Page 54 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 2 Problems 41

Figure P2-14.
period of oscillation in rolling motion is 1.3 s. Determine the moment of inertia of the
buoy about its rotational axis.
2-15 The oscillatory characteristics of ships in rolling motion depend on the position of the
metacenter M with respect to the center of gravity G. The metacenter M represents
the point of intersection of the line of action of the buoyant force and the center line
of the ship, and its distance h measured from G is the metacentric height, as shown in
Fig. P2-15. The position of M depends on the shape of the hull and is independent of
the angular inclination 6 of the ship for small values of 6. Show that the period of the
rolling motion is given by

T = 2t7

where J is the mass moment of inertia of the ship about its roll axis, and W is the
weight of the ship. In general, the position of the roll axis is unknown and J is
obtained from the period of oscillation determined from a model test.

Figure P2-15.

2-16 A thin rectangular plate is bent into a semicircular cylinder, as shown in Fig.
P2-16. Determine its period of oscillation if it is allowed to rock on a horizontal
surface.

Figure P2-16.
2-17 A uniform bar of length L and weight W is suspended symmetrically by two strings, as
shown in Fig. P2-17. Set up the differential equation of motion for small angular
oscillations of the bar about the vertical axis 0-0, and determine its period.