Introduction

The first two chapters provided the basics of small-rotation dynamics with applications to rigid bodies or near rigid bodies. The purpose of this chapter is to provide an introduction to the finite element method (FEM) of structural analysis to the limited extent necessary for the use of this textbook. That is, the present discussion of the FEM is limited mostly to one-dimensional structural elements.Specifically, all the example problems and exercises deal only with linear, beam finite elements and linear,spring finite elements. Neither of these elements require the finite series sophistication of two- orthree-dimensional elements such as plate or solid finite elements. A very brief introduction to multidimensional finite elements is presented in Endnote (). If the reader is already familiar with the FEM, then this chapter can be skipped, particularly because the next chapter provides ample further review of this topic. Since the finite element method is so extensively used for, and so particularly suited to, structural dynamics analyses, no other method of structural analysis is used for the calculations presented in this textbook. For the sake of instruction, the use of the FEM in this textbook is oriented to hand calculations rather than the use of one of the many available and routinely used commercial software programs that all do essentially the same things and differ only in style. Thus the reader should be able to gain insight into what all such FEM programs need to do.

Introduction

The purpose of this chapter is to introduce damping forces into the structural equations of motion. Simply speaking, damping forces are internal or external friction forces that dissipate the energy of the structural system. Although damping forces are usually much smaller than their companion inertia and elastic forces, they nevertheless can have a significant affect on a vibratory motion, especially after many periods of vibration, or when the system is vibrating at one of certain important frequencies called the system's natural frequencies. This chapter describes various ways of characterizing damping and explains how the damping properties of a vibratory system can be measured. Solutions for the motion of one-DOF systems are presented for force free and certain applied forces to better explain the role that damping plays in structural systems.

Descriptions of Damping Forces

When an actual, force free, structural system is set in motion by means of initial deflections or initial velocities, or both, any point within the system generally vibrates with amplitudes that are very little different over short time intervals; that is, time intervals lasting typically five or fewer periods of the vibration. Figure 5.1(a) shows the calculated amplitude–time trace of such a vibration where the period t of the vibration is 1 sec and the initial displacement has a unit value. As will soon be seen, the sinusoidal expression that describes the force free motion of a one-DOF undamped system, has to be modified, in this case by an exponential multiplier, when one representative form of system damping is present.

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).