ACM41000: Uncertainty Quantification
Dr James Herterich
Assignment 2
This is a graded assignment. The due date is March 14 at 9pm. Upload your
solutions to Brightspace.
Please upload a single PDF file. This is to include scanned written
solutions and PDF outputs of your R code and figures. Please
combine the PDFs into one.
For full marks, ensure your work is clear and well-commented.
Page 1 of 4
1. A skydiver of mass m falling to Earth is governed by simple mechanics: (i)
the gravitational pull of the Earth Fg = −mg accelerates the skydiver, (ii)
a frictional drag force Fdrag = −mDv decelerates the skydiver, where D is
the drag coefficient and v is the velocity. We take g = 9.81 m/s2. Usually
the skydiver accelerates to a terminal velocity, whereby gravity and drag
balance so that acceleration is zero and the skydiver maintains the same
velocity until the parachute is pulled. The position (height) x is given by
Newton’s second law where the forces are these two accelerations:
d2x
dt2 = −D
dx
dt − g. (1)
You cannot change the pull of gravity. So, in order to increase your velocity,
v = dx/dt, you can only control the frictional drag force. The coefficient D
can be changed either by your shape/position (eg. circle, square, eloganged
square) or how you interact with the air around you (the suit you wear).
Suppose our skydiver is testing out a new aerodynamic suit to complement
the usual aerodynamic postion adopted. This suit is desiged to reduce fric-
tional drag with the surrounding air. The skydiver wishes to determine the
drag coefficient of this suit, taking a minimal amount of additional instru-
ments along to measure altitude and velocity during a trial run. The initial
height is given by x(t = 0) = 1000 m. The instruments take measurement
every 2 s and measure altitude to an accuracy of 10 m and velocity to an
accuracy of 1 m/s. After 20 s the parachute is opened. The results are in
Table 1.
(a) Solve (1) numerically in R as a system of first-order ODEs with D a
parameter. Try D = 1 as a parameter and compare with the data.
(b) Perform a least squares fit using the Levenberg–Marquardt algorithm
to find the best fit value of the coefficient of drag D based on the data
from the trial run
(c) Are the solutions for height and velocity sensitive to the parameter D?
Give a physical explanation – or interpret from the difference between
D = 1 and the best fit D – to go with your answer.
Page 2 of 4
Table 1: Table of experimental results for Q1.
Time (s) Height (m) Velocity (m/s)
0 1000 0
2 980 -15
4 950 -23
6 890 -27
8 840 -30
10 780 -31
12 710 -32
14 650 -32
16 590 -32
18 520 -33
20 460 -33
2. Consider the double spring mass system in Figure 1. Block 2 is attached to
Block 1 by a string, and Block 1 is attached to the wall by a spring. Each
spring has a rest length, R1 and R2, respectively, and a spring constant, k1
and k2, respectively. Each Block has mass m1 and m2, respectively, and
width w1 and w2, respectively.
Figure 1: Double spring mass system for Q2.
Assume that both springs are of the same mass, m1 = m2 = 1, and same
width w1 = w2 = 1. Also assume that both strings are the same rest length,
R1 = R2 = 3, but not necessarily of the same material, k1 6= k2. We lift
mass m2 up by a distance of 1. With the system at rest, we let mass 2
drop, setting the system into action. Some data has been recorded for this
experiment, in Table 2.
The system is modelled by the following second-order differential equations
m1
d2x1
dt2 = −k1(x1 −R1) + k2(x2 − x1 − w1 −R2), (2a)
m2
d2x2
dt2 = −k2(x2 − x1 − w1 −R2), (2b)
Page 3 of 4
Table 2: Table of experimental results for Q2.
t x1 x2
0 0 1
2 6.35 12.65
4 -0.25 1.73
6 5.46 12.01
8 0.63 2.89
10 5.1 10.1
12 1.97 4.62
14 3.58 8.19
16 2.7 7.15
18 2.19 5.97
20 4.41 9.13
with initial conditions
x1(0) = 0, x2(0) = 1,
dx1
dt (0) = 0,
dx2
dt (0) = 0. (3)
(a) Write the above as a system of four first-order differential equations
(b) Solve the new system from (a) numerically in R, and take the param-
eters k1 = k2 = 5.
(c) Use Table 2 in the Levenberg–Marquardt algorithm to estimate k1 and
k2, taking an initial guess of k1 = k2 = 5.
(d) With the best-fit parameter in (c), plot the numerical solution to the
system over the data.
(e) Determine and plot the 95% confidence ellipse for this parameter fit.
Warning: be careful that the data only represents two variables
(plus time) in the four-variable system.
Page 4 of 4