The natural frequencies of a fixed-fixed uniform shaft made of homogeneous material undergoing longitudinal vibration is given by rad/s, where L is the length of the bar, G is the shear modulus, ρ is the density and i is the mode number (i=1,2,3...). The corresponding mode is sin (iΠx/L). You do not need to prove this. Obtain approximations to the first three natural frequencies( within 0.1% of the exact solution) by modelling the shaft as a series of coil springs of torsional stiffness and thin rotors with moments of inertia. Discuss your findings. Plot the first three modes and compare with the exact modes. You will need to write a MATLAB code to obtain these results.  The elastic stiffness of a shaft segment of length L/n is given by nGJ/L. Present results in terms of non-dimensional frequency parameter . For convenience, take J as unity, although the value of J will not affect the non-dimensional frequencies.