Perfect Involute Tooth Forms?

Interesting question from a builder:

I'm new to wood clocks, but have some mechanical engineering experience. I see that the tooth profiles on the pinions are nothing like those on the wheels they drive (and neither look like involute teeth). 

Why the difference in tooth profiles? It seems to me that the two ought to be the same. To add to the question, it seems to me the quasi-trapezoidal teeth on the driven wheels would be easier to cut accurately, suggesting the pinion teeth should be similar. What do you say?

Regards,

John

Aloha John, it is nice to hear from you.

Your engineering experience has taught you right.  It has taught you about high speed gearing at its finest.  

Thing is...clocks don't move fast so that level of perfection is not required.

As a matter of fact, in my own experiments, it's difficult to notice any difference at all with various tooth forms.

I discuss this at length in my book, Practical Guide.

Take a look at the various tooth forms that I use in my designs.  You'll see straight sided, involute, "random", rounded, and none of them seem to perform much better than any other.

You can make high speed, "perfectly" engineered teeth if you like, but do some experiments on your own and I think you'll convince yourself that pretty much any tooth form works well at low speeds.

The old tymers used to cut triangular tooth forms.  Worked fine for them.  Hacked them out with adz and axes.

Take a look at the teeth on this old clock on the cover of the National Association of Watch and Clock Collectors (NAWCC) journal.  Here is a perfectly fine running clock with possibly the poorest choice of tooth form available...triangles.  Ha!  

And yet................it has been working for centuries.

So, my advice is to make whatever tooth form strikes your fancy and compare it against the "perfect" involute tooth forms and see how much difference it makes in the running of your clock.

Enjoy!  Aloha.  Clayton

Tooth Forms, Including Involute Tooth Form

Bob L. emails:  Hi,  I am considering your Toucan clock plans.  I have a little past experience in gear design, and am thinking of making the gears by a different method than scroll sawing.  Because of this, I wonder if you tell me:

1)  Do the gears in the Toucan have the common "involute" profile or are they a different profile??

2)  What diametral pitch are gears in the Toucan??

Toucan by Jeff Hecht

Aloha Bob, it is nice to hear from you and I'm glad you enjoyed viewing my site.  The Toucan is a wonderful, fun and easy to build clock.  It's my top design engendering the creativity and personalization of the various builders.  You can see many of their delightful "brain children" in the Flickr' pool of builder' pix (link on the Toucan order page).  Those pix really are worth viewing for some great ideas.

And how can I say that it is easy to build?  Because when you are looking through the Flickr' pool of builder's pix the first one is of a 13 year old boy that successfully built the Toucan.  Of course, maybe he's just a genius(?).

Gear design is a very popular topic and lots of time is spent on gear design in engineering books.  I write many pages in my book about gear design as it relates to wooden clock building, and I really do need to create a blog post about the topic.

First we have to ask ourselves what the purpose of the gears really is, and what is their real function?  

We could simply have two circles turning edge to edge and get the same ratio, but without the addendums and dedendums (the uppity and downity parts of the tooth) of spur gears our circle gears could easily slip.  Slippage would cause a deficiency of the transmission of the driving force through the system, but more than that, in a clock slipping wheels would also cause the hands to show the incorrect time and even cause a desynchronization of the relationship of the minute to hour hands.  

However, that does not mean we cannot use edge driven circles in our clock designs.  Take a look at my Horologium and you'll see that the hour wheel is actually a pulley run by the drive weight cord.  The gear ratio is maintained by the barrel diameters of the drive pulley and hour wheel.  http://www.lisaboyer.com/Claytonsite/horologium.htm

So basically the "teeth" on a gear do two things for us; they efficiently transmit the driving force, and by doing so they prevent misalignment caused by slippage of the wheels.

When we are contemplating tooth forms for a particular mechanism, a couple other things that need consideration is the speed of transmission of the drive force, and the direction of that force through the gear train.  

In a clock the transmission of force is extremely slow.  In the normal grandfather clock the fastest wheel only turns one revolution per minute.  This slow transmission of forces allows for some very forgiving tooth forms.  There are medieval metal clocks that are still operational with simple triangular shaped tooth forms.  Some of the teeth in the very early wooden clocks were created with hand tools and look as though some teeth were hacked into shape.  These inefficiency of these irregularities and imperfections many times can easily be overcome with a bit more drive weight.  

It is only when we have gears rapidly moving together that we actually need the perfect involute gears with close tolerances that we've all learned about in physics class.  There are other very efficient ways of transmitting power through the system without the use of involute gears.  Take a look at my Vortex.  Vortex has straight sided teeth on all of its gears.  Straight sided gear teeth are about as far from the perfect involute curve as you can get.  Scroll down the page a little further and take a look at the side view and you'll see how I created the roller lantern pinions that perfectly match with the straight sided tooth forms of the wheels to give excellent power transmission and minimize contact friction. http://www.lisaboyer.com/Claytonsite/vortexpage1.htm

And the last thing to consider is the direction of the forces transmitted through the gears.  Clocks run in only once direction.  Clocks do not ever run in reverse.  That means that the back side of the gear tooth never gets used and can be created in any form that the designer favors, as long as it does not interfere with the incoming tooth from the other gear.  Most of the time we simply make it a mirrored reflection of the front side of the tooth, but that mirroring is not necessary since that back side of the tooth is never used in a clock.

Most all of my tooth designs are of a modified involute design because even as forgiving as these wonderful mechanisms are one of the goals of a good design is to try and decrease the amount of driving force required.  Involute-like teeth help with that, but as we've discussed above involute tooth forms have no particular spiritual holiness sent from the God of Physics.  This gives us a tremendous latitude in design.  We can make our teeth straight sided, curved sided, or perfectly involute, but I have to chuckle because whatever tooth form we choose, we still don't need the back of it.

Enjoy!  Aloha.  Clayton