Arm Formulae from Alignment Nulls

With the various computer based ways of calculating alignments, it is easy to try out different dimensions and see how they affect the geometry. So, if you wish to have less distortion at one particular point which doesn't correspond to one of the established alignment weightings, instead of rewriting the Lofgren equations,  you can just play around with the Vinyl Engine spreadsheets, and adjust the input parameters until you get the result you want.

While this makes it much easier to do than it was, (with calculators 30 years ago when I was doing it, or with paper and pencil and logarithms, when Lofgren was doing it), any budding designer might wish to know more about the way those parameters are linked, as it gives an insight into the way some alignment protractors work.

So, for a null based geometry, given the two nulls (N1 & N2), that is, the radii at which tracking error is zero:

Linear offset (Lo) equals the mean of the sum of the nulls  

Lo = (N1+N2)/2

The Dennesen and its clones, the Pro-ject and the expensive Uni-Protractor, use the fact that  Lo -N1 (the distance the slide is offset from the spindle) is a constant. This fixes a specific effective length (L) at N1 for any given mounting distance (D),  as (referring to the drawing of the basic geometry rectangle below):

L^2 = D^2 + Lo^2 -(Lo-N1)^2

or, alternatively, for a given effective length,

D^2 = L^2 - Lo^2  + (Lo-N1)^2

                                       
The distance Lo-N1 is half of the difference of the nulls.
Therefore for any alignment, this distance can be found if the nulls are known.

From the basic geometry rectangle above.

Given that for any given alignment ( LofgrenA/Baerwald, Lofgren B, Stevenson, etc)  and given minimum and maximum recorded radii, the nulls are constant, then the Linear Offset is constant, and so Lo-N1 is constant.  Therefore if D is fixed, then L  is fixed at N1, at which point the cantilever should be at 90 degrees to the radius.

Also, the offset angle OA is easily found, as the sine of the offset angle is: linear offset divided by effective length:

sin OA = Lo/L

So, knowing the nulls and one parameter, the others can be found.