on the two dimensional viscoelastic differential constitutive relation the differential equation of motion of the axially moving viscoelastic rectangular plate constituted by the kelvin voigt model with parabolically varying thickness in the y direction is derived the dimensionless complex frequencies of axially moving viscoelastic plate with different boundary conditions versus the dimensionless moving speed for various aspect ratio thickness parameter and the dimensionless delay time are analyzed by the differential quadrature method the effects of various parameters such as aspect ratio thickness parameter the dimensionless moving speed and the dimensionless delay time on the vibration characteristics of the axially moving viscoelastic rectangular plate with parabolically varying thickness are discussed

transversal waves in beams via the network simulation method abstract in this paper the network simulation method nsm is proposed as a tool for solving the transverse motion of thin beams resulting from a bending action the reliability and robustness of the proposed technique is demonstrated with two examples in which a pinned pinned beam and a clamped mass load beam are respectively submitted to different flexural forces the nsm is a numerical technique for the solution of linear and non linear problems which are defined by a mathematical model assuming that the electrical variables of voltage and current are equivalent respectively to the displacement and the spatial variation of the displacement a network model for each volume element is designed such that electrical equations are formally equivalent to the spatial discretised equations of the mechanical model the whole network model including the devices associated with the boundary conditions is solved by the numerical computer code pspice the numerical results have been compared with those obtained by the mode superposition method the agreement between them is impressive the temporal behaviour of each quantity involved in the problem is obtained without transform complications since the whole study has been conducted in the time domain with the possibility of obtaining the corresponding spectral content at the end of calculation the main contribution of the present paper therefore is to employ the nsm numerical method in order to arrive at solutions in the time domain of dynamic mechanical problems one must also emphasize that nsm has not previously been used by other authors to solve physical problems like the ones presented here

a differential quadrature procedure for three dimensional buckling analysis of rectangular plates abstract this paper presents an application of the differential quadrature dq method for three dimensional buckling analysis of rectangular plates the governing equations of the plate model are first presented in terms of displacement stress displacement relationship and boundary conditions with three dimensional flexibility these equations are then normalised and discretised using the dq procedure example problems pertaining to the buckling of rectangular plates with generic boundary conditions are selected to illustrate the efficiency and simplicity of implementing the dq procedure the convergence characteristics of the method are first conducted based on numerical studies the dq solutions are then compared where possible with exact or approximate solutions it is found that the differential quadrature method yields accurate results for the plate problems under the current investigation in addition to the above some parametric studies are carried out by varying the plates aspect ratio boundary conditions and thickness to width ratio under axial and biaxial loading

vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices this work provides a general formulation to solve vibration problems for continuous systems with damping effects including modal transient harmonic and spectrum response analyses in modal analysis the system eigenvalues and corresponding eigenfunctions can be determined the orthogonal relations of eigenfunctions are shown for transient harmonic and spectrum analyses the generic force actuator functions and response sensing operators are introduced respectively and used to derive the system response the time domain response is obtained for transient analysis the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis the solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force actuator and response sensor mode shape functions in particular one dimensional beam and two dimensional plate vibration analyses are illustrated by following the developed generic formulation this work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well

a new hybrid heuristic approach for solving large traveling salesman problem this paper presents a new metaheuristic approach called acomac algorithm for solving the traveling salesman problem tsp we introduce multiple ant clans concept from parallel genetic algorithm to search solution space utilizing various islands to avoid local minima and thus can yield global minimum for solving the traveling salesman problem moreover we present two approaches named the multiple nearest neighbor nn and the dual nearest neighbor dnn to acomac to enhance large tsps to validate the proposed methods numerous simulations were conducted to compare acomac and dorigo s acs with and without the addition of the multiple nearest neighbor nn method or the dual nearest neighbor dnn approach using a range of tsp benchmark problems according to the results of the simulation adding the nn or dnn approach to acomac or acs as initial solutions also significantly enhances the performance of acomac and acs in solving the traveling salesman problem meanwhile using acomac dnn with tsp can yield better solutions than the other stated approaches additionally acomac or acomac nn utilizing five ant clans with a total of 20 ants is verified to yield better solutions furthermore acomac with a local weighting w set to 0 6 can yield better solutions in terms of length

