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This fully revised fifth edition provides comprehensive coverage of flexible multibody system dynamics. Including an entirely new chapter on the integration of geometry, durability analysis, and design, it offers clear explanations of spatial kinematics, rigid body dynamics, and flexible body dynamics, and uniquely covers the basic formulations used by the industry for analysis, design, and performance evaluation. Included are methods for formulating dynamic equations, the floating frame of reference formulation used in small deformation analysis, and the absolute nodal coordinate formulation used in large deformation analysis, as well as coverage of industry durability investigations. Illustrated with a wealth of examples and practical applications throughout, it is the ideal text for single-semester graduate courses on multibody dynamics taken in departments of aerospace and mechanical engineering, and for researchers and practicing engineers working on a wide variety of flexible multibody systems.
The primary purpose of this book is to develop methods for the dynamic analysis of multibody systems (MBS) that consist of interconnected rigid and deformable components. In that sense, the objective may be considered as a generalization of methods of structural and rigid body analysis. Many mechanical and structural systems such as vehicles, space structures, robotics, mechanisms, and aircraft consist of interconnected components that undergo large translational and rotational displacements. Figure 1.1 shows examples of such systems that can be modeled as multibody systems. In general, a multibody system is defined to be a collection of subsystems called bodies, components, or substructures. The motion of the subsystems is kinematically constrained because of different types of joints, and each subsystem or component may undergo large translations and rotational displacements.
In this chapter, approximation methods are used to formulate a finite set of dynamic equations of motion of multibody systems that contain interconnected deformable bodies. As shown in Chapter 3, the dynamic equations of motion of the rigid bodies in the multibody system can be defined in terms of the mass of the body, the inertia tensor, and the generalized forces acting on the body. On the other hand, the dynamic formulation of the system equations of motion of linear structural systems requires the definition of the system mass and stiffness matrices as well as the vector of generalized forces. In this chapter, the formulation of the equations of motion of deformable bodies that undergo large translational and rotational displacements are developed using the floating frame of reference (FFR) formulation. It will be shown that the equations of motion of such systems can be written in terms of a set of inertia shape integrals in addition to the mass of the body, the inertia tensor, and the generalized forces that appear in the dynamic formulation of rigid body system equations of motion and the mass and stiffness matrices and the vector of generalized forces that appear in the dynamic equations of linear structural systems. These inertia shape integrals that depend on the assumed displacement field appear in the nonlinear terms that represent the inertia coupling between the reference motion and the elastic deformation of the body. It will be also shown that the deformable body inertia tensor depends on the elastic deformation of the body, and accordingly it is an implicit function of time.
This chapter provides explanations of some of the fundamental issues addressed in this book. It also provides detailed derivations of some of the important equations presented in previous chapters. The first two sections of this chapter show the detailed derivation of the quadratic velocity centrifugal and Coriolis force vector of the spatial flexible body presented in Chapter 5. The final expression of these forces is obtained using two different approaches; the kinetic energy and the virtual work. It is also shown in Section 3 of this chapter how a general expression of these forces that is applicable to any set of orientation parameters can be obtained. This is the expression used in the generalized Newton–Euler equations presented in Chapter 5 of the book. The generalized centrifugal and Coriolis inertia forces associated with any set of orientation parameters including Euler angles can be obtained from the forces that appear in the Newton–Euler equations using a simple velocity transformation.
Thus far, only the dynamics of multibody systems consisting of interconnected rigid bodies has been discussed. In Chapter 2, methods for the kinematic analysis of the rigid frames of reference were presented and many useful kinematic relationships and identities were developed. These kinematic equations were used in Chapter 3 to develop general formulations for the dynamic differential equations of motion of multi-rigid-body systems. In rigid body dynamics, it is assumed that the distance between two arbitrary points on the body remains constant. This implies that when a force is applied to any point on the rigid body, the resultant stresses set every other point in motion instantaneously, and as shown in the preceding chapter, the force can be considered as producing a linear acceleration for the whole body together with an angular acceleration about its center of mass. The dynamic motion of the body, in this case, can be described using Newton–Euler equations, developed in the preceding chapter.
In the preceding chapter, methods for the kinematic analysis of moving frames of reference were presented. The kinematic analysis presented in that chapter was of a preliminary nature and is fundamental for understanding the dynamic motion of moving rigid bodies or coordinate systems. In this chapter, techniques for developing the dynamic equations of motion of multibody systems (MBS) consisting of interconnected rigid bodies are introduced. The analysis of multibody systems consisting of deformable bodies that undergo large translational and rotational displacements will be deferred until we discuss in later chapters some concepts related to the body deformation. In the first three sections, a few basic concepts and definitions to be used repeatedly in this book are introduced. In these sections, the important concepts of the system generalized coordinates, holonomic and nonholonomic constraints, degrees of freedom, virtual work, and the system generalized forces are discussed. Although the reader previously may very well have met some, or even all, of these concepts and definitions, they are so fundamental for our purposes that it seems desirable to present them here in some detail. Since the direct application of Newton’s second law becomes difficult when large-scale multibody systems are considered, in Section 4, D’Alembert’s principle is used to derive Lagrange’s equation, which circumvents to some extent some of the difficulties found in applying Newton’s second law as demonstrated by the discussion and example presented in Sections 5 and 6. In contrast to Newton’s second law, the application of Lagrange’s equation requires scalar quantities such as the kinetic energy, potential energy, and virtual work. In Sections 7 and 8 the variational principles of dynamics, including Hamilton’s principle, are presented. Hamilton’s principle can also be used to derive the MBS dynamic equations of motion from scalar quantities. This chapter is concluded by discussing the numerical procedures and their relationship to the Lagrange–D’Alembert principle and by developing the equations of motion of multibody systems consisting of interconnected rigid components.
