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Part -1 Explanation involves the following: A- Mechanical - Derive the transfer function for a mechanical system. - Reference the transfer function to the generalized second order transfer function. - Use the transfer function in Matlab to plot the system’s transient response (for a step input) under different damping conditions. Use the Matlab results to determine system damping, and other transient characteristics. - Plot the system poles in the complex plan. Use the poles plot to infer some conclusions - Plot the frequency response under different damping conditions. Use the response to infer the following: natural frequency, cut-off frequency, roll-off rate, type of response B- Electrical - Write the state equations for a second order electrical system - use the derived state equations to: o determine the system’s eigenvalues o predict the system’s generalized transient response due to a step input o Use Matlab to plot the step - transient response of the system for different states of the system o Plot the systems response for a sinusoidal input o make concluding remarks about the response in relation to system poles. C- Simulation - Simulate the electrical circuit and investigate: o the transient response for a step input o show the response of different states due to the step input o compare this result to that obtained with Matlab o investigate the output (response) for non-zero initial conditions and compete to Matlab results o investigate the response (output) for a sinusoidal input and compare to Matlab Results. D- Discussion and Conclusions o Discuss results o explain certain phenomenon of system response o demonstrate understanding of the material Deliverables: All deliverables must be complete, well organized, have professional appearance, show all work, provide commented and organized Matlab Code, provide Matlab results and plots (label correctly), discuss results. MUST be well organized and very easy to follow. For each part of the sections above: 1- give the problem statement 2- provide schematics/drawings 3- show all theoretical work -- well presented 4- discussion of results thus far 5- provide compete and well documented Matlab code anytime used 6- provide results of software (Matlab or Simulink). It must be well presented ( don’t just show results and leave to the reader to figure out) 7- correctly labeled and titles figures. 8- draw conclusions or state comments that demonstrate your understanding Solutions must be complete, clear, professionally presented, well organized This is individual work. All Matlab code must be placed in the corresponding dropbox for this exam on D2L Part -2 Problem No.1 (Mechanical System) For the mass-spring-damper system shown, determine the expression for the motion, x(t) and plot it. K= 9 N/m, m=3kg, f = 28N and the damping constant has the following three possibilities: � = # ���� (�) 12 ���� (�) 1 ���� (� ) 10.39232 �. �/� . Assume zero initial conditions{�(0) = 0 �, �̇(0) = 0 �/� � is the displacement and �̇ is the velocity. Hint: Refer to Chapter 3 and to lectures of 4/17 through 4/27 Do not substitute values for m, K, B, or f yet. Step 1- Write the differential equation for the displacement, y(t) �̈(�) = − ? ? �̇(�) − ? ? �(�) + ? ? �(�) Step 2- With zero initial conditions and using your result in Step 1, determine the transfer function G(s). Ensure it is written in the standardized form (as was done in class.) �(�) = C(D) E(D) = ? Step 3- In this step, you will substitute the values of m and K as given above and for each case of B you are to: - calculate the poles and zeros of the transfer function - predict the step response of the system - calculate the damping ratio, the natural and damping frequencies - complete the table Using your result of G(s) found in Step 2, calculate and complete the table below. Must Show work for case (b) Table-1: Second order Translational System’s Data B (N.s/m) ��(rad/sec) ��(rad/sec) � poles zeros Expected Step Response classification Case( a) 12 Case( b) 1 Case( c) 10.3923 Work for Case (b): Step 4- Use Matlab (Submit the complete code in the designated dropbox on D2L) Write a complete and well organized and documented Matlab code as follows: Name this code: Pr1Stp4.m - First, make a data section for the system constants m, K, B, and A. Note A is the magnitude of the input (in our case f = 28N, thus A = 28.) you may want to be more creative here and make B a