On the geometrically exact beam model: A consistent, effective and simple derivation from three-dimensional finite-elasticity

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

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A Numerical Solution for Modelling Mooring Dynamics, Including Bending and Shearing Effects, Using a Geometrically Exact Beam Model

Journal of Marine Science and Engineering ◽

10.3390/jmse9050486 ◽

2021 ◽

Vol 9 (5) ◽

pp. 486

Author(s):

Tobias Martin ◽

Hans Bihs

Keyword(s):

Numerical Solution ◽

Offshore Wind ◽

Mooring Line ◽

Beam Model ◽

Tight Coupling ◽

Energy Propagation ◽

Two Phase ◽

Geometrically Exact Beam ◽

Mooring Dynamics ◽

Rotational Deformation

During the operation of moored, floating devices in the renewable energy sector, the tight coupling between the mooring system and floater motion results in snap load conditions. Before snap events occur, the mooring line is typically slack. Here, the mechanism of energy propagation changes from axial to bending dominant, and the correct modelling of the rotational deformation of the lines becomes important. In this paper, a new numerical solution for modelling the mooring dynamics that includes bending and shearing effects is proposed for this purpose. The approach is based on a geometrically exact beam model and quaternion representations for the rotational deformations. Further, the model is coupled to a two-phase numerical wave tank to simulate the motion of a moored, floating offshore wind platform in waves. A good agreement between the proposed numerical model and reference solutions was found. The influence of the bending stiffness on the motion of the structure was studied subsequently. We found that increased stiffness increased the amplitudes of the heave and surge motion, whereas the motion frequencies were less altered.

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A Fast Three-Dimensional Model for Strip Rolling

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.716.566 ◽

2016 ◽

Vol 716 ◽

pp. 566-578 ◽

Cited By ~ 1

Author(s):

Christian Overhagen ◽

Paul Josef Mauk

Keyword(s):

Finite Element ◽

Stress Gradient ◽

Three Dimensional ◽

Significant Loss ◽

Three Dimensions ◽

Beam Model ◽

Stress Distributions ◽

Three Dimensional Model ◽

Strip Rolling ◽

Lateral Direction

Rolling Models have come a long way from the first empirical relations about forward slip and bite conditions to their current state, which allows local quantities to be calculated in two and three dimensions. In this paper, state-of-the-art of analytical modelling of the rolling process is shown with a fully three-dimensional rolling model for hot and cold strip rolling with stress distributions in the longitudinal, vertical and lateral directions. For this purpose, von Karman’s strip approach is extended to account for the stress gradient in lateral direction, as was already shown in different papers. The stress gradient in the vertical (through-thickness) direction is introduced by a modern implementation of Orowan’s inhomogeneous deformation theory. The local stress distributions are compared to results from Finite-Element Calculations obtained with modern FEM codes. It will be shown, under which circumstances expensive FEM calculations can be replaced by simpler models like the one proposed here, which are more time and cost-effective without a significant loss in result precision. The rolling model is extended with a Finite Element Beam Model for work and backup roll deformation, as well as local work roll flattening and thermal crown for hot rolling. The Effects of those features on stress distribution and exit strip profile are shown for hot and cold rolling.

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Inclusion of Local Shell Behavior of Tubes Into a Two-Dimensional Beam Approximation of Deformation in Nuclear Fuel Channels

Computational Mechanics: Developments and Applications ◽

10.1115/pvp2002-1287 ◽

2002 ◽

Author(s):

S. Khajehpour ◽

R. G. Sauve´ ◽

N. Badie

Keyword(s):

Finite Element ◽

Contact Stiffness ◽

Three Dimensional ◽

Local Deformation ◽

Beam Model ◽

Two Dimensional ◽

Dimensional Modeling ◽

Dimensional Finite Element ◽

Nonlinear Force ◽

Three Dimensional Finite Element

A method has been developed to incorporate the local three-dimensional shell behavior of two concentric tubes in the two-dimensional beam modeling of the problem. The two dimensional modeling of fuel channels in CANDU pressurized heavy water nuclear reactors is used in lieu of a more accurate three dimensional finite element approach in order to reduce the on-line simulation time which greatly affects the SLAR (Spacer Location And Repositioning) maintenance operation cost during outage. However, effort must be made to include the three-dimensional shell behavior of these channels into the two-dimensional modeling. In recent studies a nonlinear force-dependent model for contact stiffness between the calandria tube and pressure tube has been developed. However, local deformation of calandria the tube at spacer locations due to in-reactor creep leads to settling of the spacer into the calandria tube that consequently reduces the gap between the two tubes. In this work, the effect of local deformation (elastic and creep) of calandria tubes on modeling of contact at spacer locations is assessed using a three dimensional finite element code. The result is incorporated into a two-dimensional beam model of the problem as a reduction in size of the spacers that separate the two tubes. It is shown that the proposed method increases the accuracy of prediction of contact time and the spacer. In general, the method described in this paper suggests a way to incorporate local shell deformation into beam models of slender shell structure.

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LOOPS, SURFACES AND GRASSMANN REPRESENTATION IN TWO- AND THREE-DIMENSIONAL ISING MODELS

International Journal of Modern Physics A ◽

10.1142/s0217751x9900213x ◽

1999 ◽

Vol 14 (29) ◽

pp. 4549-4574 ◽

Cited By ~ 3

Author(s):

C. R. GATTRINGER ◽

S. JAIMUNGAL ◽

G. W. SEMENOFF

Keyword(s):

Three Dimensional ◽

Ising Models ◽

Algebraic Representation ◽

Simple Derivation ◽

Counting Problems ◽

Intrinsic Geometry ◽

Geometry Factor ◽

Loop Expansion ◽

Loop Surface

We construct an algebraic representation of the geometrical objects (loop and surface variables) dual to the spins in 2 and 3D Ising models. This algebraic calculus is simpler than dealing with the geometrical objects, in particular when analyzing geometry factors and counting problems. For the 2D case we give the corrected loop expansion of the free energy and the radius of convergence for this series. For the 3D case we give a simple derivation of the geometry factor which prevents overcounting of surfaces in the intrinsic geometry representation of the partition function, and find a classification of the surfaces to be summed over. For 2 and 3D we derive a compact formula for 2n-point functions in loop (surface) representation.

