Our beams do not stretch. It would take about 1.5kg to stretch a 100mm length of 0.012" guitar string by 1mm. We don't have those kind of longitudinal forces in our models. Our beams merely bend.
All our beam curves are cubics. Taking the simple case of a single point load
F midway between two supports spaced at
L apart, the equation of the curve, for
x < 0.5
L, is:
y =
F(4
x3 - 3
L2x)/48
EI(The deflection
y is at a maximum for a midpoint load, and where the slope of the curve, i.e.
dx/
dy, is zero.)
The arc length
L' of each section of this curve can be derived from the differential of the equation of curvature:
L' = integral[{(1 + (
dx/
dy)
2)
0.5}
dy]
In the case of multiple spans, as in a CSB, the beam curve equations are vastly more complex, because of the adjacent bending moments, but they are still cubics, and arc lengths can be derived if one is feeling nutty enough.
Prototype bridge designers are not overly concerned with changes in arc length - they are concerned more with beam rotation and temperature expansion - these (and friction) are avoided by using suitable bearings:
- bridge-roller-bearing.png (158.48 KiB) Viewed 2968 times
Victorian engineers tended to slosh a gallon or two of heavy grease onto such bridge bearings.
In our models, as the beam bends and moves a little bit over its fulcrum points and its axleboxes, the values of the frictional forces at these points are in proportion to the reaction load at each point. This proportion is given by the coefficient of friction
u. These frictional forces will be highest toward the middle of an 0-6-0 frame, because this is where the frame loads are higher. The determinant of the frame loads is the CofG. For a typical 'balanced' 0-6-0, the two outermost fulcrum points each have approx 15% of the frame load, and the two inner ones have approx 35% each. Likewise, the frictional forces at the hornblocks are proportional to the axle load.
- continuous-friction-sources3.png (4.6 KiB) Viewed 2968 times
The direction of these frictional forces will depend on the direction the beam is moving over each support or axlebox. In all cases, the axis of the frictional force is horizontal, and will be in the opposite direction to the beam movement. Thus the frictional force(s) will tend to tighten a span as it becomes more tensioned, and weaken a span as it relaxes. Although the extent of the movement of the beam over a load point will alter the
work done by the beam, it
does not alter the frictional force at each load point, which remains proportional to the reaction force at the load point. In other words, the effect of friction will increase the degree of equalisation along the beam.
Taking the example of the 2-axle case where the left-hand axle encounters a bump and compresses its span, and likewise there is a relaxation of the right-hand span, the likely scenario for the directions of the frictional forces at the frame fulcrum points is:
- continuous-friction-sources4.png (2.46 KiB) Viewed 2903 times
In this case, the high frictional force at the middle of the frame pointing to the right (the mid-frame load is about 60%, say, for a typical 2-axle case) is countered (although to an unknown extent) by the two lesser frame frictional forces pointing to the left. The situation for an 0-6-0 is more complex, but, notwithstanding the disparity of the values of the frame load frictional forces, I agree with Will that the effects of the frictional forces generally tend to 'net out'. Because our beams are jiggling around over their load points all the time (on undulating track), I'm also inclined to agree with Will that we are dealing with sliding friction rather than static friction values, so I can understand why he doesn't get too excited about the 'finessing' of the friction situation by using lube. Some guitarists however feel inclined to provide a smidgeon of lubrication at nutbridges, to keep string tensions in good order:
The concentration of the largest frictional forces in the middle of the frame will I think tend to prevent beam creep along the frame.
