Figure 1: ribbed platic sleeve
where µ is the coefficient of static friction and is the normal force of the plug (or hole) on each rib. Using to denote the stress on a compressed rib, the retention force due to one rib is:
This is the initial value; retention force will decrease over time due to creep.
where is the stress on the rib along its width. Assuming that the maximum stress is significantly larger than the yield stress of the material (unfortunately, this is certain to be the case for us). Then we can approximate the stress-strain curve as a straight line for the elastic region, going from (0, 0) to (,), and another straight line for the yield region, going from (,) to (,). and are only approximately the yield strain and stress; for best results, they should be read from the stress vs. strain curve. Using this approximation, we find that the force of deformation is:
Combining this with the retention force computed above, we find that the maximum insertion force (which occurs as the plug bottoms out) is:
Creep probably will not reduce the insertion force very much because insertion happens fairly quickly.
where is the stress along a rib. Using the small-angle approximation , the strain along a rib due to the tilt is . Half of each rib is compressed by the torque, and half is relaxed. For small displacements, such as we are assuming, the stress changes approximately linearly with strain, but differently for the two halves. For compression, one uses , the local slope of the stress-strain curve. For relaxation, , the elastic modulus (slope of curve at 0,0) is a much better approximation. The largest displacement for which these assumptions are valid is approximately that displacement resulting in going to zero at the end of the relaxed portion of the ribs. For flexure past this point, the plug will deflect more for a given moment than predicted by the equation, because the relaxed portion of the ribs stop contributing once they are relaxed to zero strain. for compression, for relaxation, for
The angular displacement caused by a moment on a pair of opposing ribs is: for This is the initial displacement; displacement will increase over time due to creep.
Desired maximum ø (angular displacement of plug) is roughly 5 mrad and maximum (transverse displacement of tip of plug) is about 15 µm, based on light throughput. In addition, we should keep the numbers in the range for which the equations are valid, since behavior degrades outside that range. Note that M (moment on the plug) and (transverse force on the plug) both contribute to displacement of the plug tip, but in opposite directions (assuming M and are both transmitted through the fiber). For a 3 mm long plug, an angular displacement of 5 mrad will displace the tip by 7.5 µm.
Crude measurements of a fiber with an extruded nylon jacket indicate that at a radius of curvature of 100 mm, M (moment on the plug) is less than 5 N-mm and Fy (transverse force on the plug) is less than 0.1 N (possibly much less). Additional moment and transverse force will be exerted on the plug during manual insertion. We estimate that a plugger can easily keep the transverse force below 2 N (1/5 the maximum acceptable plugging force), and probably less. With our current plug design, the plugger holds a portion of the plug 10 mm long, so the maximum moment induced by plugging is 20 N-mm. Hence the maximum expected moment and transverse force on a plug are: M = 25 N-mm, = 5 N.
Assume we make the sleeve out of TFE (teflon). Suppose the sleeve has 8 ribs, each with a = 0.4 mm, b = 0.30 mm, = 0.06 mm, c = 3.0 mm. Then = 0.20. Inspecting the stress-strain curve (fig. 3, sec. III, ref. 1) we find: = 1750 psi = 12.1 MPa, = 0.03, = 3250 psi = 22.4 MPa, = 65000 psi = 448 MPa, and = 9000 psi = 62 MPa. We do not have accurate data for the coefficient of static friction (µ) under load, but it appears to be about 0.02 at 9000 psi (table II and fig. II, sec. V, ref. 1).
The equations given above for retention and insertion force ( and ) are for one rib; for eight ribs, multiply the result by eight. The equations for angular and radial displacement ( ø and ) are for a pair of ribs; using the simple approximation that stiffness scales by cos^2, eight ribs reduces the displacements by a factor of 2.
result rating notes 4.3 N marginal 7.3 N good Fd3.0 N ø 8.2 mrad (M = 25 N-mm) 2x too big ø/M0.33 mrad/N-mm, for ø<10 mrad 1.2 µm (Fy = 5 N) good /<15 µm
Creep reduces stress about a factor of 2.0 in one day, and 2.2 in a week (fig. 9a, sec. III, ref. 1); , ø and degrade proportionately. The creep factors are relative to 1 second, which seems to be roughly when the stress was measured in the stress-strain curves. Unfortunately, these figures are for tension, not compression; no data were found for compression.
x = tangential position y = radial position z = axial position a = tangential width of rib b = radial thickness of rib before deformation b = change in thickness of rib due to insertion c = axial length of rib = insertion force = retention force = radial force = force of deformation M = moment ø = angular displacement µ = coefficient of static friction = strain = yield strain (approx.see text) = maximum strain of rib = (b - b) / b = stress = yield stress (approx.see text) = maximum stress (stress on rib after insertion) = slope of stress-strain curve at (0,0)the elastic modulus = slope of stress-strain curve at (em, sm)
