=Paper=
{{Paper
|id=Vol-1904/paper5
|storemode=property
|title=Probability-theoretical model for product assembly parameters assessment
|pdfUrl=https://ceur-ws.org/Vol-1904/paper5.pdf
|volume=Vol-1904
|authors=Nikolay V. Ruzanov,Mikhil A. Bolotov,Vadim A. Pechenin
}}
==Probability-theoretical model for product assembly parameters assessment ==
https://ceur-ws.org/Vol-1904/paper5.pdf
Probability-theoretical model for product assembly parameters assessment
N.V. Ruzanov1, М.А. Bolotov1, V.А. Pechenin1
1
Samara National Research University, 34 Moskovskoe Shosse, 443086, Samara, Russia
Abstract
The proposed work provides a model for estimating the assembly parameters of the products by the example of the cone ring assembly
process. There was carried out the simulation of the process of parts mating along the surfaces with form deviation. As a result of Monte Carlo
simulation, there were obtained statistical characteristics of the parameters of the assembled units depending on various initial positions of the
parts being assembled. The proposed model could be used to assess the accuracy of the products assembled in the aircraft industry.
Keywords: assembly parameters; mating of components; Monte Carlo method; probabilistic assessment; probability density
1. Introduction
The result of manufactured products assembly depends on the actual shape of the parts used, their initial location and the
assembly process. The errors of the actuating mechanisms used during parts manufacture have a significant effect on the
accuracy of the manufactured products [1], so that even the parts of the same type do not have actual identical shape. Moreover,
even for a single batch of products it is hardly achievable to ensure the exact matching of assembly conditions for various items
manufacture.
Assembly parameters assessment will allow determining an achievable accuracy of product manufacturing, which, in turn,
will enable to solve a number of problems:
1. Determine the percentage of products meeting process requirements [2].
2. Determine rational tolerances for product parameters [3].
3. Identify critical factors affecting the quality of product assembly (assembly deviation, parts prepositioning or tool selection
for parts assembly) [4-7]
An assessment of the assembly parameters of products can be obtained through various approaches. One of them is based on
accumulation and analysis of results of the manufactured products inspection phase. Another approach is based on numerical
simulation of the product assembly process and subsequent analysis of the obtained results. The use of production statistics
requires considerable human and material resources, and therefore is hardly feasible. The first approach being rather difficult to
realize is the cause why math modeling methods are widely used to solve the specified problem.
In this paper, the authors present a model for estimation of product assembly parameters, based on assembly process
simulation by numerical methods. Moreover, there is given an example of using the proposed model for estimation of assembly
parameters of cone rings that are widespread products in the aircraft industry.
2. Model description
Product characteristics depend not only on parts comprising it but also on assembly procedures. Assembling of complex
products consisting of many parts is a multi-criteria task that takes into account the size of all the components, their mutual
arrangement in the finished product, and the alignment procedure [8].
Today, the parts' quality issues are understood deeply, so in order to improve quality characteristics of the product further, the
researches aimed at studying the product assembly procedure are becoming increasingly popular. The complexity of
implementing multiple product assembly leads to the fact that the majority of such researches is based on the employment of
numerical simulation methods in the assembly process.
One of the features of parts assembly in the aircraft industry is the deformation of parts of the product. So as to determine the
condition of parts of the product assembled, ANSYS application is used, which enables us to calculate the strength of the parts
and assemblies, to solve the problems of gas and hydrodynamics [9]. Such approaches have high computational costs, so in order
to simplify the solution of the assembly problem, many researchers consider the mating parts as absolutely rigid bodies.
The authors [10] consider mating two parts along plane surfaces, having form deviation. Researchers proposed a
mathematical model simulating the assembly along the planar surfaces; and the result of using of such model is calculation of the
clearance between the surfaces in product assembled.
Paper [11] is also devoted to the problem of parts mating along the planar surfaces. The authors of this work suggested a
model for describing the deviation of the shape of the specified surfaces and considered the result of the parts mating with
various shape deviations of the planar surfaces.
Most of the works focus on the simulation of the assembly process without setting up a formal problem. In paper [12] the
authors formalize the product assembly problem and suggest using such concepts as the initial assembly conditions, the
assembly quality assessment function, and the assembly sequence function.
The proposed work considers the assessment of the assembly parameters of the product obtained from various assembly
process models.
The mutual arrangement of the parts is a fundamental assembly parameter of the product as other geometric characteristics of
the finished product can be derived from it (for example, out-of-true running, out-of-flatness, uneven clearance, etc.). The
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methods for positioning coordinate systems that are bound to these parts are well suited to describe the mutual arrangement of
the parts.
Let us consider a product that is assembled from two parts K1 and. The product coordinate system R is compatible with the
design coordinate system R1 of the part K 1 . With such a transformation, the part K1 will be stationary relative to the coordinate
system R , and the assembly will be carried out by moving the part K 2 . The assembled state of the product can be described by
the position of the design coordinate system R2 . The errors in the assembly process will result in the onset of a set of assembled
states .
dx
dy
dz
: , (1)
where dx, dy, dz is the displacement of the coordinate origin of system R2 relative to the origin of system R1 ;
, , are the angles of turn of basis vectors of system R2 relative to vectors R1 .
The elements of this set describe the mutual arrangement of the parts in the assembled product. To solve the problem of
evaluating the assembly parameters of the products, it is necessary to get the parameters for this set. The use of the Monte Carlo
method enables us to determine the approximate value of this parameters. According to this method, the first step is to perform a
multiple numerical simulation of the assembly of the product and save the simulation results. The second step is to investigate
the obtained simulation results and calculate the required parameters.
We will use the following values to estimate the set of assembled states: the mean value, root-mean-square deviation and
probability density.
