L Analysis of the deviations of a casting and machining process using a Model of
Manufactured Parts
Frédéric Vignat, François Villeneuve, Mojtaba Kamali Nejad
University of Grenoble, G-SCOP Laboratory
46, avenue Félix Viallet - 38031 Grenoble Cedex 1 - France
[email protected]
Abstract
In previous works, the authors have developed the Model of Manufactured Part (MMP), a method for
modeling the different geometrical deviation impacts on the part produced (error stack-up) in a multi-stage
machining process. The aim of this paper is to extend this model to other manufacturing processes like
casting and forging and to propose a stochastic approach to analyze the dimensional quality of the produced
parts. This paper firstly reminds the Model of Manufactured Part. The MMP is then extended to casting or
forging processes. A method to analyze the dimensional quality of the produced parts is then developed. The
simulation uses Monte Carlo approach to generate a population of virtually manufactured parts
representative of the real produced parts. A case study is then proposed, for a part produced in a multi-stage
casting and machining process. The result of the simulation for two quality criteria is given.
Keywords:
Dimensional quality analysis, Process planning, machining and casting, Errors stack up, Model of
Manufactured Part
1 INTRODUCTION
Today, manufacturing engineers are faced with the
problem of selecting the appropriate process plan
(manufacturing processes and production equipment) to
ensure that design specifications are satisfied.
Developing a suitable process plan for release to
production is complicated and time-consuming. Currently,
trial runs or very simple simulation models (1D tolerance
charts for example [1]) are used to check the quality
criterion. The trial runs are very costly and, on the other
hand, the accuracy of simulation fails to meet today’s
requirements. These problems can be overstepped by
developing accurate models and methodologies for
simulating the manufacturing process and predicting
geometrical variations in the parts produced. More
accurate models will make it possible to evaluate the
process plan, determine the tolerance values in terms of
manufacturing capabilities during the design phase, and
define the manufacturing tolerances to be checked for
each setup. In the literature available on this subject, the
evaluation of a process plan in terms of tolerances is
called the tolerance analysis or dimensional quality
analysis. In this paper the model of manufactured part
(MMP) is proposed for simulating the manufacturing
process and statistical approaches are used for the aim
of dimensional quality analysis.
In this paper we shall focus on analysis relating to error
propagation in a multi-stage manufacturing process
including casting (or forging) and machining processes.
Huang et al [2] propose a state space model to describe
part error propagation in successive machining
operations. Surface deviation is expressed in terms of
deviation from nominal orientation, location and
dimensions. The error sources in machining operations
are classified as fixture errors, datum errors (errors of
positioning surfaces of the part from previous set-ups)
and machine tool errors. A part’s deviation is expressed
in terms of the deviation of its surfaces and is stored in a
state vector x(k ) . This vector is then modified by moving
from operation k to k  1 . Zhou et al [3] uses the same
state model but the surface deviation compared with the
nominal state is expressed using a differential motion
vector. However, these two models require specific fixture
setups (e.g., an orthogonal 3-2-1 fixture layout). More
recently, Loose et al [4] used the same state space
model with a differential motion vector but including
general fixture layouts. Although a general fixture layout
is considered, the error calculation of a fixture is based
on its locator deviations (a locator is a punctual
connection). Hence, positioning cases with Plane/Plane
contact or Cylinder/Cylinder floating contact are not
envisaged.
Huang et al [5] propose a simulation-based tolerance
stack-up analysis. Manufacturing errors are classified as
follows: work holding errors (i.e. fixture errors, datum
errors and raw part errors), machine tool errors and
cutting tool errors. A surface is modeled using uniformly
distributed sample points (point cloud), which is a basic
technique applied in CMM type inspections. By putting
