Recursive Algebraic Equations Research Articles

Free Vibration Analysis of Non-uniform Rayleigh Beams on Variable Winkler Elastic Foundation using Differential Transform Method

This study examines the effect of variable Winkler foundation on the natural frequencies of a prestressed nonuniform Rayleigh beam. In this work, the elastic coefficients of the foundations are assumed to vary along the length direction of the beam. A semi-analytical approach known as Differential Transform Method (DTM) is applied to the non-dimensional form of the governing equations of motion of the prestressed non-uniform Rayleigh beam and a set of recursive algebraic equations are obtained. Evaluating these derived equations and using some computer codes written and implemented in MAPLE 18, the non-dimensional frequencies and the associated mode shapes of the beam are obtained. The effects of variable Winkler foundation variations and axial force for various values of the slenderness ratio on the non-dimensional frequencies are investigated. The clamped-clamped and simply supported boundary conditions are considered to illustrate the accuracy and efficiency of this method. Finally, the results obtained are validated and are found to compare favorably well with those in the open literature.

In-Plane Free Vibration Analysis of a Scimitar-Type Rotating Curved Beam Using the Adomian Modified Decomposition Method

Free in-plane vibrations of a scimitar-type nonprismatic rotating curved beam, with a variable cross-section and increasing sweep along the leading edge, are calculated using an innovative, efficient and accurate solver called the Adomian modified decomposition method (AMDM). The equation of motion includes the axial force resulting from centrifugal stiffening, and the boundary conditions imposed are those of a cantilever beam, i.e., clamped-free and simple-free. The AMDM allows the governing differential equation to become a recursive algebraic equation suitable for symbolic computation, and, after additional simple mathematical operations, the natural frequencies and corresponding closed-form series solution of the mode shapes are obtained simultaneously. Two main advantages of the application of the AMDM are its fast convergence rate to a solution and its high degree of accuracy. The design shape parameters of the beam, such as transitioning from a straight beam pattern to a curved beam pattern, are investigated. The accuracy of the model is investigated using previously reported investigations and using an innovative error analysis procedure.

Free Vibrations with Large Amplitude of Axially Loaded Beams on an Elastic Foundation Using the Adomian Modified Decomposition Method

Analytical solutions describing free transverse vibrations with large amplitude of axially loaded Euler–Bernoulli beams for various end restrains resting on a Winkler one‐parameter foundation are obtained using the Adomian modified decomposition method (AMDM). The AMDM allows the governing equation to become a recursive algebraic equation, and, after some additional simple mathematical operations, the equations can be cast as an eigenvector problem whose solution results in the calculation of natural frequencies and corresponding closed‐form series solution of the mode shapes. Important to the use of the Adomian modified decomposition method is the treatment of the nonlinear Fredholm integral coefficient, which forms part of the governing equation. In addition to the calculation of natural frequencies and mode shapes, investigations are made of the effects on the free vibrations of the Winkler parameter and of increasing the axial loading.

Vibration analysis of a uniform pre-twisted rotating Euler–Bernoulli beam using the modified Adomian decomposition method

The governing equations for a pre-twisted rotating cantilever beam are derived and used for free vibration analysis of a pre-twisted rotating beam whose flexural displacements are coupled in two planes. First differential equations of motion of a rotating twisted beam, including terms due to centrifugal stiffening, are derived for an Euler–Bernoulli beam undergoing free natural vibrations. The general solutions of these equations are obtained on applying the Adomian modified decomposition method (AMDM). The AMDM allows the governing differential equations to become recursive algebraic equations and the boundary conditions to become simple algebraic frequency equations suitable for symbolic computation. With additional simple mathematical operations on the model, the natural frequencies and corresponding closed-form series solution of the mode shape can be obtained simultaneously. Two main advantages of the application of the AMDM are, for the cases considered here, its fast convergence rate to the solution with the high degree of accuracy. As the AMDM technique is systematic, it is found straight-forward to modify boundary conditions from one case to the next. Comparison of results with published data showed the present calculations to be in reasonable agreement.

