## Model Less Costly

When possible the ideal approach is to design products that rely on the formulation and analysis of mathematical models of static and/or dynamic physical systems. This is of interest because a model is more accessible to study than the physical system the model represents. Models typically are less costly and less time-consuming to construct and test. Changes in the structure of a model are easier to implement, and changes in the behavior of a model are easier to isolate and understand in a computer system (Chapter 5).

A model often provides an insight when the corresponding physical system cannot, because experimentation with the actual system could be too dangerous, cosdy, or too demanding. A model can be used to answer questions about a product that has not yet been finalized or realized. Potential problems can provide an immediate solution.

A mathematical model is a description of a system in terms of the available equations that are available from the engineering books. The desired model used will depend upon: (1) the nature of the system the product represents, (2) the objecdvcs of the designer in developing the model, and (3) the tools available for developing and analyzing the model.

Because the physical systems of primary interest are stadc and/or dynamic in nature, the mathematical models used to represent these systems most often include difference or differential equations. Such equations, based on physical laws and observations, are statements of the fundamental relationships among the important variables that describe the system. Difference and differential equation models are expressions of the way in which the current values assumed by the variables combine to determine the future values of these variables. As reviewed later it is important to relate static and/or dynamic loads on plastic products to operating temperatures.

### Model Type

A variety of models are available that can meet the requirements for any given product. The choice of a particular model always represents a compromise between the accuracy in details of the model, the effort required in model formulation and analysis, and usually the time frame that has to be met in fabricating the product. This compromise is reflected in the nature and extent of simplifying assumptions used to develop the model.

Generally the more faithful or complete the model is as a description of the physical system modeled, the more difficult it is to obtain useful general solutions. Recognize that the best engineering model is not necessarily the most accurate or precise. It is, instead, the simplest model that yields the information needed to support a decision and meet performance requirements for the product. This approach of simplicity also involves the product's shape to the fabricating method used. Most designed products do not complicate fabricating them, however there are those that can complicate the fabrication resulting in extra cost not initially included and the possibility of defective parts.

Recognize that simpler models frequendy can be justified, particularly during the initial stages of a product study. In particular, systems that can be described by linear difference or differential equations permit the use of powerful analysis and design techniques. These include the transform methods of classical theory and the state-variable methods of modem theory.

Target is to have more than one model in the evaluation. Simple models that can be solved analytically are used to gain insight into the behavior of the system and to suggest candidate designs. These designs are then verified and refined in more complex models, using computer simulation. If physical components are developed during the course of a study, it is often practical to incorporate these components directly into the simulation, replacing the corresponding model components.

### Computer Software

Mathematical models are particularly useful because of the large body of mathematical and computational theory that exists for the study and solution of equations. Based on this theory, a wide range of techniques has been developed. In recent years, computer programs have been written that implement virtually all of these techniques. Computer software packages are now widely available for both simulation and computational assistance in the analysis and design of control systems (Chapter 5).

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