scholarly journals Convergence of Hencky-Type Discrete Beam Model to Euler Inextensible Elastica in Large Deformation: Rigorous Proof

2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jean-Jacques Alibert ◽  
Alessandro Della Corte ◽  
Pierre Seppecher
Keyword(s):  
Large Deformation ◽  
Beam Model ◽  
Rigorous Proof

Development of a Nonlinear, C1-Beam Finite-Element Code for Actuator Design of Slender Flexible Robots

2017 ◽  
Author(s):  
Hidenori Murakami ◽  
Oscar Rios ◽  
Takeyuki Ono
Keyword(s):  
Finite Element ◽  
Large Deformation ◽  
Cross Section ◽  
Beam Model ◽  
Finite Element Code ◽  
Shape Functions ◽  
Lagrange Equations ◽  
Element Discretization ◽  
Flexible Robots ◽  
Actuator Design

For actuator design and motion simulations of slender flexible robots, planar C1-beam elements are developed for Reissner’s large deformation, shear-deformable, curbed-beam model. Internal actuation is mechanically modeled by a rate-form of beam constitutive relation, where actuation curvature is prescribed at each time. Geometrically, a curbed beam is modeled as a frame bundle, whereby at each point on beam’s curve of centroids a moving orthonormal frame is attached to a cross section. After a finite element discretization, a curve of centroids is modeled as a C1-curve, employing cubic shape functions for both planar coordinates with an arc-parameter. The cubic shape functions have already been utilized in linear Euler-Bernoulli beams for the interpolation of transverse displacement. To define the rotation angle of each cross section or the attitude of the moving frame, quadratic shape functions are used introducing a middle node, resulting in three angular nodal displacements. As a result, each beam element has total eleven nodal coordinates. The implementation of a nonlinear finite element code is facilitated by the principle of virtual work, which yields Reissner’s large deformation curbed beam model as the Euler-Lagrange equations. For time integration, the Newmark method is utilized. Finally, as applications of the code, a few inchworm motions induced by different actuation curvature fields are presented.

Development of an Active Curved Beam Model Using a Moving Frame Method

2016 ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Large Deformation ◽  
Timoshenko Beam ◽  
Virtual Work ◽  
Three Dimensional ◽  
Moving Frame ◽  
Beam Model ◽  
Principle Of Virtual Work ◽  
Beam Equations ◽  
Moving Frame Method ◽  
Frame Method

In order to develop an active large-deformation beam model for slender, flexible or soft robots, the d’Alembert principle of virtual work is derived for three-dimensional elastic solids from Hamilton’s principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross-sections of a deforming beam. In the derivation of the beam model, Élie Cartan’s moving frame method is utilized. The resulting large-deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate-form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. Finally, to illustrate the deformation due to internal actuation, an active Timoshenko-beam model is derived by linearizing the nonlinear planar equations. For an active, simply-supported Timoshenko-beam, the analytical solution is presented.

Development of an Active Curved Beam Model—Part II: Kinetics and Internal Activation

2017 ◽  
Vol 84 (6) ◽  
Author(s):  
Hidenori Murakami
Keyword(s):  
Large Deformation ◽  
Timoshenko Beam ◽  
Virtual Work ◽  
Constitutive Relations ◽  
Differential Forms ◽  
Three Dimensional ◽  
Beam Model ◽  
Cauchy Stress ◽  
Principle Of Virtual Work ◽  
Beam Equations

Utilizing the kinematics, presented in the Part I, an active large deformation beam model for slender, flexible, or soft robots is developed from the d'Alembert's principle of virtual work, which is derived for three-dimensional elastic solids from Hamilton's principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross sections of a deforming beam. In the derivation of the beam model, Élie Cartan's moving frame method is utilized. The resulting large deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. To illustrate the deformation due to internal actuation, an active Timoshenko beam model is derived by linearizing the nonlinear planar equations. For an active, simply supported Timoshenko beam, the analytical solution is presented. Finally, a linear locomotion of a soft inchworm-inspired robot is simulated by implementing active C1 beam elements in a nonlinear finite element (FE) code.

