# Interest of the Inertial Tolerancing Method in the Case of Watch Making Micro Assembly

**ABSTRACT** A mechanical part to tolerance is traditionally expressed as a [Min Max] interval which allows the definition of the conformity

of the characteristic. Inertial tolerancing offers a new point of view of the conformity based on the mean square deviation

to the target. This article demonstrates the efficiency of inertial tolerancing and proposes a comparison with the traditional

tolerancing method in the case of watch making micro assembly.

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**ABSTRACT:**Inertial tolerancing [1]-[3] presents is a statistical expression of a characteristic tolerance without defining the specification interval as is traditionally the case. Conformity is defined by "inertia" (Taguchi loss function [4]) around the target. Although this tolerancing method offers numerous advantages in terms of Quality/Cost ratio, it raises new problems in the customer/supplier relationship. This paper develops an acceptance sampling method design for inertial tolerancing and a sorting strategy when a lot is rejectedInternational Journal of Web Engineering and Technology 04/2013; Volume 2,(Issue 10):265-271.

Page 1

The watch making micro assembly problem, inertial tolerancing the solution?

A Maurice Pillet*, Yann Carrara**

SYMME Laboratory, Université de Savoie, B.P.806, 74016 Annecy, cedex, maurice.pillet@univ-savoie.fr,

Patek Philippe SA CP 2654 1211 Genève 2, Yann.Carrara@patek.com

Abstract

In watch making micro assembly, functional clearances are very tight and the assembly of a bridge on the

main plate must satisfy simultaneity several clearances. The statistical combination of several clearances

having a weak yield leads to an extremely weak global assembly yield. That results in numerous and

expensive final improvements and reworks.

Classical worst case tolerancing method is inappropriate because of the production cost and statistical

tolerancing is very risky in case of a decentered production. Inertial tolerancing offers an interesting

alternative to the traditional method. Whereas a tolerance is traditionally expressed as a [Min Max] interval

inertial tolerancing is based on the mean square deviation to the target. This new interpretation of the

conformity makes an important cultural change in the production control. This article demonstrates the

efficiency of the inertial tolerancing and proposes a comparison with traditional tolerancing method in the

case of watch making micro assembly.

We also show how inertial tolerancing is declined in production for guarantee the production control.

Keywords:

Statistical tolerancing, Inertial tolerancing, micro assembly, watch making.

1 INTRODUCTION

The functional quality of a mechanical assembly is often a

direct function of a functional clearance. In assemblies of

watch making, this clearance results from the combination

of several elementary components. In micro-assembly,

tolerances are very tight in regard of the dimensions and

the yields of these assemblies are often weak. In watch

making case, in the assembly of a bridge on the main

plate, the same assembly must satisfy several clearances

(fig 1). The statistical combination of several clearances

with a weak yield leads to an extremely weak global

assembly yield. That results in numerous and expensive

final reworks. [1]

Bridge Wheel Jewel Main Plate

Fig. 1: Typically problem in watch making assembly

To increase the yield (Manufacturing and assembly),

different tolerancing methods are available notably worst

case and statistical tolerancing method. The worst-case

method favours the assembly yield to the detriment of the

manufacturing yield. Theoretically by this approach, if

tolerances are respected, the assembly yield must be

100%. Unfortunately although such tolerances are not

compatible with the production and control means

currently available. Statistical tolerancing introduces by

Evans [9] through his state of the art could be appear as a

solution The statistical tolerancing method allows an

important widening of the tolerances, but does not

guarantee the conformity of the assembly. [2][3][13].

Graves[10] warns us against five fails of the statistical

tolerancing limit and proposes some precautions at the time

of statistical tolerancing design. But, even with these

precautions, statistical tolerancing remains dangerous.

Inertial tolerancing [4][5][6][7][8] proposes an alternative to

the traditional methods making it possible to obtain good

yield at the same time in assembly and production.

Tolerance does not define an interval but the maximum

mean square deviation compared to the target (equation 1).

I: Inertia

: off-centering off the mean

: Standard deviation

This new expression of the tolerance has many

advantages. It makes it possible to increase variability in

production while guaranteeing the functional clearance.

