Synthesis of Tolerances for Spacecraft Mechanism
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The document discusses the synthesis of tolerances for spacecraft mechanisms, emphasizing the significance of proper tolerance design on quality and cost. It analyzes tolerances using worst case and root sum square methods, applying the direct linearization method for a door opening mechanism as a case study. The results indicate that statistical tolerancing offers wider tolerances with lower production costs while meeting functional requirements.
![IJSRD - International Journal for Scientific Research & Development| Vol. 3, Issue 10, 2015 | ISSN (online): 2321-0613
All rights reserved by www.ijsrd.com 385
Synthesis of Tolerances for Spacecraft Mechanism
Sulficker Ali Ismail1 N. Sujithkumar2 J. Dileeplal3 Y. S. Shankar Narayan4
1
P. G. Scholar 2,4
Scientist/Engineer 3
Associate Professor
1,3
Department of Mechanical Engineering
1,3
Mar Athanasius College of Engineering, Kothamangalam, Kerala, India 2,4
Reliability Assurance
Mechanical Division, Systems Reliability Group, ISRO Satellite Centre, Bangalore, Karnataka, India
Abstract— Tolerance have a great impact on the quality and
cost of the mechanism. Proper tolerance design enables
complex mechanical assemblies consisting of numerous parts
to assemble and work together in a proper manner so that they
fulfill their design objectives. In this paper the tolerance of a
spacecraft mechanism is analyzed using worst case and root
sum square method. The tolerance synthesis of the spacecraft
mechanism is done by direct linearization method.
Key words: Tolerance Analysis, root sum square method,
Assembly Simulation by Software R
I. INTRODUCTION
Tolerance is the specified amount a feature is to vary from its
nominal. Tight tolerances assure product performance
whereas looser tolerances reduce cost. A balance have to be
made between the two. Tolerance analysis shows the effect
of component tolerances on the output variability of a
mechanism. Tolerance stackups determine the performance
and functionality of the assembly. Proper tolerance synthesis
enables complex mechanisms with numerous parts to
assemble and work together to satisfy its design objectives.
The main objective of tolerance technology is to determine
tolerances of individual parts based on the the functional
requirements developed during product design [1].
Tolerance analysis is performed when natural
process tolerances of the components are known and there is
a need to calculate design tolerance of the assembly. It
determine if the given tolerances can meet the functionality
of the product. It also provides guidance to where the
tolerances must be tighter and where it can be relaxed.
II. TOLERANCE ANALYSIS
In order to ensure the functionality of mechanical assemblies,
designers attempt to control the variation of critical assembly
dimensions. Acceptable limits on the variation are defined
according to the performance requirements. Tolerance
analysis can be conducted in two main methods: worst-case
and statistical methods [2].
The worst case tolerance analysis calculates
assembly tolerance by summing component tolerances. It
satisfy assembly tolerance with 100% probability but worst-
case tolerancing can lead to excessively tight part tolerances
and hence high production costs. As there is no part falling
outside the tolerance range the assembly tolerance calculated
will be high.
For one-dimensional assemblies,
𝑑𝑈 = ∑ 𝑇𝑖 ≤ 𝑇𝐴𝑆𝑀 (1.1)
For two or three-dimensional assemblies,
𝑑𝑈 = ∑ (
𝜕𝑓
𝜕𝑥 𝑖
𝑇𝑖) ≤ 𝑇𝐴𝑆𝑀 (1.2)
where 𝑥𝑖 are the nominal component tolerances, 𝑇𝑖 are the
component tolerances, 𝑑𝑈 is the predicted assembly
variation, 𝑇𝐴𝑆𝑀 is the specified limit for 𝑑𝑈, and 𝑓(𝑥𝑖) is the
assembly function describing the resulting dimension of the
assembly. The partial derivative
𝜕𝑓
𝜕𝑥 𝑖
represent the sensitivity
of the assembly tolerance to variations in individual
component tolerances.
The statistical tolerancing method calculates
assembly tolerance by taking the root sum square of the
component tolerances. Tolerances are commonly assumed to
a 3𝜎 process level. This method assumes the component
process tolerance follow normal distribution.
For one-dimensional assemblies,
dU = ∑[Ti
2
]
1
2⁄
≤ TASM (1.3)
For two or three-dimensional assemblies,
𝑑U = [∑ (
∂f
∂xi
)
2
Ti
2
]
1
2⁄
≤ TASM (1.4)
Statistical tolerancing approach can produce much
higher profit than the worst case approach [3]. Tolerancing
decisions can affect the quality and cost of the mechanism.
