1.4: A Short Discussion of Engineering Models

The mass-damper of Figure 1.3.1 can be used to represent approximately (i.e., to model) some actual physical systems. One such system is a surface ship moving over the water under its own propulsion or being pushed/pulled by a tugboat. Another is an automobile hydroplaning on a wet road. You can probably think of other similar real systems. However, it is important for us, as engineers, to recognize that the mass-damper system is not the actual system, but only an approximate idealized physical model of the actual system. We are able to derive from this idealized physical model the solvable mathematical model, which consists of ODE Equation 1.3.3 and known values for \(f_x(t)\) and \(v_0\). The actual physical system, on the other hand, might be so complicated that it cannot be characterized mathematically with absolute precision. For example, the ideal viscous damping model used in the derivation of ODE Equation 1.3.3 is almost certainly not an exact representation of the liquid drag forces acting on either a surface ship or a hydroplaning car.

The same general observation applies for almost any idealized physical model and associated mathematical model developed for engineering purposes: the physical model is, at best, a reasonably accurate approximation of the actual physical system. The fidelity of a model usually depends on a number of factors, including system complexity, uncertainties, the costs of modeling and mathematical/computational solutions, time constraints, modeling skills of the engineer, etc.

But a reasonably accurate approximate model often suffices for engineering purposes. Engineering systems are usually designed conservatively, with redundancies and factors of safety to compensate for severe overloads, unexpected material flaws, operator error, and the many other unpredictable influences that can arise in the functioning of a system. As engineers, we almost never require 100 % accuracy; we are usually satisfied if our mathematical/computational predictions of system behavior are qualitatively correct and are quantitatively within around \(\pm\)10 % (in a general sense) of the actual behavior.

The main point of this discussion is to emphasize that any idealized physical model used for engineering analysis and design is only an approximation of an actual physical system. Moreover, the primary subjects of this book are the fundamental dynamic characteristics of idealized physical models, because a great deal of practical experience has shown that these are also the characteristics of many real engineering systems. Therefore, this book does not consider in depth the development of idealized physical models to represent actual systems; rather, we shall focus on deriving mathematical models (mostly ODEs) that describe idealized physical models, and on solving the mathematical equations and exploring the characteristics of the solutions.

The process of developing idealized physical models to represent real systems involves both science (theory and experimental data) and “art” (experience and intuition); you probably will encounter this process in laboratory and design courses, and later in your professional practice of engineering.