On the exponential decay of the Euler–Bernoulli beam with boundary energy dissipation

We study the asymptotic behavior of the Euler–Bernoulli beam which is clamped at one end and free at the other end. We apply a boundary control with memory at the free end of the beam and prove that the “exponential decay” of the memory kernel is a necessary and sufficient condition for the exponential decay of the energy.     

Exact boundary controllability of vibrating non-classical Euler–Bernoulli micro-scale beams

This study investigates the exact controllability problem for a vibrating non-classical Euler–Bernoulli micro-beam whose governing partial differential equation (PDE) of motion is derived based on the non-classical continuum mechanics. In this paper, it is proved that via boundary controls, it is possible to obtain exact controllability which consists of driving the vibrating system to rest in finite time. This control objective is achieved based on the PDE model of the system which causes that spillover instabilities do not occur.     

Coefficient identification in the Euler–Bernoulli equation using regularization methods

In this paper, we will study the inverse problem of identification of flexural rigidity coefficient in the Euler–Bernoulli equation. This inverse problem is ill-posed. To solve it, we will use regularization methods. In particular, we will apply the mollification method and the Landweber iteration method, in particular, to find the regularized solution of the Moore–Penrose generalized inverse to a linear operator and with this, we reconstruct the coefficient. At the end of this paper, will present some examples of interest.     

Exponential stability of uniform Euler–Bernoulli beams with non-collocated boundary controllers

We study the stability of a robot system composed of two Euler–Bernoulli beams with non-collocated controllers. By the detailed spectral analysis, we prove that the asymptotical spectra of the system are distributed in the complex left-half plane and there is a sequence of the generalized eigenfunctions that forms a Riesz basis in the energy space. Since there exist at most finitely many spectral points of the system in the right half-plane, to obtain the exponential stability, we show that one can choose suitable feedback gains such that all eigenvalues of the system are located in the left half-plane. Hence the Riesz basis property ensures that the system is exponentially stable. Finally we give some simulation for spectra of the system.     

Solvability of the clamped Euler–Bernoulli beam equation

In this study, solvability of the initial boundary value problem for general form Euler–Bernoulli beam equation which includes also moving point-loads is investigated. The complete proof of an existence and uniqueness properties of the weak solution of the considered equation with Dirichlet type boundary conditions is derived. The method used here is based on Galerkin approximation which is the main tool for the weak solution theory of linear evolution equations as well as in derivation of a priori estimate for the approximate solutions. All steps of the proposed technique are explained in detail.     

The analysis of a finite element method for the three-species Lotka–Volterra competition-diffusion with Dirichlet boundary conditions

A Galerkin finite element method is developed for the two dimensional/three dimensional nonlinear time-dependent three-species Lotka–Volterra competition-diffusion equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation are proved. An error estimate for the numerical solution is obtained. Numerical computations are carried out to examine the expected orders of accuracy in the error estimates.     

Free vibration of non-uniform Euler–Bernoulli beam under various supporting conditions using Chebyshev wavelet collocation method

This paper proposes operational matrix of rth integration of Chebyshev wavelets. A general procedure of this matrix is given. Operational matrix of rth integration is taken as rth power of operational matrix of first integration in literature. But, this study removes this disadvantage of Chebyshev wavelets method. Free vibration problems of non-uniform Euler–Bernoulli beam under various supporting conditions are investigated by using Chebyshev Wavelet Collocation Method. The proposed method is based on the approximation by the truncated Chebyshev wavelet series. A homogeneous system of linear algebraic equations has been obtained by using the Chebyshev collocation points. The determinant of coefficients matrix is equated to the zero for nontrivial solution of homogeneous system of linear algebraic equations. Hence, we can obtain ith natural frequencies of the beam and the coefficients of the approximate solution of Chebyshev wavelet series that satisfied differential equation and boundary conditions. Mode shapes functions corresponding to the natural frequencies can be obtained by normalizing of approximate solutions. The computed results well fit with the analytical and numerical results as in the literature. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation method is quite good even for small number of grid points.     

On the analytical approach to the linear analysis of an arbitrarily curved spatial Bernoulli–Euler beam

The equilibrium and kinematic equations of an arbitrarily curved spatial Bernoulli–Euler beam are derived with respect to a parametric coordinate and compared with those of the Timoshenko beam. It is shown that the beam analogy follows from the fact that the left-hand side in all the four sets of beam equations are the covariant derivatives of unknown vector. Furthermore, an elegant primal form of the equilibrium equations is composed. No additional assumptions, besides those of the linear Bernoulli–Euler theory, are introduced, which makes the theory ideally suited for the analytical assessment of big-curvature beams. The curvature change is derived with respect to both convective and material/spatial coordinates, and some aspects of its definition are discussed. Additionally, the stiffness matrix of an arbitrarily curved spatial beam is calculated with the flexibility approach utilizing the relative coordinate system. The numerical analysis of the carefully selected set of examples proved that the present analytical formulation can deliver valid benchmark results for testing of the purely numeric methods.     

Two identities for the Bernoulli–Euler numbers

Two identities for the Bernoulli and for the Euler numbers are derived. These identities involve two special cases of central combinatorial numbers. The approach is based on a set of differential identities for the powers of the secant. Generalizations of the Mittag–Leffler series for the secant are introduced and used to obtain closed-form expressions for the coefficients.     

