Homework answers / question archive / 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 no order differential equation to be represented using first order motrix differential equations through the definition of n state variables

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 no order differential equation to be represented using first order motrix differential equations through the definition of n state variables

Statistics

Share With

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 no order differential equation to be represented using first order motrix 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: 4 = Aq + Bx b) The output equation: y = Cq + Dx Note: The variables in the equation above are denoted in bold as they are matrices. For an oh order system with / inputs and m outputs, the sizes of the matrices are as below: Matrix Size Name Type nx1 State vector Function of time A nxn State matrix Constant nor Input matrix Constant rx1 Input Function of time mx1 Output Function of time C mxn Output matrix Constant D mxr Direct transition (or feedforword) 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).

Section 3: Modelling System response to input (25 marks) The response of the suspension system 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" rj- 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 (sim function. Following is some sample code that shows how this function could be used. (xsig, ti] = inputsig [10) ; generate Signal duration 10 Sys1 = tf (8, A] ] define system, use of (transfer function) Sysout = 1sim (Sysl, xsig, t1) A 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 0.1 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 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 my and k, parameters given in Section 1 and C, 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 a,. Determine the frequency of the -3dB point (magnitude = 1/V2 of passband). Mint: 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 1: Mathematical Analysis of System (25 markal 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 LOCDE (linear constant-coefficient differential equation) below: d'y(!) C dy (c) k -y(0) = Car(t) in de + -x(0) (1) 3. Using the Laplace transform of the equation above, find an expression for H (s) , 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 dy (c) de2 + 24an- + y (!) =0 (2) where ( is the damping ratio and a is the undamped natural (resonant) frequency. 4. From equations (1) and (2], determine expressions for < (the damping ratio) and at (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 on and { ). 6. From the answer to part 5, determine the full mathematical expression (in terms of a, and { ) for the natural response of the system for the following cases: a. <=0 b. 0 < <1 c. <=1 d. (31 Consider a suspension system with the following parameters: i12 = 380 kg k. = 15,000 N/m 7. Determine , (in rad/s) for this suspension system and the corresponding value for for (in HE). 8. Calculate the required value of C, in order to achieve ( = 1 Note: Complete and clear working is required for all answers for this section.

Analysing a simple system A vehicle suspension system can be modelled by the block diagram shown in Figure 1 below: Body mass Spring of Jumpercl courti ciont: "Whaal moss, su G Figure 1: Block diagram of vehicle suspension system In this block diagram, the variation in the road surface height r as the vehicle moves is the input to the system. The tyre is modelled by the spring and dashpot (damping) system with spring constant k, and damping coefficient C, respectively and this results in the displacement of the wheel (x], represented by the mass my. The wheel's displacement acts as an input to the suspension system, modelled by the spring and dashpot with spring constant k, and damping coefficient C, respectively and this results in the displacement (yl, of the body, represented by the mass my. When the car is at rest, it is taken that r = 0, x = 0 and y = 0. (Note: my 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: Spring Damper Figure 2: A single mass spring-damper system In Figure 2: * 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.