In the previous chapter we addressed zeroth-order variational principles rooted in the fundamental principle of d'Alembert. In this chapter we will focus on the first-order variation of displacement. We will begin with Jourdain's Principle of Virtual Power (Jourdain 1909). The principle is based on the notion of virtual velocity. The derivation of the equations of motion for particles and rigid bodies using Jourdain's Principle closely parallels the derivation of the equations of motion using d'Alembert's Principle in the previous chapter. While Jourdain's Principle is analogous to d'Alembert's Principle, it can be seen as an independent principle of analytical dynamics. Therefore, the material in the previous chapter is not a prerequisite to this chapter, and this chapter will be treated in a stand-alone manner.

Virtual Velocities

Virtual velocities refer to all velocities of a system that satisfy the scleronomic constraints of the system. In the case of virtual velocities, time and position are frozen or stationary.

Jourdain's Principle of Virtual Power

PRINCIPLE 6.1 The virtual power of a system is stationary. That is,

δP = 0.

Additionally, the constraints of the system generate no virtual power,

δPc = 0.

This is known as Jourdain's Principle.

As with d'Alembert's Principle, it is noted that while Jourdain's Principle can be seen as providing an alternate statement of Newton's second law, for interacting bodies, a law of action and reaction (Newton's third law) is still needed. Therefore, when we use Jourdain's Principle to derive the equations of motion for systems of particles/bodies, we will invoke the law of action and reaction.

A Single Particle

Jourdain's Principle for a single point mass with a discrete set of nf external forces, ﹛ f1, …, fnf ﹜, acting on it is expressed as

where δv represents the velocity variations. During these variations, time and position are stationary. That is, δt = 0 and δr = 0.

The traditional starting point in the classical study of mechanics is with the so-called conservation principles. Newton's famous laws of motion form the basis of classical mechanics. We can consider Newton's second law as part of a more general conservation principle; specifically the conservation of translational momentum.

In this chapter we will begin with Newton's laws, which address point masses, and generalize to extended bodies (rigid bodies) using Euler's extension of Newton's laws. In both cases, point masses and rigid bodies, conservation of momentum plays the central role. We will articulate the Newton-Euler Principle as the basis for deriving the equations of motion for both a single rigid body and for systems of interconnected rigid bodies (i.e., kinematic chains). As we apply the conservation of translational and angular momentum to a single rigid body, we will define a set of fundamental inertial properties, including the center of mass and the inertia tensor. Furthermore, as we apply the conservation principles to a set of interconnected rigid bodies, we will articulate the iterative Newton-Euler method for deriving the equations of motion of kinematic chains.

A fundamental concept in Newtonian mechanics is that of an inertial (Galilean) reference frame. This is a frame of reference in which time and space are homogeneous and isotropic; that is, the laws of physics are the same for all points and orientations. All inertial references frames have constant motion (constant translational velocity) with respect to each other. Derivatives will be assumed to be taken with respect to an inertial reference frame, unless otherwise stated.

The Newton-Euler Principle

We begin by addressing a single point mass. Newton's laws of motion are as follows:

1. 1. With respect to an inertial reference frame, a point mass at rest tends to stay at rest. A point mass in motion tends to move at a constant velocity unless acted on by external forces.

2. 2. The sum of forces, f, acting on a point mass impart to it an acceleration, a, in the same direction as the resultant force vector and with a magnitude inversely proportional to the mass, M, of the body. That is, f = Ma.

In the previous two chapters we addressed zeroth- and first-order variational principles. In this chapter we will focus on the second-order variation of displacement. We will begin with Gauss's Principle (Gauss 1829). The principle is based on the notion of virtual acceleration. The derivation of the equations of motion for particles and rigid bodies using Gauss's Principle closely parallels the derivation of the equations of motion using d'Alembert's Principle and Jourdain's Principle in the previous chapters. As with d'Alembert's and Jourdain's Principles, Gauss's Principle can be seen as an independent principle of analytical dynamics. Therefore, as with the previous chapters, this chapter is presented in a stand-alone manner.

Virtual Accelerations

Virtual accelerations refer to all accelerations of a system that satisfy the scleronomic constraints. In the case of virtual accelerations, time, position, and velocity are frozen or stationary.

