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.

Fluid mechanics impinges on practically all areas of human endeavour. But it is not easy to grasp its principles and ramifications in all of its diverse manifestations. Industrial applications usually require the numerical solution of the equations of motion of a fluid on a very large scale, perhaps coupled in a complicated manner to equations describing the response of solid structures in contact with the fluid. There has developed a tendency to regard the subject as defined solely by its governing equations whose treatment by numerical methods can furnish the solution of any problem.

There are actually many practical problems that are not yet amenable to full numerical evaluation in a reasonable time, even on the fastest of present-day computers. It is therefore important to have a proper theoretical understanding that will permit sensible simplifications to be made when formulating a problem. As in most technical subjects such understanding is acquired by detailed study of highly simplified ‘model problems’. Many of these problems fall within the realm of classical fluid mechanics, which is often criticised for its emphasis on ideal fluids and potential flow theory. The criticism is misplaced, however: For example, potential flow methods provide a good first approximation to airfoil theory, and ‘free-streamline’ theory (pioneered in its modern form by Chaplygin) permits the two-dimensional modelling of complex flows involving separation and jet formation.

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

Steady free-streamline flows of water when gravitational forces can be neglected have been discussed in §3.7. Most unsteady free-streamline problems are intractable except by numerical means and generally become more so when gravitational forces are important. However, flows involving gravity where the unsteady motion is a ‘small’ perturbation of a relatively simple mean state occur frequently in the form of surface waves. In the absence of motion the free surface of a liquid in equilibrium under gravity is often ‘horizontal’. A disturbance applied locally that distorts the surface brings into play gravitational restoring forces that cause the disturbance to spread out over the surface in the form of ‘waves’. The waves carry energy away from the source region, propagating parallel to the mean free surface. The agitation produced by a passing wave and the energy flux is generally in the form of a transient disturbance of the fluid particles (around approximately closed particle paths), which are not in themselves transported to any great extent by the wave, and the influence of the wave on fluid at depths exceeding a characteristic wavelength tends to be negligible. In this section these general properties of surface gravity waves are discussed and illustrated by simple examples.

Conditions at the free surface

Consider the simplest case of water whose free surface in equilibrium can be regarded as horizontal and in the plane z = 0 of the coordinate axes (x, y, z), where z increases vertically upwards (Figure 5.1.1).

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).