4. STEPS OF NEEDLE GRIPPER DESIGN FOR DEMOULDING FLEXIBLE
4.4 Comparing the theoretical results with the measurements, evaluation
4.4.2 Possible methods for further research
To handle these problems three different approaches seem practical. These solutions would require a whole new research and cannot be further part of this work. However to have a complete explanation of the problem the methods will be mentioned and the foundations of one (the most expedient) will be introduced in more detail;
1, Empirical method; determine a proper coefficient for each foam and needle. Many fur-ther needle force measurements would be needed. With statistical methods an empirical coef-ficient could be defined for every case. This method may provide good numerical results, but it would not give proper solution to the problem.
2, A new function could be introduced for the force parameter that has a dependence on the needle diameter (ΩNE(D)). This could account for the differences in the force values due to the diameter differences. At the moment this could only be done by an empirical trial and er-ror process, so it would still not provide a good solution and answer to the problem.
3, A more convenient way is to analyse the real contact surface between the foam and the needle. This can be treated in a more exact mathematical way that could show the underlying relations of a non-homogeneous structure and could give an elegant solution to the problem.
That is the reason why for the further analysis the third method will be chosen.
4.4.2.1 Determining the contact between the foam structure and the surface of the needle
The introduction of the area contact ratio αA in Chapter 4.2.3 was made to characterise the ratio of actual contact surface that is involved in making the frictional force. This parameter has no dimension and can be determined statistically (either from analysing the foam’s struc-ture or by visual observation). αA can be considered as ‘measure’ of area covering. This pa-rameter can be related to the methods of fractal geometry. The described method is just one way of modifying the theory and it is not compleat. It is just a way of showing, how the re-sults may be made numerically more suitable.
4.4.2.2 Some basics on fractals
The idea of the treatment of the problem this way came when I was analysing the structure of the foam. To see why a foam can be considered a natural fractal some basic concept of fractal geometry will be considered. This will not be complete, but it will contain just enough for the explanation to be conclusive and to establish the concept.
The definition of a fractal (used by Mandelbrot) being a set that with its Hausdorff dimen-sion strictly greater than its topological dimendimen-sion [62] proved to be unsatisfactory that it ex-cludes a number of sets that clearly ought to be regarded as fractals. Other definitions were also proposed but they all have the same drawback. Instead of creating a definition when we refer to a set ‘A’ as a fractal the following typical characteristics are looked for [28]:
- Self similarity, sometimes approximate or statistical,
- Has a fine structure, new detail is revieled at an arbitrary small scale,
- ‘A’ is too irregular to be described in traditional geometry, both locally and globally,
- Usually the fractal dimension of ‘A’ (defined in some way), is greater than its topologi-cal dimension,
- In most cases ‘A’ is defined in a very simple way, perhaps recursively.
These are the most usual characteristics of an original fractal, so when these facts are matched in a geometrical object it can be considered as one [68].
To handle fractal Hausdorff measure and Hausdorff dimension is used, defined by:
{ }
=
∑
∞=1
A of cover -is U : U inf ) (
i
i S i
S A δ
Hδ (143)
where U is a non-empty subset of the n dimensional Euclidian space.; If {Ui} is a finite col-lection of sets of diameter at most δ that cover A, with 0 < |Ui| ≤ δ for each i then {Ui} is a δ-cover of ‘A’. The infimum HSδ(A) increases, and so approaches a limit as δ→0;
) ( lim )
(A 0 S A
δ
S Hδ
H = → (144)
This limit exists for any subset ‘A’ though the limit can be 0 or ∞ [28] and it is called the s-dimensional Hausdorff measure of ‘A’. It is evident that HSδ(A) is non-increasing with s if δ <
1 for any set ‘A’, this case HS(A) is also non-increasing. If T > s and {Ui} is a δ-cover of ‘A’
then:
∑
∑
≤ −i S i
S T T
i Ui
U δ (145)
so taking infima and δ→0 we get HS(A) < ∞ then HT(A) =0. This means that there is a critical value of s at which HS(A) ‘jumps’ from ∞ to 0 [28]. It is called the Hausdorff dimension of
‘A’ and is written as dim H A. So from these formulas:
dim s
if 0
dim s
) if
( A
A
A
H S H
>
<
= ∞
H (146)
If s= dim H A then HS(A) may be zero or infinite or can satisfy 0 < HS(A) < ∞. It is clear that if dim H A < 1 the set is totally disconnected. The difficulty of determining the Hausdorff di-mension led to other definitions of didi-mensions that are also used for analysing fractals and the values of each may be different. The ideal is the measurement at scale δ, the dimension of ‘A’
can be defined by a power law obeyed by Mδ(A) as δ→0.
c S
A
Mδ( )≈ δ− (147)
for constant s and s it can be sad that ‘A’ has ‘dimension’ s with regarded as the ‘s-dimensional ‘length’’ of ‘A’. From this, the formula can be defined as:
δ
δ
δ log
) ( limlog
0 −
= →
A
s M (148)
These formulas are easy to work with because s can be estimated as the gradient of a log-log graph plotted over a range of δ. For real phenomena the range of δ can only be finite, espe-cially when the atomic scale is reached. There of course may be no exact power law for Mδ(A) and the closes that can be determined are the lower and upper limits. For s to behave as a di-mension the measurements needs to scale with the set, so it is required Mδ(δA)=M1(A) for all δ. In general if Mδ(A) is homogeneous of degree d, then Mδ(δA)=δdM1(A) then a power law of the form [28]:
S
c d
A
Mδ( )≈ δ − (149)
corresponds to a dimension. These formulas are only approximations, but they can work well if the range is defined properly.
