Modelling and control by Anthony Rossiter

# Modelling principles and analogies

This is the first chapter in the theme on modelling and simulation of linear models. These resources look at primarily of 1st and 2nd order systems. 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.

## Modelling 1 - Resistors in series

Modelling electrical circuits with series arrangements of resistors and an introduction to Kirchhoff's voltage law.

#### Quick test question

Resistors in series have which property given a fixed applied voltage?
A. The overall current flow increases as more resistors are added in series.
B. The overall resistance increases as more resistors are added in series.
C. The overall resistance reduces as more resistors are added in series.
D. None of the above.

## Modelling 2 - fluid pipes in series

 Modelling the flow rate of fluids through series arrangements of pipes and analogies with electrical resistor circuits.
• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Fluid pipes of the same cross-sectional area and material arranged in series have which property for a fixed flow rate?
A. The ratio of the differential pressure between the ends and the overall pipe length are proportional.
B. The ratio of the differential pressure between the ends and the overall pipe length are inversely proportional.
C. As the differential pressure increases, the overall length reduces.
D. None of the above.

## Modelling 3 - heat conductors in series

Modelling the flow rate of energy (or heat) through series arrangements of heat conductors with different temperatures at each end and analogies with electrical resistor circuits.

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

#### Quick test question

For a fixed temperature difference, which statement is true?
A. Adding more layers increases the heat flow.
B. Increasing the cross-sectional area by more than the length (as a ratio) reduces the heat flow.
C. Reducing the length for a fixed cross sectional area reduces the heat flow.
D. None of the above.

## Modelling 4 - resistors in parallel

Modelling electrical circuits with parallel arrangements of resistors and an introduction to Kirchhoff's current law.

#### Quick test question

For resistors in parallel, which statement is false?.
A. For a fixed voltage supply, the overall current reduces.
B. For a fixed voltage supply, the overall current increases.
C. The overall resistance reduces.
D. None of the above.

## Modelling 5 - fluid pipes and heat conductors in parallel

 Modelling the flow rate of fluids through parallel arrangements of pipes and the flow of heat through parallel arrangements of conductors and analogies with electrical resistor circuits.
• A summary of key facts and derivations is given in the files of section 2 and section 3.
• A talk through video is on youtube.

#### Quick test question

Arranging pipes in parallel has the following effect, compared to a single pipe.
A. For a fixed pressure supply, the overall flow reduces.
B. For a fixed flow rate, the overall pressure difference reduces.
C. Has equivalent flow rate to a longer single pipe.
D. None of the above.

## Modelling 6 - Springs in parallel

 Modelling parallel arrangements of springs and analogies with electrical resistor circuits.

#### Quick test question

For springs arranged in parallel, which statement is true?
A. The overall stiffness is less than each individual spring.
B. The overall stiffness is greater than each individual spring.
C. The overall stiffness may be less and may be more than for than each individual spring. .
D. None of the above.

## Modelling 7 - Springs in series

Modelling series arrangements of springs and analogies with electrical resistor circuits.

#### Quick test question

For springs arranged in series, which statement is true?
A. The overall stiffness is less than each individual spring.
B. The overall stiffness is greater than each individual spring.
C. The overall stiffness may be less and may be more than for than each individual spring. .
D. None of the above.

## Modelling 8 - Summary of analogies

Summary of analogies between resistor circuits, fluid flow systems, heat conductors and systems of springs.
• A summary of key facts and derivations is given in the pdf file .
• A concise tutorial sheet is here.
• A talk through video is on youtube.

#### Quick test question

Which is analogous to set of insulating materials in series (such as cavity walls in a house or multiple layers of clothing)?.
A. Resistors placed in series.
B. Pipes placed in parallel.
C. Springs arranged in parallel.
D. Springs arranged in series.

## Modelling 9 - Concepts of linearity, superposition and dealing with non-linear systems

The first video gives an introduction to the concept of superposition and a link to transfer functions. The second video shows how linearisation of non-linear systems implies that superposition can still be used, within a small domain around the linearisation point and includes some simple examples.
• A summary of key facts and derivations is given in the two files. File1 and file2.
• Talk through video 1 is on youtube.
• Talk through video 2 is on youtube.

## Extra questions on modelling

• Tutorial sheet on units and basic modelling concepts.
• Tutorial sheet on circuits of resistors.
• Tutorial sheet on springs, fluid and heat.
• #### Quick test question 1

The coulomb is a unit of what quantity ?
A. Current.
B. Charge.
C. Resistance.
D. Voltage.

#### Quick test question 2

Which of the following equations does NOT correctly describe Ohm's Law?
A. v=Ri.
B. i=v/R.
C. R=vi.
D. R=v/i.

#### Quick test question 3

In a parallel arrangement of resistors R1 and R2, which of the following are false?
A. 1/R =1/R1 +1/R2.
B. i = i1 + i2.
C. v = v1 + v2.
D. none of the above.

#### Quick test question 4

In a series connection of light bulbs to the mains . When one bulb burns out, what is the effect on the other bulbs ?
A. Other bulbs go out.
B. Little or no affect.
C. The remaining bulbs go dim.
D. The remaining bulbs are brighter.

#### Quick test question 5

The Newton is a unit of what quantity ?
A. Force.
B. Velocity.
C. Displacement.
D. Acceleration.

#### Quick test question 6

In a series arrangement of springs k1 and k2 (extensions x1 and x2), which of the following are false?
A. 1/k =1/k1 +1/k2.
B. x = x1 + x2.
C. f = f1 + f2.
D. none of the above.

#### Quick test question 7

The Joule is a unit of what quantity ?
A. Power.
B. Energy.
C. Temperature.
D. Length.