13. Variation of the Theme
How is the separation of the tines effected by the modification of their length, tilt angle, width and thickness? Equation 2 was applied to calculate the results. All data have been assembled in Graph 5 so that we can recognise their correlations
Sample Calculation 3 The curves have been calculated by changing only one of the variables while keeping the others constant. The variables were: tilt angle between the tines τ = 15 °, effective length l = 9mm, width at the place of bending b = 3.5mm and thickness t = 0.4mm also at the location of bending. They are lined up along the vertical black dotted line.
For this lined up set of data the calculation showed a widening of the tines by 0.84mm as it is indicated along the horizontal black dotted line.
The brown curve shows a widening with increasing tilt angle.
Changing the width of the tines has the least effect, the curve is the flattest. Altering the material thickness has the highest impact on the tine separation, followed by the change of the effective length.
It is not an unknown fact that flex nibs are made of thin material and have comparatively longer tines.
The diagram can be used to estimate the outcome of variations. For example: Reduction of the width to 3mm increases the separation from 0.84mm to 1mm (ΔW=0.16mm) and reducing the thickness to 0.35mm enlarges the separation to 1.2mm (ΔT=0.36mm). Application of both variations at the same time would result in a tine separation of 1.36mm.
14. Permanent Damage
Earlier on I talked about applying a force to a nib so that its tines spread beyond the point of no return. Ingeneering calls this a distortion extending beyond the Yield Strength of a material. I have written a detailed paper dealing with this topic under the heading Stresses and Strains.
Within the limits of elastic deformation the force‑deformation relationship is proportional and reliable calculations are possible using the Modulus of Elasticity, also called the Young’s Modulus. Graph 6 shows a typical pattern of a ductile material such as steel.
As long as the strain (elongation per cross‑sectional area) stays under the straight line up to the Yield Strength no damage occurs. Amongst others, Gold’s curve does not have a distinct yield strength as shown in the graph. When this is the case, then the yield strength is assigned to a point where the plastic deformation equals 0.2%.
For most standard gold the yield strength is 200MPa (megapascal = 1N/mm2) with little variation. Nibs are mostly made of 14 karat gold alloy with a yield strength around 320MPa. The latter I use in the following Sample Calculation 4.
Just to add another useful comment: In the segment of the curve called Strain Hardening is where work hardening happens, and Necking occurs when the cross‑section of a component reduces before it breaks.
Sample Calculation 4
I use the dimensions and data of the Waterman Ideal 2, fine nib. Its tines spread by 1.5mm under a load of 3N. Effective length = 9mm, effective breadth = 7mm (both tines), thickness of tines = 0.4mm
F = σ×b×t2 / l×6 = 320×7mm×0.16mm2 / 9mm×6 = 6.3N
The load at the yield limit is 6.3N for a flat nib.
Antonios Zavaliangos, a contributor to the forum, has performed a test when measuring the correlation between the load and tine separation of a particular nib. The graph shows a residual spread of the tines after the nib has been loaded beyond the yield limit. With his permission, I show graph 7. As the load increases the tines spread. When the load has reached 2.5N (250g), he reduced it back to zero; however, a residual deformation of 0.45mm remains.
In the next chapter, Nibs are not Flat, I show in Drawing 7 that profiled tines bend only to a fifth of the flat profile. Based on this information the force necessary to cause permanent deformation would be 30N (3kg in old terms).
15. Nibs are not Flat
Yes, most nibs are not flat but curved; and let me say it right now: there are some beautiful exceptions. However, now, it is high time to have a attend to the matter of curvature. Looking at our familiar equation δ = F×l3 ÷ 3EI we realise that besides the effective bending length it is only the moment of inertia that influences the degree of bending of the tines. The moment of inertia considers all aspects of the nib’s profile.
To help develop a feel for the shape characteristics of a nib, let us return to Drawing 4 from further up, which shows the flat strip. The profile’s dimensions are t and b as well as the location of the bending axis x-x define its area of moment of inertia.
From experience we know, the closer one brings something to the axis of bending the easier it is to bend or break, eventually. Wrestlers know this, and if you want to snap a branch, you bend it over your knee and not out in free air. For the technically minded among the readers, I mentioned that the Parallel Axis Theorem, as well as the Radius of Gyration, consider this. All others, don’t worry, I will talk around it without mentioning those expressions again.
Did I admit before that higher mathematics is not my forte? Luckily, ingeneering handbooks are reliable sources for area momenta of inertia of many shapes. Alas, I could not find a formula for a circular, tubular section as it presents itself in the profile of nibs.
In a way, this may be good because I decided to use the profile of our example, b = 3.5mm and t = 0.4mm and kink it in the middle by α = 7.5° as shown in Drawing 7.
Remember folding a sheet of paper? Likely, as expected, the effect of kinking was significant. In our example with the flat profile, the calculated tine widening was 0.69mm; the kinked profile opens only by 0.14mm, just to a fifth. The stiffening is caused by the shift of the axis of bending from x‑x to y‑y by the amount of Δy = 0.11mm. I won’t go further into this because to work this out I had to apply one of those unspeakable methods.
I am always happy when the mathematics of physics agrees with what happens in my observations. At such times, my physics teacher would place a q.e.d. under his calculations with such determination that often the chalk would break because of his vehement final full stop.
Continuing from this observation, one could construct the curved profile from a flat profile with several kinks for the purpose of calculation. I would have used this method if I had to find my way around designing a nib. For now, we got the gist and therefore, let’s move on.
16. Effective Dimensions
The terms effective length and breadth have come up already several times. Together with the radius of curvature, they are the shape dimensions, which determine the way nibs bend and spread their tines. With a different emphasis, I have written about this topic in the paper How to… for Nibs, from where I took Drawing 8. Now, let us explore this drawing from a different angle.
Presuming that the nib material is of consistent thickness the golden dotted line marks the shortest width combined with the flattest radius. R3 is smaller than R2 and as we have proven in the previous chapter the smaller the radius (the larger the kink angle), the stiffer the profile. The green dotted line is the effective length of bending, the average distance from the tip to line b.
In some nib designs the effective dimensions are influenced by a variation of the thickness, see Drawing 9. The placement, profile and gradient of the cross‑over can vary. Thickening the tines as shown makes them flex in a confined area and they bend less or not at all along their length. This causes the slit to widen linearly rather than curved, which influences its capillary characteristics.
Drawing 10 shows two nibs with their slits opened to the same degree but one with a proportional, linear widening, the other with a curved (just slightly overdone to make my point). One can find arguments for either being of advantage. Theoretically, the slit widening with the linear increase should hold and transport the ink better than the one with the curved expansion, but only in conjunction with the capillary action of the paper. I wonder whether reality agrees with me. Let me know your experience.
Continue reading about G – Applications and Modifications – chapt. 17 – 19
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