k out of n systems reliability and availability analyses are recognized as essential for guiding decision makers in the implementation of actions addressed to improve the technical and economical performance of complex systems for industrial systems with reparable components the most interesting parameter used to drive maintenance is the stationary availability in this regard the present paper proposes an exact formula for computing the system stationary availability of a k out of n system such a formula is proved to be in agreement with the fundamental theorem of markov chains then a multi objective mathematical model is formulated for choosing the optimal system configuration design the pareto front is developed using the lexicographic goal programming lgp method and the topsis method is successively implemented to choose the k out of n configuration that represents the best compromise between the considered objective functions a numerical example is provided stationary availability formula proposed for k out of n systems validation of the proposed formula by markov chains design of a k out of n system selection of pareto solution by means of topsis multi criteria method

application of the spline element method to analyze vibration of skew mindlin plates with varying thickness in one direction this paper presents an application of the spline element method based on the mindlin plate theory to analyze the vibration of thick skew plates with varying thickness in the longitudinal direction to demonstrate the convergence and accuracy of the present method several examples are solved and results are compared with those obtained by other numerical methods good accuracy is obtained frequencies of skew mindlin plates with varying cross sections having arbitrary boundary conditions are analyzed for different thickness ratios aspect ratios ratios of the width to thickness and skew angles

nonlinear free transverse vibrations of in plane moving plates without and with internal resonances in this paper nonlinear free transverse vibrations of in plane moving plates subjected to plane stresses are investigated the hamilton principle is applied to derive the governing equation and the associated boundary conditions the method of multiple scales is employed to analyze the nonlinear partial differential equation the solvability conditions are established in the cases without internal resonance and with 3 1 or 1 1 internal resonances some numerical examples are presented to demonstrate the effects of in plane moving speeds on the frequencies the nonlinear frequencies of the in plane moving plate without internal resonances are numerically calculated the relationship between the nonlinear frequencies and the initial amplitudes is showed at different in plane moving speeds and the nonlinear coefficients respectively it is feasible to investigate resonances without the modes not involved in the resonances the effects of the related parameters are demonstrated for the case of 3 1 and 1 1 internal resonances respectively the differential quadrature scheme is developed to solve numerically the governing equation and confirm results via the method of multiple scales

improved eigenvalues for combined dynamical systems using a modified finite element discretization scheme new approaches are presented to discretize an arbitrarily supported linear structure carrying various lumped attachments specifically the exact eigendata i e the exact natural frequencies and mode shapes of the linear structure without the lumped attachments are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration this is achieved by identifying a set of minimum changes in the finite element system matrices and enforcing certain constraint conditions once the updated matrices for the linear structure are found the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices numerical experiments show that for the same number of elements the proposed scheme returns higher natural frequencies that are substantially more accurate than those given by the finite element model alternatively the proposed discretization scheme allows one to efficiently and accurately determine the higher natural frequencies of a combined system without increasing the number of elements in the finite element model

free vibration analysis of rectangular plates with variable thickness abstract an approximate method for analyzing the free vibration of thin and moderately thick rectangular plates with arbitrary variable thickness is proposed the approximate method is based on the green function of a rectangular plate the green function of a rectangular plate with arbitrary variable thickness is obtained as a discrete form solution for deflection of the plate with a concentrated load the discrete form solution is obtained at each discrete point equally distributed on the plate it is shown that the numerical solution for the green function has good convergency and accuracy by applying the green function the free vibration problem of the plate is translated into the eigenvalue problem of the matrix the convergency and accuracy of the numerical solutions for the natural frequency parameter calculated by the proposed method are investigated and the frequency parameters and their modes of free vibration are shown for some rectangular plates

