ENS6160: SIGNALS & SYSTEMS MAJOR ASSIGNMENT Due 5:00 p.m. Tuesday, 27 October 2020 This assignment is worth 20% of the unit mark. This assignment is divided into sections. Total 100 marks You will need to reference all external sources of information. IMPORTANT: Submission Instructions All assignment documents should be submitted via Blackboard. The link for submitting the assignment will be activated 2 weeks prior to the deadline. You should upload a suitably formatted report in Adobe PDF format (preferred) or as a Microsoft Word document (hard copies in the form of printed, handwritten or scanned reports will not be accepted). The report should document in detail your individual approach, work, results, and justified answers to the stated problems. While you are strongly encouraged to discuss this assignment with other students and with the teaching staff to better understand the concepts, the assignment is to be submitted under your name. It is assumed that you are certifying that the details are entirely your own work and that you played at least a substantive role in the conception stage. You should not use results from other students in preparing your solutions. You should not take credit for computer code or graphics that were generated by other students. You should also not directly share your solutions with other students. The teaching staff will check for similarities and you should be prepared to explain your answers and respond to other related questions that may be asked. You should also not use any other resources (including textbooks and websites) without explicitly acknowledging them and being able to explain their inclusion. Assignments should have a title page with your name and student number clearly stated. Do NOT use the standard assignment coversheet as it gets flagged by Turnitin as plagiarised material. You must complete the Turnitin declaration when submitting the assignment. All sources of information and resources other than those provided as part of the unit materials of this unit must be properly referenced (in-text and end-text) using the APA format. The use of materials drawn from other sources without appropriate acknowledgement is plagiarism and consequences for such actions will apply. Refer to the Academic Misconduct section at the end of the assignment. Analysing a simple system A vehicle suspension system can be modelled by the block diagram shown in Figure 1 below: Figure 1: Block diagram of vehicle suspension system In this block diagram, the variation in the road surface height as the vehicle moves is the input to the system. The tyre is modelled by the spring and dashpot (damping) system with spring constant and damping coefficient respectively and this results in the displacement of the wheel (), represented by the mass . The wheel’s displacement acts as an input to the suspension system, modelled by the spring and dashpot with spring constant and damping coefficient respectively and this results in the displacement ( ), of the body, represented by the mass . When the car is at rest, it is taken that = 0, = 0 and = 0. (Note: is normally a quarter of the vehicle mass since it is assumed the weight is distributed evenly between the 4 wheels. This system is composed of two mass-spring-damper systems ‘stacked’ one on top of the other. We shall first consider the behaviour of a single sub-system. Consider the simple mass-spring damper system shown in Figure 2 below: Figure 2: A single mass-spring-damper system In Figure 2: x is the position of input body/surface, with its rest position given by x = 0. The mass m represents the mass The height of mass m above its reference level is called y. The reference level is chosen such that when system is at rest, y = 0. Section 1: Mathematical Analysis of System (25 marks) 1. Draw a free-body diagram showing all the forces acting on the mass m shown in Figure 2. 2. From the earlier description, diagrams and the laws of Physics, show that the motion of the system in Figure 2 can be described by the LCCDE (linear constant-coefficient differential equation) below: + + = + 1 3. Using the Laplace transform of the equation above, find an expression for , the system transfer function. The mass-spring-damper system is a damped second order system. It is common to express the homogenous second order DE for such a damped system as + 2 + = 0 2 where is the damping ratio and is the undamped natural (resonant) frequency. 4. From equations (1) and (2), determine expressions for (the damping ratio) and (the natural frequency) in terms of the parameters m, k and C 5. Determine the characteristic equation and eigenvalues (characteristic values) for this system based on equation (2) above (in terms of and ). 6. From the answer to part 5, determine the full mathematical expression (in terms of and ) for the natural response of the system for the following cases: a. = 0 b. 0 < ="">< 1="" c.="" ="1" d.="" =""> 1 Consider a suspension system with the following parameters: = 380 kg = 15,000 N/m 7. Determine (in rad/s) for this suspension system and the corresponding value for (in Hz). 8. Calculate the required value of in order to achieve = 1 Note: Complete and clear working is required for all answers for this section. Section 2: System analysis using Matlab (30 Marks) In this section, the system responses should be analysed using Matlab. Refer to the document “A Brief MATLAB Guide” in order to understand how to represent LTI systems in Matlab, and hence how to determine impulse response, step response and frequency response of systems. Sample Matlab code that has been provided as part of the learning materials can also be modified to suit. Students are advised to refer to the help function within Matlab as well as online Matlab documentation for more details. Note: MATLAB is installed in the engineering computer labs and is also available to ECU students via ECU’s site licence (refer to the announcement regarding site license for details). Using the commands given in the Guide, analyse the response of the suspension system using the and parameters given in Section 1 and value calculated in question 8. 9. Plot the impulse response and step response of the system (for 2 seconds duration and time ‘step size’ of 1 millisecond) using the impulse and step functions. Include all plots (properly labelled) in your submission. 10. Determine the frequency response from 0 to 200 rad/s using the freqs command. Plot the magnitude and phase response over this frequency range. Hint: Use frequency ‘step size’ of 0.1 rad/s. Hint 1: You can plot all 4 graphs in one go using a 2 x 2 matrix of plots using subplot(22n), where n determines which of the 4 subplots gets used. Hint 2: In order to clearly see variations over a range of frequencies, it is best to use a log scale for the frequency and magnitude (phase would still be displayed using linear scale). The functions loglog (for magnitude) and semilogx (for phase) can be used instead of plot. 11. Determine the magnitude response at . Determine the frequency of the -3dB point (magnitude = 1⁄√2 of passband). Hint: Use the ‘data cursor’ tool on the plot of the magnitude response. It shows the x and y values of the plot as you move along the curve. 