U-joint/drive shaft alignment and trig functions

[Rod Neff <rodneyneff@xxxxxxxxx>]

    Basically, the shaft being driven through a u-joint or shaft "B"(drive
shaft in our case) speeds up and slows down TWICE per revolution at any
angle greater than 0 degrees, the more the angle in relation to the constant
speed driver shaft, or shaft "A"(transmission output shaft in out case) the
greater the speed up/slow down per revolution.

 Max speed of shaft B is determined by shaft A rpm times Secant of the
shafts angle, minimum speed of shaft B is determined by shaft A rpm times
Cosine of the shafts angle.

(Gleaned the above, and some of the below from my Machinist Handbook)

I put the oilhead GS on the center stand and to get full extension of the
rear, using a bevel square I determined max downward angle of the swing
arm/drive shaft to be 15 degrees from the transmission out put shaft.

So, 15 degree secant = 1.0353, 15 degree cosine=0.96592. Plug in 1000 rpm as
the multiplier and shaft B has about a 7% speed swing, again, which occurs
twice per revolution, roughly 3.5% slower, then 3.5% faster than constant
speed driver shaft A at 15 degrees or full extension of the rear shock in my
case.

Adding shaft C into the mix, (on paralever bikes this would be the shaft the
rear drive pinion gear rotates on) which is being driven by variable speed
shaft B, it is possible to deliver the constant A shaft velocity to shaft C.
  

How?   The angle of shafts A & B must be the same as the angle of shafts B &
C. And the orientation of the u-joints relative to each other must be 90
degrees out of phase.

 Its not easily apparent, but if the yolks attached to both ends of the
drive shaft are aligned, the u-joints are 90 degrees out of phase. I want to
attach the usual disclaimers to this last statement. I am pretty sure it is
correct, but please feel free to correct me if I am wrong. Basically, when
the drive shaft B is in its fast quadrant, pinion shaft C is in its slow
quadrant of the arc, canceling the B shaft speed variation, delivering
constant velocity to the pinion shaft C.

Now, understanding all of the above,  instead of canceling the shaft B speed
variations as outlined, it is also possible to double this twice per
revolution speed up/slow down event, if in our case the u-joint alignment is
exactly where it should NOT be. I haven't  had a paralever apart yet myself,
but from looking at the manual, it does look possible to put shafts B and C
in identical speed quadrants. Shaft B speeds up/slows down, shaft C speeds
up/slows down that much more in addition to the already faster/slower
velocity of shaft B. 

Which leads me to a post I read on the GS list several months ago which was
about person who was disassembling, then reassembling his drive line at
various u-joint alignment profiles to minimize vibration. If  I remember
right this person also stated he had taken several un-molested from the
factory drive lines apart and there was no rhyme or reason to the u-joint
alignment. I found this last statement hard to believe. I have known about
this phenomenon for some time, but only recently done this bit of research.
Do German automotive engineers and QC really allow random u-joint alignment
on their motorcycles?

Yes, they sure do. 

I ran all this by a local mechanic who I have known for about 20 years, he
verified that factory u-joint alignment is random. He is aware of u-joint
alignment, BMW teaches it in their mechanics classes. He didnt have an
answer as to why they dont do it right at the factory. Says when he has one
apart he puts it together correctly.

This may explain at least some of the drive line failures people have
experienced.
Also would be interested to know if these aftermarket shortened paralever
link arms are associated with drive line problems.

Why am I bothering with this? Recently installed an aftermarket rear shock
on the 98 which raised the rear end up about 5 degrees at its lowest
setting, giving me long term reliability concerns. Also, I have never been
into the paralever drive line yet, but am about to find out what it is all
about. I have to replace the clutch slave cylinder on the 00.

Going back to at least /2, and most likely before that, BMW compensated for
drive line shock by using a cam coupling/ spring dampener affair on the
transmission input shaft, this same concept is still used today. Starting
with the /7 they added this cam dampener on the drive shaft as well. And
sometime in the 80s  at least some of the drive shafts started being built
with a rubberized dampener. K bikes for instance. Not sure just exactly how
this rubber dampening works, if any one has a cut away view, please let me
know.

My little write up here is dry and probably doesnt make much sense just
reading through it. But with some time to think about it, and a few pictures
this concept should become clear.

The first diagram here will help you quickly understand the concept, the key
phrase here is "more/less distance must be traveled in a fixed period of
time" . In a nutshell, one side of the u-joint travels in a circle, the
other side travels in an elliptical path. The second diagram is how the
drive shaft yolk orientation should be.
http://www.4xshaft.com/driveline101.html

More info
http://www.novak-adapt.com/knowledge/driveshafts.htm
http://www.drivelinespecialist.com/tech/Driveline101.html

couldn't find the mention of Secant  trig function in any of the above
links, only my Machinist Handbook mention this, $10 at used book store, a
wealth of information.

Trig table
http://www.industrialpress.com/Trig.htm 

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End of oilheads-digest V1 #100
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