Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D....

Università degli Studi di BresciaDipartimento di Ingegneria Meccanica

A. Magalini, D. Vetturi, D. CambiaghiPlanck LFI Alignment Analysis Approach

SLIDE 1DESIGN 2004 – DUBROVNIK

Planck LFI Alignment Analysis Approach




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 horn) 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




Low Frequency Instrument

BEU

FPU

Waveguides

Su

pp

ort

str

uct

ure

s

1.5

m




Horn

LFI HFI

Main Frame

WaveguidesSupportStructure

Waveguides

BEU

Bipodes

Focal Plane Unit




Focal Plane Unit

CFRP struts Interfacesto satellite

30 GHzfeed horns

70 GHz feed horns

44 GHzfeed horns

HFI




To evaluate the alignment error of the sensors no reliable procedure was found by the manufacturer.

The Mechanics Department of the University of Brescia was involved in the problem

In the following it is shown the dedicated procedure developed in order to evaluate the error budget




x

y

z4

3

2

1

General structure analysis

xL2

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




Uncertainty contributions

Dimensional tolerances

Geometrical tolerances

Thermal distortions (elements 3 and 4)

Changes of position and orientation of each local reference frame

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




Matrix approach

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 VMV




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

Matrix approach




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

11

1

1

1z

y

x

PC V

V

V

V

1144332211 PCPCALALALALPC VMVMMMMMMMMV

PCV Position vector of the PC

in the absolute ref. frame

1z

y

x

PC V

V

V

V

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

M

Matrix approach




Tolerances effectsFH 44 GHz

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

x

y

z

P P z

y

xx




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

Tolerances effects




Contributions

1 3

2

1. translation in 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)

x

y

z

P

x

y

z

P

P z

y

xx




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.

Contributions




Numerical simulation

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 (Uniform or normal distribution).

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.




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





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.





Qualitative results (single FH dof)

Probability distribution

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

single dof value

pro

ba

bil

ity

Simulation

Normaldistribution

Number of simulations N=1500




Simulation tool

A dedicated numerical analisys tool has been designed in MS VBA for Excel environment.




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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D....

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