Date Set: 21st  September 2023

Date Due: 15th December 2023, 12:00 (noon) (TBC)

Method of submission and file type: ELE2 (zip)

Answer all questions.

The actual number of marks associated with this coursework is 100, which translates to 50% of the overall module assessment.

This coursework tests your understanding of modelling and simulation of engineering systems.

Instructions

•  This coursework has two parts:

1)  A PDF file for Question (1) to (7).

2)  A Python program (py file) to implement simulations for Question (8).

•  Both files (PDF and py) must be zipped and must be submitted as a single zip, using ELE2 (seehereandherefor detail).

•  Submission must be made by 12:00 (noon) on the date indicated.

•  Hand written work must be on lined A4 paper, scanned and converted into PDF.

•  Your solutions must include neatly drawn labelled diagrams (where applicable), correct use of mathematical notation, sufficient comments and workings for the

marker(s) to easily follow your process. Where practicable, you should also show how you have checked your answers.

•  Python program must be clearly described with sufficient comments to explain the steps in the program.

•  Marks are awarded for clear, legible presentation of work.

System description: Mechanical mass-spring-damper system

The coursework deals with modelling and analysis of the mechanical mass-spring- damper system shown inFigure 1.

Figure 1: Nonlinear mass-spring-damper system

The variables x1 (t) and x2 (t) represent the positions of mass m1  and m2. The force f(t) is the system input, whilst the output is the position x2 (t) .

Here the masses are m1  =  1 kg and m2   = 1 kg and the damper b  =  1 Ns/m. The spring is nonlinear, and the force (N), Fs (t), required to stretch the spring is:

Fs (t) = 2x1(2)(t)                                                                     Eq.(1)

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 inFigure 1are given by:

 +  + 2x1(2)(t) −  = 0                                          Eq.(2)

d2x2 (t) dt2

 − 

= f(t)

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:

d2 δx1 (t)     dδx1 (t)                                        dδx2 (t)                                       Eq.(4)

dt2         +      dt      + 2(1 + 2δx1 (t)) −      dt       = 0

d2 δx2 (t) dt2

 − 

= δf(t)

Eq.(5)

Note: 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 EquationEq.(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

d2x1 (t)

dt2

 + 2(1 + 2x1 (t)) −  = 0

 +  −  = f(t)

Eq.(6)

Eq.(7)

Question 4 (6 marks)

Apply Laplace transform on the linearised system in equationsEq.(6)-Eq.(7). Assume

zero initial conditions, i.e. x1 (0) = 0,̇(x)1 (0) = 0, x2 (0) = 0,̇(x)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 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 (12 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 =  −0.3522  ± 1.7214i and one at s = 0. You need to find the other one real pole. You can use calculator to find these values.)

(Remark 2: the location of the remaining pole is between -2 and 0). (4 marks)

b)  Determine if the system is stable. Justify your answer.(3 marks)

c)  Draw the “s-plane” . (5  marks)

Question 7 (3 marks)

Use Final Value Theorem to predict the value of x2 (t) (if available) when the input is a unit impulse. Justify your answer.