Question

An underwater habitat in the form of a sphere is secured to the ocean floor. It is supported by a column stiffened by cables as shown in the following figure. The mass of the column is small compared to that of the sphere and can be ignored. The structural and hydrodynamic damping are negligible. Derive an expression to determine the natural frequency of the system assuming that the cables are infinitely stiff and permit rotation at the point where cable are connected to the column Derive an expression to determine the natural frequency of the system assuming that the stiffness of the cable in the longitudinal direction is kg Ma ==PT added mass of the sphere M: mass of the sphere E: Yourn's Modulus [kN/m2] I: Moment of Inertia of the cross-section of the column [m4] P: Density of water

Answer 1

If the cables are infinitely stiff, the column along with the sphere will rotate about the point A shown. The stiffness for the remaining column can be calculated (as the column is cantilever). Once the stiffness is known, the natural frequency can be obtained by dividing it with total (i.e. mass of sphere and the added mass).

For the second case, the stiffness of each part of column and the stiffness of the cable is shown. The net stiffness can be obtained. Once the net stiffness is obtained, the natural frequency of the system can be determined.

D - Added Mars Mas paloma) M Mars of sphere 3EI 2 a) Cables are Infinitely stiff, the metion (Retation) Evill happen only about point A le + Effective length of Column le le = Į - (1/2) 31-(01) 13 (Stiffness of bolum) A " K= 3ET SET b.) (M+Ma) 321 (M+ma) >>= 10-01) (M + 2 piesa') K 3 Kq 361 327 1 3 12 {m + 2pal}}{3-1003 Ks {ks ke K (2Kg + k) SET Ke K (K,+ 2K) (2Kg + K,+K) > K- 321 43 + 2kg (₂-8/2) 31 SET - | 2K (2-8/2) (M + M) ET ) 3f1 + 2K, w = m 3 (1-D) 3.2 2K + BET 3 3:{ma Śpito.) 3 į (-012