Modelling and control by Anthony Rossiter

# MODELLING 1st ORDER SYSTEMS FROM MECHANICAL, ELECTRICAL, FLUID AND THERMAL SCENARIOS

This is the first chapter in the theme on modelling and simulation of linear models. Use the left hand toolbar to access the other chapters and themes. It is implicit in several of these chapters that students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials and Laplace. More information on these can be found in the Mathematics theme on the left hand toobar.

This chapter contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and also a tutorial sheet for users to test themselves.

## 1st order modelling 1 - mass-damper

Derives the model representing simple mass-damper systems with a focus on parallel arrangements but some brief discussion of alternatives.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Which represents a model for a mass damper (mass M, damping B, force f, displacement x, velocity v) ?
A. M(dx/dt)+Bx =f.
B. B(dv/dt)+Mv=f.
C. M(dv/dt)+Bv=f.
D. None of the above.

## 1st order modelling 2 - spring-damper

Derives the model representing spring-damper systems with a focus on parallel arrangements and some brief discussion of a series set up.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Which represents a model for a spring damper in parallel (spring k, damping B, force f, displacement x, velocity v) ?
A. B(dx/dt)+kx=f
B. B(dv/dt)+kv = f
C. k(dx/dt)+Bx=f
D. None of the above.

## 1st order modelling 3 - resistor-capacitor

Derives models for series resistor-capacitor circuits and discusses analogies with spring-damper systems. Brief consideration of parallel resistor-capacitor circuit.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## 1st order modelling 4 - resistor-inductor

Derives models for series resistor-inductor Â circuits and discusses analogies with mass-damper systems and summarises broader analogies between electrical and mechanical systems. Brief consideration of parallel resistor-inductor circuit.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## 1st order modelling 5 - fluid pipes and tanks

Derives models for the depth of simple tank systems with in-flow and out-flow  based on pipe flow models. Considers high pressure input or direct inflow and analogies with  electrical circuits.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## 1st order modelling 6 - thermal systems

Derives models for simple thermal systems containing capacitance and insulation. Summarises analogies with other systems.

## 1st order modelling 7 - time constant form and analogies

Looks at models in first 6 videos and shows how to put into time constant form. Uses this to give an alternative view on analogies. Time constant form is discussed more in the videos on responses. #### Quick test question

For a spring-damper system, what is the time constant?
A. The ratio of stiffness over damping.
B. The ratio of damping over stiffness.
C. The sum of stiffness and damping. .
D. None of the above.

## 1st order modelling 8 - tank level system

Introduces the modelling of a simple tank level system and shows this has a 1st order model.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## 1st order modelling 9 -mixing tank

Discusses the modelling of a mixing tank with no reaction. Introduces the concept of deviation variables. Again, this leads to a first order model. WARNING: There are 2 minor but obvious typos (11min and 11.38) in this show where the narrator manually writes T/Fo; should be just T because T=V/Fo.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video 1 is on youtube.

## 1st order modelling 10 - mixing tank examples and reactions

Expands on the mixing tank example of the previous video by giving a numerical example and also showing how the modelling would be affected by the presence of a reaction within the tank which also results in nonlinear behaviour. Local linearisation in conjunction with deviation variables is used.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

## 1st order modelling 11 - heat exchanger

Looks at the modelling of a heat exchanger which has liquid flowing in at one temperature, being mixed in a tank in the presence of heating and then exiting tank at same flow rate. It is shown that his model has simple first order dynamics, but is represented as having two inputs rather than one.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

## 1st order modelling 12 - tank system tutorial example

Presents a typical tutorial/exam question on a tank system for students to try. Then develops solutions so students can compare their work with the possible solution provided.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

## 1st order modelling 13 - heat exchanger tutorial example

Presents a typical tutorial/exam question on a heat exchanger system for students to try. Then develops solutions so students can compare their work with the possible solution provided.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.

## 1st order modelling 14 - mixing system tutorial example

Presents a typical tutorial/exam question on a mixing system with nonlinear reaction for students to try. Then develops solutions so students can compare their work with the possible solution provided.

• A summary of key facts and derivations is given in the pdf file.
• Talk through video is on youtube.  