the part through different machining setups, the
coordinates of these points in the local part coordinate
system are changed due to manufacturing errors. The
Monte Carlo method is used to perform the simulation.
The different possible errors are considered in this
simulation but Part/Fixture interaction is not studied and it
is assumed that part/fixture contact is perfect.
From the molding process quality evaluation side, (huang
et Al) [5][6] propose an advanced searching method for
setting the robust process parameters for injection
molding based on the principal component analysis
(PCA) and a regression model-based searching method.
In [5] the PCA is utilized to construct a composite quality
indicator to represent the quality loss function of multiquality characteristics while in [6] the quality criterion is
the volumetric shrinkage. The design of experiment and
ANOVA methods are then used to choose the major
parameters, which affect parts quality and are called as
adjustment factors. Secondly, a two-level statistically
designed experiment with the least squared error method
was used to generate a regression model between part
quality and adjustment factors. Based on this
mathematical model, the steepest decent method is used
to search for the optimal process parameters. In [6] only
volume shrinkage is studied while in [5] the molding
process quality composite indicator is function of three-
quality characteristics: maximal volumetric shrinkage rate,
maximal warpage, and maximal shear stress. For both
studies process parameters to optimize are cooling time,
plastic temp, mould temperature, filling speed, holding
pressure, holding time, filling pressure, screw stroke.
[7] presents a neural network-based quality prediction
system for a plastic injection molding process. A selforganizing map plus a back-propagation neural network
(SOM-BPNN) model is proposed for creating a dynamic
quality predictor. Three SOM-based dynamic extraction
parameters with six manufacturing process parameters
and one level of product quality is dedicated to training
and testing the proposed system. In addition, Taguchi’s
parameter design method is also applied to enhance the
neural network performance. The three SOM-based input
parameters are the injection stroke curve, injection
velocity curve, and pressure curve, while the six BPNN
input parameters are injection time, VP switch position,
packing pressure, injection velocity, packing time and
injection stroke; one BPNN output variable is weight
(quality characteristic).
The different reviewed papers do not discuss about
geometric error models for multistage manufacturing
process where stages can use casting, forging and/or
machining. In this paper a method for modeling the
different geometrical deviation impacts on the part
produced (error stack-up) in a multi-stage machining
process is proposed and is applied to various
technologies (casting and machining). Then, a stochastic
approach to analyze the dimensional quality of the
produced parts is presented.
This paper firstly reminds the Model of Manufactured Part
(the MMP); a method for modeling the different
geometrical deviation impacts on the part produced (error
stack-up) in a multi-stage machining process. The MMP
is then extended to casting or forging processes.
A method to analyze the dimensional quality of the
produced parts is then developed. The simulation is
based on a Monte Carlo approach.
A case study is then proposed, for a part produced in a
multi-stage casting and machining process. The result of
the simulation for two quality criteria is given.
from machine geometry and control to cutting
deformations.
At the end of the modeling process, a virtual
manufactured part (MMP) is created. This MMP stores
data about the deviations generated (combination of
parameters and range of variation) during the full
machining process. See figure 1.
2 MODEL OF MANUFACTURED PARTS (MMP)
We proposed in previous papers [9,10,11] a method for
modeling successive machining processes that takes into
account the geometrical and dimensional deviations
produced with each machining setup and the influence of
these deviations on further setups. The method can be
extended to other manufacturing technologies, like
casting or forging. In the present paper, an extension of
the MMP simulation method to the case of a process
combining casting and machining is developed.
rx 0 


