MECHANICAL ENGINEERING METHODS

Mini Project V

ME

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Due: August 3

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Instructions

1. You are required to solve all problems for credit; however, not all problems will be graded. A
random sample of problems will be selected for grading.

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. You will submit one report per group. Clearly print names of all group members. Attach
your computer programs to the pdf file.

3. You must show all detailed steps to get full credit. Please write legibly.

4. For ease in grading, please do NOT write on the backside of the paper.

5. If you are asked to plot graphs, all graphs should be clearly labeled; i.e. the x and y axes of
the plots should be labeled. If there are multiple curves on the same graph, indicate what
each curve corresponds to using a legible legend. Clearly show the range of the axes. The
plots should be clearly readable. Provide legends. Give caption to each plot.

6. If you have questions please do not hesitate to contact me. Good Luck!

1All assignments are due before the start of the lecture, unless otherwise stated.

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1. A football has a mass of mp = 0.413 kg. It can be approximated as an ellipsoid 6 in diameter
and 12 in long. A ‘Hail Mary’ pass is thrown upward at a 45◦ angle with an initial velocity of
55 mph. Air density at sea level is ρ = 1.22 kg/m3. Neglect any spin on the ball and any lift
force. Assume turbulent flow and a constant air-drag coefficient of CD = 0.13.

The equation of motion of the ball is given as,

dxp
dt

= up (1)

dyp
dt

= vp (2)

mp
dup
dt

= −Fcos(θ) (3)

mp
dvp
dt

= −Fsin(θ) − W; (4)

where xp and yp are the x and y co-ordinates of the centroid of the ball (x is positive to
right and y is positive upwards), up and vp are the horizontal and vertical velocities of the
ball, respectively, W = mpg is the weight of the ball and g = 9.81 m/s

2 is the gravitational
acceleration. The drag force is given as

F = CD
ρ

2
(u2p + v

2
p)
π

4
D2; θ = tan−1

(
vp
up

)
. (5)

The diameter D = 6 inch. Assuming that the quarterback is about 6 ft tall, The initial
location of the centroid of the ball can be taken as xp = 0 and yp = 6 ft.

(a) Using a Forward Euler numerical scheme with general step size ∆t, obtain the finite
difference approximation to the system of equations. Clearly write down the initial
conditions for all variables.

(b) Write a program to solve the system of equations using the above equations that will
provide solutions to xp, vp, up and vp as a function of t. Make sure you input different
parameters in consistent units. Use SI units.

(c) Solve the system of equations (with appropriate step size) until the ball hits the ground
again; i.e when yp = 3 inch (half of the maximum diameter of the ball). Plot xp versus
t, yp versus t, up versus t, and vp versus t over the duration of the air travel of the ball.

i. What is the horizontal distance traveled? If the ball is thrown from the 40 yard line
of the offense, do the offense have a chance of a touchdown (for a touchdown from
this position, the ball would have to travel at least 60 yard or more)?

ii. What is the maximum vertical distance traveled?

iii. How long does it take for the ball to hit the ground?

iv. How do you know all your plots and answers are correct?

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2. Buckling of Column: Under a uniform load, small deflections y of a simply supported beam
are given by

EI
d2y

dx2
= qx(x − L)/2; y(0) = y(L) = 0 (6)

where L = 10 feet is the length of the beam, EI = 1900 is modulus of elasticity times moment
of inertia, and q = −0.6 is load distribution. The beam extends from x = 0 to x = L. The
goal is to find y at every 0.1 foot using the Direct Method and plot y(x) versus x.

(a) Using centered-differencing, formulate the finite difference approximation. To illustrate
your work, use 5 grid points (2 of them are on the boundaries, y = 0 and y = L).
Write down finite-difference approximations for the interior points, for a general uniform
grid spacing of ∆x. Then write your algebraic equations in the matrix-vector form with
unknown vector on the left hand side and all known quantities on the right hand side.

(b) Using a computer program for N + 1 total points, solve the above equation using the
direct method with the grid spacing of 0.1 foot. Plot y(x) versus x. Compare your
solution with the Exact Solution (clearly label your graphs).

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