Contents

Mechanical systems are seen in machines, devices, equipment, and structures in a wide variety of engineering applications. Modeling of a mechanical system involves forces and motion about relevant objects, which can be either solids or fluids. Depending on specific concerns in an application, there are several types of mathematical models for mechanical systems at different levels of complexity, including lumped-parameter models, rigid-body models, and deformable- or flexible-body models. For instance, in studying the dynamic behaviors of an airplane, a lumped-parameter model may be good enough to describe its flight trajectory and a rigid-body model may be sufficient to study its three-dimensional motion in space. However, a flexible-body model is necessary to understand the vibration and stress in the airplane structure in performance evaluation and failure analysis.

This chapter assembles six problems of combined dynamic systems from engineering applications, namely, vibration analysis of a moving car, speed control of a coupled engine–propeller system (electromechanical system), modeling and analysis of a bimetallic strip thermometer (thermomechanical system), modeling and analysis of a resistive-heating element (electro-thermo-mechanical system), feedback control of a water purification system component (liquid-level system), and the working principles of sensors, electroacoustic, and piezoelectric devices. Each problem is presented in one section.

This chapter provides the reader with a brief refresher course on the mathematical apparatus crucial for modeling of dynamic systems. Sections 2.1 and 2.2 present basic concepts and terminology of vector and matrix algebra. Definitions and basic operations on complex numbers are introduced in Section 2.3. Section 2.4 is devoted to one of the important methods for solving differential equations – the Laplace transform. Sections 2.5 and 2.6 discuss the types of differential equations widely encountered in modeling of common dynamic systems and develop methods for solving these equations. Section 2.7 introduces the mathematical foundation for deriving transfer functions and creating block diagrams of various linear time-invariant dynamic systems. Section 2.8 presents a brief overview of solving differential equations numerically, with MATLAB and Wolfram Mathematica.

Fundamental Principles of Thermal Systems

Practically every modern engineered dynamic system has electrical components such as motors, sensors, controllers, or, at the very least, power sources. Therefore, an understanding of the physical processes occurring in the typical electrical circuits, and the ability to model behavior of an electrical subsystem are essential for anyone interested in dynamic systems.

Controls in engineering are efforts to change, design, or modify the behaviors of dynamic systems. Automatic control is control that involves only machines and devices, and that has no human intervention. Examples of automatic control are diverse, including room-temperature control, cruise control of cars, missile guidance, trajectory control of robots, control of appliances such as washing machines and refrigerators, and control of industrial processes like papermaking and steelmaking. In this chapter, for simplicity, an automatic control system is called a control system. Because the focus of this text is on the modeling and analysis of dynamic systems, only basic concepts of feedback control are introduced. For theories and methods about feedback control systems, one may refer to standard textbooks for control courses, including those listed at the end of the chapter.