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Modeling of three-dimensional beam nonlinear vibrations generalizing Hencky’s ideas

Mathematics and Mechanics of Solids ◽

10.1177/10812865211067987 ◽

2022 ◽

pp. 108128652110679

Author(s):

Emilio Turco

Keyword(s):

Kinetic Energy ◽

Strain Energy ◽

Nonlinear Vibrations ◽

Three Dimensional ◽

Integration Scheme ◽

Beam Model ◽

Model Formulation ◽

Discrete Approach ◽

Numerical Integration Scheme ◽

Proposed Model

In this contribution, a novel nonlinear micropolar beam model suitable for metamaterials design in a dynamics framework is presented and discussed. The beam model is formulated following a completely discrete approach and it is fully defined by its Lagrangian, i.e., by the kinetic energy and by the potential of conservative forces. Differently from Hencky’s seminal work, which considers only flexibility to compute the buckling load for rectilinear and planar Euler–Bernoulli beams, the proposed model is fully three-dimensional and considers both the extensional and shear deformability contributions to the strain energy and translational and rotational kinetic energy terms. After having introduced the model formulation, some simulations obtained with a numerical integration scheme are presented to show the capabilities of the proposed beam model.

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Stability analysis for a single-point cutting tool deflection in turning operation

Advances in Mechanical Engineering ◽

10.1177/1687814019853188 ◽

2019 ◽

Vol 11 (6) ◽

pp. 168781401985318

Author(s):

Amon Gasagara ◽

Wuyin Jin ◽

Angelique Uwimbabazi

Keyword(s):

Cutting Forces ◽

High Speed ◽

Cutting Tool ◽

Single Point ◽

Three Dimensional ◽

High Speed Steel ◽

Cutting Parameters ◽

Depth Of Cut ◽

Beam Model ◽

Tool Deflection

In this article, a new model of regenerative vibrations due to the deflection of the cutting tool in turning is proposed. The previous study reported chatter as a result of cutting a wavy surface of the previous cut. The proposed model takes into account cutting forces as the main factor of tool deflection. A cantilever beam model is used to establish a numerical model of the tool deflection. Three-dimensional finite element method is used to estimate the tool permissible deflection under the action of the cutting load. To analyze the system dynamic behavior, 1-degree-of-freedom model is used. MATLAB is used to compute the system time series from the initial value using fourth-order Runge–Kutta numerical integration. A straight hard turning with minimal fluid application experiment is used to obtain cutting forces under stable and chatter conditions. A single-point cutting tool made from high-speed steel is used for cutting. Experiment results showed that for the cutting parameters above 0.1mm/rev feed and [Formula: see text]mm depth of cut, the system develops fluctuations and higher chatter vibration frequency. Dynamic model vibration results showed that the cutting tool deflection induces chatter vibrations which transit from periodic, quasi-periodic, and chaotic type.

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Effects of Belt Flexural Rigidity on the Transmission Error of a Carriage-driving System

Journal of Mechanical Design ◽

10.1115/1.533562 ◽

2000 ◽

Vol 122 (2) ◽

pp. 213-218 ◽

Cited By ~ 7

Author(s):

Hung-Ming Tai ◽

Cheng-Kuo Sung

Keyword(s):

Boundary Conditions ◽

Numerical Method ◽

Experimental Technique ◽

Moving Boundary ◽

Flexural Rigidity ◽

Three Dimensional ◽

Transmission Error ◽

Beam Model ◽

Driving System ◽

The Relationship

This paper investigates the effects of belt flexural rigidity and belt tension on transmission error of a carriage-driving system. The beam model associated with both the clamped and moving boundary conditions at two ends is utilized to derive the governing equation of the belt. The belt flexural rigidity is obtained and verified by an experimental technique. In addition, a numerical method is proposed to determine the belt profile, transmission error and transmission stiffness. Results show that transmission error of a carriage-driving system increases when the carriage moves away from the driving pulley due to finite belt flexural rigidity. According to the analyses, application of appropriate tension on the belt can significantly reduce the error. Furthermore, the transmission stiffness for representing the entire rigidity between the carriage and pulley is investigated based on the proposed beam model. A three-dimensional plot that indicates the relationship among the transmission stiffness, belt tension and the position of the carriage is obtained. [S1050-0472(00)01102-8]

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Nonlinear Model on Torsional Behaviors of CFRP Strengthened Reinforced Concrete Beams

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.47-50.881 ◽

2008 ◽

Vol 47-50 ◽

pp. 881-885

Author(s):

Werasak Raongjant ◽

Meng Jing

Keyword(s):

Finite Element ◽

Three Dimensional ◽

Reinforced Concrete Beams ◽

Fiber Reinforced Polymer ◽

Beam Model ◽

The Finite Element Method ◽

Reinforced Polymer ◽

Linear String ◽

Dimensional Finite Element ◽

Three Dimensional Finite Element

In this paper, a reasonable three dimensional finite element beam model was developed to predict the mechanical behaviors of carbon fiber reinforced polymer (CFRP) strengthened RC box beam under combined bending, shear and torque. The comparison of calculated results with the experiment results of torque-twist relationship, the strain developments in steels and CFRP strips and the force of non-linear string element indicates that the finite element method presented in this study can simulate the behavior of beams well.

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