American Journal of Mechanical Engineering and Automation 2015; 2(5): 55-58 http://www.openscienceonline.com/journal/ajmea) Published online September 28, 2015 (http://www.openscienceonline.com/journal/
Monte Carlo Model Applied to Tolerance Analysis of Mechanical Assembly Sets Helena V. G. Navas UNIDEMI,, Faculty of Sciences and Technology, Universidade Universi NOVA de Lisboa, Lisbon, Portugal
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To cite this article Helena V. G. Navas. Monte Carlo Model Applied to Tolerance Analysis of Mechanical Assembly Sets. Sets American Journal of Mechanical Engineering and Automation. Vol. 2, No. 5,, 2015, pp. 55-58.
Abstract Tolerancing is an important issue in the quality evaluation of mechanical systems. Tolerancing of an industrial gear assembly set includes dimensioning and accuracy, fits and tolerances, clearances and interferences, accuracy of surface form, accuracy of orientation and location of component features. In this study, dimensional tolerancing of an elementary assembly unit that is part of an industrial gear assembly set was considered. A tolerances attribution was carried out aided by Monte Carlo Model and based on occurrence probability of assembled units acceptance by functional condition criteria. A JAVA program was created for fo this purpose.
Keywords Dimensional Tolerancing, Dimensional Analysis, Analysis Model, Dimensional Dime Chain, Monte Carlo Model
1. Introduction Dimensioning and tolerancing are important issues in the quality evaluation of mechanical systems. Mechanical parts are produced from blank or stock materials through a definite set of technological operations, and their dimensional and geometrical accuracy are key points for the assembly process and for the product performance. The specification of dimensional and geometrical tolerances of mechanical parts is crucial for the final definition of a manufacturing process and its detailed programme. The decisions about adjustments, surface roughness, dimensional and geometrical tolerances of mechanical parts and of respective assembly sets determine the technological process, measuring operations and required resources [1]. The appropriateness of dimensional and geometrical tolerancing depends on particular characteristics of the assembly and their mechanical part, its production rate, technological process, specific contract requirements, available human and technical means, costs, deadlines, etc. The requirements related to dimensional and geometrical precision and exactness of mechanical assemblies had a considerable increase during last decades. A growing need for componentss interchangeability and an improvement of mechanical equipment assembly techniques leaded
tolerancing to a high level as never before. Any mechanical design process must include a dimensional analysis. Correct proportions between interconnected dimensions and dimensional tolerances are obtained through dimensional analysis [2]. A rational tolerance specification of any mechanical system must be based on an analytical model, with which the dimensional tolerances are calculated [3]. Several analysis models and nd tools were developed to assist design engineers in specifying tolerances on the basis of performance and manufacturing considerations (Fig. 1 shows several models).
Fig. 1. Mathematical models of tolerance accumulation [3].
The simulation models can be especially adequate to
American Journal of Mechanical Engineering En and Automation 2015; 2(5): 55-58 55
attribution of tolerances for elements of most complex assembly sets, turning the tolerancing tasks faster and easier [4].
2. Monte Carlo Model on Dimensional Tolerancing of Gear Assemblies Sets The Tolerancing ancing of an industrial gear assembly set includes dimensioning and accuracy, fits and tolerances, clearances and interferences, accuracy of surface form, accuracy of orientation and location of component features. The functional conditions can be like follows: fol sealing between components, rotation between components, balancing between components, gearing, fastening of connecting components, sliding between components, press fit, clearance fit, etc. In this study, dimensional tolerancing of an elementary assembly embly unit that is part of an industrial gear assembly set was considered. This is an extensive list of functional conditions for the assembly set. Only one functional condition was considered - the axial clearance (F) of the gear (Fig. 2 shows the functional condition).
The Fig. 3 shows the dimension chain.
Fig. 3. Dimension chain.
The chain equation is [5] (1)
Maximum and minimum limits of the clearance Fmax. = dmax. – amin. – 2bmin. – cmin.
Fmin. = dmin. – amax. – 2bmax. – cmax.
(3)
The clearance can vary between 0,1 and 0,6 mm. In other words, the clearance F is F = 0 ++0,6 0,1
The clearance tolerance is TF = Fmáx. – Fmín. = 0,6 – 0,1 = 0,5 mm.
(4)
Table 1 contains the nominal dimensions of the chain elements. Table 1. Nominal dimensions of the chain elements. elements Chain element Nominal Dimension, mm
a 140
b 20
c 120
d 300
A tolerances attribution was carried out aided by Monte Carlo Model and based on occurrence probability of assembled units acceptance by functional condition criteria (Fig. 4 shows the tolerances attribution). A JAVA program was created for this purpose.
