Problem A new type of chair for a ski lift has been developed, and the manufacture has designed a simplified model (Figure) of chair's behavior upon loading a group of individuals.
Recall that the basic spring equation is:
W = k.x
where W is the weight or force applied to the spring, k is the spring constant, and x is the displacement (or stitch) of the spring.
You have been hired to calculate the total displacement of the system of springs and weights shown in Figure. The properties of the system are:
| Parameter |
Value |
| k1 |
10,000.0 N.m |
| k2 |
5,000.0 N.m |
| k3 |
8,000.0 N.m |
| k4 |
3,500.0 N.m |
| k5 |
4,500.0 N.m |
| W1 |
500.0 N |
| W2 |
1,000.0 N |
| W3 |
1,000.0 N |
The first step is to derive a force balance on each weight. For the first weight, WI, you need to consider every spring touching the weight, including the displacement and direction of force:
W1 = k1.x1 - k3.(x2 - x1) and for the second weight:
W2 = k2.x2 + k3.(X2 - X1) - k4.(X3 - X2) -k5.(X3 - X2)
In addition to the basic set of parameters given in the problem statement, the manufacturer has also asked you to include a plot of the displacement of each weight (i.e., each skier) as a function of W3. The plot should have a range of values for W3 on the x-axis, and then three curves showing the displacement (on the y-axis) of the different weights. You are free to use any linear system solver from chapter 6, including the solvers that are part of the SciPy and/or numpy packages.