Natural Frequencies of Marine Drilling Risers
Authors: Dareing D.W. , Huang T. 
Published: December 2007

This paper gives the mathematical basis for determining natural frequencies and corresponding mode shapes for marine drilling risers. Numerical results relate natural frequencies to riser parameters; the data cover risers operating within 600 ft of water. An approximate method is given for establishing marine-riser natural frequencies.

Introduction
The effect of wave forces on the life and performance of marine drilling risers is a concern to both contractor and operator because a broken riser means capital loss and expensive down-time. The literature reflects that much effort is being made to improve understanding of riser motion and stresses during operation to upgrade riser design and operating practices. It is hoped that riser technology eventually will lead to reliable predictive techniques, verified by experimental data. predictive techniques, verified by experimental data. While accurate dynamic-stress predictions are desirable, natural-frequency data alone may be directly useful in alleviating dynamic stresses during operations.
Large vibrational stresses are normally associated with resonance that exists when the frequency of the imposed force, from whatever source, is tuned to one of the natural frequencies of the elastic system (marine riser). When this condition exists, a natural mode will dominate the over-all dynamic behavior of the riser. It is possible to excite two or more modes simultaneously; and in the case of the marine drilling riser in the North Sea, the first two modes are the most vulnerable. One of the difficulties in developing a program for predicting the forced vibration response of a marine riser is predicting the forced vibration response of a marine riser is modeling the wave forces or the source of excitation. In addition, damping is not well defined. A predictive technique, no matter how sophisticated, is no better than the boundary conditions imposed on it. On the other hand, input frequencies (or wave frequencies) can be determined on location and used along with natural-frequency data to determine whether a resonant or a critical dynamic condition exists.
This paper presents data that relate the first five natural frequencies of marine drilling risers to typical riser and drilling parameters. The operational parameters are related through dimensionless numbers that are parameters are related through dimensionless numbers that are generated for most drilling risers operating within a water depth of 600 ft. The drilling riser is idealized as a vertical flexible beam with pinned supports. Variable tension and fluid environment make the mathematics different from classical beam theory and lead to a differential equation that is perhaps unique to the oil industry. Eigenvalues and eigenvectors are determined by solving this differential equation. In addition, the exact natural frequencies or eigenvalues are compared with frequencies obtained through an approximate method based on classical, uniformly tensioned beam equations. An example calculation illustrates the practical use of the data.

Differential Equation of Motion
The natural modes and frequencies of marine risers depend on the usual beam parameters. The usual assumptions made in developing beam-column equations also are made here. However, the fact that the riser is surrounded by fluid of one density (sea water) and contains fluid of another density (drilling mud), and has internal tension that varies along its length, makes the riser quite different from an elementary beam. It is, therefore, worthwhile to show how these parameters enter into the differential equation of motion. The following derivation is based on plane motion; the spatial coordinates are illustrated in Fig. 1a.
JPT
P. 813