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Zach wrote:The lever arm works out pretty close to a full 5 gallon tank 15 feet away from the center of gravity = 600 ftlbs
A 20 gallon tank 10 feet away = 1,600 ftlbs
Zach,
You have just hit on something that I would like to know how to calculate. I was trying to work backwards from your numbers to figure it out, but I realized that there must be (?) a height component too (well, maybe not; might be only in my mind).
I'm not expecting a physics class here, but is there a reasonably simple formula you use? I have the basic concepts of lever arm and etc.
Like say you are comparing a 25# anchor, 20 feet ahead of the center of mass, and 10 feet above it (if there is some kind of horizontal "line" in there), or a 50# anchor 5 feet ahead of the center, and 5 feet above center? Or, as you might already have figured out, a 350# engine 10 feet behind the denter and 4 feet above it vs. a 75# engine 18' behind and 8 feet above?
Just let me know if that's too much to ask, and I should just go take a physics class ;D It just sounded like something you might have a "quick" calculation method for.
I simplify it a bit, as in the case of a triton tank, the fuel at the aft half some two and a half feet forward of the front half... levers differently. My example was on a 2d level.
The math there is the distance from center of gravity, in feet, multiplied by the weight of the object. I used 7 pounds for the weight of the fuel, if memory serves.
The physics lesson about height:
But the gist of the idea is the boat has a center about which it rotates. This is a point inside a triangle, the triangle being made up of the center of gravity, the center of buoyancy, and the center spot on the boat where the sail pushes. All this works out to be a force vector, think about a plus sign where straight up is the center of buoyancy, straight down the center of gravity, forward is the force of propulsion, and backward is the force of drag.
So, somewhere is a center point, that can be calculated and measured that the boat rotates, pitches, and yaws around. You've probably noticed that the boat behaves differently under different wind conditions, wave conditions... and when she's at rest. So, whatever center point this is, only matters for that point of sail, day, moon cycle.
For us, center of gravity is close enough for the purposes of figuring where things ought to go... So to calculate foot pounds all you do is find the design drawing of the boat, and measure away from this point in feet and multiply the distance by the weight of the object.
That works real well for stuff down low in the boat...
To apply this, the thought process is that the longer the lever arm something has the more influence it has on the boat as it moves. This means that when the boat rocks, it will rise further and settle further. Strangely though, a long heavy lever arm means that it will pitch slower.
A good example of this is a small skiff with a battery and gas tank way up forward in the bow. Without weight as far forward in the boat, it doesn't have enough weight to keep the bow from bouncing like a basketball every time its force vector (that plus shaped formula...) changes. When it hits a wave, the bow pops up in the air. With no weight to keep it down, more weight is transferred to the rear of the boat... making the bow go higher. Now the height of the bow from the water surface is greater, and the boat accelerates downward with more force... This bounce, has a frequency... and a waveform... just like the waves on which the boat rides.
The more weight, and the more lever arm the slower the boat responds to influences. The slower the frequency... but the wave form can be higher.
To much weight up forward on a planing boat makes her porpoise, her bow rises... then it falls... and when it falls, she feels squirly and the prop may ventilate... and then her bow goes back up, and she may bog down. The cure for this is to either move the weight aft... or to put trim tabs, and adjust that old plus shaped force vector, and have more pushing up on the aft end to keep the bow from bouncing up as high on each of its cycles.
Another good example of this, is to think about two boats that are of the same plans, but one was planked with a much heavier, denser wood... and because of this, had some of her ballast removed to keep her on her lines. Compared to her otherwise identical weighted sister... The first, heavier planked boat will be more sea kindly... or as a racing sailor might say, ponderous. Or as a friend likes to say, sails like a diaper...
Starting to get into height...
Why is this? For one her center of gravity is higher, not only because she has less ballast down low... but also because her planks one foot in away from the center of gravity, influence a lever arm greater than the same stack of boards on her sistership. Two feet ahead, the difference is compounded. Fifteen feet ahead... Run the math on the area of each station, multiply by the specific gravity and you've got a ratio to calculate out a percentage difference...
I ran into this, when I replanked and framed 65 feet on the side of Noel with salt treated, kiln dried #1 pine... where she had red fir previously. Red fir is .53 and kiln dried pushes .67.
