Homework 3
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ME 370 Homework 3
Problem 3.1
Watch the following video about debugging Matlab code:
https://youtu.be/PdNY9n8lV1Y
. Based on the
video, give two examples of things you can do to find and eliminate errors in Matlab code.
Problem 3.2
For the system on the right, the moment in the spring is given by
𝑘𝑘
tan
−1
𝜃𝜃
where
𝑘𝑘
is spring constant and
𝜃𝜃
is the stretch in the spring. The moment in the damper is
given by
𝑐𝑐𝜃𝜃
̇
5
where
𝑐𝑐
is the damping constant. Write the EOM as a first-order system.
Problem 3.3
Just after a gust of wind, a 40 lb traffic light has a position of -1 in and a velocity of -6 in/s relative to its
rest position. Suppose the equivalent stiffness is 6.3 lb/in.
a)
Find the response of the system analytically.
b)
Suppose a student were to calculate the phase shift
ϕ
using their calculator. Explain why they
should or should not add π to the value
from the calculator.
Partial answer: a)
𝑥𝑥
= (1.26 in) sin(7.80
𝑡𝑡
-2.23)
Problem 3.4
Download the files prob3_4.m and trafficLightEOM.m from Canvas. These files will be used to find the
response of the system in Problem 3.3 numerically (i.e., using ode45). There are several errors in the files.
For all problems that involve Matlab, submit the code you used/wrote as well as the output. Matlab has a
publish option that may be useful:
https://www.mathworks.com/help/matlab/matlab_prog/publishing-
matlab-code.html
a)
Find and fix all of the errors. Also briefly explain why the error is an error (as if you were
explaining it to your friend). Circle, highlight, or otherwise clearly indicate where you modified
the code.
b)
Plot of the position of the traffic light with respect to time for both the analytical and numerical
responses. Does your plot seem correct? Why or why not?
Hint: When you download the files, make sure you can see them in the Current Folder window in Matlab.
If you cannot, either move the files to your current folder or change the working folder. See the third video
in Module 3E for additional details.
Partial answer: b)
Problem 3.5
A 1000 lb machine is attached to a wall mount with four hollow
Neoprene (Modulus of elasticity
𝐸𝐸
= 1,500 psi, Yield strength
2,000 psi) pads of outer diameter 4 in, inner diameter 2 in, and
length 3 in. What is the maximum initial speed of the cart so that
vibratory stress (the stress caused by the vibration) in each leg is
less than 10% of the yield stress? Assume the initial position of
the cart is 0.
Hint: The stress caused by an axial load is given by
𝜎𝜎
=
𝐹𝐹
/
𝐴𝐴
where
𝐹𝐹
is the axial force in the bar and
𝐴𝐴
is the cross-sectional
area. Think about what vibratory system element the legs are and
how you can calculate the force in that element. Also think about how amplitude is related to displacement.
Answer: 34.1 in/s
Problem 3.6
To properly control airplanes, the inertia needs to be known.
[Data and picture from Soule, H., Miller, M., 1934. The
Experimental Determination of the Moments of Inertia of
Airplanes.]
a)
Draw the free body diagram of the swinging gear-plus-
airplane (pendulum). Write the equation of motion. The
total weight of the pendulum is 2,591 lb. The distance
from the oscillation axis to the pendulum center of
gravity is 9.050 ft.
b)
Using your equation of motion, find the equivalent
stiffness of the pendulum.
c)
Find the mass moment of inertia of the pendulum about
the oscillation axis. When the pendulum is gently
swung, its period of oscillation is 3.759 sec.
d)
Find the mass moment of inertia of the airplane about
the airplane’s center of mass. The inertia of the airplane
about the oscillation axis is
𝐼𝐼
𝑂𝑂
=
𝐼𝐼
1
− 𝐼𝐼
2
where
𝐼𝐼
1
is the inertia of the pendulum about the
oscillation axis and
𝐼𝐼
2
=
638.1 slug
⋅
ft
2
is the inertia of the frame about the oscillation axis. The
weight of the airplane is 2,208 lb and the distance from the oscillation axis to the airplane’s center
of mass is 9.513 ft.
Hints: Derive the equation of motion and use the approximation that
sin
𝜃𝜃 ≈ 𝜃𝜃
for small angles to find the
equivalent stiffness for the pendulum. The parallel axis theorem will be useful for finding the inertia of
the airplane about the center of mass.
Answer: b) 23,450 lb
⋅
ft/rad c) 8,393 slug
⋅
ft
2
d) 1,549 slug
⋅
ft
2
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