Gauss's Principle

PRINCIPLE 7.1 The virtual acceleration work of a system is stationary. That is,

δA = 0.

Additionally, the constraints of the system generate no virtual acceleration work,

δAc = 0.

This is known as Gauss's Principle.

As with d'Alembert's Principle and Jourdain's Principle, it is noted that while Gauss's Principle can be seen as providing an alternate statement of Newton's second law, for interacting bodies, a law of action and reaction (Newton's third law) is still needed. Therefore, when we use Gauss's Principle to derive the equations of motion for systems of particles/bodies, we will invoke the law of action and reaction.

A Single Particle

Gauss's Principle for a single point mass with a discrete set of nf external forces, ﹛ f1, …, fnf ﹜, acting on it is expressed as

where δa represents the acceleration variations. During these variations, time, position, and velocity are stationary. That is, δt = 0, δr = 0, and δv = 0.

Dynamics is traditionally defined as the classical study of motion with respect to the physical causes of motion, that is, forces and moments. Kinematics, on the other hand, is concerned with the study of motion without respect to the underlying physical causes. In this sense, kinematics is really a fundamental prerequisite upon which dynamics is constructed.

For the purposes of this text, the terms dynamics and mechanics are taken to be synonymous. The choice of which term is used is based more on the academic community than on a strict technical distinction. The engineering community typically adopts the term dynamics and the physics and applied mathematics communities typically adopt the term mechanics. The term dynamics is predominantly used in this text.

Historical Background

Interest in the dynamics of linked multibody systems has existed throughout much of recorded human history. As an example, representations of the human form in art have included anthropomorphic constructions made up of mechanical elements like those depicted in Giovanni Braccelli's Bizzarie di Varie Figure published in 1624 (see Figure 1.1). Braccelli's art coincided with the birth of the mechanical philosophy of René Descartes, Pierre Gassendi, and others. The mechanical philosophy sought to describe physical phenomena in terms of intricate mechanisms. Decades after the birth of the mechanical philosophy, a systematic theory of mechanics began to flourish with Newtonian mechanics.

Analytical dynamics (historically referred to as analytical mechanics) is identified with a number of formulations of classical mechanics that arose after Isaac Newton published his Philosophiae Naturalis Principia Mathematica in 1687. The cornerstones of Newtonian mechanics are his laws of motion, which were applied to point masses. From a modern perspective, Newton's second law can be seen more generally as a conservation law, specifically, as a law of conservation of momentum. As such, Newtonian mechanics, and its extension to extended (rigid) bodies by Leonhard Euler, are based on two fundamental conservation principles: (1) the conservation of translational momentum and (2) the conservation of angular momentum.

In contrast to the conservation principles underlying the Newton-Euler mechanics, the analytical dynamics that followed were based on variational principles. The conservation of vector quantities like translational and angular momentum was replaced with principles rooted in the variation of scalar quantities like work and energy.

This book addresses the analytical dynamics of multibody systems and is intended for a one- to two-semester advanced graduate-level course in analytical dynamics. The emphasis is on a solid theoretical foundation with examples that concretely illustrate the theory. I have included a chapter on the fundamental mathematics that is helpful in navigating the principles of dynamics. This includes coverage of linear systems and differential geometry. A chapter on kinematics, the study of the geometry of motion, follows. The first chapter, on dynamics, addresses conservation principles, fundamentally the conservation of momentum embodied in the Newton-Euler Principles. Historically, analytical mechanics (dynamics) has referred to the so-called variational principles, rooted in the calculus of variations. Three chapters cover zeroth-, first-, and second-order variational principles, respectively. Lagrangian and Hamiltonian mechanics are among the more well-known formulations arising from variational principles covered in this book. I also cover important, but lesser known, higher-order principles, including Jourdain's Principle of Virtual Power, Gauss's Principle of Least Constraint, and Hertz's Principle of Least Curvature, as well as Kane's formulation and the Gibbs-Appell formulation.

As an aside, it is worth noting that modern theoretical physics emerged out of the classical variational principles. Einstein's general theory of relativity is commonly formulated using Lagrangian mechanics. Dirac was the first to use the Lagrangian in quantum mechanics and provided separate formulations of quantum mechanics and general relativity based on the Hamiltonian formalism. He also provided a generalized formulation of constrained Hamiltonian systems. Additionally, Feynman's path integral formulation of quantum mechanics has its classical ancestry in Hamilton's Principle of Least Action.

