Vortex induced vibration (VIV) should be taken into account for the vibration analysis of a cantilevered marine riser. This paper is dedicated to the VIV for the nonlinear vibration analysis of a cantilevered pipe discharging fluid with a nozzle at sea. In this study, three dimensional (3-D) motion equations are derived based on modified Hamilton Principle in a rigorous way. The Morison formula is used to calculate the external fluid-induced static drag force and viscous damping forces in the in-line (IL) and cross-flow (CF) directions. A double-van der Pol oscillator model is adopted employing a set of calibrated wake coefficients in Facchinetti et al. (2004) for the VIV effect. A new hydrodynamic force model is proposed considering the coupling effect of IL and CF motions. The mathematical model together with proposed solution methods are validated after comparisons with experimental results.