A uniform bar has these specifications: length I, mass density per unit volume p, and torsion al stiffness IpG, where Ip is the polar moment of inertia of the cross section and G the shear modulus.
The end x = 0 is fastened to a torsion al spring of stiffness K Ib· in./rad, and the end 1 is fixed.
Determine the transcendental equation from which natural frequencies can be established.
Verify the correctness of this equation by considering special cases for K = 0 and K = ∞.