9. When one Applies Sufficient Force…
… one can bend any nib! When I say force, what sort of force do I refer to? Allow me to make you reacquainted with the ideas and rules about forces. I will dig around in my old school books and see how they approached the topic, there.
Remember the central aspect of forces? They are vectors, which are defined by having a magnitude and direction. Therefore, in technical drawings, they are depicted as arrows. For completion’s sake: There are other units which have only a magnitude but no direction, like the weight of a bar of chocolate, an hour of walking in the park, temperature, etc. They are called scalars.
When dealing with forces, one must adhere to a few rules of which I will mention some which are significant for our perusal.
- A force can be moved along the line of its direction, the line of its action.
- For any force to exist or to build up, it needs an opposing force of the same magnitude. Good old Newton’s Law still applies.
- Forces oriented along the same line of action, their magnitudes can be directly added or alternatively subtracted when pointing in opposite directions.
What happens when forces do not follow the same line of direction? As long as they aim at the same point, their combined (resulting) force (the resultant) can be determined with the help of a vector diagram, either graphically (which is usually accurate enough), or mathematically, applying the formulae of good old Pythagoras. This resultant force acts upon the point of action in the same way as the individual forces.
Just for completion’s sake: If the forces do not aim at the same point or do not even exist in the same dimension (plane), they still can be combined, but then, things get more complicated.
Let me explain: I show here figure 1 from somewhere else again to save you from scrolling up and down. We have two forces V and H combined into their resultant F. The principle can be reversed, namely: A force F can be separated into its vertical and horizontal (or any other directions) components with the same diagram. So, if V is the writing pressure, what is H? It is the force, which needs to be overcome against friction between nib and paper. In this diagram the friction needs to be overcome through pushing, if it’s a dragging force, then the diagram would be a mirror image flipped around the vertical axis.
The same applies to three-dimensional systems as shown in figure 2. Here the force F is divided into its vertical component V and its horizontal projection H_{R} with its two parts H_{1} and H_{2}. V and H_{2} have the resultant V_{R}. V and H_{1} would have the resultant V_{L}, correspondingly.
Forces and their lines of action are not arbitrary. They follow lines or planes (which can be arbitrary) of freedom of a mechanical system in which they act.
For example: The application of a force F to a fountain pen may not be in the recommended direction for the nib to move across the paper, which could be the direction of H_{1} or H_{2 }because, in both situations, the nib would experience a sideload. For analysing the force distribution analogue with the scenario of a fountain pen, the flipping of the force F around the vertical axis of V (as described before) would correspond more with the real situation.
Why I haven’t drawn the diagrams that way? Good question! Because they look too odd for the explanation of the vector addition of forces. Having said this, why don’t you draw them, as an exercise and send them to me.
I will add them here. Besides, it will become clearer! Promise.
By the way: When I studied (in the late sixties), we were taught to write the letters symbolising forces in an italic and sort of Gothic font. Since I could not find it in my drawing program and also was not sure whether this tradition still applies, I wrote them in italic, only.
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Now, let’s apply our new knowledge to the situation of a fountain pen as shown in figure 3. The active, vertical force for writing is F_{V}. It is the sum of the forces F_{MV} and F_{G}. They can be merely added because they act along the same line of action.
Wait, don’t panic: The forces F_{MV} is the vertical component of the force F_{M}, the tangential force caused by the momentum with the radius r_{M} around the point where the hand rests on the surface. The force F_{G} is caused by the weight of the unsupported part of the hand and the weight of the pen. That was not too hard?
Figure 4 shows how the vertical force F_{V} is split up into F_{T}, the force which acts to separate the tines and F_{R}, the force needed to overcome the friction between the nib tip and paper.
Let me point out something, which is already well known by experienced flex nib writers: “When you reduce the angle α between pen and paper, the nib responds more readily to writing pressure variations.”
Figure 5 demonstrates this, it is a variation of figure 4. You can notice that with the same vertical force F_{V} but a reduced angle α the tine spreading force F_{T} increases, simultaneously, the friction force F_{R} reduces. For the technically minded: F_{T} increases with the cos of α while F_{R} reduces with the sin of α.
