Monday, 21 October 2013

The tube map dynamic model

The London tube map is a strange and wonderful thing. Compared to most maps, it is not information dense. Yet it contains exactly enough information to allow passengers to easily navigate a complex system of interchanges. Compared to most maps, it is not geometrically similar. The one criteria that usually determines the usefulness of a map has been flouted: indeed a geometrically similar tube map would be more difficult to follow, and hence less useful.

The London tube map is an example of how the measure of a model's quality and information content depends on its purpose. There is no one model that is the best choice in all contexts. It is also a useful reminder that a model is distinct from the concept it represents.

Simple models as a conceptual overview

Just as maps are used to represent the earth's surface, mathematical models can be used to represent dynamic systems. As demonstrated by the London tube map, there are some instances where similarity is undesirable, because the complexity required to represent it detracts from the purpose of the model. A similar way of thinking can be applied to dynamic models of mechanical systems (such as a wave energy converter). A linearised model represented in the frequency domain may be useful because of the simplicity gained by making it a less faithful representation of the real world system. Here the purpose is to give an overview of the interaction between system components, and to investigate alternative design routes.

In both the tube map and the frequency domain model, simplicity results in a more useful tool. The interaction of the key components are easy to visualise, and the models can be incorporated into the mental models of its users, resulting in a common framework on which to base communication.

Simple models as starting points

Simple models are also useful as starting points when the concept being modelled is not initially well understood. Here an iterative approach is useful. Each modelling cycle consists of building a model and comparing its output to the real world. The purpose of the first cycle is to identify the requirements of subsequent models. The resources required for the iterative modelling process can be minimised by gaining as much knowledge as possible about the system using low resource models, as this allows efficient resource allocation on subsequent high resource models. The outcome of this iterative process is a final model that represents a subset of the modelled entity with the accuracy required for a specific purpose.

In the early stages of modelling a dynamic system, a common way of simplifying the problem is to use theory that is valid only for a subset of the problem. The model is then applied to a larger subset, which includes cases where the modelling assumptions do not apply. Such a model can be a useful step in an iterative modelling process, as its purpose may include identification of behaviour not represented adequately due to this extrapolation. Such a model is not suitable for answering specific engineering questions, as the degree of accuracy is unknown.

Frequency domain modelling of wave energy converters

Frequency domain models of dynamic systems are useful as the first iteration of modelling. As they can be solved quickly, they are low resource models that allow fast progression through early design cycles. Linear models are a good basis for building more complex models, as once in time domain form, non-linear terms can be added.

Frequency domain models are also useful for giving an overview of wave power theory. Their simplicity aids understanding and communication. As frequency domain models can be solved algebraically, the equation of motion can be manipulated into different representations. The equation of motion is easy to combine with other information that is conventionally represented as functions of frequency: wave spectra and hydrodynamic forces. A single model representing the sea state, wave body interactions, and dynamic response is then available to investigate how different parameters influence power capture. Algebraic solution gives the opportunity to explore the interaction of the basic components, in order to build up a better understanding of the system.

Image credits:

'London underground circuit map radio' by Yuri Suzuki:
Photo copyright Hitomi Kai Yoda

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