Introduction

In this chapter, the response of linear dynamic systems is studied in more detail. We begin with a study of efficient techniques for solving first-order and second-order single-degree-of-freedom (SDOF) equations of motion. Taking advantage of the simplicity of the first-order equation, we show how to obtain analytical expressions for the system response to harmonic excitation and how to write the steady-state response in terms of the magnitude and phase difference. It leads to the definition of the frequency-response function (FRF) and its analytical expression in terms of a complex variable (Ref. [1]). We then show how two- or more degree-of-freedom equations of motion can be treated similarly. The focus will be on multiple DOF mass–spring–damper systems (lumped parameter systems). The last part of this chapter deals with the bending vibration of flexible beams, which are described by partial differential equations rather than by ordinary differential equations for mass–spring–damper systems (Ref. [2]). The partial differential equations are decoupled by the separation of variable method to yield two sets of ordinary differential equations. From these equations, the common features of flexible structures are introduced, including the natural modes, frequency equations, and modal amplitudes. A discussion of the orthogonality property of the natural modes and the expansion theorem is then presented.

Single Degree of Freedom

A system of considerable importance in vibrations is the mass–spring–damper system, shown in Fig. 5.1.

Introduction

In the previous chapter, we paid attention to the solution of a scalar ODE. A scalar ODE describes the dynamics of a single-input single output (SISO) dynamic system. In general, a system can have several inputs and several outputs. Such a multi-input multi-output (MIMO) dynamic system can be described by a set of coupled ODEs involving several input and output variables. We must solve these equations simultaneously to obtain the dynamic response of the system. We can find such a solution by using matrix theory, which conveniently rearranges the set of coupled equations in a compact form. Instead of having several coupled scalar ODEs, we now have one single matrix ODE. Matrix operations can be performed on the matrix differential equations, and the final solution can be expressed in a simple form. It is important to realize that applying such matrix operations is equivalent to operating on the scalar ODEs individually, although with the scalar approach it is very easy to miss the general picture. Thus, matrix theory is one fine example in which one gains by simply rewriting an old problem in a new form that can be analyzed more effectively. Another reason why the matrix formulations are so useful is that a set of coupled differential equations of any order can be rewritten as a single matrix differential equation of first order. The same statement applies for a single scalar ODE of any order as well.

Introduction

This chapter describes several computational algorithms to compute the predictive control law that has some feature of adaptive control. All algorithms make use of the multi-step-ahead output prediction as derived in Chap. 10 based on the finite-difference model. The generalized predictive control (GPC) algorithm (Ref. [1–4]) is based on system output predictions over a finite horizon known as the prediction horizon. In determining the future control inputs, it is assumed that control is applied only over a finite horizon known as the control horizon. The GPC is computed with the Toeplitz matrix formed from the step-response time history of the system in conjunction with a cost function with weighted input and output. The control input is obtained by minimization of the cost function. There are three design parameters involved: the control weight, the prediction horizon, and the control horizon. A proper combination of these parameters is required in order to guarantee stability of the predictive control law.

In contrast to the GPC approach, another approach is the deadbeat predictive control (DPC) (Ref. [5–9]). The DPC feedback law is supposed to bring the output response to rest after a few specific time steps. Similar to GPC, DPC has a control design parameter and an identification parameter related to the order of the system. The control design parameter, which is similar to the GPC control horizon, gives the number of time steps for the system to become deadbeat (rest). The DPC guarantees closed-loop stability for a controllable system.

Introduction

In the analysis of continuous systems, the formulations describing the system response are governed by partial differential equations, as presented in the last chapter. The exact solution of the partial differential equations satisfying all boundary conditions is possible for only relatively simple systems such as a uniform beam. Numerical procedures must be used to approximate the partial differential equations and predict the system response.

The finite-element method is a very popular technique for the numerical solution of complex problems in engineering. It is a technique for solving partial differential equations that represent a physical system by discretizing them in their space dimensions. The discretization is performed locally over small regions of simple but arbitrary shape, i.e., finite elements. For example, in structural engineering, a structure is typically represented as an assemblage of discrete truss and beam elements. The discretization process converts the partial differential equations into matrix equations relating the input at specified points in the elements to the output at these same points. To solve equations over large regions, the matrix equations for the smaller subregions are summed node by node to yield global matrix equations.

Our objective in this chapter is to present the fundamental principles of the finite-element method. It is not our goal to summarize all the finite-element formulations available, but rather to establish only the basic and general principles that provide the foundation for a preliminary understanding of the finite-element method.

This book is based on a series of lecture notes developed by the authors. The first author has used part of the notes for two graduate-level classes in System Identification and Control of Large Aerospace Systems at the Joint Institute for Advancement of Flight Sciences, George Washington University at NASA Langley Research Center for the past 10 years. The second author has used part of the notes for senior and first year graduate-level courses in Dynamics and Control of Mechanical Systems and System Identification at Princeton University and Dartmouth College since 1995. There are many reasons that motivated the writing of this book; some of them are outlined below.

First, the lecture notes received overwhelming response from the students taking these courses, with many urging us to turn these materials into a textbook. When developing the notes, we tried to place emphasis on the fundamentals and clarity of presentation. Second, the subject matter is important in practice, but it is challenging both for students to learn and for us to teach because it is an integration of several disciplines: structural dynamics, vibration analysis, modern control, and system identification. The primary goal is for students to learn what these tools are without having to take a separate course for each subject and how they are brought together to solve a vibration control problem.

