Introduction

The focus of this textbook is on the vibrations of engineering structures, not mechanisms. However, this chapter focuses on pendulums as representative of mechanisms. Pendulums are rarely a part of an engineering structure. However, because the motions of pendulums are familiar to everyone, they do provide a comparatively simple means for both visualizing and explaining some basic aspects of more general vibratory systems. Pendulums also provide an opportunity to consolidate the lessons on dynamics set forth in the first chapter without the complication of dealing with flexible structures. As an aside, pendulums also provide a relatively simple introduction to the quite challenging topic of nonlinear vibrations. Thus, despite their limited relevance to engineering structures in general, this introductory study of structural dynamics begins with the study of the back-and-forth motion of pendulums.

The static equilibrium position (SEP) of any dynamical system is the deflected position of that system in response to all the applied static loads and their support reactions, if any. A stable pendulum system is defined as any system that, when displaced from its static equilibrium position, tends to return to that SEP as a result of the presence of a gravitational force field or other force field. An example of body forces other than gravitational forces that stabilize a structural system is the centrifugal force field acting on a rotating helicopter blade.

No actual structure is rigid. All structures deform under the action of applied loads. When the applied loads vary over time, so, too, do the deflections. The time-varying deflections impart accelerations to the structure. These accelerations result in body forces called inertial loads. Since these inertia loads affect the deflections, there is a feedback loop tying together the deflections and at least the inertial load part of the total loads. When the applied loads result from the action of a surrounding liquid, then the deflections determine all the applied dynamic loads. Therefore, unlike static loads (i.e., slowly applied loads), differential equations based on Newton's laws are required to mathematically describe time-varying load–deflection interactions. Inertial loads can also have the importance of being the largest load set acting on parts of a structure, particularly if the structure is quite flexible.

In order to appreciate how significant time-varying forces can be, consider, for example, the time-varying loads that act on a typical large aircraft. After the aircraft starts its engines, it generally must taxi along taxiways to a runway and then travel along the runway during its takeoff run. Taxiways and runways are not perfectly flat. They have small alternating hills and valleys. As will be examined in a simplified form later in this book, these undulations cause the aircraft to move up and down and rock back and forth on its landing gear, that is, its suspension system.

This textbook is designed to be the basis for a one-semester course in structural dynamics at the graduate level, with some extra material for later self-study. Using this text for senior undergraduates is possible also if those students have had more than one semester of exposure to rigid body dynamics and are well versed in the basics of the linear, stiffness finite element method. This textbook is suitable for structural dynamics courses in aerospace engineering and mechanical engineering. It also can be used in civil engineering at the graduate level when the course focus is on analysis rather than earthquake design. The first two chapters on dynamics should be particularly helpful to civil engineers.

This textbook is a departure from the usual presentation of this material in two important ways. First, from the very beginning, descriptions of system dynamics are based on the simpler-to-use Lagrange equations. To this end, the Lagrange equations are derived from Newton's laws in the first chapter. Second, no organizational distinctions are made between multidegree of freedom systems and single degree of freedom systems. Instead, the textbook is organized on the basis of first writing structural system equations of motion and then solving those equations mostly by means of a modal transformation. Beam and spring stiffness finite elements are used extensively to describe the structural system's linearly elastic forces. If the students are not already confident assemblers of element stiffness matrices, Chapter 3 provides a brief explanation of that material.

Introduction

A knowledge of the rudiments of dynamics is essential to understanding structural dynamics. Thus this chapter reviews the basic theorems of dynamics without any consideration of structural behavior. This chapter is preliminary to the study of structural dynamics because these basic theorems cover the dynamics of both rigid bodies and deformable bodies. The scope of this chapter is quite limited in that it develops only those equations of dynamics, summarized in Section 1.10, that are needed in subsequent chapters for the study of the dynamic behavior of (mostly) elastic structures. Therefore it is suggested that this chapter need only be read, skimmed, or consulted as is necessary for the reader to learn, review, or check on (i) the fundamental equations of rigid/flexible body dynamics and, more importantly, (ii) to obtain a familiarity with the Lagrange equations of motion.

The first part of this chapter uses a vector approach to describe the motions of masses. The vector approach arises from the statement of Newton's second and third laws of motion, which are the starting point for all the material in this textbook. These vector equations of motion are used only to prepare the way for the development of the scalar Lagrange equations of motion in the second part of this chapter. The Lagrange equations of motion are essentially a reformulation of Newton's second law in terms of work and energy (stored work).

Introduction

As discussed in the last part of Chapter 5, digital computer software capabilities have currently reached a point where numerical solutions to very large, linear, structural dynamics problems can be successfully achieved. As an indication of the growth in size of structural models being used in dynamic analyses, note that it is now not uncommon for structural dynamic analyses to employ the same detailed FEM models prepared for the purposes of static stress analyses. As a result of this marked increase in the number of DOF used in analyses, and just as importantly, as part of the clear trend toward automating everything, the integration of the equations of motion is rarely done by any means other than by digital computer-based numerical methods. Although these reasons are sufficient for looking at numerical integration techniques, there are still other important reasons. The foremost of these other reasons is that numerical integration is the only practical approach when material nonlinearities (e.g., plasticity) or geometric nonlinearities are part of the system's mathematical model.

