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Da Vinci's Dilemma

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Da Vinci's Dilemma
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                                    By   Dean A Beeman

                                       Copyright 2014 All Rights Reserved

Available as a Download

DaVinci/ModelComplete.jpg

                                                
So far we've explored men, the sea, and America. In my opinion those subjects are worthy things to explore. Trying to fully understand any one of them is a fool's game.

And until now our boats has been, in one way or another, successful. The philosophy of each one has been pretty clear, and the designers, builders and sponsors were more or less successful. Perhaps the sailors weren't so lucky, but that's a sailor's life.

This time I'm going to build a power boat that was designed by a landlubber. As a boat it was a total failure. As a matter of fact it's famous, but never existed. Which is just as well since, as designed, it couldn't float or even operate. It's interesting because those characteristics aren't the point.

At the dawn of machines (the 15th century or so), it was common for what we now call engineers to create a rendering of a final product without revealing the mysteries of how it worked, or for that matter, that it worked at all. (If you think that anything has changed since then, consider how many fortunes have been made in Silicon Valley on precisely the same premise and evidence.)

To demonstrate 21st century thinking I'm sticking with hype and hope but throwing practicality out the window.

So we're going to build a power boat as an object of 15th century entrepreneurship and its 21st century legacy, but not as a power boat. Or even a boat, for that matter.

In case you've been following the construction of ships that aren't really ships and boats that aren't really boats, I'm on my way to the extreme. (A sailor's sense of humor is something we should have addressed before now.)

And since I'm a Buonarroti fan I tried to somehow fit a gravy boat into this whole subject. No luck. So we're stuck with Da Vinci.


Background

Sometime in the early 15th century an Italian engineer by the name of Mariano di Jacopo detto il Taccola (Taccola) rendered his concept of a boat powered by rotary paddles. That boat, as drawn, was powered (more or less) by what we now know as a winch.

Much later in that century a young man named Leonardo Da Vinci theorized that such a simple mechanical device could be improved if its driving mechanism could be replaced by a machine- in other words, moving parts that acted together to perform the same function. Enter the whole subject of gears.

Improvements and documentation on the basic theory of the gear and the screw date back at least to Archimedes. (And while the ancient Chinese, Greeks and Egyptians knew ( and in many cases documented) most of what was found in Da Vinci's notebooks, they didn't paint the Mona Lisa, so they've been screwed out of the credit for things like the helicopter and the armored tank. (As a matter of fact, prehistoric humans who observed a frog, a dragonfly or a beetle stand unacknowledged, too.)

But single details in a mural or charcoal images on the wall of a cave just aren't the same as thousands of pages of detailed art.

We've discussed the value of works-in-progress, and the enforcement of fair play associated with the whole concept of copyright. Since it's unlikely that any human will ever match the volumes of such work recorded by Da Vinci, civilized people have a responsibility to preserve them.

The original sketches that make up the Codex Atlanticus reside in the Biblioteca Ambrosiana, Milan, Italy. I'm going to have some light fun with one of those sketches (and a few others), but that doesn't mean that you and I shouldn't make a contribution to their preservation by recognizing the value of whatever copyright resides with that library.

My references for Da Vinci's sketches can be found online, or in various texts. I own a copy of "Leonardo's Machines- Da Vinci's Inventions Revealed", C. David & Charles Limited 2006, originally published by Guinti Editore, C. 2005, Florence-Milan. While the images are enlightening, the text is often one gear short of a workable machine.

Anyway, in folio (page) 945r in the Codex Atlanticus, Da Vinci presents a sketch of what appears to be a boat and its mechanism. The probable details of its mechanism could, according to some authors, be found in folio 30v and perhaps folio 24r.

Let's start with the finished drawing from folio 30v (this machine is reportedly similar to a set of drawings by Francesco di Giorgio). Some writers have implied that this winch is the basis for the power that later appeared on Da Vinci's paddleboat.

In Da Vinci's formal schematic, an opposing pair of internal, saw-toothed gears are driven by a pair of opposing pawls. If the drive shaft is turned in one direction (by a lever in this case), the first drive pawl catches a gear tooth and moves that gear in one direction, while the other pawl rides over its saw-tooth. Move the lever in the other direction and the second (riding) pawl now drives its gear in the opposite direction of the first while the first pawl rides.

Unfortunately, in order for two fixed gears on a single drive shaft to drive a third gear forward you need a fourth gear to reverse one of the first two. Or split the drive shaft, resulting in the same, but more dramatic, outcome.

If you doubt that, find a copy of the drawing ( folio 30v) and draw arrows associated with the lever direction on the finished figure to the right of the exploded view. The drive gears will continuously move in opposite directions, but the weight on the pulley isn't going anywhere.

