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Objectifs. Prsentation de techniques non-dterministes pour la modlisation et le calcul des variations gomtriques;Complmentarit avec les techniques de calcul des tolrances (arithmtiques ou au pire des cas);Prsentation de travaux de recherche antrieurs ou en cours autour du thme; Princi
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1. Modlisation gomtrique des systmes poly-articuls Approches statistiques et techniques de simulation Monte-Carlo
Nabil ANWER - MCF
LURPA/ENS Cachan
Universit Paris Nord 13
anwer@lurpa.ens-cachan.fr
2. Objectifs Prsentation de techniques non-dterministes pour la modlisation et le calcul des variations gomtriques;
Complmentarit avec les techniques de calcul des tolrances (arithmtiques ou au pire des cas);
Prsentation de travaux de recherche antrieurs ou en cours autour du thme;
Principe de calcul de logiciels commerciaux (Tolmate, CeTol, );
Mthodes pour lanalyse des erreurs cinmatiques.
3. Variations gomtriques des pices mcaniques (rpartition des caractristiques) Comportement statistique
dune spcification (lot de pices) Comportement statistique dune spcification (pice)
4. Variations gomtriques des pices mcaniques (rpartition des caractristiques)
5. Variations gomtriques des ensembles mcaniques (position dun point, jeu)
6. Techniques de rsolution
7. Rappels de statistiques
8. Introduction Statistiques
Dcrivent des populations
Estiment des paramtres
Testent des hypothses
Statistiques descriptives
Visent explorer les donnes et en tirer un certain nombre de mesures et d'indices, ou des reprsentations graphiques
Variables alatoires
Probabilits et lois de distribution
9. Statistiques descriptives Vocabulaire :
Population : Ensemble des objets de ltude (pices mcaniques)
Individu : lment de la population (pice mcanique)
chantillon : Partie de la population
Taille : Nombre dindividus dans la population/chantillon
Variable : Application associant chaque individu un caractre (valeur dune cote mesure).
On associe un caractre une variable statistique X qui donne la valeur du caractre pour un individu ( ex : la variable X donne la taille d'un lve; la variable Y donne le poids d'un lve)
Variables qualitatives Vs. Variables quantitatives
10. tude dune variable Effectifs et Frquences
n : taille de la population
xi : i-me modalit de la variable X
ni : nombre dindividus ayant xi comme modalit (effectif de la modalit xi)
fi : frquence de la modalit xi (fi = ni/n)
Pour un caractre donn une population peut tre rpartie en classes
Le centre d'une classe [ai, ai+1[ est la valeur (ai + ai+1)/2
ex : le centre de la classe [150, 170[ est (150 + 170)/2 = 320/2 = 160
L'amplitude d'une classe [ai, ai+1[ est (ai+1 - ai)
11. Combien de classes doit-on raliser ? Absence de rgles universelles, solutions empiriques et pragmatiques.
l'objectif est de conserver la distribution sa forme gnrale
Critre de Brooks-Carruthers, le nombre de classes Kt :
Kt < 5 log10 n
Critre de Huntsberger-Sturges, le nombre de classes Kt :
Kt = 1 + (10 log10 n)/3
Formule empirique :
Intervalle de classe ht :
Wt = Xmax - Xmin (tendue)
ht = Wt/Kt
12. Exemple
13. Paramtres caractristiques
14. Variables alatoires Une variable alatoire X est une variable qui prend ses valeurs au hasard parmi un ensemble de n valeurs possibles (n fini ou infini).
Une valeur particulire de X est dsigne par xi
n valeurs x1, x2, x3, ... , xn dune variable alatoire X peuvent tre caractrises par :
15. Loi de distribution dune variable alatoire continue Densit de probabilit :
Proprits :
16. Exemples de distributions Distribution Normale : Distribution Uniforme :
17. Moyenne et cart-type (Estimation) Dans la pratique, m et s dune variable alatoire X sont rarement connus. Ils sont estims partir des observations dont on dispose sur un chantillon.
