On the Transitivity Equations of Rigid-Body Dynamics

1992 ◽  
Vol 59 (4) ◽  
pp. 955-962 ◽  
Author(s):  
John G. Papastavridis
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Body Dynamics ◽  
Analytical Mechanics ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Body Dynamics

This paper presents direct derivations of the various forms of the famous transitivity equations of rigid-body kinematics from a simple and unified viewpoint. These forms are indispensable in the derivation of the Eulerian (gyro) equations of motion via the Lagrangean (analytical mechanics) method. Both true (holonomic) and quasi (nonholonomic) coordinate forms are presented. A vectorial derivation of the Eulerian equations from the Central Equation and from Hamilton’s Principle is also included in the Appendix.

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.

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.

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.

scholarly journals On the rotational equations of motion in rigid body dynamics when using Euler parameters

2015 ◽  
Vol 81 (1-2) ◽  
pp. 343-352 ◽  
Author(s):  
Karim Sherif ◽  
Karin Nachbagauer ◽  
Wolfgang Steiner
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Body Dynamics ◽  
Euler Parameters ◽  
Body Dynamics

scholarly journals Slipping–rolling transitions of a body with two contact points

2021 ◽  
Author(s):  
Mate Antali ◽  
Gabor Stepan
Keyword(s):  
Rigid Body ◽  
Contact Point ◽  
Static Friction ◽  
Rigid Body Dynamics ◽  
Contact Forces ◽  
Friction Forces ◽  
Contact Points ◽  
Body Dynamics ◽  
Coulomb Model ◽  
Rigid Surfaces

AbstractIn this paper, the general kinematics and dynamics of a rigid body is analysed, which is in contact with two rigid surfaces in the presence of dry friction. Due to the rolling or slipping state at each contact point, four kinematic scenarios occur. In the two-point rolling case, the contact forces are undetermined; consequently, the condition of the static friction forces cannot be checked from the Coulomb model to decide whether two-point rolling is possible. However, this issue can be resolved within the scope of rigid body dynamics by analysing the nonsmooth vector field of the system at the possible transitions between slipping and rolling. Based on the concept of limit directions of codimension-2 discontinuities, a method is presented to determine the conditions when the two-point rolling is realizable without slipping.

New solutions of classical problems in rigid body dynamics

2015 ◽  
Vol 69 ◽  
pp. 40-44
Author(s):  
H.M. Yehia ◽  
E. Saleh ◽  
S.F. Megahid
Keyword(s):  
Rigid Body ◽  
Rigid Body Dynamics ◽  
Body Dynamics
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