Capstan angle

[Phillip Ford]

    Well, it's two against one, so you guys must be right.  The 
capstan and key form one solid unit.  This unit is rotating about a 
fixed point.  An applied force on this unit will create a torque 
about that fixed point.  What determines this torque is the distance, 
in a direction normal to the force, to the rotation point, multiplied 
by the force.  See the link that you gave for clarification.  If a 
vertical force of 10 lbs is applied at the top of the capstan, which 
is 5 inches horizontally from the balance point, then the torque at 
the balance point produced by this force is 50 in-lbs (and the torque 
required on the front end of the key to balance this is also 50 
in.-lbs - so, if your measuring point is 10 inches from the balance 
point, then you have a downweight of 5 lbs).  The angle of the 
capstan, or where it touches the key,  has nothing to do with it.
    If you want to take the capstan itself as a free body, then if the 
capstan is vertical and a vertical force of 10 lbs is applied to the 
top of it, then the reacting force at the key contact point is a 
vertical force of 10 lbs.  The capstan will in turn apply this 10 lbs 
force to the key.  Torque about the balance point is 10 lbs x 5 
inches = 50 in-lbs.
    Now angle the capstan way back so that it contacts the key at a 
point 4 inches from the balance point.  Apply a vertical force to the 
top of the capstan, still at a point 5 inches from the balance point. 
The reacting force on the capstan at the key contact point is a 
vertical force of 10 lbs AND a moment of 10 lbs x 1 inch (the 
distance between the applied force and the reacted force), or 10 
in-lbs (if you want to think of it in this way, this is the moment 
needed to keep the capstan from falling over as a result of the force 
being applied off to one side).  These reactive forces are in turn 
applied to the key.  So, we have a vertical force of 10 lbs being 
applied at 4 inches from the balance point, and a moment of 10 in-lbs 
being applied at 4 inches from the balance point.  Moment about the 
balance point is 10 lbs x 4 inches + 10 in-lbs.  Moment about the 
balance point is STILL 50 in-lbs.
    Now angle the capstan way forward so that it contacts the key at a 
point 6 inches from the balance point.  Apply a vertical force to the 
top of the capstan, still at a point 5 inches from the balance point. 
The reacting force at the key contact point is a vertical force of 10 
lbs. AND a moment of 10 lbs x (-1) inch (the distance between the 
applied force and the reacted force, it's negative because you are 
moving away from the balance point), or -10 in-lbs.  These reactive 
forces are in turn applied to the key.  So, we have a vertical force 
of 10 lbs being applied at 6 inches from the balance point, and a 
moment of -10 in-lbs being applied at 6 inches from the balance 
point.  Moment about the balance point is 10 lbs x 6 inches - 10 
in-lbs.  Moment about the balance point is STILL 50 in-lbs.
    If you don't want to believe me it's a simple thing to prove to 
yourself.  Set up a lever (or a key) and put an arm (such as a 
capstan) at wildly different angles, contacting the lever at greatly 
different points, but make sure that the weight that you put on is 
always the same distance from the fulcrum, or balance point.  See how 
much weight it takes to balance on the other side.  You'll see that 
it doesn't matter where you attach the arm or how you angle it.  The 
only thing that matters is the distance of the weight from the 
fulcrum.
    I would also caution you against being deceived by anecdotal 
evidence of people who say that they have 'improved' the key ratio by 
angling the capstans back.  What they have accomplished is to move 
the contact point between the top of the capstan and the wippen heel 
a little closer to the balance point, so that the key ratio has 
changed.  That change resulted from moving the contact point, or 
point of force application, a little closer to the balance point. 
Not from angling the capstan so that it contacts the key closer to 
the balance point.

>Hi David, and Phil
>
>I got to agree with Dave here.  In fact, if the key is oriented 
>horizontally and the ratio equated to the key being balanced thus, 
>then the key ratio is exactly the straight line from the front of 
>the key to the middle of the balance rail pin to the middle of the 
>hole of the capstan.

I try to measure to the capstan contact point.  This is easier said 
than done if the capstan is angled, because it's hard to tell exactly 
where the contact point is, and it changes with the capstan height. 
If you're using the key hole as your measuring point that might, at 
least partially, explain some of the discussions that we've had in 
the past about why various methods of measuring the action ratio seem 
to be inconsistent.  I'll have to look back at my Stanwood articles 
to see how he defines key ratio.

Regards,

Phil Ford

>   Thats the balance ratio in anycase.  The ratio of the key in any 
>other position can be calculated in the same way really after one 
>adjusts for the offset from horizontal. Or... one can use Torque to 
>figure it out for any given angle... which is more ususal I 
>suppose.   
>In essence tho.. the key ratio is dependant at any given time on its 
>angular orientation.  But if half way through the key stroke the key 
>is horizontal... then its perhaps easier to just view it that way to 
>begin with for simplicity.
>
>
>Cheers
>RicB
>
>
>Sorry, but you are incorrect about the key ratio.  The key ratio depends
>completely on where the capstan enters the key, nothing more nothing
>less.  The ratio of the wippen lever, on the other hand, will depend on
>where the top of the capstan contacts the wippen heel.  
>
>David Love
>davidlovepianos@comcast.net <mailto:davidlovepianos@comcast.net>
>
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