Application of differential transformation finite element method in aperiodic vibration of non-prismatic beam

The paper presents the combination of multi-step differential transformation method (MsDTM) and finite element method (FEM) in vibration analysis of non-prismatic beam. A new algorithm, named differential transformation finite element method (DTFEM), combines the advantages of both methods. It enables to solve a wide class problems and obtain solutions in the form of piecewise functions that are very useful in further analyses, especially in time-consuming computations of dynamic systems. In contrary to standard DTM, with the proposed method, we can successfully calculate vibrations of non-prismatic structures governed by partial differential equations with variable coefficients. The application of FEM reduces partial differential equation to system of ordinary differential equations depending on time. After applying multi-step DTM the system is converting into a set of recursive algebraic equations which can be solved in a symbolic way. The final approximate solution is composed of a sequence of single solutions obtained in every time interval.The algorithm of DTFEM was described and illustrated with an example of non-prismatic simply supported beam subjected to uniformly distributed load increasing nonlinearly in time. The results in the form of beam displacements, velocities and accelerations were compared with numerical approach based on Newmark method. Two cases of damping model were taken into account: structural damping model and external damping model represented by two viscous dampers. A very good agreement was found in both cases. Presented analyses show that DTFEM is more accurate, with the same time step, than Newmark method. Therefore, in some cases, it may be possible to obtain sufficiently accurate results faster. The main advantage of the proposed method is the solution’s form which is a function of time and not a discrete set of results.

The Combination of Multi-Step Differential Transformation Method and Finite Element Method in Vibration Analysis of Non-Prismatic Beam

The paper presents a new algorithm being the combination of multi-step differential transformation method (MsDTM) and finite element method (FEM) as a powerful tool for solving variety dynamic problems. The proposed algorithm, named as differential transformation finite element method (DTFEM), transforms partial differential equation into a set of recursive algebraic equations. The final form of a solution is a piecewise function which in general case may be a symbolic function. High effectiveness and accuracy of DTFEM is demonstrated on the example of forced vibrations of non-prismatic Euler–Bernoulli beam. Computed time histories of displacements, velocities and accelerations are highly consistent with results obtained by Newmark method.

Numerical Solutions of Delay Volterra Integral Equations Using Single-term Walsh Series Approach

In this article, we study the numerical solutions of the delay Volterra integral equations and the system of delay Volterra integral equations using Single Term Walsh Series (STWS) method. The delay Volterra integral equation and its system is converted into a recursive linear algebraic equations and its systems. Solving this recursive algebraic equations and its systems, we find the discrete solutions to the delay Volterra integral equations and its systems respectively. Numerical examples are presented to show the computational efficiency of this STWS method for delay Volterra integral equations and its systems.

A Numerical Solution of Hereditary Equations with a Weakly Singular Kernel for Vibration Analysis of Viscoelastic Systems / Vienâdojumu Ar Vâjo Singulâro Kodolu Skaitliskais Risinâjums Iedzimto Viskoelastîgo Sistçmu Vibrâciju Analîzei

Abstract Viscoelastic, or composite materials that are hereditary deformable, have been characterised by exponential and weakly singular kernels in a hereditary equation. An exponential kernel is easy to be numerically implemented, but does not well describe complex vibratory behaviour of a hereditary deformable system. On the other hand, a weakly singular kernel is known to describe the complex vibratory behaviour, but is nontrivial to be numerically implemented. This study presents a numerical formulation for solving a hereditary equation with a weakly singular kernel. Recursive algebraic equations, which are numerically solvable, are formulated by using the Galerkin method enhanced by a numerical integration and elimination of weak singularity. Numerical experiments showed that the present approach with a weakly singular kernel is well fitted into a realistic vibratory behaviour of a hereditary deformable system under dynamic loads, as compared to the same approach with an exponential kernel.

Simulation of tapered rotating beams with centrifugal stiffening using the Adomian decomposition method

The Adomian modified decomposition method (AMDM) is employed for the free transverse vibration analysis of rotating non-uniform Euler-Bernoulli beams using several boundary conditions, rotation speeds, and beam lengths. The equation of motion includes the axial force due to centrifugal stiffening. The AMDM allows the governing differential equation to become a recursive algebraic equation and after additional simple mathematical operations on the model, the natural frequencies and corresponding closed-form series solution of the mode shape can be obtained simultaneously. The AMDM technique is systematic, so it is straightforward to modify the boundary conditions for specific cases. Comparisons with previously reported results demonstrated that the results obtained by the proposed method were in close agreement.

