Mean-Axis Conditions for Finite Elements Undergoing Large Rotations and Translations

Author(s):  
Venkata R. Sonti ◽  
Om P. Agrawal

Abstract The mean-axis conditions for three types of finite elements that undergo large rotation and translation are derived. The cases include two Lagrangian planar elements (linear triangle and linear rectangle) and a three dimensional rectangular plate element. Each case includes two elements connected at the nodes. Three separate coordinate systems are used in describing the systems. The numerical values of different matrices including the mass matrix are presented for specific cases. It may be seen that the non-linear mass matrix is expressed in terms of a set of time-invariand matrices. The total mass matrix of the body i is obtained by assembling the individual mass matrices of the finite elements of body i.

1996 ◽  
Vol 11 (4) ◽  
pp. 371-380 ◽  
Author(s):  
Alphose Zingoni

Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.


Author(s):  
Alan S. K. Kwan

The stiffness relationship and the distributed mass matrix for a geometrically nonlinear three dimensional straight axial element is derived for use in prestressed cablenet structures. The justification for the use of a linearised stiffness relationship is provided through a theoretical derivation. Results using this simple element have shown a high degree of correlation with results to those available in the literature obtained with more complex curved finite elements, analogous membrane models and other techniques.


Author(s):  
John Wiechel ◽  
Sandra Metzler ◽  
Dawn Freyder ◽  
Nick Kloppenborg

Reconstructing the mechanics and determining the cause of a person falling from a height in the absence of witness observations or a statement from the victim can be quite challenging. Often there is little information available beyond the final resting position of the victim and the injuries they sustained. The mechanics of a fall must follow the physics of falling bodies and this physics provides an additional source of information about how the fall occurred. Computational, physics-based simulations can be utilized to model the free-fall portion of the fall kinematics and to analyze biomechanical injury mechanisms. However, an accurate determination of the overall fall kinematics, including the initial conditions and any specific contributions of the person(s) involved, must include the correct position and posture of the individual prior to the fall. Frequently this phase of the analysis includes voluntary movement on the part of the fall victim, which cannot be modeled with simulations using anthropomorphic test devices (ATDs). One approach that has been utilized in the past to overcome this limitation is to run the simulations utilizing a number of different initial conditions for the fall victim. While fall simulations allow the initial conditions of the fall to be varied, they are unable to include the active movement of the subject, and the resulting interaction with other objects in the environment immediately prior to or during the fall. Furthermore, accurate contact interactions between the fall victim and multiple objects in their environment can be difficult to model within the simulation, as they are dependent on the knowledge of material properties of these objects and the environment such as elasticity and damping. Motion capture technology, however, allows active subject movement and behaviors to be captured in a quantitative, three-dimensional manner. This information can then be utilized within the fall simulation to more accurately model the initial fall conditions. This paper presents a methodology for reconstructing fall mechanics using a combination of motion capture, human body simulation, and injury biomechanics. This methodology uses as an example a fall situation where interaction between the fall victim and specific objects in the environment, as well as voluntary movements by the fall victim immediately prior to the accident, provided information that could not be otherwise obtained. Motion capture was first used to record the possible motions of a person in the early stages of the fall. The initial position of the fall victim within the physics based simulation of the body in free fall was determined utilizing the individual body segment and joint angles from the motion capture analysis. The methodology is applied to a real world case example and compared with the actual outcome.


1998 ◽  
Vol 5 (2) ◽  
pp. 111-117
Author(s):  
Ara Arabyan ◽  
Yaqun Jiang

A pipe element developed earlier for the analysis of combined large bending and torsional deformations of blood vessels under static loading is extended to model behavior in the presence of large rotations and time-varying external forces. As in the case of the earlier element, the enhanced element supports ovalization and warping of its cross-section. The enhancements presented in this paper are comprised of a mass matrix and gyroscopic effects resulting from fast rotation rates and large deformations. The effectiveness of the element is demonstrated by two examples, which simulate the three-dimensional behavior of a highly flexible pipe under dynamic loading conditions.


Author(s):  
W. Fan ◽  
W. D. Zhu ◽  
H. Ren

In this investigation, a new singularity-free formulation of a three-dimensional Euler-Bernoulli beam with large deformation and large rotation is developed. The position of the centroid line of the beam is integrated from its slope, which can be easily expressed by Euler parameters. The hyper-spherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied. Hence, each node of a beam element has only four nodal coordinates, which is significantly fewer than an absolute node coordinate formulation (ANCF) and the finite element method (FEM). Governing equations of the beam and constraint equations are derived using Lagrange’s equations for systems with constraints, which are solved by an available differential algebraic equation solver. The current formulation can be used to calculate the static equilibrium and dynamics of an Euler-Bernoulli beam under arbitrary concentrated and distributed forces. While the mass matrix is more complex than that in an absolute nodal coordinate formulation, the stiffness matrix and generalized forces are simpler, which is amenable for calculating the equilibrium of the beam. Several numerical examples are presented to demonstrate the performance of the current formulation. It is shown that the current formulation can achieve the same accuracy as the FEM and ANCF with a fewer number of coordinates.


