Answer Happy

A floating platform is comprised of a rigid board of length L
that sits atop two
identical floats, each of mass m/2 and cross-sectional area A, as
shown in Figure A3-
1. The mass of the board is negligible compared with the mass of
the floats.
Determine the natural frequencies and natural modes for the
floating platform.
Assume small rotations of the platform.
A floating platform is comprised of a rigid board of length L that sits atop two identical floats, each of mass m/2 and cross-sectional area A, as shown in Figure A3- 1. The mass of the board is negligible compared with the mass of the floats. Determine the natural frequencies and natural modes for the floating platform. Assume small rotations of the platform. m/2 m/2 Figure A3-1 The effects of buoyancy of each floats may be accounted for through equivalent vertical springs of stiffness k that is proportional to the cross-sectional area of the floats. (The moment of inertia of a mass about an axis is defined as the mass multiplied by the square of the distance from the mass to the axis). (a) Derive the equations of motion of the platform and comment on the structure and the solutions of the equations. (b) Determine the stiffness and mass matrices of the platform. (c) Calculate the circular frequencies of the platform and comment on the modes of vibration. (d) If the two floats have the same mass but different cross-sectional area A, will the solutions be the same and why?