natural frequencies and ope loop responses of an elastic beam fixed on a moving cart and carrying an intermediate lumped mass abstract in this paper the motion of a bernoulli euler cantilever beam clamped on a moving cart and carrying an intermediate lumped mass is considered the equations of motion of the beam mass cart system are analyzed through unconstrained modal analysis and a unified characteristic equation for calculating the natural frequencies of the system is established the changes of natural frequencies and the corresponding mode shapes with respect to the changes in the ratios between the beam mass the lumped mass and the cart mass and to the concentrated position of the lumped mass are investigated with the frequency equation which can be generally applied to this kind of system the exact and assumed mode solutions including the dynamics of the base cart are obtained and the open loop responses of the system by arbitrarily designed forcing functions are given by numerical simulations the results match well with physical phenomena even in the extreme cases where the mass is attached to the bottom and to the top of the beam

a proof of convergence for ant algorithms a proof of convergence for ant algorithms is developed ant algorithms were modeled as branching random processes the branching random walk and branching wiener process to derive rates of birth and death of ant paths substitution is then carried out in birth death processes which proves that a stable distribution is surely reached this indicates that ant algorithms converge with probability one this analogy models ant algorithms complexity parameters such as the number of cycles the degrees of freedom of problem and the number of ants

nonlinear dynamics of axially moving plates abstract the nonlinear dynamics for forced motions of an axially moving plate is numerically investigated using von karman plate theory and retaining in plane displacements and inertia the equations of motion are obtained via an energy method based on lagrange equations this yields a set of second order nonlinear ordinary differential equations with coupled terms the equations are transformed into a set of first order nonlinear ordinary differential equations and are solved via the pseudo arclength continuation technique the near resonance nonlinear dynamics is examined via plotting the frequency response curves of the system results are shown through frequency response curves time histories and phase plane diagrams the effect of system parameters such as the axial speed and the pretension on the resonant responses is also highlighted

modelling real time dynamic substructuring using partial delay differential equations real time dynamic substructuring is a new component testing method for simulating the dynamics of complex engineering systems the physical component is tested within a computer generated virtual environment using real time control techniques delays in communication which occur between the component and the virtual environment can potentially destabilize the simulation in this paper the mechanism for this instability is examined using a beam oscillator system as a case study we will show how the stability and the amplitude response of the system change with the time delay numerical simulations of the reduced system as well as a full delayed beam equation are performed a series of experimental tests is carried out on a beam oscillator system comparison of the theoretical numerical and experimental results is presented and these agree remarkably well

an exact modal analysis approach to vibration analysis of structures with mass spring subsystems and rotational joints abstract a novel exact modal analysis approach is presented for vibration analysis of plane continuous structures which are coupled with discrete mass spring subsystems and include elastic rotational joints modelling local flexibility using the theory of generalised functions to handle the discontinuities of the response variables every continuous member with any number of mass spring subsystems and joints is treated as a two node element for which a 6 6 exact dynamic stiffness matrix is obtained in closed form as a result the global dynamic stiffness matrix is built by a standard finite element assembling procedure with size depending only on the number of nonzero nodal degrees of freedom of member to member nodes upon deriving pertinent orthogonality conditions for the modes the system response under arbitrary loads is obtained by modal impulse and modal frequency response functions under the assumption of proportional damping the solutions are exact and can be used as benchmark for classical finite element solutions the approach is formulated for various mass spring subsystems acting in transverse and axial directions relative to every member

