NYU Tandon School of Engineering
ME-GY 6003 Applied Mathematics in Mechanical Engineering
CBE-GY 6153 Applied Mathematics in Engineering
Fall 2021
Term Exam 1
Name
Murtaza Fatepurwala
ID N18321595
Instructions:
ˆ This exam consists of a single problem in multiple sections of equal weight.
ˆ Answer all sections on the front sides only of 8-1/2” x 11” paper.
ˆ Your analysis must be legible and logical. Clearly label all work so that it is obvious
which problem section you are solving.
ˆ Show all of your work. Answers without work shown will receive no credit.
ˆ You must work independently. Do not give nor receive assistance from anyone.
ˆ This project is open book, open notes. Use of the internet is limited to downloading
the exam, uploading your finished work and the following approved websites:
– Wolfram Alpha: Wolfram Alpha
– General Engineering Data: Engineer’s Toolbox
ˆ All plots must be made using appropriate software (e.g GNUplot, Matlab, Python or
similar) and must include all proper graphical elements (e.g. axis labels, units, etc.).
ˆ The exam must be submitted as a single PDF file with your name both on the exam
pages and in the file name, uploaded to Brightspace no later than October 12, 2021 at
11:59 pm EST.
ˆ Violation of any of these rules, or the intent of these rules, will result in a score of zero.
Honor Statement: I certify that while taking this exam I have adhered to all University
policies regarding academic integrity. I have only used my textbook, class notes, Power Point
slides as resources. I have limited my use of the internet to downloading and uploading my
exam and approved websites. I am aware that according to the course syllabus, cheating on
this exam (including using pre-worked solutions from the internet) will result in an exam grade
of zero.
Signature
In 1989, Hurricane Hugo hit South Carolina with wind speeds estimated as high as 60 m/s. As
a result, there was substantial damage to local forests. An interesting observation was that a
majority of trees lay pointing in exactly the same direction, and were broken at approximately
the same height from their base. Similar events happened when Hurricane Fran hit North Carolina (1996) and when Hurricane Michael hit Georgia (2018). This seems to be a reproducible
phenomena. For more information about the bending of trees in high winds, see Hugo Physics,
by W. Kubinec, Physics Teacher, March, 1990, which has been posted on Brightspace.
Figure 1: Image Courtesy www.wabe.org
You are to examine a mathematical model for the bending of trees in strong winds. In this
model the tree is represented as a cylindrical beam of length L. Wind hitting the branches of
a tree imparts a force to the trunk of the tree. As a first approximation, dynamic effects are
neglected, and the force may be considered time-invariant. Thus, the deflection y(x) of the tree
may be modeled by solving the (static) beam deflection equation. The boundary conditions
are those of a cantilevered beam: the deflection and slope are zero at the rooted end, and the
bending moment and shear force are zero at the free end, which is at the top of the tree.
Tall trees in the forest have most of their crown (branches and needles) in the upper 50% of
their length, so you may ignore the force of the wind on the lower portion of the tree. Further,
assume that the wind hitting the tree’s crown results in a uniform load per unit length w0 ,
(
0
0 ≤ x < L/2
w(x) =
w0 L/2 ≤ x ≤ L
a.) Sketch the model and indicate the coordinate system and any relevent parameters.
b.) Determine the slope of the tree, y ′(x), by solving the deflection equation. Find all constants of integration by applying the appropriate the boundary conditions. Hint: Ensure
the continuity of the third derivative at the point x = L/2.
c.) Determine the deflection of the tree, y(x), by following a process similar to that in part
(b.) above.
d.) Compute the maximum deflection of the tree, y(L). Which parameter has the greatest
influence on the maximum deflection?
The total force F on the tree can be approximated by:
ρAv 2
6
3
where ρ = 1.23 kg/m is the density of air, v is the wind speed (m/s), and A is the projected
area of the tree’s crown (m2 ). The crown in roughly cylindrical and its area is a rectangle with
A = (2R)(L/2) = RL, where R is the average radius of the cylinder. The moment of inertia
for a uniform cylindrical beam is I = 1/4πr 4, where r is the radius of the cylinder. The total
force per unit length is then w0 = F/(L/2)
F =
Assume that the amount that a tree will bend depends on each of the following parameters:
Symbol
r
R
L
E
v
Description
radius of trunk
radius of crown
height of tree
modulus of elasticity
average wind speed
Typical Values
0.15 - 0.25 m
3-4m
15 - 20 m
11 - 14 N/m2
40 - 60 m/s
e.) Mathematically show how each parameter affects the deflection, and explain in physical
terms why this makes sense. For example, a large value of E results in less bending since
E appears in the denominator of y(L). This means that hard wood like oak bends less
than a soft wood such as palm.
f.) Plot y(x) for 40 m/s and for 60 m/s winds for an average tree by choosing representative
values of each parameter, e.g. r = 0.2 m, R = 3.5 m, L = 17.5 m, E = 12.5 x109 N/m2 .
Compare the two curves and comment.
g.) In the derivation of the beam equation it was assumed that the deflection of the beam
was small. What is the largest possible value of y(L) that is predicted by this model if
all parameters are chosen from the table? Is this prediction realistic?
h.) The beam equation always predicts that a beam will bend, even if the load and flexural
rigidity are excessive. One approach to determine breaking is to evaluate where the stress
function y ′′(x)/I(x) reaches a maximum. Determine this stress function and estimate
where the beam might break.
i.) Real tree trunks are not uniformly wide, they taper as they approach the top of the tree.
Substitute r(x) = 0.2 − x/15L into the equation for I and plot the stress function along
the length of the tree. Where does the maximum stress occur? Does this location depend
on the speed of the wind? On the radius of the crown? On the height of the tree?
j.) A mathematical model is sensitive to an assumption if small changes in the assumption
lead to widely different predictions for the model. Repeat part (i.) using r(x) = 0.2 −
x/20L and r(x) = 0.2 − x/10L to describe the radius of a tree trunk that tapers. Is this
model sensitive to the taper of the trunk?