To evaluate the impact of tolerance on mechanism quality,
designers need to simulate the influences of tolerances with
respect to the functional requirements [4]. Marler
implemented a 2D method for linearized vector-based
tolerance analysis, the direct linearized method (DLM). The
DLM provided a relatively simple method for linearizing the
vector loop equations of a tolerance model using a first order
Taylor expansion, and then extracting the sensitivities and
solving for dependent assembly variables using matrix
algebra [5]. Statistical tolerancing allows more realistic
tolerance calculations, leading to larger tolerances with lower
production costs without loss in product quality.
III. METHODOLOGY
Tolerance analysis is the process of taking known tolerances
and analyzing the combination of these tolerances at an
assembly level. Direct linearization method (DLM), is used
for tolerance analysis of the mechanical assembly. The model
is constructed of common engineering elements: vector
chains, kinematic joints, assembly datums, dimensional
tolerances, and assembly tolerance limits [6].
The DLM applies matrix algebra and root sum
squares error analysis to vector loop based models to estimate
tolerance stackup in assemblies. Dimensional variations
account for small changes in size due to manufacturing
processes. Kinematic variations describe the propagation of
variation through an assembly by small adjustments between
mating parts. The key features of this model are:
1) A set of rules is provided to assure a valid set of vector
loops. The loops include only those controlled
dimensions which contribute to assembly variation. All
dimensions are datum referenced.
2) A set of modeling elements will be introduced to assist
in identifying the adjustable dimensions within the](https://image.slidesharecdn.com/ijsrdv3i100292-160107095952/85/Synthesis-of-Tolerances-for-Spacecraft-Mechanism-1-320.jpg)
![Synthesis of Tolerances for Spacecraft Mechanism
(IJSRD/Vol. 3/Issue 10/2015/076)
All rights reserved by www.ijsrd.com 386
assembly that change to accommodate dimensional
variations.
3) In addition to describing variation in assembly gaps, a
comprehensive set of assembly tolerance requirements
is introduced, which are useful to designers as
performance requirements.
4) Algebraic manipulation to derive an explicit expression
for the assembly feature is eliminated. The new system
operates equally well on implicit assembly equations.
The loop equations are solved the same way every time.
It is well suited for computer automation.
5) Differentiation of a complicated assembly expression is
replaced by a single matrix operation, which determines
all necessary tolerance sensitivities simultaneously.
IV. CASE STUDY
The DLM is applied on the door opening mechanism (Fig. 1)
for tolerance analysis. The components of hinge assembly are
out-board bracket, hinge shaft, spherical bearing, in-board
bracket, stopper pin, and latch spring. Dimensions a to k are
the dimensions of the component features which contribute to
assembly variation. These are shown in the Fig. 2. Tolerances
are estimates of the manufacturing process variations.
Fig. 1: Components of door opening mechanism
Fig. 2: Dimensions of door opening mechanism
The general tolerances for dimensions are taken as
per ISO 2768 [7] and is tabulated in Table I.
Tolerances to functional requirements of the pay
load door opening mechanism was calculated using direct
linearization method [8]. The steps in DLM are broadly
classified into two:
1) Steps in creating an assembly tolerance model
Create an assembly graph
Locate the datum reference framefor each part
Create vector loops
Define performance requirements
2) Steps in analyzing an assembly tolerance model
Generate assembly equations from vector loops
Calculate derivatives and form matrix equations
Solve for assembly tolerance sensitivities
Dimension Nominal Tolerance
a 4mm ±0.1mm
b 6mm ±0.02mm
c 8mm ±0.2mm
d 10.5mm ±0.2mm
e 12mm ±0.2mm
f 5mm ±0.1mm
g 40mm ±0.3mm
h 10mm ±0.2mm
i 2mm ±0.1mm
j 12.5mm ±0.2mm
k 43mm ±0.3mm
Table I: Dimensions And Tolerances Of Door Opening
Mechanism
After the sensitivity matrix is obtained, worst case
and RSS expression for tolerance accumulation is formed.
Table II gives the assembly variables obtained by both worst
case (WC) and root sum square (RSS) methods.
