Downbearing

[David B. Stang]

Ric,
I think the String Angle from front termination  =  0.66 deg
and the angle from the aliquot/hitchpin          =  1.33 deg

You got them swapped. Just by observation, the angle from the front should be smaller 
than the back since the speaking length is larger than the back length.

(By the way, I was less bothered by the commas than by the 10 digits of precision.
Any more than 2 or 3 digits doesn't mean much.)

I have attached a diagram to show how I think you arrived at these numbers. The
red line represents the string. The angle at the bridge is c. The string displacement 
is the length Z. a is the front angle, b is the back angle.
     Z = X sin(a)
     Z = Y sin(b)
     d = a + b      where d = (180 - c)
therefore
     X sin(a) = Y sin(b)
so
     sin a  =  (Y/X) sin( d - a )   =  (25/50) sin( 2deg - a )

solving for a (transcendental equation):
     a = arcsin (  (1/2) sin( 2 - a ) )

this gives a = 0.66 deg, b = 1.33 deg , Z = .58 mm

(Actually for such small angles, it's a pretty good approximation simply to say
     a = (1/2)b
     d = a + b
which gives you the same result to the 4th decimal place)

The force can be considered as vectors similar to the triangle of lengths,
where the "X" and "Y" contributions are added:

     force = 160 lbs ( sin(a) + sin(b) )  = 5.58 lbs

-- someone correct me if I'm wrong. --

Being kind of a Newbie, I'm still not always clear on what people mean when they 
say "downbearing". I wonder if everyone is on the same page. Downbearing is really a 
force, right?  So when someone refers to "downbearing" as an angle, or as a length, 
what are they talking about??  Which angle?? Which length??  If the point is to recreate 
the same downbearing on a rebuild, then it doesn't matter I guess, as long as you 
measure the same thing before and after.  But if you make a general statement like 
"good downbearing in the treble is .1-inch", are you talking about the string displacement 
at the bridge from the horizontal, or what?? Does such a general statement make any sense, 
since Downbearing is really a force, and the same displacement on different pianos would 
result in different forces?? 

Thanks. And thanks for the opportunity to refresh some of my trig.
Dave



Ric Brekne wrote:

>Hi folks
>
>I'm working on a spreadsheet for calculating a few different things
>about downbearing and just wanted to check  a few numbers with you
>before going further then I am now.
>
>Say you have the following string situation.  Assume a bridge width of 0.
>
>Speaking length 50mm
>back length   25 mm
>Angle at the bridge 178 degrees.  (i.e. the angle created by the strings
>deflection upwards.)
>Tension on the string lengths 160 lbs
>
>Are the following results then valid ?
>
>Undeflected string length total 74,98984643
>String angle from the front termination 1,333363422 degrees
>String angle from the aliquot / hitchpin  0,666636578 degrees
>String deflected Height  0,581737034 mm
>Downwards force on the bridge. 5,584676 lbs
>
>Thanks for any and all corrections.
>
>RicB

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