A remarkable nature of the effect of boundary conditions on closed-form solutions for vibrating inhomogeneous Bernoulli–Euler beams

In this paper three sets of boundary conditions are considered for reconstructing the stiffness of the inhomogeneous Bernoulli–Euler beams. The essence of the paper consists in postulating the mode shape of the vibrating beam as a static deflection of associated uniform, homogeneous beam. This unconventional way of problem formulation turns out to lead to series of new closed-form solutions. For each combination of the boundary conditions several cases of the inertial coefficients are considered. All exact solutions for natural frequencies are represented as rational expressions of the involved coefficients. Solutions are written in terms of two positive integers: `m' representing the degree of the polynomial in the inertial term and `n' indicating power in the postulated mode shape. A remarkable conclusion is reached: For specified `m' and `n', the natural frequencies of the inhomogeneous beams with different boundary conditions coalesce. This remarkable nature does not imply that these beams share the same frequencies. In fact, these are different beams for each set of boundary conditions the expression for the stiffness is different. The paper should be considered as a first step towards analysis of uncertainty, inherently present in structures.     

Summation formulae of products of the Apostol–Bernoulli and Apostol–Euler polynomials

In this paper, a further investigation for the Apostol–Bernoulli and Apostol–Euler polynomials is performed, and some summation formulae of products of the Apostol–Bernoulli and Apostol–Euler polynomials are established by applying some summation transform techniques. Some illustrative special cases as well as immediate consequences of the main results are also considered.     

Adaptation of homotopy analysis method for the numeric–analytic solution of Chen system

In this paper, a new reliable algorithm based on an adaptation of the standard homotopy analysis method (HAM) is presented, which is the multistage homotopy analysis method (MSHAM). The freedom of choosing the auxiliary linear operator and the auxiliary parameter are still present in the MSHAM. The solutions of the non-chaotic and the chaotic Chen system which is a three-dimensional system of ordinary differential equations with quadratic nonlinearities were obtained by MSHAM. Numerical comparisons between the MSHAM and the classical fourth-order Runge–Kutta (RK4) numerical solutions reveal that the new technique is a promising tool for solving the non-linear chaotic and non-chaotic Chen system.     

Convergence of the Euler–Maxwell two-fluid system to compressible Euler equations

The combined non-relativistic and quasi-neutral limit of two-fluid Euler–Maxwell equations for plasmas is rigorously justified in this paper. For well-prepared initial data, the convergence of the two-fluid Euler–Maxwell system to the compressible Euler equations is proved in the time interval where a smooth solution of the limit problem exists.     

The analysis of a finite element method for the Navier–Stokes equations with compressibility

Summary. A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier–Stokes equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate for the numerical solution is obtained. Received September 9, 1997 / Revised version received August 12, 1999 / Published online July 12, 2000     

Dynamic responses of Euler–Bernoulli beam subjected to moving vehicles using isogeometric approach

Dynamic analysis of beam structures subjected to moving vehicles using an isogeometric Euler–Bernoulli formulation is presented in this paper. The method utilizes B-Splines or Non-Uniform Rational–Splines (NURBS) as the basis functions for both geometric and analysis implementation. The rotation-free technique has been incorporated into the formulation by using only one deflection variable with excluding the rotational degrees of freedom adopted for each control point. Then, it enables to use a few degrees of freedom (Dofs) to achieve a highly accurate solution. The validations of the proposed method included a complicated moving vehicle and rough pavement effects are compared to the precisely analytical results. Compared with most existing methods of finite element method (FEM) and readily analytical solutions, the present technique indicated the effectiveness of present isogeometric method and its well accurate prediction for suitable simulating the interaction model of the bridge structures and complicated vehicles.     

Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation

This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli–Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting, which is so general as to include different types of random variables and second-order random fields as sources of uncertainty. The second-order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. The direct and adjoint stochastic systems are uncoupled, which permits to exploit parallel computing architectures to solve the set of deterministic problem that arise from the stochastic collocation method. As a result, problems with a relative large number of random variables can be solved with a reasonable computational cost. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems.     

Output tracking for an Euler–Bernoulli beam equation with moment boundary control and shear boundary disturbance

This paper is concerned with performance output tracking for an Euler–Bernoulli beam equation with moment boundary control and shear boundary disturbance. An infinite-dimensional disturbance estimator is designed to estimate the total disturbance. By compensating the total disturbance, a servomechanism corresponding to the reference signal and servomechanism-based output feedback control law are designed. It is proved that under such control law, the performance output tracks exponentially the reference signal and the involved states of closed-loop system are bounded. The most important contribution is to deal with the shear boundary term stemmed from the error system between the disturbance estimator and the original system. The admissibility does not hold for such shear boundary term, while the corresponding boundary terms in the existing literature was proved to be admissible. Two key steps are presented to cope with such problem: First, the semigroup generation and exponential stability for a coupled beam system are verified by Riesz basis approach; second, the admissibility of a control operator for semigroup governed by such coupled beam system is proved. Moreover, Sobolev embedding theorem is introduced to simplify the proof of the boundedness of the closed-loop systems with respect to the available literature. Some numerical simulations are presented to illustrate the effectiveness.     

A posteriori error analysis using the constitutive law for the Crouzeix–Raviart element

In this work, we study the error in the approximation of the solution of elliptic partial differential equations obtained with the nonconforming finite elements method; we adopt the error in a constitutive law approach.