But to his credit, Da Vinci sketched his theory in great detail.

On to the paddleboat, folio 945r.

As best I can recall, my grades in Art as a grammar school subject were not discussed with my parents. (There were equally-depressing grades in other subjects that I keenly recall.)

So my rendering of folio 945r is right up there with a stick figure drawing of a cloud. But I'm providing it so you'll get a feeling for the image that you might want to study on the web:

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DaVinci/PaddleboatSketch.jpg
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There is roughly a 4:1 speed advantage by having a large drum (in the foreground) drive a smaller gear on the paddle's drive axle (at the top). That might be OK if the smaller gears were hardened steel, but given the resistance (water) on the very large paddles these wooden gears would have shredded in short order, even with a fourth gear.

Assuming that the resistance could be managed by lowering the water resistance (smaller paddles, less depth, etc.), we're still stuck with the machine. Step down on one pedal and the drive gears go forward, driven by the rotation of the drive drum, Step down on the other pedal and the drive drum reverses and, like the winch, the drive gears reverse, too. The saw tooth gears are very similar to the previous drawing. So this boat isn't going anywhere, either.

You will discover that there are many models and drawings available that attempt to resolve this dilemma. Most simply leave off a gear or two, so something can rotate. Others add a seesaw set of drive pedals or eliminate the belt and pedals altogether. If you wander through a number of online images and study the meshing of gears you'll soon get the gist of these excuses.

Frankly I think we can solve the engineering problem with some minor slight-of-hand. We'll see how that goes.

A more practical design than the one I've chosen can be found in folio 1063 and elsewhere. In that case the machine gears are replaced by direct drive crank handles. That version might work on a very small boat (the size that people pedal around resort ponds), but not on the Arno.

And as presented in yet another example, these cranks actually drive large iron flywheels that in turn power the paddle wheels. That inertial version might work a little better, but it isn't nearly as much fun to build.

I wouldn't call any of the other boats in this series a work of art. But in this case we don't have much choice since it wasn't a boat to begin with. So I'm going to build it for the sole purpose of enjoying how a bunch of wooden pieces (mostly scrap from other models) can be assembled into a very pleasant but fatal contribution to the subject of power boats. And the timeless nature of well-intentioned ideas.

I also considered (given those subjects), John Fitch's Perserverance. But since we've already given him credit (in the Steamboat) for the same contribution, it's time to move on.

I've also decided to have some fun with dangling participles.

Before I leave the weighty subjects that we’ve been considering, it occurred to me that Da Vinci would have been much more successful if he had the benefit of some 20th century communications tools. You are all familiar with (or at least have imagined) the dreaded Conference or Board Room exercises in bloviated logic.

I happen to believe that 15th century nobles, like their modern counterparts, would have been much more supportive of Da Vinci if he had been able to create a PowerPoint request for funding:


DaVinci/Flowchart.jpg

                                     Layout

The most difficult part of recreating any of Da Vinci's machine concepts is the studded wheel (external gear). Sooner or later, a studded gear drives a cage gear (the small gears in the drawings). These all require more-or-less perfect circles and spacing of teeth. Or you're stuck with a very bumpy machine.

While most of his designs consist of large (main) gears with teeth that are spaced at a fairly simple and easy-to-draw 10 degrees (more or less), in those designs you still have to install 36 teeth in an inner circle and another 36 on the outer rim of the same circle. We've been through the tolerances of wood, and that many teeth on a 6' wheel might be one thing, but getting that many onto a one inch wheel will be a squeeze.

But any size difference in the gears results in differing angles due to the meshing of their teeth with the teeth of the main gear, and any radius that is not a multiple of 10 degrees will eventually cause a set of gears to jam, even with wide tolerances (spacing). And I intend to power the paddles.

And one inch is about the right size for, say, a tree ornament.

Since Da Vinci and his peers either didn't know or didn't care about scale, I see no reason why I should. But gears require consistency of angle and, if you get into the subject, about a dozen other characteristics.

Let's start with all the important components:

DaVinci/DaVinciParts4.jpg

Looking at the circle (lower right), I’m starting with a gear that has a diameter of 1”. That particular circle has been carved up into 10 degree radii. I have extended these so you can use a compass and create the same pattern at whatever size you choose. 

Now if you doubt that 10 degrees was a staple of Da Vinci’s sketches, here is that same circle overlaid on Vitruvius (the Vitruvian Man):

DaVinci/VitruvianMan2.jpg

(This sketch has been copied, edited and otherwise reproduced so many times that copyright protection is another fool's game.)

Back to business, the smaller circle in my schematic is ¼ the size of the larger, and the red lines on that circle are drawn at 40 degrees. So any set of teeth on one wheel should (will) mesh with the gear teeth on the other.
 

