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CONTRIBUTION CONCERNING THE INFLUENCE OF GEOMETRICAL PARAMETERS ABOUT THE BEHAVIOUR OF CORRUGATED DIAPHRAGMS WITH TRIANGULAR PROFILE
IFAC MCPL 2007 The 4th International Federation of Automatic Control Conference on Management and Control of Production and Logistics September 27-30, Sibiu - Romania
CONTRIBUTION CONCERNING THE INFLUENCE OF GEOMETRICAL PARAMETERS ABOUT THE BEHAVIOUR OF CORRUGATED DIAPHRAGMS WITH TRIANGULAR PROFILE
Florin Negoescu , Eugen Axinte Technical University Gh.Asachi Jassy Abstract: In this paper the authors purpose a researching influence the geometrical parameter of the corrugated diaphragms about behaviour at the solicitation with a uniform pressure. It is an importance problem in establishing optimal form and dimension, knowing influence of the geometrical parameter who describes the diaphragm. The results are presented under form one matrix mathematic model and for validation are used the Snedecor test. Copyright © 2007 IFAC Keywords: corrugated diaphragm, characteristic, elastic deformation, sensible element, transducers
1. INTRODUCTION
- bigger sensitivity - give possibilities to keep constant effective surface of the diaphragm when centre of the diaphragm are moving along to plan in which are the diaphragm situated; - biggest arrows without appearance of permanent deformations, sow gives the possibilities to measure superior pressure; - easy to modelate the characteristic by simply modifying the height H of the undulations, witch is allowed to counterbalance the unlinears of the parameter to measure, which mean straighten the scale of the appliance.
Regarding the construction of the measure appliances, we can certainty affirm, that because the property of corrugated diaphragms to measure the value of inflection at relative small efforts, as elastical elements, those are used in top technological areas, such as aero-spatial industry, military industry, control and automation for industrials process. Applicable areas for that type of diaphragm are sow large, so that the modest enumeration for contrivances in which construction are used, may contain some tenthly appliances and are include example part of the most variety technical fields, beginning with the regulators of temperature used at hatcheries and end with measure indicators of high and speed of aeroplanes. It is important to remark the fact that, the diaphragms may also be used as separating tool between two medias and for elastic tightens, allowed the transmission of movement from medias with pressure or vacuum. The advantage of those diaphragms, are (Negoescu et al., 2002): - characteristics closer to linear, sow a shorter variation for effective surface; - give possibilities for designer to adopt relatively easy the characteristics of some given dates;
Fig. 1. Corrugated diaphragm
437
The function of the appliances has as sensitive element metallically diaphragms it’s based on elastically deformation of those under the action of a pressure or external forces. Metallically diaphragm is embedding on appliance contour, and the pressure or force are apply on one of the face of diaphragm, act who made elastical deformation of that. On other face of the diaphragm it is a solid centre, whose displacement are dependent by the displacement in the centre of diaphragm. Here buy, the displacement of the solid centre represent useful information, being utterance whit measured size.
which answers must date to a series of which problems appear to the of an elaboration his process the product. In the strategy experimentation am contained two successive directions of thing: The associative programming of the experiences and analyse yes experimental. 2. THE EXPERIMENTAL PROGRAMMING AND THE RESEARCH SETUP The objective ascertainment of the researches he did envisaging the current stage of knowledge and on the strength of an incident to date the applicability of the results and the material existing possibility. The method of the plans of experimentation consists in ascertainment the plan of which his thing presupposes at least of experience, taking count of the quality of the results.
In the projection of metallically corrugated diaphragms, designer it is interested not only by the size of deformation, but also by the linearity of relation between deformations – pressure. Although that linearity of the relation is useful on all area, that thing it is right only for certain portion of that. Pursuant to SAMA Standard (Scientific Apparatus Makers Association), who is affiliated at Instrument Society of America, usually it is measured at an unliniarity and experimented as linearity; represent the maximum deviation between the obtained curve and a straight line. Principal problem clung of studying the corrugated diaphragm it is the determination of the principal characteristics, witch mean to determine a relation between the arrow or deformation „W” and the force or pressure „P” witch product her, in the case of given form of corrugate. W = f(P)
2.1. The definition of the objectives The objectives in this experiment are: - To establish the influence of geometrical parameters on the value of the arrow of deformation’s diaphragms under the action of pressure - To determinate the influence of geometrical diaphragm parameters on the linear characteristics - To establish the effects of the facts and the interactions of those - To compare the experimental characteristic with theoretical one and the one obtain by numerical analyses
(1)
Is one of the most important base parameters of the analytical and elastic diaphragms the characteristic of elastic diaphragms can be express, according the action load, by a relation as (1), and the graphic representation of characteristic can be linear, call constant or unliniar. Some of those elastic diaphragms presenter a linear characteristic only on a certain part linear areas, specific to the corrugate metallic diaphragms depending on geometrical parameters whence characterised it is necessary the following resources.
