Partitioned Transient Analysis Procedures for Coupled-Field Problems: Accuracy Analysis

1980 ◽  
Vol 47 (4) ◽  
pp. 919-926 ◽  
Author(s):  
K. C. Park ◽  
C. A. Felippa
Keyword(s):  
Rigid Body ◽  
Time Integration ◽  
Governing Equations ◽  
Implicit Integration ◽  
Practical Applications ◽  
Coupled Field ◽  
Stable Partition ◽  
Rigid Body Motions ◽  
Coupled Field Problems ◽  
The Right

Partitioned solution procedures for direct time integration of second-order coupled-field systems are studied from the standpoint of accuracy. These procedures are derived by three formulation steps: implicit integration of coupled governing equations, partitioning of resulting algebraic systems and extrapolation on the right-hand partition. It is shown that the combined effect of partition, extrapolation, and computational paths governs the choice of stable extrapolators and preservation of rigid-body motions. Stable extrapolators for various computational paths are derived and implementation-extrapolator combinations which preserve constant-velocity and constant-acceleration rigid-body motions are identified. A spectral analysis shows that the primary error source introduced by a stable partition is frequency distortion. Finally, as a guide to practical applications, the advantages and shortcomings of five specific partitions are discussed.

Coordinate Mappings for Rigid Body Motions

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Rigid Body ◽  
Lie Group ◽  
Equations Of Motion ◽  
Time Integration ◽  
Exponential Mapping ◽  
Group Setting ◽  
Dual Quaternions ◽  
Integration Schemes ◽  
Rigid Body Motions ◽  
Lie Group Integration

Geometric methods have become increasingly accepted in computational multibody system (MBS) dynamics. This includes the kinematic and dynamic modeling as well as the time integration of the equations of motion. In particular, the observation that rigid body motions form a Lie group motivated the application of Lie group integration schemes, such as the Munthe-Kaas method. Also established vector space integration schemes tailored for structural and MBS dynamics were adopted to the Lie group setting, such as the generalized α integration method. Common to all is the use of coordinate mappings on the Lie group SE(3) of Euclidean motions. In terms of canonical coordinates (screw coordinates), this is the exponential mapping. Rigid body velocities (twists) are determined by its right-trivialized differential, denoted dexp. These concepts have, however, not yet been discussed in compact and concise form, which is the contribution of this paper with particular focus on the computational aspects. Rigid body motions can also be represented by dual quaternions, that form the Lie group Sp̂(1), and the corresponding dynamics formulations have recently found a renewed attention. The relevant coordinate mappings for dual quaternions are presented and related to the SE(3) representation. This relation gives rise to a novel closed form of the dexp mapping on SE(3). In addition to the canonical parameterization via the exponential mapping, the noncanonical parameterization via the Cayley mapping is presented.

Partitioned Transient Analysis Procedures for Coupled-Field Problems: Stability Analysis

1980 ◽  
Vol 47 (2) ◽  
pp. 370-376 ◽  
Author(s):  
K. C. Park
Keyword(s):  
Stability Analysis ◽  
Transient Analysis ◽  
General Procedure ◽  
Implicit Integration ◽  
Solution Vector ◽  
Analysis Technique ◽  
Coupled Field ◽  
Explicit Procedure ◽  
Single Field ◽  
Coupled Field Problems

A general partitioned transient analysis procedure is proposed, which is amenable to a unified stability analysis technique. The procedure embodies two existing implicit-explicit procedures and one existing implicit-implicit procedure. A new implicit-explicit procedure is discovered, as a special case of the general procedure, that allows degree-by-degree implicit or explicit selections of the solution vector and can be implemented within the framework of the implicit integration packages. A new element-by-element implicit-implicit procedure is also presented which satisfies program modularity requirements and enables the use of single-field implicit integration packages to solve coupled-field problems.

Approximation of Finite Rigid Body Motions

2009 ◽  
Author(s):  
Andreas Mu¨ller
Keyword(s):  
Rigid Body ◽  
Euler Equations ◽  
Time Derivative ◽  
Time Integration ◽  
Body Motion ◽  
Motion Group ◽  
Time Stepping ◽  
Approximation Formulas ◽  
Rigid Body Motions ◽  
Local Parameters

It is well-known that there is no integrable relation between the twist of a rigid body and its finite motion, since the angular velocity components are non-holonomic velocity coordinates. Moreover, the reconstruction of the body’s motion requires to solve a set of differential equations on the rigid body motion group. This is usually avoided by introducing local parameters (e.g. Euler angles) so that the problem becomes an ordinary differential equation on a vector space (e.g. kinematic Euler equations). In this paper the original problem on the motion group is treated. A family of approximation formulas is presented that allow reconstructing large rigid body motions from a given velocity field up to a desired order. It is shown that a k-th order accurate reconstruction requires the first k – 1 time derivative of the velocity. As an application the reconstruction formulas are used for the rotation update in a momentum preserving time stepping scheme for time integration of the dynamic Euler equations.

Analytical models for the rigid body motions of skew bridges

1987 ◽  
Vol 15 (8) ◽  
pp. 923-944 ◽  
Author(s):  
Emmanuel A. Maragakis ◽  
Paul C. Jennings
Keyword(s):  
Rigid Body ◽  
Analytical Models ◽  
Skew Bridges ◽  
Rigid Body Motions

scholarly journals The Rotating Rigid Body Model Based on a Non-twisting Frame

2020 ◽  
Vol 30 (6) ◽  
pp. 3199-3233 ◽  
Author(s):  
Cristian Guillermo Gebhardt ◽  
Ignacio Romero
Keyword(s):  
Kinetic Energy ◽  
Rigid Body ◽  
Conservation Laws ◽  
Rotation Rate ◽  
Governing Equations ◽  
New Model ◽  
Body Model ◽  
Integration Schemes ◽  
Rigid Body Model ◽  
Unit Vectors

Abstract This work proposes and investigates a new model of the rotating rigid body based on the non-twisting frame. Such a frame consists of three mutually orthogonal unit vectors whose rotation rate around one of the three axis remains zero at all times and, thus, is represented by a nonholonomic restriction. Then, the corresponding Lagrange–D’Alembert equations are formulated by employing two descriptions, the first one relying on rotations and a splitting approach, and the second one relying on constrained directors. For vanishing external moments, we prove that the new model possesses conservation laws, i.e., the kinetic energy and two nonholonomic momenta that substantially differ from the holonomic momenta preserved by the standard rigid body model. Additionally, we propose a new specialization of a class of energy–momentum integration schemes that exactly preserves the kinetic energy and the nonholonomic momenta replicating the continuous counterpart. Finally, we present numerical results that show the excellent conservation properties as well as the accuracy for the time-discretized governing equations.

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

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