A Virtual Body and Joint for Constrained Flexible Multibody Dynamics: A Generalized Recursive Formulation

1999 ◽  
Author(s):  
D. S. Bae ◽  
J. M. Han ◽  
J. H. Choi
Keyword(s):  
Rigid Body ◽  
Reference Frames ◽  
Equations Of Motion ◽  
Recursive Formula ◽  
General Purpose ◽  
Rigid Body Dynamics ◽  
Flexible Body ◽  
Relative Coordinate ◽  
Body Dynamics ◽  
Flexible Body Dynamics

Abstract This research extends the generalized recursive formulas for the rigid body dynamics to the flexible body dynamics using the backward difference formula (BDF) and the relative generalized coordinate. When a new force or joint module is added to a general purpose program in the relative coordinate formulations, the modules for the rigid bodies are not reusable for the flexible bodies. Since the flexible body dynamics handles more degrees of freedom than the rigid body dynamics does, implementation of the flexible dynamics module is generally complicated and prone to coding mistakes. In order to overcome the implementation difficulties, a virtual rigid body is introduced at every joint and force reference frames. A virtual flexible body joint is introduced between two body reference frames of the virtual and original bodies. Since the multibody system dynamics are formulated by highly nonlinear algebraic and differential equations and there are many different types of joints, a tremendous amount of computer implementation is required to develop a general purpose dynamic analysis program using the relative coordinate formulation. The implementation burden is relieved by the generalized recursive formulas. The notationally compact velocity transformation method is used to derive the equations of motion in the joint space. The terms in the equations of motion which are related to the transformation matrix are classified into several categories each of which recursive formula is developed. Whenever one category of the terms is encountered, the corresponding recursive formula is invoked. Since computation time in a relative coordinate formulation is approximately proportional to the number of the relative coordinates, computational overhead due to the additional virtual bodies and joints is minor. Meanwhile, implementation convenience is dramatically improved.

A Virtual Body and Joint for Constrained Flexible Multibody Dynamics

1999 ◽  
Author(s):  
D. S. Bae ◽  
J. M. Han ◽  
J. H. Choi
Keyword(s):  
Rigid Body ◽  
Multibody Dynamics ◽  
Reference Frames ◽  
Flexible Multibody Dynamics ◽  
General Purpose ◽  
Rigid Body Dynamics ◽  
Flexible Body ◽  
Body Dynamics ◽  
Flexible Multibody ◽  
Flexible Body Dynamics

Abstract A convenient implementation method for constrained flexible multibody dynamics is presented by introducing virtual rigid body and joint. The general purpose program for rigid and flexible multibody dynamics consists of three major parts of a set of inertia modules, a set of force modules, and a set of joint modules. Whenever a new force or joint module is added to the general purpose program, the modules for the rigid body dynamics are not reusable for the flexible body dynamics. Consequently, the corresponding modules for the flexible body dynamics must be formulated and programmed again. Since the flexible body dynamics handles more degrees of freedom than the rigid body dynamics does, implementation of the module is generally complicated and prone to coding mistakes. In order to overcome these difficulties, a virtual rigid body is introduced at every joint and force reference frames. New kinematic admissibility conditions are imposed on two body reference frames of the virtual and original bodies by introducing a virtual flexible body joint. There are some computational overheads due to the additional bodies and joints. However, since computation time is mainly depended on the frequency of flexible body dynamics, the computational overhead of the presented method could not be a critical problem, while implementation convenience is dramatically improved.

Dynamics of Flexible Body Negotiating a Curve

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
Huailong Shi ◽  
Liang Wang ◽  
Ahmed A. Shabana
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Body Dynamics ◽  
The Body ◽  
Flexible Body ◽  
Motion Trajectory ◽  
Centrifugal Forces ◽  
Coriolis Forces ◽  
Body Dynamics ◽  
Body Reference

When a rigid body negotiates a curve, the centrifugal force takes a simple form which is function of the body mass, forward velocity, and the radius of curvature of the curve. In this simple case of rigid body dynamics, curve negotiation does not lead to Coriolis forces. In the case of a flexible body negotiating a curve, on the other hand, the inertia of the body becomes function of the deformation, curve negotiations lead to Coriolis forces, and the expression for the deformation-dependent centrifugal forces becomes more complex. In this paper, the nonlinear constrained dynamic equations of motion of a flexible body negotiating a circular curve are used to develop a systematic procedure for the calculation of the centrifugal forces during curve negotiations. The floating frame of reference (FFR) formulation is used to describe the body deformation and define the nonlinear centrifugal and Coriolis forces. The algebraic constraint equations which define the motion trajectory along the curve are formulated in terms of the body reference and elastic coordinates. It is shown in this paper how these algebraic motion trajectory constraint equations can be used to define the constraint forces that lead to a systematic definition of the resultant centrifugal force in the case of curve negotiations. The embedding technique is used to eliminate the dependent variables and define the equations of motion in terms of the system degrees of freedom. As demonstrated in this paper, the motion trajectory constraints lead to constant generalized forces associated with the elastic coordinates, and as a consequence, the elastic velocities and accelerations approach zero in the steady state. It is also shown that if the motion trajectory constraints are imposed on the coordinates of a flexible body reference that satisfies the mean-axis conditions, the centrifugal forces take the same form as in the case of rigid body dynamics. The resulting flexible body dynamic equations can be solved numerically in order to obtain the body coordinates and evaluate numerically the constraint and centrifugal forces. The results obtained for a flexible body negotiating a circular curve are compared with the results obtained for the rigid body in order to have a better understanding of the effect of the deformation on the centrifugal forces and the overall dynamics of the body.

