Pages: 15

This chapter provides introductory material and an overview of the textbook. The chapter opens with a motivating example of mathematical modeling, in which the concept of conservation of mass is used to represent a physical system (a mixing tank) by a set of equations. Terminology related to differential equations is covered next including the concepts of order, initial and boundary conditions, and ordinary versus partial. Next, terminology related to modeling is presented, including the dynamic and steady-state models and forcing functions. Finally, the objectives of the text (analytical solutions of differential equations, mathematical modeling, and numerical solutions of differential equations) are listed and described.

Pages: 24

In Chapter 2, the modeling of one-dimensional motion in a gravitational field is used to further illustrate how mathematical models can be developed starting with a fundamental physical principle (Newton’s second law) and supplemented with constitutive relations. Two models are developed (with and without air resistance) and allow the introduction of two techniques (anti-differentiation and separation of variables) for solving ordinary differential equations. The chapter also contains a section on counting equations and unknowns to determine degrees of freedom, which is useful for complex models composed of multiple equations. The chapter concludes with a review of partial fraction expansion both for real and complex roots in the denominator.

Pages: 61

Chapter 3 covers the well-known classical techniques for solving ordinary linear differential equations. The chapter opens with some motivating examples (taken from later chapters) of differential equations resulting from mathematical modeling of various physical systems. The concept of linearity, in the context of ordinary differential equations, is defined next. The sections describe the integrating factor method for first-order linear equations and the characteristic equation method for second-order homogeneous equations with constant coefficients. The methods of undetermined coefficients and variation of parameters are then developed for solution of nonhomogeneous linear differential equation. Next, a section on using Taylor series to obtain approximate solutions to differential equations that are nonlinear or have variable coefficients is provided. The chapter concludes with a section on transient and final response, which provides an additional bridge between an analytical solution and a physical response.

Pages: 47

Chapter 4 presents techniques for using Laplace transforms to obtain the analytical solutions of linear differential equations with constant coefficients. The chapter opens with the definition of the Laplace transform, which is then used to generate transforms of common functions and to derive a variety of useful transform properties. Next, procedures are developed and examples are provided for solving linear differential equations with constant coefficients. The concept of the transfer function is presented and related to the characteristic equation of Chapter 3 to study the qualitative response of systems. Finally, the power of the transform in handling multiple coupled differential equations is demonstrated.

Pages: 48

This chapter presents in detail the response of systems described by linear first and second-order ODEs with constant coefficients. Section 5.1 shows how to convert first-order ordinary differential equations (ODEs) into a standard form so that the underlying physical model can be generalized to the same parameters. The time constant and the system gain are introduced for the first-order ODE, and the system response for several common forcing functions is analyzed. Finally, the transfer function for a first-order system is expressed in terms of the generalized parameters. Section 5.2 covers the response of second-order systems and introduces the characteristic time, the damping ratio, and the system gain. As in the previous section, response to several forcing functions is analyzed, and the transfer function is expressed in terms of generalized parameters. Section 5.3 introduces coupled first-order ODEs, which lead to a second-order ODE. Section 5.4 introduces coupled second-order ODEs to show the rich range of responses that can emerge from relatively simple underlying physics. Finally, Section 5.5 presents a few examples as to how a system can be engineered to obtain a desired response and how the response of an ODE to a forcing function can be used to identify the key parameters of the underlying system.

Pages: 42

This chapter focuses on mechanical systems formed by springs, masses, and dashpots (also sometimes referred to as pistons or dampers). Many systems are formed by these three elements. The chapter begins by looking at the physical law that is used, as well as element equations and other experimental facts necessary to develop the models. In this and the following four chapters, many models will be developed. The solution of these models must be obtained to be of use to engineers/scientists; this is why we presented the analytical solution of differential equations early in the book. Having an equation describing the response of a system to a forcing function is useful; however, quite often, a graph of the response may be even more useful. In several examples in this chapter, the analytical solution is used to graph the response. As discussed in Chapter 1, another way of obtaining the graph of the response is by solving the model using the computer – this is what we call simulation. Chapter 11 discusses simulation and shows the comparison with the graph obtained from the analytical solution. The reader is strongly encouraged to read and study Chapter 11.

Pages: 30

Chapter 7 extends the concepts from Chapter 6, which covered translational motion in systems composed of masses, springs, and dampers, to systems where the elements rotate, such as those composed of gears, wheels, levers, pendulums, or rotating shafts. Applications are confined to systems in which each rotating element is a rigid body that is rotating about a single fixed axis. The chapter opens with a discussion of Euler’s second law, moment of inertia, and torque. Next, torsion springs, rotational dampers, and gears are covered, along with their interconnection rules. Numerous examples are provided, and the chapter concludes with a section on systems in which both translational and rotational elements are present. The focus of Chapter 6 was on systems where the elements moved in pure translation. In this chapter, we extend the concepts from Chapter 6 to systems in which the elements rotate, such as those composed of gears, wheels, levers, pendulums, or rotating shafts. We confine ourselves to systems in which each rotating element is a rigid body that is rotating about a single fixed axis.

Pages: 58

Chapter 8 focuses on the application of mass balances to various industrial processes. The chapter opens by describing the physical law (conservation of mass) that forms the basis for the models developed, including those for both single and multicomponent systems. Relations for flow rates, concentrations, and flow elements are covered next followed by several modeling examples. Additional relations for processes involving separations or chemical reactions are provided in Section 8.5, which is followed by another section of example problems. Section 8.7 provides applications to bioengineering processes, beginning with a description of compartment modelling and followed by presentation of models for biological reactions, including fermentation.

Pages: 40

The focus of Chapter 9 is on thermal systems, or systems that undergo changes in energy content as a result of heat transferring to or from them. The governing physical principle that guides the development of mathematical models for these systems is the concept of conservation of energy. The chapter opens with a discussion of the various modes of heat transport and how energy may be stored or accumulated in a body. Next, dynamic, lumped parameter systems are covered where the independent variable is time, and the mathematical models are initial value problems. Applications to flow systems and systems with internal heat generation are included. The last part of the chapter covers systems in steady state where the independent variable is a position coordinate – which results in boundary value problems.

Pages: 61

Chapter 10 focuses on the modeling of electrical circuits. The chapter opens with a description of electrical quantities such as voltage, current and charge. Kirchhoff’s current and voltage laws are covered next, followed by describing the current–voltage relations of circuit elements, including resistors, capacitors, and inductors. Models for single loops circuits are developed by simple application of Kirchoff’s voltage law, while more complex circuits are analyzed using the mesh and nodal analysis methods. The importance of initial conditions is stressed and reinforced with several examples. Additional applications include heat dissipation in circuits and a brief but complete use of an RC circuit to approximate high- and low-pass filters.

Pages: 39

Chapter 11 covers the numerical simulation of ordinary first- and second-order differential equations. The chapter opens by discussing the difference between a numerical solution to a differential equation and an analytical solution, followed by a discussion of Euler’s method, which is the simplest technique for obtaining numerical solutions to differential equations. Next comes the introduction of block diagrams, which are used as a way of entering differential equations into several computer software packages used for their numerical solution. Finally, one of these packages, Simulink®, is used to obtain numerical solutions to several mathematical models developed in earlier chapters. The chapter concludes with a section that provides some additional topics for using Simulink, including consolidating blocks, trial-and-error calculations, and switches.