PART I: FREE VIBRATION

Introduction

This last chapter considers the dynamic response of axially loaded structural systems in which the motion is not necessarily confined to the local vicinity of an underlying equilibrium position and dynamic behavior is not necessarily harmonic. In a number of places throughout this book, a statement has been made to the effect that largeamplitude behavior will be described later. We now finally consider such situations, largely in terms of revisiting examples detailed in earlier examples, but now, not relying on certain restrictions, for example, linear, or small-amplitude, behavior. Both free and forced vibrations will be considered.

By way of a simple introduction, we go back to the softening cable example described in Section 3.5, and specifically consider the context of Fig. 3.12. This is a free vibration started (with initial conditions) some distance from any of the three available stable equilibrium points present at this level of loading. Because there is no damping, the total energy is conserved, and thus phase trajectories can be viewed as contours of constant total energy. Figure 16.1(a) illustrates the energy levels as a contour plot, and thus we can view the phase trajectory of Fig. 3.12 living in the second darkest shade within the contours of Fig. 16.1(a). Parts (b)–(d) give specific examples of time series generated (numerically) by different initial conditions. We see that the time series in part (b), which corresponds to the phase trajectory shown in Fig. 3.12, is far from sinusoidal. The time series shown in part (c) has relatively small amplitude with a natural frequency close to that predicted by linear theory; see Fig. 3.2 (but still slightly nonlinear; note the expanded y-axis).

Introduction

Often times, plates and panels are subject to in-plane loads. Their dynamic characteristics are influenced in ways not dissimilar to those of axially loaded beams. However, the modeling of these 2D systems is more challenging, involving for example more boundary conditions. Typical plates also exhibit considerable postbuckled stiffness, even to the extent that buckled plates can fulfill useful design purposes, that is, the elastic critical load and ultimate failure load are quite different.

This chapter introduces some basic concepts from the theory of thin, rectangular plates. The bending of plates has received considerable attention over the years and is well established in the literature. In its simplest context, we might consider a long plate supported on only two opposite sides as analogous to a wide beam. More sophisticated analyses would then incorporate large deflections of relatively thick plates including various shapes and higher-order effects. It is assumed that the reader is somewhat familiar with simple plate bending theory, both in terms of the governing differential equations and energy considerations. Thus this chapter will focus on the interaction of in-plane forces, large deflections, and dynamic response.

Brief Review of the Classical Theory

Consider a flat rectangular plate as shown in Fig. 10.1. The plate has a uniform thickness h, and coordinates x and y describe the middle surface of the plate. The z coordinate is directed vertically upward from this plane.

Introduction

We first consider a number of discrete-link models in which system properties are concentrated at specific locations. The motivation for considering simple mechanical models is that most of the concepts of dynamics and stability issues encountered with continuous systems (e.g., beams, plates) can be observed with discrete systems but are somewhat easier to analyze. In fact the governing equations will tend to be algebraic rather than differential (at least in space), and it is natural to start with a look at systems in which the behavior of the system is completely described by just a single degree of freedom.

An Inverted Pendulum

Consider the simple hinged cantilever illustrated in Fig. 5.1. This system consists of a concentrated mass supported by a massless but rigid bar of length L. A torsional spring supplies a linear restoring force that is proportional to the angle of rotation of the hinge (in either direction), with spring coefficient K. The angle of rotation θ thus describes the location of the mass at any given instant of time. Typically, the vertical force is simply P = Mg, but here we assume that an axial load of magnitude P acts independently of the fundamental approaches introduced in Chapter 2 for writing the governing equation of motion.

Introduction

In this chapter, we consider the dynamics of a thin elastic strut including axial effects, arising primarily from one of two situations:

• the axial load is applied externally (including postbuckling), or

• deformation is sufficient to cause coupling between axial and bending behavior (the membrane effect).

In the first section, attention is focused on a traditional approach to setting up the equations of motion (by means of D's principle) for the simple case based on engineering beam theory (Euler-Bernoulli) with the addition of axial loads. The resulting partial differential equation of motion is then separated into temporal and spatial ordinary differential equations and the response analyzed for various magnitudes of the axial load. Then an energy approach is used together with Rayleigh's method. In this case, additional terms are retained in the potential energy to allow postbuckled effects to be analyzed and the effect of initial geometric imperfections are included. An alternative approach is developed based on a simple application of Hamilton's principle, and in this instance stretching effects are also included and a solution developed by use of Galerkin's method. This approach will be similar to that used in the previous chapter on strings but now bending strain energy enters into the analysis (as well as compressive axial loading). In the final part of the chapter we consider the dynamics of struts that are loaded by gravity through self-weight. The next chapter will then continue the study of axially loaded members but with the scope opened to include a wider class of problem.

Basic Formulation

In this section, we develop the governing equation of motion for a thin, elastic, prismatic beam subject to a constant axial force.

In most practical applications, Euler-Bernoulli beam theory is often sufficient to provide useful information about the relation between axial loading and free vibrations. However, there are a number of instances in which the axial loading, or some related effect, results in relatively highly deflected states of the system, especially when the structure under consideration is very slender. For example, a pipeline or cable is characterized by having one of its dimensions very much greater than the other two, and the loads to which it is subject may often result in large deflections, even in cases in which self-weight is the only appreciable loading. Elastic bending stiffness does not necessarily dominate the effects of gravity, for example. In these cases, a more sophisticated description of the geometry is needed, and it is these types of flexible structures that form the basis for this chapter. In Chapter 7, we saw how initial postbuckling could be handled by retaining extra terms in the various energy expressions. But now, we allow (static) deflections to become large by using an arc-length description of the geometry and then consider small-amplitude oscillations about these nonlinear equilibrium configurations. In the final section, a FE solution is also shown for a specific case (essentially with the same approach as used toward the end of Chapter 9). It also turns out that experimental verification is relatively easy, especially if thermoplastics like polycarbonate are used.