In the classical finite-element (FE) formulation for beams, plates, and shells infinitesimal rotations are used as nodal coordinates. As a result, beams, plates, and shells are not considered as isoparametric elements. Rigid body motion of these nonisoparametric elements does not result in zero strains and exact modeling of the rigid body inertia using these elements cannot be obtained. In this chapter, a formulation for the large reference displacement and small deformation analysis of deformable bodies using nonisoparametric finite elements is presented. This formulation, in which infinitesimal rotations are used as nodal coordinates, leads to exact modeling of the rigid body dynamics and results in zero strains under an arbitrary rigid body motion. It is crucial in this formulation that the assumed displacement field of the element can describe an arbitrary rigid body translation. Using this property and an intermediate element coordinate system, a concept similar to the parallel axis theorem used in rigid body dynamics can be applied to obtain an exact modeling of the rigid body inertia for deformable bodies that have complex geometrical shapes. More discussion on the use of the parallel axis theorem in modeling the inertia of rigid bodies with complex geometry is presented in Chapter 8 of this book. It is recommended that the reader reviews the basic materials presented in Chapter 8 in order to recognize that the coordinate systems used to develop the large displacement FE/FFR formulation presented in this chapter are the same as the coordinate systems used to model the complex geometry in the case of rigid body dynamics.
There are two main concerns regarding the use of the classical finite-element (FE) formulations in the large deformation and rotation analysis of flexible multibody systems. First, in the classical FE literature on beams, plates, and shells, infinitesimal rotations are used as nodal coordinates. Such a use of coordinates does not lead to the exact modeling of a simple rigid body motion. Second, lumped mass techniques are used in many FE formulations and computer programs to describe the inertia of the deformable bodies. As will be demonstrated in this chapter, such a lumped mass representation of the inertia also does not lead to exact modeling of the equations of motion of the rigid bodies.
While a body-fixed coordinate system is commonly employed as a reference for rigid components, a floating coordinate system is suggested for deformable bodies that undergo large rotations. When dealing with rigid body systems, the kinematics of the body is completely described by the kinematics of its coordinate system because the particles of a rigid body do not move with respect to a body-fixed coordinate system. The local position of a particle on the body can then be described in terms of fixed components along the axes of this moving coordinate system. In deformable bodies, on the other hand, particles move with respect to the selected body coordinate system, and therefore, a distinction is made between the kinematics of the coordinate system and the body kinematics.
In the virtual prototyping, durability analysis, and design processes, accurate computer modeling of a large number of physics and engineering systems is necessary. For such systems that consist of interconnected bodies, developing credible computer models requires the use of accurate geometry description as well as the analysis techniques described in this book. Nonetheless, virtual prototyping, durability analysis, and product design are currently performed in many industry sectors using three different incompatible systems: computer-aided design (CAD) system for creating the geometry, finite element (FE) software for developing the analysis mesh, and multibody system (MBS) software for constructing and numerically solving the differential/algebraic equations (DAEs) of constrained systems. The use of the three-software technology has resulted in unreliable stress and durability results, significant waste of engineering time and efforts, misrepresentation of significant model details, and significant economic loss.
The primary purpose of this book is to develop methods for the dynamic analysis of multibody systems that consist of interconnected rigid and deformable components. In that sense, the objective may be considered as a generalization of methods of structural and rigid body analysis. Many mechanical and structural systems such as vehicles, space structures, robotics, mechanisms, and aircraft consist of interconnected components that undergo large translational and rotational displacements. Figure 1.1 shows examples of such systems that can be modeled as multibody systems. In general, a multibody system is defined to be a collection of subsystems called bodies, components, or substructures. The motion of the subsystems is kinematically constrained because of different types of joints, and each subsystem or component may undergo large translations and rotational displacements.
Basic to any presentation of multibody mechanics is the understanding of the motion of subsystems (bodies or components). The motion of material bodies formed the subject of some of the earliest researches pursued in three different fields, namely, rigid body mechanics, structural mechanics, and continuum mechanics. The term rigid body implies that the deformation of the body under consideration is assumed small such that the body deformation has no effect on the gross body motion. Hence, for a rigid body, the distance between any two of its particles remains constant at all times and all configurations. The motion of a rigid body in space can be completely described by using six generalized coordinates.