Assignment: 15% Weighting

[Due Date: 11.59 pm, 11 May 2025)

Assignment Instructions:

 This assignment is intended to be solved by every student independently, discussion is allowed but no copy and paste!

 Solve the problems with clear handwriting or typing or combination of both and upload one PDF file.

 List every single workout/step with your answer. Only typing to the point answer will not receive full marks.

 You may need to use MATLAB to solve some questions. please copy and paste your own MATLAB codes, MATLAB figures and Simulink/Simscape diagrams in the answer script.

 Copying assignments is extremely prohibited and if proven both parties will receive '0' marks.

 There are questions that every individual may choose different numerical values in such cases copying will be easily detected.

 Remember, individual students will have different assumptions of the constants variables values, so, avoid copy-paste, otherwise, the answer will receive a zero mark. Make sure to describe the detailed procedures in your answer as applicable to get partial marks in case the answer is not completed.

 Make sure you type the answer next to the question number, e.g., P.1(i), P.1(ii), and so on.

Submission Instructions:

 Complete the attached assignment and submit through this submission site.

 Make sure you name the file "Student's surname_username_assignment15pc"

 If you have any questions, please let me know.

 Complete the " Disclaimer_sign and upload_SP2 2025.docx ".

P. No.    Problem Statement

P.1 Consider the following quarter-car model for an active suspension system.

i) Describe all the symbols (Ms, Mu, Fc, x1, x2, zr, ks, ku, bs, bu) used in the model and draw the appropriate free-body diagrams (FBDs) for the masses.

ii) Derive the differential equations that describe the system dynamics.

iii) Formulate the state-space representation based on the differential equations derived in (ii).

iv) Choose appropriate parameter values (please assume your own numerical values — do not copy from any source), and plot the displacement of the sprung mass in response to the following road disturbances:

(A) a square wave road disturbance. (Please assume your own square wave, do not copy from any source.)

(B) a trapezoidal wave road disturbance. (Please assume your own trapezoidal wave, do not copy from any source.)

(C) a step change in road profile zr (Please assume your own step change, do not copy from any source.)

Please provide detailed derivations, MATLAB codes, and corresponding MATLAB plots.

P.2 Consider the following mass-spring-damper system with an initial displacement only (no external force present):

i) Derive the transfer function of the system based on its dynamic equations.

ii) Write MATLAB code to simulate and plot the displacement response of the system over time.

iii) Create Simulink and Simscape models of the system in MATLAB. Simulate and compare the displacement characteristics using both models. (Please assume your own initial displacement and parameter values - do not copy from any source.)

Include all derivations, MATLAB codes, Simulink/Simscape diagrams, and output plots in your submission.

Hint: Please be careful with the initial conditions.

For example: x(0) = [cℎ00se y0ur 0wn value], x(0) = 0, and so on.

P.3 Consider a type-0, 3rd order transfer function (please assume your own numerical values, do not copy from any source) of a dynamic system and perform. the following operations.

i) Draw the block diagram of the system with a proportional controller (cascaded with the system block) and a negative unity feedback path? (Please assume your own value of the gain, do not copy from any source.)

ii) Find the steady-state error of the system analytically and using MATLAB code for step input?

P.4 Find the response type of the following transfer functions using MATLAB? (Please assume your own values for the numerators (k1, k2, k3, and k4), do not copy from any source.)

i) Find the poles of the following transfer functions using MATLAB code, plot the step responses using MATLAB code, and identify the response type, e.g., overdamped, underdamped, or so?

ii) Compare the step responses of case (b) and (d) in same MATLAB plot and comments on the comparison.

P.5 Consider a 3rd order transfer function (please assume your own numerical values, do not copy from any source) with at least two complex poles on the left half s plane and one pole at (-10,0). Hence perform. the following operations.

i) Write the MATLAB code to translate the transfer function and plot the step response?

ii) Plot the root locus. Based on the root locus plot, what are the dominant poles of the transfer function found in (i)?