Calculating GM or Metacentric Height -
The following formula calculates GM in feet based on the natural roll period of the vessel measured in seconds or T and its beam in feet. A natural roll period is the time between two successive peak roll angles on the same side of a vessel. The most accurate method is to use an average of several roll periods.
GM = (.44 X Beam / T)²
For example, let’s say that a small vessel has a beam of 18 feet and its roll period or “T” is 6 seconds. The calculated metacentric height of this vessel would be:
(.44 X 18 feet / 6 seconds)² = 1.74 feet
With a GM of 1.74 feet, the vessel would be rather tender. With small vessels, the minimum metacentric height should be at least 3 feet and generally over 4 feet is preferred.
Calculating Natural Roll Period Using a Known GM:
We can predict a vessel’s natural roll period or “T” if its GM has been previously determined. Use the following formula:
T = (.44 X Beam) / Square Root of GM
If our vessel’s GM was determined to be 4 feet with a beam of 20 feet then its predicted natural roll period is:
(.44 X 20) / 2 = 4.4 seconds
Remember that a vessel has two centers of gravity. We have discussing heeling and listing which involves a vessel’s transverse stability. The other deals with its longitudinal stability which also has a center of gravity (LCG) and a center of buoyancy (LCB). Like transverse stability, we are adding and removing weight in terms of passengers, cargo, fuel, and stores. Instead of dealing with a listing or heeling vessel we are concerned with the trimming the vessel fore and aft. Imagine that a vessel is perfectly balance longitudinally on a point called the Longitudinal Center of Flotation (LCF). When weight is added either forward or after of this point, the vessel either trims down at the bow or at the stern. Usually a vessel sits deeper at the stern (drag) to increase the effectiveness of propulsion. So when a vessel is properly trimmed it preserves its design drag plus ensures that it has proper reserve buoyancy and metacentric height.
Calculate the metacenter (GM) for the following vessels:
A 50-foot charter boat with an 18 foot beam and 4.3 second rolling period?
A 95-ton steam vessel with a 22-foot beam with a rolling period of 6.5 seconds?
A vessel with a 16-foot beam with a 3.2 second rolling period?
A 26-foot UTV with a 12-foot beam and a 2.1 second rolling period?
An UPV with a 19-foot LOA, 8-foot beam with a 3.0 second rolling period?
Answers: 1. 3.39 feet 2. 2.21 feet 3. 4.84 feet 4. 6.32 feet 5. 1.37 feet