Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 1/22 Statistical method for...

Università degli Studi di BresciaDipartimento di Ingegneria Meccanica

1/22

Statistical method for the prediction of

tolerancing effects in design

Andrea Magalini, David Vetturi, Luca Pagan

Università degli Studi di Brescia, Dipartimento di Ingegneria Meccanica

Alenia S.p.A. - LABEN (Vimodrone, Milano, Italy)

ICEM 12 - 12th International Conference on Experimental Mechanics29 August - 2 September 2004, Politecnico di Bari, Bari, Italy


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Foreword

Statistical method for the prediction of tolerancing effects in designA. Magalini, D. Vetturi, L. Pagan

Design to manufacturing deviations from the nominal configuration (restrictions imposed by tolerancing)

Uncertainty related to the final effective three-dimensional configuration of a mechanical assembly

Uncertainty estimation: statistical problem

Metrological analogy (Guide to the Expression of Uncertainty in Measurement - GUM)),...,,( 21 KXXXfY


3/22

Planck LFI is an instrument designed to be operative in space.

It is aimed to detect micro waves comingfrom deep space.

With reference to this memory, it is characterised by a number of sensors (feed horns) which must be carefully aligned with a target point.

The following pictures give an overall sight of the instrument and some details of the sensor pack.

The Low Frequency InstrumentIntroduction



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The Low Frequency InstrumentOverall Sight


BEU

FPU

Waveguides

Su

pp

ort

str

uct

ure

s

1.5

m


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The Low Frequency InstrumentOverall Sight


Horn

LFI HFI

Main Frame

WaveguidesSupportStructure

Waveguides

BEU

Bipodes


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The Low Frequency InstrumentFocal Plane Unit FPU


Bipodes Interfacesto satellite

30 GHzfeed horns

70 GHz feed horns

44 GHzfeed horns

HFI


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The Low Frequency InstrumentFocal Plane Unit FPU


Feed Horns (1) Main Frame Top (2)

Main Frame (3)

Bipodes (4)

Interfaces to satellite


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x

y

z4

3

2

1xL2

yL2

zL2

xL1

yL1

zL1

xL3

yL3

zL3

xL4

yL4

zL4

xA4

yA4

zA4

xA3

yA3

zA3

xA2

yA2

zA2

xA1

yA1

zA1

Local reference frame Coupling reference frame

1 - Feed Horn

2 - Frame top (OMT for 70 GHz FH)

3 - Main Frame

4 - Exapode

PC

Geometrical ApproachAssembly Modelling



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Geometrical ApproachUncertainty Contributions

Dimensional tolerances

Geometrical tolerances

Fit tolerances

Thermal distortions*

Changes of position and orientation of each local reference frame

Fit modalitiesChanges of position and orientation of each local coupling reference frame

*thermal effects are here disregarded



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The orientation of each reference frame and the position of its origin can be described in any reference frame by one 4x4 matrix, keeping into account rotations and translations at the same time.

1000z

y

x

ABT

T

T

RM

A B

xB

yB

zB

OB

xA

yA

zA

OA

P

PAV

PBV

OV

1PA

PA

PA

PA z

y

x

V

11z

y

x

BO

BO

BO

O T

T

T

z

y

x

VA

A

A

1PB

PB

PB

PB z

y

x

V

PAPB VMVrr

Geometrical ApproachHomogeneous Matrixes



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ML1 describes the L1 ref. frame in the A1 ref. frame MA1 describes the A1 ref frame in the L2 ref frame ML2 describes the L2 ref. frame in the A2 ref. frame MA2 describes the A2 ref. frame in the L3 ref. frame ...

x

y

z

4

3

2

1xL2

yL2

zL2

xL1

yL1

zL1

xL3

yL3

zL3

xL4

yL4

zL4

xA4

yA4

zA4

xA3

yA3

zA3

xA2

yA2

zA2

xA1

yA1

zA1PCML1

MA1ML2

MA2ML3

MA3ML4

MA4

1PCV

PCV Position vector of the

PC in the L1 ref. frame




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x

y

z

4

3

2

1xL2

yL2

zL2

xL1

yL1

zL1

xL3

yL3

zL3

xL4

yL4

zL4

xA4

yA4

zA4

xA3

yA3

zA3

xA2

yA2

zA2

xA1

yA1

zA1PCML1

MA1ML2

MA2ML3

MA3ML4

MA4

1PCV

1PCV Position vector of the

PC in the L1 ref. frame

PCV Position vector of the PC

in the absolute ref. frame

11

1

1

1z

y

x

PC V

V

V

V

1144332211 PCPCALALALALPC VMVMMMMMMMMV

1z

y

x

PC V

V

V

V

Position of the PC (x,y,z)FH orientation(angles given following any chosen rule)

M




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Local frame L1: origin in correspondence of the P point x axis correspondent to the FH real axis y axis passing from the centers of the pins

locations

FH 44 GHz

x

y

z

P P z

y

xx

Tolerancing Specifications EffectsReference frames definition

Drawings are courtesy of Alenia S.p.A. - LABEN



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Coupling frame A1:

origin in the theoretical P point (on the FH)

x axis correspondent to the FH theoretical axis

y axis passing from the centers of the pins locations

Tolerancing Specifications EffectsReference Frames Definition



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1. translation along x of the L1 origin

2. rotations around A1 z & y axis

3. pins location in the yz plane: translations in x e z of the L1 origin and rotation around the A1 x axis (general dimensional tolerance)

Tolerancing Specifications EffectsUncertainty Contributions

1 3

2

x

y

z

P

x

y

z

P

P z

y

xx



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The shown uncertainty causes produce a modification of the orientation of the L1 ref. frame as regards the A1 ref. frame (2 & 3) and a translation of the L1 origin in the A1 ref. frame (1).

There are further uncertainty contributions relating the uncertainty connected to the theoretical position of the PC: they influence the components of the position vector for the PC in the L1 ref. frame.

Tolerancing Specifications EffectsUncertainty Contributions



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Using the Monte-Carlo simulation technique, for each uncertainty contribution, a series of N values is generated within the range of values defined by the considered tolerance (suitable distribution function).

A series of N vectors, representing the N different situations is found, in consequence of the considered tolerances.

For each situation (vector), the rotation-translations matrixes are computed.

Numerical Statistical AnalysisMonte Carlo Simulation



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Three angles, defining the orientation of the feed horn, are extracted from the total M rotation-translations matrix.

The position of the PC in the absolute ref. frame is obtained by multiplying the M matrix per the VPC1 vector.

The three found rotations and the the three found translations give the FH space location.




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The six degrees of freedom (position of the PC, orientation of the FH axis) of the FH are considered as random variables.

For each random variable a series of N possible numerical values has been obtained by the previously explained method.

Starting from the N values (constituting a sample), available for each variable, a proper probability distribution can be computed for this (its parameters are estimated). So a mean value and a standard deviation are calculated for the six degrees of freedom.




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Qualitative ResultsA FH d.o.f.

Phase Center position - z axis

f(x)

Simulation

Histogram

Theoretical



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General statistical method based on the Monte Carlo simulation technique for the uncertainty estimation

Can be adopted for: uncertainty estimations, sensitivity analysis, quality evaluation of design

Generality: non-linear problems, multi-output problems, whatever kind of probability distribution function

Improvements: adaptive Monte Carlo, closed-loop geometry structures

Conclusions and Discussion


Method


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... thank you

Authors are grateful to:• ALENIA S.p.A. - LABEN (Vimodrone, Italy) • CENTROTECNICA S.a.S. (Milano, Italy)


Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 1/22 Statistical method for...

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