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2015 | Buch

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Concepts and Formulations for Spatial Multibody Dynamics

verfasst von: Paulo Flores

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Applied Sciences and Technology

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

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Über dieses Buch

This book will be particularly useful to those interested in multibody simulation (MBS) and the formulation for the dynamics of spatial multibody systems. The main types of coordinates that can be used in the formulation of the equations of motion of constrained multibody systems are described. The multibody system, made of interconnected bodies that undergo large displacements and rotations, is fully defined.
Readers will discover how Cartesian coordinates and Euler parameters are utilized and are the supporting structure for all methodologies and dynamic analysis, developed within the multibody systems methodologies. The work also covers the constraint equations associated with the basic kinematic joints, as well as those related to the constraints between two vectors.
The formulation of multibody systems adopted here uses the generalized coordinates and the Newton-Euler approach to derive the equations of motion. This formulation results in the establishment of a mixed set of differential and algebraic equations, which are solved in order to predict the dynamic behavior of multibody systems. This approach is very straightforward in terms of assembling the equations of motion and providing all joint reaction forces.
The demonstrative examples and discussions of applications are particularly valuable aspects of this book, which builds the reader’s understanding of fundamental concepts.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Definition of Multibody System

Abstract

This chapter presents a general view of multibody system concept and definition by describing the main features associated with spatial systems. The mechanical components, which can be modeled as rigid or flexible, are constrained by kinematic pair of different types. Additionally, the bodies can be actuated upon by force elements and external forces due to interaction with environment. This chapter also presents some examples of application of multibody systems that can include automotive vehicles, mechanisms, robots and biomechanical systems.

Paulo Flores

Chapter 2. Fundamental Concepts in Multibody Dynamics

Abstract

In this chapter, the fundamental ingredients related to formulation of the equations of motion for multibody systems are described. In particular, aspects such as degrees of freedom, types of coordinates, basic kinematics joints and types of analysis in multibody systems are briefly characterized. Illustrative examples of application are also presented to better clarify the fundamental issues for spatial rigid multibody systems, which are of crucial importance in the formulation development of mathematical models of mechanical systems, as well as its computational implementation.

Paulo Flores

Chapter 3. Global and Local Coordinates

Abstract

This chapter described the global and local coordinate systems utilized in the formulation of spatial multibody systems. Global coordinate system is considered in the present work to denote the inertia frame. Additionally, body-fixed coordinate systems, also called local coordinate systems, are utilized to describe local properties of points that belong to a particular body. Furthermore, the process of transforming local coordinates into global coordinates is characterized by considering a transformation matrix. In the present work, Cartesian coordinates are utilized to locate the center of mass of each rigid body, as well as the location of any point that belongs to a body.

Paulo Flores

Chapter 4. Euler Angles, Bryant Angles and Euler Parameters

Abstract

This chapter deals with the different approaches for describing the rotational coordinates in spatial multibody systems. In this process, Euler angles and Bryant angles are briefly characterized. Particular emphasis is given to Euler parameters, which are utilized to describe the rotational coordinates in the present work. In addition, for all the types of coordinates considered in this chapter, a characterization of the transformation matrix is fully described.

Paulo Flores

Chapter 5. Angular Velocity and Acceleration

Abstract

In this chapter, a complete characterization of the angular velocity and angular acceleration for rigid bodies in spatial multibody systems are presented. For both cases, local and global formulations are described taking into account the advantages of using Euler parameters. In this process, the transformation between global and local components of the angular velocity and time derivative of the Euler parameters are analyzed and discussed in this chapter.

Paulo Flores

Chapter 6. Vector of Coordinates, Velocities and Accelerations

Abstract

This chapter describes the how the vector of coordinates are defined in the formulation of spatial multibody systems. For this purpose, the translational motion is described in terms of Cartesian coordinates, while rotational motion is specified using the technique of Euler parameters. This approach avoids the computational difficulties associated with the singularities in the case of using Euler angles or Bryant angles. Moreover, the formulation of the velocities vector and accelerations vector is presented and analyzed here. These two sets of vectors are defined in terms of translational and rotational coordinates.