F_{T}_{ }= F_{V } × cosα F_{R} = F_{T}_{ } × sinα
As I said, this is not an unknown fact; here I offer a technical explanation. In the picture, you can see that the thickness of the feed is a limiting factor.
Ingeneers of fountain pens for flex nibs have catered for this and reduced the thickness of the feed as can be seen in photo 12 thus, permitting the pen to be tilted even more, as shown in figure 5.
There are more good things to say about this construction and the nib, and I will return to it later, more than once.
10. Spreading of Tines
After we isolated the forces acting on the tip of the nib let’s have a look at how they separate the tines. Why? Isn’t it obvious? It appears obvious, but you sense already, there is more behind it than meets the eye.
I have written already about the reason why the tines open in the chapter about Fountain Pen Nib Mechanics; therefore, I only repeat what is necessary to know with regard to flex nibs. Let’s just quickly recapitulate, adhering to the old motto: “Repetitio est mater studiorum.”
In drawing 1 I show a simplified nib. The opening of the tines is a consequence of the nib being bent (during manufacturing) by the tine angle τ along the axis in line with the slit. Cut out a model from a piece of paper and play with it. You can see that increasing τ results also in an increase of W even though the deflection of the tines is the same. Good thought, however, there seems to be always an, however.
Drawing 2 is an ingeneering drawing showing the geometric relationship between dimensions. The vertical bending caused by the force F_{b} by the amount b applies the F_{d} = F_{b} × cosτ onto the tines which results in the deflection d of the tines as follows:
d = b / cosτ
meaning that the tine deflection d increases more rapidly than the amount of bending b due to the leverage of cosτ.
The widening of the slit follows the equation:
W/2 = b × tanτ
or W = 2 b × tanτ
Again, the larger the angle τ, (the tilt angle between the tines) the wider the gap between the tines at the same angle of bending.
The benefit of this geometry sounds promising, however, when considering the forces, it does not look quite as bright. Now, it is the other way round. Force F_{d} reduces by cosτ. Or in other words, the larger the angle τ, the more a nib resists opening the tines.
Furthermore, approximating the radius to a straight line is somewhat misleading. A flat sheet bends much more readily than a curved one (take a piece of paper and check it out). Curving the tines induces stiffness that’s why all cars look like potatoes with ridges because they can be manufactured from much thinner metal sheets (also more work-hardening) and still provide equal rigidity (but not the same protection). I will address the curved tine topic a bit further down.
To summarise: I demonstrated the interaction between the writing angle and bending force, the tilt angle of the tines and their resulting widening in a simplified manner. However, the correlation between shape and forces very much depends on the construction of the nib such as their cross-sectional radii and variation of curvature, the material and overall dimensions, self-evidently.
Before I go into that, let’s have a mental rest and again consider the question about the basics of:
11. Bending
I cannot remember having reverted to my old ingeneering handbooks as much as I do now, not that I have forgotten the content but just to check on how they explain things there. There is a marked difference between knowing something and passing it on, comprehensively and comprehensibly.
Let’s begin.
The literature on the fundamentals of bending does not distinguish between elastic deflection or plastic deformation. (If you are unsure, sidetrack to Stresses and Strains for a moment.) I guess, the resistance a material musters to counteract distortion does not influence the principles of bending; however, material characteristics determine the outcome.
In ingeneering principles on statics and dynamics, anything that bends is called a beam, which comes in all kinds of versions. These categories have been established to help sort out the formulas for calculating the forces and dimensions, formulas, which have been calculated and derived by the clever people during hours of drudgery, so that we can focus on the actual task at hand. For us, the fountain pen people, it is enough to know that seen in bending terms, nibs fall into the category of the cantilever beam.