18. Thorme de la limite centrale La moyenne dune variable alatoire X calcule sur des chantillons de mme taille n est une variable alatoire note
19. Thorme de la limite centrale Illustration par lexemple (10000 essais)
20. Comment lier les variations gomtriques aux distributions statistiques ? Hypothse :
Connaissance priori de la distribution sous-jacente
Limites :
Forte hypothse de normalit (mythe de la loi normale)
Tests de normalit pas souvent effectues
Modle utilis :
t=ks
t : caractristique observe ou mesure (ex. tolrance, jeu, position dun point, variable articulaire)
: cart-type de la distribution sous-jacente
k : coefficient qui dpend de la nature de la distribution et de la proportion dacceptation en gnral 99,73% (approche 6s) (risque)
21. Lien fort avec les capabilits : point de vue MSP
22. Capabilits : court terme vs. long terme
23. Capabilits : indices
24. Tolrancement statistique Amlioration du modle arithmtique
25. tude dun mcanisme lmentaire
26. Mthode au pire des cas Forme mini-maxi :
a ( b + c+ d + e + f ) + (ta + tb + tc + td + te + tf )/2 ? X maxi
a (b + c+ d + e + f) - (ta + tb + tc + td + te + tf)/2 ? X mini.
Forme moyenne et IT :
a ( b + c+ d + e + f) = X moyen
ta + tb + tc + td + te + tf ? ITx
Les valeurs encadres a, b, c, d, e, f sont supposes connues. La condition respecter ne concerne que les tolrances.
Application numrique :
la rpartition uniforme des tolrances (ta = tb = ) donne :
ta = tb = tc = td = te = tf = ITx /6 = 0,12/6 = 0,02.
27. Synthse sur les mthodes au pire des cas Il est peu probable que les pices soient aux limites des tolrances et que toutes les pices soient maximales/minimales en mme temps.
En milieu industriel, lapproche au pire des cas est juge trop svre, des mthode statistiques sont utilises ds que le nombre de pices de la chane de cotes devient important.
Les mthodes statistiques de rpartition des tolrances des mthodes de calculs prvisionnels (le tolrancement statistique se fait en bureau d'tudes, bien avant que les fabricants ne ralisent les pices).
Il nest donc pas toujours possible de tenir compte des rsultats de production pour faire des optimisations (approches robustes).
28. Modles statistiques
29. Modles statistiques
30. Calcul
31. Conclusions
32. Prise en compte du dcentrage
33. Autres modles (Anselmetti)
34. Modle gnrale pour le tolrancement statistique De nombreuses conditions fonctionnelles s'crivent comme combinaison linaire des composantes indpendantes :
Les coefficients ai peuvent tre dus une symtrie, une projection ou un effet de bras de levier
En gnral,
35. Problme de lindpendance La condition d'indpendance n'est pas toujours vrifie :
- Assemblage comportant deux pices identiques tires du mme lot (deux entretoises ou deux flasques de chaque ct d'un mcanisme symtrique, plaque tires dans la mme tle).
- Pices symtriques issues du mme moule (s'il y a un cart de fermeture du moule, les deux pices subiront la mme variation).
- Influence d'un paramtre extrieur qui modifie la dimension des pices (usure, temprature, dformation...).
Liens dus au processus de fabrication (deux gorges identiques ralises par le mme outil, usinage en commande numrique avec le mme outil, mme montage d'assemblage...).
36. Influence de plusieurs spcifications sur une mme pice
37. Prise en compte des contacts
38. Conclusions Le tolrancement statistique permet une rduction considrable des cots.
Les modles statistiques se basent sur des hypothses fortes de pseudo-normalit et manquent de formalisme rigoureux.
Le cas de variables indpendantes est trs souvent rencontrs en milieu industriel.
La superposition de plusieurs spcifications rend le problme plus complexe.
La prise en compte des contacts se base sur des approches exprimentales pour lesquelles les identifications de modles sont amliorer.
Le bouclage contrle/fabrication/conception est le seul moyen de garantir la robustesse et loptimum.
