Modern Dimensional Analysis Model Laws Used to Model Additive Manufacturing Processes
Abstract
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Abstract
1. Introduction
2. The Most Used Dimensional Methods
- the Buckingham’s theorem;
- the partial differential equations applied to fundamental differential relations of the analyzed phenomenon, when the initial variables, by suitable grouping, offer these dimensionless quantities;
- the complete, but at the same time the simplest, equation(s) which describe the phenomena, which will be transformed into dimensionless forms, finally offering the desired groups.
- the protocol in obtaining the desired set of groups is rather chaotic, arbitrarily and strongly depending on the ingenuity and experience of the involved specialist;
- for the involved specialist, solid knowledge in the field of the analyzed phenomenon and higher mathematics are required as well;
- only rarely (occasionally) can the complete ML be obtained, mainly due to the fact that there are only a limited number of the involved mathematical relations related to the phenomena;
- for common engineers or specialists, involved in prototype–model correlation analysis, CDA does not represent an easy approach.
- the involved specialist, instead of being a thorough connoisseur in the phenomenon as well as in higher mathematics, only has to identify the set of the involved variables, together with their dimensions, which have (or can present) a certain extent influence on the analyzed phenomena;
- it has a unitary, simple, and user-friendly protocol, which assures at once to automatically eliminate all insignificant/irrelevant variables;
- in all cases, MDA assures obtaining the complete set of the dimensionless variables, as well as the complete ML; this is practically impossible using the aforementioned methods, excepting some particular cases;
- this ML is very flexible, suitable for several particular cases, corresponding to simplified approaches of the phenomena;
- by a priori setting of the directly related variables to the conceived experimental investigations on model, hereafter named independent variables, MDA assures additional flexibility, which represents a significant advantage, non-existent in all the methods mentioned above; their a priori choice is possible both for the prototype and model;
- this set (of the independent variables) assures defining the most suitable model, which will offer for the involved model the most simple, lower-cost testing conditions, safety, as well as repeatable experimental investigations;
- the rest of the variables, hereafter named dependent variables, can be chosen priori only for the prototype; their magnitudes for the model are strictly obtained by applying a given (suitable) element of the ML;
- among the dependent variables there are also a small number of prototype variables, whose magnitude cannot be obtained more easily (with low cost or accessible experimental measurements) and whose determination is actually the purpose of this dimensional analysis; thus, these aforementioned prototype’s variables are obtained by applying the ML;
- furthermore, MDA removes the restriction of the geometric similarity of the model with the prototype, e.g., the shape of the cross-sections can be different at the model from the prototype; in this case, instead of choosing the cross-sectional dimensions as independent variables, one will substitute them by the second order moment of inertia of the cross-section;
- if the material is considered as an independent variable, chosen by means of Young modules, then one can accept different materials for the model, relative to the prototype;
- in the case of choosing the flexural stiffness (rigidity) instead of the cross-sectional dimensions, than neither the shape of the cross-section, nor the type of material must be identical in the prototype and model; the single condition remaining is that their scale factor must remain the same (to be constant), where this scale factor is defined as the ratio of the flexural stiffnesses, that is, , with the aforementioned indexing (2 for model, and 1 for prototype).
3. Most Used Dimensional Methods in Additive Manufacturing
4. Modern Dimensional Analysis Involved in Additive Manufacturing
- all the variables, together with their dimensions, that can influence the phenomena to a certain extent are selected;
- by taking into consideration the experimental requirement of as simple, repeatable, and less-expensive model as possible, the independent variables are selected; their dimensional exponents constitute the so-called Matrix A, a non-singular square one, i.e., ;
- the rest of the variables (the dependent ones) constitute Matrix B, without any requirement; from matrix B, a number of variables can be neglected at any time in order to model a simpler case;
- completing these matrices with matrix and an adequate unit matrix results in the so-called Dimensional Set (Table 1);
- Each line , of matrices C–D, will offer by a simple calculus one element of the requested complete ML, corresponding to one of the dimensionless variable.
- One can observe that, for the requested ML, there are many complicated and sophisticated groupings of elements from the involved differential equations in order to obtain the required dimensionless variables and no requirement for deep connoisseurs in the analyzed phenomena;
- it must be emphasized that these elements of the ML do not represent actual physical laws, but only correlations between the variables related to the prototype and the model, which must be respected in everything;
- the unique protocol automatically eliminates the irrelevant variables (their columns in matrix C will contain only zeros);
- based on experimental investigations (strictly on the attached model), it became possible to obtain the foresighted (anticipated) parameters for the prototype by means of the deduced ML;
- the deduced ML also represents a very flexible set of information, because one can ignore (eliminate) some elements, if we are looking for a simplified model–prototype correlation, without any changing in the expressions of the other elements of the ML; this also constitutes another distinctive advantage of the MDA, which is not proper to any of the above-mentioned dimensional methods;
- In addition, depending on the concrete conditions available, the strategy of the experimental investigations can be adapted to the new conditions, by reseating and reconsidering the sets of independent variables, relative to the dependent ones;
- The deduced ML for a given type of structural element (e.g., straight bars) can be extended without any difficulties to the real structure, taking into consideration their homologous points of time as well as loads for the demanded variable;
- –
- the beam’s dimensions , as well as the area defined by the ribs;
- –
- the applied force;
- –
- the Young modulus, which for PLA is ;
- –
- the useful volume of the beam, that is related to the filling degree.
