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

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

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:

𝑑2𝑥1(𝑡) 𝑑𝑥1(𝑡) 𝑑𝑥2(𝑡)

2 + 2 + 𝑥2(𝑡) − 2 = 0

𝑑𝑡2 𝑑𝑡 1 𝑑𝑡 Eq.(2)

𝑑2𝑥2(𝑡) 𝑑𝑥2(𝑡) 𝑑𝑥1(𝑡)

2 + 2 − 2 = 𝑓(𝑡)

𝑑𝑡2 𝑑𝑡 𝑑𝑡 Eq.(3)

 

Question 3 (10 marks)

Linearise the system for the point 𝑥1(0) = 1, 𝑥2(0) = 0, 𝑓(0) = 0, and show that the linearised differential equations are given by:

𝑑2𝛿𝑥1(𝑡) 𝑑𝛿𝑥1(𝑡) 𝑑𝛿𝑥2(𝑡)

2 + 2 + (1 + 2𝛿𝑥1(𝑡)) − 2 = 0

𝑑𝑡2 𝑑𝑡 𝑑𝑡 Eq.(4)

𝑑2𝛿𝑥2(𝑡) 𝑑𝛿𝑥2(𝑡) 𝑑𝛿𝑥1(𝑡)

2 + 2 − 2 = 𝛿𝑓(𝑡)

𝑑𝑡2 𝑑𝑡 𝑑𝑡 Eq.(5)

 

Hints: 𝛿𝑥1(𝑡) = 𝑥1(𝑡) − 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 𝛿𝑥1(𝑡) with 𝑥1(𝑡), 𝛿𝑥2(𝑡) in Eq.(4) and Eq.(5) with 𝑥2(𝑡) and 𝛿𝑓(𝑡) with 𝑓(𝑡) (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:

𝑑2𝑥1(𝑡) 𝑑𝑥1(𝑡) 𝑑𝑥2(𝑡)

2 + 2 + (1 + 2𝑥1(𝑡)) − 2 = 0

𝑑𝑡2 𝑑𝑡 𝑑𝑡 Eq.(6)

𝑑2𝑥2(𝑡) 𝑑𝑥2(𝑡) 𝑑𝑥1(𝑡)

2 + 2 − 2 = 𝑓(𝑡)

𝑑𝑡2 𝑑𝑡 𝑑𝑡 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. 𝑥1(0) = 0, 𝑥̇1(0) = 0, 𝑥2(0) = 0, 𝑥̇2(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 𝑓(𝑡) and the output is 𝑥2(𝑡).

(5 marks)

b) What is the order of the system? Justify your answer.

(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).

 

 

b) Determine if the system is stable. Justify your answer.

 

c) Draw the “𝑠-plane”.

 

(12 marks)

 

(3 marks)

 

(5 marks)

 

 

Question 7 (5 marks)

Use Final Value Theorem to predict the value of 𝑥2(𝑡) (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 𝑓(𝑡) is a unit impulse.

For each method (20 marks × 𝟐 = 𝟒𝟎 marks):

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

(10 marks)

 

ii. Select an appropriate step size ℎ = Δ𝑡. Explain your reasoning.

(2 marks)

 

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

(2 marks)

 

iv. Calculate the error (𝐸𝑡) and percentage error (|𝜖𝑡|%) 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 𝑥2(𝑡) 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).

 

Attachments:

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