Il reste intgrer les aspects 3D (tolrancement statistique radial)
La caractrisation statistique des zones de tolrances travers les travaux en gomtrie probabiliste et en simulation est une nouvelle alternative.
39. Caractrisation statistique des dfauts Apports de la mtrologie
Apports des techniques de simulation
40. Caractrisation statistique des dfauts
41. Caractrisation statistique des dfauts
42. Approches de simulation Techniques de monte carlo
43. Prsentation de la mthode Exemple illustratif
44. Prsentation de la mthode Estimation de
Soit p(u) une fonction de densit de probabilit uniforme sur [a, b]
Soit Ui la i me variable alatoire uniforme de densit p(u)
Alors, si n est grand :
45. Principe de simulation monte carlo Principe
Pour effectuer des simulations probabilistes sur ordinateur, on utilise un gnrateur de nombres pseudo-alatoires (une suite (xn)n de nombres rels compris entre 0 et 1) qui imitent une ralisation d'une suite de variables alatoires indpendantes et identiquement distribues suivant la loi uniforme sur [0;1].
Loi uniforme sur [a,b]
Si U est une variable uniforme sur [0;1] alors
Loi normale N(m,s)
Si U1 et U2 sont deux variables uniformes indpendantes sur [0;1] alors
46. Approches base de simulation Tirage alatoire de caractristiques
Connaissance priori des distributions des caractristiques (point, droite, plan)
Estimation de la rsultante
Utilisation de logiciels statistiques (Minitab)
47. Analyse statistique des zones de tolrances
48. Analyse statistique des zones de tolrances
51. Statistiques sur la normale
52. Empilage de pices 2D
53. Analyse statistique des zones de tolrances
54. Analyse statistique des zones de tolrances
58. Analyse statistique des zones de tolrances
59. Analyse statistique des zones de tolrances
60. Position Error in Assemblies and Mechanisms Travaux de Jonathan Wittwer
61. Position Error in Assemblies
62. Direct Linearization (DLM) [That is the Direct Linearization Method, or DLM]
Its a good thing this session is before lunch, or these next few slides would put you all to sleep.
This method is based upon finding the sensitivities of the various parameters on the output error.
First, we start with the closed loop and expand it into two equations the summation of the vector components in the x and y directions.
This is a nonlinear system of equations since the unknown variables are theta 3 and theta 4.
However, we can linearize them by taking a taylors series expansion and dropping higher order terms.
X is a vector of primary random variables and U is a vector secondary random variables.
A and B are matrices of partial derivatives.
We then solve for the variations in the unknown variables.
We could stop here if all we wanted to know was the variation in angles theta 3 or theta 4[That is the Direct Linearization Method, or DLM]
Its a good thing this session is before lunch, or these next few slides would put you all to sleep.
This method is based upon finding the sensitivities of the various parameters on the output error.
First, we start with the closed loop and expand it into two equations the summation of the vector components in the x and y directions.
This is a nonlinear system of equations since the unknown variables are theta 3 and theta 4.
However, we can linearize them by taking a taylors series expansion and dropping higher order terms.
X is a vector of primary random variables and U is a vector secondary random variables.
A and B are matrices of partial derivatives.
We then solve for the variations in the unknown variables.
We could stop here if all we wanted to know was the variation in angles theta 3 or theta 4
63. Solving for Assembly Variation But that would be too easy, so we need to include the open loop equation also.
Here, we expand the open loop equation to describe the position of the point on our mechanism.
We then take the taylors expansion of the open loop equations, substitute in for the secondary variables, perform a little linear algebra magic,
And wallah, we have the position variation in terms of a friendly sensitivity matrix and our known variables.
The sensitivity matrix can then be used to obtain both deterministic and probabilistic results.But that would be too easy, so we need to include the open loop equation also.
Here, we expand the open loop equation to describe the position of the point on our mechanism.
We then take the taylors expansion of the open loop equations, substitute in for the secondary variables, perform a little linear algebra magic,
And wallah, we have the position variation in terms of a friendly sensitivity matrix and our known variables.
