Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0410286

Three Dimensional Nonlinear Dynamics of Slender Structures: Cosserat Rod Element Approach

D. Q. Cao·Dongsheng Liu·Charles H. -T. Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, the modelling strategy of a Cosserat rod element (CRE) is addressed systematically for 3-dimensional dynamical analysis of slender structures. We employ the exact nonlinear kinematic relationships in the sense of Cosserat theory, and adopt the Bernoulli hypothesis. For the sake of simplicity, the Kirchoff constitutive relations are adopted to provide an adequate description of elastic properties in terms of a few elastic moduli. A deformed configuration of the rod is described by the displacement vector of the deformed centroid curves and an orthonormal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parametrized by a rotational vector. The approximate solutions of the nonlinear partial differential equations of motion in quasi-static sense are chosen as the shape functions with up to third order nonlinear terms of generic nodal displacements. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A simple example is presented to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures have been presented through numerical simulations by Matlab software.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

D. Q. Cao, Dongsheng Liu, Charles H. -T. Wang. 2004-10-12. Three Dimensional Nonlinear Dynamics of Slender Structures: Cosserat Rod Element Approach. https://arxiv.org/abs/math/0410286

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

O-segments on topological measure spaces

Let $μ$ be a finite non-atomic measure on a set $X$. A \emph{segment} for $μ$ is an increasing family $u(t)$, $t\in[0,1]$, of measurable sets with $u(0)=\emptyset$, $u(1)=X$ and $μ(u(t))=t\,μ(X)$; by a theorem of Fryszkowski, finitely many measures $f\,dμ$ admit a common segment. We study \emph{open segments}, i.e.\ segments consisting of open sets, when $X$ carries a topology. We show that weak outer regularity of $μ$ is equivalent to the approximability of measurable sets by open sets, and that it holds for every finite Borel measure on a metrizable space. Our main result states that if $μ$ is finite, non-atomic and weakly outer regular, then for all $f,g\in L^1(μ)$ with $(\int f\,dμ,\int g\,dμ)\neq(0,0)$ there is a common open segment for $f\,dμ$ and $g\,dμ$; in particular, for every $f\in L^1(μ)$ there is an open segment for $μ$ which is simultaneously a segment for $f\,dμ$. Relative versions allow one to prescribe an initial set. We also characterize open segments as upper semicontinuous functions with uniform distribution and show that on separable metrizable spaces a single measure admits such a function which is continuous.

math.FA↗
arXiv2026

A Note On Majorizing Norms on Majorized Spaces

We introduce a new type of norm for ordered vector spaces majorized by a proper (convex) cone that generalizes the notions of order unit norm and base norm. Then we give sufficient conditions to ensure its completeness. In the case of normed Riesz spaces, we also present necessary completeness conditions and describe the completed norm. Finally, we define a particular family of well-behaving sets. Spaces majorized by the cone generated by these sets display surprisingly interesting structural properties. In particular, we generalize two famous standard results in Banach lattice theory. Namely, every principal ideal in a Banach lattice is an AM-space and every Banach lattice having a generating cone with a compact base admits an equivalent L-norm.

math.FA↗
arXiv2026

The Cesàro Operator, Rhaly operators, and the Toeplitz Algebra

Barr\'ıa and Halmos [\emph{Trans. Amer. Math. Soc.} (1982), (10) Example] asked whether the Cesàro operator belongs to the Toeplitz algebra. We answer this question affirmatively and prove that the Cesàro operator is the product of a Toeplitz operator and a continuous function of a self-adjoint Toeplitz operator. Rhaly operators are a natural generalization of the Cesàro operator. We give necessary and sufficient conditions for a Rhaly operator to belong to the Toeplitz algebra or the Hankel algebra. The main approach is to use the Fejér kernels to get localized estimates for operators. This yields a necessary condition for an operator to belong to the Hankel algebra. As an application, we find a class of Rhaly operators that are asymptotic Toeplitz operators and essentially commute with the unilateral shift, but are not in the Hankel algebra. This answers (17) Question in the same paper.

math.FA↗