ANGLING FOR 90° - tangential pivot tonearms

[directdriver]

Mark, I owe you more than a beer. I'll add a bowl of alphabet soup.

[Mark Kelly]

And now to Floating B.

Please note that I'm using the spindle as the origin in these calculations
where I used the point B in the last. This simplifies the maths a bit. If it
causes confusion, the best option is to beg the mods to allow me to edit the
previous post and I'll change the co-ordinate system so they're both the
same.

We have a series of radial lines from A to C of length R. I will give points
for 3 values of R, namely 50, 100 and 150 but to do this properly it is best
to calculate many more points (that's what spreadsheets are for).

We have corresponding line segments CD which are orthogonal to the radii AC.
This is the length of the main part of the arm, we will set it to some fixed
value and I'm going to use 150 so the two examples are similar. If we then
draw the line DA for each point D we have a right angle triangle with the
two right sides known, the hypotenuse is found by Pythagoras. For the radii
chosen, the hypotenuses are 158.1, 180.3 and 212.1. The angle ADC will be
ARCSIN (AC / AD) so we have 0.321, 0.588 and 0.785 respectively*. Each
segment CD is part of a longer segment CB where B is the intersection with
the base line. Note that with this geometry the point B is different for
each radius, unlike the Birch geometry where it is fixed.

We now set the pivot point P. Let's make the length of the arm from pivot P
to each point D 60 mm. If the length AP is made to equal the sum of PD and
DC, then the arm will have zero overhang, so we set point P at (-210,0).
This defines a second triangle with three known sides so we can use the
cosine rule <http://en.wikipedia.org/wiki/Law_of_cosines> to calculate the
angles APD which are thus 0.449, 0.923 and 1.463. This in turn allows us to
define point D for any radius. The X coordinate will be -210 - PD*COS APD,
the Y coordinate will be PD sin APD so the three points D are (-155.9,
26.0), (-173.8, 47.8) and (-203.6, 59.6).

In turn this means we can calculate the angle PAD which is simply ARCTAN
(Yd/Xd) for each point D giving -0.165, -0.269 and -0.285 respectively. Each
segment CD is part of a longer segment CB where B is the intersection with
the base line. Note that with this geometry the point B is different for
each radius, unlike the Birch geometry where it is fixed.
We can now calculate the angle ABC which must be equal to ADC + PAD. The
slope of the line BC is the tan of this angle so we have 0.157, 0.330 and
0.546 respectively. Since we have a slope and a point on each line (the
points D) we can calculate the intercept with the base line which will be
the point B by substituting into y = mx +b the same was we did previously,
obtaining intercepts of -321.4, -318.55 and -312.68 respectively.

Although this looks like the geometry is quite different from that of Birch,
it isn't. The difference is that we've put both B and P on the base line
whereas with Birch geometry P is always below the base line. We can show the
equivalence by calculating the equivalent "Birch line" and finding the
intercepts which will fall near the opposite Thales locus. In this case that
locus is at (-315, 2) but I'm not going to clog things up by showing you how
to find this.

The beauty of this geometry is that we do not need to use the Thales locus
so we can completely eliminate the errors in the Birch geometry. We do this
by getting rid of the remaining variable length, which is the length BD.
Instead we assign a new set of points E which fall a defined distance past D
on the line CB. Since we know the equation of the line and the points D, for
a given length DE the points E are defined by Xe = Xd- DE * slope, Ye = Yd -
DE * ( 1 - slope^2). These points will define a track which will constrain
the motion of the segment DC so that the angle DCA is always 90 degrees.

The fun bit is designing the interface between the arm and the track.

You now owe me two beers.

* Angles in radians.

[directdriver]

What have I gotten myself into?! I should have listened to the moderator EC8010 and focused on unipivot designs.

[walterwalter]

It is rather amazing, that after one hundred years of disc record players history , new purely mechanical arm designs are still coming, and they still are able to come closer and closer to perfection. The last Schroeder design looks like almost final stage, but who knows...

