a) (4 pts) You have been asked to characterize the frequency response of a system with
transfer function G(s). You app ly an input sinusoid, 2sin t, to th e system and measure
the output in the plot below. What is M(jw ) and cp(jw) at w = l rads - 1?
_1_5 "•... .. .. Input
L_-----'---__ _j__ _ __JL___-----'---_4:-L•~. • _ __j __ _j_ __ ____j__-'::::::Jº==ut:p:':u:t:::::'.J
o 2 3 4 5 6 7 8 9 10
b) (4 pts) You are designing a mass-spring -damper syst em with closed loop poles as marked
below. Your colleague suggests that you tweak the system 's spr ing and damper such
that the poles move to a new location (red x's) as shown. Does the percent overshoot
in a step response increase , decrease , or stay the same with th is design change? Ju stify
c) ( 4 pts) Given the following state transit ion matrix, what is x( t) given initial conditions
x(O) = [1 1]' and no input to your system?
<I>)( t= [1 + 2c t + 2c 2t 2 + o.se t + o.5c 2t]
0.5 + e- t + e- 2t 1 + 2e-t + 2e- 2t
d) (4 pts) You decide to test the cruise control system for an autonomous vehicle with a
ramp input while measuring output position x(t). Your input and output are plott ed
as shown below. What is the steady state error in this system?
0.2 0.4 0.6 0.8 1
1.2 1.4 1.6
e) (4 pts) You have been given a complex system described by the following differentia
l equation . Linearize this system around an equilibrium point (w0 = 1 rad s- 1,
To =0.1 N m) and provide the linearized differential equation.
14w +w3 = lOT
Question 3: Microswimmers
Microswimmers are microscopic swimming robots that are actuated by a torque generated
from an externa ! magnetic field. A schematic of a microswimmer at rest and in motion is
provided in the figure below.
Microswimmer at rest Microswimmer in motion T
__ ___, Horizontal
The microswimmer is made of two rigid links with rotational inertias J1 and J 2 , respectively.
A torque T is applied to link 1 using a magnetic field. Because the microswimmer is very
small, it experiences Stokes' drag, which means that the drag on each link is linearly related
to the link velocity (Td rag = c0); each link has a different drag coefficient, c1 and c2 . Link 1
is connected to link 2 by a torsional spring with spring constant K,. Link 2 is not magnetic
and so it <loes not see any input torque. The rotational motion of the microswimmer can be
described by the angle that each link makes with the horizontal, shown as 01 and 02 in the
diagram. For this problem, you can ignore any translational motion ; please focus
purely on rotational motion of the links described by 01 and 02 .
a) (5 pts) Derive the equat ions of motion for this system using whichever method you
b) (5 pt s) Microswimmers have interesting dynam ics because their charact eristic lengths
are very small. This allows us to assume that they have no inertia and are only affected
~y vi~~ous forces in the system. Therefore, for the rest of th is problem we can say that
01 = 02 = O. Using Laplace transform s, write the transfer function for the system with
02(s) as the out put , i.e. º:l:r
c) (? pts) Using your equat ions of motion from part (a) and the assumpt ion that 01
02 = O, derive a state space model for this system assuming an input torque and an
output of 02.
d) (5 pts) Your labma t es, Sean and Kevin, are helping you collect experimental data.
Sean has been developing a model of the t orsion spring and he estima tes that, for
t his system, ¡,;, = 0.01 N m rad - 1
. Kevin has sent you the Bode plot below in which
he charac terized the motion of link 1 for different inpu t torq ue frequencies . Note that
this is describing G(s) = ~(~(prio vided below) and not the same transfer function you
derived earl ier!. Use the plot to determine the values of c1 and c2 . (As you are pulling
points from the Bode plot, you may round off to the neares t factor of 10.) The units
of the drag coefficients are N m s rad- 1
G(s) _ 01(s) _ 0.1s+ K:
- r(s) - c1c2s2 + K:(c1 + c2)s
e) (4 pts) Your boss wants to know the frequency at which the microswimmers' response
to a sinusoidal inpu t torque star ts to have an amplitude equal to or smaller than
the inpu t torque 's amplitude - a frequency known as the 'fallout frequenc y'. For the
system (G(s) = 01 (s)/r (s)) described by the Bode plot above, at what frequency <loes
f) ( 5 pts) To more easily characteri ze your microswimmer, your labmate Emma suggests
tha t you glue down link 2 (so that 02 = 02 = O) and measure the step response to a step
inpu t in magnetic torque. When link 2 is fixed, the t ransfer function can be rewritten
G(s) = 01(s) = 1
r(s ) c1s + K:
How would you change K, to decrease the system 's time constant? Ju stify your answer.
