The Trim Equation
To truly make sense of this, you will have to had to have read my essays in Dynamics – Surfing the Force. Perhaps its not completely necessary to have read them all, but to have at least looked at some of the illustrations is bound to help.
I was hoping to pop this essay at the end of the Dynamics – Surfing the Force, but the posts in the thread where straying off topic, as post often do, so I felt it best to start a Part II, so to speak.
So here’s the trim equation,
Rtrim = W [sin(roll angle)/cos(pitch angle)] i + W[sin(roll angle)/cos(pitch angle)] [ 1/ tan(yaw anlge)] j
Lets surf. The example is a surfer in trim, perhaps in the tube.
For those who have been exposed to vectors, this should all be pretty straightforward. (Please check my work!) For those who haven’t had the pleasure, this can still be understood, but it will require that you at least understand how I define pitch, roll and yaw, see figure 1. Remember that the bottom plane of the board is perpendicular to the propulsion vector. (If you can remember your HS geometry you should be fine.)
By the way, I figure that the way you find out if you are right, at least in the math, is by putting ‘it’ out there and letting those sufficiently interested, go over what you’ve done and point how wrong you actually are, or in general how much of a mess you’ve made. Since you need a forum (a bunch of potentially interested people) for such a thing, Swaylock’s seemed like the place, given the topic (and my resources.)
I’ve made some assumptions in order to keep the confusion to a minimum.
First, the amount of surface area that produces the propulsion remains constant. Second, the surfer is in trim, neither dropping nor rising on the face –i.e. he’s holding a horizontal line. Third, drag is not considered, nor are fins. But they’re obviously there. These are admittedly important things to leave out. But I feel that, by not including them at this point, it will not take away from the point to be made, which should be apparent soon enough. They will likely be other assumptions that I have made and have forgotten to state, and perhaps you’ll be kind enough to point them out, but my strategy has been to simplify in order to convey the bigger picture.
If you make the assumption the surfer and surfboard weight is balanced by the vertical component of propulsion, then what’s left is basically the sum of the propulsion vector components in the x and y directions (see diagram.) That is the surfer will be moving both towards-the-beach and down-the-line. Under these conditions, the orientation of the surfboard that of pitch, roll and yaw is indicated in the diagram (again, you’ll have to use a little geometry to get there.)
Now with the vertical component of P, that is Pzk being equal to W (which is, as you may have guessed, mass times the acceleration due to gravity), and with some substitution, you get the following for the propulsive force left after you balance its vertical component with the weight,
Here the simplified vector equation is
R = P + W
we have trim so,
Pz = P cos(pitch angle) = W
so, we now have
P = W/ cos(pitch angle),
which eventually gives,
Rtrim = W [sin(roll angle)/cos(pitch angle)] i + W[sin(roll angle)/cos(pitch angle)] [ 1/ tan(yaw angle)] j
or,
force in the towards-the-beach direction = W [sin(roll angle)/cos(pitch angle)]
and
force in the down-the-line direction = W[sin(roll angle)/cos(pitch angle)] [ 1/ tan(yaw angle)]
By the way, if you’re wondering why yaw wouldn’t impact the force towards-the-beach, it does, see diagram, the effect is just sort of buried.
So, you’re in the tube, but you sense you need a little more down-the-line speed, what do you do? From the above relationships, it would appear you’ve got a few options.
You might consider decreasing you yaw, or trying to somehow pull the nose towards the face of the wave. This would reduce tan(yaw angle), which amounts to increasing 1/ tan(yaw angle), which would give you a little speed in the down-line-direction. (By the way to see how the cosine, sine and tangent functions change with the angle see figure 3, its crude the actually relationships are curves, but its enough.)
Or you might consider increasing your roll. Now the problem with that strategy is that roll will also send you towards the beach a lot faster, something which is likely to get you in trouble. You, being the computer you are, decide the roll option is not the best one and grab your rail pulling up to prevent it.
What’s left, increase your pitch, thereby decreasing cos(pitch angle)? Maybe, but possible a little dangerous, for your current pitch is just about right to balance your weight against the propulsive force. You decide not to go for too much more pitch, but it’s a nice option to have.
Your solution? Grab the rail and try and swing the nose towards the wave face (decrease yaw.) You accomplish this by ever so gently trying to torque the board about its center of mass by placing a little more weight on you rear foot, and given you’ve grabbed the rail in front of your center of mass, pull a little towards the wave.
But you know that by screwing with the yaw which, given that you’re in trim, will likely impact both your towards-the-beach and down-the-line velocity (see diagram for relationship), you pull extra hard on that rail, and kept the pressure up on that rear foot. You want to keep roll to a minimum.
Are these the only options, in my simplified example probably. But obviously there are more.
Cameras click.
Does this prove all of wild and crazy notions regarding propulsion, propulsive flow versus apparent flows, etc. as developed in Dynamics – Surfing the Force? Hardly. If you buy into my model this is what you get.
What’s next? You actually have the nut, though perhaps not trival, its would not require all that much to start adding the other (vector) terms, such as drag, maybe the propulsive component of kant and toed lateral fins, or the consequences of tail flips and rocker on the propulsive component, even concaves and vees and the other strange things that are done to boards, that is you buy my nonsense about propulsion in surfing.
Much of what now needs to be done however is building functions –e.g. collect data on drag as a function of orientation, or on propulsion as a function of orientation.
But that’s the nut.
Kevin
PS
I’m sure I’ve made a mistake. Perhaps someone would be kind enough to find it and let me know where.
(edit: I had ‘I sure I’ve…’ my apologies. I’ve corrected it. )
