On Wed, Jul 14, 2010 at 10:42 AM, John Delacour <JD at pianomaker.co.uk> wrote: > At 09:41 -0400 14/7/10, jimialeggio wrote: > Jim I. writes: > > ... I guess it turns out the the calc uses vectors rather than the high >> school trig we are usually use. >> >> >> Geeks who understand vectors seem to do all kinds of cool and bizarre >> things with them... >> > JD comments: > > Sure, but vectors (as such) don't really come into this; it's simply the > same old basic trig applied to a very elementary mechanical law. Most of us > were taught both before we ever heard of a vector. > Nick writes: True, but recall that vectors are simply triangular dimensions that are understood to be forces (acting in a direction) rather than linear dimensions. As forces with direction they are now called vectors. The classic 3-4-5 right triangle can be be understood linearly as 3 feet by 4 feet and completed by the 5 foot hypotenuse, or it can be understood as a 3 pound tension, a 4 pound tension and completed by a 5 pound tension at the hypotenuse. JD writes: > > The distance of the string/bridge contact above the horizontal would be > calculated by multiplying the speaking length by the SINE of the angle, so > the longer the string, the greater that distance is for a given angle, but > by elementary mechanics the force (the tension) will be shared in the same > proportion between the x axis and the y axis, so the length of the string is > immaterial; only the angle and the tension are needed for the calculation. > In this case we are dividing the force between the y axis and a line > sloping up from the x axis, so we use the SINE rather than the TANGENT. > Nick again: Agreed, but note that Wolfenden and Nalder call out for the TANGENT to be factored with the rear string length. This length is measured from the front bridge pins to the rear string rest. Such a plan is quite practical in that it calculates the gap required at the rear string rest (duplex bump, etc.). When the gap is known, the "string/bridge contact above the horizontal" can be determined. A typical problem to solve might be (regardless of how long the speaking length is): GIVEN: a 6" rear string length and a desired angle of 1.5 degrees (the Wolfenden catch-all standard), what gap is required at the rear string rest bump in the unstrung belly? SOLUTION: 6 x TAN 1.5 = 6 x 0.02619 = 0.157" gap. Either way, with angles as small as those of downbearing, the SINE or TANGENT will work. For example, SINE of 1.5 degrees = 0.02618, while that of the TANGENT = 0.02619. As angles get larger these two functions depart noticeably (tangents grow larger) and serve decidedly different uses in mechanics. As to vectors, the shallow downbearing triangle thus formed is now understood in terms of forces rather than linear dimensions. The smallest useful model of this triangle is 1 lb (and a teeny tiny bit more) of tension in the hypotenuse, 1 lb. in the adjacent leg, and 0.02619 lb. compressive pressure downward (which is the same number as the angle function*). Thus, when the tension increases from 1 lb to 160 lbs. we multiply this 160 by the angle function of choice (say TANGENT) and we come up with a proportionately larger vertical force. This vertical force, which began in the small model as 0.026 lbs., has now proportionally increased to 4.19 pounds of anticipated bearing for this one string. This is the vector analysis in action, and it is completely tied to the trig function underpinnings. For you math purists, since the angle is so shallow the use of right angle trig still holds. You will have to employ the Law of Cosines to be perfectly exact, but for all the trouble the answer will agree quite closely to the simpler method above. * The Lowell gage is based on this concept, and the Wixie gage amounts to the same thing except that the readout is in angles and percentages of angles (I think this is correct). So, for you Wixie users, consider half angle factors as 0.009. Since rounded-off angle additions merely mount up, a 1 degree angle factor = 0.018 and 1.5 degree angle = 0.026. For those familiar with the bubble gage, and wish to relate the bubble to the Wixie: When the Wixie readout is, say, a 1.25 degree angle (assuming you don't have trig function calculator with you) a bubble type conversion number can be had by adding 0.018 (for the 1 degree part) to the 0.25 percentage part. The 0.25 percentage part conversion to bubble is had by multiplying 0.018 by 0.25 which yields 0.004. Adding 0.018 to 0.004 = 0.022 which is the TANGENT of 1.25 degrees (+ or -). To imagine the number of tick marks crossed by the bubble in the vial, divide 0.022 by 0.003 and you get 7.33 tick marks. The bubble gage should agree with the Wixie here. Nick > > JD > > > -- Nick Gravagne, RPT AST Mechanical Engineering -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://ptg.org/pipermail/pianotech.php/attachments/20100714/2ce68564/attachment.htm>
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