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There are some fairly simple equations that describe the geometry of all pivotal tonearms in terms of effective length, pivot-to-spindle mounting distance, offset angle, and alignment null-points. I have posted these equations a number of times:Sometimes tonearm manufacturers publish parameters such as effective length, pivot-to-spindle mounting distance and offset angle without providing the alignment null-points. If you would like to calculate the alignment null-points there is a quadratic equation that can be solved as shown below:
For example, suppose you know the following parameters:
Pivot-to-Spindle Mounting Distance = 258-mm
Effective Length = 273.4-mm
Offset Angle = 19.98-degreesUsing the above equations, you can easily determine the alignment null-points to be:
N1 = 70.1-mm
N2 = 116.7-mmA graph of this alignment would look like this:
Edits: 04/16/17Follow Ups:
Hi: Would you happen to know the equations used to generate the shown traces? Many thanks.
There is nothing shown in your post.
Sorry, the post appears fine on my end. I hope this re-post is readable. I was wondering if you happened to have the equations used to generate the traces for tracking error and distortion you show in your figure? Many thanks.
You need to download my spreadsheet from the link below. I don't use equations to plot the curves. Instead, the curves are plotted from individual points in Columns C and D.I used the Law of Cosines to solve the pivot/stylus/spindle triangle for the complement of the angle at the stylus, e.g., the angle between the pivot/stylus axis and the tangent to the groove. These values are listed in Column B. Actual tracking error is determined by subtracting the cartridge offset from the groove tangent as shown in Column C.
Tracking Distortion (Column D) is approximated by dividing tracking error by groove radius, which is also called the weighted tracking error curve. These numbers are scaled so that they approximate actual percent tracking distortion at a stylus tip velocity of 10 cm/sec. In other words, the number 1.0 in Column D represents 1-percent distortion at a stylus tip velocity of 10-cm/s.
Alignment null-points are located at the groove radii in Column A where tracking error in Column C approaches zero. Cells F3 and G3 display interpolated null radii.
Best regards,
John Elison
Edits: 10/16/15
"Tracking Distortion (Column D) is based on dividing tracking error by groove radius; this is also called weighted tracking error."
Ah, so this is related to the 2nd order distortion heard when misalignment happens but may not be a 1:1 correlation? I will go ahead and download this and look at it more carefully when I get a chance. Thanks.
The equation used in my spreadsheet to approximate tracking error distortion is a simplification of the following equation developed first by Erik Lofgren and later by E.G. Baerwald in their respective papers on tonearm alignment. I have chosen a constant, that when multiplied by tracking error and divided by groove radius approximates 2nd order harmonic distortion for 33.3-RPM LPs with a stylus tip velocity of 10-cm/s.
I can read Sanskrit and ancient Akkadian cuneiform but not that stuff. You're not going to be in Naples, Florida anytime soon, I suppose?
Edits: 10/15/15
I have no plans to go to Naples, Florida.
Sorry!
John Elison
John, great stuff. do you have a spreadsheet with the graph that I can use? Thanks, Pat
If so, please email to banpuku@mac.com
The Enjoy-the-Music website has it at the link below.
The Analog Dept also offers my spreadsheet as a free download.
So just curious the Black Patti recording any idea what it goes for?
I think you have the wrong thread, or at least the wrong person. I don't even know what the Black Patti recording is!
Sorry!
John Elison
My hero.
If there a way to calculate effective length for an arm with no manufacturer's data?
Dave
To calculate requires data. You can measure effective length.
Well John, I meant to show you this Tuesday, but forgot. Worked it out a while back when you told me you used the Law of Cosines. I think it's your formula where 1 - sin^2 is replaced by cos^2. It sure beats my Law of Sines formula!
Steve
Yes, that is simpler. My math skills are a bit rusty. I've made the change to my equation table.
Did you realign your tonearm?
Thanks,
John Elison
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