dynamic analysis of a uniform cantilever beam carrying a number of elastically mounted point masses with dampers abstract the free and forced vibration analyses of a uniform cantilever beam carrying a number of spring damper mass systems with arbitrary magnitudes and locations were made by means of the analytical and numerical combined method ancm first of all a method was presented to replace each spring damper mass system by a massless equivalent spring damper system with effective spring constant keffand effective damping coefficient ceffso that the ancm is available for the title problem next the equation of motion for the constrained beam with spring damper mass systems attached was derived by using the natural frequencies and normal mode shapes of the unconstrained beam without carrying any attachments incorporating the expansion theorem finally the eigenvalues and the forced vibration responses of the constrained beam were determined by conventional numerical methods to confirm the reliability of the presented theory all the numerical results obtained from the ancm were compared with the corresponding ones obtained from the conventional finite element method fem and good agreement was achieved the influence of the damping magnitude of each spring damper mass system on the eigenvalues and the forced vibration responses of the constrained beam was studied because the order of the overall property matrices for the equation of motion of the constrained beam derived from the ancm is much lower than that from the conventional fem the storing memory and the cpu time required by the ancm are much less than those required by the fem

vibration of beams with arbitrary discontinuities and boundary conditions abstract a general solution of the vibration of an euler bernoulli beam with arbitrary type of discontinuity at arbitrary number of locations is presented in this paper to account for the discontinuity term induced by various additional elements on the beam heaviside s function is used to express the modal displacement of the whole beam by a single function this general modal displacement function is then solved by using laplace transformation this general solution consists of four types of basic modal shapes induced by four corresponding types of discontinuity terms at the discontinuity points various discontinuity terms are obtained and expressed by the boundary values of the modal displacement in a recursive way consequently the modal displacement can be determined by examining only the conditions on the boundary in such a way the present solution reduces the vibration of beams with arbitrary discontinuities to the same order of the case without discontinuity point to demonstrate the efficiency and applicability of the present method three application examples are presented calculation example shows that the lead zirconate titanate pzt actuator should be placed as close to the fixed end as possible to achieve the best excitation effect on a cantilever beam a new method to calculate the driving point anti resonance frequency is also proposed numerical results suggest that the variation of driving point anti resonance frequency can be used to determine the location and size of crack in beams due to the generic nature of the solution and the problem the present method can be utilized in smart structures modeling and structural health monitoring of beam type structures

on the eigenfrequencies of a cantilever beam with attached tip mass and a spring mass system this paper deals with the derivation of the frequency equation of a special combined dynamic system it consists of a clamped free bernoulli euler beam with a tip mass where a spring mass system is attached to it the derivation of the frequency equation is essentially carried out by means of the lagrange multipliers method frequency equations of some simpler systems are also obtained from the general expression by using limiting processes these equations are then numerically solved for various combinations of physical parameters the comparison with the results from the solution of the corresponding bernoulli euler frequency equations shows that the frequency equations established are very accurate

real time dynamic substructuring in a coupled oscillator pendulum system real time dynamic substructuring is a powerful testing method which brings together analytical numerical and experimental tools for the study of complex structures it consists of replacing one part of the structure with a numerical model which is connected to the remainder of the physical structure the substructure by a transfer system in order to provide reliable results this hybrid system must remain stable during the whole test a primary mechanism for destabilization of these type of systems is the delays which are naturally present in the transfer system in this paper we apply the dynamic substructuring technique to a nonlinear system consisting of a pendulum attached to a mechanical oscillator the oscillator is modelled numerically and the transfer system is an actuator the system dynamics is governed by two coupled second order neutral delay differential equations we carry out local and global stability analyses of the system and identify the delay dependent stability boundaries for this type of system we then perform a series of hybrid experimental tests for a pendulum oscillator system the results give excellent qualitative and quantitative agreement when compared to the analytical stability results

dynamic analysis of axially moving orthotropic plates integral transform solution the generalized integral transform technique gitt is employed to obtain a hybrid analytical numerical solution for dynamic response of axially moving orthotropic plates considering two combinations of simply supported s and clamped c boundary conditions ccss and cccc through the double integral transform for the transverse displacement field in the x and y directions the implementation of gitt approach for analyzing the forced vibration equation eliminates the space variable and leads to a coupled system of second order ordinary differential equations odes in the dimensionless temporal variable the mathematica s built in function ndsolve is employed to numerically solve the resulting transformed ode system excellent convergence behavior of the proposed eigenfunction expansions is demonstrated for calculating the transverse displacement at various points of axially moving orthotropic plates the first sixteen mode shapes of vibration for axially moving plates are graphically illustrated the proposed approach is verified by comparing the transverse vibration displacement of central point displacement profiles along the symmetric lines and natural circular frequencies of the plate for zero moving velocity case with the corresponding results obtained by abaqus finite element program moreover parametric studies are performed to analyze the effects of flexural rigidity ratio dimensionless axial speed and aspect ratio on the vibration amplitude of moving plates

