Currently at the Woods Hole Oceanographic Institution. Journal of Sound and < ibration (2002) 250(4), 675}709 doi:10.1006/jsvi.2001.3949, available online at http://www.idealibrary.com on VIBRATION OF A COMPLIANT TOWER IN THREE-DIMENSIONS S. M. HAN AND H. BENAROYA Department of Mechanical and Aerospace Engineering, Rutgers, the State ;niversity of New Jersey, Piscataway, 98 Brett Road, NJ 08854-8058, ;.S .A . (Received 2 February 2001, and in ,nal form 10 August 2001) The three-dimensional motion of an o!shore compliant tower using both rigid and #exible beam models is studied in this paper. The tower is modelled as a beam supported by a torsional spring at the base with a point mass at the free end. The torsional spring constant is the same in all directions. When the beam is considered rigid, the two-degree-of-freedom model is employed. The two degrees constitute the two angular degrees of spherical co-ordinates, and the resulting equations are coupled and non-linear. When the beam is considered as elastic, three displacements are obtained as functions of the axial co-ordinate and time; again with coupled and non-linear equations of motion. The free and the forced responses due to deterministic loads are presented. The free responses of the rigid and elastic beams show rotating elliptical paths when viewed from above. The rate at which the path rotates depends on the initial conditions. When a harmonic transverse loading is applied in one direction, the displacement in that direction shows subharmonic resonance of order 1/2 and 1/3 while the displacement in the perpendicular direction is a!ected minimally. Next, in addition to the harmonic load in one direction, a transverse load is applied in the perpendicular direction. The transverse load varies exponentially with depth but is constant with time. It is found that the transverse load a!ects the transverse displacements in the perpendicular direction minimally.  2002 Elsevier Science Ltd. 1. INTRODUCTION The purpose of this paper is to show how to model and predict three-dimensional responses of a structure in an ocean environment. O!shore structures are used in the oil industry as exploratory, production, oil storage, and oil landing facilities. Detailed speci"cations and descriptions can be found in Hydrodynamic of O+shore Structures by Chakrabarti [1]. O!shore structures are designed to be self-supporting and su$ciently stable for o!shore operations such as drilling and production of oil. In general, there are two types of stationary o!shore structures: "xed and compliant. Fixed structures are designed to withstand environmental forces without any substantial displacement. Compliant structures, on the other hand, are designed to allow small but not negligible deformation and de#ection. While the stability of "xed structures is provided by structural rigidity, the stability of compliant structures is provided by tension due to buoyancy chambers. For these compliant structures, the dynamic responses need to be explored fully. The "xed structures are economically feasible only up to water depths in the range of 300}500 m. Fixed platforms are indeed the most popular and proli"c structures for water depths of 100}200 m. However, they become impractical for deep water because they must 0022-460X/02/090675#35 $35.00/0  2002 Elsevier Science Ltd.