Free Vibrations of a Square Cylinder of Varying Mass Ratios

Two-dimensional space-time finite-element simulations are carried out to study the effect of oscillator mass ratio, m* on free in-line and transverse vibrations of a rigid square cylinder. The effect of viscous damping or resistance is not considered and thus cylinders of mass ratio 1, 5, 10 and 20 execute free undamped vibrations. Results are presented for 50≤Re≤250 where Re is the Reynolds number. For m* = 1, synchronization between cylinder oscillation and vortex-shedding is 1:1 (periodic flow). Thus, the Cl-Y phase portraits must be single looped closed curves. For m* = 5-20, synchronization is 1:1 (periodic flow) prior to galloping. For m* = 5, the synchronization during galloping is quite different from 1:3. Along ‘increasing Re’, the entire galloping branch of m* = 5 cylinder is quasi-periodic. However, along ‘decreasing Re’, it is mostly quasi-periodic and it is periodic for Re≤190. An 1:3 sub-harmonic synchronization is seen for the m* = 10 and 20 cylinders only in the periodic regime of galloping occurring at higher Re. For m*= 1, only one kink is seen in the response curve signifying transition from initial to lower branch. Two kinks are captured for m* = 5 cylinder. For m*= 10 and 20, an additional third kink is resolved in the galloping branch that signifies transition from quasi-periodic flow/body motion to periodic flow/body motion.