Consistent Tangent Stiffness of a Three-Dimensional Wheel–Rail Interaction Element

2021 ◽  
Author(s):  
Quan Gu ◽  
Jinghao Pan ◽  
Yongdou Liu
Keyword(s):  
Finite Element ◽  
Quadratic Convergence ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Light Rail ◽  
Software Framework ◽  
Tangent Stiffness ◽  
Rail Vehicle ◽  
Interaction Element ◽  
Nonlinear Equations Of Motion

Consistent tangent stiffness plays a crucial role in delivering a quadratic rate of convergence when using Newton’s method in solving nonlinear equations of motion. In this paper, consistent tangent stiffness is derived for a three-dimensional (3D) wheel–rail interaction element (WRI element for short) originally developed by the authors and co-workers. The algorithm has been implemented in finite element (FE) software framework (OpenSees in this paper) and proven to be effective. Application examples of wheelset and light rail vehicle are provided to validate the consistent tangent stiffness. The quadratic convergence rate is verified. The speeds of calculation are compared between the use of consistent tangent stiffness and the tangent by perturbation method. The results demonstrate the improved computational efficiency of WRI element when consistent tangent stiffness is used.

scholarly journals Nonlinear Structural Dynamic Finite Element Analysis Using Ritz Vector Reduced Basis Method

1996 ◽  
Vol 3 (4) ◽  
pp. 259-268 ◽  
Author(s):  
M.S. Yao
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Finite Amplitude ◽  
Basis Vector ◽  
Reduced Basis ◽  
Ritz Vector ◽  
Element Analysis ◽  
Structural Dynamic ◽  
Nonlinear Equations Of Motion

The large number of unknown variables in a finite element idealization for dynamic structural analysis is represented by a very small number of generalized variables, each associating with a generalized Ritz vector known as a basis vector. The large system of equations of motion is thereby reduced to a very small set by this transformation and computational cost of the analysis can be greatly reduced. In this article nonlinear equations of motion and their transformation are formulated in detail. A convenient way of selection of the generalized basis vector and its limitations are described. Some illustrative examples are given to demonstrate the speed and validity of the method. The method, within its limitations, may be applied to dynamic problems where the response is global in nature with finite amplitude.

Finite Element Models for Flexible Cosserat Solids

2020 ◽  
Author(s):  
Olivier A. Bauchau ◽  
Minghe Shan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Displacement Vector ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
The Finite Element Method ◽  
Discrete Equations ◽  
Computation Efficiency ◽  
Elasticity Problems ◽  
Element Method

Abstract The application of the finite element method to the modeling of Cosserat solids is investigated in detail. In two- and three-dimensional elasticity problems, the nodal unknowns are the components of the displacement vector, which form a linear field. In contrast, when dealing with Cosserat solids, the nodal unknowns form the special Euclidean group SE(3), a nonlinear manifold. This observation has numerous implications on the implementation of the finite element method and raises numerous questions: (1) What is the most suitable representation of this nonlinear manifold? (2) How is it interpolated over one element? (3) How is the associated strain field interpolated? (4) What is the most efficient way to obtain the discrete equations of motion? All these questions are, of course intertwined. This paper shows that reliable schemes are available for the interpolation of the motion and curvature fields. The interpolated fields depend on relative nodal motions only, and hence, are both objective and tensorial. Because these schemes depend on relative nodal motions only, only local parameterization is required, thereby avoiding the occurrence of singularities. For Cosserat solids, it is preferable to perform the discretization operation first, followed by the variation operation. This approach leads to considerable computation efficiency and simplicity.

Nonlinear Dynamics of Rotating Structures and Helicopter Blades

2018 ◽  
Author(s):  
Matteo Filippi ◽  
Alfonso Pagani ◽  
Erasmo Carrera
Keyword(s):  
Numerical Solutions ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Constitutive Law ◽  
Rotating Blades ◽  
Helicopter Blades ◽  
Strain Components ◽  
Theory Approximation ◽  
Nonlinear Equations Of Motion ◽  
Lagrange Strain

This work explores the effects of geometrical nonlinearities in the vibration analysis of rotating structures and helicopter blades. Structures are modelled via higher-order beam theories with variable kinematics. These theories fall in the domain of the Carrera Unified Formulation (CUF), according to which the nonlinear equations of motion of rotating blades can be written in terms of fundamental nuclei, whose formalism is an invariant of the theory approximation. The inherent three-dimensional nature of CUF enables one to include all Green-Lagrange strain components as well as all coupling effects due to the geometrical features and the three-dimensional constitutive law. Numerical solutions are considered and opportunely discussed. Also, linearized and full nonlinear solutions for vibrating rotating blades are compared both in case of small amplitudes and in the large deflections/rotations regime.