Fig. 4. Assembly tolerance analysis by Monte Carlo simulation [3].
Fig. 2. Functional condition (the axial clearance). clearance)
F = - b + d – c– b – a
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(2)
The model considers an expectative statistical distribution for each functional dimension, taking into consideration also specific characteristics of each component and its probable fabrication process [6]. The model presupposes an attribution of tolerances and tolerance zones for all dimension of a dimension chain. The program simulates a fabrication of each component, an assembly and a subsequent checking of functional condition. The cycle repeats until a given number of assembled units are simulated. In the study case, 1000 assembly units were “fabricated” and “assembled” at each attempt. Afterwards, the percentage of rejected assemblies was calculated and compared with the required rejection rate value [7]. The components tolerances were adjusted for the specified value of rejected assemblies. The iterations presuppose the adjustment attempt of components tolerances according to the clearance deviation from the desired value. If the obtained clearance ance is too tight, the components tolerances are increased. If the clearance exceeds the specified value, the components tolerances are reduced. This procedure can have several iterations until the axial clearance specification is fulfilled. In this study, the normal distribution was used for functional
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Helena V. G. Navas:
Monte Carlo Model Applied to Tolerance Analysis of Mechanical Assembly Sets
dimensions deviations [8]. The acceptance rate of each component fabrication process was 99,73%, it means 2700 rejected components per million (± 3σ). The analysis of components characteristics and it probable fabrication process was done. It was decided that the dimension a (the width of the gear) must have a precision corresponding to international tolerances levels IT 5 or IT 6 [9] (these levels are used for high precision mechanical construction). The other dimensions can have a precision corresponding to levels between IT 9 and IT 11 [9] (these levels are used for common mechanical construction). The first attempt was done with the following tolerances and tolerance zones: a = 140h6 = 1400−0,025 b = 20h11 = 200−0,13 c = 120h11 = 1200−0,22 d = 300 D11 = 300 ++0,51 0,19 Table 2 contains other values used on the first attempt. Table 2. Deviations of the functional dimensions and their frequencies (attempt Nº 1). Interval Frequency
Dimension Deviations, µm
a b1 b2 c d
- 3σ 0,021 -25,00 -130,0 -130,0 -220 +190
- 2σ - 1σ + 1σ 0,136 0,682 -18,75 -12,50 -97,5 -65,0 -97,5 -65,0 -165 -110 +270 +350
+ 2σ 0,136 -6,25 -32,5 -32,5 -55 +430
+ 3σ 0,021 0 0 0 0 +510
Fig. 5 shows the dimension deviation distribution bar chart.
a = 140h6 = 1400−0,025 b = 20h10 = 20 0−0,084 c = 120h10 = 1200−0,14 d = 300 E10 = 300 ++0,32 0,11
3. Conclusions The advantages of this methodology as well of the simulation models on the whole increase with the growing of mechanical system complexity. The methodology obtains an overall view of the system, permitting parallel and simultaneous tolerancing solutions. In opposition, when an assembly set contains several elementary assembly units and/or when there are several functional conditions then the use of traditional procedures turns the tolerancing tasks slow and complex, turning overall view impossible and search for parallel solutions difficult. The disadvantage of the Monte Carlo method is that it requires large samples to achieve reasonable accuracy [3].
Acknowledgments The present author would like to thank the Faculty of Science and Technology of the New University of Lisbon (UNL) and the Portuguese Foundation for Science and Technology (FCT) through the Strategic Project no. UID/EMS/00667/2013. Their support helps make our research work possible.
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f (x)
0,682
0,136
0,136
0,021
0,021 x
Fig. 5. Dimension deviation distribution bar chart.
The results obtained on the first attempt were 909 accepted assemblies and 91 rejected assemblies. The rejection rate was too high, so more attempts with tighter tolerances for functional dimensions fabrication were done [10]. The process is convergent, because the rejection rate becomes more and more reduced as the fabrication tolerances are decreased. The null rejection rate was obtained on the last attempt. 1000 assembly sets were “fabricated” and “assembled” and all of them were accepted. This attempt was done with the following tolerances and tolerance zones:
American Journal of Mechanical Engineering and Automation 2015; 2(5): 55-58
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Spotts, M. F.: Dimensioning and Tolerancing for Quantity Production, Prentice-Hall (1983).
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ISO 286-1: ISO System of Limits and Fits – Part 1: Bases of Tolerances, Deviations and Fits, International Organization for Standardization (1998).
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