Modeling and Experimentation of a Ribbed Caudal Fin for Aquatic Robots

2017 ◽  
Author(s):  
Oscar Rios ◽  
Ardavan Amini ◽  
Hidenori Murakami
Keyword(s):  
Large Deformation ◽  
Beam Theory ◽  
Fe Model ◽  
Beam Model ◽  
Caudal Fin ◽  
Nonlinear Beam ◽  
Beam Elements ◽  
Side Walls ◽  
Thin Beams ◽  
Shear Deformable

Presented in this study is a mathematical model and preliminary experimental results of a ribbed caudal fin to be used in an aquatic robot. The ribbed caudal fin is comprised of two thin beams separated by ribbed sectionals as it tapers towards the fin. By oscillating the ribbed caudal fin, the aquatic robot can achieve forward propulsion and maneuver around its environment. The fully enclosed system allows for the aquatic robot to have very little effect on marine life and fully blend into its respective environment. Because of these advantages, there are many applications including surveillance, sensing, and detection. Because the caudal fin actuator has very thin side walls, Kirchhoff-Love’s large deformation beam theory is applicable for the large deformation of the fish-fin actuator. In the model, it is critical to accurately model the curvature of beams. To this end, C1 beam elements for thin beams are developed by specializing the shear-deformable beam elements, developed by the authors, based upon Reissner’s shear-deformable nonlinear beam model. Furthermore, preliminary experiments on the ribbed fin are presented to supplement the FE model.

Coupled Deformation Modes in the Large Deformation Finite-Element Analysis: Problem Definition

2006 ◽  
Vol 2 (2) ◽  
pp. 146-154 ◽  
Author(s):  
Bassam A. Hussein ◽  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Large Deformation ◽  
Cross Section ◽  
Flexible Structures ◽  
Beam Theory ◽  
Problem Definition ◽  
Beam Model ◽  
Element Analysis ◽  
Elastic Line ◽  
The Cross ◽  
Deformation Modes

In the classical formulations of beam problems, the beam cross section is assumed to remain rigid when the beam deforms. In Euler–Bernoulli beam theory, the rigid cross section remains perpendicular to the beam centerline; while in the more general Timoshenko beam theory the rigid cross section is permitted to rotate due to the shear deformation, and as a result, the cross section can have an arbitrary rotation with respect to the beam centerline. In more general beam models as the ones based on the absolute nodal coordinate formulation (ANCF), the cross section is allowed to deform and it is no longer treated as a rigid surface. These more general models lead to new geometric terms that do not appear in the classical formulations of beams. Some of these geometric terms are the result of the coupling between the deformation of the cross section and other modes of deformations such as bending and they lead to a new set of modes referred to in this paper as the ANCF-coupled deformation modes. The effect of the ANCF-coupled deformation modes can be significant in the case of very flexible structures. In this investigation, three different large deformation dynamic beam models are discussed and compared in order to investigate the effect of the ANCF-coupled deformation modes. The three methods differ in the way the beam elastic forces are calculated. The first method is based on a general continuum mechanics approach that leads to a model that includes the ANCF-coupled deformation modes; while the second method is based on the elastic line approach that systematically eliminates these modes. The ANCF-coupled deformation modes eliminated in the elastic line approach are identified and the effect of such deformation modes on the efficiency and accuracy of the numerical solution is discussed. The third large deformation beam model discussed in this investigation is based on the Hellinger–Reissner principle that can be used to eliminate the shear locking encountered in some beam models. Numerical examples are presented in order to demonstrate the use and compare the results of the three different beam formulations. It is shown that while the effect of the ANCF-coupled deformation modes is not significant in very stiff and moderately stiff structures, the effect of these modes can not be neglected in the case of very flexible structures.

Comments on the large deformation elastic beam model developed by D.Y. Gao

2020 ◽  
Vol 110 ◽  
pp. 103607 ◽  
Author(s):  
Jitka Machalová ◽  
Horymír Netuka
Keyword(s):  
Large Deformation ◽  
Elastic Beam ◽  
Beam Model

A new Timoshenko beam model for simulation of mechanical system with large deformation

2016 ◽  
Vol 59 (11) ◽  
pp. 1639-1645
Author(s):  
Guan Zhou ◽  
Ren Hui ◽  
WanZhong Zhao ◽  
ChunYan Wang ◽  
Bing Xu
Keyword(s):  
Mechanical System ◽  
Large Deformation ◽  
Timoshenko Beam ◽  
Beam Model ◽  
Timoshenko Beam Model
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