2.2. The establishing of the experiment type The utilization of the experimental plans method of enforces use the notions as the (Alexis, 1999): Factors (his variable what state reacts on the system studied), Answered (the which size is measured knew the effect of the factors about the system), Level of the factor (the values on which takes them the factor on complete sweep of the experiences) and Factor or significant interaction (his factor which interaction through modify drives to the change of the answer of the system researched).
In choosing the materials destination to execute the elastic diaphragms have to be taken in consider all the facts which characterize operating system (request of the material, work medium, the behaviour of the material and eventually change in time of some of those characteristics), the possibilities of supply, the character of production.
For this work, the planning and drive the experiments he accomplished taking count of a series of appearances considerate accepted on world plans. To the matrix construction used the experiment with notation Yates (-1, accordingly the inferior level of a respective factor + 1, of superior level of a factor) her originality consist in the fact as the permits the procurement directly the general formulae ale of average effects, carry is calculated totalized the all answers of the experiences, preciously of the signs of the factors
A process is a good known except through experimentation, because the theoretical models the by-paths create on the strength of a hypotheses simplification, I carry only that drive to the of a appearance abbots against reality. The must experimental research is effectuated across 438
from each experience and divide the total in number attempts effectuate (Pillet, 1992). For the qualitative of a determination sizes on experimental path he accomplished a number finite of experimentations with a sufficient accuracy, maintaining same conditions of menstruation. The gathering of the date and the statistical their remaking he achieved taking count of the general valid methods and using specific programs of computer.
Table 2. The matrix of experimentation The Factors 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
To accomplish the porpoise objectives was betake to realise the tests pursuant to the requests of factorial orthogonal complete experiment at two levels, such 2n (24 = 16 tests). The experiments had place for each combination of those two level, for those four independent variables take in consideration (the active diameter of diaphragm D, the thickness of material h, the amplitude of corrugated H and the number of those n), as the table 1. For the quantification of the admeasurements on the experimental path, are shall accomplished a number finite of measurements with a sufficient accuracy maintaining same conditions of mensuration. Established most probable value of moderate size, carry estimates the true value is shall utilized the law of natural repartition, and the accuracy of mensuration shall be evaluate of the average quadratic his error the standard error.
D h H n
The factor of test The diameter’s diaphragm The thickness of material The amplitude’s corrugated The number’s corrugated
Level 1 -1
Level 2 +1
Unit of gauge
60
43
mm
0,06
0,12
mm
0,52
0,87
mm
6
4
-
H -1 -1 -1 -1 +1 +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 +1
h -1 -1 +1 +1 -1 -1 +1 +1 -1 -1 +1 +1 -1 -1 +1 +1
n -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1
0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12
Elastic Deformation [mm] 1,98 0,98 1,84 0,78 1,84 0,87 1,68 0,6 0,83 0,32 0,41 0,16 0,88 0,33 0,45 0,17
An importance problem a scientific research, he is the ascertainment of the optima conditions for development of the technological process, with the observance functional requirement and of quality. Improvement of the performances produced is possible by use the physical his mathematical model. The mathematical model is used frequent because simplify the interpretation of the effects of the factors and interactions and permits find the comfortable solutions. The interpretation of the factor effects is easier if the model, which characterizes the process, is written in a matrix form.
Table 1. The factors and levels of test The symbol
D -1 -1 -1 -1 -1 -1 -1 -1 +1 +1 +1 +1 +1 +1 +1 +1
Pressure [bar]
2.3. Building the mathematical model The aim of the model is expressed the unknown variables ale of the model depending on one knower, thus that to is satisfied the criteria’s of performance, that is to is solved the system. Analyse with help of the model enable us to establish what modifications modify the system performances. This modelling permits a formulation the observations about the influence on which the input parameters the interactions among these have it about value of the force of press keep material, because don't work with the effective values ale of input factors, but with these levels, which thing remove the subjectivism entered of the models classical (Vigier, 1993).
