Analyzing engineering systems or engineering problems is a three-step process. The first step is formulation of the problem. Engineers need to look at the physical system and develop a representative model.
Modeling can be as simple as drawing free body diagram with forces acting on a body or as complex as developing a mechanical model of a human ear. The model obtained needs to be converted into equations with knowns and unknowns. One example of modeling is assuming elastic bodies as mass, spring, and damper systems, drawing the free body diagrams, and formulating the force equations. Another example is finding the transfer function of a system by converting the system equations into Laplace equations.
The second step of the process is solving the mathematical equations that were developed in the first step. In this step, we often use initial conditions, boundary conditions, or any other conditions stated in the problem statement. For example, we can solve the electrical or electronics equations to find the Thevenin equivalent voltage source, or solve the heat transfer equation and find the temperature of a particular length of a brass rod.
The third step is interpreting the results, explaining what the results mean, and, if necessary, modifying the design to solve the problem. One example of this step is the conclusion that the resistor will burn, because the joule heating is more than its power rating. Another example gives the conclusion that the beam will fail, because stress is more than its yield strength. For a more complex system, engineers must interpret the system’s stability based on the location of its poles and zeroes.
Even though all the steps are important, most engineering books limit the discussion on both the modeling techniques and how to interpret the results to just a few pages. The solution methods make up the bulk of the book. For example, books on control systems engineering spend only a few pages on formulating the transfer functions of physical systems and spend most of their pages solving the math problems. Lots of pages explain how to plot root locus by finding poles, zeroes, asymptotes, breakaway points, departure angles, etc. Similarly, only a small percentage of a typical electronics engineering graduate program is devoted to developing SPICE models for electronics components.
In the past two decades advances in computing power have led to tremendous progress in problem-solving tools. Software tools such as Maple can solve differential equations and spit out the results in no time. While plotting a good root locus with pencil and paper takes at least 20 minutes, MATLAB can do the same in a matter of microseconds without any mistakes. A step or an impulse response can also be generated in no time. Finite elements analysis packages such as Ansys, Abaqus, or Comsol can solve challenging 3D electromagnetic problems quite easily.
The most important point is that the results reported by these tools are only as good as the model the engineer uses. The model should represent as close to the physical system as possible for the results to reflect the true conditions of the real system. Since it’s been becoming easier to solve, it makes all the more sense to spend more time improving the modeling skills. The importance of accurate modeling is increasing with the emergence of newer and faster problem-solving tools.
By all means, engineers should have the necessary mathematical skills to solve numerical problems. Without underestimating the value of solving skills, it will be more beneficial to new engineers if future textbooks devote a majority of their pages to modeling and interpreting the results.
Raghavendra Angara, PhD, is a senior mechatronics R&D engineer. He belongs to the ASME, ISA, IEEE, and ASQ.