>Against positive downbearing, against a string offset angle, and pin >inclination, how is it physically possible for a string to climb, >creep, or otherwise get up a pin so it is no longer in contact with >the bridge cap, and stay there? I don't buy it. I'd like to see >anyone take a reasonably normal piano and make a string stay up a >pair of bridge pins without touching the cap. Most everyone seems to >take it on faith that this happens naturally and ubiquitously, but >no one seems able or willing to demonstrate that it is indeed >possible by doing it, and proving it by sliding something under the >string between bridge pins. > >Ron N OK, just for fun (or for the sake of argument, if you prefer) I'll take a crack at this. I thought I would throw a little math at this. My assumptions: String tension T = 150 lbs. Side bearing angle 8 degrees Down bearing angle 1 degree Bridge pin angle (relative to cap surface) 15 degrees Friction between string and bridge pin is given by static friction formula FR = u N where: u = coefficient of friction between string and bridge pin. This will depend on the material of the string and the material of the bridge pin and on surface finishes of each. For high polished steel on highly polished brass it would be on the order of 0.2. For rusty steel on rusty steel it might be on the order of 1.0 or more. N = normal force (force perpendicular to bridge pin) exerted by the string The side bearing force is given by SB = T sin (8 deg) = 150 (.139) = 20.9 LB The down bearing force is given by DB = T sin (1 deg) = 150 (.017) = 2.7 LB Each of these forces will have components normal to the bridge pin and parallel to the bridge pin. FOR THE SIDE BEARING: Normal force N1 = 20.9 cos (15 deg) = 20.2 LB (note that this is towards or into the bridge pin) Parallel force P1 = 20.9 sin (15 deg) = 5.4 LB (note that this force is down toward the bridge cap) FOR THE DOWN BEARING: Normal force N2 = 2.7 sin (15 deg) = 0.7 LB (note that this is away from the bridge pin and is counteracting the normal force from the side bearing) Parallel force P2 = 2.7 cos (15 deg) = 2.6 LB (note that this is down toward the bridge and is adding to the parallel force from the side bearing) TOTAL FORCES: N = N1 + N2 = 20.2 - 0.7 = 19.5 LB P = P1 + P2 = 5.4 + 2.6 = 8.0 LB If the force parallel to the pin is higher than the friction generated by the normal force then the string will want to move down the pin. If the friction force is higher than the force parallel to the pin then the string will want to stay where it is (even if it's above the bridge cap). FRICTION FORCE: This is going to depend on the friction coefficient that you assume: I think a reasonable number might be 0.6. For this assumption friction force is: FR = u N = 0.6 (19.5) = 11.7 LB In this case the string will want to stay where it is, and not move down the bridge pin. For this set of assumptions (which I think are probably representative of pianos out there) the string will want to stay above the bridge cap. I can imagine another set of assumptions (perhaps also representative of other pianos out there with different side bearing angle, different down bearing angle, different bridge pin angles, and different coefficient of friction from different bridge pin materials or surface smoothness) where the numbers would work out the other way and the string would want to move down the pin. A few things to note: As the piano ages the string and the bridge pin are probably going to get some corrosion which is going to raise the coefficient of friction, making the string less likely to move down the pin. The higher the string is above the bridge cap the greater the side to side displacement (the greater the side bearing angle) because of the angled bridge pins, and the greater the down bearing angle, which will increase all of the normal and parallel forces. However, the friction force will go up fastest, making the string less likely to move down the pin. This is somewhat counterintuitive to me, but there you are. None of these forces will make the string move up the pin. So, the string won't 'climb' the pin. However, if due to humidity increase the bridge cap swells and raises the string, then the string isn't going to follow it back down (at least in the situation given in the example above) when the cap shrinks back down on a humidity decrease. So, the idea of tapping the strings down periodically is perhaps not unsound practice. On a practical note I have to agree with David I that my experience in prepping new pianos is that on tapping the strings down (and I'm talking about a light tap) I often see a noticeable downward movement of the string and will often hear a noticeable improvement in the tone. I don't have an explanation for why strings on newly strung pianos would end up some distance off the bridge cap, but my perception is that they do. I don't do too much prepping of new pianos these days, but I'll be happy to try to stick a feeler gage or piece of paper under some strings the next time I'm around an unprepped new piano. I'll report back. Phil Ford San Francisco
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