Nonlinear Elastic Beam Research Articles

Analysis and simulations of a nonlinear elastic dynamic beam

A model for the dynamics of a Gao elastic nonlinear beam, which is subject to a horizontal traction at one end, is studied. In particular, the buckling behavior of the beam is investigated. Existence and uniqueness of the local weak solution is established using truncation, approximations, a priori estimates, and results for evolution problems. An explicit finite differences numerical algorithm for the problem is presented. Results of representative simulations are depicted in the cases when the oscillations are about a buckled state, and when the horizontal traction oscillates between compression and tension. The numerical results exhibit a buckling behavior with a complicated dependence on the amplitude and frequency of oscillating horizontal tractions.

Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end

This work studies large deflections of slen- der, non-prismatic cantilever beams subjected to a combined loading which consists of a non-uniformly distributed con- tinuous load and a concentrated load at the free end of the beam. The material of the cantilever is assumed to be non- linearly elastic. Different nonlinear relations between stress and strain in tensile and compressive domain are considered. The accuracy of numerical solutions is evaluated by com- paring them with results from previous studies and with a laboratory experiment.

Nonlinear Modeling of the Dynamical Behavior of the Three-Dimensional Elastic Beam

AbstractThe paper outlines a dynamic model of the nonlinear elastic beam. The investigated model in this paper is based on a linear model and considers additional relevant geometrical nonlinear couplings. The developed model considers terms up to the 3rd order. Furthermore, complex couplings between the guided motion of the beam in combination with external forces and the geometrical nonlinear relations are considered. A finite element method is used for discretization. The complex behavior is shown using known benchmark as well as with new complex simulation examples. Simulations of the flexible beam are used to show the coupling effects between the rotating motion including external effects and the axial, transverse deformations with geometrical nonlinear terms.

Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity

We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions $$\left\{ \begin{gathered} u^{(4)} (t) = f(t,u(t),u'(t),u''(t),u'''(t)),a.e.t \in [0,1], \hfill u(0) = a,u'(0) = b,u(1) = c,u''(1) = d, \hfill \end{gathered} \right. $$ where the nonlinear term f(t, u0, u1, u2, u3) is a strong Caratheodory function. By constructing suitable height functions of the nonlinear term f(t, u0, u1, u2, u3) on bounded sets and applying the Leray-Schauder fixed point theorem, we prove that the equation has a solution provided that the integration of some height function has an appropriate value.

Non-trivial solutions for nonlinear fourth-order elastic beam equations

The existence of at least one non-trivial solution to a boundary value problem for fourth-order elastic beam equations, under a non-standard growth condition of the nonlinear term, is established. Our approach is based on a local minimum theorem for differentiable functionals.

Undamped nonlinear beam excited by additive [formula omitted]-regular noise

Stability of a nonlinear elastic beam excited by a space-time dependent nonrandom force or a L 2 -regular random noise f ( x , t ) (white in time, with general spatial covariance) is considered. It is supposed that the transversal displacement y of the beam with length l is governed by the quasilinear partial differential equation (PDE) ρ 0 h y t t + D y x x x x − [ N 0 + ε ∫ 0 l ( y x ′ ) 2 d x ′ ] y x x + k y = f ( x , t ) , 0 < x < l , t ≥ 0 . As a main result we show that its expected energy is linearly bounded at time t in spite of the presence of additive L 2 -regular space-time noise and cubic-type nonlinearities. Appropriate partial-implicit discretization with similar qualitative behavior, consistency and stability are discussed as well. The technique of Lyapunov-type functionals processing the information on the related energy E is exploited. We compare our results to those for the linear beam. In both cases, the expected energy E [ E ] ( t ) is nondecreasing and growing linearly in time t . In fact, for all nonrandom t ≥ s and additive L 2 -regular noise f with covariance operator Q ( f ) with finite trace, it satisfies the trace formula E [ E ] ( t ) = E [ E ] ( s ) + Trace ( Q ( f ) ) ( t − s ) / 2 .

