The equilibrium position of an Euler-Bernoulli beam is investigated. Using the discrete Euler-Lagrange and Lagrange-d’Alembert principles, the behavior of the beam is simulated using variational integrators and, in particular, AVIs (Asynchronous Variational Integrators). Special emphasis is placed on the discrete mechanics and geometric structure underlying stress resultant beam models.

where J5m(x), Em(x) denote the polynomials of Bernoulli and Euler in the usual notation. It is perhaps not so familiar that (1.1) and (1.2) characterize the polynomials. More precisely, as Nielsen has pointed out [3, p. 54], if a normalized polynomial satisfies (1.1) for a single value k>l, then it is identical with Bm(x); similarly if a normalized polynomial satisfies (1.2) for a single odd k>...

In the present work, the nonlinear equation of motion of Bernoulli-Euler beam is used for investigating the scalability of the shooting method. A parallel implementation of the complete process of computing the nonlinear frequency-response function is presented and its efficiency is studied. The process involves algebraic operations of sparse and dense matrices and also solutions of linear and ...

Using an abstract condition of Riesz basis generation of discrete operators in the Hilbert spaces, we show, in this paper, that a sequence of generalized eigenfunctions of an Euler– Bernoulli beam equation with a tip mass under boundary linear feedback control forms a Riesz basis for the state Hilbert space. In the meanwhile, an asymptotic expression of eigenvalues and the exponential stability...

This paper describes a comprehensive nonlinear multiphysics model based on the Euler-Bernoulli beam equation that remains valid up to large displacements in the case of electrostatically actuated Mathieu resonators. This purely analytical model takes into account the fringing field effects and is used to track the periodic motions of the sensing parts in resonant microgyroscopes. Several parame...

Abstract: This work describes the modeling and control of an experimental platform associated to a single link flexible robot, whose motion is restricted to an horizontal plane, thus neglecting the gravity effects. The modeling problem is addressed using the so-called Euler-Bernoulli beam equation. The design, construction and integration of an experimental set-up developed for this work is als...

Outline of introduction: The practical use of these models is based on the fact that they can be kept realistic enough. Of course, this doesn’t mean to include every possible detail, but rather of every major mechanism. Bernoulli 1760 nonlinear ordinary differential equation. Effect of cowpox inoculation on the spread of smallpox. The initial plan is to discuss nonextended models in Section 1, ...

The linear theory of thermoelastic damping (TED) has been extensively developed over the past eight decades, but relatively little is known about the different types of nonlinearities that are associated with this fundamental mechanism of material damping. Here, we initiate the study of a dissipative nonlinearity (also called thermomechanical nonlinearity) whose origins reside at the heart of t...

This paper addresses the problem of deformation control of an Euler-Bernoulli beam with in-domain actuation. The proposed control scheme consists in first relating the system model described by an inhomogeneous partial differential equation to a target system under a standard boundary control form. Then, a combination of closed-loop feedback control and flatness-based motion planning is used fo...

Screened potential, modified by non standard electron cloud distributions responsible for the shielding effect on fusion of reacting nuclei in astrophysical plasmas, is derived. The case of clouds with depleted tails in space coordinates is discussed. The modified screened potential is obtained both from statistical mechanics arguments based on fluctuations of the inverse of the Debye-Hückel ra...