Question 4: Jumping Robots
Insects along with vert ebrat es like frogs jump off of compliant subst ra tes frequent ly. You
have been tasked with design of a robot th at can do the same. The rob ot is modeled as a
spring-mass syst em connected to the subst rate that is also modeled as a spring-mass system
as shown in the figure below.
Masses and spr ing constants have been changed to make the mat h easier, and equat ions of
motion are given below.
m, x·, = k2X2 - k2x, - k, x,
m2x·2 = k2X1 - k2X2
a) (5 pts) Find the na tur al frequencies of t his system (assuming that the robot is attached
to the subst ra te).
b) (5 pts) Find the mode shape (e.g., amplitude ratio A2/Ai) for the lowest natura l frequency
of t his syst em.
c) (5 pts) A colleague on your team suggests that with cer ta in initial conditions, you can
get both masses moving in the same direction. Assuming that you found A2/ A1 = 2
for the first nat ural frequency and A2/ A1 = - 4 for the second natur al frequency (not
the correct answers for b), wha t initi al posi tion do you need for the robot mass (x2 (0))
if x 1(0) = 0.1 cm to excite the mode in which the masses move in the same direction?
Moving in the same directi on would imply tha t the robot gets a little boost from the
substrate anda higher jump !
d) (3 pts) Another colleague suggests that you look at the force tra nsmissibility (X2 (jw)/ F(jw ))
if a downward force from the robot is applied on m1 . Adding this force res ults in the
following equatio ns of motion ( slight ly modified from above). Tha t same colleague (yes
- a Stanfo rd grad) te lls you that they have charact erized the transm issibility of t his
system, but you have taken DSC at CMU and you do not believe that they measured
the correct syst em. Why is this tran smissibility plot incorr ect?
mid:"1 = k2x2 - k2x1 - k 1x1 - F
m2x·2 = k2x 1 - k2x2
"' ~ 200
o '--- .....-. - ~ -----.....1--...i..--..i....--1....-...... ..,__.....J
o 0.5 1.5 2 2.5 3 3.5 4 4.5 5
frequency rat io w/wn,1
Question 5: Motor Contro l
You and your roommate have become especially inventive over the last six weeks. As a
bioengineering major at CMU, your roommate has invented a new test for Covid-19. Your
roommate would like to use the output from her sensor to control a physical pointer indicating
the level of Covid antibodies in her sample. This pointer can be made through posit ion
control of a motor andas a proud CMU mechanical engineer, you offer to design t his pointing
system. The transfer function of your motor is given below.
G(s) = V(s) = s(s + 4)
a) (3 pts) Is the motor BIBO stab le? Just ify your answer.
b) (5 pts) Find the step response of the motor, B(t), to a 3V step input. Find the form
of the time response of the system without solving for the coefficients of the partial
fraction expansion (e.g. B(t) =A+ Be - t + C sin t + ... ).
c) (5 pts) Now you place the motor in a unity gain feedback loop with a proport ional
controller. This system is linked directly to the sensor that your friend has developed
as shown below. Calculate the closed loop transfer function T(s) = ~ where R(s) is
the input to your friend's sensor.
10 Kp G(s)
sensor controller motor
d) (5 pts) Find the closed loop poles needed to achieve a percent overshoot of 4.3 % and
a set tling time T8 =0.5 s. Mark these poles on the complex plane.
e) (5 pts) Can you achieve a percent overshoot of 4.3 % and a settl ing time T8 =0.5 s given
a proportional controller? Justify your answer.
f) (5 pts) Now that you have completed DSC, you reason t hat a PD controller is the way
sensor cont roller motor
Find t he gains Kp and Kd required to meet the specificat ions (%OS = 4.3 % and
T. = 0.5 s). Design this PD controller analyt ically (you can ignore the effect of any
zeros introduced by the PD controll er). You do not need Matlab to solve th is problem.
g) (5 pts) A friend from ECE suggests that you use the following circuit as your controller.
Calculate the transfer function V0 (s)/½(s) . What kind of controller is th is?
V¡ R 1
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