Stabilization of transmission coupled wave and Euler–Bernoulli equations on Riemannian manifolds by nonlinear feedbacks

We consider the transmission system of coupling wave equations with Euler–Bernoulli equations on Riemannian manifolds. By introducing nonlinear boundary feedback controls, we establish the exponential and rational energy decay rate for the problem. Our proofs rely on the geometric multiplier method.     

A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects

A new Bernoulli–Euler beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equation and complete boundary conditions for a Bernoulli–Euler beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure- and surface energy-dependent size effect. In addition, Poisson’s effect is incorporated in the current model, unlike existing beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as special cases. The current model reduces to the classical Bernoulli–Euler beam model when the microstructure dependence, surface energy, and Poisson’s effect are all suppressed. To demonstrate the new model, a cantilever beam problem is solved by directly applying the general formulas derived. Numerical results reveal that the beam deflection predicted by the new model is smaller than that by the classical beam model. Also, it is found that the difference between the deflections predicted by the two models is very significant when the beam thickness is small but is diminishing with the increase of the beam thickness.     

Vibration analysis of Euler–Bernoulli nanobeams by using finite element method

In the present study, an efficient finite element model for vibration analysis of a nonlocal Euler–Bernoulli beam has been reported. Nonlocal constitutive equation of Eringen is proposed. Equations of motion for a nonlocal Euler–Bernoulli are derived based on varitional statement. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Vibration characteristics, such as fundamental frequencies, are illustrated in graphical and tabulated form. Numerical results are presented to figure out the effects of nonlocal parameter, slenderness ratios, rotator inertia, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of nanobeams.     

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.     

Equivalence transformations of the Euler–Bernoulli equation

We give a determination of the equivalence group of the Euler–Bernoulli equation and of one of its generalizations, and thus derive some symmetry properties of this equation.     

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.     

Stabilization of a thermoelastic Mindlin–Timoshenko plate model revisited

This paper is a continuation of our work in Grobbelaar-Van Dalsen (Appl Anal 90:1419–1449, 2011) where we showed the strong stability of models involving the thermoelastic Mindlin–Timoshenko plate equations with second sound. For the case of a plate configuration consisting of a single plate, this was accomplished in radially symmetric domains without applying any mechanical damping mechanism. Further to this result, we establish in this paper the non-exponential stability of the model for a particular configuration under mixed boundary conditions on the shear angle variables and Dirichlet boundary conditions on the displacement and thermal variables when the heat flux is described by Fourier’s law of heat conduction. We also determine the rate of polynomial decay of weak solutions of the model in a radially symmetric region under Dirichlet boundary conditions on the displacement and thermal variables and free boundary conditions on the shear angle variables.     

Characterization of the Bernoulli–Navier model for a rectangular section beam as the limit of the Kirchhoff–Love model for a plate

In this paper, we compare the Kirchhoff–Love model for a linearly elastic rectangular plate \({\Omega^{t\varepsilon}=(0,L)\times(-t,t)\times(-\varepsilon,\varepsilon)}\) of thickness \({2\varepsilon}\) with the Bernoulli–Navier model for the same solid considered as a linearly elastic beam of length \({L}\) and cross section \({\omega_1^{t\varepsilon}=(-t,t)\times(-\varepsilon,\varepsilon)}\). We assume that the solid is clamped on both ends \({\{0,L\}\times[-t,t]\times[-\varepsilon,\varepsilon]}\). We show that the scaled version of the displacements field \({{\bf{\zeta}}^t}\) in the middle plane, solution of the Kirchhoff–Love model, converges strongly to the unique solution of a one-dimensional problem when the plate width parameter \({t}\) tends to zero. Moreover, after rescaling this limit, we show that, as a matter of fact, it is the solution of the Bernoulli–Navier model for the beam. This means that, under appropriate assumptions on the order of magnitude of the data, the Bernoulli–Navier displacement field is the natural approximation of the Kirchhoff–Love displacement field when the cross section of the plate is rectangular and its width is sufficiently small and homothetic to thickness.     

An Euler–Bernoulli Beam with Dynamic Frictionless Contact: Penalty Approximation and Existence

In this work, we consider the dynamic frictionless Euler–Bernoulli equation with the Signorini contact conditions along the length of a thin beam. The existence of solutions is proved based on the penalty method. Employing energy functional with the penalty method, we bound integral of contact forces over space and time. Hölder continuity of the fundamental solution plays an important role in the convergence theory.     

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.     

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.     

Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L ² norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.     

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.     

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.     

Asymptotic behavior of some elastic planar networks of Bernoulli–Euler beams

We study the energy decay of a network of elastic Bernoulli–Euler beams in star-shaped and tree-shaped network configurations. We show that the vibrations of star-shaped or tree-shaped network of beams are not exponentially stable in the energy space. So, we give explicit polynomial decay estimates valid for regular initial data. These estimates depend on the diophantine approximations properties.     

Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate

We study an initial–boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff & Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established for a quasilinear non-parabolic thermoelastic Kirchhoff & Love plate in multiple dimensions.     

Stability of a Kirchhoff–Roe scheme for two-dimensional linearized Euler systems

By applying Helmholtz decomposition, the unknowns of a linearized Euler system can be recast as solutions of uncoupled linear wave equations. Accordingly, the Kirchhoff expression of the exact solutions is recast as a time-marching, Lax–Wendroff type, numerical scheme for which consistency with one-dimensional upwinding is checked. This discretization, involving spherical means, is set up on a 2D uniform Cartesian grid, so that the resulting numerical fluxes can be shown to be conservative. Moreover, semi-discrete stability in the \(H^s\) norms and vorticity dissipation are established, along with practical second-order accuracy. Finally, some relations with former “shape functions” and “symmetric potential schemes” are highlighted.