This characteristic is particularly interesting in the complex

case of the assembly of micromechanical parts.

2 YIELD INCIDENCE OF

METHOD

We propose to study the application of inertial tolerancing

in the case of an assembly of a bridge on a main plate (Fig.

1). This assembly comprises k wheels (k generally lying

between 2 and 5), each wheel clearance implying a chain

of n components (n generally lying between 3 and 5)

Assumptions

• The component count c is identical for each wheel

• The functional clearance is identical for each wheel and

equal to t

• In the case of a centered production, we wish to obtain a

capability index Cp = 1 that is to say ( = t/3)

• Productions are normally distributed

22

I

(1)

THE TOLERANCING

Page 2

In the case of traditional tolerancing method, Cp is

defined by equation 2.

2t

Cp

6

(2)

In the case of inertial tolerancing method Cp we define by

the relation 3

I

Cp

(3)

In the case of traditional tolerancing method, Cpk is

defined by equation 4.

. 3

In the case of an inertial tolerancing method, we defined

Cpi by the relation 5

I

Cpi

. 3

,

t

USL LSLMin

Cpk

(4)

I

0

(5)

I0: Inertia tolerance

I: Inertia of the batch

Calculation of inertial tolerance for each component

Several approaches can be considered to calculate the

inertial tolerance according to the assumptions of

capability, and respect of the centering on the target. Two

approaches were proposed which will be tested on this

example.

1. Classical inertial tolerancing

2. Adjusted inertial tolerancing

We will compare these both approaches with the worst

case and statistical tolerancing method for the yield of

assembly and the variations authorized in production.

Case #1: Classical inertial tolerancing

To calculate the component inertial tolerances (IP), the

assembly distribution is considered centered, six standard

deviations contained in the functional tolerance

interval (t).

2t

I

CC

6

(6)

With the precedent assumptions

ntnII

CP

3

(7)

IC: Clearance Inertia

IP: Part inertia

With the assumption of a capability Cp = 2 and Cpi = 1,

the maximum decentring is equal to:

I

2

P

P

(8)

n

I

I

4

I

C

P

PPP

4

3

3

2

2

22

(9)

The most unfavourable assembly corresponds to the

situation where all decenterings are added. With the

assumption of independence we can write:

n

PC

(10)

6244

2

C

2

P

2

P

tI

n

nInI

n

C

C

(11)

With a tolerance oft on the clearance, the z index is

calculate by the relation (12)

n

t

n

t

t

n

I

nt

z

C

36

6

12

1

6

4

3

2

(12)

The yield is calculated by z), where represent the

standard normal cumulative distribution function.

N 2

t/2

Inertial tolerance 0.236t

percentage of increase in the

production dispersion

compared to worst case

tolerancing

(position centered)

3.551

3

t/3

0.192t 0.167t 0.149t

4

t/4

5

t/5

Worst caseTolerance

141% 173% 200% 224%

Z =

n

36

Yield

3.000 2.536 2.127

1.000 0.999 0.994 0.983

Global Yield on k assembly

K

2

3

4

5

1.000

0.999

0.999

0.999

0.997 0.989 0.967

0.996 0.983 0.951

0.995 0.978 0.935

0.993 0.972 0.919

Table 1 - Yield with classical inertial tolerancing

Case #2: Statistical tolerancing

The same table can be obtained for statistical tolerances

under similar conditions: Cp = 2, Cpk = 1:

nttP

/

n

ttP

6

P

6

The maximum decentring is for Cpk = 1:

t

t

PPP

3

n

t

n

t

n

22

The most unfavourable assembly corresponds to the

situation where all decenterings are added [12]. With the

assumption of independence we can write:

2

nt

n

PC

6

2

P

t

n

C

The z index is calculated by the relation (13)

n

t

n

tt

z

36

6

2

(13)

The table 2 shows the risks of the statistical tolerancing

method. With a Cp = 2 on each component and Cpk = 1, as

soon as there are 4 wheels in the mechanism the yield of

the assembly can be close to zero although the individual

yield of each component is 0.9973.