[Esperado]

Nice Thread.
In 1970, i was working in the research and department office of Scientelec (A well known French hifi manufacturer at this time).
There were too a famous manufacturer of professional turntables and PU cartridges Clement (He equipped the state French Radio "ORTF").
Clement
He produced a Radial arm turntable:
Platine bras radial
http://www.audiofolia.com/Download/RDS201.pdf
The first, as far as i know, to have a strait free arm on a chariot , measuring the angle error with a light, and moving the arm's charriot with a motor.
Back to my story. I wanted a radial arm for my company too. So, i had offered two patents (around 1970) to my company. The first one was a pure copy of the clement's one, with just the idea that, instead of moving the arm, it was the record plate witch moved under the arm.
The second idea was to fix the arm at the periphery of a rotating plate, and to make-it turn in order to correct the error.
The arm itself is free on the two axes, the error of angle is detected under the main plate by a light and two photo cells. Each time the arm is moved by the grove to the inside, it create an error, the the motor make rotate the A plate until the error is 0.
On my point of view, any system where the head is making an angle is messy, as it create a "centripetal ?" force.
To conclude, i would like to point out the most intelligent radial arm turntable ever:
Technics SL7.
Technics SL-7 Owners Manual, Service Manual, Schematics, Free Download | Vinyl Engine
The arm, his chariot and his motor was in the cover: you do not have to wait for it to be parked before opening the cover. Brilliant. This plate was able to detect the size of the record, and was fully automatic if you like: one button press.
That the one i use.

[directdriver]

Esperado: "The second idea was to fix the arm at the periphery of a rotating plate, and to make it turn-in order to correct the error.
The arm itself is free on the two axes, the error of angle is detected under the main plate by a light and two photo cells. Each time the arm is moved by the grove to the inside, it create an error, the the motor make rotate the A plate until the error is 0."

Conceptually, the rotating plate with point A resembles Birch geometry with point P, except one is motorized and the other mechanical. I'm curious as to how you make all the red lines converge to or close to point B on the Birch design? Or is it even necessary?

[directdriver]

walterwalter: "It is rather amazing, that after one hundred years of disc record players history, new purely mechanical arm designs are still coming, and they still are able to come closer and closer to perfection. The last Schroeder design looks like almost final stage, but who knows..."

I am amazed that nobody is talking about it. Frank is a modest guy and his design is flying under the radar but people need to talk more about such revolutionary design instead of talking about upgrading their next air-pump!

[Esperado]

Originally Posted by directdriver

I'm curious as to how you make all the red lines converge to or close to point B on the Birch design? Or is it even necessary?

They only converge ...at infinity ;-)
Conceptually, there is nothing in common. Whatever the distance of the head from the center , there is always a position of the plate A (as you can see on my design) for the Diamond of the head to be on the tangent. You don't have to know where, as the error correction system will turn the A plate in the good direction untill no error (the light hit the center of the record plate).
Exactly like the Technics or the Clement. The arm is totally free on the two horizontal and vertical axes.The difference is that, on the traditional radial arms, the error system can be in the arm itself, looking for good perpendicularity between the arm and his chariot.
Here the error will be measured by 2 cels at the vertical of the record plate axe. If the light hit the Right cell, the motor will turn the A plate clockwise, if it hit the left cell, anticlockwise. When the light is right in the middle (no error), the motor is stopped.
The advantage of this system is just that it is easier and cheaper, on a mechanical point of view to rotate something than to generate a linear movement.
Note that the second parallel arm under the plate will have an advantage, resonance of the arm (with the cell suspension) will not be the same in vertical and horizontal direction.

Originally Posted by walterwalter

...they still are able to come closer and closer to perfection.

For me, the Technics was absolutely perfect: you cannot hope something better. Heavy, easy to run, and the size was just 30cm X30cm, the exact size of the 33RPm. More, you where obliged to run the turntable closed, so, no dust. And it was not a so expensive turntable.

[walterwalter]

Quote:
Here the error will be measured by 2 cels at the vertical of the record plate axe. If the light hit the Right cell, the motor will turn the A plate clockwise, if it hit the left cell, anticlockwise. When the light is right in the middle (no error), the motor is stopped.
The advantage of this system is just that it is easier and cheaper, on a mechanical point of view to rotate something than to generate a linear movement.
Reply:
Design is elegance itself, among electronically controlled linear trackers.

[Mark Kelly]

Originally Posted by Esperado

They only converge ...at infinity ;-)
Conceptually, there is nothing in common.

No that's not true. DDs supposition is correct, the geometry is a variation on the Birch geometry.

If you construct an extension of the arm past the circumference of the circle you will find that these extensions approximately converge at the opposite Thales locus. The servo system basically reduces the error inherent in the Birch geometry. Frank's guidance system serves a similar pupose.

It would be possible to dispense with the servo system entirely and use the convergence of the arm extensions to establish something very close to tangential tracking.

Esperado if you have the exact length of the arm, the radius of the chariot (between pivots) and the distance from the platter spindle to the chariot's centre I will show you how it conforms to Birch geometry. It would be interesting to measure the actual error in the servo system and see how the two compare. What is the distance between the two lights cells?