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Explanation & Answer
Hi, Here are the answers dude. Please feel free to let me know if you need any other help regarding this. I am attaching the python code in the word document. Also, you can see them in the pdf as well. I am also uploading a concepts files including the the concepts and some reference sites for the ease of your understanding.
Concepts
1. Free Body Diagram (FBD)1, 2
The FBD or Free Body Diagram looks in to the bodies at equilibrium,
skier in this case to look in the forces acting upon them by different
elements.
For example, for the first skier the downward forces are W1 and
downward pool by spring 3 (with spring constant k3), which has a
downward stretch of x2(that of the Skier 2) less the downward stretch of
the skier 1 (x1).
The Upward forces are the pull in the spring 1 (with spring constant k1).
Since, the skier is at equilibrium, i.e., there is no acceleration acting upon
it, hence the downward forces and upward forces equate each other.
W1 = k1 . x1 – k3.(x2 – x1)
2. Hooke’s Law for Spring3
The hook’s law for spring suggest that any force acting on a spring and
the displacement (stretch or shrinkage) are proportional. They can be
equated with a constant depending on different properties of the spring,
known as the spring constant.
F = K . x
Where,
1
F = Force acting on the Spring
x = displacement
K = The spring constant
https://en.wikipedia.org/wiki/Free_body_diagram
2
https://www.kpu.ca/sites/default/files/Faculty%20of%20Science%20%26%20Horticulture/Physics/Ch5%201%20-%20Static%20Equilibrium_0.pdf
3
https://en.wikipedia.org/wiki/Hooke%27s_law
3. Symmetric Matrix4
The Symmetric matrix is a square which remains unchanged when
transposed. Some properties of a symmetric matrix are –
i.
The transposed matrix and the original matrix are the same.
ii.
The elements on the main diagonal of the matrix remains
unchanged while transposing.
iii.
The elements are equal with following equation
aij = aji
4. Solution of Linear equations using numpy.linalg.solve(C, K)5, 6
The numpy linear equation solver takes two matrices as the input and
output the solution vector. For the solution of the following system of
the linear equation, the inputs are: first input – the coefficient matrix
and the second input – the constant matrix.
𝑐11
(𝑐21
𝑐31
𝑐12
𝑐22
𝑐32
𝑐13 𝑥1
𝑘1
𝑐23) (𝑥2) = (𝑘2)
𝑐33 𝑥3
𝑘3
C.X = K
𝑐11
Where, C = (𝑐21
𝑐31
𝑐12
𝑐22
𝑐32
𝑐13
𝑥1
𝑘1
𝑐23), K = (𝑘2) and X = (𝑥2)
𝑐33
𝑥3
𝑘3
5. Matplotlib Plot Functions7
The matplotlib pyplot functions are used to create the plots of the displacements as
functions of the 3rd Weight.
4
https://en.wikipedia.org/wiki/Symmetric_matrix
5
https://docs.scipy.org/doc/numpy-1.13.0/reference/routines.linalg.html
6
https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.linalg.solve.html#numpy.linalg.solve
7
https://matplotlib.org/api/pyplot_api.html
The Ski Lift Chair Problem
With FBD (Free Body Diagram) of the balances of the W1 and W2 and Hooke’s
law applied on the Springs k1, k2 and k3 the following equations of equillibrium
are given –
For First Weight, Eqn. (1):
W1 = k1 . x1 – k3.(x2 – x1)
For Second Weight, Eqn. (2):
W2 = k2.x2 + k3.(x2 – x1) – k4. (x3 – x2) – k5. (x3 – x2)
Hence, using the same FBD and Hook’s law on the third weight.
K4.(x3-x2)
K5.(x3-x2)
W3
For Third Weight, Eqn. (3) :
W3 = k4. (x3 – x2) + k5. (x3 – x2)
Now, rearranging...