scholarly journals New Analytical Model Used in Finite Element Analysis of Solids Mechanics

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1401 ◽  
Author(s):  
Sorin Vlase ◽  
Adrian Eracle Nicolescu ◽  
Marin Marin
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Elastic Solid ◽  
The Finite Element Method ◽  
Governing Equations ◽  
Element Analysis ◽  
Elastic Multibody System

In classical mechanics, determining the governing equations of motion using finite element analysis (FEA) of an elastic multibody system (MBS) leads to a system of second order differential equations. To integrate this, it must be transformed into a system of first-order equations. However, this can also be achieved directly and naturally if Hamilton’s equations are used. The paper presents this useful alternative formalism used in conjunction with the finite element method for MBSs. The motion equations in the very general case of a three-dimensional motion of an elastic solid are obtained. To illustrate the method, two examples are presented. A comparison between the integration times in the two cases presents another possible advantage of applying this method.

DYNAMICS OF SHALLOW CABLES UNDER TURBULENT WIND: A NONLINEAR FINITE ELEMENT APPROACH

2011 ◽  
Vol 11 (04) ◽  
pp. 755-774 ◽  
Author(s):  
NICOLA IMPOLLONIA ◽  
GIUSEPPE RICCIARDI ◽  
FERNANDO SAITTA
Keyword(s):  
Finite Element ◽  
Equilibrium Shape ◽  
Equations Of Motion ◽  
Plane Motion ◽  
Computational Effort ◽  
Nonlinear Finite Element ◽  
Initial Shape ◽  
Linear Behavior ◽  
Element Technique ◽  
Nonlinear Equations Of Motion

In classic cable theory, vibrations are usually analyzed by writing the equations of motion in the neighborhood of the initial equilibrium configuration. Furthermore, a fundamental difference exists between out-of-plane motions, which basically corresponds to the linear behavior of a taut string and in-plane motion, where self-weight determines a sagged initial profile. This work makes use of a continuous approach to establish the initial shape of the cable when it is subjected to wind or fluid flow arbitrarily directed and employed a novel nonlinear finite element technique in order to investigate the dynamics present around the initial equilibrium shape of the cable. Stochastic solutions in the frequency domain are derived for a wind-exposed cable after linearization of the problem. By applying the proper orthogonal decomposition (POD) technique with the aim of reducing computational effort, an approach to simulate modal wind forces is proposed and applied to the nonlinear equations of motion.

scholarly journals Energy of Accelerations Used to Obtain the Motion Equations of a Three- Dimensional Finite Element

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 321 ◽  
Author(s):  
Sorin Vlase ◽  
Iuliu Negrean ◽  
Marin Marin ◽  
Maria Luminița Scutaru
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
The Finite Element Method ◽  
Method Of Calculation ◽  
Elastic Systems ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element ◽  
Multi Body ◽  
Elastic Elements

When analyzing the dynamic behavior of multi-body elastic systems, a commonly used method is the finite element method conjunctively with Lagrange’s equations. The central problem when approaching such a system is determining the equations of motion for a single finite element. The paper presents an alternative method of calculation theses using the Gibbs–Appell (GA) formulation, which requires a smaller number of calculations and, as a result, is easier to apply in practice. For this purpose, the energy of the accelerations for one single finite element is calculated, which will be used then in the GA equations. This method can have advantages in applying to the study of multi-body systems with elastic elements and in the case of robots and manipulators that have in their composition some elastic elements. The number of differentiation required when using the Gibbs–Appell method is smaller than if the Lagrange method is used which leads to a smaller number of operations to obtain the equations of motion.

Modal Analysis of Ballooning Strings With Small Sag

1999 ◽  
Author(s):  
Rongjun Fan ◽  
Sushil K. Singh ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
High Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Theoretical Predictions ◽  
Nonlinear Equations Of Motion

Abstract During the manufacture and transport of textile products, yarns are rotated at high speed and form balloons. The dynamic response of the balloon to varying rotation speed, boundary excitation, and disturbance forces governs the quality of the associated process. Resonance, in particular, can cause large tension variations that reduce product quality and may cause yarn breakage. In this paper, the natural frequencies and mode shapes of a single loop balloon are calculated to predict resonance. The three dimensional nonlinear equations of motion are simplified via small steady state displacement (sag) and vibration assumptions. Axial vibration is assumed to propagate instantaneously or in a quasistatic manner. Galerkin’s method is used to calculate the mode shapes and natural frequencies of the linearized equations. Experimental measurements of the steady state balloon shape and the first two natural frequencies and mode shapes are compared with theoretical predictions.