vibrations of a plate with an attached two degree of freedom system abstract this note deals with a theoretical analysis of the dynamical behavior of a system made up of a plate with a two degree of freedom 2 dof system elastically mounted this study was performed by means of an analytical model based on lagrange s multipliers the results are verified with the values obtained using fem the analysis is of interest from both academic and technological viewpoints the case of a plate structure supporting a 2 dof system has not been previously considered in the technical literature using the lagrange s multipliers approach this system can be considered as a dynamic absorber

reliability evaluation and optimal design in heterogeneous multi state series parallel systems this paper addresses the heterogeneous redundancy allocation problem in multi state series parallel reliability structures with the objective to minimize the total cost of system design satisfying the given reliability constraint and the consumer load demand the demand distribution is presented as a piecewise cumulative load curve and each subsystem is allowed to consist of parallel redundant components of not more than three types the system uses binary capacitated components chosen from a list of available products to provide redundancy so as to increase system performance and reliability the components are characterized by their feeding capacity reliability and cost a system that consists of elements with different reliability and productivity parameters has the capacity strongly dependent upon the selection of constituent components a binomial probability based method to compute exact system reliability index is suggested to analyze the problem and suggest an optimal near optimal system structure an ant colony optimization algorithm has been presented the solution approach consists of a series of simple steps as used in early ant colony optimization algorithms dealing with other optimization problems and offers straightforward analysis four multi state system design problems have been solved for illustration two problems are taken from the literature and solved to compare the algorithm with the other existing methods the other two problems are based upon randomly generated data the results show that the method can be appealing to many researchers with regard to the time efficiency and yet without compromising over the solution quality

free vibration of a uniform beam with multiple elastically mounted two degree of freedom systems abstract the free vibration of a beam with one or more elastically mounted two degree of freedom systems that translate and rotate is considered in this note the assumed modes method is applied to formulate the equations of motion and the natural frequencies of the system are found by solving for the roots of a given characteristic determinant if the number of attached spring mass systems is small one can exploit the sherman morrison woodbury determinant formula and reduce the characteristic determinant to one of smaller size which will be easier to code and more computationally efficient to solve

the exact solution for free vibration of uniform beams carrying multiple two degree of freedom spring mass systems abstract the literature regarding the exact solutions of natural frequencies and mode shapes of a uniform beam carrying multiple two degree of freedom 2 dof spring mass systems is rare thus this paper aims at studying this problem using the numerical assembly method nam first of all the equivalent springs for replacing the effect of a 2 dof spring mass system are determined next the coefficient matrix for a 2 dof spring mass system attached to the uniform beam is derived based on the compatibility of deformations and equilibrium of forces including moments the coefficient matrices for the left end and right end of the beam are also derived based on the various boundary conditions of the beam combining the coefficient matrices for all the 2 dof spring mass systems attached to the beam and the coefficient matrices for the boundary conditions of the beam one obtains the overall coefficient matrix of the constrained beam i e the beam carrying any number of 2 dof spring mass systems the product of the overall coefficient matrix and the vector for all the integration constants yields a set of simultaneous equations let the coefficient determinant of the last simultaneous equations equal to zero one obtains the frequency equation the roots of the last frequency equation denote the natural frequencies of the constrained beam substituting the roots of the frequency equation into the set of simultaneous equations one may determine the associated mode shapes of the constrained beam in this paper the exact solution refers to the one obtained from the continuous model instead of the discrete mode besides the accuracy of the analytical and numerical combined method ancm given by the existing literature is dependent on the total number of vibration modes considered but this is not true for the accuracy of the nam adopted here to confirm the reliability of the presented theory all the numerical results obtained from nam are compared with the corresponding ones obtained from the conventional finite element method fem and good agreement is achieved