12. Discuss the response of the system. Why do the impulse and step responses have that particular shape? How well will this system fulfil its purpose of a vehicle suspension? Note: The function of a suspension system is to ‘filter out’ the effect of bumps, potholes and other such road surface irregularities, but allow the vehicle to ‘follow the road’ as the height of the road surface varies. 13. Repeat the analysis above (steps 9 – 11) for the following damping ratios a. = 0.4 b. = 0.7 c. = 1.5 d. = 2.0 Hint 3: It would be more efficient to put all the necessary commands into a script file (a .m file) so you can edit the parameters and then run all the commands at once. 14. Based on the results of the Matlab analysis above, which of the 5 values of damping ratio would be best for application as a suspension system. Justify your selection. Section 3: Modelling System response to input (25 marks) The response of the suspension system designed in Section 2 can be modelled by providing an input x that resembles the road input (the assumption is that the tyre assembly does not impact the system response and therefore x is the same as road ‘input signal’ r). The M-file for the custom-written function inputsig has been provided, with explanatory comments in the file. 15. Use the inputsig function to generate a sinusoid of 1 Hz with magnitude 0.5 and duration 2 seconds, then simulate the suspension system’s response to this function and plot the input and output response vs time (output plot below input plot for easy comparison). Hint 4: The system response to an input signal can be simulated using the lsim function. Following is some sample code that shows how this function could be used. [xsig, t1] = inputsig(10); % generate signal duration 10 Sys1 = tf(B,A); % define system, use tf (transfer function) Sysout = lsim(Sys1, xsig, t1); % simulate LTI system response Note: Students are advised to refer to the help function within Matlab as well as the online Matlab documentation for more details. 16. Repeat step 15 (plots of input vs output) for input duration of 1 second with the input signal being: a. a sinusoid of frequency 10 Hz and magnitude 0.05 b. square wave of frequency 4 Hz and magnitude 0.1 c. pulse of frequency 3 Hz and magnitude 0.05 17. Compare the system response to the 4 types of input and explain the output signal for each and the differences. What would these 4 represent in terms of ‘road input’? 18. Repeat step 16 with a combination of: a. Sinusoid of 1Hz and magnitude 0.5 plus a sinusoid of frequency 10 Hz and magnitude 0.05 b. Sinusoid of 1Hz and magnitude 0.5 plus a square wave of frequency 4 Hz and magnitude 0.1 c. Sinusoid of 1Hz and magnitude 0.5 plus a pulse of frequency 3 Hz and magnitude 0.05 d. Sinusoid of 1Hz and magnitude 0.5 plus a sinusoid of frequency 10 Hz and magnitude 0.05 AND a square wave of frequency 4 Hz and magnitude 0.1 19. Describe the differences in input and output waveforms in the cases in step 18. Hence comment on the effectiveness of the ‘designed’ suspension system. Section 4: State Space Analysis (20 marks) A brief introduction to state space analysis State space analysis enables a system that would be described using an n th order differential equation to be represented using first order matrix differential equations through the definition of n state variables. The state space representation of a system is given by two equations: a) The state equation: $% = &$ + '( b) The output equation: ) = *$ + +( Note: The variables in the equation above are denoted in bold as they are matrices. For an n th order system with r inputs and m outputs, the sizes of the matrices are as below: Matrix Size Name Type q n x 1 State vector Function of time A n x n State matrix Constant B n x r Input matrix Constant x r x 1 Input Function of time y m x 1 Output Function of time C m x n Output matrix Constant D m x r Direct transition (or feedforward) matrix Constant 20. Research and discuss the advantages of state space analysis and the types of problems for which state space analysis is most suited (in no more than 500 words). 21. Discuss the approach to, advantages of and possible difficulties in using state space techniques to analyse the complete suspension system including the tyre and wheel, as shown in Figure 1. Give examples as applicable. Note: You will need to properly reference your sources (both in-text and end-text). All referencing must conform to the APA referencing standard (as per ECU referencing standards). ~ End of Assignment Instructions ~ MAJOR ASSIGNMENT MARKING SCHEME Section 1: Mathematical Analysis (25 marks) Description Marks Free body diagram 3 Deriving DE formula 4 System transfer function 2 Expressions for damping coefficient and natural frequency 2 Characteristic equation and eigenvalues 4 Natural response (4 cases) 8 Calculation of and 2 TOTAL 25 Section 2: Matlab System Analysis (30 marks) Description Marks Impulse and step response, frequency plot, and specific magnitude values 12 Description of responses and relating it to the system’s desired function 3 Analysis of various damping ratios 12 Selection and justification of preferred damping ratio 3 TOTAL 30 Section 3: Modelling System response to input (25 marks) Description Marks Simulation of input and output responses (4 basic types) 10 Explanation of responses 4 Simulation of ‘combination’ inputs 8 Discussion output response and suitability of system 3 TOTAL 25 Section 4: Investigation of State Space Analysis Techniques (20 marks) Description Marks Discussion of state space analysis advantages and disadvantages, and suitable problems 12 Discussion on the use of state space analysis for complete suspension system 5 Referencing 3 TOTAL 20 Academic Misconduct Edith Cowan University has firm rules governing academic misconduct and there are substantial penalties that can be applied to students who are found in breach of these rules. Academic misconduct includes, but is not limited to: Plagiarism Unauthorised collaboration Cheating in examinations Theft of other students’ work Additionally, any material submitted for assessment purposes must be work that has not been submitted previously, by any person, for any other unit at ECU or elsewhere. The ECU rules and policies governing all academic activities, including misconduct, can be accessed through the ECU website. If you are unsure of the meaning of terms like plagiarism or would like to understand this topic better, you are encouraged to access the AIM-Student community site in Blackboard.