Fibonacci Numbers - Compute the Ratio - Mass Spring System - Application of Computing Coursework - Science and Maths Assignment Help

Assignment Task:

Science and Maths Assignment Help

Q1 The Fibonacci numbers are computed according to the following relation:

Fn+1 = Fn + Fn−1

with F0 = F1 =1., compute the ratio.

i) Create a script m-file to compute the ratio Fn Fn-1 for the first 50 Fibonacci numbers.

ii) Plot a graph of this ratio as a function of n. What do your results show?

iii) It is claimed that this ratio approaches the value of the golden mean 1 + 5 /2.

What is the smallest value of n that gives 0.1% accuracy of the golden mean?

Q2 A simple mass spring system subject to damping, is vibrating under the action of a periodic force F cos t . The equation of motion is

d2xdt2 + 2kωdxdt+ω2x = F cost

where x is the displacement of the mass, k is the damping parameter and

 is the natural frequency of the system.

i) Create a function m-file to solve this second-order differential equation with the system of first-order differential equations. ii) Consider the unforced motion when F = 0 and  = 3.8 with the initial conditions   0 = 1,  0 = 0. Use the numerical solver ode45 to solve the differential equation for the cases k = 0.0, 0.1, 0.2, 0.4, 1.0, 2.0 over the range 0 to 4. Plot the six graphs on one figure. Interpret the results physically for the different values of k. iii) Consider the forced motion when F = 2.0 and  = 3.8 with the initial conditions  0 = 1,  0 = 0. Use the numerical solver ode45 to solve the differential equation for the cases when k = 0.0, k = 1.0 over the range 0 to 4. Plot the two graphs on one figure. State the physically important difference between the results of parts (ii) and (iii)

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