main-content

This text closes the gap between traditional textbooks on structural dynamics and how structural dynamics is practiced in a world driven by commercial software, where performance-based design is increasingly important. The book emphasizes numerical methods, nonlinear response of structures, and the analysis of continuous systems (e.g., wave propagation). Fundamentals of Structural Dynamics: Theory and Computation builds the theory of structural dynamics from simple single-degree-of-freedom systems through complex nonlinear beams and frames in a consistent theoretical context supported by an extensive set of MATLAB codes that not only illustrate and support the principles, but provide powerful tools for exploration. The book is designed for students learning structural dynamics for the first time but also serves as a reference for professionals throughout their careers.

Chapter 1. Foundations of Dynamics

Abstract
To establish a starting point for our exploration of structural dynamics, we first review the fundamental ideas of the dynamics of particles and rigid bodies. The theories in this book are built on Newton’s laws of motion for particles and Euler’s extension of those laws to rigid bodies. In each derivation of the equations of motion for a structural element or system, we will identify a typical particle and track its motion as part of the development of the theory. In most cases, we will identify a rigid body (e.g., the cross section in a beam) that plays an important role in describing the motion. Hence, a review of the foundations of dynamics is in order. This brief tour through the basic concepts will also allow us to set up some of the notation used throughout the book.

Chapter 2. Numerical Solution of Ordinary Differential Equations

Abstract
The equations of motion that emanate from Newton’s laws for particles and rigid bodies are second order ordinary differential equations, and are often nonlinear. In our study of the dynamic response of structural systems we will encounter problems with complex forcing functions (e.g., earthquakes) and nonlinearity in the response (e.g., material yielding). While there are methods available to solve linear ordinary differential equations in a classical sense, many practical problems are not amenable to those techniques. We will rejoice when a classical solution is available (and squeeze as much insight from it as we can), but we will also prepare for the more common situation where numerical analysis of the equations of motion is required. While it may seem like a bit of an interruption to turn our attention to the numerical solution of ordinary differential equations, we do so to lay the groundwork for developing the tools that we will need to make progress with many of the problems of structural dynamics. In this chapter we consider a few of the more common numerical integrators and examine their behavior in some detail.

Chapter 3. Single-Degree-of-Freedom Systems

Abstract
Structural dynamics is the study of systems that are designed to oscillate under the influence of applied loads. The essential features of the response of an oscillating system can be observed in simplest form in a single-degree-of-freedom (SDOF) oscillator. Hence, we start our exploration of structural dynamics here, both to refresh our understanding of how to solve the differential equations that govern these types of systems and to see some of the basic behavioral features of oscillating systems. Among those features, we will find that a linear SDOF system possesses a natural frequency, which is determined by the physical properties of the system, and the system will resonate if excited by a sinusoidal force with a driving frequency equal to the natural frequency. We will examine the effects of viscous damping on the response of the system and consider various loading conditions, including earthquake excitation. Finally, we will extend the formulation to include nonlinear response of the restoring force elements and develop a framework for the numerical analysis of nonlinear systems that will be useful throughout the book.

Chapter 4. Systems with Multiple Degrees of Freedom

Abstract
In this chapter we consider structures with multiple degrees of freedom. Since we often label the number of degrees of freedom N, we call them NDOF systems. In this and the next several chapters we will analyze idealized discrete structures, wherein the parts associated with mass are considered rigid and the parts associated with restoring forces have no mass. This simplification will result in equations of motion that are coupled ordinary differential equations. While we will relax the assumptions that decouple mass and stiffness in the later chapters of the book, this idealization gives a great deal of insight into the dynamic response of structural systems.

Chapter 5. Nonlinear Response of NDOF Systems

Abstract
In this chapter, we lay out an approach to nonlinear analysis of the NDOF system with two objectives: (1) connect with the nonlinear models we introduced for the SDOF systems in Chap. 3 and (2) set up an approach to formulating the equations of motion in a way that will extend gracefully to trusses, frames, and other complex structures. In particular, we will establish a strategy based upon the principle of virtual work that leads to an algorithm for directly assembling the equations of motion. This approach will serve as a foundation for formulating equations of motion throughout the rest of the book. Once formulated, we will solve the equations of motion by extending the numerical tools already developed for the SDOF system.

Chapter 6. Earthquake Response of NDOF Systems

Abstract
The motion of the NDOF system in an earthquake includes the motion at the base, which we will take as a prescribed motion. In essence, we assume that the ground motion just happens to the structure and that the structure does not influence the ground motion (which is not completely true, but it is a good place to start with the problem). In this chapter, we add earthquake ground motions as a source of excitation, show how they impart dynamic forces in the shear building, and analyze the response those forces induce.

