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.

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 TUTORIAL PRESENTATION OF ALTERNATIVE SOLUTIONS TO THE FLEXIBLE BEAM ON RIGID CART PROBLEM

2005 ◽  
Vol 29 (3) ◽  
pp. 357-373 ◽  
Author(s):  
R. G. Langlois ◽  
R. J. Anderson
Keyword(s):  
Rigid Body ◽  
Multibody Dynamics ◽  
Flexible Multibody Dynamics ◽  
Beam Theory ◽  
General Purpose ◽  
Rigid Body Dynamics ◽  
Body Motion ◽  
Flexible Beam ◽  
Euler Beam ◽  
Flexible Multibody

A classical planar problem in forward flexible multibody dynamics is thoroughly investigated. The system consists of a damped flexible beam cantilevered to a rigid translating cart. The problem is solved using three distinctly different conventional approaches presented in roughly the chronological order in which they have been applied to flexible dynamic systems. First, a modal superposition formulation based on Bernoulli-Euler beam theory is developed. Second, an alternative solution is developed drawing exclusively on methods for rigid body dynamics combined with a knowledge of the theoretical modal behaviour of continuous beams. Third, a formulation based on the conventional finite element method using four-degree-of-freedom planar beam elements is adapted to include the rigid body motion of the cart. The relative merits of the three formulations are discussed and numerical simulation results generated using each of the three formulations are compared with each other and with a solution from a general-purpose flexible multibody dynamics formulation that is briefly outlined. The relative accuracy and efficiency of the methods and the challenges associated with generalizing each formulation are discussed.

A Uniform Rotationless Formulation of Flexible Multibody Dynamics: Conserving Integration of Rigid Bodies, Nonlinear Beams and Shells

2007 ◽  
Author(s):  
Peter Betsch ◽  
Nicolas Sa¨nger
Keyword(s):  
Multibody Dynamics ◽  
Flexible Multibody Dynamics ◽  
Rigid Bodies ◽  
Rigid Body Dynamics ◽  
Flexible Bodies ◽  
Finite Dimensional ◽  
Nonlinear Beams ◽  
Body Dynamics ◽  
Flexible Multibody ◽  
Additional Constraints

A uniform framework for rigid body dynamics and nonlinear structural dynamics is presented. The advocated approach is based on a rotationless formulation of rigid bodies, nonlinear beams and shells. In this connection, the specific kinematic assumptions are taken into account by the explicit incorporation of holonomic constraints. This approach facilitates the straightforward extension to flexible multibody dynamics by including additional constraints due to the interconnection of rigid and flexible bodies. We further address the design of energy-momentum schemes for the stable numerical integration of the underlying finite-dimensional mechanical systems.

Flexible Multibody Dynamics Formulation by Hamilton’s Equations

2010 ◽  
Author(s):  
Martin M. Tong
Keyword(s):  
Multibody Dynamics ◽  
Multibody System ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Flexible Body ◽  
Flexible Multibody System ◽  
Generalized Coordinates ◽  
Flexible Multibody ◽  
The Matrix

Numerical solution of the dynamics equations of a flexible multibody system as represented by Hamilton’s canonical equations requires that its generalized velocities q˙ be solved from the generalized momenta p. The relation between them is p = J(q)q˙, where J is the system mass matrix and q is the generalized coordinates. This paper presents the dynamics equations for a generic flexible multibody system as represented by p˙ and gives emphasis to a systematic way of constructing the matrix J for solving q˙. The mass matrix is shown to be separable into four submatrices Jrr, Jrf, Jfr and Jff relating the joint momenta and flexible body mementa to the joint coordinate rates and the flexible body deformation coordinate rates. Explicit formulas are given for these submatrices. The equations of motion presented here lend insight to the structure of the flexible multibody dynamics equations. They are also a versatile alternative to the acceleration-based dynamics equations for modeling mechanical systems.

Flexible Multibody Dynamics Based on a Fully Cartesian System of Support Coordinates

1993 ◽  
Vol 115 (2) ◽  
pp. 294-299 ◽  
Author(s):  
N. Vukasovic ◽  
J. T. Celigu¨eta ◽  
J. Garci´a de Jalo´n ◽  
E. Bayo
Keyword(s):  
Multibody Dynamics ◽  
Reference Frames ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Body Motion ◽  
Cartesian System ◽  
Flexible Multibody ◽  
Local Reference ◽  
Rigid Multibody

In this paper we present an extension to flexible multibody systems of a system of fully cartesian coordinates previously used in rigid multibody dynamics. This method is fully compatible with the previous one, keeping most of its advantages in kinematics and dynamics. The deformation in each deformable body is expressed as a linear combination of Ritz vectors with respect to a local frame whose motion is defined by a series of points and vectors that move according to the rigid body motion. Joint constraint equations are formulated through the points and vectors that define each link. These are chosen so that a minimum use of local reference frames is done. The resulting equations of motion are integrated using the trapezoidal rule combined with fixed point iteration. An illustrative example that corresponds to a satellite deployment is presented.

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.

Procedure for non-smooth contact for planar flexible beams with cone complementarity problem

2020 ◽  
pp. 146441932095745 ◽  
Author(s):  
Xinxin Yu ◽  
Marko K Matikainen ◽  
Ajay B Harish ◽  
Aki Mikkola
Keyword(s):  
Rigid Body ◽  
Multibody Dynamics ◽  
Complementarity Problem ◽  
Arbitrary Shape ◽  
Penalty Method ◽  
Nonlinear Complementarity Problem ◽  
Flexible Multibody Dynamics ◽  
Detection Algorithm ◽  
Contact Detection ◽  
Flexible Multibody

Contact description plays an important role in modeling of applications involving flexible multibody dynamics. Example of such applications include contact between a belt and pulley, crash-worthiness analysis in aerospace and automotive engineering. Approaches such as the linear complementarity problem (LCP), nonlinear complementarity problem (NCP) and penalty method have been proposed for contact detection and imposition of contact constraints. Contact description within multibody dynamics, however, continues to be a challenging topic, particularly in the case of flexible bodies. This paper describes and compares the use of two contact descriptions in the framework of flexible multibody dynamics; (1) the use of nonlinear cone complementarity approach (CCP) and (2) the penalty method. Both contact models are presented together with a master-slave detection algorithm. The modified form of node-to-node approach presented facilitates creation of pseudo-nodes where gap function can be calculated. This reduces the cumbersome effort of contact search. Since large deformations can be an important phenomenon in flexible multibody applications, beam elements based on the absolute nodal coordinate formulation (ANCF) are implemented in this study. To make a comparison of two approaches, the damping component is included in the penalty method by using the continuous contact model introduced by Hunt and Crossley. Numerical results are based on the simulation of ANCF beam contact with rigid ground, rigid body with an arbitrary shape and pendulum contact. Although kinematic results show a good agreement between both approaches when the coefficient of restitution is zero, the unphysical interpenetration appears in the penalty method. Nonlinear minimization problem solved by CCP approach helps to prevent the penetration during contact event. Furthermore, the proposed contact detection algorithm is proved to be capable of being used for multiple contact between beam and arbitrary shape rigid body; different contact types, such as side-by-side and corner-by-side, can be performed without prediction.

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.

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