The sensitivity matrix can then be used to obtain both deterministic and probabilistic results.
64. Worst-Case vs. Statistical These equations are somewhat familiar they are just another way of writing the equations that Dr. Chase presented in his presentation this morning.
For the worst-case situation, the error is simply the sum of the tolerances multiplied by the magnitude of their respective sensitivities.
For statistical cases, the error is the root sum square.These equations are somewhat familiar they are just another way of writing the equations that Dr. Chase presented in his presentation this morning.
For the worst-case situation, the error is simply the sum of the tolerances multiplied by the magnitude of their respective sensitivities.
For statistical cases, the error is the root sum square.
65. Deterministic Methods: Worst-Case Direct Linearization:
Uses the methods just discussed.
Vertex Analysis:
Finds the position error using all combinations of extreme tolerance values.
Optimization:
Determines the maximum error using tolerances as constraints. Here, this is a little better. Less math, but more words.
In order to validate the worst-case direct linearization method, it will be compared to two other deterministic methods: Vertex analysis and Optimization.
The vertex analysis finds
[This kind of analysis is usually used to validate other worst-case approaches. For every set of initial conditions, the set of nonlinear equations is solved. Although the results of a vertex analysis are often used in statistical studies, the data itself is deterministic because it is based on fixed tolerance values.]
Optimization determines
The optimization routine ideally searches out the whole design space to discover the absolute maximum error.
The third method is to use Optimization to determine
Using an optimization that searches the whole design space is the ultimate test of worst-case deterministic methods ]Here, this is a little better. Less math, but more words.
In order to validate the worst-case direct linearization method, it will be compared to two other deterministic methods: Vertex analysis and Optimization.
The vertex analysis finds
[This kind of analysis is usually used to validate other worst-case approaches. For every set of initial conditions, the set of nonlinear equations is solved. Although the results of a vertex analysis are often used in statistical studies, the data itself is deterministic because it is based on fixed tolerance values.]
Optimization determines
The optimization routine ideally searches out the whole design space to discover the absolute maximum error.
The third method is to use Optimization to determine
Using an optimization that searches the whole design space is the ultimate test of worst-case deterministic methods ]
66. Deterministic Results So, when we look at the results for the position error of a mechanism, it is important to compare these different methods.
This graph represents the position error for the 4-bar at a given configuration using deterministic methods, where the path slope is shown by this blue line.
The dots are the results of the vertex analysis
The dashed box is the result of the W-C DLM analysis
The Star is the result of the optimization routine.
First, our optimization routine did not find a way out of the room. In other words, the actual absolute maximum error corresponds to one of the vertex points.
The amount that the vertex points and optimization are outside the worst-case box represents the error in linearizing the equations.
The Worst Case DLM seems very conservative in some places, especially since the vertex points seem to be clustered around a slope of 45 degrees.So, when we look at the results for the position error of a mechanism, it is important to compare these different methods.
This graph represents the position error for the 4-bar at a given configuration using deterministic methods, where the path slope is shown by this blue line.
The dots are the results of the vertex analysis
The dashed box is the result of the W-C DLM analysis
The Star is the result of the optimization routine.
First, our optimization routine did not find a way out of the room. In other words, the actual absolute maximum error corresponds to one of the vertex points.
The amount that the vertex points and optimization are outside the worst-case box represents the error in linearizing the equations.
The Worst Case DLM seems very conservative in some places, especially since the vertex points seem to be clustered around a slope of 45 degrees.
67. Statistical Methods Monte Carlo Simulation
Thousands to millions of individual models are created by randomly choosing the values for the random variables.
Direct Linearization: RSS
Uses the methods discussed previously.
Bivariate DLM
Statistical method for position error where x and y error are not independent. Now we come to the statistical methods:
First, a very common method is Monte Carlo Simulation, where thousands
This method is the statistical corollary to the Vertex Analysis method. Instead of fixed values for the tolerances, the variables are chosen randomly based upon a given probability density function (such as uniform or normal). The model is then solved using some nonlinear solver.
