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MATH2115: Engineering Practice
创建日期：2024-05-28 16:43:07
所属分类：数学mathematics
标签： MATH2115
Engineering Practice

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MATH2115: Engineering Practice 3 - Mathematical Modelling for Engineers

Assignment 2 2023
There are six (6) questions over two (2) pages.
Show all working for your solutions
Any pictures produced must be drawn accurately by software, preferably Desmos.
Furthermore, pictures must clearly show all relevant details, and no unnecessary details.
Pay close attention to the given units, and adjust your working accordingly.
Question 1: (1+3=4 Points) Consider a non-uniform 10m long cantilever beam, with flexural rigidity of
EI (x) =
300
x+ 20
kNm2
a) (1 Point) What is the flexural rigidity at each end of the beam?
b) (3 Points) Calculate the deflection function for this beam under a uniform distributed load of 10N/m
over the whole beam.
Question 2: (1+3+1=5 Points) Consider a uniform 10m long beam, with flexural rigidity of 12,000Nm2
that has a fixed supports on both ends.
A weight is hung under the beam—attached via cable at the x = 3 point—applying a downward force of
200N . At the same time a winch is connected to the beam—via a cable attached at the x = 7 point—pulling
upward with a force of 200N .
a) (1 Point) What load function, f(x), represents this load?
b) (3 Points) Calculate the deflection function for the beam.
c) (1 Point) Include a picture of the beam when it is under load.
Question 3: (3+1 = 4 Points) Consider a non-uniform cantilever beam that is 1m long and whose flexural
rigidity is given by the function EI (x) = 30/x+2 kNm2.
a) (3 Points) Calculate the influence function (a.k.a., “Green’s function”) for the beam.
b) (1 Point) Use the influence function (and probably Desmos) to find the deflection of the beam
under a load of f(x) = 100 sin(x) cos(x) N/m. Include a picture of the deflected beam; the maximum
deflection must be able to be visually identified from the picture to about 0.1mm.
Question 4: (3 Points, Challenge) Further consider the beam in question 3.
Suppose I have a 20kg weight shaped as a right-angled triangle† with side lengths 30 cm, 40 cm and 50 cm.
The force applied by the weight at any point is proportional to the height of the shape at that point.
Calculate load functions for all six possible orientations of the triangular weight where the right-most point
of the weight is flush with the right-most end of the beam. For each orientation use Green’s function (and
probably Desmos) to find the maximum deflection in millimetres. Include a picture of the deflected beam;
the maximum deflection must be able to be visually identified from the picture to about 0.1mm.
Note: You’ll need to look up how to convert between weight and force.
Warning: make sure that the total force applied by your load functions is 20 kg.
†Don’t ask why I have a right angled triangle that weights as much as a bag of concrete; I’m a computer nerd & mathematician
in one, so I’m weird.
Question 5: (1+3=4 Points) Consider the function f(x) = 100 e−x restricted to the domain 0 ≤ x ≤ 2.
a) (1 Point) Extend f(x) to make an even function f∗(x).
b) (3 Points) Calculate the Fourier cosine series for f∗(x).
Question 6: (1+3=4 Points) Suppose we have a 2m long rod whose temperature is given by the function
u(x, t) for x on the beam and time t.
The left hand end of the rod is heated so that it is 100◦, and the heat on the rod as a whole is proportional
to e−x.
a) (1 Point) What is the initial condition u(x, 0) for this rod?
b) (3 Points) Use separation of variables to solve the heat equation for this rod if the ends of the rod
are insulated so that ux(0, t) = 0 = ux(2, t).
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