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The dynamics of mechanical systems was first stated by Isaac Newton in his of 1687. Newton’s laws form the basis for the derivation of the equations of motion for particles. Modern dynamics is introduced through the use of vectors, free-body diagrams and reference frames. The use of particle mass to represent a body is an idealized concept that provides the simplest model in dynamics. It is important to note that Newton’s second law, in its present form, has been used to derive current dynamic principles such as work and energy. Dynamics is made easy by transforming its vectorial representation to scalar forms by means of using dot products. This was the basis for variational principles. We shall review the dot product and cross product of vectors to see how some of these rules apply. The aim of this chapter is to review the basic principles in Newton’s second law to formulate the equation of motion of problems involving system of particles.

Kinematics is the study of position, velocity and acceleration without the need of forces. Kinematics, however, for years used kinematical relations to design linkages, gears and a number of complex engineering problems. Although in principle we can analyze the motion of a system of degrees of freedom, we often rely on simple models to obtain realistic answers. The concept of a rigid body is a good example where we assume that the body doesn’t deform under the influence of external forces or simply the deformation is negligible. In this chapter, we set the foundation for the kinematical quantities needed to describe a rigid-body motion.

In this chapter, great emphasis is given to automatic generation of the matrices and arrays needed to express the kinematics of a general treelike multibody system. The coefficients that we develop will be in an explicit form ready for computer implementation. In the sequel, we introduce an array that contains the topological information about the structure interconnection and make use of the previously defined kinematics of rigid bodies and extend its definition to general open-structure problems. A more structured formulation for the partial angular velocity and partial velocity arrays and their derivatives is presented. At the end of the chapter the student and engineer are expected to be able to formulate the kinematics of any open mechanical system or chain following the procedures presented herein.

The modeling of forces in the context of multibody dynamics is undoubtedly challenging and important. Although the reader might be familiar with some of those concepts, their contribution in a general system of dynamics is the main focus of this chapter. We make use of concepts developed in Chapter 3 to extend the formalism of forces in the presence of a treelike configuration of interconnected rigid bodies. To this end, a brief review of the concepts of forces and their classification is presented; then a general form of the representation of those forces in the dynamics of multibody systems is developed.

The purpose of this chapter is to give the readers an overview of the task required in the formulation of the equations of motion, so that they can judge the utility of the methods as they apply to multibody dynamics. Most important, the equations are expressed in explicit forms, making use of the previously developed arrays in kinematics for computer application. The equations presented in this chapter are unique and present the explicit form of all the terms needed in the dynamics of MBS. The ‘final’ form of the equations can be achieved through most of the methods and should be viewed as a tool in getting the equations of motion without the use of any standard approach in dynamics.

Dynamics of particles stems from theoretical physics where most of the pioneering work started with Isaac Newton and Leibniz’s discovery of calculus. Calculus of variation was then introduced by Leonard Euler which became an important tool in physics. Joseph-Louis Lagrange introduced the Lagrange function (, ) which depends on a set of generalized coordinates and velocities of a system. The general work of Euler and Lagrange also led to the principle of least action, where differential equations are obtained by minimizing the action over an interval of time. The Hamiltonian formalism of mechanics was then introduced in 1830, where the Hamiltonian function represents the total energy of the system. The Hamilton principle is related to the Lagrange through a transformation called the Legendre transformation. Equations of motion can be derived from the Hamilton principle. While the history of mechanics was based on particle dynamics, Lagrange functions are extended to continuous system- or rigid-body dynamics. Gibbs-Appell introduced a function that deals with acceleration, velocity and position from which the equations of motion are derived. This chapter introduces the formulation for the various ways of deriving the equations of motion and highlights the benefits of each.

In the preceding chapters, treatment of the generalization of kinematics and the equations of motion was done in the presence of no constraints. In general, we assumed that the system is not subjected to any external condition or forces restricting its global motion, in which case the generalized coordinates are looked upon as the independent coordinates of the system. If a multibody system becomes subjected to a kinematic or geometric condition during the cycle of its motion, the multibody system is said to be . If the dynamical system has independent generalized coordinates and constraint equations, the total number of degrees of freedom is given by — . In this chapter, we study the dynamics of multibody systems in the presence of constraints and explore various techniques used to develop the proper governing equations of motion. Dynamics of multibody system requires some intuition where one needs to be able to differentiate between an open treelike system and identify the constraints. Conditions that must be imposed on rigid bodies are usually treated through the generalized forces and closed loops, prescribed motions and any geometrical or kinematical constraints are treated through the use of additional constraint equations to be solved together with the equations of motion.

Constraints often are represented by their position, velocity and acceleration form. When imposing a set of constraints, holonomic or nonholonomic on dynamical systems it is important that the solution must satisfy the constraints at all levels. To prevent numerical irregularities and divergence of a solution, stability of the equations of motion must be preserved. In this chapter we discuss the numerical stability of the governing equations of motion when integrated to yield the time history of a system. These equations are composed of a set of differential and algebraic equations. Since this is a numerical problem, it will be in the interest of students and practicing engineers to see how the problem is resolved. First, we introduce the Baumgarte technique, used to stabilize and reduce the error due to conventional representation of the constraint equations. Second, a method by Amirouche and Ider, which makes use of a modification of the constraints at singular configurations is introduced to regularize the stability of the dynamical equations. The aforementioned problems are still research topics, and further work is needed.

In the context of dynamics and control of multibody systems, we are required to linearize the dynamical equations of motion, so that proper control laws can be applied to study the efforts of vibration on the control of flexible structures. The linearization sets the stage for an important topic in the dynamics of flexible (elastic) bodies undergoing large rotations. The objective of this chapter is to highlight the essential elements in linearization and to identify in vibration, the eigenfunctions needed for a description of the elastic deformation associated with the dynamics of multibody systems. In addition to the above, we present a review of the vibration of continuous beams as they are vital to the analysis of flexible bodies presented in the next two chapters.

In this chapter, we focus our attention on the formulation of the equations of motion of multibody systems with flexible terminal links. It is our interest to see how flexibility (elastic deformation) affects the dynamics, hence providing some insight into how to control it. Vibration of terminal links is more pronounced when high speed of light structure is required. For instance, the accurate positioning of a robotic manipulator requires that we assume a certain vibrating behavior in the control algorithm to be able to predict more precisely the position of its endpoint. In today’s technology, it is not possible to foresee any design of a control block for industrial / space manipulators that would not include the effects of flexibility.