Introduction

The objective of a control system is to influence the dynamic system to make it behave in a desirable manner (Ref. [1–9]). Typical objectives of a control system are regulation and tracking. In a regulation problem, the system is controlled so that its output is maintained at a certain set point. In a tracking problem, the system is controlled so that its output follows a particular desired trajectory. A special case of the regulation problem is the stabilization problem, in which a control system is designed to bring the system to rest from any nonzero initial conditions (i.e., the desirable set point is zero). For a flexible structure that may be subjected to unwanted vibration, this is usually the most important goal of a controlled system. Stabilization is the focus of this chapter. In particular, we consider a special but very important class of control systems, namely state-feedback control, in which the control input is some function of the system states. For the moment we assume that there are enough sensors to measure the state of the system at any point in time to be used in computing the control input. If the state of the system cannot be measured directly, then a state observer is needed to estimate the system state from the measurements. The estimated state is then used in a state-feedback-control law. This subject of state estimation will be dealt with in the next chapter.

This chapter collects open problems that in one way or the other relate to the material discussed in this book. They represent the complement of the material, in the sense that they attempt to describe what we do not know. We should keep in mind that it is most likely the case that only a tiny fraction of the knowable is known. Hence, there is a vast variety of questions that can be asked but not yet answered. The author of this book exercised subjective taste and judgment to collect a small subset of such questions, in the hope that they can give a glimpse of what is conceivable. Most of the problems are elementary in nature and have been stated elsewhere in the literature.

Two of the twenty-three problems have been solved since this book first appeared in 2001. These are P.8 Union of disks and P.9 Intersection of disks, both solved in [1]. Since the described approaches to the two problems are different from the eventual solution and perhaps useful to understand generalized versions of the problem, we decided to leave Sections P.8 and P.9 unchanged.

Empty convex hexagons

Let S be a set of n points in ℝ 2 and assume no three points are collinear. A convex k-gon is a subset of k points in convex position.

This chapter describes an algorithm for simplifying a given triangulated surface. We assume this surface represents a shape in three-dimensional space, and the goal is to represent approximately the same shape with fewer triangles. The particular algorithm combines topological and numerical computations and provides an opportunity to discuss combinatorial topology concepts in an applied situation. Section 4.1 describes the algorithm, which greedily contracts edges until the number of triangles that remain is as small as desired. Section 4.2 studies topological implications and characterizes edge contractions that preserve the topological type of the surface. Section 4.3 interprets the algorithm as constructing a simplicial map and establishes connections between the original and the simplified surfaces. Section 4.4 explains the numerical component of the algorithm used to prioritize edges for contraction.

Edge contraction algorithm

A triangulated surface is simplified by reducing the number of vertices. This section presents an algorithm that simplifies by repeated edge contraction. We discuss the operation, describe the algorithm, and introduce the error measure that controls which edges are contracted and in what sequence.

Edge contraction

Let K be a 2-complex, and assume for the moment that |K| is a 2-manifold. The contraction of an edge ab ∈ K removes ab together with the two triangles abx, aby, and it mends the hole by gluing xa to xb and ya to yb, as shown in Figure 4.1. Vertices a and b are glued to form a new vertex c.

The three sections in this chapter apply what we learned in Chapter 1 to the construction of triangle meshes in the plane. In mesh generation, the vertices are no longer part of the input but have to be placed by the algorithm itself. A typical instance of the meshing problem is given as a region, and the algorithm is expected to decompose that region into cells or elements. This chapter focuses on constructing meshes with triangle elements, and it pays attention to quality criteria, such as angle size and length variation. Section 2.1 shows how Delaunay triangulations can be adapted to constraints given as line segments that are required to be part of the mesh. Section 2.2 and 2.3 describe and analyze the Delaunay refinement method that adds new vertices at circumcenters of already existing Delaunay triangles.

Constrained triangulations

This section studies triangulations in the plane constrained by edges specified as part of the input. We show that there is a unique constrained triangulation that is closest, in some sense, to the (unconstrained) Delaunay triangulation.

Constraining line segments

The preceding sections constructed triangulations for a given set of points. The input now consists of a finite set of points, S ⊆ ℝ2, together with a finite set of line segments, L, each connecting two points in S. We require that any two line segments are either disjoint or meet at most in a common endpoint.

The primary purpose of this chapter is the introduction of standard topological language to streamline our discussions on triangulations and meshes. We will spend most of the effort to develop a better understanding of space, how it is connected, and how we can decompose it. The secondary purpose is the construction of a bridge between continuous and discrete concepts in geometry. The idea of a continuous and possibly even differential world is close to our intuitive understanding of physical phenomena, while the discrete setting is natural for computation. Section 3.1 introduces simplicial complexes as a fundamental discrete representation of continuous space. Section 3.2 talks about refining complexes by decomposing simplices into smaller pieces. Section 3.3 describes the topological notion of space and the important special case of manifolds. Section 3.4 discusses the Euler characteristic of a triangulated space.

Simplicial complexes

We use simplicial complexes as the fundamental tool to model geometric shapes and spaces. They generalize and formalize the somewhat loose geometric notions of a triangulation. Because of their combinatorial nature, simplicial complexes are perfect data structures for geometric modeling algorithms.

Simplices

A finite collection of points is affinely independent if no affine space of dimension i contains more than i + 1 of the points, and this is true for every i. A k-simplex is the convex hull of a collection of k + 1 affinely independent points, σ = conv S.