Differential Equations and Matrices

This is an individual coursework.
Percentage of module mark: 30
Intended Learning Outcomes:
1.
Ability to formulate and analyse mathematical model
s of engineering problems
2.
Ability to apply mathematical techniques and modern
mathematical software to
solving engineering problems, in particular: soluti
on of ordinary differential equations
using analytical techniques and Matlab
Hand Out: 28th October 2013
Hand in: 10th January 2014
Question 1: 30 marks
Question 2: 35 marks
Question 3: 35 marks
Total: 100 marks
All students have been assigned a unique number
N
which is used for setting parameter
values in all questions.
Please look up your number in the table provided wi
th this assignment!
REMEMBER: coursework hand-in now has a hard deadlin
e. Be late, get zero
(ABsent / failure to submit). Plan to hand in at le
ast a day early. Better still, a week.
For full marks, all working must be shown.
Where boxes are provided, please fit all the workin
g and the answers in the boxes.
You are allowed to use Matlab unless specified othe
rwise explicitly. You are not allowed to
use software not covered in the module (e.g. Maple
or Simulink)
Please submit an electronic version via Moodle. Onl
y two files up to 20 Mb can be uploaded
to Moodle, multiple files can be combined in a ZIP
archive if necessary. It is however
recommended to only submit one file.
208MAE, coursework 1 (2013/2014)
Page 2 of 20
1 Beam bending
Problem:
A uniformly thick beam of length
L
with square cross-section with width
a
has a distributed
force,
?
?
?
?
?
?
+
–
=
–
x
e
L
x
Q
x
q
2
2
sin
1
)(
p
N/m,
applied
0
=
x
to
L
x
=
(
Q
is a constant). A point force
P
is applied at
L
x
=
.
P
x
y
x
= 0
q
x L
=
Find expressions for shear and moment
analytically
(not using Matlab).
Sketch distributions of shear forces and moment alo
ng the beam length (using Matlab)
Parameters:
Values of
L, EI, Q
and
P
for each student are calculated as follows (here
N
is the number
assigned to you in the table):
N
L
*
01.0
1
+
=
(m)
600
/)
300(*
50
N
P
–
=
(kN)
300/
5
N
e
Q
=
(kN/m)
5
10
6
×
=
EI
(Pa m
4
)
Marking scheme:
Finding correct expressions for shear and moment
ANALYTICALLY
(not using Matlab,
please show your working)
(20 marks)
Plots of shear force and moment
(10 marks)
208MAE, coursework 1 (2013/2014)
Page 3 of 20
Solution:
1. Correct parameters for my student number
N
=
L
=
P
=
Q
=
EI
=
2. Shear
Equation for shear:
General solution of shear equation (please put your
working here):
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