IJAMS
Design Tolerance Suitable for Manufacturing and Assembly in
Concurrent Engineering
Pavel G. Ikonomov
Mechanical & Aerospace Engineering Department
School of Engineering & Applied Science
46-128/A Engr. IV, Mailcode 159710
420 Westwood Plaza, Los Angeles, CA, 90024
Tel: 310-794-4082, Fax: 310-206-4830
pavel@seas.ucla.edu
Emiliya Dimitrova Milkova
923 Levering Ave.
Los Angeles, CA, 90024
E-mail: emiliya@hotmail.com
Suren Dwivedi
Endowed Chair Professor
Department of Mechanical Engineering
University of Louisiana at Lafayette
Lafayette, LA 70504
dwivedi@louisiana.edu
Concurrent Engineering technology reduces the length of the design-manufacturing cycle while
at the same time has a higher requirement for quality and tolerances. Most of the inadequacies
of tolerance assignment at the design stage are discovered late, at the production or assembly
stage. A computer model to be used by the designer for suitable tolerance assignment to meet
manufacturability and assemblability requirements is needed. In order to carry out tolerance
assignment we propose the introduction of a statistical tolerance model at the design stage.
Evaluation of manufacturing tolerances will inspect the validity of design tolerances and define
whether re-design or adjustments of the design are needed. Similarly evaluation of assembly
tolerances that will define assemblability of the proposed design will be done.
1. Introduction
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PAVEL G. IKONOMOV, EMILIYA DIMITROVA MILKOVA
By providing manufacturability we can help the designer to modify the design, if necessary
to balance the needs for efficient machining against the needs for quality product. The process
plan precedence influences greatly how we achieve machining tolerance requirements. Fixture
problems and change of the designated datum also influence accuracy. There are numerous papers
dealing with design, process planning and scheduling problems. For the purpose of simplicity we
will assume that the process planning problem is solved regarding the sequence optimization,
machine tool and fixture selection. An automatic way to generate and evaluate alternative
operation plans for a given design was proposed by Gupta[1].
Even though the planning process is integrated, the main problem to be solved is how to
reach the prescribed accuracy and assemblability of the designed part. If we can prove that the
designer’s decision is correct, the green light can be given for the transformation from design
stage to production stage. In most cases the design model is assumed to have flaws, so redesign
will be necessary. We have emphasized here the importance of tolerance checking and assistance
to the designer in choosing correct and possible to manufacture tolerances. Similar problems arise
with assemblability.
Real
World
Computer
Interface
Part A
Manufacturing
tolerance
simulation
Nominal
Shape
Properties
Designer
Output
Computer
Part
A
Input
Tolerance
Manufacturing and
Assembly
tolerance simulation
Manufacturing
tolerance
simulation
Part
B
Dat
a
flow
Statistical
Tolerance
Part B
Tolerance
Statistical
Tolerance
Nominal
Shape
Properties
Design Tolerance Suitable for Manufacturing and Assembly in Concurrent Engineering
102
2. Design and manufacturing systems
2.1
Concept of the concurrent engineering
Concurrent Engineering technology shortens the design-production cycle while at the
same time requirements for quality and especially tolerances are high. Shifting of the
manufacturing design, assembly design and verification to the early stage of the design of the
product will have an immense impact on the way product is developed from conceptual stages to
realization. [2]
2.2
Design system for tolerance assignment
The designer works interactively with the computer to design parts in accordance with the
worst-case tolerance. When those tolerances cannot be met statistical manufacturing tolerances
and assembly statistical tolerances are applied instead. This ensures that the designed parts can be
machined and assembled at the actual manufacturing process to follow. The proposed evaluation
system for tolerance for manufacturing and assembly works in an interactive way, see Figure.1.
For easy perception we divide the working space into Real world and Computer interface. The
Real world is the way designers see the part and its assembly on the monitor screen. The
computer interface displays data and is used for interaction between the designer and data. The
Database contains manufacturing and assembly data and is built with a specific data structure.
3. Tolerance analysis at design stage
Concurrent engineering places a great significance on the role of the designer. Small errors
in the design phase will have a significant impact on the manufacturing and assembly phases. It is
desirable to clear all possible obstacles affecting tolerance requirements. There are many
considerations regarding achievement of the described accuracy of the machine part. Here we will
specify some general requirements, solutions and the needs of tolerance evaluation. There is not
possible to predict all variation of manufacturing process, but we shall be able to help designer
with tolerance setting based on existing variation (mathematical) or statistical tolerance practices
103
PAVEL G. IKONOMOV, EMILIYA DIMITROVA MILKOVA
manufacturing and assembly stages tolerances are checked. The designer develops the part
according to function requirements. He checks the designed product for conformance with worstcase tolerance. The designed product model is submitted for manufacturing. Here
manufacturability check is performed on the product model against manufacturing tolerance. Next
the part is produced following worst-case tolerance or statistical tolerances. Statistical tolerances
are used when costs for achieving worst-case tolerance is very high or it is not possible to
produce all the parts with the prescribed tolerance. In some cases when it is not possible to
produce the part, the design model is returned for redesign. After manufacturing the product, it is
checked for manufacturing tolerance and then sent to the assembly. At assembly phase part is
checked against worst-case tolerance or statistical tolerance. Statistical tolerance is used when
costs for achieving worst-case tolerance is very high or it is not possible to assemble all the parts
with prescribed tolerance. In some cases when it is not possible to assemble the parts, the design
model is returned for redesign. Assembly checking is performed on the parts and if they satisfy
assembly tolerance then the products are certified.