She had listed to port, on account of a generator... Afterwards, she sat even in the water. Her beam is 16 feet... so the difference in weight of the two bundles of boards it took to plank her, was enough to counter the weight of an 800lb generator 4 feet from her centerline. Neat, eh?
Now as far as height goes, your lever arm is now a force vector! (Well, it always is but now you can really tell a difference by the seat of the pants.)
Lets say I build a fly bridge on a boat that doesn't have one. For our purposes lets put it so it extends exactly equal from the center of gravity on both sides, symmetrical so it counter balances its self. Does the boat react the same as if it were not there? Nope... Your center of gravity has moved up in the boat because of the tower. This means she may not feel as tender when you step aboard... but once she starts to roll, she has less reserve to keep her from rolling. When I insulated Noel (fiberglass batts... pretty standard on the ye olde wooden cabin boats.) there was a decent change in her motion on board.
Sailboat terms first... Lets put 1 pounds on top of a 30 foot mast. Lets make the boat a shoal draft, deck stepped mast. Lets call the center of gravity two feet below the cabin top, so the lever arm is 32 feet long. Lets say the boat has 1000 pounds of lead, 4 feet below the center of gravity.
What do we know about our boat? We have a levering force of 4,000lbs available from that lump of lead. (In real life it is less than that, because the lever arm is less for the top of the keel than the bottom...) Our lever arm for that 1 lb pidgeon sitting on top of the boat, has 32 feet of lever. This means that the boat feels 32 pounds of force. Our levering force of the keel now has competition from a bird. The levering force of the keel is now 3,968 pounds. This is the real reason why racing sailors get pissed off when pigeons try to land on the windex. This is why folks pay so much for light weight rigging and spars. It isn't the sum total of the weight that takes away the righting moment of the keel... but 2 foot above the deck, that piece of cable weighs twice what it actually weighs. 30 feet up, its 30 times as heavy to the keel... So if the first foot weighs .236, the second foot weighs .472, and the 30th foot weighs 7.08 pounds. It doesn't take long to make a real heavy rig.
That is what makes one boat stiff, and another heel over. (Given they are otherwise identical...) Strangely though, weight aloft can be a good thing.
It is fairly typical for small tugs around here to have two 55 gallon jugs on their roof, some 15 feet above the bottom of their flat bottoms. We have a lot of inshore barges... hence flat barge like tugs. Grin.
Anyway, by putting one half full of water and the other half full of diesel they have a lot of weight as high up as you can put it. All this weight, slows down the rate which they rock and roll in the waves. A flat barge in our sounds which are quite wide and shallow (Deep being 15 feet...) the frequency of the waves is quite close together. So, a lot of weight up high slows down the rocking... and makes things less like puke machines. If they were offshore, all that weight up top would make them a wee bit more likely to turn turtle.
Back to sailboats, lets say someone converts a boat that has a 600 pound wood mast and puts a light weight aluminum stick on her. While the aluminum sticked boat is going to be more fun to race around, as you can push her harder before she heels over into the slipping sideways, weather helm inducing frenzy that makes for a fun sail. At anchor, the wooden stick is going to slow down her roll in all directions... though she may roll further with each move every little wake and skiff isn't going to move her much.
Go under the boat for a bit, and wing keels and deep fins start making sense. Take the same weight lump of lead and move it down a foot. If it weighs 1000lbs, and its 1 foot below the boat it has 1,000 pounds of righting moment. Put it on a wee skinny fiberglass stick 10 feet down and it makes 10,000lbs of righting force.
A funny...
I saw an interesting thing one night, enjoying a beer by the dock... the water looked like a lake, not a breath of wind and slack tide. With a couple of loud squawk a pigeon announced its arrival. Sitting down on a dock line of a 50 some foot sport fishing boat. The dock line was about 15 feet long... this little bird sat down in the middle. So with 7 feet to each side, the lever arm was equal... But because it divided the two points, its 1 pound acted like 2. Moving the boat, as the bird slowly lowered towards the water... startling its self and taking off into the night. I was one beer away from bursting a laughing muscle at such a sight... This same principle is how us lowly mortals can haul up a sail tighter than the wee small day sailing mast mounted winches will haul up a halyard. With an S turn on the cleat, enough friction is present with us tugging on it with one hand... That we can tug with 2:1 pressure on the halyard by pulling straight away from the mast, taking up the slack faster than the sail can take it back.