In the case of inertial tolerancing the worst yield is equal to

0.919 (k=5 and n = 4) for the same widening of the

Page 3

production spread in the centred case as statistical

tolerances.

N

Worst caseTolérance t/2

statistical tolerance

percentage of increase

in the production

dispersion compared

to worst case

tolerancing

2 3

t/3

4

t/4

5

t/5

0.447t

0.707t 0.577t 0.5t

141% 173% 200% 224%

Z=

n

36

1.757 0.804 0.000 -0.708

Yield 0.961 0.789 0.500 0.239

Global Yield on k assembly

2

3

4

5

Table 2 - Yield with statistical tolerancing

K

0.923

0.886

0.851

0.818

0.623

0.492

0.388

0.306

0.250

0.125

0.063

0.031

0.057

0.014

0.003

0.001

Case #3 - Adjusted inertial tolerancing

Adragna [11] showed that it is possible to calculate inertia

on each component in order to guarantee yield of the final

assembly.

Adragna proposed an inertial adjusted coefficient (IC) in

order to guarantee that the assembly Cpk index will never

be lower that a CpkMin value. The IC coefficient is

calculated by the simple following relation:

9

2

Min

n

CpkI

C

(14)

This IC coefficient has the same rule as the inflated

coefficient for the traditional inflated statistical tolerancing:

t

I

2

1

. . 6

nI

C

P

(15)

In our application we want to guarantee in all cases the

assembly yield greater that 0.997 (Cpkmin = 1)

9

1

n

IC

9

. 3

2

n

n

t

IP

(16)

With the assumption of a capability Cp = 2 and Cpi = 1,

the maximum decentring is equal to:

tIP

P

9

6

2

2

n

n

(17)

9

.

3

6

2

2

P

2

P

n

n

t

I

P

(18)

The most unfavourable assembly corresponds to the

situation where all decentrings are added. With the

assumption of independence, we can write:

PC

n

9

1. 6

2

P

n

t

n

C

With a tolerance of t on the clearance, the z index is

calculate by the relation [10]

n

n

9

n

9

t

n

9

n

nt

6

t

t

z

C

C

316

1. 6

.

3

2

(19)

This relation gives the following outputs:

n 2 3 4 5

IC 1.106 1.155

0.2132

t

1.202 1.247

IP

0.1667t 0.1387t 0.1195t

percentage of increase in the

production dispersion

compared to worst case

tolerancing

(position centrered)

128% 150% 166% 179%

n

n

9

z

316

4.184 3.928 3.747 3.610

Global Yield on k assembly

2

3

4

5

Table 3 - Yield with adjusted inertial tolerancing

Yield 1.0000 1.0000 0.9999 0.9998

k

1.000 1.000

1.000 1.000

1.000 1.000

1.000 1.000

1.000

1.000

1.000

1.000

1.000

1.000

0.999

0.999

The yield is guaranteed for each assembly. It is possible

also to guarantee the global yield as we will detail it in the

end of the article.

In the situation of this article, yield is better than 0.997. In

fact the situation Cp= 2 and Cpi = 1 is not the worst case

situation

3 DISCUSSION

The assembly of micromechanics components often leads

to small yield of production and assembly because of the

relative importance of the variability. In this context, it is

important to choose a tolerancing method able to

guarantee the functional specification for the product, but

also to give the greatest possible variability for the

production.

Worst case tolerances guaranteed

specification but is very restrictive for the production

Statistical tolerances gives freedom to the production but

does not guaranteed a good assembly capability.

We show in this paper through generic example of

assembly in watch making the great interest to use inertial

the functional

Page 4

tolerances. It guarantees at the same time good assembly

capability while leaving the most possible freedom to the

production.

4 THE STATISTICAL PROCESS

INERTIAL TOLERANCING

The control chart introduced by Shewhart as a SPC’s tool

(Statistical Process Control), is actually the most popular

chart used in industry. However, inertial tolerancing

method introduces a new concept in the conformity

approach and we are developed a special control chart for

control the production inertia.