Worst case
tolerance
Manufacturing
tolerance
Assembly
tolerance
A1
Functional
Requirements
Worst
case (ISO)
tolerance
Functional
Requirements
Manufacturing
tolerance
A11
Design Check
Assembly
tolerance
Product
Model
Workpiece
Workpiece
Real A12
Manufacturing Manuf. Part and
Statistical data
Check
Other Parts
Other Parts
Practical
test
Product
certified
Manuf.
Product
certified
Real
A13
Assembly
Check
Practical
test
Product
certified
Design Tolerance Suitable for Manufacturing and Assembly in Concurrent Engineering
104
This is the reason we propose application of the statistical tolerance in the design phase.
Mathematical interpretation of the statistical tolerance that is well known shall be introduced to
the designer as an alternative to worst-case tolerancing. Further an ISO statistical tolerance
Functional
restriction
Functional
restriction
Design
Req.
Worst
case (ISO)
tolerance
Design
Requirements Estimation A111
Design
Tolerance
Statistical
manuf.
data
Statistical
assembly
data
Manufacturing
restrictions
Assembly
restrictions
A11
Manufacturing
restrictions
Initial
Product
Model
Assembly
restrictions
Manuf.
Product
EstimationA112
model
Product Model with
Manufacturing
Statistical
certified
tolerance
Statistical or ISO tolerance
manufacturing data
Estimation A113
Product
Assembly
Tolerance
model
certified
6 sigma
group
compensation method
Statistical assembly data
6 sigma
group
compensation method
6 sigma
group
compensation method
Figure 3. Statistical Tolerance.
standard shall be developed.
Product
model
certified
on the real design-production process, shown in Figure 2. This will ensure proper tolerance
assignment at the design stage and production process without design flaws, as
105
PAVEL G. IKONOMOV, EMILIYA DIMITROVA MILKOVA
the consideration regarding manufacturing and assembly requirements has been done at the design
stage
From the above it follows that the designation of the adequate tolerance based on the
existing manufacturing and assembly practices will ensure manufacturability and assemblability of
the designed products. On the other hand, such a system gives the designer greater flexibility to
assign bigger tolerance, so cheaper products can be produced.
4. Variation method for accuracy calculation
Manufacturing
or assembly
requirements
Calculate
variation
tolerance
∆ro
machine
error
∆dro
surface dim.,
form error
Reshetov’s method
machine accuracy
estimation
Variation method for accuracy calculation of machine
surfaces is based on Reshetov - Portman’s method for
machine tools accuracy estimation [3]. It is based on the
mathematical model of the main error of the mechanical
system.
The errors of position of the machine units and elements are
input parameters, and the errors of dimension position and form of machined surface are output
parameters.
The model uses form shaping system code to represent different machining processes. This
code is further used in calculation of the accuracy of machined surfaces. The models of form
shaping system of cutting tool for single point, linear tool and surface tool are constructed and
also machine layout can be achieved. From those models calculation of form-shape layout output
is done. Positional error vector and nominal radius vector of the surfaces are calculated using the
following equations.
∆ r0 =
A r
0,l
l
(1)
S1
S2
S2=S1
S3=S4
Real situation
S3
Gap
S4
V
V
Interference
Design view
Figure 4. Dimensional restriction.
Design Tolerance Suitable for Manufacturing and Assembly in Concurrent Engineering

m  ∂
∂ r0 
=
d r0 ∑
 ∂q ∆q0i
i=1
0i 

106
(5)
Where ∆rb is defined as a sum of vector of dimensional and positional errors.
eb is the matrix (4x4) of the positional error of the coordinate system, which is coupled with base
surface relative to the main system [3]. The elements ∆q are used to estimate dimensional and
positional errors. Form errors can be estimated as standard deviation from base surfaces. Second
method is by constructiion of new surfaces that includes distortion of the form of the nominal
surfaces. For details refer to the origin source [3]. This method for evaluation of accuracy can be
applied to calculation of tolerances from the designer when machining specifications are known.
