(A) Based on the second-order FRF provided, determine appropriate values for ω n, ζ, and K. ω n =500 rad/s; ζ=0.1; K=0.

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1 ME35 Homework # Due: 1/1/1 Problem #1 (3%) A co-worker brings you an accelerometer spec sheet with the following frequency response function (FRF):. s G accelerometer = [volt +.1 jω.1 ω m ] (A) Based on the second-order FRF provided, determine appropriate values for ω n, ζ, and K. ω n =5 rad/s; ζ=.1; K=.15 [ volt s m ] (B) In terms ω n, ζ, and K, write the equations needed to determine the magnitude and phase angle of G(jω). G(jω) = K ; (1 ω ω ) +( ζ ω) n ω n G(jω) = atan ( ζ ω n ω/(1 ω ω n )) (C) Based on the equations developed in part (B), write a MATLAB program to plot magnitude G(jω) and phase G(jω). The resulting figure should show both axes on linear scales, with rad/s on the horizontal axis and either magnitude or degrees on the vertical axis. Your code should allow you to vary the values for ω n, ζ, and K, as well as setting bounds for the input frequency. Show your code. clear all; clc; %please enter the minimum input frequency f_bond_l = input('please enter the minimum input frequency: '); if f_bond_l < %please enter the maximum input frequency f_bond_h = input('please enter the maximum input frequency: '); if f_bond_h < else if f_bond_h <= f_bond_l disp('maximum input frequency should be larger than minimum input frequency!'); %please enter the omega omega = input('please enter the omega: '); if omega < disp('no negative omega '); %please enter the damping ratio zeta = input('please enter the damping ratio: '); if omega < disp('no negative damping ratio '); %please enter the sensitivity k = input('please enter the sensitivity: '); 1/7

2 Phase (deg) Gain ME35 Homework # Due: 1/1/1 for i=1: w(i)=f_bond_l+(i-1)*fix(f_bond_h-f_bond_l)/; %magnitude and phase angle mag(i) = k/sqrt((1-w(i)^/omega^)^+((*zeta*w(i))^/(omega^))); phase(i) = -atan((*zeta*w(i)/omega),(1-w(i)^/omega^)); %draw plots subplot(,1,1) plot(w,mag) title('plot of accelerometer gain') xlabel(''); ylabel('gain'); subplot(,1,) plot(w,phase/pi*18) title('plot of accelerometer phase shift') xlabel(''); ylabel('phase (deg)'); % (D) Using the values determined for ω n, ζ, and K in part (A), run your code from part (C) to plot the accelerometer gain and phase for frequencies between 1 and 1, rad/s. 8 x 1-3 Plot of accelerometer gain Plot of accelerometer phase shift (E) Using the semilogx command, rewrite your MATLAB code from part (C) to plot the input frequency on a logarithmic scale, going from 1 to 1 rad/sec. Also adjust the magnitude values to be shown in decibels. Show your updated code. clear all; clc; %please enter the minimum input frequency f_bond_l = input('please enter the minimum input frequency: '); if f_bond_l < %please enter the maximum input frequency f_bond_h = input('please enter the maximum input frequency: '); if f_bond_h < else if f_bond_h <= f_bond_l disp('maximum input frequency should be larger than minimum input frequency!'); /7

3 Phase (deg) Gain(dB) ME35 Homework # Due: 1/1/1 %please enter the omega omega = input('please enter the omega: '); if omega < disp('no negative omega '); %please enter the damping ratio zeta = input('please enter the damping ratio: '); if omega < disp('no negative damping ratio '); %please enter the sensitivity k = input('please enter the sensitivity: '); for i=1: w(i)=f_bond_l+(i-1)*fix(f_bond_h-f_bond_l)/; %magnitude and phase angle mag(i) = k/sqrt((1-w(i)^/omega^)^+((*zeta*w(i))^/(omega^))); mag_d(i)=*log1(mag(i)); phase(i) = -atan((*zeta*w(i)/omega),(1-w(i)^/omega^)); %draw plots subplot(,1,1) semilogx(w,mag_d) title('plot of accelerometer gain') xlabel(''); ylabel('gain(db)'); subplot(,1,) semilogx(w,phase/pi*18) title('plot of accelerometer phase shift') xlabel(''); ylabel('phase (deg)'); % (F) Using the values determined for ω n, ζ, and K in part (A), run your code from part (E) to plot the accelerometer gain and phase for frequencies between 1 and 1, rad/s. - Plot of accelerometer gain Plot of accelerometer phase shift (G) Plot by hand the gain (in db) and phase (in degrees) versus ω for the accelerometer FRF given above. Use a semi-log plot, with the input frequency spanning a few decades on either 3/7