It's also how I get aboard Noel at high tide... I step on the dock line, and let my 200 pounds, turn into 400 and move her 160,000 pounds over to the dock when my tug won't bring her over. Then again, there are days when stepping on a dock line. It is a little funny feeling when its howling, to step on a line have it come down... then be lifted back up and have to cling to a piling like a bilge rat...
Tikvah, I apologize for the thread-jacking :blush: On the other hand, I envisioned Zach posting a couple of wee-short formulae and being done with it ;) (ha ha)
Tim, is there a good way for me to make this and the prior two posts a new thread? I mean, besides me copying and pasting the content? Maybe it could go in Boat Nerdery? Should I start a thread there to receive this content? I will continue for now presuming we'll move this to hide my faux pas.
Anyway, thank you, Zach. I had a basic grasp on some of the concepts you explained, but more examples made them clearer. The weight aloft has for some reason always been clearer to me than the idea of moving a heavy weight both up and forward or aft. Maybe that's because the mast is more-or-less at the center of the "plus" (fore and aft and athwartships, but not up and down)?
In the engine example, are you saying we can get away with calculating only the mass of the engine times the feet aft of the center, and that with only a few feet of up/down we don't need to calculate that?
So a 350# engine 7 feet aft of the center and on a line with it is like 2450#
And a 75# engine 14 feet aft of the center and 3 feet above it in the lazarette (which we don't count?) is like 1050#?
Somehow the latter still feels to me like it should matter that it is not as low in the boat as the former.
And also "weight in the ends" counts as "worse," right? I mean, more than just the weight we calculated?
Ahh, I think there are several different things being written about here.
Center of gravity, and the difference that weight and location make, starts out easy: force x distance of every little scrap of the item can be figured and balanced; the balance point is the CG. It's calculable in 3-D. For an object like a boat, this CG has to be calculated in 3-D. Recall that boats/ships which are modified or added to can change their seakeeping qualities in awful ways if the modifications are not considered in 3-D. In the interests of overall weight reduction, weight aloft in a boat is very important. A pound at 50 feet has to be balanced by 5 pounds at 10 feet: save the pound aloft and you save 6 pounds in the keel.
Inertia is that tendency of a mass to keep moving in the direction and speed that it was. This is what makes a light boat more responsive than a heavy one. Quetzal, with her 16000 lbs, needs time and a lengthy gust to get moving; a 350 lb Flying Dutchman will accelerate quickly enough to slide her wet-bottomed crew aft (fun but unsettling!).
Polar moment of inertia is that tendency of a mass to keep rotating about its pole (a handy axis through its center of gravity) in the direction and speed that it was. This is the bit that's important with 'weight in the ends', 'weight aloft', 'weight location in a keel', and even 'weight out at the sides' of a boat. A little additional weight at a distance from the CG will change the polar moment alot more than that weight would do at a lesser distance. (More than anybody'd want to know: http://en.wikipedia.org/wiki/Moment_of_inertia.) An increased polar moment means that the boat will pitch or roll more slowly. You want the boat to pitch advantageously and that might mean more or less. In light air, dinghy sailors might well choose to sit closer to the ends to keep the sails from being disturbed by pitching. More often, we want the boat to lift to waves so as to not punch into 'em. A sailboat that's lost her rig in a storm is much more uncomfortable for her crew; the ballast snaps her back to vertical.
For us on this list, our boats are likely to be very massive and will tend to behave as their designers intended; most of us don't change enough stuff to affect things really badly. Now, add a chain rode, a couple CQRs, and an anchor winch...or change out that nasty heavy aluminum stick and put in a nice light taller carbon fiber one. Or load a couple thousand pounds of stuff for long distance cruising.
Ah ha! I had a moment of silence till tikvah pointed me towards this... thought it disappeared in cyber space!
Yup yup, you've got it right Rachel on the center of gravity on our size boats, and even stuff the size of Noel's 83 feet... So long as the majority of the weight is within a foot or two of the waterline to each way, and on centerline its not a huge deal. Lower is better, and closer to the center is better...