Today, numerical integration is a well-developed field with many textbooks available to provide a comprehensive overview on both simplistic and sophisticated levels. See, for example, Refs. [9.1, 9.2]. Therefore it is appropriate for this textbook to provide only a brief introduction to the popular numerical integration techniques that are particularly suitable for the numerical integration of the ordinary differential equations that result from the modal transformation applied to a finite element model or are suitable for the direct integration of the matrix equation of motion in terms of the original generalized coordinates.

Introduction

The previous four chapters emphasized the advantages of using discrete mass mathematicaxsl models wherein both the structural mass and the nonstructural mass is “lumped” at selected (usually a relatively few) finite element nodes or at short distances from those finite element nodes. The alternative in mass modeling is the seemingly more realistic mathematical model where the mass is distributed throughout each structural element. Such distributed or continuous mass models are not nearly as useful as discrete mass models. However, continuous mass models do have enough instructional value and occasional engineering value that they cannot be wholly ignored. Their instructional value resides in (i) seeing the results of dealing with what is essentially an infinite DOF system; (ii) the reinforcement, and perhaps deeper understanding, obtained through repetition of the same analysis procedures used with discrete mass systems in a different context; and (iii) discovering the very few types of structures which can be usefully described by this much more concise type of modeling. Therefore the purpose of this chapter is to discuss some of those situations where the use of continuous mass models is of some, albeit small, value in the study of structural dynamics.

Again, continuous mass models are practical only in quite restricted circumstances. All cases examined here are limited to structures that are modeled as a single structural element (e.g., one beam or one plate).

The main purpose of Chapter 11 was to introduce information space (I-space) concepts and to provide illustrative examples that aid in understanding. This chapter addresses planning under sensing uncertainty, which amounts to planning in an I-space. Section 12.1 covers general-purpose algorithms, for which it will quickly be discovered that only problems with very few states can be solved because of the explosive growth of the I-space. In Chapter 6, it was seen that general-purpose motion planning algorithms apply only to simple problems. Ways to avoid this were either to develop sampling-based techniques or to focus on a narrower class of problems. It is intriguing to apply sampling-based planning ideas to I-spaces, but as of yet this idea remains largely unexplored. Therefore, the majority of this chapter focuses on planning algorithms designed for narrower classes of problems. In each case, interesting algorithms have been developed that can solve problems that are much more complicated than what could be solved by the general-purpose algorithms. This is because they exploit some structure that is specific to the problem.

An important philosophy when dealing with an I-space is to develop an I-map that reduces its size and complexity as much as possible by obtaining a simpler derived I-space. Following this, it may be possible to design a special-purpose algorithm that efficiently solves the new problem by relying on the fact that the I-space does have the full generality. This idea will appear repeatedly throughout the chapter.

Chapter 3 only covered how to model and transform a collection of bodies; however, for the purposes of planning it is important to define the state space. The state space for motion planning is a set of possible transformations that could be applied to the robot. This will be referred to as the configuration space, based on Lagrangian mechanics and the seminal work of Lozano-Pérez [659, 663, 660], who extensively utilized this notion in the context of planning (the idea was also used in early collision avoidance work by Udupa [947]). The motion planning literature was further unified around this concept by Latombe's book [591]. Once the configuration space is clearly understood, many motion planning problems that appear different in terms of geometry and kinematics can be solved by the same planning algorithms. This level of abstraction is therefore very important.

This chapter provides important foundational material that will be very useful in Chapters 5 to 8 and other places where planning over continuous state spaces occurs. Many of the concepts introduced in this chapter come directly from mathematics, particularly from topology. Therefore, Section 4.1 gives a basic overview of topological concepts. Section 4.2 uses the concepts from Chapter 3 to define the configuration space. After reading this, you should be able to precisely characterize the configuration space of a robot and understand its structure.

So far in Part II it has been assumed that a continuous path sufficiently solves a motion planning problem. In many applications, such as computer-generated animation and virtual prototyping, there is no need to challenge this assumption because models in a virtual environment usually behave as designed. In applications that involve interaction with the physical world, future configurations may not be predictable. A traditional way to account for this in robotics is to use the refinement scheme that was shown in Figure 1.19 to design a feedback control law that attempts to follow the computed path as closely as possible. Sometimes this is satisfactory, but it is important to recognize that this approach is highly decoupled. Feedback and dynamics are neglected in the construction of the original path; the computed path may therefore not even be usable.

Section 8.1 motivates the consideration of feedback in the context of motion planning. Section 8.2 presents the main concepts of this chapter, but only for the case of a discrete state space. This requires less mathematical concepts than the continuous case, making it easier to present feedback concepts. Section 8.3 then provides the mathematical background needed to extend the feedback concepts to continuous state spaces (which includes C-spaces). Feedback motion planning methods are divided into complete methods, covered in Section 8.4, and sampling-based methods, covered in Section 8.5.

Motivation

For most problems involving the physical world, some form of feedback is needed.