3.3.2 Inertial control chart

The inertial control chart is based on two limits [11]. The

first control limit (LSC) is computed with a -risk (false

alarm risk), in the same way proposed by Shewhart

control chart:

CONTROL IN

n

I LC

historical

2

1

(14)

In which Ihistorical is equal to ST the standard variation

short terms, n is the sampling size and

square with n degree freedom and the risk of false

alarm.

The second control limit (LSC) is computed with a risk

(risk of no detecting of the off-centered process):

2

1

is the chi

2

Max

ILC

(15)

In which

freedom and the risk of no detecting.

2

is the chi square distribution with degree of

12

2

4

Cp

Cp

n

(16)

It is possible to introduce in the chart the conformity limit:

the inertia limit IMax

Then, the inertial control chart can be presented in the

following form:

Fig. 2: The inertial control chart

The inertia can be control by an graphic. This graphic

show the decentering and the standard deviation in a

circle representation of the different limits. Then, the

inertia control chart give the time evolution of the inertia

and the graphic give information on the last sample

showing simultaneity standard deviation and mean of the

population.

The graphic help the operator to make a decision of

regulation.

Fig. 2: The graphic

5 APPLICATION IN THE PATEK PHILIPPE COMPANY

The aim of this example is to show how inertial tolerancing

is introduced in production. The application is the assembly

of wheels in a mechanical watch movement (fig. 1).

Each wheel in the gear train of a movement needs a certain

axial clearance between its first jewel in the main plate and

its second jewel in the bridge (fig. 3). The jewels act as

bearings.

The functional clearance (FC) of the wheel assembly is the

axial clearance. The specification of the clearance is

defined by the combination of n elementary components

(EC) (equation 17). n is generally comprised between 3 and

5.

n

ECFC

(17)

Fig. 3: Functional clearance in a wheel assembly

The tolerancing of the functional clearance can change for

each wheel depending on the product itself and on how the

wheel interacts with their neighbours. The tolerance interval

on the clearance lies between 15 and 30 µm (t).

By using worst case tolerancing, the tolerance interval on

each elementary component is defined by (equation 18).

n

P FC

tt

(18)

2tFC (µm)

2tP (µm) with n=3

2tP (µm) with n=4

2tP (µm) with n=5

Table 4 – Tolerance interval in worst case tolerancing

15 20 25 30

5.0

3.8

3.0

6.7

5.0

4.0

8.3

6.3

5.0

10.0

7.5

6.0

Inertia

Green zone

Orange zone

Red zone

Black zone

LSC

LSC

IMax

Green zone

Black zone

red zone

Orange zone

Last point

(out of control)

Page 5

By using classical inertial tolerancing, the table 5 shows

the part inertia (IP) for different functional tolerance

interval (t) (see equation 7).

2tFC (µm)

IC (µm)

IP (µm) with n=3

IP (µm) with n=4

IP (µm) with n=5

Table 5 – Part inertia in classical inertial tolerancing

15 20 25 30

2.5 3.3 4.2 5.0

1.4

1.3

1.1

1.9

1.7

1.5

2.4

2.1

1.9

2.9

2.5

2.2

With inertial tolerancing, quotation of each elementary

component must be adapted (Fig. 4).

Fig. 4: Quotation with inertial tolerancing

By using worst case tolerancing, if all the parts are within

the tolerance limits (2t) or by using inertial tolerancing, if

the batch of parts is within the defined inertia, the wheels

after assembly will meet the functional clearance required.

Here starts the interaction between product development

and production. With the tolerance defined in table 4,

capability of the production processes available to

produce the watch parts are low, Ppk < 1. There are two

reasons, the process itself often has a low short term

capability (Cp < 1.5) and a drift on the process has a

major impact on quality with such small tolerances. In fact,

to correct such drifts, corrections of only 1 to 3 microns is

needed and this is hardly achievable with traditional

production processes.

In this application, two processes are needed to

manufacture the elementary component:

1. Push fit for the jewels in the main plate and

bridges.

2. Turning and rectification for the wheel axis.

To define a process, the first characteristic to measure is

the short term capability or short-term standard deviation

.

s

In our application, a good short term standard deviation

for both processes is 1.0 µm. With a stable process, the

standard deviation does not change during production of

a batch. With this assumption, we can calculate the

maximum off-centering allowed for the elementary

component, using inertial tolerancing (table 6).

