MATH2115 - Curvature and Analyzing Beam Deflection Assignment

Assignment Task

Question 1

Consider the equation

y²  y² − 5  = x²  x² − 4

Calculate the equation for curvature  for  this  curve. What is the curvature of the curve at the point √2, −1 ?

Question 2

Suppose we have a 4 metre long non-uniform beam. The beam is simply supported on the left and elastically supported on the right.

The beam is thinner on the right end than it is on the left end. The width at any point, x, the beam is given by the function W (x) = 3/(x+2). The flexural rigidity of the beam is directly proportional to the width of the beam at that point, with the flexural rigidity equal to 100,000, Nm² at the thickest part of the beam.

Two blocks are placed on the beam.  Each box is 3 metres long, 1 metre high, and applies 21 N of downward force in total. The first box is placed directly on the beam flush with the left hand side of the beam. The second box is placed on top of the first box, but is flush with the right hand side of the beam.

In this scenario:

• The load for each box is evenly distributed over it’s contact surface (i.e., evenly distributed over the length at which it touches something).
• The spring of the elastic support has stiffness given by the constant k₀ = −1.

Find the deflection function y(x) for the beam and load.

Question 3

Consider A 2 m rod that has a thermal conductivity of κ = 1/10.

The rod is cooled on both ends (to ensure that the temperature at the ends is always 0^◦). The middle of the rod is heated for a period after which the temperature on the rod is H(x) = 300x − 150x².

The heating is then immediately removed from the rod, and the rod is left to cool. (You may assume that this happens when t = 0).

After how many (whole) seconds is the first time the temperature over the entire rod uniformly equal to 0^◦ within ±0.005?

Question 4

Consider a 2 m long medium in which a wave might propagate. The medium has a speed of propagation of c = 1/10. The ends of the medium restrict the wave so that the wave always has an amplitude of 0 at those endpoints.

Initially, there is no wave, but the medium is disturbed so that the rate of change of the wave over time is initially given by the function Solve the wave equation for this medium and this wave.

Include in your submission a computer drawn picture (using Desmos or similar computer software) of a picture of the Fourier series for f (x). Make sure to use enough terms to clearly show the series tracing the shape of f (x). Include this picture in your submission.

Question 5

Consider a uniform 10m long beam, with flexural rigidity of 120,000Nm², and a linear density of ρ = 192.

The beam is simply supported (i.e., it has simple supports at both ends), and a weight is hung under the beam—attached via cable at the x =  3  point—applying a downward force of 100 N .  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 100 N .

The deflection function of the beam, in mm, is given by

y(x)  =  1/36*5  (x − 3)³H(x − 3) − (x − 7)³H(x − 7)   − 2x³ + 42x

However, the cables used weren’t rated for this force, and they both snapped at precisely the same time. One moment there was a load causing the beam to deflect, and the next moment there wasn’t.With the sudden removal of the load, the beam vibrated. We do not know why the beam was configured this way, nor why weak cables were used. We think a bet and maybe alcohol was involved, but nobody will admit to any knowledge about the beam (or how it came to be configured this way) at all.

Solve the beam vibration problem for this beam after the load disappeared. You should consider that the instant the cables broke was at time t = 0. You can presume that the initial partial derivative with respect to time is u_t(x, 0) = 0. Have Desmos (or other computer software of your choice) produce an animation of the vibrating beam.

Make an argument about what the value of curvature for the curve in Question 1 at the point (0, 0). Make sure to use all relevant mathematical mathematical properties of the curve.

Question 7

Watch the video embedded in the Assignment page on Canvas. For reference:

• The ruler was clamped at the 3 cm and 30 cm
• The weight was 0.36 kg
• The weight was hung at the 17 cm
• The deflection at the 16 cm point was was 1.5 mm.
• The deflection at the 20 cm point was 1 mm.
• We are supposing that the ruler has a constant flexural rigidity (i.e., that it is a uniform beam).

Use beam deflection calculations to estimate the flexural rigidity of the ruler.

Question 8

Show, mathematically, that for a uniform cantilever beam under a uniform distributed load, the maximum deflection will always happen at the unsupported end.

You may assume for the purposes of this question that the unsupported end is on the right.

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