When the liniarity is big (or unliniarity is low) the relation signal out – signal in can be consider a straight line, and so the sensibility can be consider constant. On the other hand, when the linearity is low, (or the unliniarity is big) the sensibility depend by the in – signal. Prosecution of the experiments take place pursuant to the matrix of experiments which was presented in table 2.
Table 3. The effects and interaction of the factors The mean M 0,8825
The effect’s factors al the level 1 D n H h 0.43875 0,03 0,12125 0,35625
The interactions between factors at level 1 D*H D*h n*H n*h H*h 0,0437 0,1575 0,0205 0,025 0,0062 0,0037 D*n
439
The model is matrix type and has the form:
+ [E H 1
E D 2 ] ⋅ D + [E n1
E H 2 ] ⋅ H + [E h1
En 2 ]⋅ n +
Eh2 ]⋅ h +
I D1n 2 I ⋅ n+tD ⋅ D1H 1 I D 2n 2 I D2H1
I D1H 2 ⋅H + I D 2 H 2
I + tD ⋅ D1h1 I D 2 h1
I D1h 2 I ⋅ h + t n ⋅ n1H 1 I D 2 h 2 In2H1
I n1H 2 ⋅H + I n 2 H 2
I n1h 2 I ⋅ h+ tH ⋅ H 1h1 I n 2h 2 I H 2 h1
1.5 1 0.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
I H 1h 2 ⋅h I H 2 h 2
Experimental point
Fig. 2. The graph of experimental values and theoretical conformable arrow of deformation To estimate the relative effects of facts are compare their effects. This comparation is realised much easier if the results are transposed as some graphics in which, in regard whit the medium line are indicated the effects of the facts and the interactions between those, present in figures 3, 4 ,5 and 6.
Wt = 0,8825 + [0,43875 - 0,43875] ⋅ Dn + - 0,03] ⋅ n +
+ [0,12125 - 0,12125] ⋅ H + + [0,35625 - 0,35625] ⋅ h +
Ef-D
0,04375 - 0,04375 +tD ⋅ ⋅n+ - 0,04375 0,04375 - 0,025 0,025 +tD ⋅ ⋅H + 0,025 - 0,025
2
0
Therefore, as per give numerical the model becomes:
+ [ 0,03
W~
2.5
(1)
I + tD ⋅ D1n1 I D 2 n1
I + t n ⋅ n1h1 I n 2 h1
W
Elastic deformation [mm]
Wt = M + [E D1
Ef-H
Ef-n
Ef-h
Int-D-n
Int-D-H
Int-D-h
Int-n-H
Int-n-h
Int-H-h
4
5
6
7
8
9
10
0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05
(2)
0,1575 - 0,1575 +tD ⋅ ⋅h+ - 0,1575 0,1575
1
- 0,00625 0,00625 + n⋅ ⋅H + 0,00625 - 0,00625 t
2
3
Fig. 3. The graph of the effects and the interactions to the level 1
- 0,00375 0,00375 +tn⋅ ⋅h + 0,00375 - 0,00375
h
H
n
D
M
1.4 Elastic deformation [mm]
0,0225 - 0,0225 +tH ⋅ ⋅h - 0,0225 0,0225
The results obtain following the mould and the dross values are express in table 3, and in figure 2 are graphic express the theoretical and experimental answers in the points analysed. Analysing the graphic present in figure 2 can be ascertain a very good preachment between the values determinate experimental (measured) and the theoretical ones so, in consequently, the values of the drowses are low, fact which evidence the low influence of interactions on answer.
