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Roll Stability = Comfort
Posted July 7, 2003
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OK - I'm kidding: there is math involved with calculating the stability of your trawler. But I'll keep the formulas simple and easy to understand. This month in PassageMaker magazine (July/August) was an excellent article titled Let's Talk About Roll Period by Charles Neville. The article was well written but it did not give us amateur naval architects the chance to really find out if our trawlers where indeed stable or provided a "comfort factor" that was within an acceptable range. The material presented here is a collection of research provided by a number of resources that are identified at the end. The formulas have been calculated using the numbers from our American Marine 49' Alaskan but can be developed using your numbers where available.
The comfort of trawlers is a complex subject because subjective human experience and reporting will determine a vessel's reputation more than technical analysis. A trawler that operates from the Bahamas where there is an almost immediate transition to open ocean conditions will be considered less comfortable than the same trawler operating in the ICW which makes a gradual transition through river, harbor, and coastal waters.
Knowing your trawler's natural roll period is an excellent indication of its stability, and there are several rules of thumb we can use to evaluate roll period. In general an acceptable, but minimum, natural roll period (seconds) should be equal to a vessels maximum waterline beam in yards. For example a trawler with a 15-foot beam should have a roll period of 5 seconds (15 feet/3 feet per yard = 5 yards or 5 seconds). Cruise ships with beams of 100 feet have roll periods of approximately 30 seconds (100 feet/3 feet per yard = 33 yards or 33 seconds). Additionally, a vessel's natural roll period may be calculated using this formula:
Natural Roll Period = (.44 x beam) / Square Root of GM
GM is the distance between the vessel's center of gravity (G) and the metacenter (M), where metacenter is defined as the point through which the center of buoyancy acts when a vessel heels. GM is provided in a vessel’s design and can be general defined as eight percent of a vessels maximum beam. GM can be estimated using 0.08 * beam.
Values used for M/V
Tortuga:
Beam - 15.1 feet
LOA - 48.7
LWL - 46.1
Displacement - 57000 pounds
Natural Roll Period = (0.44 x 15.1) / sqrt(0.08 * 15.1) = 7.3 seconds
Waves have periods, which is the time it takes for successive crests to pass a given point. If the natural roll period of a vessel equals or is an even interval of wave period, then synchronous rolling or pitching is likely to occur. Synchronous rolling and pitching, at the least, makes a trawler' motion painful, and at its worst is sufficiently dangerous to capsize her. What is synchronous rolling and pitching? It is wave action enhancing a boats roll such that the vessel is unable to come to an upright position and stop. It is similar to pushing a swing, where a person pushes at the outer ends of the swing, increasing the motions amplitude, and preventing the swing from slowing down and reaching equilibrium.
All vessels have stability curves that show the righting force at increasing angles of heel. There is always a maximum angle of heel, after which there is no righting force and a capsizing force comes into action. A righting force exists so long as CG and CB are separated, but once a vessel rolls to an angle where CB and CG (the two red vectors in the diagram above) are no longer separated but are in line, a vessel reaches the point of neutral stability. There is no restoring force and the vessel will roll over. When synchronous rolling occurs a vessel can be pushed to this ultimate angle very quickly, and without much warning, over the vessel goes!
To assist in identifying these potentially dangerous conditions the Marine Prediction Center (MPC) produces a Wave Period Charts twice each day (00Z and 12Z). These charts are found on the MPC web page (http://www.opc.ncep.noaa.gov) and are broadcast via Coast Guard Wefax. It is important for mariners to examine both Sea State Charts (height measured in feet and meters in three-foot increments) and Wave Period Charts (wave periods in seconds) to determine if a present or projected track will provide safe and comfortable conditions for a vessel, its crew, and cargo.
Because of the shape of the hull, the athwartship (left or right) center of buoyancy (the "B" red vector line) has moved from its upright position on the centerline to a point left and outboard of the centerline as shown in the first diagram. Nothing has been done to change the weight of the vessel, of course, and it acts vertically downward through the vessel's center of gravity. The equal and opposite force of buoyancy acts vertically upward from the heeled center of buoyancy. The horizontal distance between the two forces is labeled GZ and is called the righting arm. One can see that the restoring moment, called the righting moment by trawler designers and termed RM, is equal to the product of the vessel's weight times the righting arm:
RM = disp x GZ
Since the forces of buoyancy and gravity are equal and act along parallel lines, but in opposite directions, a rotation is developed. This is called a couple, two moments acting simultaneously to produce rotation. This rotation returns the trawler to where the forces of buoyancy and gravity balance out. By definition, the point on the vessel's centerline vertically above the heeled center of buoyancy is the metacenter, labeled M in the first illustration. Look at the first drawing and you can see that the righting arm, GZ is directly proportional to the metacentric height, GM. Double GM, and you will double GZ and thereby double the righting moment. The formula for this relationship is:
GZ = GM sin 0
To compute RM or GZ as standalone numbers does not make sense. A complete analysis would require generating the entire series of numbers for all expected angles of heel for you trawler, then plot these numbers with the RM value to create a graph. This graph would then represent the ability for your trawler to right itself given specific roll conditions.
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For small angles of heel (0o through 7o to 10o, metacenter doesn’t move), the value for the trawler’s righting arm (GZ) may be found by using trigonometry:
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What does all of this mean - a trawler can be made more stable by lowering G, by raising M, or both; anything that increases the distance (GM) between the two. In other words, keep the center of buoyancy (the red "B" vector) as close to the center of gravity (red "G" vector) as possible. With our new understanding of buoyancy and center of gravity in place, we can begin to see what type of motor yacht our readers would favor for offshore cruising. Clearly these are not planing cruisers with beams of 12' and top speed in excess of 15 knots. They are large, heavy trawlers with high static and dynamic stability, which will produce an easy ride with minimal crew fatigue.