5. Worst-case (standard) tolerance
When solving tolerancing problems, one must choose between worst-case tolerancing and
statistical tolerancing. Worst-case tolerance is the standard tolerance that guarantee 100% inter-
works in case of top-down assembly approach, when the same CAD system is used for all
members of the enterprise. In case of concurrent engineering usage of the same type of CAD
107
PAVEL G. IKONOMOV, EMILIYA DIMITROVA MILKOVA
software is impossible, so we have to find some common solution for each participant. We
impossible assembly
Dimensional
restriction
α
t
Interference
Gap
Geometrical tolerance
restriction (positional
tolerance)
Figure 5. Geometrical tolerance
restriction.
propose virtual assembly as a solution. The aim of this virtual assembly is to support participants
in concurrent engineering, suppliers and contractors, during their design processes. The needs of
the designers we will explain with the aid of some examples.
We specify two levels of checking for assemblability regarding tolerances. [2]
• The first level is the ordinary dimension restriction, related to size and form, usually defined on
the part drawing (Figure 4).
• The second level check is for geometrical restriction derived from relations between mating
parts or from geometrical tolerance requirements (Figure 6). Simply stated this is to ensure that
geometrical restriction from one part will not interfere with physical dimension of the other
part. That means to meet geometrical tolerance requirements.
Lets analyze the problem of the Virtual assembly depicted in Figure 4. Suppose we have a
simple assembly, requiring mating surface S1 with S2 and S3 with S4. As can be seen alignment
of the two parts are correct, but surfaces S1 and S2 leave a gap and S3 and S4 interfere. The
designers specify assembly, use visualization tools and verify visually the correctness of the
assembly, without considering tolerances.
Design Tolerance Suitable for Manufacturing and Assembly in Concurrent Engineering
108
5.2.2 Geometrical tolerance restriction
Geometrical restrictions are more complicated in their matter. They are derived from
functional requirement of the parts. Ordinarily they are represented with geometrical tolerances
and aim to give unconditional geometrical assembly between parts. Their nature is static, but the
complicated three dimensional spatial relationships between related parts is an intricate problem.
Lets analyze the problem of the Virtual assembly depicted in Figure 6. Suppose we have a simple
assembly, requiring mating surfaces of the upper and lower part. It is possible to find a solution
for dimensional and geometrical restriction, but still both parts cannot be assembled. The
designers shall have a mechanism to prevent such mistakes. The only way to do that is to consider
the mating parts dimensions, geometry and tolerances. We had proposed Virtual gauge base
Virtual assembly solution that is based on geometrical tolerance. An extended Virtual assembly
model using STEP standard Express-G model schema shown in earlier publication. [2]
6. Statistical tolerances6.1 Statistical tolerances method
Statistical tolerance methods differ from worst-case tolerance in that they allow tolerances
to be increased which lead to a more economical process. Statistical tolerances are determined by
a target, tx, and maximum standard deviation, σ x,. They are
Manufacturing
or assembly
denoted by tx≤3σ x. They are more commonly denoted by tx=
requirements
Worst case
±∆x. When using the <ST> symbol, it is most common to have
Calculate
tolerance
Increased
statistical
∆x = 3σ x. However, this relationship is not formalized in the
statistical
tolerance
tolerance
standards and the six sigma approach uses
6σ
method
σ
2
i
= ∑ i =1 σ i
m−1
2
∆x = 6σ x. [4]
(6)
109
δ
PAVEL G. IKONOMOV, EMILIYA DIMITROVA MILKOVA
mean
δ
=
t∆
∆
λ 'mean( m−1)
(9)
λ’i =1/3 for equality distribution law, λ’i =1/6 for Simpson law, λ’i =1/9 for normal distribution
law. More accurate calculation gives.
k
λ=
t
:
:
i
2
i
(10)
Tolerance increasing can be calculated.
R=
1
m −1
t λ
∆
'
(11)
mean
The basic advantage of the statistical method is the possibility to increase tolerances,
comparatively to the worst-case tolerance method. That makes possible for the designer to
designate higher tolerances for design parts that have tolerances that cannot be met by the
traditional worst-case tolerance method. It also makes sure that parts can be produced with
existed machines and assembly can be carried out. When there is an additional increase of the
tolerance or special requirements for manufacturing and assembly, the group method and the
compensation method can be used.
6.2
Group (selective) method and method of compensation (regulation) link
6.2.1 Group method
Worst case
tolerance
Manufacturing
or assembly
requirements
Calculate
Increased
∆
Kmean
= ∆ Khole − ∆ Kpin = ( k − 1)δ hole +
+δ
holen
2
+ z1 − ( k − 1)δ
min
pin
δ
−
(14)
pin
2
Design Tolerance Suitable for Manufacturing and Assembly in Concurrent Engineering
Table 1. Relative increasing R of the
medium value of the tolerance
related to risk percentage λ’mean.