Just to ensure we all have the same understanding, please consider the definition. “A cantilever is a beam fixed immovably at one end. The load is taken up by the bearing in the form of a momentum and shear stress.” See the example in figure 7:
I could have chosen a model more appropriate to writing but, like you may have sensed already, the forces and bending characteristics of nibs are very intricate but learning the basics helps to develop a feel for what’s going on inside a nib’s material. The picture simply expresses: “When applying a downward force, anywhere along the cantilever, it bends by the distance δ.”
In drawing 3 you see a symbolic depiction of figure 6 as commonly used in ingeneering to quickly clarify the beam characteristic, isolate the load distribution and to determine the method of calculation.
These are the commonly used parameters for calculating beam:
- Thickness t of the material
- breadth b
- length l from the firm anchor to the point of action of the force F
- distortion distance δ – for example, the total flexing of the nib, including tines and all
The equation for calculating the deflection contains the modulus of elasticity of the material and the shape characteristics of the beam. Since we are not calculating anything in particular but want to compare the effects forces have on a nib in general, we can neglect these factors. Furthermore, we want to find out how varying the dimensions of a nib affect its behaviour. Keeping the material and its condition the same, the slimmed-down equation is shown in equation 1:
δ ≈ (F × l^{3})÷(b × t^{3} ÷ 4) … …. Equation 1
Discussion of equation 1:
The deflection distance δ increases
- proportionally with the force F
- by the length cubed
distortion distance δ decreases
- inverse proportional to the width b
- by the inverse of the thickness cubed
Applied the above to a nib, this means: lengthening the slit or increasing the thickness have significant, cubed but reversed effects.
Two things to consider:
• The length of the slit is not necessarily the effective length of bending. In chapters of other pages, I have looked at this aspect, specifically, in the chapter Fountain Pen Nibs Mechanics and Design of Fountain Pen Nibs.
• I have read in forums when people reduced the thickness through hammering or roll-forming. Then the material work-hardens thus changes its condition. Annealing reduces the internal stress of the component but also returns the material to a softer state. Otherwise, in order to reduce the thickness with this effect, the excess material would need to be machined away, through grinding, for example.
12. Bending of a Profile
I have mentioned before that next to the modulus of elasticity, the profile of the component influences the style of the bend, significantly. We all have observed that when one holds a sheet of paper at one edge, it will flop down. After you fold or curve the sheet, it will stick out horizontally, just like a cantilever beam.
In ingeneering, this geometric shape characteristic that stiffens the paper is called The Area Moment of Inertia, (Wikipedia), casually called inertia and abbreviated as a capitalＩ. It considers the profile and dimensions of a component but not its material. One can see this in equation 2, showing the cantilever formula when resolved for the distortion distance δ
δ = F×ℓ^{3} ÷ 3EＩ . . . Equation 2
Equation 2 starts with F, the bending force followed by ℓ, which is the length of the unsupported beam from the bearing to the point where the force is applied. E is the modulus of elasticity (considering the material), andＩis the area moment of inertia (considering the shape). Unfortunately, printed in this font, l and I look very similar, I hope the colouring helps reduce the confusion.
Area moments of inertia can be calculated and since it involves higher mathematics, not one of my fortes, I am glad that some bright people have worked out the final formulas for basic shapes (like equation 3), which are collected in ingeneering handbooks.
The good thing is that one can assemble more complicated profiles by merely adding and subtracting these basic shapes, however, the bending axis x-x must be considered (Tricky; I won’t go into it. The name is Steiner’s Theorem (Wikipedia), just in case…)
I am aware that not many nibs are flat, but as we progress you will notice that once you get a feel for the bending behaviour of a flat strip, applying it to a curved nib will come easier. For now, let’s start with a flat strip as in drawing 4. Its profile is defined by t (thickness) and b (breadth). The orientation of the bending axis x-x determines which side is b and which t. One also needs to know the location of the bending axis, it can be outside the profile (like with most nibs). The moment of inertia, as shown in handbooks, is mostly presented with the axis centred between and parallel to two opposing sides; moving the bending axis, changes the inertia.
