|United Nations University - Work in Progress Newsletter - Volume 14, Number 1, 1992 (UNU, 1992, 12 pages)|
|A World of CHAOS|
|UNU's New Logo - Part of Its Recently Launched New Visual Identity|
|The Interplay of Many Ideas|
|Chaotic Movement in the Economy|
|The Origin of Chaos|
|Chaos and Social Order|
|Population Dynamics and the Arms Race|
|Enlarging Knowledge: The Road to Universality|
|Improving Earthquake Prediction|
|How Ships Capsize|
|Waking from Dreams of Determinism|
|The Body's Chaotic Rhythms|
|The Beating Heart|
|Mapping the Brain|
By Steven R. Bishop
In planning any structure or system, an engineer will naturally seek to minimize the possibilities of failure during all imaginable future operations - in the materials used, the design configurations adopted, appropriate use of site, and so on. But no matter how sound the long-term format, failure more often than not results from the unexpected - in a sudden wind, an unanticipated vibration, a sharp jolt from an explosion or tremour. Failure, in an engineering sense, is not a steady state - systems can fail after just a few of what engineers call forcing cycles, or periodic application of some external impulse.
Thus elements of chaos are very much bound up in engineering dynamics. Engineers have begun to apply newly discovered aspects of chaotic dynamics to a range of their concerns, from offshore oil platforms to tensile strength of beams to flash fires In buildings. In his report to the Tokyo chaos symposium, Dr. Steven Bishop of the Dynamics Research Centre at University College, London, UK, talked about work he had done on the unexpected capsize of fishing vessels. - Editor
Arguably the most exciting mathematical development of recent times is the discovery of chaos and the fractal nature of many natural systems. We now know that chaos is a typical ingredient of engineering dynamics, being inherently nonlinear. This has been observed in mechanical and structural systems governed by Newton's laws of motion applicable to machine operations, vibration of structures, and the response of vehicles and other compliant objects. The associated motions are irregular; they are unpredictable and can lead to unexpected failure of what was thought to be otherwise a safe and deterministic system.
In my discussion here, I am going to preclude conservative systems, which are perhaps of greater relevance in celestial mechanics or particle physics, and consider the world typically encountered in engineering, in which some dissipation of energy tends to make the system settle down towards a steady state. However, rather than focusing on the long-term, possibly chaotic, steady state solutions which produce chaotic attractors that help identify the system, engineers have to express the integrity of a system through the transient motions from short bursts of excitation.
To understand any particular failure mechanism, the emphasis must be placed on transient dynamics -failure is a transient phenomenon and not a steady state. Typically, systems can fail after just a few forcing cycles. To illustrate this property, we consider the question of ship capsize.
Why Ships Capsized
In Canada during the 1970s it was reported that several fishing vessels capsized in sea conditions which were not thought to be particularly hazardous. To account for these unusual events, the Canadian Government commissioned a report on the nature of the stability of the common design models of fishing vessels used at the time. The overall aim of this report was to investigate whether computer programmes could be used to formulate changes to the existing requisitions for determining the necessary stability characteristics of these small vessels. The particular computer programmes used for the purpose of the study calculated the nonlinear time history in the six degrees of freedom (position and velocity coordinates for three usual dimensions), taking into account added mass, wave forces, etc.
In the report three known dangerous capsize mechanisms were considered:
1. resonant rolling;
2. subharmonic rolling;
3. stability loss at wave crest.
It is the third of these mechanisms that provides the most surprising capsize, giving little or no information of any incipient failure of the system. The hull profile plays an important role on the stability in certain positions of the vessel relative to the wave profile and it was this point that was investigated more closely.
When Capsize Happened
The conditions under which the vessels capsized suddenly without prior warning were:
1. a following sea;
2. the vessel travelling at roughly the same speed as the forward velocity of the waves;
3. a wave length about equal to the ship's length;
4. a steep wave.
For simulation purposes, the time history was started with the vessel on the crest of a wave with the save speed slightly greater than the vessel's forward velocity. The speeds were such that after a time of about 100 seconds the ship was once again on a save crest.
Initially, a transient motion occurs before the vessel settles down to an equilibrium state. Then as the wave crest approaches amidships at about 94 seconds (see illustration), the vessel begins to roll to starboard and suddenly capsizes, turning upside down when approximately 104 seconds have elapsed.
With the hull balanced on the wave crest, and the stern emerging from the wave surface, both of the vessels studied in the report experienced a loss of waterplane inertia and hence stability resulting in capsize. The use of bilge keels to increase the damping had little effect on the capsize under these conditions with a wave crest amidships. It is possible that it is this mechanism of failure that caused the capsize, since there was no evidence that any resonant or subharmonic rolling motion occurred prior to failure.
Prediction of Failure
The prediction of capsize, as we have seen, is associated with the study of transient dynamics, and not with chaotic attractors. However, the theory of chaos and methods developed for the study can be used to provide a framework for our understanding of engineering dynamical systems.
Why a Ship Capsizes
The illustrations above show the sequence as a wave overtakes a ship, causing it to roll to starboard and eventually capsize. The time between wave crests is 100 seconds. Sequence begins at 95 seconds into the roll of the wave when the crest has reached amidships. With each-succeeding second, the wave upsets the ship's equilibrium further until, at about 101 seconds, the vessel reacts to the weight of the wave crest, rolls heavily to starboard, and capsizes.
The figures above are taken from a time sequence showing a ship capsizing in a steep following sea, with the vessel travelling roughly the same speed as the forward velocity of the waves. Less than 10 seconds elapse from the time the wave crest approaches amidships (t=95 sec.) until the ship is turning upside down (t=103 sec.).