A formal method for improving the transient behaviour of a non-linear flexible link

This article presents a procedure for improving the transient response of a boundary-controlled non-linear flexible elastic beam through a feed-forward action. It is shown that it is possible to obtain a formal analytic solution of an infinite-dimensional linear system, which approximates the non-linear dynamics, under time-varying boundary conditions in terms of the free vibration modes of the structure, whose calculation is a relatively easy task. The feed-forward action is then obtained by inverting the relationship between input and output variables. A numerical example is presented and some simulation results are discussed.

On the Dirichlet problem for a nonlinear elastic beam equation

Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation.

A sum operator equation and applications to nonlinear elastic beam equations and Lane–Emden–Fowler equations

This paper is concerned with an operator equation A x + B x + C x = x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane–Emden–Fowler equations.

Assessment of two analytical methods in solving the linear and nonlinear elastic beam deformation problems

PurposeIn the last two decades with the rapid development of nonlinear science, there has appeared ever‐increasing interest of scientists and engineers in the analytical techniques for nonlinear problems. This paper considers linear and nonlinear systems that are not only regarded as general boundary value problems, but also are used as mathematical models in viscoelastic and inelastic flows. The purpose of this paper is to present the application of the homotopy‐perturbation method (HPM) and variational iteration method (VIM) to solve some boundary value problems in structural engineering and fluid mechanics.Design/methodology/approachTwo new but powerful analytical methods, namely, He's VIM and HPM, are introduced to solve some boundary value problems in structural engineering and fluid mechanics.FindingsAnalytical solutions often fit under classical perturbation methods. However, as with other analytical techniques, certain limitations restrict the wide application of perturbation methods, most important of which is the dependence of these methods on the existence of a small parameter in the equation. Disappointingly, the majority of nonlinear problems have no small parameter at all. Furthermore, the approximate solutions solved by the perturbation methods are valid, in most cases, only for the small values of the parameters. In the present study, two powerful analytical methods HPM and VIM have been employed to solve the linear and nonlinear elastic beam deformation problems. The results reveal that these new methods are very effective and simple and do not require a large computer memory and can also be used for solving linear and nonlinear boundary value problems.Originality/valueThe results revealed that the VIM and HPM are remarkably effective for solving boundary value problems. These methods are very promoting methods which can be wildly utilized for solving mathematical and engineering problems.

Improved Decentralized Method for Control of Building Structures under Seismic Excitation

A decentralized control method with improved robustness and design flexibility is proposed for reducing vibrations of seismically excited building structures. In a previous study, a control scheme was developed for multistory building models using nonlinear, decentralized control theory. This control method has now been improved in this study in that less information about material properties and geometrical parameters of the building is needed and the selection of control design parameters is more flexible. The nonlinear behavior of the proposed control system is studied and its stability property is proven mathematically. To evaluate the effectiveness and robustness of the proposed method, three illustrative structural models, i.e., an eight-story elastic shear beam model, a two-story nonlinear elastic shear beam model, and a 20-story elastic benchmark model are studied. The 1940 El Centro and the 1995 Kobe earthquakes are used in these examples. The performance of the current control design, as applied to these examples, has shown to be more effective in reducing structural responses and improving robustness.

Necessary and sufficient condition for the existence of positive solutions of a coupled system for elastic beam equations

In this work, by using fixed point theorem and monotone iterative technique, the author establishes a necessary and sufficient condition of the existence of positive solutions for a class of nonlinear singular elastic beam differential system under reasonable conditions. Moreover, some uniqueness results of positive solution and the estimate of convergent rate of the iterative sequence of solution are obtained.

Large deflections of nonlinearly elastic non-prismatic cantilever beams made from materials obeying the generalized Ludwick constitutive law

This paper studies large deflections of nonlinearly elastic cantilever beams made from materials obeying the generalized Ludwick constitutive law. An exact moment-curvature formula which can be applied to study arbitrarily loaded and supported beams of rectangular cross-sections is developed. Several advantages of the generalized Ludwick’s model are illustrated. Numerical examples considered in this materially and geometrically nonlinear analysis clearly indicate rich nonlinear behavior of the beams.