TPlane  R LO ( O, X , Y , Z )  ry 0 
 0 tz 

( O, X, Y , Z )
2.1 MMP and machining process
In the MMP, the errors generated by a manufacturing
process are considered to be the result of two
independent phenomena: positioning and machining.
These deviations are accumulated over the successive
setups (See figure 1). The result is expressed in terms of
deviation of the actual surfaces compared with those of
the nominal part. In order to capture the error stacks, an
intermediate virtual part (MWP) is put through the
different setups. In setup k, the machined surface
deviation is the combination of positioning errors and
machining errors. Positioning errors are caused by
surface deviations from a previous setup (datum errors)
and fixture surface deviations in setup k. Machining errors
are machined surface deviations compared with the
nominal position in the machine tool in setup k. These
errors stem from multiple and various sources ranging
Setup 0
Setup 1
MWP(1)
Setup 2
MWP(2)
MMP
Nominal
MMP
MWP(3)
Figure 1 : Tolerance stack-up model
The geometrical model used to describe the deviations of
the surfaces of the MMP is based on the Small
Displacement Torsor (SDT) concept proposed by Bourdet
et al [12] for assembly simulation.
A SDT is defined by two vectors representing the values
of three small rotations rx, ry,
rz denoted by R and
three small translations tx, ty, tz denoted by L
concerning a surface. The SDT concept has been
extended to manufacturing process simulation by
Villeneuve et al [13]. It is based on an ideal part made up
of ideal (perfect form) surfaces. The surfaces of the ideal
part are deviated and these deviations are measured in
relation to their nominal position. For each surface, the
deviations are expressed by a SDT whose structure
depends on the surface type. In Fig. 2, SDT which
expresses the relative position of the associated plane
and the nominal one in a local coordinate system (with
origin O) is mentioned in Equation (1)
(1)
NB: Vectors are written in bold letters
This SDT can be expressed at any point Vi of the plane
by Equation (2).
TPlane  R