Author(s):  
Johannes Gerstmayr

The present paper deals with the modeling of three dimensional (3D), nonlinear beam finite elements which do not employ rotational parameters. The beam elements are based on the so-called absolute nodal coordinate formulation (ANCF), which has been introduced in the late 1990’s. Early implementations of 3D ANCF beam finite elements incorporated problems regarding Poisson, thickness and shear locking. In the present paper, two alternative models for the work of elastic forces are presented. The first approach, which is intended to provide reference solutions, is close to the original approach. However, the effect of Poisson and thickness locking is eliminated by proper integration of the contributing terms in the virtual work of elastic forces. In the second approach, a corotationally linearized model is developed, which is based on a simple formulation for the elastic forces. The latter model only takes into account small deformations with respect to a corotating reference frame, but it is different from the conventional floating frame of reference formulation, because it has a constant mass matrix. The second approach is intended to be advantageous in practical applications where only small deformations with respect to large rigid body motions need to be taken into account, such as in robotics or machine dynamics. Numerical results are presented, which show that the new approaches agree with the solution of static and dynamic problems using classical finite elements or analytical methods.


2019 ◽  
Vol 150 (2) ◽  
pp. 569-606 ◽  
Author(s):  
Dat Cao ◽  
Luan Hoang

AbstractThe Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.


2013 ◽  
Vol 745 ◽  
pp. 101-118 ◽  
Author(s):  
Gaetano Giunta ◽  
Yao Koutsawa ◽  
Salim Belouettar

A Unified Formulation for deriving several higher-order theories and related finite elements for beams is presented within this paper.Three-dimensional structures with piezo-electric layers are considered.Static and free vibration analyses are carried out.Models' main unknowns are the displacements and the electric potential.They are approximated above the beam cross-section via Lagrange's polynomials in a layer-wise sense.Finite elements stiffness and mass matrices are derived in a nucleal form using d'Alembert's Principle.This nucleal form is representative of the generic term in the approximating expansion of the displacements and electric potential over the cross-section.It is, therefore, invariant versus the theory expansion order and the element nodes' number.In such a manner, higher-order displacements-based theories that account for non-classical effectssuch as transverse shear deformations and cross-section in- and out-of-plane warping are straightforwardly formulated.Results are given in terms of displacements, electrical potential and stresses.Comparison with three-dimensional finite elements models are provided, showing thataccurate results can be obtained with reduced computational costs.


2021 ◽  
Vol 10 (4) ◽  
pp. 253-268
Author(s):  
Ruhi Aydin

In the static analysis of beam-column systems using matrix methods, polynomials are using as the shape functions. The transverse deflections along the beam axis, including the axial- flexural effects in the beam-column element, are not adequately described by polynomials. As an alternative method, the element stiffness matrix is modeling using stability parameters. The shape functions which are obtaining using the stability parameters are more compatible with the system’s behavior. A mass matrix used in the dynamic analysis is evaluated using the same shape functions as those used for derivations of the stiffness coefficients and is called a consistent mass matrix. In this study, the stiffness and consistent mass matrices for prismatic three-dimensional Bernoulli-Euler and Timoshenko beam-columns are proposed with consideration for the axial-flexural interactions and shear deformations associated with transverse deflections along the beam axis. The second-order effects, critical buckling loads, and eigenvalues are determined. According to the author’s knowledge, this study is the first report of the derivations of consistent mass matrices of Bernoulli-Euler and Timoshenko beam-columns under the effect of axially compressive or tensile force.


2019 ◽  
pp. 50-71
Author(s):  
Vedat Sar

This paper proposes a model that explains the response to complex psychological trauma and dissociation. This tri-modal reaction model tries to account for the mental striving of the traumatized individual in dealing with unbearable pain when fighting for overall survival.Rather than conceptualizing this process in consecutive phases, the response of the individual to developmental trauma is described in three modes which often co-occur: Acute reaction, chronic process, and alienation. Each mode operates in a window of overmodulation and undermodulation of emotions. This tri-modal model resembles medical conceptualizations of injury, response, and illness as they occur to the body. Psychotherapeutic intervention to trauma-related conditions has to consider the possible co-presence of the three modes. Such three-dimensional understanding of respo nse to trauma has also implications for mental integration. Namely, the latter is a multidimensional phenomenon rather than a linear sum of parts.


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