Paulo Flores

Chapter 7. Kinematic Constraint Equations

Abstract

This chapter presents a general methodology for the formulation of the kinematic constraint equations at position, velocity and acceleration levels. Also a brief characterization of the different type of constraints is offered, namely the holonomic and nonholonomic constraints. The kinematic constraints described here are formulated using generalized coordinates. The chapter ends with a general approach to deal with the kinematic analysis of multibody systems.

Paulo Flores

Chapter 8. Basic Constraints Between Two Vectors

Abstract

This chapter deals with the characterization of the basic constraints between two vectors. This issue plays a crucial role in the formulation of constraint equations for mechanical joints. In particular, relations between two parallel and two perpendicular vectors are derived. Moreover, formulation for a vector that connects two generic points is presented. The material described here is developed under the framework of multibody systems formulation for spatial systems.

Paulo Flores

Chapter 9. Kinematic Joints Constraints

Abstract

The kinematic joints constraints for several types of mechanical joints are derived here. Special attention is given to the spherical joint, revolute joint and spherical-spherical joint. In this process, the fundamental issues associated with kinematic constraints are developed, namely the right-hand side of the acceleration constraint equations and the contributions to the Jacobin matrix. The material presented in this chapter is developed under the framework of multibody systems formulation for spatial systems.

Paulo Flores

Chapter 10. Equations of Motion for Constrained Systems

Abstract

In this chapter, the formulation of motion’s equations of multi-rigid body systems is described. The generalized coordinates are the centroidal Cartesian coordinates, being the system configuration restrained by constraint equations. The present formulation uses the Newton-Euler’s equations of motion, which are augmented with the constraint equations that lead to a system of differential algebraic equations. This formulation is straightforward in terms of assembling the equations of motion and providing all reaction forces.

Paulo Flores

Chapter 11. Force Elements and Reaction Forces

Abstract

In the present chapter some of the most relevant applied forces and joint reaction forces are introduced. There are many types of forces that can be present in multibody systems, such as gravitational forces, spring-damper-actuator forces, normal contact forces, tangential or frictional forces, external applied forces and moments, forces due to elasticity of bodies, and thermal, electrical and magnetic forces. However, only the first six types of forces are relevant in the multibody systems of common application.

Paulo Flores

Chapter 12. Methods to Solve the Equations of Motion

Abstract

This chapter presents several methods to solve the equations of motion of spatial multibody systems. In particular, the standard approach, the Baumgarte method, the penalty method and the augmented Lagrangian formulation are revised here. In this process, a general procedure for dynamic analysis of multibody systems based on the standard Lagrange multipliers method is described. Moreover, the implications in terms of the resolution of the equations of motion, accuracy and efficiency are also discussed in this chapter.

Paulo Flores

Chapter 13. Integration Methods in Dynamic Analysis

Abstract

This chapter describes the main integration algorithms utilized in the resolution of the dynamics equations of motion. Particular emphasis is paid to the Euler method, Runge-Kutta approach and Adams predictor-corrector method that allows for the use of variable time steps during the integration process. The material presented here, relative to numerical integration of ordinary differential equations, follows that of any undergraduate text on numerical analysis.

Paulo Flores

Chapter 14. Correction of the Initial Conditions

Abstract

This chapter presents a general approach to deal with the correction of the initial conditions at the position and velocity levels. This procedure is of paramount importance to avoid constraints violation during the numerical resolution of the equations of motion. The material presented here closely follows the standard methodologies available in the literature. Thus, in this chapter, a simple and efficient approach to correct the initial conditions at the position and velocity levels is revised.

Paulo Flores

Chapter 15. Demonstrative Example of Application

Abstract

In this chapter a simple pendulum is considered as a demonstrative example of application of the methodologies described in the previous paragraphs. This example allows for the comparison of the different methods to solve the equations of motion in terms of accuracy and efficiency. Finally, the main concluding remarks of the material presented here are summarized and analyzed.

Paulo Flores

Titel: Concepts and Formulations for Spatial Multibody Dynamics
verfasst von: Paulo Flores
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-16190-7
Print ISBN: 978-3-319-16189-1
DOI: https://doi.org/10.1007/978-3-319-16190-7

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