1.2 1 0.8 0.6 0.4 0.2 0 1
2
1
2
1
2
1
2
Level
Fig. 4. The graph of the effects about value of the elastic deformation
440
n1
n2
H1
H2
h1
Table 4 The aria of analyses
h2
2
F
Elastic deformation [mm]
1.8
D n H h D*n D*H D*h N*H N*h H*h
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 D1
D2
D1
D2
D1
D2
Fig. 5. The graph of the interactions between factors about value of the elastic deformation
Elastic deformation [mm]
H1
H2
h1
h2
h1
h2
n2
n1
n2
H1
H2
fα
p=0,99
p=0,95
16,26 16,26 16,26 16,26 16,26 16,26 16,26 16,26 16,26 16,26
6,61 6,61 6,61 6,61 6,61 6,61 6,61 6,61 6,61 6,61
The significance Significant Insignificant Significant Significant Insignificant Insignificant Significant Insignificant Insignificant Insignificant
3. RESULTS AND DISCUSSIONS Interpreted the results obtain and present under the form of matrix model (2), of graphics and of Snedecor test can be remarked the following conclusions tied by the influence of the studied facts on the value of deformation arrow. Using the compare coefficient Fisher if is accepted a level of trust of 99%, can be told that, for a risk of 1% between taken considered facts, have a significance influence on the arrow value which with are deforming the diaphragm when on their surface are applied a uniform distributed pressure, active diameter of diaphragm, (most important) thickness of material, the amplitude of corrugates and interact between the diameter and thickness of the material. The fact that interact between the diameter of the diaphragm and the thickness of material have a important effect, may be explained by the modification of material volume which are obedient to external forces (deforming pressure)
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 n1
485,04 2,26 37,04 319,78 4,82 1,57 62,50 0,09 0,03 1,27
fα
Fig. 6. The graph of the interactions between factors about value of the elastic deformation From the graphs 3, 4 ,5 and 6 can be observed that exist important interactions between diameter-number of corrugates and between diameter-the thickness of material, so the effects of each one, depend by the levels of other facts, but the other interactions are low, almost negligible. The wickets interaction is between the number of corrugates and the thickness of material.
Even if we low the trust level up to 95% can be remark that for a risk of 5% the effect of „n” factor also of the other interacts not becomes important so can be insignificant ,fig.7 The experience plans give the possibilities to determinate the influent facts and to know optimal level for everyone of them. After we eliminated the essential facts the matrix purpose model become:
2.4. The acceptance testing and validate the model To see which facts taken in consideration have important influence to the value of arrow of deformation is applied the test Snedecor (Ionescu, and Amarandei, 2004), take account that the numbers of liberty grades of model is 11 and the number of dross liberty degrees is 5, and the check and the arrange of the influence facts is present in table 4.
W = 0,8825 + [0,43875 - 0,43875]⋅ D +
+ [0,12125 - 0,12125]⋅ H + + [0,35625 - 0,35625]⋅ h + 0,1575 - 0,1575 +tD ⋅ ⋅h - 0,1575 0,1575
441
The target for the experiment is to find a optimal configuration for a set of facts taking analyse, configuration which are lead to maximise the elastic deformation it is recommends for independent facts that type of matrix D = [1 0], H=[1 0], h=[1 0].
REFERENCES Alexis, J. (1999) Metoda Taguchi în practica industrială, Editura tehnică, Bucureşti; Ionescu, R., Amarandei, D. (2004) Planificarea experimentelor eficienta si calitate. Bucuresti, Editura AGIR,; Negoescu, F., Braha V., Nagîţ, Gh. (2002), Membrane metalice ondulate - Elemente de proiectare şi tehnologii de fabricare. Editura Tehnica – Info, ISBN 9975-63-176-2, Chişinău; Pillet, M. (1992), Introduction aux plans d’expériences par la méthode Taguchi. Les éditions d’organisations, Paris, ; Vigier, M. (1993), Pratique des plans d’expériences. Méthodologie Taguchi. Les Editions D’Organisations, Paris.
It can be told that the rigidity of the diaphragm grow if the diameter and the thickness of the materials have small values, and the amplitude of corrugates grow, in time to sensibility of diaphragm for same value at the mentioned parameters go low. So are confirmed the fact that sensibility go low in the same time with growing the rigidity.
Fisher teor. Fisher p=0,99 Fisher p=0,95 500 450 400 350 300 250 200 150 100 50 0
Ef-D
Ef-n
Ef-H
Ef-h Int-D-n Int-D-H Int-D-h Int-n-H Int-n-h Int-H-h
Fig. 7. The graph representation of the Fisher test
Growing the rigidity in the same time with growing the thickness of material is explained by the growing of volumes of material which presumed for deformation bigger forces. The movement of the diameter of diaphragm make growing the rigidity because grow stability of the form, knowing are the fact that in case of thin elements, decrease bigger dimension lead to big rigidity of the form. The optimization of metallic corrugated diaphragms, is adverted to touch the performances technically maxims, only that satisfy the concomitance the all criteria enforced is difficult, sometimes just impossible, because one factors can be antagonistic his subjective. With all these an optimum design is admissible and she assures to obtain the maximum performances, in certain conditions.
