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Tom
Big Wall climber
San Luis Obispo CA
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This is my take on the equation for an exponential spiral, which differs from John's by the use of the tangent function, instead of the sine function.
EDIT:
Here's the exponential spiral:
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Brutus of Wyde
climber
Old Climbers' Home, Oakland CA
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"They do tend to flip sideways when walking. 2/3 weeks ago, at IC, one of my friends used the #2 as a 1st piece, bud did not extend it (not even the double sling). After being jostled by the rope during the full ascent, it flipped. (Note he's an experienced trad leader). Well I pulled the cam out when I got up hanging on a camalot... It took me 15 minutes, and it was possible only because there was enough room to slide my nut tool where the lobes press against the wall to slide them (otherwise they bite the rock). The crack width was a perfect #2... "
I had a #0.5 flip as well, as a result of pushing the cam deeper into the crack... Reset it, on lead, using the nut tool in the outside holes of the trigger bar. It was far easier to clean this cam than any other I have ever skewqed up a placement with. the trigger bar holes are a beautiful feature I would like to see on more cams.
Brutus
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healyje
Trad climber
Portland, Oregon
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Yeah, once you place a Max Cam it is imperative that it not move - they have to be slung appropriately or they will like end up in a less than desirable state after any jostling. They likewise should never be "pushed" further or adjusted after placing one. If you want it somewhere or someway different then pull it and re-place it. They have some unique attributes but you can not and should not treat them like a typical symmetric cam.
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Shu Pong
climber
Arlington, Washington
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That whole tangent sine thing made me start thinking. because they can't both be right. So I first tried to figure it out mathematically, before I finally realized that I have no idea how to do it that way. So I used matlab to make a plot of one spiral using tangent(red) and another spiral using sine(blue). At first I did it with a 15 degree cam angle, although it's obvious that they are not the same, it's still hard to tell which one is right. So I bumped up the cam angle to 45 degrees and suddenly it becomes quite obvious that tangent is definitely the correct function to use. If you use sine it is still a logarithmic spiral but it does NOT have the cam angle that you think it does.
The plot below is generated with the equation:
r = e^(mθ)
m is the trig ratio we use to input the cam angle
I used m = sin45 for the blue spiral and
I used m = tan45 for the red spiral
At 45 degrees, the two are different enough to easily see which one really has a cam angle of 45 degrees
What we find is tangent wins, the blue one is still a good curve but it actually has a cam angle of about 35.26 degrees, not 45.
The problem is when you are down in the 13-14 degree range the difference between tangent and sine is quite small and hard to notice the difference.
In the first post there are some measurements for the max cam that come out to a 40 degree cam angle on the small cam and 16.5 on the large one. This means the actual cam angles would be in the neighborhood of about 32.7 degrees and 15.8 degrees. You can see how much smaller the difference is for smaller cam angles.
It's funny because I was using sine also, it was the formula given in my a*#ignment. I just finally figured this out last night and my assignment was due today. It didn't actually make much difference in the assignment, so not much to redo, but I showed the prof who promptly spent the first 10 minutes of class studying the plot I had given him.
So thanks Tom for posting your formula or I probably never would have noticed the error.
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Tom
Big Wall climber
San Luis Obispo CA
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Jun 10, 2006 - 08:34am PT
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That whole tangent sine thing made me start thinking. because they can't both be right.
No, they can't. The difference between the sine and tangent of 14 degrees is about 3%. Using the sine would result in a smaller camming angle than what was intended; the tangent of 13.6 degrees equals the sine of 14 degrees. Similarly, if you take the sine of 45 degrees (0.7071) and then take the arctangent, you get 35.2644 degrees, in accordance with Shu Pong's diagram above.
The correct equation uses the tangent of the angle, not the sine. This can be seen by deriving the equation from first principles; all that is required is we define the curve to have a constant radius-to-tangent angle for the entire curve. When we do that, the exponential spiral results. And the tangent of the angle appears as the constant (shape defining) parameter.
Refer to my earlier post, for a diagram showing how the variables are defined.
The first equation below is valid for any 2-D plane curve expressed in polar coordinates, regardless of whether the radius-to-tangent angle is a constant, or it varies. Also, if deriving this equation from scratch, it would be a rather subtle mistake to use the sine function instead of the tangent function.
Notice that for the special case of the radius-to-tangent angle being 90 degrees, the tangent function becomes infinite, the exponent in brackets becomes zero, the exponential function (e^0) becomes one, and the radius is a constant for the entire curve - a circle.
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deuce4
Big Wall climber
the Southwest
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Topic Author's Reply - Jun 13, 2006 - 12:23am PT
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Here's an article of mine explaining constant camming angle cams (written when I was more of a geek):
http://bigwalls.net/climb/camf/index.html
(I just reworked the webpage as the original html code was from the early 90's and the illustrations did not appear correctly-fixed now). Note: still haven't fixed the superscripts in the text--see the illustrations for proper format of the equations.
Back in the day, I did study friction of various rocks, and found that generally granite had a coefficient of friction with aluminum of well over 0.30, and typically, if I recall correctly on the order of 0.32 or more. Sandstone was hard to measure as there are shear planes associated with the granular stucture, but the friction (with aluminum) was generally lower. Walt and I did some tests in Sedona one year with various alloys, when we also tested an assortment of sandstone bolts with my hydraulic puller.
It turns out mining engineering texts are a good reference for the frictional characteristics of rock, as many of the spring bolts used for mining are dependent on friction.
What happened after the era of Friends is that manufacturers started to play with variable cam angles (ones that varied along the arc of the cam) in order to increase range. I think Metolius were the first to experiment with this technique.
ShuPong's illustrations are nice and look like they're drawn by a fine engineer.
hope this helps
cheers
JM
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Makwizard
Trad climber
durham
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Oct 16, 2008 - 05:32pm PT
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I am currently conducting research at Duke University to analyze and redesign cam lobes. The end goal is to improve a cam's ability to hold in soft rock and flaring cracks. I have just begun my research but will be continually posting updates of my findings on my [url="http://www.duke.edu/~mak25/research.html"]cam research page[/url].
Thanks.
[url="http://www.duke.edu/~mak25"]my homepage[/url].
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couchmaster
climber
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Oct 16, 2008 - 11:18pm PT
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Looking forward to it!
Thanks
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reallyy big star
Social climber
some, place
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Feb 12, 2018 - 06:16am PT
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the free-body diagram of my soul
requires words, no lines or greek symbols:
my destiny is a vector;
normal to my dreams.
thus god recalled my soul;
but no one turned it in.
for all whom employ it
in its broken state
enjoy a pleasingly misguided journey.
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