Using superposition of undamped modes to model non-orthogonally damped systems

Abstract

In many structural vibration problems non-orthogonal damping leads to coupling between modes, rendering simple normal mode superposition solutions not sufficiently accurate. The flow-induced vibration of drilling and production risers used in the offshore petroleum industry is an example of such a structure. The uneven distribution of hydrodynamic damping often results in non-orthogonal damping. The most common solution technique, pioneered by Lord Rayleigh, is to construct the solution as a simple superposition of normal modes, achieved by simply ignoring the off-diagonal terms introduced by the modal damping matrix. This approach is often not sufficiently accurate when used to predict the fatigue damage rate of marine risers exposed to common ocean currents. In many cases an accurate solution may be constructed by using complex modes which include the coupling created by the non-orthogonal damping. Though accurate the complex mode approach is computationally not very efficient. This problem is made more severe by the inherent non-linear nature of flow-induced vibration, in which the magnitude of the periodic excitation from vortex shedding is dependent on the displacement response amplitude. For each riser design it is common to make thousands of fatigue life predictions corresponding to widely varying ocean current profiles. Therefore, the numerical techniques used to make the predictions need to be both efficient and accurate. Simple normal mode superposition is efficient but often not sufficiently accurate due to the influence of non-orthogonal damping. Mode superposition using complex modes is accurate but much less numerically efficient. The goal of this research was to develop an analysis method with the efficiency of simple normal modes and the accuracy of complex modes. A solution using undamped normal modes is presented with realistic examples. A linear, tensioned, homogeneous beam under various damping distributions is chosen as a numerical example. Based on the pseudo force iteration method, a frequency domain, corrected normal mode superposition is formulated. The computed response is compared to that computed from direct numerical integration and complex mode superposition. Comparisons show that the corrected undamped normal mode superposition solution is accurate and very efficient.