442
CONTRIBUTION CONCERNING THE INFLUENCE OF GEOMETRICAL PARAMETERS ABOUT THE BEHAVIOUR OF CORRUGATED DIAPHRAGMS WITH TRIANGULAR PROFILE
Florin Negoescu , Eugen Axinte Technical University Gh.Asachi Jassy Abstract: In this paper the authors purpose a researching influence the geometrical parameter of the corrugated diaphragms about behaviour at the solicitation with a uniform pressure. It is an importance problem in establishing optimal form and dimension, knowing influence of the geometrical parameter who describes the diaphragm. The results are presented under form one matrix mathematic model and for validation are used the Snedecor test. Copyright © 2007 IFAC Keywords: corrugated diaphragm, characteristic, elastic deformation, sensible element, transducers
1. INTRODUCTION
- bigger sensitivity - give possibilities to keep constant effective surface of the diaphragm when centre of the diaphragm are moving along to plan in which are the diaphragm situated; - biggest arrows without appearance of permanent deformations, sow gives the possibilities to measure superior pressure; - easy to modelate the characteristic by simply modifying the height H of the undulations, witch is allowed to counterbalance the unlinears of the parameter to measure, which mean straighten the scale of the appliance.
Regarding the construction of the measure appliances, we can certainty affirm, that because the property of corrugated diaphragms to measure the value of inflection at relative small efforts, as elastical elements, those are used in top technological areas, such as aero-spatial industry, military industry, control and automation for industrials process. Applicable areas for that type of diaphragm are sow large, so that the modest enumeration for contrivances in which construction are used, may contain some tenthly appliances and are include example part of the most variety technical fields, beginning with the regulators of temperature used at hatcheries and end with measure indicators of high and speed of aeroplanes. It is important to remark the fact that, the diaphragms may also be used as separating tool between two medias and for elastic tightens, allowed the transmission of movement from medias with pressure or vacuum. The advantage of those diaphragms, are (Negoescu et al., 2002): - characteristics closer to linear, sow a shorter variation for effective surface; - give possibilities for designer to adopt relatively easy the characteristics of some given dates;
Fig. 1. Corrugated diaphragm
437
The function of the appliances has as sensitive element metallically diaphragms it’s based on elastically deformation of those under the action of a pressure or external forces. Metallically diaphragm is embedding on appliance contour, and the pressure or force are apply on one of the face of diaphragm, act who made elastical deformation of that. On other face of the diaphragm it is a solid centre, whose displacement are dependent by the displacement in the centre of diaphragm. Here buy, the displacement of the solid centre represent useful information, being utterance whit measured size.
which answers must date to a series of which problems appear to the of an elaboration his process the product. In the strategy experimentation am contained two successive directions of thing: The associative programming of the experiences and analyse yes experimental. 2. THE EXPERIMENTAL PROGRAMMING AND THE RESEARCH SETUP The objective ascertainment of the researches he did envisaging the current stage of knowledge and on the strength of an incident to date the applicability of the results and the material existing possibility. The method of the plans of experimentation consists in ascertainment the plan of which his thing presupposes at least of experience, taking count of the quality of the results.
In the projection of metallically corrugated diaphragms, designer it is interested not only by the size of deformation, but also by the linearity of relation between deformations – pressure. Although that linearity of the relation is useful on all area, that thing it is right only for certain portion of that. Pursuant to SAMA Standard (Scientific Apparatus Makers Association), who is affiliated at Instrument Society of America, usually it is measured at an unliniarity and experimented as linearity; represent the maximum deviation between the obtained curve and a straight line. Principal problem clung of studying the corrugated diaphragm it is the determination of the principal characteristics, witch mean to determine a relation between the arrow or deformation „W” and the force or pressure „P” witch product her, in the case of given form of corrugate. W = f(P)
2.1. The definition of the objectives The objectives in this experiment are: - To establish the influence of geometrical parameters on the value of the arrow of deformation’s diaphragms under the action of pressure - To determinate the influence of geometrical diaphragm parameters on the linear characteristics - To establish the effects of the facts and the interactions of those - To compare the experimental characteristic with theoretical one and the one obtain by numerical analyses
(1)
Is one of the most important base parameters of the analytical and elastic diaphragms the characteristic of elastic diaphragms can be express, according the action load, by a relation as (1), and the graphic representation of characteristic can be linear, call constant or unliniar. Some of those elastic diaphragms presenter a linear characteristic only on a certain part linear areas, specific to the corrugate metallic diaphragms depending on geometrical parameters whence characterised it is necessary the following resources.
