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2011 | Buch

Vectors and tensors Kapitel lesen Erstes Kapitel lesen

Flexible Multibody Dynamics

verfasst von: O. A. Bauchau

Verlag: Springer Netherlands

Buchreihe : Solid Mechanics and Its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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Über dieses Buch

The author developed this text over many years, teaching graduate courses in advanced dynamics and flexible multibody dynamics at the Daniel Guggenheim School of Aerospace Engineering of the Georgia Institute of Technology.

The book presents a unified treatment of rigid body dynamics, analytical dynamics, constrained dynamics, and flexible multibody dynamics. A comprehensive review of numerical tools used to enforce both holonomic and nonholonomic constraints is presented. Advanced topics such as Maggi’s, index-1, null space, and Udwadia and Kalaba’s formulations are presented because of their fundamental importance in multibody dynamics. Methodologies for the parameterization of rotation and motion are discussed and contrasted. Geometrically exact beams and shells formulations, which have become the standard in flexible multibody dynamics, are presented and numerical aspects of their finite element implementation detailed. Methodologies for the direct solution of the index-3 differential-algebraic equations characteristic of constrained multibody systems are presented. It is shown that with the help of proper scaling procedures, such equations are not more difficult to integrate than ordinary differential equations.

This book is illustrated with numerous examples and should prove valuable to both students and researchers in the fields of rigid and flexible multibody dynamics.

Inhaltsverzeichnis

Frontmatter

Basic tools and concepts

Frontmatter
1. Vectors and tensors
Abstract
Vectors and tensors are basic tools for the formulation of kinematics and dynamics problems. This chapter introduces notations and the fundamental operations on vectors and tensors that will be used throughout this book.
O. A. Bauchau
2. Coordinate systems
Abstract
The practical description of dynamical systems involves a variety of coordinates systems. While the Cartesian coordinates discussed in section 2.1 are probably the most commonly used, many problems are more easily treated with special coordinate systems. The differential geometry of curves is studied in section 2.2 and leads to the concept of path coordinates, treated in section 2.3. Similarly, the differential geometry of surfaces is investigated in section 2.4 and leads to the concept of surface coordinates, treated in section 2.5. Finally, the differential geometry of three-dimensional maps is studied in section 2.6 and leads to orthogonal curvilinear coordinates developed in section 2.7.
O. A. Bauchau
3. Basic principles
Abstract
This chapter reviews the basic principles of dynamics. Newton’s laws are the foundation of mechanics and dynamics and deal with the behavior of particles subjected to forces. Section 3.1 presents Newton’s three laws and the principle of work and energy. Section 3.2 introduces the concept of conservative forces that play a fundamental role in dynamics. The principle of conservation of energy is discussed in section 3.2.1.
O. A. Bauchau
4. The geometric description of rotation
Abstract
The most natural way of describing rotations is rooted in their geometric representation, which is the focus of this chapter. More abstract approaches, however, also exist and will be presented in chapter 13.
O. A. Bauchau

Rigid body dynamics

Frontmatter
5. Kinematics of rigid bodies
Abstract
Newton’s laws deal with the dynamic behavior of a single particle and Euler’s laws generalize the analysis to the case of a system of particles. Rigid bodies form a special case of “systems of particles,” and their dynamic behavior is studied in depth in chapter 6. This chapter focuses on the kinematics of rigid bodies, i.e., the description of the motion of rigid bodies without consideration of the forces that create this motion.
O. A. Bauchau
6. Kinetics of rigid bodies
Abstract
In section 3.4, the dynamic response of a system of particles subjected to both internally and externally applied loads is studied and leads to Euler’s first and second laws [14, 15]. Rigid bodies can be viewed as systems of particles subjected to both internal and external forces. The former forces are those that maintain the shape of the rigid body. By definition, a rigid body is one for which the distance between any two of its particles remains constant at all times. The displacement field of the rigid body must satisfy the kinematic constraints developed in section 5.1 and its velocity field those described in section 5.2.
O. A. Bauchau

Concepts of analytical dynamics

Frontmatter
7. Basic concepts of analytical dynamics
Abstract
Newtonian mechanics deals with the response of particles to externally applied loads and Euler generalized these concepts to systems of particles. For simple systems of particles, it is convenient to use Cartesian coordinates to represent the configuration of the system, butmore often than not, other types of coordinates are used as well. For instance, path or surface coordinates were introduced in chapter 2. The manipulation of finite rotation also plays an important role in dynamics and was studied in depth in chapter 4.
O. A. Bauchau
8. Variational and energy principles
Abstract
This chapter investigates applications of the principles of analyticalmechanics developed in chapter 7 to dynamical systems. First, the principle of virtual work presented in section 7.5 for static problems will be generalized to dynamic problems, leading to d’Alembert’s principle, see section 8.1. Next, Hamilton’s principle is presented in section 8.2 as an integral version of d’Alembert’s principle. Finally, Lagrange’s formulation is presented in section 8.3, leading to Lagrange’s equations of motion.
O. A. Bauchau