The concept of stability is intrinsically a dynamical one. This is recognized even by the simplistic classical definition, which ignores the random disturbances of the real world and just inquires what would happen if a system were displaced to an adjacent position in phase space. So we are lucky, indeed, to have this well-conceived book written by a leading researcher who has mastered both nonlinear dynamics and the static bifurcations of elastic stability theory. The latter theory works well for conservative systems, for which powerful energy theorems are available, but needs augmenting by dynamical methods in the presence of loading that is either nonconservative or time dependent.

Lawrence Virgin has of course just the right background, having chosen (in his usual thoughtful way) to work first at University College London, then with Earl Dowell at Duke University. He is currently the Chair of the Department of Civil and Environmental Engineering at Duke (which has an active interdisciplinary program in nonlinear dynamics) and has enjoyed productive collaborations with Raymond Plaut (Virginia Polytechnic Institute and State University). His previous book, Introduction to Experimental Nonlinear Dynamics (also published by Cambridge University Press), brought a welcome sense of realism into the often esoteric field of nonlinear dynamics by focusing on experimental investigations, and I am delighted to see a similar emphasis in this new book titled Vibration of Axially Loaded Structures.

Understanding the buckling and vibration of structures under axial compression is of very great importance to structural and aerospace engineers, to whom this book is primarily addressed.

General Comments

Rationale and Scope

• The material covered by this book spans the areas of vibration and buckling. Both of these areas can be considered as subsets of structural mechanics and play a central role in the disciplines of civil, mechanical, and aerospace engineering.

• Although vibration and buckling are key elements in the teaching of advanced engineering, they are typically taught separately. However, the interplay of dynamics and stability in structural mechanics and its coverage in a single text provide an opportunity to present material in an interesting way.

• The quest for stronger, stiffer, and more lightweight structural systems is making the material covered in this book increasingly important in practical applications.

• By using axially loaded structures as a consistent theme, the book covers a wide variety of types of structure, methods of analysis, and potential applications without trying to cover too much. Experimental verification appears throughout.

• The level of material is appropriate for upper-level, advanced undergraduate classes, and graduate students, but researchers and practicing engineers will find plenty of interest too.

• The text is liberally illustrated by figures, and close to 500 technical references are given.

Acknowledgments

The material presented in this book contains a synthesis of material from the general literature together with results from my own research program. In terms of the latter, this is by no means a solo endeavor, and there are a number of people I would like to thank.

The previous chapter focused attention on the behavior of axially loaded prismatic thin beams in which the external loading consisted primarily of loads applied at the end of the member or in which significant axial effects were induced (e.g., the membrane, or stretching, effect). In this chapter, the scope of the analysis is opened to include a wider variety of situations in which axial loading and dynamic effects are considered, but in which boundary conditions, for example, do not necessarily fall into simple categories. We will also typically have to rely more on approximate techniques. Furthermore, there are cases in which the geometry of the structure means that a system's vibration and stability characteristics may depend on a number of parameters not considered in the previous chapter that focused on relatively straightforward prismatic members. We start this chapter by looking at a couple of cases in which this happens.

A Beam on an Elastic Foundation

It is not uncommon for a beam to have some kind of continuous support along its length. We can think of this as an elastic foundation and assume the foundation stiffness is linear. A practical example of this might be the sleepers under a railroad track, where a significant axial-loading effect is caused by thermal expansion. Referring to the schematic shown in Fig. 8.1 and again assuming the ends of the beam are pinned, we can extend the analysis of the previous chapter.

LEARNING OBJECTIVES FOR THIS CHAPTER

1. 14–1 To characterize the steady-state behavior of a system through analysis of the system transfer function.

2. 14–2 To evaluate the steady-state disturbance sensitivity of a system.

3. 14–3 To understand the trade-off between transient and steady-state performance specifications in control system design.

4. 14–4 To select gains of a proportional–integral–derivative controller based on open-loop system performance.

5. 14–5 To design an appropriate cascade compensator based on steady-state and transient performance specifications.

INTRODUCTION

In Chap. 1, it was pointed out that negative feedback is present in nearly all existing engineering systems. In control systems, which are introduced in this chapter, negative feedback is included intentionally as a means of obtaining a specified performance of the system.

Control is an action undertaken to obtain a desired behavior of a system, and it can be applied in an open-loop or a closed-loop configuration. In an open-loop system, shown schematically in Fig. 14.1, a process is controlled in a certain prescribed manner regardless of the actual state of the process. A washing machine performing a predefined sequence of operations without any information and “with no concern” regarding the results of its operation is an example of an open-loop control system.

In a closed-loop control system, shown in Fig. 14.2, the controller produces a control signal based on the difference between the desired and the actual process output. The washing machine previously considered as an open-loop system would operate in a closed-loop mode if it were equipped with a measuring device capable of generating a signal related to the degree of cleanness of the laundry being washed.