AMDM for free vibration analysis of rotating tapered beams

The free vibration of rotating Euler-Bernoulli beams with the thickness and/or width of the cross-section vary linearly along the length is investigated by using the Adomian modified decomposition method (AMDM). Based on the AMDM, the governing differential equation for the rotating tapered beam becomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies and the closed form series solution of the corresponding mode shapes can be easily obtained simultaneously. The computed results for different taper ratios as well as different offset length and rotational speeds are presented in several tables and figures. The accuracy is assured from the convergence and comparison with the previous published results. It is shown that the AMDM provides an accurate and straightforward method of free vibration analysis of rotating tapered beams.

Vibration and stability of a double-beam system interconnected by an elastic foundation under conservative and nonconservative axial forces

The Adomian modified decomposition method (AMDM) is employed in this study to investigate the free vibration and stability of a cantilever double-beam system, which is continuously joined by a Winkler-type elastic layer. The free end of each beam is restrained by a translational spring and subjected to a combination of compressive axial and follower loads. Based on the AMDM, the governing differential equations for the double-beam system are represented as a recursive algebraic equation. By using boundary condition equations, the natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The double-beam system becomes unstable in the form of either divergence or flutter with the increasing loads. Then the critical loads are discussed under different boundary conditions and the nonconservative parameters. Furthermore, the effect of the value of the spring stiffness on the critical loads for either flutter or divergence instability of the double-beam systems is studied.

Free Vibrations of Spinning Beams Under Nonclassical Boundary Conditions Using Adomian Modified Decomposition Method

This study employs the Adomian modified decomposition method (AMDM) for the dynamic analysis of Euler–Bernoulli beams spinning about their longitudinal axes under various boundary conditions. Based on the AMDM, the governing differential equations for the spinning beam become a recursive algebraic equation system. By using the boundary condition equations, the natural frequencies can be readily obtained. The computed results under different classical and nonclassical boundary conditions as well as spinning speeds are presented. The accuracy is assured from comparison with published results. It is shown that the AMDM offers an accurate and straightforward method of free vibration analysis of spinning beams with arbitrary boundary conditions.

Stability Analysis of Cantilever Cracked Beams Subjected to Follower Forces

A stability analysis of a cracked cantilever beam subjected to a follower compressive load is presented by using Adomian modified decomposition method(AMDM). The free end of the cantilever beam is restrained by a translational spring and subjected to a follower force. The crack is modeled as an equivalent massless rotational spring. Based on the AMDM the governing differential equation for this cracked beam becomes a recursive algebraic equation. By using boundary conditions and continuity/jump condition equations at crack location, the dimensionless natural frequencies and corresponding closed-form series solution of mode shapes can be obtained simultaneously. The accurate and effective of the proposed method are verified by comparing the results using the AMDM to those given in the literature. The effect of the location and depth of the crack on the critical compressive load for flutter or buckling instability of the beam is studied. Furthermore, the critical spring stiffness is calculated to determine the different type of instability of the beam under the follower compression. The numerical calculations results show that the maximum changes of the natural frequencies are observed when the crack is located near clamped end. It is also found that the stability of beam may be improved due to crack.

Application of Adomian Modified Decomposition Method to Free Vibration Analysis of Rotating Beams

The Adomian modified decomposition method (AMDM) is employed in this paper for dynamic analysis of a rotating Euler-Bernoulli beam under various boundary conditions. Based on AMDM, the governing differential equation for the rotating beam becomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions as well as different offset length and rotational speeds are presented. The accuracy is assured from the convergence and comparison published results. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of rotating beams with arbitrary boundary conditions.