- –
- the second order moment of inertia, instead of the given cross-sectional dimensions;
- –
- by fusion of the initial variables, a more flexible model can be obtained, e.g., instead of E and Iz, one can use their product , that is, the stiffness module; if necessary, the density or specific gravity can also be involved.
- On the other hand, in order to increase the number of dimensions and thus reduce the number of the dimensionless variables, the so-called dimension splitting can be applied, e.g., instead of length “m”, its components along (mx, my, and mz) can be used.
- By choosing and as independent variables, there is evidence that that the remaining parameters , can be divided into others, such as ; ; without affecting the final ML;
- In addition, for a correct merging of the initial sizes in , they (that is, and ) can no longer appear in other elements, as they are omitted in ;
- When we abstract from these internal dimensions, we obtain the solid cross-sectional beam; in this sense, we simply neglect these internal dimensions together with their corresponding elements of the deduced ML.
B | A | ||||||||
---|---|---|---|---|---|---|---|---|---|
v | a* | b* | c* | A1 | F | Vutil | L | E*Iz | |
m | 1 | 1 | 1 | 1 | 2 | 0 | 3 | 1 | 2 |
N | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
π1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | −1 | 0 |
π2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | −1 | 0 |
π3 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | −1 | 0 |
π4 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | −1 | 0 |
π5 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | −2 | 0 |
π6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 2 | −1 |
π7 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | −3 | 0 |
5. Conclusions
- The elaborated MLs, validated by meticulous experimental investigations, offer several useful correlations between the prototype’s (i.e., the final product) and the assigned reduced scale model’s geometrical parameters and mechanical behaviors;
- by performing experimental investigations on the assigned model, applying the ML, it was possible to predict the real-scale prototype’s response to the applied mechanical or thermal loading;
- the ML also assures the stiffness optimization of the final product by means of changing the geometry and orientation of the involved ribs, the filling percentage, and the usable material.
- one other kind of stiffness optimization is the well-known honeycomb cross-section, which can also be more easily modelled with the deduced MLs by substituting the involved new length variables instead of the used ones;
- all the above-analyzed MLs can be successfully applied both for solid cross-sectional beams manufactured from PLA, and metallic ones; metal beams have simpler and more flexible MLs, as verified in advance by the authors;
- The authors’ future goals consist of extending the research area on the optimization of mold-forms used in plastic material components fabrication, mainly for in demand spare parts.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Matrix B | Matrix A | |||||||||||||
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | |||||||
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | |||||||
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | |||||||
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | |||||||
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | |||||||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
B | A | ||||||||
---|---|---|---|---|---|---|---|---|---|
v | a* | b* | c* | A1 | F | L | Gamma | E*Iz | |
m | 1 | 1 | 1 | 1 | 2 | 0 | 1 | −3 | 2 |
N | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |
π1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0.2 | −0.2 |
π2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0.2 | −0.2 |
π3 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0.2 | −0.2 |
π4 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0.2 | −0.2 |
π5 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0.4 | −0.4 |
π6 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | −0.4 | −0.6 |
π7 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.2 | −0.2 |
B | A | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
v | a* | b* | c* | A1 | a | b | L | F | E | |
mx | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 0 | 0 |
my | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | −1 |
mz | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | −1 |
N | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
π1 | 1 | 0 | 0 | 0 | 0 | 0 | −1 | 0 | 0 | 0 |
π2 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | −1 | 1 |
π3 | 0 | 0 | 1 | 0 | 0 | 0 | −1 | 0 | 0 | 0 |
π4 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | −1 | 0 | 0 |
π5 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | −1 | −1 | 1 |
π6 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | −1 | 1 |
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Asztalos, Z.; Száva, I.; Scutaru, M.-L.; Vlase, S.; Gálfi, B.-P.; Renáta-Ildikó, S.; Popa, G. Modern Dimensional Analysis Model Laws Used to Model Additive Manufacturing Processes. Appl. Sci. 2024, 14, 6965. https://doi.org/10.3390/app14166965
Asztalos Z, Száva I, Scutaru M-L, Vlase S, Gálfi B-P, Renáta-Ildikó S, Popa G. Modern Dimensional Analysis Model Laws Used to Model Additive Manufacturing Processes. Applied Sciences. 2024; 14(16):6965. https://doi.org/10.3390/app14166965
Chicago/Turabian StyleAsztalos, Zsolt, Ioan Száva, Maria-Luminița Scutaru, Sorin Vlase, Botond-Pál Gálfi, Száva Renáta-Ildikó, and Gabriel Popa. 2024. "Modern Dimensional Analysis Model Laws Used to Model Additive Manufacturing Processes" Applied Sciences 14, no. 16: 6965. https://doi.org/10.3390/app14166965