# Chapter two

Modelling and behaviour

## Section four: modelling second order systems

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

A dx^{3}/dt^{3} + B d^{2}x/dt^{2} + 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 both 2nd and higher 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 2nd order model?

The sections on system responses will develop the principles and analogies further to illustrate how we create models of dynamic behaviour, including systems beyond 2nd order.

### 1. Simple electrical circuits

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

Resistor-inductor-capacitor modelling notes (PDF, 262 KB).

RLC circuit modelling video.

### 2. Simple linear mechanical systems

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

Mass-spring-damper video and notes (PDF, 439 KB).

Mass-spring-damper with a pulley notes (PDF, 535 KB).

Car suspension system notes (PDF, 361 KB).

Mass-spring-damper and gears video.

### 3. Electro-mechanical

Real systems often combine components from multiple disciplines, electro-mechanical being very common.

DC servo with a flexible shaft notes (PDF, 331 KB).

DC servo with simple dynamics notes (PDF, 307 KB).

DC servo with gears notes (PDF, 361 KB).

DC servo modelling video.

DC servo with modifications video.

### 4. Fluid systems

In a manufacturing system, there may be multiple tanks which are interconnected and this can lead to higher order dynamics.

Multiple tank systems notes (PDF, 770 KB).

Modelling a two tank system video.

### 5. Miscellaneous case studies

A number of examples exploiting first principles modelling to derive models of various orders.

Speed of a car notes (PDF, 248 KB).

House temperature with radiator dynamics notes (PDF, 195 KB).

Aeroplane roll notes (PDF, 224 KB).

Diabetes and blood sugar levels notes (PDF, 184 KB).