Flexible multibody dynamics

Frontmatter
9. Constrained systems: preliminaries
Abstract
In the previous chapter, variational and energy principles were derived for dynamical system. Various formulations were addressed including d’Alembert’s principle, Hamilton’s principle and Lagrange’s formulation. In all cases, developments were limited to unconstrained dynamical systems. This means that the number of generalized coordinates used to represent the configuration of the system was equal the number of degrees of freedom of the system.
O. A. Bauchau
10. Constrained systems: classical formulations
Abstract
Chapter 8 presented variational and energy principles for unconstrained dynamical system. This chapter generalizes these formulations to enable the treatment of constrained systems. D’Alembert’s principle is treated in section 10.1. The generalization of Hamilton’s principle and Lagrange’s formulation to systems with holonomic constraints is presented in section 10.2 and section 10.3 generalizes the same formulations to systems with nonholonomic constraints.
O. A. Bauchau
11. Constrained systems: advanced formulations
Abstract
Multibody systems are characterized by two distinguishing features: system components undergo finite relative rotations and these components are connected by mechanical joints that impose restrictions on their relative motion. Finite rotations introduce geometric nonlinearities, hence, multibody systems are inherently nonlinear. Mechanical joints, such as the lower pair joints presented in section 10.4, result in algebraic constraints leading to a set of governing equations that combines differential and algebraic equations.
O. A. Bauchau
12. Constrained systems: numerical methods
Abstract
The classical and advanced formulations presented in chapter 10 and 11, respectively, provide the theoretical background for the analysis of constrained dynamical systems. In this chapter, practical numerical algorithm are described and compared.
O. A. Bauchau

Parameterization of rotation and motion

Frontmatter
13. Parameterization of rotation
Abstract
The effective description of rotations has led to the development of numerous parameterization techniques presenting various properties and advantages, as described in the following review papers [239, 240, 241, 242, 243, 244, 245].Whether originating from geometric, algebraic, or matrix approaches, parameterization of rotation is most naturally categorized into two classes: vectorial and non-vectorial parameterizations. The former refers to parameterization in which a set of parameters, sometimes called rotational “quasi-coordinates,” define a geometric vector, whereas the latter cannot be cast in the form of a vector. These two types of parameterizations are sometimes denoted as invariant and non-invariant parameterization, respectively.
O. A. Bauchau
14. Parameterization of motion
Abstract
While the parameterization of rotation discussed in chapter 13 has received wide attention, much less emphasis has been placed on that of motion. This is probably due to the fact that the analysis of motion is often described in terms of rather abstract mathematical formulations.
O. A. Bauchau

Flexible multibody dynamics

Frontmatter
15. Flexible multibody systems: preliminaries
Abstract
Multibody systems are characterized by two distinguishing features: system components undergo finite relative rotations and these components are connected by mechanical joints that impose restrictions on their relative motion. Broadly speaking, multibody systems can be divided into three categories, rigid multibody systems, linearly elastic multibody systems, and nonlinearly elastic multibody systems. This classification and its implication on modeling techniques for multibody systems are discussed in section 15.1.
O. A. Bauchau
16. Formulation of flexible elements
Abstract
This chapter deals with the formulations of flexible elements such as flexible joints, cables, beams, and plates and shells, which are presented in sections 16.1, 16.2, 16.3, and 16.4, respectively. In all cases, geometrically exact formulations are derived, i.e., the displacements and rotations of the elements are arbitrarily large, although strain components are assumed remain small, a feature that significantly simplifies the governing equations of motion of these structural components.
O. A. Bauchau
17. Finite element tools
Abstract
Numerous textbooks [324, 197, 198] present detailed development of the theoretical and numerical concepts underpinning the finite element method. Similar developments are clearly beyond the scope of this text. The present chapter focuses on specific details of the finite element method that are relevant to its application to flexible multibody systems. Techniques for interpolation of displacement and specially rotation fields are presented in sections 17.1 and 17.2, respectively.
O. A. Bauchau
18. Mathematical tools
O. A. Bauchau
Backmatter
Metadaten
Titel
Flexible Multibody Dynamics
verfasst von
O. A. Bauchau
Copyright-Jahr
2011
Verlag
Springer Netherlands
Electronic ISBN
978-94-007-0335-3
Print ISBN
978-94-007-0334-6
DOI
https://doi.org/10.1007/978-94-007-0335-3

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