In the case of my 350 pound engine 5 feet or so aft of the center of gravity on Pylasteki vs the 75 pound outboard hung on the port side of the transom she rose up in the back enough to almost flatten out her shear line to the waterline, but being that the little 6 horse was only a foot off centerline... she didn't heel over much. What I wouldn't want to do, is strap 75 pounds to one side of the cabin top... or load 3 or 4 water cans way out to the side deck on only one side as she will list much more than if down on the settee inside. This is the reason why rail meat works better the further up and over you can get it... and the reason why you can swing the boom way out perpendicular the boat and shimmy out on it to get unstuck when aground.
Listing is one of the things that is fun on Noel at times, as a sheet of 3/4 marine plywood weighs 66 pounds... and if you have 10 sheets on board, laid along one side of the cabin there is a noticeable change in the angle of the world. Being that its about 6 feet from the waterline to the deck... that plywood is centered up on 9 feet off the water. Noel has no ballast, other than a pair of huge cummins engines and a steel shoe on her keel, so her center of gravity is at or above the waterline. She rocks and rolls more with less plywood on board, than more... but that is just because of a thousand pounds off to one side. Her final stability would be compromised...
Back to fore and aft, and height...
She became much more tender. When my 200 pounds steps on to the port side bow chock, the boat behaved like a tank with the engine and fuel tank in place. With the inside gutted, and the wet balsawood cleared out of the deck... she acted like my little 23 foot AMF day sailer. Walk down the side decks and she heeled over, step on the stem and the rear end propped up out of the water. Unlike my 23 footer... if you grab hold of a shroud and swing your weight to the outside, she only heeled over 10 degrees or so before stopping and the keel said... "No way..."
A lot of this was the lever arm of having a lot of weight aft, the other half was as she came out of the water her water plane was reduced. So, with less in the water it takes less to push down her ends. Her bow goes down much faster than the stern anyway (pointy end...) as the rear end has a larger water plane. Think about pushing down a slab sided barge, vs a wine glass hull with a flared bow. If you move up 3 inches you lose quite a bit of beam (by %) for every inch you go up in the bow. With her nearly gutted, the top of the rudder was only an inch or so below the water.
Back to the engine:
So, lets say the engine is 1 foot below the center of gravity, that would be 350 pounds of ballast. If you want to noodle over something, think about inch pounds for a minute. If you are tightening a wee small fastener, or bending a thin piece of balsawood you are dealing with weights less than 1 foot. So, you could calculate the inch pounds of loading by multiplying the weight by the number of inches below the center of gravity. In this case, thats 12 inches. 4200 inch pounds... To get foot pounds you divide by 12, and come back to 350.
Lets move port or starboard of center, like an outboard on the back.mounted off 1 foot off center, it has a 75 foot pounds of levering force to heel the boat over.
Whoopdeedoo... Except lets think about the force of something like a rudder. Noels rudders take 23,000 inch pounds to turn at 35 degrees from straight ahead. (Over 35 the rudder starts to stall.)
Why inch pounds? Because you can calculate out the area in inches on something funky shaped like a trapezoidal rudder. This is fun, as that means held hard over at speed it takes 3,833 foot lbs of force to hold her hard over. From here, you can calculate what sort of steering gear it takes to hold everything in place
What does this mean for a sailboat? Well, if you have a tiller that is 4 feet long, and it takes 50 pounds of force to push her hard over... You have 200 foot pounds of turning force, acting against the center of that polar diagram.
Applied: Pylasteki had a pretty decent amount of weather helm. She had a wood rudder. Wood in water is buoyant... meaning that it floats. Part of what makes a monohull have weather helm, is that when heeled over you are having to hold the rudder down as it wants to float up. The further over you heel, the less weight is carried by the pintles and gudgeons and the more the tiller has you pick it up. So... where is one place that heavy parts on a boat sound kosher? Rudders!
For me, I'm plotting a flat plate stainless rudder out of 316L on a piece of aqualoy 22 shaft... Part of this is to put some more weight down low, so while I am putting in the lower bearing, I'm thinking to open up the false keel and slide in a few ingots of lead and get a bit more ballast down low, far enough back to give her a bit of a lever arm so when I step on the bow from the dock she feels like a tank again. (Grin)
