Generalizing Modal Analysis
Why do we need Mathematical Modeling?
We live in a world full of technological marvels - from cars to airplaines, computers to cell phones, batteries to power plants - all of these technologies that impact our lives in significant ways depend on science. All of these inventions came as a result of scientists who studied how the world works and then made this knowledge available to others. Mathematics is the language used by scientists to describe what they learn about the world, because mathematics provides a precise means to express this knowledge in a way that can be verified or used by others. The study of how the world works is called physics.
Useful inventions often rely on a complex combination of parts and multiple physical physical phenonena. Often there are several desirable characteristics for a particular invention. For example, people like to have cars that are reliable, get good gas mileage, and are designed for safety in the case of an accident, among other desirable qualities. It can be very expensive to build a complex systems such as a car. It is impractical to build a new car prototype each time you would like to experiment with a change that might improve saftey or gas mileage, so typically mathematical models are built which simulate the behavior of a system and can be used to explore whether design changes will lead to improvements.
A good mathematical model is an important tool for a scientist or engineer because it help them explore and understand the behavior of the system better. Mathematical models are often easier to build and easier to modify and explore than physical prototypes. A good mathematical model can help you tell if a design change results in a system that works better, or if the change makes the system perform worse. If you want to improve something, it sure helps if you understand it first and a good mathematical model gives you a way to explore how the system behaves. V. I. Arnold, who was a prominant mathematician, put it this way; "Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." (discussion on teaching of mathematics in Palais de Découverte in Paris on 7 March 1997).
Developing a good mathematical model is an important first step, but in order to understand the model, you need to understand the model solutions. Many of the laws of physics, such as Newton's laws of motion ($F = ma$), are expressed as differential equations. A physical system can often be modelled as, or reduced to, a system of coupled ordinary differtial equations (ODEs). The solutions to a system of ODEs depends on the starting state of the system (or initial conditions) and its interaction with its surroundings (boundary conditions). A different set of initial or boundary conditions can lead to solutions to the system may look very different, and gaining a good understanding of how a system behaves under many different conditions can be a difficult challenge. Techniques for exploring the behavior of a mathematical model can be divided into quantitative techniques and qualitative techniques.
The goal of a quantitative technique is to calculate an accurate solution to the mathematical model for a system. An accurate solution to a good mathematical model can be useful for understanding the behavior of the actual system the model is based on. The accuracy of the solution may be very important when designing, improving, or determining the safety of the system, since the solutions are often used to answer specific question about the system, or about the effect a design change would have on the system. But an accurate solution doesn't address an important question - How does the behavior of the system change under a different (initial or boundary) conditions and can the behavioral trends be characterized?
The goal of a qualitative technique is to understand the behavioral trends of a system under a range of conditions. Many qualitative techniques require a system to be simplified in one way or another in order to identify behavioral trends. In other words, accuracy may be sacrificed in order to gain a broader understanding of how the system behaves.
Both accuracy and an understanding of behavioral trends can be valuable information in practice, and a tools that combines both quantitative accuracy and qualitative understanding of trends can be very powerful.
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