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.

THIS IS ONE OF THE LONGER chapters in this book and should be revisited many times at various levels. To cover in detail the subject of the properties of gases and liquids would take an entire book. Reference [1] is a classic example of such a book. We begin our study of properties by defining a few basic terms and concepts. We follow this by examining ideal-gas properties that originate from the equation of state and calorific equations of state. Various approaches for obtaining properties of nonideal gases, liquids, and solids follow. We emphasize the properties of substances that have coexisting liquid and vapor phases. The concept of illustrating processes graphically, using thermodynamic property coordinates (i.e., T– and P– coordinates), is developed.

IN THIS CHAPTER, we extend the overarching mass and energy conservation principles to reacting systems. In addition to conserving mass in an overall sense, the mass of individual elements must also be conserved in the transformation of reactants to products. Complicating energy conservation (the first law of thermodynamics) is the need to account for the potential energies associated with the making and breaking of chemical bonds. To accomplish this, we introduce the concept of standardized enthalpies. We introduce other new concepts related to mass conservation (e.g., various measures of stoichiometry) and energy conservation (e.g., adiabatic flame temperatures and fuel heating values). The chapter also applies the theoretical developments to practical steady-flow systems (e.g., furnaces, boilers, and combustors). Although the chapter focuses on combustion, the ideas developed here apply to other reacting systems as well.

CHAPTER 10 CONSIDERED MIXTURES OF ideal gases. In this chapter, we will apply a mixture analysis to investigate air–water mixtures, referred to as moist air. Since the water content in the air is relatively low, the partial pressure of the water is low. At low partial pressures, the water vapor can be approximated as an ideal gas and the moist air is an ideal-gas mixture. This chapter will first define some terms commonly used for moist air: specific humidity, relative humidity, and dew point. The analysis of moist air will then be used in several common applications: evaporative coolers, humidifiers, air conditioners, dehumidifiers, and cooling towers.