The problem will be dependent of a parameter S which is given as: s - 1.6 An all
ID: 3027557 • Letter: T
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The problem will be dependent of a parameter S which is given as: s - 1.6 An allen-wrench (also called unbrako fastener) is usually defined as a L-shaped bar with a hexagonal head, used to turn screws with hexagonal sockets (see Figure 1). However, for simplicity we will consider the case of an allen-wrench with quadratic head of which we are going to calculate the torsional rigidity la order to do this we have to solve the following partial differential equation: {partial differential ^2 u/partial differential x_1^2 + partial differential ^2 u/partial differential x_2^2 = -2 x = (x_1,x_2) Ohm, on partial differential Ohm , where n is the quadrat Ohm = [-s,s]^2(see Figure 2) Find the corresponding weak formulation of (1), and explain why this formulation has a unique solution.Explanation / Answer
ux1x1 + ux2x2 = -2 in r < s with u(x1, x2) vanishing on r = s.
In polar coordinates,
urr +1/r ur = -2
We omit the term because we consider u only depends on r.
Consider v = ur and multiply by r to get
rvr + v = -2r
=> (rv) = -2r,
Integrate both sides with respect to r to get
rv = -r2 + c
=> ur = -r + cr1
=> u = -r2/2 + c log(r) + d,
where c and are constants.
But term log(r) introduces a singularity at r = 0 therefore take c = 0.
Also u = 0 at r = s.
=> 0 = -s2/2 + d
=> d = s2/2
So our desired solution is u(r) = -r2/2 + s2/2
This is the only possible solution for the problem due to the uniqueness theorem for the Neumann problem.
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