Assembly
Variable
Nominal WC RSS
u1 6.2mm
±0.14075m
m
±0.10552m
m
u2 14mm
±0.14075m
m
±0.10552m
m
u3
14.648m
m
±0.68149m
m
±0.3529m
m
u4 1mm ±1.2mm
±0.52915m
m
∅1 11ᴼ ±3.057ᴼ ±2.0821ᴼ
Table II: Assembly Variables
The assembly variable u4 is restricted to 0.2 mm by
the designers for achieving the performance requirement of
the assembly. The current tolerance does not satisfy the above
mentioned criteria. Therefore, the tolerances of the
dimensions need to be changed.](https://image.slidesharecdn.com/ijsrdv3i100292-160107095952/85/Synthesis-of-Tolerances-for-Spacecraft-Mechanism-2-320.jpg)
![Synthesis of Tolerances for Spacecraft Mechanism
(IJSRD/Vol. 3/Issue 10/2015/076)
All rights reserved by www.ijsrd.com 387
The tolerance allocation is determined by using
percent contribution charts. The tolerance with largest
contribution is changed and is again checked for assembly
variations. The results are tabulated in Table III.
Dimension
± Tolerance
% Relaxation
WC RSS
a 0.1 0.1 0.00%
b 0.02 0.02 0.00%
c 0.2 0.2 0.00%
d 0.03 0.1 233.33%
e 0.03 0.05 66.67%
f 0.04 0.1 150.00%
g 0.03 0.05 66.67%
h 0.2 0.2 0.00%
i 0.04 0.1 150.00%
j 0.2 0.2 0.00%
k 0.03 0.05 66.67%
Table III: Modified Dimensional Tolerance
The modified tolerance allocation shows that the
statistical tolerancing gives wider tolerance than the worst
case approach.
V. ASSEMBLY SIMULATION BY SOFTWARE R
The purpose of this simulation is to model manufacturing
variations and predict their effects on the assembly. A sample
of 100000 of each part is selected and assembled together.
The assembly variation of all the 100000 assembly is
calculated and the number of rejects is calculated. The result
obtained is plotted as a histogram and is shown in Fig. 3.
Fig. 3: Histogram of assembly variation of u4
The mean (µ) and standard deviation (σ) of the
assembly variation is found to be 0 and 0.05677851 mm
respectively. The probability of assemblies having variation
between -0.2 mm and 0.2 mm is 0.9995662. Which means out
of the 100000 assemblies the number of defects will be 43.38.
VI. CONCLUSION
In this paper, the direct linearization method is used for
tolerance analysis of the payload door opening mechanism by
both worst case and statistical methods. Statistical method
provides relaxed tolerance for components of the assembly
compared to that by worst case method. Component
tolerances are analysed to satisfy the functional requirements,
which will make the assembly compatible with the design.
The number of rejects of assemblies by statistical tolerancing
is calculated by simulation using software R.
The statistical tolerencing provides wider tolerances
than worst case tolerancing while satisfying the assembly
requirements. Wider tolerances are comparatively easy to
process and relative cost for manufacturing is less.
VII. SCOPE FOR FUTURE RESEARCH
In addition to the dimensional tolerances, the geometric
tolerances can be added to the variation analysis. Geometric
tolerances include important variation sources caused by
irregularities in form, shape and size. This may make
significant contributions in a tolerance analysis.
Most materials change length as they change
temperature. As a result of this change, the dimensions and
tolerances of a product become at variance with the design
values. Hence, thermal effects can be included in the analysis
when designing a product like spacecraft component that will
undergo temperature cycling.
REFERENCES
[1] B. R. Fischer, Mechanical Tolerance Stackup and
Analysis, CRC Press, 2011.
[2] K. W. Chase and A. R. Parkinson, “A survey of research
in the application of tolerance analysis to the design of
mechanical assemblies”, Research in Engineering
Design, vol. 3 (1), pp. 23-37, 1991.
[3] A. Shan, R. N. Roth and R. J. Wilson, “A new approach
to statistical geometrical tolerance analysis”,
International Journal of Advanced Manufacturing
Technology, vol. 15 (3), pp. 222-230, 1999.
[4] A. J. Qureshi, J. Y. Dantan, V. Sabri, P. Beaucaire, and
N. Gayton, “A statistical tolerance analysis approach for
over-constrained mechanism based on optimization and
monte carlo simulation”, Computer-Aided Design, vol.
44 (2), pp. 132-142, 2012.