L Vi (Vi, X,Y,Z)  R

L O  R  OVi (Vi, X,Y,Z)
Associated surface
Real surface
Z
tz
ry
Y
O
rx
Vi
X
Nominal surface
Figure 2: Deviation of a plane
(2)
The MMP does not only represent a model of one
manufactured part containing a description of the process
in terms of geometrical deviations and accumulated
defects. In fact, because it indicates the variation range of
the generated defects it represents the series of parts
produced. The model describes the defects, classifies
them and indicates their variation range.
The SDT describe the MMP surface deviations, i.e. the
MMP parameters, which can be classified according to
four categories.
In this equation, TD2,Pi represents the casting surface
deviation, TD1,D2 the positioning error between die 1 and 2
and TP,D1 the positioning error between nominal part and
die 1, which can be set to zero as far as this value has no
importance for the rest of the process.
The deviations parameters are classified according to
four categories:

Machining deviations -DM- ( rxi , ry i , tzi …)

Fixture surface deviations -DH-( rxiSj , txiSj …)
Link parameters -LHP-( lrx iSj , ltxiSj …)



Casting deviations -DM- ( rxi , ry i , tzi …)

Die surface deviations -DS- ( rxiDj , txiDj …)

Link parameters -LDD- ( lrx iDj , ltxiDj …)

Actual surface deviations relative to the nominal
part ( rx P , Pi , ry P , Pi …)
Actual surface deviations relative to the nominal part
( rx P , Pi , ry P , Pi …)
The machining deviation parameters (DM) are limited by
constraints (CM) representing machine and tools
capabilities. The DH parameters are limited by
constraints (CH) representing the fixture quality.
Due to the type of link (floating or slipping), the link
parameter values (LHP) are determined by a specific
algorithm (CHP) including constraints and, in certain
cases, a positioning function.
For each MWP surface made, the positioning and
machining deviations are added. The deviation relative to
the nominal part is determined and expressed as TP,Pi for
surface i of the MWP. This torsor will be kept in the MWP
data for possible further use in another setup for an
assembling procedure or for the purposes of tolerance
analysis.
For the example surface deviations of a plane relative to
the nominal part is expressed as Equation (3).
 rx P ,P 3
0 


