Lie Group Modeling of Nonlinear Helical Beam Elements

2014 ◽  
Author(s):  
Ralph Jödicke ◽  
Uwe Jungnickel ◽  
Andreas Müller
Keyword(s):  
Lie Group ◽  
Beam Model ◽  
Deflection Curve ◽  
Finite Frames ◽  
Beam Elements ◽  
Group Modeling ◽  
Stiffness Matrices ◽  
Torsion Energy ◽  
Shearing Energy ◽  
Helical Beam

A viscoelastic beam model is presented based on SE(3) group theory. We discretize a rod with beams between finite frames on the rod and regard the configurations of these frames as elements of the SE(3) Lie group. Two subsequent frames are connected by a beam. The curvatures and strains are assumed to be constant on the trajectory between them. If the deflection curve of the beam is modeled as a helix, the resulting beam model is geometrical exact for large bending deformations. The stiffness matrices of the discrete beam elements result from the potential extensional and shearing energy as well as from the potential bending and torsion energy. The benefit of this SE(3) modeling for translational elastic coordinates and for translational forces in comparison to an SO(3) × ℝ3 variant is demonstrated.

scholarly journals Solution of Contact Problems for Nonlinear Gao Beam and Obstacle

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
J. Machalová ◽  
H. Netuka
Keyword(s):  
Solution Method ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Contact Problems ◽  
Beam Model ◽  
Contact Conditions ◽  
Beam Elements ◽  
Solution Methods ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

Contact problem for a large deformed beam with an elastic obstacle is formulated, analyzed, and numerically solved. The beam model is governed by a nonlinear fourth-order differential equation developed by Gao, while the obstacle is considered as the elastic foundation of Winkler’s type in some distance under the beam. The problem is static without a friction and modeled either using Signorini conditions or by means of normal compliance contact conditions. The problems are then reformulated as optimal control problems which is useful both for theoretical aspects and for solution methods. Discretization is based on using the mixed finite element method with independent discretization and interpolations for foundation and beam elements. Numerical examples demonstrate usefulness of the presented solution method. Results for the nonlinear Gao beam are compared with results for the classical Euler-Bernoulli beam model.

scholarly journals BEAM ELEMENT UNDER FINITE ROTATIONS

2021 ◽  
Vol 30 ◽  
pp. 87-92
Author(s):  
Emma La Malfa Ribolla ◽  
Milan Jirásek ◽  
Martin Horák
Keyword(s):  
Cross Sections ◽  
Shooting Method ◽  
Small Strain ◽  
Beam Model ◽  
Equilibrium Equations ◽  
Finite Rotations ◽  
Nonlinear Beam ◽  
Linear Elastic ◽  
Beam Elements ◽  
Large Rotations

The present work focuses on the 2-D formulation of a nonlinear beam model for slender structures that can exhibit large rotations of the cross sections while remaining in the small-strain regime. Bernoulli-Euler hypothesis that plane sections remain plane and perpendicular to the deformed beam centerline is combined with a linear elastic stress-strain law.The formulation is based on the integrated form of equilibrium equations and leads to a set of three first-order differential equations for the displacements and rotation, which are numerically integrated using a special version of the shooting method. The element has been implemented into an open-source finite element code to ease computations involving more complex structures. Numerical examples show a favorable comparison with standard beam elements formulated in the finite-strain framework and with analytical solutions.

scholarly journals A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model

2021 ◽  
Author(s):  
Stefan Hante ◽  
Denise Tumiotto ◽  
Martin Arnold
Keyword(s):  
Lie Group ◽  
Group Structure ◽  
Equations Of Motion ◽  
Second Order ◽  
The Body ◽  
Benchmark Problems ◽  
Beam Model ◽  
Step Size ◽  
Time Step ◽  
Time Step Size

AbstractIn this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

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.

Output Displacement Analysis of Symmetric Four-Bar Mechanism with Right Angle Flexible Hinge

2013 ◽  
Vol 798-799 ◽  
pp. 321-324
Author(s):  
Lei He ◽  
Chuan Yi Lv ◽  
Xian Hai Yang
Keyword(s):  
Structural Characteristics ◽  
Beam Deflection ◽  
Beam Model ◽  
Deflection Curve ◽  
Element Analysis ◽  
Output Displacement ◽  
The Finite Element Analysis ◽  
Beam Output ◽  
The Right ◽  
Curve Equation

in order to analyze the guide beam output displacement of the right-angle flexure hinge symmetrical four-bar mechanism, a half beam model was established based on the special structural characteristics and deformation characteristics. The formula of guiding displace were derived using approximate beam deflection curve equation. The finite element analysis of the model was carried out by ANSYS, it analyzed the cause of the theoretical value and simulation value for different reasons.