2tFC (µm) 15 20 25 30

σ (µm) 1.0 1.0 1.0 1.0

δP (µm) with n=3

δP (µm) with n=4

δP (µm) with n=5

Table 6 – Maximum off-centering with a stable process by

using classical inertial tolerancing

1.0

0.8

0.5

1.6

1.3

1.1

2.2

1.8

1.6

2.7

2.3

2.0

To achieve the same quality (at 6 sigma) with a production

normally distributed and using worst case tolerancing, the

off-centering of the production must be reduced (table 7).

We can see that in half of the examples, off-centering is not

allowed.

2tFC (µm)

σ (µm)

15 20 25 30

1.0 1.0 1.0 1.0

δP (µm) with n=3

δP (µm) with n=4

δP (µm) with n=5

Table 7 – Maximum off-centering with a stable process by

using worst case tolerancing

(-0.5) 0.3

(-1.1) (-0.5)

(-1.5) (-1.0)

1.2

0.1

(-0.5)

2.0

0.8

0

To reduce overall production costs, liberty for off-centering

is essential in production process. But this liberty must be

controlled.

Every production process has its own short-term capability.

Knowing each process capability, we can distribute this off-

centering liberty on each elementary component. Thus, we

give the maximum of liberty to the weakest process.

This is very easy with adjusted inertial tolerancing (table 8).

2tFC (µm)

IC (µm)

IP (µm) with n=3

IP_A (µm) coefficient 1

IP_B (µm) coefficient 1

IP_C (µm) coefficient 2

IP_A (µm) coefficient 1

IP_B (µm) coefficient 1

IP_C (µm) coefficient 3

Table 8 – Part inertia in adjusted inertial tolerancing

15 20 25 30

2.5 3.3 4.2 5.0

1.4

1.0

1.0

2.0

0.8

0.8

2.3

1.9

1.4

1.4

2.7

1.0

1.0

3.0

2.4

1.7

1.7

3.4

1.3

1.3

3.8

2.9

2.0

2.0

4.1

1.5

1.5

4.5

Finally, to pilot and control a production process, an inertial

control chart is used.

For example, to control the push fit process of a jewel in a

main plate, we follow the production with a control panel

(Fig. 5). Top left chart shows the last three in-line process

measured samples. Top right is the historical inertial chart

and below are two charts to analyze drifts or out-of-control

points of the inertia (either average or standard deviation).

Dark curves are for final positioning and light curves are for

pre-positioning.

Page 6

Fig. 5: Production control panel

6 CONCLUSION

The assembly of micromechanics components often leads

to small yield of production and assembly because of the

relative importance of the variability. In this context, it is

important to choose a tolerancing method able to

guarantee the functional specification for the product, but

also to give the greatest possible variability for the

production.

Inertial tolerancing is a well adapted solution in this case.

In this paper we have presented an application in watch

making micro assembly where the worst case tolerance

(between ±1.5 to ± 5 µm) are tight in regards of the

production processes available for micro-machining.

Inertial tolerancing make possible a statistical approach

without the well know risks of the quadratic statistical

tolerancing on then functional

tolerancing accept a de-centering of the process

proportional to the historic standard deviation. It

guarantee the functional clearance without excessive

requirements in the process capability.

clearance. Inertial

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17-19, 2003.

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your organization – Journal of Quality technology –

Vol. 33, N°3, 293-303, July 2001

3. K.W. Chase. Tolerance allocation methods for

designers. ADCATS

YoungUniversity, 1999.

4. Pillet M., Inertial tolerancing in the case of assembled

products, Recent advances in integrated design and

manufacturing in mechanical engineering, No. ISBN

1-4020-1163-6, 2003, pp. 85-94,

5. Pillet M., Inertial Tolerancing, The Total Quality

Management Magazine, Vol. 16, No. Issue 3 - Mai

2004, 2004, pp. 202-209,

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Chollet, and J. Jacot.

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Lausanne(EPFL),Lausanne(Swiss),2007

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2.

Report 99-6, Brigham

Defining assembly

Polytechnique de

des Systèmes

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