After the chapters on the variational principles, I have included a chapter on an alternate formulation of classical dynamics that has found significant utility in the control of robotic systems. The so-called task space formulation of dynamics was pioneered by Khatib under the name of operational space dynamics. It provides a transformation of the configuration space description of system dynamics into a more convenient task-oriented description. An applications-oriented chapter is included on biomechanical systems. This provides a basic overview of musculoskeletal and neuromuscular biomechanics with extensive coverage of application examples using actual anthropometric and muscle property data. The final chapter provides a brief survey of some analytical dynamics software.

In the previous chapter we addressed principles based on the conservation of vector quantities, specifically translational and angular momentum. In this chapter we will address principles rooted in the variation of scalar quantities like work and energy. We will begin with d'Alembert's Principle of Virtual Work, which is an extension of Bernoulli's static principle to dynamics. The principle is based on the notion of virtual displacement. We will refer to d'Alembert's Principle as a zeroth-order variational principle to denote that it is based on the variation of the zeroth-order derivative of displacement. This is in contrast to higher-order variations related to velocity and acceleration, which will be discussed in subsequent chapters.

Virtual Displacements

Virtual displacements refer to all displacements of a system that satisfy the scleronomic constraints of the system. Scleronomic constraints refer to constraints that are not explicitly dependent on time, as opposed to rheonomic constraints, which are explicitly dependent on time. In the case of virtual displacements, time is frozen or stationary.

D'Alembert's Principle of Virtual Work

PRINCIPLE 5.1 The virtual work of a system is stationary. That is,

δW = 0.

Additionally, the constraints of the system perform no virtual work:

δWc = 0.

This is known as d'Alembert's Principle.

It is noted that while d'Alembert's Principle can be seen as providing an alternate statement of Newton's second law, for interacting bodies, a law of action and reaction (Newton's third law) is still needed. Therefore, when we use d'Alembert's Principle to derive the equations of motion for systems of particles/bodies, we will invoke the law of action and reaction.

A Single Particle

D'Alembert's Principle for a single point mass with a discrete set of nf external forces, ﹛ f1, …, fnf ﹜, acting on it is expressed as

where δr represents the displacement variations. During these variations time is stationary. That is, δt = 0.

This chapter provides a brief and very selective survey of some software useful for solving problems in analytical dynamics. I have chosen to include only software intended for academic and industrial use that has been validated for modeling accuracy. Consequently, I have omitted the many dynamics engines used for video games and entertainment purposes.

General Purpose Mathematical Software

A number of powerful tools for mathematical analysis exist. Two of the most popular are Mathematica by Wolfram Research Inc. and MATLAB by The MathWorks Inc. They provide capabilities for both symbolic manipulation of expressions (i.e., computer algebra systems) and numerical computation and evaluation of expressions. While the basic Mathematica and MATLAB applications do not provide high-level programming abstractions specific to multibody system modeling and analysis, they are extremely useful for the low-level symbolic and numerical processing required in solving problems in analytical dynamics (see Figure 10.1 for a screenshot of Mathematica). This has a benefit in the learning process, as they require the student to understand the central concepts of analytical dynamics and to set up the solution procedure in a detailed manner, while alleviating much of the tedium of the algebraic and numerical work.

Packages and Extensions for Multibody Dynamics

Wolfram Research and MathWorks both provide domain-specific extensions for analyzing multibody systems. This gives the user access to high-level abstractions for modeling, simulating, and analyzing multibody systems. These high-level abstractions can be accessed through a programming language or through a diagrammatic interface supporting both acausal and causal modeling. Acausal modeling involves component-based schematic representations of the physical system, whereas causal modeling involves block diagram representations of the underlying mathematical description of the system.

Simscape Multibody (formerly SimMechanics) by The MathWorks Inc. allows users to model multibody systems using a library of blocks associated with bodies, joints, constraints, spatial transformations, force elements, and sensors (see Figure 10.2 for a screenshot of Simscape Multibody). Entire CAD assemblies can be imported along with associated geometry, inertial properties, joint parameters, and constraints. Simscape Multibody then formulates and integrates the equations of motion for the multibody system. Animations are generated during simulation to allow visualization of the motion of the system.