The paddle is a paddle. If you don't like mine, have at it.

                                                     

                                                  The Math


If you locate an original drawing and simply count and extrapolate what can be seen in Da Vinci's sketch, the gear count is as follows:


            Main drive gear:                        36 spindles (spokes) and teeth

            Intermediate Drive Gear

                                    Interior             18 teeth

                                    Outer               12 teeth

            Cage (small) gear                       6 spindles


The diameter of the main drive gear is 77% (roughly) of the diameter of the intermediate gear. This is to compensate for the meshing of the main gear with the interior gears of the intermediate gear. Once the intermediate gear is turned by the main gear the tooth count on the outer and cage gear must be multiples of each other, but could be any number that is evenly divided by 360. Da Vinci apparently chose 6 and 12. The ratio of the intermediate gear to the diameter of the small cage gear is about 4:1.


Keep in mind that he was still a young man (30 years or so) who was educating himself (probably through the discipline of his sketches), so numerical uniformity was not high on his list of interesting subjects. And given the ratios, his paddleboat was capable of reaching a speed (by my math) of at least 30mph (if the components didn't disintegrate, the paddles moved in the right direction and the operator was athletic).


So while some optimists have given him (very) dubious credit for inventing the automobile, I think he was closer to the first F1 power boat. (Getting back to Guy Lombardo, the man might have been whacky enough about power boats to try this.)


                                    Tools and Materials


Looking at the sketch of this model, the math leads to the task of drilling about 176 holes that will be filled with a like number of pegs, spokes and axles. Compared to building a large square-rigger that's a small number of holes (unless you cheat.)


So if you go back to the tutorial on how to drill an accurate hole, you might decide that a major risk in building this model is finger cramps.


I'll be using tools that I haven't mentioned earlier:

            -A 5/16th leather punch. This is optional- you can create the same small circle cuts by roughing-in a circle and turning the pieces on a rotary tool or drill.

            -An X-Acto circle cutter. Also optional for the same reason.

            -A micro-drill chuck for a cordless screwdriver. Also not needed, but handy for quick tests of prototypes.


Next up are the materials.


By now you know that I have at least a square foot of leftover 1/64th aircraft plywood (from the Higgins Boat) and scrap pieces of 1/32 and 1/16 mahogany and basswood slab (the Honey Fitz).


From a collection of small veneer samples I also have a square of what I believe is chestnut burl (unbacked) and a scrap of rosewood veneer. A little homework on Italian hardwood production and Da Vinci's early life and career might be helpful here. (If I were to build an exact model I'd substitute chestnut and beech for basswood and mahogany.)


A #80 x ½" miniature machine screw with nut and washers. This is the mandrel I'm using to turn the larger wheels.


One length of 3/16 square basswood and whatever you choose as a 3 ½ x 6"  base.


Some brass rod, ( 1/32 and 1/16) and brass tubing with those inside diameters that will sleeve the rods.


4 brass pad eyes.


From the display cases that I've built I have a bunch of small leftover acrylic pieces that I'm using to create templates. Any clear plastic will probably work just as well


And a small scrap of thin leather.


Keep in mind that we're building what Da Vinci envisioned: a machine that will fit into a boat. Any old boat will do if you want to go that route.

   

                                       The Model


In this series we've built templates for drilling, cutting and, in more than one case, logic.


In order to build this model we're going to create templates that are a bit flexible. (Or invest a few thousand dollars in a computer-aided laser.) I'm going for a monetary investment approaching zero.


As a guide I've created and I'm building from a master schematic. This is based on the earlier discussions, and will print on standard letter-size paper:

DaVinci/DaVinciLayout.jpg

 

                                            Construction



It may be difficult to see them until the page prints, but I've added drilling points to the smallest circle. I'm using that cage gear to dictate the circumferences of the larger gears simply because it's the most difficult to get (more or less) right.


At whatever scale you choose, the drilling dots on the smaller circle have a radius on-center that is ¼ the length of the radii in the inner portion of the larger circle. Geometry being what it is, the circumferential distance between those dots equals the like distance between the intersections of the inner circle.


In other words the gears will mesh and the machine will operate.


                                    Process


In the other models a photographic flow seemed like a good idea. For this one, all you really need is a picture of the tools and templates:

DaVinci/Process.JPG

 

I'm going to build the smallest gears first. These won't be needed until much later in the build, but this is a good introduction to the difficulty you're facing in getting every spoke and gear exactly where Da Vinci's concept placed them.


We've been through the wonders of the human eye. Since drill bits will wander based on the grain of the wood, and even the most careful center-punch will be slightly off-center, my advice is to experiment with this set of gears until they are nearly-perfect.