[5] J. D. Marler, “Nonlinear tolerance analysis using the
direct linearization method”, ADCATS Report,
Bringham Young University, 1988.
[6] V. Srinivasan, Dimensioning and Tolerancing
Handbook, IBM Research and Columbia University,
New York, 1999.
[7] American National Standards Institute, Y14. 5M,
Dimensioning and Tolerancing, American Society of
Mechanical Engineers, New York, 1994.
[8] P. Faerber, “Tolerance analysis of assemblies using
kinematically derived sensitivities,” M .S. Thesis,
Brigham Young University, Provo, Utah, 1999.](https://image.slidesharecdn.com/ijsrdv3i100292-160107095952/85/Synthesis-of-Tolerances-for-Spacecraft-Mechanism-3-320.jpg)
Synthesis of Tolerances for Spacecraft Mechanism
- 1. IJSRD - International Journal for Scientific Research & Development| Vol. 3, Issue 10, 2015 | ISSN (online): 2321-0613 All rights reserved by www.ijsrd.com 385 Synthesis of Tolerances for Spacecraft Mechanism Sulficker Ali Ismail1 N. Sujithkumar2 J. Dileeplal3 Y. S. Shankar Narayan4 1 P. G. Scholar 2,4 Scientist/Engineer 3 Associate Professor 1,3 Department of Mechanical Engineering 1,3 Mar Athanasius College of Engineering, Kothamangalam, Kerala, India 2,4 Reliability Assurance Mechanical Division, Systems Reliability Group, ISRO Satellite Centre, Bangalore, Karnataka, India Abstract— Tolerance have a great impact on the quality and cost of the mechanism. Proper tolerance design enables complex mechanical assemblies consisting of numerous parts to assemble and work together in a proper manner so that they fulfill their design objectives. In this paper the tolerance of a spacecraft mechanism is analyzed using worst case and root sum square method. The tolerance synthesis of the spacecraft mechanism is done by direct linearization method. Key words: Tolerance Analysis, root sum square method, Assembly Simulation by Software R I. INTRODUCTION Tolerance is the specified amount a feature is to vary from its nominal. Tight tolerances assure product performance whereas looser tolerances reduce cost. A balance have to be made between the two. Tolerance analysis shows the effect of component tolerances on the output variability of a mechanism. Tolerance stackups determine the performance and functionality of the assembly. Proper tolerance synthesis enables complex mechanisms with numerous parts to assemble and work together to satisfy its design objectives. The main objective of tolerance technology is to determine tolerances of individual parts based on the the functional requirements developed during product design [1]. Tolerance analysis is performed when natural process tolerances of the components are known and there is a need to calculate design tolerance of the assembly. It determine if the given tolerances can meet the functionality of the product. It also provides guidance to where the tolerances must be tighter and where it can be relaxed. II. TOLERANCE ANALYSIS In order to ensure the functionality of mechanical assemblies, designers attempt to control the variation of critical assembly dimensions. Acceptable limits on the variation are defined according to the performance requirements. Tolerance analysis can be conducted in two main methods: worst-case and statistical methods [2]. The worst case tolerance analysis calculates assembly tolerance by summing component tolerances. It satisfy assembly tolerance with 100% probability but worst- case tolerancing can lead to excessively tight part tolerances and hence high production costs. As there is no part falling outside the tolerance range the assembly tolerance calculated will be high. For one-dimensional assemblies, 𝑑𝑈 = ∑ 𝑇𝑖 ≤ 𝑇𝐴𝑆𝑀 (1.1) For two or three-dimensional assemblies, 𝑑𝑈 = ∑ ( 𝜕𝑓 𝜕𝑥 𝑖 𝑇𝑖) ≤ 𝑇𝐴𝑆𝑀 (1.2) where 𝑥𝑖 are the nominal component tolerances, 𝑇𝑖 are the component tolerances, 𝑑𝑈 is the predicted assembly variation, 𝑇𝐴𝑆𝑀 is the specified limit for 𝑑𝑈, and 𝑓(𝑥𝑖) is the assembly function describing the resulting dimension of the assembly. The partial derivative 𝜕𝑓 𝜕𝑥 𝑖 represent the sensitivity of the assembly tolerance to variations in individual component tolerances. The statistical tolerancing method calculates assembly tolerance by taking the root sum square of the component tolerances. Tolerances are commonly assumed to a 3𝜎 process level. This method assumes the component process tolerance follow normal distribution. For one-dimensional assemblies, dU = ∑[Ti 2 ] 1 2⁄ ≤ TASM (1.3) For two or three-dimensional assemblies, 𝑑U = [∑ ( ∂f ∂xi ) 2 Ti 2 ] 1 2⁄ ≤ TASM (1.4) Statistical tolerancing approach can produce much higher profit than the worst case approach [3]. Tolerancing decisions can affect the quality and cost of the mechanism. To