2.2. The establishing of the experiment type The utilization of the experimental plans method of enforces use the notions as the (Alexis, 1999): Factors (his variable what state reacts on the system studied), Answered (the which size is measured knew the effect of the factors about the system), Level of the factor (the values on which takes them the factor on complete sweep of the experiences) and Factor or significant interaction (his factor which interaction through modify drives to the change of the answer of the system researched).
In choosing the materials destination to execute the elastic diaphragms have to be taken in consider all the facts which characterize operating system (request of the material, work medium, the behaviour of the material and eventually change in time of some of those characteristics), the possibilities of supply, the character of production.
For this work, the planning and drive the experiments he accomplished taking count of a series of appearances considerate accepted on world plans. To the matrix construction used the experiment with notation Yates (-1, accordingly the inferior level of a respective factor + 1, of superior level of a factor) her originality consist in the fact as the permits the procurement directly the general formulae ale of average effects, carry is calculated totalized the all answers of the experiences, preciously of the signs of the factors
A process is a good known except through experimentation, because the theoretical models the by-paths create on the strength of a hypotheses simplification, I carry only that drive to the of a appearance abbots against reality. The must experimental research is effectuated across 438
from each experience and divide the total in number attempts effectuate (Pillet, 1992). For the qualitative of a determination sizes on experimental path he accomplished a number finite of experimentations with a sufficient accuracy, maintaining same conditions of menstruation. The gathering of the date and the statistical their remaking he achieved taking count of the general valid methods and using specific programs of computer.
Table 2. The matrix of experimentation The Factors 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
To accomplish the porpoise objectives was betake to realise the tests pursuant to the requests of factorial orthogonal complete experiment at two levels, such 2n (24 = 16 tests). The experiments had place for each combination of those two level, for those four independent variables take in consideration (the active diameter of diaphragm D, the thickness of material h, the amplitude of corrugated H and the number of those n), as the table 1. For the quantification of the admeasurements on the experimental path, are shall accomplished a number finite of measurements with a sufficient accuracy maintaining same conditions of mensuration. Established most probable value of moderate size, carry estimates the true value is shall utilized the law of natural repartition, and the accuracy of mensuration shall be evaluate of the average quadratic his error the standard error.
D h H n
The factor of test The diameter’s diaphragm The thickness of material The amplitude’s corrugated The number’s corrugated
Level 1 -1
Level 2 +1
Unit of gauge
60
43
mm
0,06
0,12
mm
0,52
0,87
mm
6
4
-
H -1 -1 -1 -1 +1 +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 +1
h -1 -1 +1 +1 -1 -1 +1 +1 -1 -1 +1 +1 -1 -1 +1 +1
n -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1
0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12 0,12
Elastic Deformation [mm] 1,98 0,98 1,84 0,78 1,84 0,87 1,68 0,6 0,83 0,32 0,41 0,16 0,88 0,33 0,45 0,17
An importance problem a scientific research, he is the ascertainment of the optima conditions for development of the technological process, with the observance functional requirement and of quality. Improvement of the performances produced is possible by use the physical his mathematical model. The mathematical model is used frequent because simplify the interpretation of the effects of the factors and interactions and permits find the comfortable solutions. The interpretation of the factor effects is easier if the model, which characterizes the process, is written in a matrix form.
Table 1. The factors and levels of test The symbol
D -1 -1 -1 -1 -1 -1 -1 -1 +1 +1 +1 +1 +1 +1 +1 +1
Pressure [bar]
2.3. Building the mathematical model The aim of the model is expressed the unknown variables ale of the model depending on one knower, thus that to is satisfied the criteria’s of performance, that is to is solved the system. Analyse with help of the model enable us to establish what modifications modify the system performances. This modelling permits a formulation the observations about the influence on which the input parameters the interactions among these have it about value of the force of press keep material, because don't work with the effective values ale of input factors, but with these levels, which thing remove the subjectivism entered of the models classical (Vigier, 1993).