vector of three quantities. Your code loops the number of quantities in vector B. And in each iteration, it picks the proper B case based on iteration number (index). However, you can run the program separately each time by providing a new value of B based on the case under test. - define the numerator and denominator vectors **Repeat all of the below for each value of B. This is indicated on the next page. ** Case (a) B = num = [ ] ? den = [ ]? -create and display the transfer function G= tf(num,den) notice: no semicolon, thus G will be displayed - pause the code here so the user can observe the transfer function. Of course, instruct the user to “ hit any key to continue …” - determine the poles and zeros of the transfer function. (There are several commands that one may use including commands that map the poles and zeros onto the complex plane. for example: pzmap(G) followed by the grid command will only use the commands: pole(G) zero(G) - pause the code here so the user can observe the transfer function. Of course, instruct the user to “ hit any key to continue …” - write the poles and zeros down as you will need to enter them into Table -2 poles are: zeros are: - Determine the step response due to f = 28N step(A*G) - title and label the axis properly. Also include a grid - of course do the pause steps to allow user to complete and paste the figure. Or you may use the figure handle as we did in class, this way all figures can be accessed. - copy and paste the figure into Word (Place below this line ) - do not close the figure yet - right click on the figure and show different transient characteristics (rise time, overshoot, steady state value, and settling time) - copy the figure with the data and paste into Word. Data on this figure will need to be placed in Table -2 later. (Place the figure below this line) Frequency Response - Use the command: bode(G) to plot the frequency response - in the magnitude plot, move the cursor to the lowest possible frequency (close to 1Hz) and write down the corresponding gain (this is called the Low Frequency Gain.) Ao(dB) = ? (write this value as it will be needed for Table -2 later) - move the cursor until you reach a gain that is 3dB below Ao(dB). Record the frequency at that point. This is the cut-off frequency wc = ? Write down this value as it will be needed later for Table -2 - For case(b) only, move the cursor to the maximum value of the magnitude frequency response, the frequency at this point corresponds to the natural frequency. wn Record this frequency and provide the Matlab figure. ( Insert the figure below this line) - - �K = = ? Case (b) B = Repeat the steps of case (a) above ** Case (c ) B = Repeat the steps of Case (a) above ** Matlab summary results. Insert your results in the table below. Transient response characterization refers to the response type: overdamped, underdamped, or critically damped. Table 2: Matlab Summary Results for the 2nd Order Mechanical system Case( a) B = 12 (N.s/m) Case( b) B = 1 (N.s/m) Case( c) B = 10.3923 (N.s/m) poles zeros tr (rise time) (sec.) ts (settling time) (sec.) % Overshoot steady state value (m) A0(dB) �L (rad/sec.) �K (rad/sec.) Transient Response Characterization Problem No. 2 (Electrical System) Given the electrical system shown. Assume zero initial energy stored in the dynamic elements L and C. A- LaPlace Domain Step 1- Derive the transfer function G(s) �(�) = �L(�) �OK(�) Make sure the transfer function is in the standard form per the class lecture Step 2- From the equation, determine the following in terms of the circuit elements R,L and C: The natural frequency: �K The damping Ration � Step 3- Given your equation in Step 1, Write a code using Matlab to determine the step response of the voltage across the capacitor for the following cases of R,L and C . Also, from the step response, complete Table 3 for each of the cases. Ensure to plot properly labeled and titled step response figures for each of the cases. Also provide the well organized and commented code in the designated dropbox. Name this code as: Pr2A_St3.m Additionally, in your code, using the equations of Step 2, calculate �K and � for each case of the component values. Cases: R = 8 ohms, L =0.2H, case a) C = 0.04F Case b) C =0.0125F Case c) C = 0.0004F Table 3: Second Order Electrical System Data Case (a) Case (b) Case (c) �� � steady state value tr (rise time) (sec.) ts (settling time) (sec.) % Overshoot Step Response Type Insert the transient response