ME 450 HOMEWORK 7 Transfer Function Analysis There are several Matlab functions that can be applied to transfer functions: pzmap, impulse, step, and bode. For this problem, we will focus on pzmap. We have already done several examples with the step function on other homework assignments and in the lectures. In the solution that will be posted after the homework is turned in, you can see the applications of the other functions. I encourage you to look at it closely. Problem 1 (5 points) : For the given transfer function, plot the poles and zeros in the complex plane using pzmap and comment on the stability of the system. G(s) =- s2 +10s +89 s Problem 2 (5 points) : For the given transfer function, plot the poles and zeros in the complex plane using pzmap and comment on the stability of the system. G2(s) =- 53 + 4s2 + 29s s2 + 25 Problem 3 (5 points) : For the given transfer function, plot the poles and zeros in the complex plane using pzmap and comment on the stability of the system. G3(s) = 32+15s +56 S+10 Problem 4 - Bode Plots (10 points) A student in ME450 used MATLAB to plot the Bode diagram of four Transfer functions and generate the step response. Unfortunately, the student forgot to label the plots. Match the transfer function, with the correct plots using the table below: Transfer Function G1 = ( 0.1s + 1)2 G2 = 0.5s2 + S G3=52 + 0.5s + 3 1 (0.1s + 1) G4 = (0.2s + 1) Step Response Bode Diagram You can easily check this with MATLAB. However, please explain your answer. For instance, in the Bode diagram you can look at the maximum phase, the initial slope of the magnitude plot, etc. In the step response, you can look at the initial slope, the steady state-value. "I used MATLAB" is not an acceptable explanation. 1 1 5 0 Magnitude (dB) 0 Magnitude (dB) -50 20 -100 40 10-1 100 10 102 10-1 10º 101 10- (A) Frequency (rad's) (B) Frequency (rad/s) O O Phase (deg Phase (deg) -100 -100 -200 -200 10-1 10º 101 102 10-1 100 101 107 Frequency (rad's) Frequency (rad/s) 0 Magnitude (dB) -2 AN Magnitude (dB) 50 0 -4 -50 -6 10-1 100 10 102 10-1 10º 101 102 (C) Frequency (rad's) (D) Frequency (rad's) 0 Phase (deg -100 Phase (deg -10 -150 -20 -200 10-1 100 101 102 10-1 100 101 102 Frequency (rad's) Frequency (rad's) Problem 5 - Match the Step Responses (10 points) (I) 1000 1 Step Response Step Response 900 0.95 800 0.9 700 0.85 Amplitude 600 Amplitude 500 0.75 400 0.7 300 200 0.6 100 0.5% 0 05 20 40 60 80 100 120 140 160 180 200 O 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (seconds) Time (seconds) Step Response Step Response (III) 1 (IV) 0.6 0.9 0.8 0.7 Amplitude 0.6 Amplitude 0.5 0.3 04 02 0.3 02 0.1 0.1 0 0 0 02 1 1.2 0 10 15 20 25 30 Time (seconds) Time (seconds) 2 Problem 6 - Two Mass System (40 pts) Consider the following two mass-spring-damper system: k2 C2 m2 k C1 m1 x,*, 1 Figure 1 - System for Simulink Problem The equations of motion for the system shown in Figure 1 are: mx +ci +kx - cx2 - k1x2 = f m 2 x 2 + (c1 + c 2 )¿2 + (k + k2 )x 2 - c1 x1 - k1 x1 = 0 a) Implement the system of equations above in Simulink using the following parameters: m1 =10; % Mass 1 [kg] m2 =100; % Mass 2 [kg] c1 =100; % Damping Coefficient 1 [Ns/m] c2 =1000; % Damping Coefficient 2 [Ns/m] k1 = 1e4; % Spring Coefficient 1 [N/m] k2 = 1e5; % Spring Coefficient 2 [N/m] Tend = 10; % Simulation Stop Time [s] You can assume that the initial conditions are all zero. Define the model parameters in a separate .m file and use the ode45 Solver inside of Simulink. This does not mean use ode45.m. Make sure to decrease the maximum step size if the plots are not smooth. b) Simulate the response of the system assuming that f (t) is a step function of magnitude 5 N. Plot the response of the systems (the two positions X1(t) and x2(t)) in two separate figures. c) Simulate the response of the system assuming that f (t) is a sinusoidal function: f(t) = 3 sin(10t). Plot the response of the systems (the two positions x1(t) and x2(t)) in two separate figures. 3 f