evaluate the impact of tolerance on mechanism quality, designers need to simulate the influences of tolerances with respect to the functional requirements [4]. Marler implemented a 2D method for linearized vector-based tolerance analysis, the direct linearized method (DLM). The DLM provided a relatively simple method for linearizing the vector loop equations of a tolerance model using a first order Taylor expansion, and then extracting the sensitivities and solving for dependent assembly variables using matrix algebra [5]. Statistical tolerancing allows more realistic tolerance calculations, leading to larger tolerances with lower production costs without loss in product quality. III. METHODOLOGY Tolerance analysis is the process of taking known tolerances and analyzing the combination of these tolerances at an assembly level. Direct linearization method (DLM), is used for tolerance analysis of the mechanical assembly. The model is constructed of common engineering elements: vector chains, kinematic joints, assembly datums, dimensional tolerances, and assembly tolerance limits [6]. The DLM applies matrix algebra and root sum squares error analysis to vector loop based models to estimate tolerance stackup in assemblies. Dimensional variations account for small changes in size due to manufacturing processes. Kinematic variations describe the propagation of variation through an assembly by small adjustments between mating parts. The key features of this model are: 1) A set of rules is provided to assure a valid set of vector loops. The loops include only those controlled dimensions which contribute to assembly variation. All dimensions are datum referenced. 2) A set of modeling elements will be introduced to assist in identifying the adjustable dimensions within the
- 2. Synthesis of Tolerances for Spacecraft Mechanism (IJSRD/Vol. 3/Issue 10/2015/076) All rights reserved by www.ijsrd.com 386 assembly that change to accommodate dimensional variations. 3) In addition to describing variation in assembly gaps, a comprehensive set of assembly tolerance requirements is introduced, which are useful to designers as performance requirements. 4) Algebraic manipulation to derive an explicit expression for the assembly feature is eliminated. The new system operates equally well on implicit assembly equations. The loop equations are solved the same way every time. It is well suited for computer automation. 5) Differentiation of a complicated assembly expression is replaced by a single matrix operation, which determines all necessary tolerance sensitivities simultaneously. IV. CASE STUDY The DLM is applied on the door opening mechanism (Fig. 1) for tolerance analysis. The components of hinge assembly are out-board bracket, hinge shaft, spherical bearing, in-board bracket, stopper pin, and latch spring. Dimensions a to k are the dimensions of the component features which contribute to assembly variation. These are shown in the Fig. 2. Tolerances are estimates of the manufacturing process variations. Fig. 1: Components of door opening mechanism Fig. 2: Dimensions of door opening mechanism The general tolerances for dimensions are taken as per ISO 2768 [7] and is tabulated in Table I. Tolerances to functional requirements of the pay load door opening mechanism was calculated using direct linearization method [8]. The steps in DLM are broadly classified into two: 1) Steps in creating an assembly tolerance model Create an assembly graph Locate the datum reference framefor each part Create vector loops Define performance requirements 2) Steps in analyzing an assembly tolerance model Generate assembly equations from vector loops Calculate derivatives and form matrix equations Solve for assembly tolerance sensitivities Dimension Nominal Tolerance a 4mm ±0.1mm b 6mm ±0.02mm c 8mm ±0.2mm d 10.5mm ±0.2mm e 12mm ±0.2mm f 5mm ±0.1mm g 40mm ±0.3mm h 10mm ±0.2mm i 2mm ±0.1mm j 12.5mm ±0.2mm k 43mm ±0.3mm Table I: Dimensions And Tolerances Of Door Opening Mechanism After the sensitivity matrix is obtained, worst case and RSS expression for tolerance accumulation is formed. Table II gives the assembly variables obtained by both worst case (WC) and root sum square (RSS) methods. Assembly Variable Nominal WC RSS u1 6.2mm ±0.14075m m ±0.10552m m u2 14mm ±0.14075m m ±0.10552m m u3 14.648m m ±0.68149m m ±0.3529m m u4 1mm ±1.2mm ±0.52915m m ∅1 11ᴼ ±3.057ᴼ ±2.0821ᴼ Table II: Assembly Variables The assembly variable u4 is restricted to 0.2 mm by the designers for achieving the performance requirement of the assembly. The current tolerance does not satisfy the above mentioned criteria. Therefore, the tolerances of the dimensions need to be changed.