I've used spray contact cement to create a sandwich of 1/32nd mahogany  and aircraft plywood., To keep the dried mahogany from shedding I've soaked it with a heavy coat of spray lacquer. (Any thin shellac should work.)


Once the piece dried, I used the leather punch to punch out 8 wheel blanks. (One of those blanks is underneath the drilling template.)You only need 4 of these, but the tolerances are such that any drilling error will ruin a blank, and I'm going to choose the best set of 4 from the drilled collection.


If you cut out some outer portion of the tiny wheel from the schematic and school-glue it to an acrylic (or other clear) scrap, you have the pattern for the first drilling template. (In my opinion it's important to see through the template into the blank.)


Once you center-punch and drill (1/32nd bit) the template holes, place your template on a blank (as in the photo) and look at the pattern. If it isn't more-or-less perfect you aren't ready to move on.


Drill a set of wheels and insert short lengths of 1/32nd brass rod into the holes. Once the cage gear is populated, insert a longer (1/16th) rod into the center hole and slowly rotate the gear. Any non-uniform spacing will be obvious (and fatal). A finished set of gears is above the small template.


I trimmed the rods with the nail clipper, sanded them smooth to the wheel and applied CA glue dots to each spoke. The glue should wick into the joint.


Once that set of gears is ready it's time to move on to the larger, but no-less precise larger gears.


The template for the larger gears is not that complicated, but at this scale must be just as precise. I constructed mine by solvent-welding a 1/16 x  ¾ x 2 ½" acrylic scrap to a 1 ½" x 2 ½" base. The scrap is on-edge at a more-or-less perfect right angle.


Placing an identical scrap to the first, I drilled 1/32nd register holes through both scraps. This results in a template with a movable face. After placing rods in those holes I sanded the outside edges flat and square to each other. This results in an outer register edge that is fairly precise.


Using the (yellow) circle cutter in the photo I scored a couple of blanks. I then rough-cut the circles, drilled 1/32nd centers, bolted the wheels together using the machine bolt, and then turned both wheels. Either a rotary tool or a drill will work as a lathe.


The larger template is used for both larger wheels. In the photo I have placed pins through the holes that I drilled to get uniform tooth spacing. The bottom (green) pin is inserted through a hole that corresponds to the spacing of the outer teeth, and the side (black) pin registers to the inner teeth.



In both cases you drill a hole using the template, rotate the wheel, place a rod through the hole you just drilled, rest that rod on the template face and drill another hole.


A blank wheel and a drilled wheel (with the inner teeth trimmed) are below the template.

For both large wheels I have trimmed the teeth to 1/8".


After the drive wheel was finished I mounted it on the shorter pedestals using the pad eyes and a short length of 1/16th rod. To keep the outer teeth from catching the pedestal I cut and placed a tubing sleeve (spacer) over that rod.


I haven't provided dimensions for that gear because there are none. My overall length is 1".


The center holes for the intermediate gear axle ( 1/16th) should line up with the drive gear axle. Build those wheels, and use lengths of tubing to space them in the center and at the posts.


At this point the drive gear should freely drive both intermediate gears.


To create a little slop in the final (cage gear) assembly I used 1/32nd rod inserted through 1/16th holes. The slight-of-hand that I mentioned is one length of tubing that keeps these gears centered and another set that extends out to the paddles.


Once the cage gear axle and gears are placed through the pad eyes at the top of the posts all three gears should rotate freely.


The paddles can be solid, but I built mine using 1/16th mahogany for the shafts, 1/32nd mahogany for the blades, and 1/16th basswood for the hub. The holes through the hub are 1/32nd, to accept the axle.


The pedals can be constructed using any 1/16th wood. I constructed mine using the scrap of rosewood.


After everything more or less worked I capped the drive wheel with chestnut and the intermediate wheels with mahogany.


A scrap of thin leather is fitted into the slots in the pedals. Da Vinci chose not to show hinges, so I haven't added them.



Note that the only items that are glued are the gear teeth and spokes. One of the important elements of a machine is the ability to take it apart, replace a part, reassemble it and put it back into service.


So my version of Da Vinci's machine can be completely disassembled and rebuilt.


I plan to mount the model in a case alongside a photographic reproduction of the original sketch. (Next photo.)


Now may be the time to address the question of whether or not the machine works.

Da Vinci's machine, as drawn, would have worked perfectly. Mine does too.


The second and less-important question is whether or not either machine could power a boat. If you want to ride in a boat that spins in a perfect circle I have a machine that works perfectly.


Here is that model and that image:

DaVinci/ModelComplete.jpg

 

If you want your boat to sail in a straight line, Da Vinci's dilemma is now yours.


Early October 2014