Sunday, January 23, 2011

Rocker Table Pt2 - Designing the Jigs.

Rocker Jig Design

At this stage I already had in mind the type of rocker line that I wanted so I made the jigs as a first step. If you are making an adjustable table then this can naturally wait until later when you've decided on all the design parameters.

I wanted to make a continuous rocker. That is, a rocker line that is a segment of the circumference of a circle - no flat spots or rapidly curving tips. I tried an M8 board that had a continuous, generous rocker and it handled really nicely in the choppy conditions we get locally. Also, after listening to Brokites video and seeing that they use a continuous rocker as well this seemed like a worthy first attempt.

It was also very easy to create the exact line using MS Excel as continuous rockers are just part of a circle. What I like about the idea (and lets see if it actually works in practice) is that if the board has sufficient flex in it the curve between your feet will flatten out when you ride because of the pressure exerted by the water in this region. This should help up wind ability and popping. It all hinges on getting the right amount of flex - which at the moment is a bit of a black art for me....but trial error will reveal all.

i) Work out the radius of the circle that traces out the rocker line you want.
There is a bit of maths need but it straight forward even if this explaination is a bit long winded.

It's based around the equation of a circle with centre at the origin (coordinates (0,0)).

The general equation is x^2+y^2= R^2,

where x - x coordinate, y- the y-coordinate and R is the radius ( '^2' denotes squared)

The rocker line will be that part of the circumference of the circle the sits at the very bottom and lies symmetrically about the y-axis. We only need to work out the coordinates of a single point on that circle to work out the radius. Once we've got this we can then plug it in to excel and get it to spit our all the points along that curve.

The point we know is on there has a x coordinate of half the board length L/2 and has a y coordinate that is equal to the y coordinate of the lowest point on the circle (0,-R) PLUS the amount of rocker you want to have in it, call this 'r').This point is at the tip of one end of the board. Put this in the formula for the circle and you get

(L/2)^2 + (-R + r)^2 =R^2 - (1)

where L = length of the board, R is the radius the circle on which the continuous rocker line will lie, r = amount of rocker at the tip of the board.

A little rearranging of this and you get

R = L^2/(8r) +r/2 - (2)

Remember that r is going to be about 0.04 m ( 4 cm) and L=1.3m. This means that the L^2/8r is going to dominate.

EXAMPLE: Using (2) for a 130cm board with 4 cm rocker then the radius is 5.3013 m.

We now have everything we need to get excel to plot out the rocker line.

ii) Generate the coordinates in excel. In excel create a column of x coordinates starting at about 5cm longer than the half board length L/2 you board. So for a 130 cm board start at 70. Reduce this buy 0.1 each row down so that you have the x-coordinate for each millimeter of the board. Take this down to -70

In the column next to the first x-coordinate (say its cell A1) enter the following formula

= R - SQRT(R^2-A1*A1)

where 'R' is the radius in centimeters calculated above using (2). This will generate all the x, y coordinates you need to plot your rocker line so that the center of the board sits at the coordinate (0,0).

iii) Chart it in Excel - Highlight all the data and generate a line graph and there you have your rocker line.

iv) Rescale the chart in both x and y direction so that when you print the chart so that 1 interval on the y axis is exactly one centimeter when you measure it after printing. Same thing with the x- axis. The only way I could think to do this was to zoom way out (20%) so the graph shrinks on the page to a business card size. then just grab the corners of the chart and drag them out. Deselect the chart and in the page setup menu set it to print 1 page high and 6 pages wide. Print off the first page and measure. Resize, measure, resize, measure........until you get it exactly right. Then print off 2 copies the chart and tape them together.

v) Do the radius calculation for the rocker line down the center of the board. For my board, I wanted to have about 10mm of concave in the middle of the board an then have it reduce down to nothing in the tips. If you were going to have the same amount of concave all the way through the board then this is dead easy. You just use the R you calculated above and subtract the amount of concave you want. You'll need to adjust this if your mold is more that about 5 cm wider than your board because part of this increased height of the center jig that pushes the concave in will be clawed back as the mold continues to slope down from the edge of your board to the edge of your jig. In my case the mold was about 30cm wider and the board on each side so even though I only wanted 10mm of concave I need to reduce the radius by 2.5 times that so the right amount of concave was introduced. This can be tweaked once the jigs are completed by placing additional height on the centre jig to increase the concave. Reducing the concave is not so easy unless you lob off a bit from the bottom of the centre jig.

To get the concave that I wanted ( diminishing to 0 near the tips) there's one extra step because the circle that you want is not going to be centred at the origin (0,0). The centre will sit above the origin now otherwise the centre rocker line will end up being roughly parallel to the rocker line at the edge and you'll have the same concave at the tips as at the centre. Fortunately, you can use a trail and error method to find this new origin or use goal seek in excel if your familiar with it.

The new origin will be on the y-axis so we know the equation for the new circle is

x^2 + (y-c)^2 = Rc^2 - (3)

where c is the y coordinate of the centre of the new circle and Rc is the radius on the circle for the centre jig.

This equation has 2 unknown quantities in it c and Rc so we need 2 points on the circle to calculate these two unknowns. We can't you the coordinates of the other tip because putting them into (3) will give you the same equation so it won't be helpful in solving for 2 unknowns. You need to use one tip and then the lowest point in the centre rocker line. This is obviously going to still be on the negative y-axis (so x=0) but its y coordinate will be R minus the amount of concave you want with this figure adjusted for the reasons discussed above.

So using the equation of a circle again but this time with the centre at (0,c) we get

(L/2)^2 + (-R+r-c)^2 = Rc^2 - (3)


0^2 + (-R+con-c)^2 = Rc^2 - (4) , where con, measured at the center of the board MUST the amount of concave adjusted for oversized mold. (i.e con = concave * oversized_factor)

Putting (3) and (4) together you get

(L/2)^2 +(R-r+c)^2 -(R-con+c)^2 = 0 - (5) where the only unknown quantity is c.

You could keep going with the algebra but the easiest way is use Excel to solve this for you. Put the left hand side of this formula in a single cell with the values for L, R,r and conc in cells of there own. (make sure you use the same units for everything i.e. all centimeters or all metres..) and see what number it calculates. Then adjust c until the value is very very close to zero (e.g 0.001 is about right). You could also use the goal seek function if you're familiar with it.

Example: For a 130cm board, with 4 cm rocker (r=4cm), concave = 1cm, oversized mold factor =2 (con = 2cm) and R (calculated in the example above) = 530cm we get from (5) that c= 529.15cm. That is, the radius of the circle that passed through the points we need it to is almost 2x as large as the one for the out jigs.

This gives you the value for 'c' and so you can work out Rc above and like before you now have everything you need to plot out the coordinates, resize the graph, print it out and tape the sheets together.

Next post will get into the actual construction of the rocker table


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