Nonlinear Finite-Element Method for Plates and Its Application to Dynamic Response of Reactor Fuel Subassemblies

1974 ◽  
Vol 96 (4) ◽  
pp. 251-257 ◽  
Author(s):  
T. Belytschko ◽  
A. H. Marchertas
Keyword(s):  
Finite Element ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Shell Element ◽  
Nonlinear Material ◽  
Time Step ◽  
Cross Sectional ◽  
Difference Formula ◽  
Lumped Masses

A finite-element procedure for transient analysis of plates and shells in three-dimensional space, and applicable to large displacements and nonlinear material properties, is described. This procedure employs a convected coordinate formulation enabling the use of simple strain-nodal displacement and nodal force-stress relations. The plate/shell element considers linear in-plane displacements and cubic transverse displacements. The orientation of lumped masses is described by unit vectors so that arbitrarily large rotations can be treated. Discretized equations of motion are integrated explicitly in time with a difference formula. Membrane artificial viscosity is utilized to stabilize occasional oscillations. The computational efficiency of the procedure is quite good: one element-time step takes 2 msec on an IBM 360/195 computer. Comparison of results with experimental data of impulsively loaded plates shows good agreement. The program was applied to a hexagonal fuel subassembly loaded internally. Various results are presented on its response and it is shown that, for that type of loading, two-dimensional cross-sectional models may be adequate.

A Static and Dynamic Model of Geared Transmissions by Combining Substructures and Elastic Foundations—Applications to Thin-Rimmed Gears

2006 ◽  
Vol 129 (2) ◽  
pp. 184-194 ◽  
Author(s):  
M. N. Bettaïeb ◽  
P. Velex ◽  
M. Ajmi
Keyword(s):  
Finite Element ◽  
Contact Stiffness ◽  
Equations Of Motion ◽  
Hybrid Approach ◽  
Three Dimensional ◽  
Element Model ◽  
3D Finite Element ◽  
Elastic Foundations ◽  
Beam Elements ◽  
Set Up

The present work is aimed at predicting the static and dynamic behavior of geared transmissions comprising flexible components. The proposed model adopts a hybrid approach, combining classical beam elements, elastic foundations for the simulation of tooth contacts, and substructures derived from three-dimensional (3D) finite element grids for thin-rimmed gears and their supporting shafts. The pinion shaft and body are modeled via beam elements which simulate bending, torsion and traction. Tooth contact deflections are described using time-varying elastic foundations (Pasternak foundations) connected by independent contact stiffness. In order to account for thin-rimmed gears, a 3D finite element model of the gear (excluding teeth) is set up and a pseudo-modal reduction technique is used prior to solving the equations of motion. Depending on the gear structure, the results reveal a potentially significant influence of thin rims on both quasi-static and dynamic tooth loading.

Generalized Vector-Network Formulation for the Dynamic Simulation of Multibody Systems

1986 ◽  
Vol 108 (4) ◽  
pp. 322-329 ◽  
Author(s):  
M. J. Richard ◽  
R. Anderson ◽  
G. C. Andrews
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
New Approach ◽  
System Description ◽  
Nonlinear Equations Of Motion ◽  
Systematic Formulation ◽  
Multi Body ◽  
Entire Procedure

This paper describes the vector-network approach which is a comprehensive mathematical model for the systematic formulation of the nonlinear equations of motion of dynamic three-dimensional constrained multi-body systems. The entire procedure is a basic application of concepts of graph theory in which laws of vector dynamics have been combined. The main concepts of the method have been explained in previous publications but the work described herein is an appreciable extension of this relatively new approach. The method casts simultaneously the three-dimensional inertia equations associated with each rigid body and the geometrical expressions corresponding to the kinematic restrictions into a symmetrical format yielding the differential equations governing the motion of the system. The algorithm is eminently well suited for the computer-aided simulation of arbitrary interconnected rigid bodies; it serves as the basis for a “self-formulating” computer program which can simulate the response of a dynamic system, given only the system description.

Sign in / Sign up

Export Citation Format

Share Document