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.