TP ,P 3  ry P ,P 3
0 
 0
tzP ,P 3  LCS 3
Where :
tzP,P3 = 7.07 lrx1S2 + 0.7 ( -ltz1S2+ ltz2S2) -7.07
rx1+7.07rx1S2 + 0.7 (-tz1- tz1S2+ tz2 + tz2S2) + tz3
(3)
2.2 MMP and casting process
For a casting process, the errors generated are
considered to be the sum of two independent
phenomena: die relative positioning and casting surface
generation. The die relative positioning is treated as the
MWP/Fixture is. It is the result of the unification of the
parallel elementary link between the dies. For each of
these elementary links, the die relative positioning
deviation will be the summation of the surface 1 of die 1
SDT, the link SDT for the die 1/die 2 assembly and the
surface 1 of die 2 SDT as indicated in Equation (4) (see
Figure 3).
TD1, D 2  TD1, Ds1  TDs1D1, Ds1D 2  TD 2, Ds1
(4)
The deviations of the casting surfaces (temperature,
distortion …) are globalised in a single torsor TDj,Pi for
surface i generated by die j.
For each part surface made by a casting process, the
deviations are accumulated. The deviation relative to the
nominal part is determined and expressed as TP,Pi. For
example, for surface i generated by die 2, see Equation
(5):
TP , Pi  TP , D1  TD1, D 2  TD 2, Pi
(5)
Figure 3: Casting process model. Dashed lines represent
the nominal surfaces, plain lines the real surfaces.
The casting deviation parameters (DM) are limited by
constraints
(CM)
representing
casting
process
capabilities. The DS parameters are limited by constraints
(CS) representing the dies quality.
Due to the type of link (floating or slipping), the link
parameter values (LDD) are determined by a specific
algorithm (CDD) including constraints and, in certain
cases, a positioning function.
3 DIMENSIONAL QUALITY ANALYSIS
The compliance to the necessary quality requirements in
a manufacturing process can be analyzed by treating the
variation of the relative position of the manufactured parts
produced surfaces.
In this section we present the different quality criteria to
be considered, which depends on the analyzed surfaces.
The way to express the criteria as equations is exposed.
Then, a stochastic simulation, based on a Monte Carlo
approach, is applied.
3.1 Quality criteria and equations
Using the MMP, TPi,Pj expresses the variations of surface
Pj relative to surface Pi belonging to the part P. It can be
calculated from the TP,Pi representing the manufacturing
deviations of each surface of the part relative to its
nominal position (see equation 6).
TPi,Pj  TP ,Pi  TP ,Pj
(6)
Depending on the quality requirement, the TPi,Pj can be
analyzed in term of small rotations or small translations at
significant points Vv (where v is the Vertex index) of the
analyzed surface Pj. These significant points are the
vertices of the boundary of the toleranced surface.
For example, when Pj is a plane, the criterion should be
to evaluate the deviations of some points Vv of the
plane’s boundary along the normal of the reference
plane. To calculate this deviation at point Vv, first TPi,Pj is
expressed at point Vv and then projected along the
verification direction
n̂ as indicated in Equation (7).
TPi, Pj  R LVv Vv, GCS
(7)
deviation  nˆ  LVv
When Pj is a cylinder, the criterion consists to evaluate
the deviations of the end points Vv of the Pj axis into a
plane perpendicular to the reference axis n̂ . To calculate
these deviations, TPi,Pj is expressed at point Vv and then
Using Measurement Results
In this strategy, a sufficient number of parts have to be
produced and measured. The manufacturing conditions
(temperature, machine tool, etc.) should be the same as
the simulation condition. The surface deviation ranges
are obtained from the measurement data. Based on the
measurement results obtained, the co-relation between
the parameters is then sought. This strategy is very close
to reality, but it is complicated to express the co-relation
between the parameters.
Using Measurement Results (independent parameters)
As explained for the previous strategy, the parameter
deviation ranges are obtained by measurements. As
opposed to the previous strategy, the parameters here
are assumed to be independent variables that varies in
the interval defined by the measured range [-3σ +3σ].
With this strategy, the constraints obtained will be as in
Equation (9) in the case of a plane. The deviation
variation range obtained is close to reality but considering
independent parameters implies that these can
simultaneously attain their extreme limits. This is highly
improbable in reality.
projected onto the plane perpendicular to n̂ as indicated
in Equation (8) (See Figure 4).
rx min  rx  rx max
TPi, Pj  R LVv Vv, GCS
tz min  tz  tz max
(8)
deviation  nˆ  LVv  nˆ
n̂
Upper Circle
of cylinder Pj
L Vv
deviation
Deviated axis
Lower Circle
of cylinder Pj
Reference axis
Figure 4: TPi,Pj deviations of a cylindrical surface
3.2 Monte Carlo simulation
The process simulation is based on a Monte Carlo
approach. The various parameters occurring in the
analysis equation are randomly generated. The objective
is to generate uniform distributions for the significant
characteristics of the considered surfaces inside a
domain limited by constraints.
For example, the machining or casting deviation
parameters (DM) are limited by constraints (CM)
stemming from the machine tool or the casting process
capabilities. These constraints limit either one or a set of
parameters.
There are different strategies for defining these
constraints as described here after. Whatever the
strategy, the multiple root causes of variation
(temperature, surface interaction,… ) are globalised in
the machining deviation parameters.
(9)
ry min  ry  ry max
Considering a variation Zone with dependent parameters
In this strategy, a variation zone is used to represent the
deviation range of a surface or its feature (axis, centre,
etc.). Desrochers et al. propose a 3-D representation of
the variation zones [14]. The SDT parameter variations
must be bound by the limits of the 3-D variation zones
they represent. These boundary areas are hyper-surfaces
of the space spanned by the six small displacement
parameters (rx, ry, rz, tx, ty, tz). Illustrated Figure 5 is the
case of a planar variation zone showing how such
constraints can be handled. In Figure 5, the variation
zone is defined as the volume ranging between two
parallel planes with a distance e between them. Any
candidate plane (shaded in Figure 5) must therefore lie
inside this zone. If four boundary points (A, B, C and D)
are used with reference point O at the barycenter, it is
possible to express their projection on the limiting planes,
yielding to the linear set of inequalities in Equation (10),
where a, b and e are known.
 e
e
 2 
2
  e   a b 1
e
   a  b 1  rx   