The standard formula for the area moment of inertia of this flat profile is given in equation 3:
I = b×t^{3} ÷ 12 … ….. Equation 3
Sample Calculation 1
We know that tines part when they are tilted at an angle along the direction of the slit. There are exceptions about which I will talk later.
For now and to develop a feel for it, I have calculated some area moment of inertia for the same profile tilted at increasing angles, as shown in drawing 5. Please notice that the results refer to situations when the profile is bent by the same force as indicated, always straight up, and the bending axes remain horizontal.
In nibs, the bending axes are more or less parallel to the profile of the individual tine, which is different from the drawing. Still, pondering the results is useful.
Sample Calculation 2
Getting serious now, let us see if we can calculate the parting of tines of a nib. I use the data from table 1, referring to the Waterman Ideal 2 with a fine nib made of a gold alloy. To spread the tines by 1.5mm, the writing pressure was 2N.
I will approximate the tine‑curvature as being a flat strip, which is tilted by τ = 30°, with a thickness t = 0.3mm, an effective bending width b = 2.5mm and the bending length ℓ = 10mm. The equation also asks for the modulus of elasticity E, which for gold alloys ranges from 72 to 83Gpa (gigapascal = 1000N/mm^{2}). I have chosen 75GPa because I don’t know the alloy. By the way, E of stainless steel is around 200GPa.
Drawing 6 shows the correlation between forces and dimensions.
The load, see table 8 is 3N, per tine is half the total load = 1.5N
The vector amount bending the tine, which is T = F/2 × cos30° = 1.06N
The momentum of inertia: equation 3 … I = b×t^{3} ÷ 12 = 2.5×0.027 ÷ 12 = 0.0056mm^{4}
Applying equation 3: δ = T×ℓ^{3} ÷ 3EI
δ = 0.87N×1000mm^{3 }÷ (3×75,000N/mm^{2}×0.0056mm^{4})
δ = 0.69mm per tine
According to our calculation, one tine moves upward by δ = 0.69mm, or sidewards by W/2 = δ × sin30° = 0.345mm and the two tines separate twice this amount, W = 0.69mm. What we expected was a separation of 1.5mm, about twice as much. What to say? Since all dimensions have been assumed, we would have been extremely lucky to hit the jackpot with the first go. I performed several more calculations following the shown pattern while I changed the thickness because it has a cubed impact on the widening and is the most inconspicuous. For easier comparison I show the results in table 7:
Table 7 | Tine Separation | |
nib thickness [mm] |
under a load of 2N [mm] |
under a load of 3N [mm] |
0.2 | 2.33 | 2.82 |
0.225 | 1.6 | 1.96 |
0.25 | 1.17 | 1.57 |
0.275 | 0.88 | 1.07 |
0.3 | 0.69 | 0.84 |
This is the way an ingeneer would research and derive parameters if only some are available. Equation 2 contains many variables which can be solved for all of them. The only variable which cannot be simply measured mechanically is the modulus of elasticity E, however, knowing it helps to narrow down the material’s range of alloys.
E = T×l^{3} ÷ 3Iδ …. here you go. equation 4
Table 8 shows where the result fits into Table 2 from the chapter Fountain Pen Flex Nib Classification:
Table 7 | |||||
Nib | load-free line width [mm] | load for 2mm line-width [N] |
tine separation [mm] | spring constant [N/mm] | |
1 | Waterman Ideal 2 fine | 0.5 | 3 | 1.5 | 3 |
Waterman calculated example |
0.5 | 3 | 1.57 | 2.8 |
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We got very far on this page. Beginning with the concept of force and its ways for calculating it, we continued with exploring the writing-angle followed by the spreading of the tines in much detail. This included the concept of bending, the bending of a profile and to finish off, we performed several calculations which provided us with a feeling for the interaction of material, shape and forces. Did you think we ended up here?
PS: In the example, we used two flat, rectangular components for the tines; two pointy, triangular shapes would have been more appropriate. However, this would have caused more trouble in the calculation that it would have benefitted our understanding.
PPS: Usually, the tines are curved. Let’s have a look at the effect of curve tines in the next chapter.
Above all: Enjoy!
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