The second method is the Root-Sum-Square direct linearization methods, where the position
This method is the statistical corollary to the Worst-Case DLM method, where standard deviations are used instead of tolerance limits.
Third is the bivariate DLM method. This is a statistical .
This is what Id call the optimum statistical method for determining position error. It ends up being the best description of the actual error zone.Now we come to the statistical methods:
First, a very common method is Monte Carlo Simulation, where thousands
This method is the statistical corollary to the Vertex Analysis method. Instead of fixed values for the tolerances, the variables are chosen randomly based upon a given probability density function (such as uniform or normal). The model is then solved using some nonlinear solver.
The second method is the Root-Sum-Square direct linearization methods, where the position
This method is the statistical corollary to the Worst-Case DLM method, where standard deviations are used instead of tolerance limits.
Third is the bivariate DLM method. This is a statistical .
This is what Id call the optimum statistical method for determining position error. It ends up being the best description of the actual error zone.
68. Bivariate Normal Position Error The key to using the bivariate distribution is to analyze the assembly variances, including the correlation between the x and y position variance.
Vx is the variance in the x-direction
Vy is the variance in the y-direction
Vxy is the covariance, where the sensitivities are combined in this manner.
These variances form a symmetric matrix, or variance tensor. The eigenvalues of this tensor are the principle variances that represent the major and minor diameters of the ellipse. The angular rotation of the ellipse can also be determined.
The key to using the bivariate distribution is to analyze the assembly variances, including the correlation between the x and y position variance.
Vx is the variance in the x-direction
Vy is the variance in the y-direction
Vxy is the covariance, where the sensitivities are combined in this manner.
These variances form a symmetric matrix, or variance tensor. The eigenvalues of this tensor are the principle variances that represent the major and minor diameters of the ellipse. The angular rotation of the ellipse can also be determined.
69. Statistical Method Results The result is an elliptic position zone oriented at some angle.
The mass of little xs is the result of Monte Carlo
The ellipse is the result of the bivariate model
The box is the result of the RSS DLM method.
So, what significance can you see from this graph?
Notice that the bivariate model is verified by the Monte Carlo results.
The DLM statistical method without considering the correlation between x and y results in a box.
The 3-sigma tolerance zone is most closely approximated using the bivariate model, whereas the straight R.S.S. method is more conservative.
It turns out that Red box represents a true 3-sigma, while the ellipse slightly underpredicts it. This was found by comparison to Monte Carlo. After 100,000 points, the ellipse contained 98.84% and the Box contained 99.73%.The result is an elliptic position zone oriented at some angle.
The mass of little xs is the result of Monte Carlo
The ellipse is the result of the bivariate model
The box is the result of the RSS DLM method.
So, what significance can you see from this graph?
Notice that the bivariate model is verified by the Monte Carlo results.
The DLM statistical method without considering the correlation between x and y results in a box.
The 3-sigma tolerance zone is most closely approximated using the bivariate model, whereas the straight R.S.S. method is more conservative.
It turns out that Red box represents a true 3-sigma, while the ellipse slightly underpredicts it. This was found by comparison to Monte Carlo. After 100,000 points, the ellipse contained 98.84% and the Box contained 99.73%.
70. Comparison of both deterministic and probabilistic methods. Here is an example of how the maximum normal position error varies for one complete revolution of a four-bar crank.
This plot which analyzes the mechanism at 100 different points was generated in seconds.
Here is an example of how the maximum normal position error varies for one complete revolution of a four-bar crank.
This plot which analyzes the mechanism at 100 different points was generated in seconds.
71. Benefits of Bivariate DLM Accurate representation of the error zone.
Easily automated. CE/TOL already uses the method for assemblies.
Extremely efficient compared to Monte Carlo and Vertex Analysis.
Can be used as a substitute for worst-case methods by using a large sigma-level First, .
Second, .
Fourth, .
AND this is as far as Ive gotten so far, so theres not much of a conclusion.First, .
Second, .
Fourth, .
AND this is as far as Ive gotten so far, so theres not much of a conclusion.