Stabilization of the dynamic elastica only by damping torque

We consider a system of partial differential equations called the dynamic elastica, which describes the motion of a geometrically nonlinear (i.e., largely deflecting) elastic beam and is derived based on Euler–Bernoulli’s assumption. The aim of the paper is to examine the effect of damping torque (i.e., external torque generated as a negative feedback of the angular velocity of the beam’s centerline) on the stability of the elastica.

On the optimal control problem governed by the nonlinear elastic beam equation

We begin with using a dual variational method in proving the existence of solutions to a beam equation with Dirichlet boundary conditions and with a non-monotone nonlinear term. Later we investigate the dependence on a control parameter for the state equation. We find the optimal process for the optimal control problem in which the dynamics is governed by the nonlinear beam equation.

On the nonlinear elastic beam equation

We consider the existence of solutions and the continuous dependence on a parameter results for fourth-order Dirichlet problem connected with the elastic beam equation. We introduce a dual variational method.

Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right

The purpose of this paper is to establish several local existence theorems concerned with n positive solutions for a fourth-order two-point boundary value problem, where n is an arbitrary positive integer and the nonlinear term may be singular. In mechanics, the problem describes deflection of an elastic beam rigidly fastened on the left and simply supported on the right.

Non-prismatic Non-linearly Elastic Cantilever Beams Subjected to an End Moment

This paper analyzes large deflection profiles of slender, inextensible cantilever beams of prismatic and non-prismatic longitudinal shapes with rectangular cross-sections subjected to a concentrated moment at the free end. The stress—strain relationship of the material is represented by the Ludwick constitutive law. Different non-linear stress—strain relations in tensile and compressive domain are considered. The main purpose of this paper is to investigate the influence of geometrical and material non-linearities on the shape of the deflection curve. The solution of a strongly non-linear set of equations is obtained numerically. In the case when non-linear stress—strain relationship in tensile and compressive domain is identical, an analytical solution is given in terms of infinite series. Several examples for a variety of different shapes of beams are presented considering both linearly and non-linearly elastic materials in the elastic domain.

Dynamic lateral torsional post-buckling of a beam–mass system: Theory

In this paper, the theory of a Cosserat point is used as a numerical model of a nonlinear elastic beam–mass system and simulations of dynamic lateral torsional buckling are compared with results of experiments. The Cosserat equations are solved using the Newmark time-integration scheme and an analytical expression for the tangent stiffness of the resulting Newton–Raphson iteration procedure is developed. Also, the effects of material damping, aerodynamic drag and gravity have been included. The simulations reproduce the experimental result that two different nonlinear modes of vibration occur at the same excitation amplitude and frequency. One mode: Torsion I is a post-buckling mode associated with out-of-plane motion of the beam–mass system, which is dominated by oscillating torsion of the beam. The second mode: Bending II is dominated by a nonlinear second bending mode in the weak bending plane. The simulations of Torsion I and Bending II and another mode Torsion II are in reasonably good quantitative agreement with the experimental results. However, the model is not able to accurately simulate the response of Torsion III (which was similar to Torsion I and II but with a larger amplitude of vibration). Comparison is also made with the commercial code ANSYS.

Design sensitivity analysis of multi-span non-linear elastic beam–columns with elastic supports

The incremental problem arising from the design perturbation of a materially non-linear elastic beam–column with intermediate supports and elastic end supports is formulated via a second variation of the potential energy functional, augmented with constraint conditions. Design variables are distributed parameters (bending stiffness and transverse load), discrete (compression force, end points dead loads and elasticity of supports) and domain (length and intermediate support points). The state variables perturbations are the solution of an incremental BVP (equilibrium differential equation, static, kinematic and jump conditions) derived from an incremental virtual work equation. A sensitivity formula is obtained by the adjoint method.