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Introduction

Next to the quasi-static wave loads, the ship's hull is subjected to wave-induced vibrations. The stiffness of the ship structure and the ship mass distribution can be regarded as a simple mass-spring system, which is associated with a natural frequency. Exposed to wave loads, the hull girder may vibrate with its natural frequencies. The vibration may be excited by nonlinear impulsive wave loads such as bow flare, bottom- or stern slamming, which leads to sudden vibrations. This is referred to as whipping. The vibration may also be excited by oscillating wave loads, which may lead to resonance vibrations. This is referred to as springing. Springing can be caused by both linear and nonlinear excitation, where the encounter frequency or the sum of two encounter frequencies coincides with a natural frequency of the hull girder.

The damping is typically low, so after a whipping event the vibration decays slowly and may last for many seconds and even minutes. Low damping may also cause significant vibration levels even though the resonance excitation is low. Whipping and springing may therefore occur more or less continuously and simultaneously and can therefore be difficult to distinguish. Their relative importance may also depend on design (flexibility and shape), loading condition (low or high draft and trim) and wave condition (sea state in conjunction with ship speed and heading). The two phenomena can be referred to as wave-induced vibrations. The associated loads are often referred to as high frequency loads, which can be compared to the conventional wave loads, which are referred to as wave frequency loads.

Ships have many vibration modes and corresponding natural frequencies. The governing vibration mode is the vertical 2-node vibration mode, which is associated with the lowest natural vibration frequency in most cases. The natural frequency may be in the order of 0.4 to 1 Hz, while the wave frequency loads are an order of magnitude less (0.05 to 0.2 Hz). The 2-node vibration mode is most easily excited and gives the largest vibration bending moment amidships. The vibration mode and the corresponding distribution of the vertical vibration bending moment are illustrated in Figure 1-1 for a homogeneous ship with a normalized scale. It resembles the envelope curve of vertical wave bending moment. Other modes may also contribute, in particular, for very large vessels. However, based on full-scale measurements, these are considered as less relevant.

Both, whipping and springing increase the fatigue loading, while only the whipping is considered to increase the extreme loading significantly. While the extreme loading and ultimate hull girder capacity are related to safety, the wave-induced vibrations also contribute to fatigue loading, which is next to safety mainly a concern related to maintenance and repair costs. Whipping and springing are thereby also related to economy. Both, fatigue and ultimate strength are addressed herein.

Regarding extreme loading, the whipping is superimposed on the static loading (still water loading) and the wave frequency loading. These three load components make up the total loading, which should be less than the ultimate capacity (collapse strength) of the hull girder.

Regarding fatigue loading, the springing and whipping are both superimposed on the wave frequency loading. The vibration and wave frequency response are related to two different frequency regimes, and they are widely spread on the frequency scale. For such broad banded processes, the fatigue cycles are counted by Rainflow counting, which is the recognized approach to establish the fatigue loading history. For the total stress history (wave frequency stress + high frequency stress) in a ship structure, the fatigue damage can be calculated, referred to as the total damage. Also for the wave frequency stress, the fatigue damage can be determined, referred to as the wave damage. The difference between the total and wave damage makes up the vibration damage. In practice it is the vibration on top of the wave frequency loading that makes up the significant part of the vibration damage. From full-scale measurements the vibration damage was found to be of comparable magnitude as the wave damage, but the relative magnitude depends on ship type, size and trade.

The basis for the herein given empirical relations has been full-scale measurements and model tests. Load histories from full-scale measurements comprise all kinds of hull girder vibrations and account for real life experience. In model tests realistic assumptions have been made in testing and evaluation of the results to avoid too conservative estimates of the effect of whipping and springing. Furthermore, damage experience from the fleet of container ships, classed by DNV GL, has been considered.