Dynamic Amplification and Extrapolation of Fundamental Laws

Study of the unpredictability of wave based phenomenon is of great importance in seakeeping.

We are accustomed to apply Newton’s Equations every now and then. But how much do we know its validity? If we seek the answer to this question, even the masterminds have to rack their brains! It is a universal and harsh truth that these are applicable to only a meager number of bodies around us.

Coming to the context of ships, Newton’s ‘linear’ postulates falter in front of the large diabolical forces and unpredictable rapidly-varying disturbances, it is subject to. If we take the pain to estimate the forces over a stipulated span of time on a side shell plating of the outer hull, we could notice that neither the strength, intensity, distribution nor the fetch of the generated stress can be estimated by our conventional methods. Dramatically enough, every single physical parameters associated with the region wields as an abrupt time-varying function. For example, the stress distribution would be following abrupt frequency response changes of intensity. This is where we introduce a non-dimensional parameter known as the Dynamic Amplification Factor. Mathematically, it can be expressed as the ratio of the maximum permissible load, Um to the static load applied to the same body, Us. Or in other words, it takes into account the “static-equivalent load” under the same conditions.

Rarely, in ships do we find static loads acting. The loading like that of storms, rain, earthquakes, floods or calmer ones like live traffic loads on a bridge, the loads encountered by a ship under all abrupt sea states are essentially highly accelerated and variant, viz. Dynamic loads. Have you ever wondered at the rogue deep sea waves lashing out at a ship? Do you think that the aggregate stresses embarked upon the ship at its multitude of structural locations could be simply assessed by a petty Newtonian Load equation? The answer is a big no. A lot of tedious work has to done by designers/naval architects, engineers, seafarers to analyse the nature of the loading at all strategic locations of any structure to execute its probabilistic failure computations.

The most preliminary application of Dynamic Amplification Factor on a ship would be to study the forces enacting on a single isolated object on board while a ship hurls above a big sea wave. This situation is analogous to the case of a simple elevator! We know that whenever an elevator goes upward with a certain acceleration, weight experienced by an object in it increases. Similarly, when a ship is aloft a wave, any object on board experiences an “added pseudo force”. 

Hence, the classical equation, F=mg gets modified in the form of

F= m*g*(DYNAMIC AMPLIFICATION FACTOR)

The basic concept lies in the fact that when a body is subject to dynamic force, its effects are more pronounced due to the materials inability to promptly react to the sudden disturbance. This is where the induction of the DAF equalizes the effects of static and dynamic loading under the given conditions. 

The determination of the Dynamic Amplification Factor, however is a cumbersome task where it involves a compact knowledge of the nature of the frequency response curves.

Some of the methods involved are complicated and require high levels of expertise. Averaging of the stress variations from time-to-time in a system like that of a ship to deduce the DAF is a challenge to a naval architect. The designer/builder hones his skills again in turn to implement this in the detailed structural design and material analysis. Thus this simple ‘extrapolation’ of the fundamental laws have pronounced implications in the ship structure analysis.