free vibration analysis of mindlin plates with linearly varying thickness abstract a method based on the variational principles in conjunction with the finite difference technique is applied to examine the free vibration characteristics of isotropic rectangular plates of linearly varying thickness by including the effects of transverse shear deformation and rotary inertia the validity of the present approach is demonstrated by comparing the results with other solutions proposed for plates with uniform and linearly varying thickness natural frequencies and mode shapes of mindlin plates with simply supported and clamped edges are determined for various values of relative thickness ratio and the taper thickness constant

a general approach to formulating the frequency equation for a beam carrying miscellaneous attachments beams carrying various lumped attachments have been studied extensively over the years in this paper a simple approach is proposed that can be used to readily determine the eigenvalues of an arbitrarily supported single span or multi span beam carrying any combination of lumped mass rotary inertia grounded translational or torsional spring grounded translational or torsional viscous damper an undamped or damped oscillator with or without a rigid body degree of freedom rather than solving a generalized eigenvalue problem to obtain the eigenvalues of the system a frequency equation is formulated instead whose solution can be easily solved either numerically or graphically the proposed scheme is easy to code and can be easily modified to accommodate beams with arbitrary supports and carrying any number of miscellaneous lumped attachments

the natural frequencies and mode shapes of a uniform cantilever beam with multiple two dof spring mass systems abstract because of the complexity of the mathematical expressions the literature concerning the free vibration analysis of a uniform beam carrying a single two degrees of freedom d o f spring mass system is rare and the publications relating to that carrying multiple two d o f spring mass systems have not yet appeared hence the purpose of this paper is to present some information in this area first of all the closed form solution for the natural frequencies and the corresponding normal mode shapes of the uniform beam alone or the bare beam with the prescribed boundary conditions are determined analytically next a method is presented to replace each two d o f spring mass system by two massless equivalent springs with spring constants k v eq iandk v eq k and then the foregoing natural frequencies and normal mode shapes for the bare beam are in turn used to derive the equation of motion of the loading beam i e the bare beam carrying any number of two d o f spring mass systems by using the expansion theorem finally the natural frequencies and the associated mode shapes of the loading beam are obtained from the last equation by using the numerical method to confirm the reliability of the present method all the numerical results obtained in this paper are compared with the corresponding ones obtained from the conventional finite element method fem and good agreement is achieved because the order of the property matrices for the equation of motion of the loading beam derived from the present method is much lower than that derived from the fem the computer time required by the former is much less than that required by the latter besides the equation of motion derived from the present method may always run on the cheaper personal computers but that from the fem may run only on the more expensive larger computers if the degree of freedom of the loading beam exceeds a certain limit

on the natural frequencies and mode shapes of a multispan timoshenko beam carrying a number of various concentrated elements abstract the purpose of this paper is to utilize the numerical assembly method nam to determine the exact natural frequencies and mode shapes of the multispan timoshenko beam carrying a number of various concentrated elements including point masses rotary inertias linear springs rotational springs and spring mass systems first the coefficient matrices for an intermediate pinned support an intermediate concentrated element left and right end support of a timoshenko beam are derived next the overall coefficient matrix for the whole structural system is obtained using the numerical assembly technique of the finite element method finally the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions respectively the effects of distribution of in span pinned supports and various concentrated elements on the dynamic characteristics of the timoshenko beam are also studied

variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay abstract in this work we give a variation of constants formula for partial functional differential equations with infinite delay we assume that the linear part is not necessarily densely defined and its resolvent operator satisfies the hille yosida condition we establish a reduction of the problem to a finite dimensional space which allows us to prove the existence of almost periodic solutions