We will be developing a code to solve a complex problem modelling a hanging cable. The steps below will help you build a progressively more sophisticated code.

Step 1.

Write a Newtons solver to find a solution to the following system of four equations

t=z[0]

u=z[1]

v=z[2]

w=z[3]

vec[0]=t**4+u**4-1

vec[1]=t**2-u**2+1

vec[2]=v**4+w**4-1

vec[3]=v**2-w**2+1.

Use an exact calculation for the Jacobian. You code should use python functions to evaluate the above vector of functions and the Jacobian. Your Newtons solver should use a loop. Report the solutions you found.

Step 2.

Write a function to find a numerical version of the Jacobian based on forward differences with Delta x = 0.02. Use the new code to resolve the problem in Step 1. Comment on the difference with Step 1.

Step 3.

Find the two solutions to the equation vec=0 where the equations are:

t=z[0]

u=z[1]

v=z[2]

w=z[3]

vec[0]=2*(u-t)**2-4*(u-t)+v**2+3*w**2+6*w+2

vec[1]=(u-t)**2+v**2-2*v+2*w**2-5

vec[2]=3*(u-t)**2-12*u**2+v**2+3*w**2+8

vec[3]=u**2-v**2+t

Comment on why you might prefer to use the numerical code for the Jacobian.

Step 4.

Write down the total energy (potential and elastic) of a hanging cable made up of three weightless springs connecting two objects of mass m1 and m2. The end points of the cable are (x0,y0) and (x3,y3). The masses are located at (x1,y1) and (x2,y2). The springs have Hookes constants k01, k12, k23 and natural lengths L01, L12 and L23. The equilibrium position of the cable is obtained by minimising the total energy of the system. Formulate the problem mathematically, and state the equations to be solved.

Step 5

In scientific coding it is useful to develop test cases which you can use to test your code. Case 1. Consider the case where m1=m2=1kg, k01,k23=100N/m and k12=0.1 with Lij=0.9 for all springs. Provide a calculation of the approximate equilibrium positions and give a short physical description. Case 2. Consider the case where all springs have kij=100, and length Lij=0.2 and the masses are taken to have m1=m2=0.01. Provide an approximate calculation of the equilibrium positions and give a short physical description.

Step 6

Consider the parameters given below. The dimensions of the lengths are in metres, and spring constants are in N/m. Write efficient code to evaluate your equations.The function representing the four equations has value Test f [ 37.2 4.98 -67.6 25.1 ]

At Test x (x1,y1,x2,y2)= [1.1 0.9 1.7 0.8].

Use this to confirm your equations are working. Report your values of the test function to five significant figures.

global x0,y0,x3,y3,m1,m2,g,k01,k12,k23,L01,L12,L23

x0=0.

y0=1.

x3=3.

y3=1.

g=9.8

m1=1

m2=3

L01=0.9

L12=0.8

L23=0.7

k01=90

k12=100

k23=80

Step 7

Test your code by considering the exact problems you studied in Step 5. Provide the results that show your code is working. Comment briefly on the difference between your exact results and the approximations of Step 5.

Step 8

Use you code to find the equilibrium position of the masses for the problem in Step 6. Report and plot the positions of your masses and provide a qualitative physical description of the configuration. My code for plotting: xpos=numpy.array( [x0,x1,x2,x3] ) ypos=numpy.array( [y0,y1,y2,y3] ) plt.figure(1) plt.ylim(-0.05, 1.05) plt.xlim(-0.05, 3.05) plt.plot(xpos,ypos,'o') plt.show()

Step 9

Criticize your code, and explain how you would modify your code if the cable had 20 masses.