A Recursive Implementation Method With Implicit Integrator for Multibody Dynamics

2001 ◽  
Author(s):  
H. J. Cho ◽  
H. S. Ryu ◽  
D. S. Bae ◽  
J. H. Choi ◽  
B. Ross
Keyword(s):  
Equations Of Motion ◽  
Flexible Body ◽  
Differential Algebraic Equation ◽  
Dynamic Correction ◽  
Body Dynamics ◽  
Mode Method ◽  
Numerical Integrators ◽  
Flexible Body Dynamics ◽  
Body Concept ◽  
Implementation Method

Abstract A recursive implementation method for the equations of motion and kinematics is presented. Computational structure of the kinematic and dynamic equations is exploited to systematically implement a dynamic analysis program RecurDyn. A differential algebraic equation solution method with implicit numerical integrators is discussed. Virtual body concept is introduced for the flexible body dynamics. The accuracy of the flexible body solutions is estimated by an error measure and is improved by the dynamic correction mode method. Several examples are solved to demonstrate the efficiency of the proposed methods.

Quasi-Variational Principles of Single Flexible Body Dynamics

2010 ◽  
Author(s):  
Haiyan Song ◽  
Jiansheng Zhou ◽  
Lifu Liang ◽  
Zongmin Liu
Keyword(s):  
Variational Principle ◽  
Variational Principles ◽  
Deformable Body ◽  
Rigid Body Dynamics ◽  
Flexible Body ◽  
Cross Link ◽  
Body Dynamics ◽  
Flexible Body Dynamics ◽  
Multi Body

The theoretical analysis of flexible multi-body system is a long-term and complicated problem. So the single flexible body dynamics should be studied firstly. Quasi-variational principle of non-conservative single flexible body dynamics is established under the cross-link of particle rigid body dynamics and deformable body dynamics. Some important problems are studied in quasi-variational principle of non-conservative single flexible body dynamics. The vibration problem of unrestrained beam can be solved very well by using quasi-variational principle.

Development of a New Multi-Flexible Body Dynamics (MFBD) Platform: A Relative Nodal Displacement Method for Finite Element Analysis

2005 ◽  
Author(s):  
Dae Sung Bae
Keyword(s):  
Rigid Body ◽  
Reference Frame ◽  
Large Deformations ◽  
Shape Function ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Flexible Body ◽  
Element Analysis ◽  
Body Dynamics ◽  
Flexible Body Dynamics

Recently the analysis of multi flexible body dynamics has been a hot issue in the area of the computational dynamics research. There have been two main streams of research. One is the extension of conventional FEA theory for the multi flexible body systems, using either the total Lagrangian or updated Lagrangian method. The other is the extension of the multi body dynamics theory. The latter is the topic of this research. One essential requirement of a shape function in FEA theory is ability to represent the rigid body motion. This research proposes to use the moving reference frame to represent the rigid body motion. Therefore, the shape function does not need to have ability to represent the rigid body motion. The moving reference frame covers the rigid body. Since the nodal displacements are measured relative to its adjacent moving nodal reference frame, they are still small for a truss structure undergoing large deformations if the element sizes are small. As a consequence, many element formulations developed under small deformation assumptions are still valid for structures undergoing large deformations, which significantly simplifies the equations of equilibrium. Several numerical examples are analyzed to demonstrate the efficiency and validity of the proposed method.

scholarly journals Equations of Motion for Train Derailment Dynamics

2007 ◽  
Author(s):  
D. Y. Jeong ◽  
M. L. Lyons ◽  
O. Orringer ◽  
A. B. Perlman
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Equations Of Motion ◽  
Computer Code ◽  
Relative Effect ◽  
Small Time ◽  
Rigid Body Dynamics ◽  
Train Derailment ◽  
Body Dynamics ◽  
Sensitivity Studies

This paper describes a planar or two-dimensional model to examine the gross motions of rail cars in a generalized train derailment. Three coupled, second-order differential equations are derived from Newton’s Laws to calculate rigid-body car motions with time. Car motions are defined with respect to a right-handed and fixed (i.e., non-rotating) reference frame. The rail cars are translating and rotating but not deforming. Moreover, the differential equations are considered as stiff, requiring relatively small time steps in the numerical solution, which is carried out using a FORTRAN computer code. Sensitivity studies are conducted using the purpose-built model to examine the relative effect of different factors on the derailment outcome. These factors include the number of cars in the train makeup, car mass, initial translational and rotational velocities, and coefficients of friction. Derailment outcomes include the number of derailed cars, maximum closing velocities (i.e., relative velocities between impacting cars), and peak coupler forces. Results from the purpose-built model are also compared to those from a model for derailment dynamics developed using commercial software for rigid-body dynamics called Automatic Dynamic Analysis of Mechanical Systems (ADAMS). Moreover, the purpose-built and the ADAMS models produce nearly identical results, which suggest that the dynamics are being calculated correctly in both models.

Explaining the importance of Euler’s method in rigid-body dynamics

2020 ◽  
pp. 030641902093412
Author(s):  
Flavius PR Martins ◽  
Agenor T Fleury ◽  
Flavio C Trigo
Keyword(s):  
Rigid Body ◽  
Engineering Students ◽  
Equations Of Motion ◽  
Euler Method ◽  
Rigid Body Dynamics ◽  
Frames Of Reference ◽  
First Year ◽  
Euler’S Method ◽  
Body Dynamics ◽  
Spinning Top

The purpose of this study is essentially pedagogical and aims to provide an additional argument in clarification of a question often raised by first-year undergraduate mechanical engineering students concerning the reason for using two frames of reference—one fixed in space and one fixed in the rigid-body—to describe its motion. The reasoning employed to illustrate the inappropriateness of using a single reference frame entails showing that the equations of motion, thus obtained, are far more complex than the equations resulting from application of the traditional Euler Method. This point is illustrated through the well-known frictionless symmetrical spinning top problem.

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