Modelling is the process of determining a description of a system based on known physical phenomena. If we can capture the behaviour of the real-life system accurate enough in a set of physical phenomena, we have our verified model. A few questions arise. What is accurate enough and how does such a model look like? In this chapter we will see that the look of the model depends on the used abstraction level.
A model is defined as simplified representation of a system to describe the behaviour of the system. So, the definition includes the aim of the model, which is to desribe a system. We should include enough physical phenomena in the model to be able to reproduce, interpolate and extrapolate real-life behaviour. To extrapolate, we should have thoughts about what physical phenomena we are discarding and up to what level they may be af influence. For example, for describing the resistance of a resistor, we may use Ohms law U = IxR. We know that the temperature dependency of the resistance is not included in Ohms law. Yet, we may understand that in the used system or circuit, the temperature will be of a minor influence.
The modelling is normally done in several abstraction levels. Starting from the real world with all its complexities and influences, we may try to reproduce a set-up in a laboratory with a limited set of influences. This set-up is then a physical model of the system under test. The set-up may be scaled, as a scale model. In that case, our understanding of the scaling of phenomena will enable us to translate the scale model to the real world.
Next, we may go to an abstract model description where physical phenomena and geometric structures are drawn as components. This is a descriptive model also called structural model. Examples are lumped element models, electronic circuits, wiring diagrams, and flow charts/state diagrams. These are normally drawings on paper. Some software packages can capture these descriptive models, but one must realize that in order to calculate some output, we still need a deeper level of modelling.
The next level of abstraction is the mathematical model. This is a set of formulas with signals and boundary conditions. The most common mathematical models are transfer functions and differential equations. The differential equation description together with the boundary conditions is a very generic way to describe a mathematical model because many physical phenomena can be described in integrals and deviations. What we normally see, is that differential equation representations of very different systems can be fairly similar. For example, the differential equation of a damped mass on a spring, is equal to that of an electronic RLC network. Hence, the behaviour is the same. This equivalence is described in the chapter about lumped element models.
Mathematical models that are defined as a differential equation consist of the following elements:
Whether the model is solved analytically (by soling the differential equation) or by means of numerical integration, all three elements have to be determined.
There are several methods to solve the mathematical models. “Solving” means that the implicit representation as coupled physical phenomena is trasnformed into an explicit representation (mathematical) of the cause-effect relation. We can distinguish the methods for solving models into analytical methods and numerical methods. The analytical methods give an explicit equation where still all the physical constants and parameters are visible as mathematical variables. This method is insightful and therefore preferred. With lumped element models this analytical solution approach is sometime possible. With numerical integration, we loose the connection to the model, so we must be aware of the inputs and outputs, and realize that the transient solution is only one single realization of possible output.
In both cases, the model can be mathematically evaluated for a specific input signal (excitation) using the Modelling by numerical integration of differential equations. This is a method to evaluate a mathematical model that is described as a differential equation. With analytical models, like the lumped element method, it is also possible to express key characteristics in explicit forms with system parameters. For example, when a mass-spring system is described in a discrete mass $M$ and a spring with spring constant $k$, it is possible to find an explicit expression of the resonance in terms of $k$ and $M$, which would not be possible with a series of numerical simulations.