figures below B- State Equations The goal is to write the state space model for the same electric circuit (shown below for convenience) in terms of E, R, L and C (assume zero initial energy is stored in the system). Let the states be as follows: �R = �L ��� �V = �X Where the output y is : case (i) y = vc case (ii) y= iL case (iii) the output is a vector of two quantities: � = Y �L �X Z case (iv) the output is a vector of three quantities: � = [ �L �X �\ ] Procedure: Step 1- Write the KVL differential equation. Refer to Chapter 3 and to the class notes on a series RLC circuit. Step 2- Convert the differential equation to state equations. Show work ^ �̇ R �̇ V _ = Y ZY Z + Y Z � Step 3- For case (i) where y = vc, complete the below � = [ ]Y Z + [ ]� Step 4- For case (ii) where y = iL, complete the below � = [ ]Y Z + [ ]� Step 5- For case (iii) where y is a vector of two quantities: � = Y �L �X Z, complete the below � = Y �L �X Z = Y ZY Z + Y Z � Step 6- for case (iv) where y is a vector of three quantities: � = [ �L �X �\ ]. write the output equations below Step 8- Matlab Simulations (See hints below) name this code: Pr2B_St8.m Matlab® Requirements: (see some Matlab hints at the end of this document) Just like with Part A of this problem, we will have three cases of the values R, L and C. Cases: R = 8 ohms, L =0.2H, case a) C = 0.04F Case b) C =0.0125F Case c) C = 0.0004F Do the following for each of the component cases Step 1- Using the state equations, plot the step response of the system for each case of the outputs above (as noted in steps 3,4,5, and 6.) Make sure to label and title each figure clearly. Include the plots in this document. Include the Matlab code-- (submit into the designated dropbox.) place the figures here Step 2- Sinusoidal Input (Name this code Pr2B_sim.m and place the code in the designated dropbox.) plot the output vc - this is case (i) for each of the situations below A) E(t) = 10sin(2 π*100 t) u(t) and zero initial conditions B) E(t) = 10sin(2 π*100 t) u(t) and iL(0) = 100mA, vc(0) = 1 volt. Some Matlab® hints: To create a state space model (like a struct) Model = ss(A,B,C,D) To find the step response of the above model (given zero initial conditions) step(Model) What if the input is a sine wave, E(t) = 20sin(2 π*100 t) Then: (BE CAREFUL here, you already included the magnitude 20 in the system matrices created above. Thus, either change this E magnitude of 20 to 1 in the original system equations or change the 20 to 1 in the below.) Need to create a time vector: t = 0:0.0001: 0.2; u= 20*sin(2*pi*100*t); y=lsim(Model,u,t) If the nonzero initial conditions: Then, add a vector for the initial conditions: X0=[1; 0.1] ; % This means vc(0) =1, iL(0) = 100mA Then t = 0:0.0001: 0.2; u= 20*sin(2*pi*100*t); y=lsim(Model,u,t,X0) in Matlab® command window, type help lsim Problem No. 3 (Multisim Simulation) Use Multisim to simulate the series RLC circuit as follows: time, t goes from 0 to 0.2 seconds. Step 1- Simulate the RLC circuit and plot the transient response of yc , vL and iL . Assume zero initial conditions. In your transient simulations, ensure to set initial conditions to 0. Note, each of the plots is done by a new simulation. Step 2- Let the initial conditions be iL(0) = 100mA, vc(0) = 1 volt. simulate and plot the transient response of yc , vL and iL . In your transient simulations, ensure to set initial conditions to User defined. Also, set the initial conditions on the circuit in Multisim. Problem No.4 (Conclusions) Provide a well written and concise explanations of the results of your Matlab and Multisim simulations. - discuss the overall process of each - discuss whether the results agreed or not - Also, discuss and explain each of the following. Give scientific explanation (short and to the point) o natural frequency of a mechanical system, o damping frequency of a system o damping ratio § type of transient response if the damping ratio is greater than 1 § type of transient response if the damping ratio is equal to 1 § type of transient response if the damping ratio is greater than zero but less than 1 § type of transient response if damping ratio is equal to zero § type of response if damping ratio is negative. o General and good description of the pole locations in a second order system for each of the following cases of damping ratio, � . case -1: � > 1 case -2: � = 1 case -3: 0 < � < 1 case -4: � = 0 case-5: � < 0

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