9. When one Applies Sufficient Force
… one can bend any nib! When I mean 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.
Remember the central aspect of forces? They are vectors, which have a magnitude and direction. Therefore, in technical drawings, they are depicted as arrows. When dealing with forces, one must adhere to a few rules of which I will mention some significant for our perusal.
- A force can be moved along the line of direction, the line of its action.
- For any force to exist or to build up, it needs an opposing force of at least the same magnitude. Good old Newton’s Law still applies.
- Forces oriented in the same line of direction, 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 good old Pythagoras’ formulae. This resultant 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 not even exist in the same dimension (pane), they still can be combined, but then, things get more complicated.
Let me explain: I show here Figure 1 from further up again to save you from scrolling up and down. We gave the two forces V and H with 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 a similar 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.
The same applies to three-dimensional systems as shown in Figure 3. 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}.
Forces and their lines of action are not arbitrary. They follow lines or planes 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 movement across the paper, which could be the direction of H_{2}. In that case, one would use for analysing the force distribution the projection of F onto the plane above H_{2}, leading to the resultant V_{R}. It will become clearer! Promise.
By the way: “When I studied (in the late sixties), we were taught to write 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 force for writing is F_{V}. It is the sum of the forces F_{MV} and F_{G}. They are merely added because they act along the same line of direction.
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?
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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 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.”
I demonstrate this in Figure 5. You can notice that with the same vertical force F_{V} the tine spreading force F_{T} increases. For the technically minded: increases with the cos of α.
F_{T}_{ }= F_{V } × cos α
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.
Designers of fountain pens for flex nibs have catered for this and reduced the thickness of the feed thus, permitting the pen being tilted even more, like in Photo 12.
There are more good things to say about this design and the nib, and I will return to it later.
10. Spreading of Tines
After we isolated the forces acting on the tip of the nib let’s have a look how they separate the tines.
I have written already about the reason why the tines open in the chapter about Nib Mechanics; therefore, I only repeat what is necessary to know in regards to flex nibs. Let’s just quickly recapitulate 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 of the nip. You can see that increasing τ results also in an increase of W even the deflection is the same. Good thought, however, there seems to be always a, however.
Drawing 2 is an ingeneering drawing showing the geometric relationship between dimensions. The vertical bending by the amount b causes a deflection d of the tines as follows:
d = b / cos τ
meaning that the tine deflection increases more rapidly than the amount of bending due to the leverage of cos τ.
The widening of the slit follows the equation:
w/2 = b × tan τ
or w = 2 b × tan τ
Again, the larger the angle τ, the wider the opening with 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 the cos τ_{F} or τ_{R}, respectively.
Furthermore, approximating the radius to a straight line is misleading. A flat sheet bends much more readily than a curved. Curving the tines causes stiffness that’s why all cars look like potatoes because they can be manufactured from much thinner metal sheets and still provide the 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 bending force, tilt angle and the tine widening in a simplified manner. However, the correlation between shape and forces very much depends on the design of the nib such as the radius and variation of curvature, the material and overall dimensions.
Before I go into that, let’s have a mental rest and again consider the question about the:
11. Bending
I cannot remember having reverted to my old ingeneering handbooks as much as I do now, just to check on how they explain things there. Let’s begin.
The literature on the fundamentals of bending does not distinguish between elastic bending or plastic deformation. I guess, the resistance a material musters to counteract distortion does not influence the principles of bending; however, material characteristics determine the outcome.
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In ingeneering, anything that bends is called a beam, which comes in all categories. They have been established to help sort out the formulas for calculating the forces and dimensions. For us, it is enough to know that in bending terms, nibs fall into the category of the cantilever beam.
Just to ensure we all have the same understanding 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.”
Here is an example in Figure 7: I could have chosen a model more appropriate to writing but the forces and bending characteristics of nibs are very intricate but learning the basics helps to develop a feel for what’s happening with nibs.
When applying a force, the cantilever bends by the distance δ. I did not show the momentum or shear stress, not needed.
In Drawing 3 you see a symbolic image as commonly used in ingeneering 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 – for example the distance from where the nib protrudes from the section to the point
- distortion distance δ – for example, the total flexing of the nib, including tines and all
The equation in question contains the modulus of elasticity of the material and individual shape characteristics of the beam. Since we are not calculating something, we can neglect these factors. At the moment, we want to find out how varying the dimensions of a nib effect 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 distortion 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 would mean: lengthening the slit or increasing the thickness have significant, cubed effects.
Two things to consider:
- The length of the slit is not necessarily the effective length of bending. In the next section, I will look at this aspect, specifically.
- If you reduce the thickness through hammering or roll-forming, the material work-hardens and its condition. Annealing would reduce the internal stress of the component and would also return the material to a softer state. Otherwise, in order to reduce the thickness, 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.
In ingeneering this shape characteristic that stiffens the paper is called The Area Moment of Inertia, abbreviated as a capital I. 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), and I is the area moment of inertia (considering the shape).
Area moments of inertia can be calculated and since it involves higher mathematics, not one of my fortes. Therefore, I am glad that some bright people have worked out the final formulas (like in Equation 3) for basic shapes, 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.
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 easy. For now, let’s start with a flat strip as in Drawing 4. Its profile is defined by t and b. The orientation of the bending axis x-x determines which side is b and which t. One also needs to know its location because of the moment of inertia changes.
The 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 indicated force, and the bending axes are horizontal.
In nibs, the bending axes are more or less parallel to the profile, which is different to the drawing. Still, pondering the results is useful.
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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 1mm, 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 the average, 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 per tine is half the total load = 1N
The vector amount bending the tine, which is T = F/2 × cos30° = 0.87N
The momentum of inertia… I = b×t^{3} ÷ 12 = 2.5×0.027 ÷ 12 = 0.0056mm^{4}
Applying Equation 2
δ = T×l^{3} ÷ 3EI
δ = 0.87N×1000mm^{3 }÷ (3×75,000N/mm^{2}×0.0056mm^{4})
δ = 0.69mm per tine
One tine would move upward by δ = 0.69mm, or sideward by W/2 = δ × sin30° = 0.345mm and the tines would separate twice this amount W = 0.69mm. Aren’t we lucky, that sounds about right, doesn’t it?
Table 7 shows where the result fits into Table 2 from higher up:
Table 7 | |||||
Nib | load-free width [mm] | load for 1.5mm line-width [N] | tine separation [mm] | spring constant [N/mm] | |
1 | Waterman Ideal 2 fine | 0.5 | 2 | 1 | 2 |
Waterman calculated example |
0.5 | 2 0.5+0.69 = 1.19mm |
0.76 | 1.5 | |
2 | No name medium | 0.7 | 1 | 0.8 | 1 |
3 | No name fine | 0.5 | 0.8 | 1 | 0.8 |
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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.
PPS: Usually, the tines are curved. Let’s have a look at the effect of curve tines after the next chapter.
Ω
Amadeus W.
Ingeneer
Continue reading about E – More Specifically – chapt. 13 – 16
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