λ’mean
1/3
1/6
1/9
1/3
1/6
1/9
Risk %
0.27
0.27
0.27
1
1
1
R
1.41
2
2.45
1.65
2.35
2.85
110
∆
= z1 + δ hole = z1 + δ pin
min
Kmean
min
(15)
The group method allows an increase in accuracy for high-precision group products, like bearing,
engines, machine cutting tools, etc. [5]
6.2.2 Method of compensation (regulation)
Worst case
tolerance
Manufacturing
or assembly
requirements
Calculate
statistical
tolerance
Increased
statistical
tolerance
Compensation
statistical
method
111
PAVEL G. IKONOMOV, EMILIYA DIMITROVA MILKOVA
∆k changes to the mean value as a result increased tolerance of δ’ of the compensation link [5]. If
(a1 + a2 + a3 + a4 + a5 ) parts with EXA
parts with EXB
(b1 + b2 + b3 + b4 + b5 )
Desired fit
Part X A
a5
b5 b4 b3 b2
a4 a3 a2
a1
b1
Part XB
Figure 7: Parts machined with
approximately equal
capabilities.
we have symmetrical value of the tolerances δ’ the equation is simplified to:
∆ = δ2
max
k
(18)
From the above it is clear that the designer shall pay attention not only to the standard tolerance
practice based on ISO standard, but common factory and industry practice in order to develop
products that meet manufacturability and assemblability requirements.
7. Meeting of manufacturing principle for Interchangeability
7.1 Direct selection
Design
Tolerance
Suitable
for
Manufacturing
and
Assembly
in
Concurrent
Engineering
112
(a1 + a2 + a3 + a4 + a5 ) parts with EX A
Desired fit
c3
b5
b4
a5
a4
c2
c1
b3
b2
Part XA
a3
a1
Part XB
b1
The distribution of
process B
c3 parts with EX B3
c2 parts with EX B2
c1 parts with EXB1
a2
Sum distribution
Figure 9: Different capabilities. Both
processes has wider than design
tolerance.
n=
PC + PC
TX + TX
A
B
A
B
(19)
For example, if designed tolerance is one tent of proposed process capabilities, there are 10
groups. The amount of mismatch can reach 10% depends of symmetry of process capabilities.
parts with EX A
(a1 + a2 + a3 + a4 + a5 )
Desired fit
113
PAVEL G. IKONOMOV, EMILIYA DIMITROVA MILKOVA
7.2.2 Unequal Capabilities, one better than the Design Tolerance
For different capabilities of the processes by which parts are machined, and when one narrow
than the design tolerance. The process with widest capability is placed at nominal dimension of the
part, for example A, after machining of the parts of type A, they are graded into groups. At last
the process of making B-parts is positioned against A-groups to achieve desired fit. The
mismatching can be kept at negligible level.
7.2.3 Unequal Capabilities, both wider than the Design Tolerance
For different capabilities of the processes by which parts are machined, and when both are
wider than the design tolerance. The process with widest capability is placed at nominal dimension
of the part, for example A, after machining of the parts of type A, they are graded into groups. At
last the process of making B-parts is positioned against A-groups in such way that the sum
distribution obtained becomes as similar to the distribution of A-parts. In Figure 9 the numbers
c1, c2, and c3 must be chose so that the bi becomes equal as possible to ai and the expectation
sum distribution of B-parts give desired fir with distribution of A-parts. The mismatching can be
kept at small depends on the knowledge about distribution.
8. Conclusions
We proposed a model for design tolerance suitable for manufacturing and assembly in
concurrent. In order to carry on tolerance requirement we propose introduction of statistical
tolerance model to design stage. For checking of machining and assembly tolerance we proposed
usage of worst-case tolerance when 100% inter-exchangeability of the design parts is required. To
reduce tolerance requirements in order to design products according to the actual manufacturing
and assembly practice, usage of statistical tolerance, 6 sigma, group method and method of
compensation (regulation) is proposed. Necessary mathematical equations and considerations are
provided.
Using statistical tolerance will improve designers work, reduce tolerance requirements and
needs for redesign in concurrent engineering environmental that place great importance on the
Design
Tolerance
Suitable
for
Manufacturing
and
Assembly
in
Concurrent
Engineering
114
[4] W. A. Taylor, Process Tolerancing: A solution to the dilemma of worst-case
statistical tolerancing, 1997, http://www.variation.com/var-soft.html
[5] B. C. Balakshin, Theory and practice of machine building, Vol.2, pp. 65-88, 1982.
[6] Q. Bjorke, Computer-Aided Tolerancing, pp. 6.4-6.9, 1978.
[7] Latombe Jean-Claude, Robot motion planning, pp.5-43, 1991.