Chapter 7. Special Methods for Large Systems

Abstract
In this book we focus on computational approaches to structural dynamics because the calculations required to find the response of a system to dynamic loads involves tedious and repetitive steps that are ideally suited to the computer. To make the codes in the book as readable as possible we avoid clever strategies that might be used in commercial codes to speed up calculations. For the small problems we solve as examples in the book, speed of computation is seldom an issue. However, real-world systems can be very complex and the time it takes to execute structural dynamics computations grows with problem size, particularly for nonlinear problems. In this chapter, we examine two approaches to deal with large problems. Both methods involve projecting the problem onto a smaller subspace to improve the speed of calculation. One downside is that such methods are approximate and, therefore, require care in application.

Chapter 8. Dynamic Analysis of Truss Structures

Abstract
The shear building gives a good introduction to discrete NDOF structural systems, encompassing many of the phenomena that are important to structural dynamics. As a structural model, however, it is very limited. There are several directions to branch out from the shear building to get to more practical structures. For our first branch, we will consider the analysis of truss structures. Trusses represent an important class of structures used widely in practice. The main reason for considering them next is because the mechanics of truss elements is very simple, and that simplicity will allow us to focus on the geometry and topology of multiply connected structures. Truss analysis will afford us an opportunity to expand on ideas presented in the previous chapters.

Chapter 9. Axial Wave Propagation

Abstract
Up to this point we have considered only discrete dynamical systems. The primary assumption in a discrete system is that elements providing restoring forces have no associated mass and that the masses are rigid. Once we remove these restrictions, we are in the realm of continuous systems. In a continuous system, both the mass and the elasticity are distributed throughout the body. All materials have this feature, so it is a more realistic model than the discrete system idealization. The price of this additional fidelity is more mathematical complication. The remainder of the book concerns continuous systems. In this chapter we will explore the simplest version of a continuous system in solid mechanics—the axial bar problem.

Chapter 10. Dynamics of Planar Beams: Theory

Abstract
In this chapter we establish the theory of beam dynamics, confining our attention to planar motion of beams. These developments will provide the theoretical foundation needed for the remainder of the book. In the following two chapters we explore methods of solution of the linearized equations of motion, both classical solutions based on separation of variables and finite element solutions. In Chap. 13 we extend the finite element solution method to nonlinear beams and in Chap. 14 we tackle the problem of analyzing planar frames.

Chapter 11. Wave Propagation in Linear Planar Beams

Abstract
In Chap. 10 we derived the equations governing the dynamic response of planar beams. A simpler set of equations emanated from the linearization of the equations of motion. Further, we introduced assumptions to yield Timoshenko, Rayleigh, and Bernoulli–Euler beam theories. Timoshenko beam theory is the most complete, including rotary inertia effects and shear deformation. Rayleigh beam theory neglects shear deformation but includes rotary inertia. Bernoulli–Euler beam theory neglects both shear deformations and rotary inertia. While these differences are usually not very important in the context of static response, they have significant consequences in dynamics. In this chapter we explore the phenomenon of wave propagation in beams through classical solutions of the linearized equations of motion of these three theories.

Chapter 12. Finite Element Analysis of Linear Planar Beams

Abstract
In the previous chapter we were able to find solutions to the beam dynamics problem using the separation of variables technique. The process went very much like it did for the axial bar problem. But the beam differential equation involves fourth order derivatives with respect to x, and consequently the spatial part of the solution was more complicated. While the classical techniques provide a good route to solving wave propagation problems in beams, we want to solve more general problems (e.g., problems with time-dependent forcing functions and possibly non-prismatic cross sections). In this chapter we will approach the problem in a different way. We will start by introducing the principle of virtual work and develop an approximate numerical approach using the Ritz method with finite element base functions.

Chapter 13. Nonlinear Dynamic Analysis of Planar Beams

Abstract
In this chapter we develop an approach to solve the fully nonlinear equations governing the dynamic response of beams derived in Chap. 10. Of course, the only route forward is numerical analysis. We will use Newmark’s method for the temporal integration and Newton’s method to solve the resulting nonlinear algebraic equations. The main additional task in setting up and solving the nonlinear problem is to compute the residual and tangent for Newton’s method. As was the case for the truss, the most complicated part of the derivation is in the internal resistance, which is present in both a static and dynamic analysis. This chapter shows how the techniques of the previous chapter can be generalized to deal with nonlinearity, opening up the possibility of modeling scenarios that involve large motions and instability.