This section presents a number of formulas to determine factors that affect stability, speed and therefore crew comfort.
DISP / LENGTH RATIO = disp/2240/(.01*lwl)3
M/V Tortuga DLR = 57000 / 2240 / (0.1
* 46.5)3
DLR = 254.46
Probably the most used and best understood evaluation factor. Low numbers (resulting from lightweight and long waterlines) are associated with high performance and quick response. The general trend for new boats is towards lower ratios that favor higher performance. The trade off is that a light boat will have more violent motion in storms. This requires constant attention to steering, resulting in crew fatigue. The ratio decreases with boat length, since heavy boats need less ballast and will be lighter than smaller boats with the same stability. Cruising designs begin around 200 and can go up to the high 300's. Many racing boats are below 100. A suggested approach is to start with a minimum of 230, optimal of 280 - 320, and maximum of 370. This will give us trawlers with a nice blend of weight (for good load carrying capability and seaworthiness) and reasonable performance. These numbers may seem high to some, but they are in agreement with the experience of many blue water sailors.
HULL SPEED = 1.34*lwl.5 ,knots
M/V Tortuga Hull Speed = 1.34 *
46.10.5
Hull Speed = 9.1 knots
Generally regarded as the highest practical velocity for a displacement boat assuming a reasonable power input (2-3 hp per ton). As a trawler's speed increases, the wave it creates becomes longer, creating a trough that moves aft. At hull speed, the trough will be as long as the waterline length, creating a "hole" that the vessel just fits. An enormous amount of power (50-100 hp / ton) is required to "climb out" of this hole and transition to higher speeds (planing). The suggested value is 7.6 knots.
M/V Tortuga LB Ratio = 48.65 /
15.1
LB Ration = 3.22
This ratio measures the fineness of the hull. Fine hulls, having ratios of 3.0 - 4.0 and higher, are long and slender which promotes easy motion, high speed (low drag), and good balance when heeled (sail boat that is). Many newer designs favor wider hulls which have larger interior volume, sail flatter, and have high reaching and down wind speed potential. One note of caution when making comparisons, longer boats tend to be finer then short ones. The suggested value is 3.4, which is fairly fine. Fine hulls tend to be well balanced and have low inverted stability.
CAPSIZE RISK = beam/(disp/(.9*64))0.333
M/V Tortuga CR = 15.1 / (57000 /
(0.9 * 64))0.333
CR = 1.52
An empirical factor derived by the USYRU after an analysis of the 1979 FASTNET Race. The study was funded by the Society of Navel Architects and Marine Engineers (SNAME). They concluded that sail boats with values greater than 2 should not compete in ocean races. Values less than 2 are "good". The formula penalizes boats with a large beam for their high inverted stability, and light-weight boats because of their violent response (low roll moment of inertia) to large waves, which are both very important during violent storms. It does not indicate or calculate static stability. Some modern coastal cruisers and many racing designs have problems meeting this criteria. An interesting note, the study concluded that static stability was relatively unimportant in predicting dynamic capsize. Beam and weight were much more important factors. Wide boats give waves a longer lever arm to initiate roll and light weight boats require less energy to roll over; both undesirable attributes in a cruising boat. The suggested value of 1.7 is very low. Anything over 2 is an unsafe trawler.
COMFORT FACTOR = disp/(.65*(.7*lwl+.3*loa)*beam1.33)
M/V Tortuga CF = 57000 / (0.65 *
(0.7 * 46.1 + 0.3 * 48.7) * 15.11.33
CF = 50.58
An empirical term developed by yacht designer Ted Brewer. Large numbers indicate a smoother, more comfortable motion in a sea-way. The equation favors heavy boats with some overhang and a narrow beam. These are all factors that slow down the boat's response in violent waves. This design philosophy is contrary to many modern "racer / cruisers", but it is based on a great deal of real blue water data, not just what looks good in a boat show. A value of 30 - 40 would be an average cruiser. Racing designs can be less than 25, and a full keel, Colin Archer sail hull design, could be as high as 60. The suggested value of 36 indicates that comfort is a high priority on cruisers.
MOMENT OF INERTIA (I) = disp1.744/35.5
M/V Tortuga MI = 570001.744
/ 35.5
MI = 5,546,480 (Still working on determining a suggested value)
An empirical term developed by SNAME. Large values resist rolling forces. The moment of inertia is very sensitive to the distance items are from the CG. A heavy rig can greatly increase I, with little impact on displacement.
Horsepower = Displacement / ((150)2 / (Hull Speed)2)
M/V Tortuga = 2500 / (22500 /
29.16)
HP @ Hull Speed = 3.24
Determines the theoretical horsepower required for a displacement hull shape to reach hull speed. This formula assumes that there is no current or wind resistance and the hull is clean and free of objects that could cause excess drag. As can be seen from this equation doubling the speed requires 4 times the horsepower so, if you set the speed down one half of hull speed (2.7 knots), the energy required will be reduced to just a little more than 3/4 horsepower.
References
http://www.johnsboatstuff.com/Articles/design.htm
http://www.sailnet.com/collections/articles/index.cfm?articleid=carrmi0031
http://www.boatdesign.com/postings/pages/stability.htm
http://www.johnsboatstuff.com/technica.htm
http://www.johnsboatstuff.com/Articles/estimati.htm
http://frances.phy.cmich.edu/people/osborn/Physics110/book/Chapters/Chapter7.htm
http://www.curtin.edu.au/curtin/centre/cmst/publicat/2001-35.pdf
http://www.dynagen.co.za/eugene/hulls/index.html
http://www.fas.org/man/dod-101/navy/docs/swos/dca/stg4-01.html
http://home.clara.net/gmatkin/design.htm
