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# Write your own Python code to implement both numerical methods to simulate the system.

INSTRUCTIONS TO CANDIDATES

PART A (Modelling)

Question 1 (2 marks)

Provide a real-life example where a general mass-spring-damper system can be found.

Question 2 (10 marks)

Using a free-body-diagram, show that the differential equations representing the system in Figure 1 are given by:

π2π₯1(π‘) ππ₯1(π‘) ππ₯2(π‘)

2 + 2 + π₯2(π‘) − 2 = 0

ππ‘2 ππ‘ 1 ππ‘ Eq.(2)

π2π₯2(π‘) ππ₯2(π‘) ππ₯1(π‘)

2 + 2 − 2 = π(π‘)

ππ‘2 ππ‘ ππ‘ Eq.(3)

Question 3 (10 marks)

Linearise the system for the point π₯1(0) = 1, π₯2(0) = 0, π(0) = 0, and show that the linearised differential equations are given by:

π2πΏπ₯1(π‘) ππΏπ₯1(π‘) ππΏπ₯2(π‘)

2 + 2 + (1 + 2πΏπ₯1(π‘)) − 2 = 0

ππ‘2 ππ‘ ππ‘ Eq.(4)

π2πΏπ₯2(π‘) ππΏπ₯2(π‘) ππΏπ₯1(π‘)

2 + 2 − 2 = πΏπ(π‘)

ππ‘2 ππ‘ ππ‘ Eq.(5)

Hints: πΏπ₯1(π‘) = π₯1(π‘) − 1, where 1 is the linearisation point.

PART B (Analytical and numerical methods)

Remark: The remaining questions will be based on the linearised system given in Equation Eq.(6) and Equation Eq.(7).

For the remaining questions, replace the linear variable πΏπ₯1(π‘) with π₯1(π‘), πΏπ₯2(π‘) in Eq.(4) and Eq.(5) with π₯2(π‘) and πΏπ(π‘) with π(π‘) (i.e. ignoring the linearisation point). In other words, the remaining questions are based on the result of the linearisation process, which is given by:

π2π₯1(π‘) ππ₯1(π‘) ππ₯2(π‘)

2 + 2 + (1 + 2π₯1(π‘)) − 2 = 0

ππ‘2 ππ‘ ππ‘ Eq.(6)

π2π₯2(π‘) ππ₯2(π‘) ππ₯1(π‘)

2 + 2 − 2 = π(π‘)

ππ‘2 ππ‘ ππ‘ Eq.(7)

Question 4 (6 marks)

Apply Laplace transform on the linearised system in equations Eq.(6)-Eq.(7). Assume zero initial conditions, i.e. π₯1(0) = 0, π₯Μ1(0) = 0, π₯2(0) = 0, π₯Μ2(0) = 0.

Question 5 (7 marks)

Using the results from Question (4),

a) Obtain the transfer function for the linearised system (i.e. equations Eq.(6) and

Eq.(7)). Note that the input of the system is π(π‘) and the output is π₯2(π‘).

(5 marks)

(2 marks)

Question 6 (20 marks)

Using the transfer function from Question (5)

a) Obtain and clearly list all the poles and zeros of the linearised system. (Remark: there should be two zeros and four poles. Two of the poles are at s = −0.12256 ± 0.74486i and one at s = 0. You need to find the other one real pole.)

(Remark 2: the location of the remaining pole is between -2 and 0).

c) Draw the “π -plane”.

(12 marks)

(3 marks)

(5 marks)

Question 7 (5 marks)

Use Final Value Theorem to predict the value of π₯2(π‘) (if available) when the input is a unit impulse. Justify your answer.

Question 8 (40 marks)

Using Equation (6) and Equation (7), create a simulation of the system using both:

a) Euler method, and

b) Heun (Runge-Kutta second order) method Assume that the input force π(π‘) is a unit impulse.

For each method (20 marks × π = ππ marks):

i. Write your own Python code that creates the simulation of the system

(10 marks)

ii. Select an appropriate step size β = Δπ‘. Explain your reasoning.

(2 marks)

iii. Select an appropriate end time for the simulation. Explain your reasoning.

(2 marks)

iv. Calculate the error (πΈπ‘) and percentage error (|ππ‘|%) and discuss the results.

Remark: Use the final value from Question (7) as the exact numerical solution

(3 marks)

v. Plot the output responses (i.e. plot the output position π₯2(π‘) against time in seconds) and discuss the results in terms of general system behaviour (e.g. underdamped or overdamped etc).

(3 marks)

Important remark for Question 8:

- Write your own Python code to implement both numerical methods to simulate the system.

- Use a single Python file to implement both methods.

- You can use an existing/example program to check your solution.

- However, do not submit the existing/example program as your submission.

- To avoid compatibility issues when marking the Python code, please use Spyder

4.2.5 (Python 3.8). (You can download Spyder through Anaconda navigator – see the webpage: https://www.anaconda.com/products/individual).

(5/5)

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