When the liniarity is big (or unliniarity is low) the relation signal out – signal in can be consider a straight line, and so the sensibility can be consider constant. On the other hand, when the linearity is low, (or the unliniarity is big) the sensibility depend by the in – signal. Prosecution of the experiments take place pursuant to the matrix of experiments which was presented in table 2.
Table 3. The effects and interaction of the factors The mean M 0,8825
The effect’s factors al the level 1 D n H h 0.43875 0,03 0,12125 0,35625
The interactions between factors at level 1 D*H D*h n*H n*h H*h 0,0437 0,1575 0,0205 0,025 0,0062 0,0037 D*n
439
The model is matrix type and has the form:
+ [E H 1
E D 2 ] ⋅ D + [E n1
E H 2 ] ⋅ H + [E h1
En 2 ]⋅ n +
Eh2 ]⋅ h +
I D1n 2 I ⋅ n+tD ⋅ D1H 1 I D 2n 2 I D2H1
I D1H 2 ⋅H + I D 2 H 2
I + tD ⋅ D1h1 I D 2 h1
I D1h 2 I ⋅ h + t n ⋅ n1H 1 I D 2 h 2 In2H1
I n1H 2 ⋅H + I n 2 H 2
I n1h 2 I ⋅ h+ tH ⋅ H 1h1 I n 2h 2 I H 2 h1
1.5 1 0.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
I H 1h 2 ⋅h I H 2 h 2
Experimental point
Fig. 2. The graph of experimental values and theoretical conformable arrow of deformation To estimate the relative effects of facts are compare their effects. This comparation is realised much easier if the results are transposed as some graphics in which, in regard whit the medium line are indicated the effects of the facts and the interactions between those, present in figures 3, 4 ,5 and 6.
Wt = 0,8825 + [0,43875 - 0,43875] ⋅ Dn + - 0,03] ⋅ n +
+ [0,12125 - 0,12125] ⋅ H + + [0,35625 - 0,35625] ⋅ h +
Ef-D
0,04375 - 0,04375 +tD ⋅ ⋅n+ - 0,04375 0,04375 - 0,025 0,025 +tD ⋅ ⋅H + 0,025 - 0,025
2
0
Therefore, as per give numerical the model becomes:
+ [ 0,03
W~
2.5
(1)
I + tD ⋅ D1n1 I D 2 n1
I + t n ⋅ n1h1 I n 2 h1
W
Elastic deformation [mm]
Wt = M + [E D1
Ef-H
Ef-n
Ef-h
Int-D-n
Int-D-H
Int-D-h
Int-n-H
Int-n-h
Int-H-h
4
5
6
7
8
9
10
0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05
(2)
0,1575 - 0,1575 +tD ⋅ ⋅h+ - 0,1575 0,1575
1
- 0,00625 0,00625 + n⋅ ⋅H + 0,00625 - 0,00625 t
2
3
Fig. 3. The graph of the effects and the interactions to the level 1
- 0,00375 0,00375 +tn⋅ ⋅h + 0,00375 - 0,00375
h
H
n
D
M
1.4 Elastic deformation [mm]
0,0225 - 0,0225 +tH ⋅ ⋅h - 0,0225 0,0225
The results obtain following the mould and the dross values are express in table 3, and in figure 2 are graphic express the theoretical and experimental answers in the points analysed. Analysing the graphic present in figure 2 can be ascertain a very good preachment between the values determinate experimental (measured) and the theoretical ones so, in consequently, the values of the drowses are low, fact which evidence the low influence of interactions on answer.
