A wedge of specific weight \(\gamma\) floats stably on the free surface of a fluid of specific weight \(\gamma_{w}\) (Figure P12.8). The wedge is given a vertical displacement \(\delta\) from this equilibrium position.

(a) Derive the differential equation governing the resulting free oscillations of the wedge. Neglect viscous effects and the added mass of the fluid.

(b) What is the equation of the trajectory in the phase plane which describes the resulting motion. Sketch the trajectory.

(c) Assume \(\delta\) is small and use the method of renormalization to determine a two-term approximation for the frequency-amplitude relationship.

Transcribed Image Text:

FIGURE P12.8 %