Some fractals created from simple algorithm can be seen on Fig. 51 to Fig. 53. The first one is imbedded in a 3D manifold, thus has a 2<s<3 dimension, the other two ‘Sierpinski’s
car-pet’, defined by the same algorithm but with different quantities, are planar. They are also fractals of these kinds that have been produced by random algorithms. Those fractals may have other dimensions.
Fig. 51. Menger Sponge, s∼2.7168 [62] Fig. 52.Sierpinski carpet 1,
s∼1.8957 [62] Fig. 53. Sierpinski carpet 2, s∼1.8957 [62]
4.4.2.3 Natural fractals
All of the natural object cannot be real fractals. They may be scaled down by some factors repeatedly to a certain size, but it cannot be repeated infinitely. A polyurethane soft foam is not a real fractal, but a fractal like object, that can be called a natural fractal. The natural frac-tals also show the same characteristic of real fractal, but only to a certain scaling. The log-log curve of a natural fractal is not a strait line (as was mentioned for a real fractal), but a curve that can be concave or convex in the function of log δ [60]. These fractals were used succes-sively for defining; filters, exhaust, carbon blacks, etc. [55].
4.4.2.4 The fractal likeness of Polyurethane foams
The foam has fine structure that can be difficultly discussed by conventional geometrical objects. It can also be considered self similar to some point. Real foam contains a few large cavities of average diameter of 5-10 mms. These cavities can be found in every 200 mm3 of soft foams. The usual void size of the foam has been discussed in Chapter 2.3.3.3, and they proved to be 1-2 magnitudes smaller than these large cavities (few millimetres to 0.1..0.5 mms). In Chapter 2.3.3.4 the presence of micro cells were shown. The microcells build up the struts and ribs and walls of the foam. The cavity inside them is 10-1µm. So by these facts the scaling can be done up to the factor of 10-7. This can be considered as a very large scaling range. The foam can be considered as a non regular Menger sponge, but having different di-mension than the original. A section of the foam can be considered as a non regular Sierpinski carpet that may also be random is some range. The contact area may be formulated by a sec-tion of the foam as a natural fractal. Also different fractals may be generated for a better cor-relation to the original foam section Fig. 54.
4.4.2.5 Possible modification of equation (140)
With these results the original equation (140) may be rewritten by the following way:
η ξ
F D L
FFNe =Ω (150)
Where: ΩF is the new force parameter that has a dimension [N/mmγ] Dξ is the modified di-ameter with dimension [mmξ] and Lη is the modified length with dimension [mmη]. The rela-tion between the dimensions are γ=ξ+η. This is the most general case. In the special case and also the test showed that η=1, this means that there is a liner correlation between the forces and the needle length, but there can be some minor differences that cannot be seen from the diagrams. That is why we leave η in the expression, so in the general case η ≤ 1and more pos-sibly ξ « 1, this is due to greater dependence of force on the diameter.
Fig. 54. Computer generated fractal of foam, with dimension s∼1.5000,s∼1.75000 [62]
From the equation (138) αA can be redefined by a new parameter that is not dimensionless.
The new parameter would be αF having a dimension of [mmω]. This parameter should be de-fine the way that when it is multiplied with Ω the dimension of the whole expression should be [N/mmγ], where from the theory (1 < γ < 2). So from the exponents it can be determined that (ω-2 = γ ). The relation of the values of course in general is: αA ≠αF , ΩNE≠ΩF.
In a more radical theory the original values of the material parameters should be determined according to the fractal dimension of the foam section. (This can only be done statistically because the section is different according to where the foam is analysed, but the difference range is not very large). This way GE, σT should be related to a surface, defined by a planar natural fractal. This way they would have a dimension of [N/mmγ] for these parameters and for ΩF as well. The calculations would be strait forward, this way.
For some numerical results concerning the fractal method I have added Table 9, to present the accuracy that can be achieved this way. The force parameter and fractal dimensions in (150) were calculated with regression analysis of the measured needle forces. The values for some foams can be found in Table 19 in Appendix A.8.5. The Maple program calculating the parameters can be found in Appendix A.8.5.
Table 9Needle forces calculated by the fractal method Needle geometry
∅0.5x40 ∅0.6x60 ∅0.8x50 ∅0.9x70 ∅1.6x40
Foam FFNe Fout Err. FFNe Fout Err. FFNe Fout Err. FFNe Fout Err. FFNe Fout Err.
1 1.035 0.98 -6.12 1.797 2.0 10 1.835 1.79 -2.8 2.747 2.652 -3.7 1.917 2.04 -5.88
6 1.31 1.28 -2.7 2.5 2.48 -0.94 2.19 2.22 0.77 3.97 4.32 7.93 2.42 2.8 13.3
7 1.21 0.99 -22.3 1.91 2.32 14.1 1.76 2.04 13.37 3.33 3.32 -0.28 1.87 2.08 10.03
12 1.4 1.49 -6.52 2.62 2.57 2.09 2.46 2.58 4.65 4.34 4.58 5.11 2.66 2.95 9.84
15 1.53 1.4 -9.5 2.73 2.89 5.52 2.59 2.68 3.38 4.31 4.61 6.5 2.83 3.22 12.06 Where : FFNe [N]is the force calculated by the fractal method, Fout[N] is the measured pull-out force and
Err. is the error in [%].
From the table it can be seen that the errors are smaller with the fractal method. With just five exceptions they are all under 10% and the rest of the errors are also not very large. The numerical values have nice results, however for some errors the continuum method gave bet-ter result e.g.: foam 7 needle 0.8x50, foam 12 needle 1.6x40). In the rest of the cases the er-rors for the fractal method are smaller than for the continuum method.
A complete theory such as this would require more considerations and calculations. Deter-mining the powers of D and L (ξ,η) from a complete theoretical point of view has not been fully solved yet. The further development of the theory will be done in new researches.