4 P hase (deg) M agnitude (db ) Phase (deg) Magnitude (db) Phase [rad] Magnitude ME35 Homework # Due: 1/1/1 side of the natural frequency. Be sure to indicate the low frequency gain and the high frequency slope. Also indicate the resonant frequency, as well as the gain where ω = ω n apoxi -55 db Wr Wn -db/dec Frequency [rad/s] (H) Use the bode command in Matlab to create a Bode plot for the accelerometer FRF given above. - Bode Diagram Problem # (%) /7

5 Phase (deg) Magnitude (db) ME35 Homework # Due: 1/1/1 You are given the above Bode diagram of the frequency response function (FRF) of an unknown system A. Please answer the followings based on the above plot. (A) If τ=1 and K=1, calculate the 1 st order FRF and steady-state output of this system when it is excited with x = 1 sin(.1t) + 1sin (1t). (hint: use FRF of 1 st order system) FRF= 1 ; y= 1sin(.1t-.1)+.1sin(1t-1.57) 1+jω (B) Based on part (A), comment on the different response of the system to very low and very high frequency input signals? for low frequency input, neither magnitude nor phase change too much; for high frequency input, magnitude decreases drastically and phase lags approaching -9 degrees Bode Diagram Now you are given the above Bode diagram for the frequency response function (FRF) of another unknown system B. Please answer the followings based on the above plot. (C) If this system is excited with x = 5 sin(.1t) + 5 cos(1t) + 5 sin(1t) + 5 cos(1t) what will be the steady-state response (hint: read out the closest value from the graph). y=.1*5sin(.1t+9 )+5cos(1t+81 )+7.9*5sin(1t+5 )+1*5cos(1t+9 ) (D) Compared to previous FRF of system A, comments on how does system B respond differently to very low and very high frequency input signals? for low frequency input, magnitude drastically decrease while phase keep leading 9 degrees; for high frequency input, magnitude doesn't change too much while phase approaching degree. Problem #3 (3%) You are given a thermocouple without any additional documentation other than that it works. (A) What parameters do you need to know to fully characterize this thermocouple? sensitivity and time constant 5/7

6 O utput (V olt) O utput V oltage (V olt) ME35 Homework # Due: 1/1/1 To find the parameters, you took the thermocouple at room temperature of 5 C and immediately place it in a container of iced water, i.e. at C and recorded the time history of the output voltage as below: Tim e (sec) (B) Based on the above response, estimate the parameters you stated in part (A). Show your work. (Mark up the relevant points and your best estimation of the corresponding values on the above figure and turn it in with your work) K=delta output/delta input=(1.5-1)/(-5)=-.5 (v/degc) use y(t) = y o + (y y o )(1 e t τ ) for t, where y =1.5; y o =y()=1, plug in (., ) solve for τ, finally we got τ=.1 (sec) You are given another gadget with a voltage input port and a voltage output port without any documentation. You feel adventurous and connect the input port to a function generator and the output port to a scope. You are also cautious and applied a step input of 5 mv into the input port and stored the following output response: Tim e (sec) (C) What is the best guess of the order of the device, 1 st or nd order? Why? nd system, vibration /7

7 Amplitude Phase (deg) Magnitude (db) ME35 Homework # Due: 1/1/1 (D) Find/identify the natural frequency, damping ratio, and static sensitivity form the above step response. Show your work and remember the proper units. (Mark up the points and your best estimation of the corresponding values on the above figure and turn it in with your work) K=5/.5=1; Td=1.-.8=.; ω d =*pi/ Td=1.5 rad/s; use OS method:ζ = 1 π ln (y f y o OS ) ζ = 1 5 ln =.; ω π d = ω n 1 ζ, we got: ω n =1.5 rad/s (E) Give the parameter that you identified in part (D), determine the corresponding differential equation and the corresponding frequency response function (FRF). 1 FRF= 1 ( ω 1.5) +j.(ω/1.5 ) PDE= 1 d y +. dy + y = 1x (x:input; y:output) 1.5 dt 1.5 dt (F) Use MATLAB and plot the corresponding Bode diagram and the step response for a 5 mv step input. Bode Diagram Step Response Time (sec) 7/7