Free vibration analysis of a type of tapered beams by using Adomian decomposition method

The Adomian decomposition method (ADM) is employed in this paper to investigate the free vibrations of tapered Euler–Bernoulli beams with a continuously exponential variation of width and a constant thickness along the length under various boundary conditions. Based on ADM the governing differential equation for the tapered beam becomes a recursive algebraic equation. By using the boundary condition equations, the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The computed results for different boundary conditions and non-uniformity ratios are presented. The accuracy is assured from the convergence and comparison published results.

Optimal control of diffusion-convection-reaction processes using reduced-order models

Two approaches for optimal control of diffusion-convection-reaction processes based on reduced-order models are presented. The approaches differ in the way spatial discretization is carried out to compute a reduced-order model suitable for controller design. In the first approach, the partial differential equation (PDE) that describes the process is first discretized in space and time using the finite difference method to derive a large number of recursive algebraic equations, which are written in the form of a discrete-time state-space model with sparse state, input and output matrices. Snapshots based on this high-dimensional state-space model are generated to calculate empirical eigenfunctions using proper orthogonal decomposition. The Galerkin projection with the computed empirical eigenfunctions as basis functions is then directly applied to the high-dimensional state-space model to derive a reduced-order model. In the second approach, a continuous-time finite-dimensional state-space model is constructed directly from the PDE through application of orthogonal collocation on finite elements in the spatial domain. The dimension of the derived state-space model can be further reduced using standard model reduction techniques. In both cases, optimal controllers are designed based on the low-order state-space models using discrete-time and continuous-time linear quadratic regulator (LQR) techniques. The effectiveness of the proposed methods are illustrated through applications to a diffusion-convection process and a diffusion-convection-reaction process.

Structure of Regular Solutions of the Three-Body Schrödinger Equation near the Pair Impact Point

We investigate the six-dimensional Schrodinger equation for a three-body system with central pair interactions of a more general form than Coulomb interactions. Regular general and special physical solutions of this equation are represented by infinite asymptotic series in integer powers of the distance between two particles and in the sought functions of the other three-body coordinates. Constructing such functions in angular bases composed of spherical and bispherical harmonics or symmetrized Wigner D-functions is reduced to solving simple recursive algebraic equations. For projections of physical solutions on the angular bases functions, we derive boundary conditions at the pair impact point.

A Born–Green–Yvon integral equation theory for self-interacting lattice polymers

A Born–Green–Yvon (BGY) integral equation is constructed for the end-to-end distribution function of an isolated polymer on a lattice. The polymer is modeled as a self-avoiding walk for which nonbonded sites interact via an attractive nearest-neighbor contact potential. The BGY equation is solved analytically using a Markov approximation for the required three-site distribution function and a delta-function pseudopotential to model the lattice contact potential. The resulting recursive algebraic equation is readily evaluated for a polymer on any Bravais lattice with equal length base vectors. Results are presented for the mean-square end-to-end separation as a function of chain length and contact energy for polymers on several two-, three-, and four-dimensional lattices. The variation of the scaling exponent 2ν with contact energy is used to locate the theta energies for these lattices.

A fast recursive solution for induction motor transients

A fast and accurate algorithm for determining induction-motor transient behavior is presented. The fast components of the stator transients are decoupled from the machine flux linkage equations by a linear transformation. The resulting differential equations are then solved by using recursive algebraic equations, which require considerably less computational effort than existing methods. An example shows that the method is very accurate and yet uses less than 40% of the CPU time required by the exact model. The method could be useful in studies that involve simultaneous responses of many machines, such as power system stability studies, where optimal computational efficiency is desirable.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis and identification of distributed systems via the combined method utilizing the finite element technique and orthogonal function expansions

In recent years, the application of the finite element method to the identification and control problem has become more and more interesting, especially for systems with irregular boundaries. The present work proposes a combined mathematical model utilizing finite element discretization in terms of space elements and orthogonal function expansions of nodal time responses, instead of the conventional finite difference approximation for time derivatives. The present approach directly reduces partial differential equations to a set of recursive algebraic equations and thus saves analysis computation time. Based on this mathematical model, the identification of system parameters in distributed systems has been carried out utilizing the least-squares method. The identification procedure is straightforward, and—as shown by illustrative examples—the results of the present approach are excellent.