1.2 1 0.8 0.6 0.4 0.2 0 1
2
1
2
1
2
1
2
Level
Fig. 4. The graph of the effects about value of the elastic deformation
440
n1
n2
H1
H2
h1
Table 4 The aria of analyses
h2
2
F
Elastic deformation [mm]
1.8
D n H h D*n D*H D*h N*H N*h H*h
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 D1
D2
D1
D2
D1
D2
Fig. 5. The graph of the interactions between factors about value of the elastic deformation
Elastic deformation [mm]
H1
H2
h1
h2
h1
h2
n2
n1
n2
H1
H2
fα
p=0,99
p=0,95
16,26 16,26 16,26 16,26 16,26 16,26 16,26 16,26 16,26 16,26
6,61 6,61 6,61 6,61 6,61 6,61 6,61 6,61 6,61 6,61
The significance Significant Insignificant Significant Significant Insignificant Insignificant Significant Insignificant Insignificant Insignificant
3. RESULTS AND DISCUSSIONS Interpreted the results obtain and present under the form of matrix model (2), of graphics and of Snedecor test can be remarked the following conclusions tied by the influence of the studied facts on the value of deformation arrow. Using the compare coefficient Fisher if is accepted a level of trust of 99%, can be told that, for a risk of 1% between taken considered facts, have a significance influence on the arrow value which with are deforming the diaphragm when on their surface are applied a uniform distributed pressure, active diameter of diaphragm, (most important) thickness of material, the amplitude of corrugates and interact between the diameter and thickness of the material. The fact that interact between the diameter of the diaphragm and the thickness of material have a important effect, may be explained by the modification of material volume which are obedient to external forces (deforming pressure)
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 n1
485,04 2,26 37,04 319,78 4,82 1,57 62,50 0,09 0,03 1,27
fα
Fig. 6. The graph of the interactions between factors about value of the elastic deformation From the graphs 3, 4 ,5 and 6 can be observed that exist important interactions between diameter-number of corrugates and between diameter-the thickness of material, so the effects of each one, depend by the levels of other facts, but the other interactions are low, almost negligible. The wickets interaction is between the number of corrugates and the thickness of material.
Even if we low the trust level up to 95% can be remark that for a risk of 5% the effect of „n” factor also of the other interacts not becomes important so can be insignificant ,fig.7 The experience plans give the possibilities to determinate the influent facts and to know optimal level for everyone of them. After we eliminated the essential facts the matrix purpose model become:
2.4. The acceptance testing and validate the model To see which facts taken in consideration have important influence to the value of arrow of deformation is applied the test Snedecor (Ionescu, and Amarandei, 2004), take account that the numbers of liberty grades of model is 11 and the number of dross liberty degrees is 5, and the check and the arrange of the influence facts is present in table 4.
W = 0,8825 + [0,43875 - 0,43875]⋅ D +
+ [0,12125 - 0,12125]⋅ H + + [0,35625 - 0,35625]⋅ h + 0,1575 - 0,1575 +tD ⋅ ⋅h - 0,1575 0,1575
441
The target for the experiment is to find a optimal configuration for a set of facts taking analyse, configuration which are lead to maximise the elastic deformation it is recommends for independent facts that type of matrix D = [1 0], H=[1 0], h=[1 0].
REFERENCES Alexis, J. (1999) Metoda Taguchi în practica industrială, Editura tehnică, Bucureşti; Ionescu, R., Amarandei, D. (2004) Planificarea experimentelor eficienta si calitate. Bucuresti, Editura AGIR,; Negoescu, F., Braha V., Nagîţ, Gh. (2002), Membrane metalice ondulate - Elemente de proiectare şi tehnologii de fabricare. Editura Tehnica – Info, ISBN 9975-63-176-2, Chişinău; Pillet, M. (1992), Introduction aux plans d’expériences par la méthode Taguchi. Les éditions d’organisations, Paris, ; Vigier, M. (1993), Pratique des plans d’expériences. Méthodologie Taguchi. Les Editions D’Organisations, Paris.
It can be told that the rigidity of the diaphragm grow if the diameter and the thickness of the materials have small values, and the amplitude of corrugates grow, in time to sensibility of diaphragm for same value at the mentioned parameters go low. So are confirmed the fact that sensibility go low in the same time with growing the rigidity.
Fisher teor. Fisher p=0,99 Fisher p=0,95 500 450 400 350 300 250 200 150 100 50 0
Ef-D
Ef-n
Ef-H
Ef-h Int-D-n Int-D-H Int-D-h Int-n-H Int-n-h Int-H-h
Fig. 7. The graph representation of the Fisher test
Growing the rigidity in the same time with growing the thickness of material is explained by the growing of volumes of material which presumed for deformation bigger forces. The movement of the diameter of diaphragm make growing the rigidity because grow stability of the form, knowing are the fact that in case of thin elements, decrease bigger dimension lead to big rigidity of the form. The optimization of metallic corrugated diaphragms, is adverted to touch the performances technically maxims, only that satisfy the concomitance the all criteria enforced is difficult, sometimes just impossible, because one factors can be antagonistic his subjective. With all these an optimum design is admissible and she assures to obtain the maximum performances, in certain conditions.
442