Yeah I was suspecting something similar but since I do not have the required knowledge in this area I differed to Dr. Ru's authority. Maybe he just did not take my query seriously and forgot what was written??HarryvL wrote: ↑Fri Jun 15, 2018 4:56 amMark, that is indeed very confusing. I had another close look at the paper and see that I was referring to Table 1, whereas you were referring to "nomenclature" at the end. You are right. The paper contains an error in that part. The description for "I" is simply wrong and the explanation of the author does not help. I is the second moment of inertia of the cross section of the hairspring and the unit is m^4
Hello again HarryvL. I got back to that paper again, “The Mechanics of Spiral Springs and It's Application in Timekeeping" and have another concern. I am having trouble understanding how they get their equations for the moment of force at point P(r, theta) on the spring. These equations have Fcx and Fcy multiplied by the shortest distance of point P to their line of action. So the moment of these forces at point P is accounted for. These equations also add the term for torque Tc at point C (center of the collet) directly into the equation for the moment about P. Shouldn’t this be scaled since the perpendicular distance to the force that generates Tc from point C is not the same as the perpendicular distance from point P? The diagram does not indicate they are purposely picking a point P on the spiral with the same distance that C has from the line of the force generating Tc. So these equations make no sense to me at this point?HarryvL wrote: ↑Fri Jun 15, 2018 4:56 amMark, that is indeed very confusing. I had another close look at the paper and see that I was referring to Table 1, whereas you were referring to "nomenclature" at the end. You are right. The paper contains an error in that part. The description for "I" is simply wrong and the explanation of the author does not help. I is the second moment of inertia of the cross section of the hairspring and the unit is m^4
I tried doing that and I do get a torque Tp but it isn’t Tc.HarryvL wrote: ↑Fri Jun 29, 2018 9:47 pmHi Mark, I had a quick look, but I believe the formula is right. It perhaps helps if you imagine the central torque being generated by two equal but opposite forces spearated by a fixed distance d, working on an imaginary lever connecting them to the point at which you want to know the total moment. You will then see that it doesn't matter how far or how close this couple is from the point of interest or even in which direction the force pair points. This explains why you can simply add couples to the moment equilibrium equation without considering their point of application. Try to draw it !!
I hope this helps.
Sorry, I just noticed this post. I take it you mean https://en.wikipedia.org/wiki/Couple_(mechanics) and independence of reference point. OK, so there exists a couple called Tc but forces Fcx and Fcy are still needed to keep things in static equilibrium?HarryvL wrote: ↑Fri Jun 29, 2018 10:15 pmPS: I don't think Tc is generated by the spring action on the collet. I think it is from an external source, like the inertia of the balance wheel? In any event, think of a couple as a "couple or pair" of forces as described above and it will become clear that point of applications does not matter.
OK thank you. However, I am still missing something because that article I linked to says a coupled-pair has equal but opposite and parallel but non-collinear forces (vectors), which to me means the system is in static equilibrium. So why is Fcx and Fcy present or what are they there for?