# Simple initial value problem

Part one: Simple initial value problemAssignment (please read the problem description before doing the assignment)(1) Find out the equation of motion of the cylinder. The derivation procedure of the equationmust be included in the report.(2) Develop a numerical method for predicting the vibration of the cylinder. The procedure ofderiving the FDM formula must be included in the report.(3) Develop a MATLAB program for solving the problem. The MATLAB program code must beincluded in the report.(4) Keeping the damping coefficient of c = 100 N·m/s, conduct numerical simulations for KC=2to 20 with an interval of 2 and find out the variation of the power with the KC number.(5) Keeping the KC number to be constant of KC=10, conduct numerical simulations for c=0 to2000 N·m/s with an interval of 50 and find the variation of the power with c.(6) In the report, show the time histories of the vibration for all the calculated cases.(7) Discuss how KC and c affects the power generation.Problem descriptionAn energy extraction device includes a cylinder elastically mounted on a spring and theelectricity generator. The cylinder vibrates only in a direction parallel to the flow and drives anelectricity generator. The electricity generator extracts the energy from the cylinder in the same wayas a damper. It is modelled as a damper with a damping constant c when the vibration is studied. Theflow velocity is a combined steady and oscillatory flow( ) sin(2 / ) u t ? U0 ?Um ?t Twhere U0 is the steady flow velocity, Um is the amplitude of the oscillatory flow velocity, T is theperiod of the oscillatory flow and t is the time.The KC number is defined as