To calculate the average value, let us use the following formula (2):
dx
dy
dz 1 n
(2)
, a i 1 ai ,
n
The root-mean-square deviation is calculated with the formula (3):
dx
dy
(3)
dz
a
,
1 n
ai a ,
n i 1
Obtaining an analytic expression for a probability density function is a complicated task, so we can use numerical methods to
evaluate the state with the approximate values of the function with the required accuracy, based on the distribution histograms
and their possible approximation.
Distribution histogram method provides empirical estimates K 2 of the density of distribution of the random value [13]. The
algorithm for obtaining the distribution density histogram is shown in Fig. 1.
To create a histogram, the observed range of the random variable is divided into several intervals, and then the number of the
random value hits in each interval is calculated. The Sturges' rule (4) is used to determine the number of the intervals:
n 1 log 2 N , (4)
The next step is standardization of the received values to meet the condition (5)
f d 1,
(5)
The final step is to approximate the probability density function on the basis of the midpoint of the intervals and values
calculated in the previous step.
The mean and root-mean-square deviation values, the approximate value of the probability density function describe a set of
assembled states and can be used to solve further tasks.
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Fig. 1. Algorithm for numerical calculation of probability density. Algorithm for numerical definition of the probability density.
3. Simulation results
To test the proposed methodology, there has been carried out the evaluation of the assembly parameters of the parts that have
the cone surfaces. Mating of such surfaces is widely used in the aviation industry and the characteristics of the entire product
depend on the assembly quality of these parts.
The mathematical model of the cone surface of a part can be represented in a parametric form (6):
F F (u , v)
x(u , v) R2 R1 * R1 * H H u F (u , v) *cos(v)
H R2 R1
, (6)
R2 R1 R1 * H
y (u , v) H * R R H u F (u , v ) *sin(v)
2 1
z (u , v) u
where F is the function of deviation of the shape of the actual part from its nominal value;
R1 , R2 , H are the radii of the cone surface and its height;
u, v - surface parameters
x(u, v), y(u, v), z(u, v) - surface point coordinates
A model of a part that has a cone surface is shown in fig. 2.
Fig. 2. A model of a part with a cone surface.
Fig. 3 shows a mechanical system model consisting of two parts that have cone surfaces. For parts K1 and K 2 the local
design coordinate systems R1 and R2 are set. Parts mating is performed along the surfaces B1 and B2 . Each surface is set in
the local coordinates of the part and is described by the formula (6).
As an assembly procedure, let us consider a translational movement of the second part. This procedure simulates the process
of cone rings assembling under press-in technology. The part K 2 is lowered onto part K1 until the parts are in contact. For
simplicity, let us consider the parts to be absolutely rigid, so that their deformation can be left out of account. In the course of the
work, 1000 experiments were carried out on the modeling of the mating of cone surfaces and the amount of data required to
carry out the evaluation was collected. Fig. 4 demonstrates the assembled states of the mechanical system.
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Fig. 3. A mechanical system model for assembling two cone rings.
Fig. 4. The assembled states of the mechanical system for the various initial simulation conditions.
The statistical characteristics of the obtained set are specified in Table 1
Table 1. Parameters of the set of assembled states .
Parameter X Y Z
Mean value -0.0066 -0.0087 -0.0147
Root-mean-square deviation 0.2576 0.2557 0.0018
Fig. 5, 6 and 7 demonstrate histograms of the distribution of assembled states along the corresponding coordinates.
Fig. 5. Distribution density of the x coordinate of the system Fig. 6. Distribution density of the y coordinate of the system
assembled state. assembled state.
Fig. 7. Distribution density of the z coordinate of the system assembled state.
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Based on the simulation results shown in Fig. 4, there was obtained a probability density histogram for the distribution of the
assembled states of the system. The obtained chart is an approximate value of the probability density function for the assembled
state of the mechanical system. Fig. 8, 9, 10 demonstrate the approximate value of a section of this function for various z
heights.
Fig. 8. Section of an approximate probability density value at height z = - Fig. 9. Section of an approximate probability density value at height z = -
0.0189. 0.0154.
Fig. 10. Section of an approximate probability density value at height z = -0.0108.
The obtained values describe a set of assembled states of the product and can be used to solve further tasks.
4. Conclusion
Assessment of the assembly parameters of the products allows to solve a number of important production tasks of the aircraft
industry related to the efficiency and the quality of the manufacturing process of the parts. Such an estimation is difficult to
implement without processing the product assembly results. One way to get the geometric parameters of an assembled product is
numerical simulation of the process of its assembly. The results of multiple simulations can be used to evaluate some assembly
parameters.
In the framework of this research, there was proposed a model for estimating the mutual arrangement of parts of the product,
which is one of the main assembly parameters. It enables us to obtain many other geometric parameters, such as out-of-true
running, out-of-flatness, uneven clearance, etc. The proposed estimation is based on the calculation of the parameters of a set of
assembled products: mean and root-mean-square deviation values, approximate probability density. These parameters may be
useful for other production tasks. Mean and root-mean-square deviation values for the mutual arrangement of parts can be used
to solve the problem of determining rational tolerances for product parameters. The probability density function can be applied
to determine the percentage of products that meet the technology requirements. All of these parameters can be used to increase
the efficiency of the technological process by identifying the most critical factors influencing the final product quality.
The proposed model was used to assess the mutual arrangement of the parts that have cone surfaces. The next step is to test
the results of the numerical simulation in practice.
Acknowledgments
This work was supported by the Ministry of Education and Science of the Russian Federation in the framework of the
implementation of the Program ‘‘Research and development on priority directions of scientific-technological complex of Russia
for 2014– 2020”.
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