|
TonyGonzales84 -> RE: String Tension Variation as a Function of Scale Length (Jul. 21 2020 23:50:37)
|
This is where I bravely tuck my tail and flee! Run awaaaaay!
I am halting my little (unplanned) journey into the world of modern acoustic research. The original question I was looking into was something like, given identical strings tuned to an identical frequency, how does a guitar’s string’s being attached to the flexible and massy guitar top affect the classic, theoretical and idealized tension-length relationship for a string with both ends attached to a rigid support? This relationship is (T2/T1) = (L2/L1)^2 where T is the string’s tension (approximated as constant for each distinct case), and L is its (scale) length (also constant for each distinct case). (T2 and L2 are those of the second guitar, T1 and L1 are of the first guitar.) Not having made progress in answering this question, I recommend continuing using the classic relation.
Solutions, analytical and numeric, very well may exist, knowledge being readily known by the specialists, but I have not been able to find such solutions nor the data that may feed into deriving them. With tongue firmly in cheek, I mention that there could be a hint to why in the deafening silence in the many textbooks on partial differential equations, on wave physics, and on mechanical vibrations that often derive and discuss strings with flexible supports, but with massless springs – not our case! -- No one is showing the string attached to mass-spring systems!
My original attack was to focus exclusively on deriving the fundamental frequency and use a Rayleigh-Ritz type of analysis, to avoid dealing with the full equations of motion. Numeric results along this line of attack would show trends, hopefully allowing understanding of how close our case is to the classical fixed-fixed relation. I might, in the near geologic future (again, tongue firmly planted in cheek), investigate very simple cases similar to the equivalent mass-spring system I assumed in my Rayleigh-Ritz attack…who knows?
Most papers I was able to quickly dig up (including those titles directly provided by Konstantin and Richard – thanks to both!, and the papers I then dug up after reviewing those) result in working in the frequency domain, quantifying and investigating the response of the full guitar when “played.” This work is very interesting, including nonlinear effects of geometry and of materials properties (nylon’s nonlinear elastic behavior, and even the effects due to the winding on the bass strings), not to speak of damping characteristics (again, for investigating response, not for free vibrations – the fundamental frequency problem of our interest).
Many roads coming out of the papers I have found, and of titles provided by Konstantin, lead to ResearchGate, which evidently requires an institutional email address in order to obtain full download privileges. This is a typical and not unreasonable barrier to entry, but one that I need to figure how to get past. I did find my name in there, as a co-author (in a former life) – a potential path to possibly explore later.
I have learned a lot and had fun too!
|
|
|
|