- 3. Synthesis of Tolerances for Spacecraft Mechanism (IJSRD/Vol. 3/Issue 10/2015/076) All rights reserved by www.ijsrd.com 387 The tolerance allocation is determined by using percent contribution charts. The tolerance with largest contribution is changed and is again checked for assembly variations. The results are tabulated in Table III. Dimension ± Tolerance % Relaxation WC RSS a 0.1 0.1 0.00% b 0.02 0.02 0.00% c 0.2 0.2 0.00% d 0.03 0.1 233.33% e 0.03 0.05 66.67% f 0.04 0.1 150.00% g 0.03 0.05 66.67% h 0.2 0.2 0.00% i 0.04 0.1 150.00% j 0.2 0.2 0.00% k 0.03 0.05 66.67% Table III: Modified Dimensional Tolerance The modified tolerance allocation shows that the statistical tolerancing gives wider tolerance than the worst case approach. V. ASSEMBLY SIMULATION BY SOFTWARE R The purpose of this simulation is to model manufacturing variations and predict their effects on the assembly. A sample of 100000 of each part is selected and assembled together. The assembly variation of all the 100000 assembly is calculated and the number of rejects is calculated. The result obtained is plotted as a histogram and is shown in Fig. 3. Fig. 3: Histogram of assembly variation of u4 The mean (µ) and standard deviation (σ) of the assembly variation is found to be 0 and 0.05677851 mm respectively. The probability of assemblies having variation between -0.2 mm and 0.2 mm is 0.9995662. Which means out of the 100000 assemblies the number of defects will be 43.38. VI. CONCLUSION In this paper, the direct linearization method is used for tolerance analysis of the payload door opening mechanism by both worst case and statistical methods. Statistical method provides relaxed tolerance for components of the assembly compared to that by worst case method. Component tolerances are analysed to satisfy the functional requirements, which will make the assembly compatible with the design. The number of rejects of assemblies by statistical tolerancing is calculated by simulation using software R. The statistical tolerencing provides wider tolerances than worst case tolerancing while satisfying the assembly requirements. Wider tolerances are comparatively easy to process and relative cost for manufacturing is less. VII. SCOPE FOR FUTURE RESEARCH In addition to the dimensional tolerances, the geometric tolerances can be added to the variation analysis. Geometric tolerances include important variation sources caused by irregularities in form, shape and size. This may make significant contributions in a tolerance analysis. Most materials change length as they change temperature. As a result of this change, the dimensions and tolerances of a product become at variance with the design values. Hence, thermal effects can be included in the analysis when designing a product like spacecraft component that will undergo temperature cycling. REFERENCES [1] B. R. Fischer, Mechanical Tolerance Stackup and Analysis, CRC Press, 2011. [2] K. W. Chase and A. R. Parkinson, “A survey of research in the application of tolerance analysis to the design of mechanical assemblies”, Research in Engineering Design, vol. 3 (1), pp. 23-37, 1991. [3] A. Shan, R. N. Roth and R. J. Wilson, “A new approach to statistical geometrical tolerance analysis”, International Journal of Advanced Manufacturing Technology, vol. 15 (3), pp. 222-230, 1999. [4] A. J. Qureshi, J. Y. Dantan, V. Sabri, P. Beaucaire, and N. Gayton, “A statistical tolerance analysis approach for over-constrained mechanism based on optimization and monte carlo simulation”, Computer-Aided Design, vol. 44 (2), pp. 132-142, 2012. [5] J. D. Marler, “Nonlinear tolerance analysis using the direct linearization method”, ADCATS Report, Bringham Young University, 1988. [6] V. Srinivasan, Dimensioning and Tolerancing Handbook, IBM Research and Columbia University, New York, 1999. [7] American National Standards Institute, Y14. 5M, Dimensioning and Tolerancing, American Society of Mechanical Engineers, New York, 1994. [8] P. Faerber, “Tolerance analysis of assemblies using kinematically derived sensitivities,” M .S. Thesis, Brigham Young University, Provo, Utah, 1999.