# Chapter two

Modelling and behaviour

## Section two: modelling first order systems

This chapter is on the theme of linear models, for example:

A d³x/dt³ + B d²x/dt² + C dx/dt + D x = K u

where x(t) is the state, u(t) the input and A, B, C, D, K are model parameters.

This section focuses on applying core modelling principles to a variety of 1st order systems.

How do I model simple electrical and mechanical systems and are there analogies between similar arrangements of different components?

What systems from a broader range of disciplines can also be described by a 1st order model?

The sections on 1st order responses will develop the principles and analogies further to illustrate how we create models of dynamic behaviour.

### 1. Simple electrical circuits

Use of Kirchoff's voltage law (voltage balance) and component equations to derive first order models for simple series circuits.

Resistor-capacitor video and notes (PDF, 665 KB).

Resistor-inductor video and notes (PDF, 678 KB).

### 2. Simple mechanical systems

Modelling a system of mechanical components arranged in parallel using force balance across components.

Spring-damper video and notes (PDF, 747 KB).

Mass-damper video and notes (PDF, 770 KB).

### 3. Fluid systems

Fluid systems are driven by pressure differences across pipes/orifices causing flow, and storage components such as tanks. Balance is done with respect to volumes and/or molar quantities.

General fluid systems notes (PDF, 718 KB).

Tank-level systems notes (PDF, 624 KB).

Mixing tanks video and notes (PDF, 772 KB).

Mixing tank with reaction video and notes (PDF, 774 KB).

Pipes and tanks modelling video.

Tank level modelling video.

Tank system modelling tutorial video.

Mixing tank tutorial video.

### 4. Thermal systems

This could be a heat exchanger in manufacturing or as simple as domestic heating.

Heat exchanger modelling video and notes (PDF, 652 KB).

House temperature notes (PDF, 654 KB).

Thermal system modelling video.

Heat exchanger modelling tutorial video.

### 5. Analogous arrangements

It is always useful to summarise the analogies between systems from different disciplines as this can help understand behaviours and support design.

Analogous first order systems notes (PDF, 903 KB).

Modelling analogies and time constant form video.

Analogies with savings models notes (PDF, 166 KB).