Introduction to Modeling and Simulation

2Second‐Order Systems

2.1 Introduction

We begin our study of modeling and simulation with a treatment of second‐order, continuous‐time systems. Second‐order systems form an important class of systems for a couple of reasons. First, the state space is two‐dimensional and so graphical methods can be used for visualization and analysis. Second, many physical laws, such as Newton's second law, , give rise to second‐order systems [29].

We first discuss trajectories and phase portraits for autonomous second‐order systems. We then discuss methods for sketching phase portraits of second‐order systems based on the concept of the direction field, and show how to classify second‐order systems in terms of the eigenvalues of the system coefficient matrix. We also discuss the existence of limit cycles, which are isolated periodic solutions. We present a fundamental result, known as the Poincaré–Bendixson theorem, which can be used to determine the existence of periodic solutions to second‐order systems. Finally, we discuss coupled second‐order systems that frequently arise from physical models of mechanical and electrical systems.

To start, suppose that the input/output system model in Figure 1.1 is described by a second‐order linear differential equation of the form

(2.1)

with input , output ...