  ry   2 
 2 
  e   a  b 1    e 
  tz 
 2   a
b 1    2 
 e 
e
 
 
 2 
2
D
(10)
C
Z
tz
rx
X
A
ry
Y
2b
B
2a
Figure 5: planar variation zone
e
Depending on the type of constraints and dependence,
there are different strategies for random generation of
parameters.
First case, the parameters are independent. Each
parameter is then generated within its range using a
classical generator like extended cellular automation
generator.
Second case, the parameters are dependent and it is
possible to make a non-correlated variable substitution.
The first case generation is then applied to the
substitution variables and the reverse substitution is
applied.
For example, in a cylindrical variation zone, a variable
substitution is made so that 4 variables are generated
without correlation according to Equation (11). The 4
defect parameters describing the cylinder “real” position
are then calculated using (12).
ru and rl between 0 and rvariation zone
with a probality density of f r  r   2 r
rvariation zone2
u and l between 0 and 2
(11)
with a probality density of f      1
2
rx   ru sin( u )  rl sin( l )
ry  ru cos( u )  rl cos( l )
tx   ru cos( u )  rl cos( l )  2
ty   ru sin( u )  rl sin( l )  2
(12)
Third case, the parameters are dependent and it is not
possible to make a non-correlated variable substitution. A
new larger variation zone is created that circumscribe the
previous one and allow using the first or second case.
Inside this new zone, a random generation is made using
the first or second case. Last, only the parameters
verifying the initial constraints are selected.
For example, in a planar variation zone, each nominal
vertex of the convex boundary of the plane is randomly
generated along the normal of the nominal plane. The
values are randomly generated between –e/2 and e/2
with a uniform density. A plane is then positioned from
the generated vertices using a mean square root criteria.
The 3 parameters rx, ry and tz are then calculated from
the plane characteristics. After the generation of the set
of parameters, a verification of the constraints is
performed. If one of the constraints is not verified, the set
is rejected. The procedure is repeated until the required
number of valid set is reached.
4 CASE STUDY
The method proposed in this paper to validate a
candidate process is applied to a part realized by
permanent mold casting followed by 2 machining set-up.
In the first manufacturing stage, the workpiece is molded
by die casting (see figure 6). The permanent mold is
made up of two dies in green and blue positioned by a
planar joint and two centering pins with clearance. The
mold also comprises a core represented in red and
positioned in the two dies by planar and cylindrical joints
with clearance. The connection between the two dies of
the mold is over constrained (degree 1) and the
connection between these two dies and the core is highly
over constrained (degree 5). Thus a clearance between
the surfaces participating in the over constraint has to be
designed. The management of the non penetration
conditions determining the possible relative position of
the 3 elements of the mold is made by the non
penetration constraints on the LDD parameters as
explained §2.2.
Figure 6: mold assembly: 2 dies (green and blue) and a
core (red)
The second stage (set-up 1) consists in a milling
operation. During this stage a plane and two cylinders (in
red on figure 7) are machined. The casting is positioned
using the cones realized on the 4 bosses at the casting
stage (figure 7). To prevent the fixture from over
constraint, only 3 cones are used making up a classical
3-2-1 positioning system (respectively on bosses 4, 6 and
9). This fixture realizes a clearance free positioning
fixture.
Figure 7: set-up 1
The third stage (set-up 2) is realized on a 4 axis
machining center. During this stage 4 planes and 4
associated cylinders (in green) are machined. The
workpiece is positioned by a planar joint without
clearance on surface plane 13 and two centering with
clearance on cylinders 14 and 15, these surfaces
machined in set-up1 (see figure 8).
Figure 8: set-up 2
0.2
Some of the criteria chosen to assess the quality of the
manufacturing process are the localization of the axis of
the machined cylinders Ø12 relative to respectively the
boss 4, 6, 9 and 11 (set-up 2) and the localization of the
axis of the machined cylinder 14 (Ø80) (set-up 1) relative
to the raw surface (Ø68) (see figure 9). These criteria will
especially assess the alignment of the machined surfaces
relative to the raw ones. These criteria are calculated
using the method explained §3.1 and for each criterion
the result is, for each cylinder, two vectors projected onto
a plan perpendicular to the nominal cylinder axis. These
projected vectors represent the position variation of the
center of the upper and lower circle of the finished
surface relative to the raw reference.
0.0
0.2
1.0
0.5
0.0
0.2
0.0
0.2
Figure 11: histogram for the relative position of the center
of the circle Ø12 relative to boss 9
0.4
0.2
0.0
0.2
0.4
0.15
Figure 9: verification criteria
To ensure a wide cover of the variation domain, 106 parts
are randomly generated as explained §3.2 and the
criteria are calculated for each of these parts. The results,
coordinates of the vectors, can be represented on a 3D
histogram (see figure 10, 11 and 12). Considering the
distribution of the points, it is possible to calculate the
ratio of parts complying with the requirements and then to
make decision about the candidate processes. From the
3 figures, it is possible to conclude that the process is
suitable (all of the deviation remain inside the 0.5
tolerance on the criteria).
0.10
0.05
0.00
0.4
0.2
0.0
0.2
0.4
Figure 12: histogram for the relative position of the center
of the upper circle of cylinder 14 relative to the raw
0.2
However, some differences appear between the 3
figures. Boss 4 is positioning three translations of the 3-21 positioning fixture of set-up1 while boss 9 only one
vertical translation. It is thus normal that the deviation is
higher in the X-Y plan for boss 9 compared to boss 4.
For cylinder 14, the raw surface is made by the core
which is positioned with clearance in the two dies. It is
thus normal that more deviation occurs. Over the simple
evaluation of the process, this analysis can give to the
process planner some improvement directions.
0.0
0.2
1.5
1.0
0.5
0.0
0.2
0.0
0.2
Figure 10: histogram for the relative position of the center
of the circle Ø12 relative to boss 4
5 CONCLUSION AND PERSPECTIVES
In the present paper, a method to assess the dimensional
quality of a multistage process is proposed. It is applied
to an example of part realised by permanent mold casting
and 2 machining stages. The method is based on the
Model of Manufactured Part (MMP). The MMP allows
data collection of deviations due to the process. It saves
the combination of the defect parameters and the
constraints on these parameters. The method is also
based on a stochastic resolution where a great number of
virtual parts (106 in the example) are manufactured and
measured regarding the quality requirements.
The present method has to be extended to others
manufacturing processes like sand casting, plastic
injection… to allow a full coverage of the manufacturing
means. A future work has been started to extend the
capability of the model to product usage in order to study
the deviations consequences on the product behaviour
without passing through tolerances. Some works have
also been already made about measurement of the
manufacturing deviations but it has mainly focused on the
range of deviation resulting from several machining
processes. The present method needs also information
on the distribution of the defects in the variation zone in
order to realise more precise stochastic simulations.
[8]
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