A NUMERICAL INVESTIGATION ON THE FREE VIBRATION OF CARBON NANOPEAPODS AS VARIABLE FREQUENCY BEAM RESONATORS

2013 ◽  
Vol 27 (21) ◽  
pp. 1350147
Author(s):  
R. D. FIROUZ-ABADI ◽  
S. M. ALAVI
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Structural Mechanics ◽  
Free Vibrations ◽  
Variable Frequency ◽  
Beam Model ◽  
Beam Resonator ◽  
Beam Elements ◽  
Mechanics Model ◽  
Resonance Frequencies

This study aims at investigation of the resonance frequencies of carbon nanopeapods constructed by a single wall carbon nanotube and encapsulated buckyball molecules ( C 60). A nanopeapod can be used as a nanoscale variable frequency beam resonator according to the number and positions of the encapsulated fullerenes. Using the molecular structural mechanics method the covalence bonds are simulated by equivalent beam elements and the van der Waal interactions between the buckyballs and nanotube are modeled as linear springs. Also, an equivalent beam model is proposed for the nanopeapod with sectional properties which are obtained by the molecular structural mechanics model. The beam-like modes of free vibrations are obtained using both models and the effect of position and number of buckyballs on the resonance frequencies are investigated.

Vector Form Intrinsic Finite Element Method for Analysis of Train–Bridge Interaction Problems Considering The Coach-Coupler Effect

2019 ◽  
Vol 19 (02) ◽  
pp. 1950014 ◽  
Author(s):  
Y. F. Duan ◽  
S. M. Wang ◽  
J. D. Yau
Keyword(s):  
Finite Element ◽  
Resonance Phenomenon ◽  
Vector Form ◽  
Beam Elements ◽  
Suspension Systems ◽  
Stiffness Matrices ◽  
Internal Forces ◽  
Dual Resonance ◽  
Multi Body ◽  
Intense Vibration

In this paper, the vector form intrinsic finite element (VFIFE) method is presented for analysis the train–bridge systems considering the coach-coupler effect. The bridge is discretized into a group of mass particles linked by massless beam elements and the multi-body coach with suspension systems is simulated as a set of mass particles connected by parallel spring-dashpot units. Then the equation of motion of each mass particle is solved individually and the internal forces induced by pure deformations in the massless beam elements are calculated by a fictitious reverse motion method, in which the structural stiffness matrices need not be updated or factorized. Though the vector-form equations resulting from the VFIFE method cannot be used to compute the structural frequencies by the eigenvalue approach, this study proposes a numerical free vibration test to identify the bridge frequencies for evaluating the bridge damping. Numerical verifications demonstrate that the present VFIFE method performs as accurately as previous numerical ones. The results show that the couplers play an energy-dissipating role in reducing the car bodies’ response due to the bridge-induced resonance, but not in their response due to the train-induced resonance because of the bridge’s intense vibration. Meanwhile, a dual-resonance phenomenon in the train–bridge system occurs when the coach-coupler effect is considered in the vehicle model.

Vibration analysis of plane frames by customized stiffness and diagonal mass matrices

2011 ◽  
Vol 225 (12) ◽  
pp. 2848-2863 ◽  
Author(s):  
M Rezaiee-Pajand ◽  
R Khajavi
Keyword(s):  
Vibration Analysis ◽  
Simple Structure ◽  
Frame Structures ◽  
Numerical Examples ◽  
Beam Elements ◽  
Stiffness Matrices ◽  
Mass Matrices ◽  
Plane Frames ◽  
Accurate Performance ◽  
Vibration Problems

This article presents a formulation for the vibration analysis of plane frames. The strain gradient notation is utilized to determine the mass and stiffness matrices. The obtained matrices can easily be parameterized due to their simple structure. Both Euler-Bernoulli- and Timoshenko-beam elements are investigated in this study. The parameterized stiffness and mass matrices are optimized for accurate performance in the vibration analysis of frame structures. Some numerical examples are solved to show the advantages of the presented scheme. Results of these sample vibration problems indicate that the proposed technique increases the accuracy of analysis, when these new stiffness and diagonal mass matrices are used.

scholarly journals Assessment of RC elements strengthened with NSM FRP rods by experimental tests

2020 ◽  
Vol 323 ◽  
pp. 01008
Author(s):  
Roberto Capozucca ◽  
Erica Magagnini
Keyword(s):  
Experimental Tests ◽  
Concrete Cover ◽  
Beam Model ◽  
Rc Beams ◽  
Near Surface ◽  
Beam Elements ◽  
Damage Degree ◽  
Reinforced Polymer ◽  
Near Surface Mounted ◽  
Vibration Tests

Near surface mounted (NSM) technique of strengthening with FRP rods inserted in grooves on the concrete cover of damaged RC beams has been improved in recent years. The aim of this paper is the examination of the static and dynamic behaviour of undamaged and damaged reinforced concrete (RC) beams with free-free ends. RC beams strengthened with NSM Glass and Carbon fiber reinforced polymer (G-CFRP) rods have been experimentally analysed. The damage of the RC beam model was obtained by the cracking of concrete under bending tests. The detection of damage and monitoring of RC beams with and without strengthening were carried out by vibration tests assuming free-free ends at different degree of damage. Envelope diagrams of Frequency Response Functions (FRFs) obtained by the dynamic experimental tests are shown and the changes of natural frequency values are correlated to the damage degree of beam elements. Experimental results are discussed with particular emphasis on the aspect of the loss of bond.

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