1. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 1DESIGN 2004 – DUBROVNIK Planck LFI Alignment Analysis Approach

2. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 2DESIGN 2004 – DUBROVNIK Planck LFI is an instrument designed to be operative in space It is aimed to detect micro waves coming from 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

3. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 3DESIGN 2004 – DUBROVNIK Low Frequency Instrument BEU FPU Waveguides Support structures 1.5 m

4. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 4DESIGN 2004 – DUBROVNIK Horn LFI HFI Main Frame Waveguides Support Structure Waveguides BEU Bipodes Focal Plane Unit

5. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 5DESIGN 2004 – DUBROVNIK Focal Plane Unit CFRP struts Interfaces to satellite 30 GHz feed horns 70 GHz feed horns 44 GHz feed horns HFI

6. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 6DESIGN 2004 – DUBROVNIK

7. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 7DESIGN 2004 – DUBROVNIK 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

8. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 8DESIGN 2004 – DUBROVNIK x y z 4 3 2 1 General structure analysis x L2 y L2 z L2 x L1 y L1 z L1 x L3 y L3 z L3 x L4 y L4 z L4 x A4 y A4 z A4 x A3 y A3 z A3 x A2 y A2 z A2 x A1 y A1 z A1 Local reference frame Coupling reference frame 1 - Feed Horn 2 - Frame top (OMT for 70 GHz FH) 3 - Main Frame 4 - Exapode PC

9. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 9DESIGN 2004 – DUBROVNIK Uncertainty contributions Dimensional tolerances Geometrical tolerances Thermal distortions (elements 3 and 4) Changes of position and orientation of each local reference frame Matching modalities Changes of position and orientation of each local coupling reference frame

10. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 10DESIGN 2004 – DUBROVNIK 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 A B xBxB yByB zBzB OBOB xAxA yAyA zAzA OAOA P

11. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 11DESIGN 2004 – DUBROVNIK M L1 describes the L1 ref. frame in the A1 ref. frame M A1 describes the A1 ref frame in the L2 ref frame M L2 describes the L2 ref. frame in the A2 ref. frame M A2 describes the A2 ref. frame in the L3 ref. frame... x y z 4 3 2 1 x L2 y L2 z L2 x L1 y L1 z L1 x L3 y L3 z L3 x L4 y L4 z L4 x A4 y A4 z A4 x A3 y A3 z A3 x A2 y A2 z A2 x A1 y A1 z A1 PC M L1 M A1 M L2 M A2 M L3 M A3 M L4 M A4 Position vector of the PC in the L1 ref. frame Matrix approach

12. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 12DESIGN 2004 – DUBROVNIK x y z 4 3 2 1 x L2 y L2 z L2 x L1 y L1 z L1 x L3 y L3 z L3 x L4 y L4 z L4 x A4 y A4 z A4 x A3 y A3 z A3 x A2 y A2 z A2 x A1 y A1 z A1 PC M L1 M A1 M L2 M A2 M L3 M A3 M L4 M A4 Position vector of the PC in the L1 ref. frame Position vector of the PC in the absolute ref. frame Position of the PC (x,y,z) FH orientation (angles given following any chosen rule) M Matrix approach

13. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 13DESIGN 2004 – DUBROVNIK Tolerances effects FH 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 x x

14. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 14DESIGN 2004 – DUBROVNIK 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

15. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 15DESIGN 2004 – DUBROVNIK 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 x x

16. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 16DESIGN 2004 – DUBROVNIK 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

17. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 17DESIGN 2004 – DUBROVNIK 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.

18. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 18DESIGN 2004 – DUBROVNIK 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 Numerical simulation

19. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 19DESIGN 2004 – DUBROVNIK 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. Numerical simulation

20. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 20DESIGN 2004 – DUBROVNIK Qualitative results (single FH dof) Number of simulations N=1500

21. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 21DESIGN 2004 – DUBROVNIK Simulation tool A dedicated numerical analisys tool has been designed in MS VBA for Excel environment.

22. Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica A. Magalini, D. Vetturi, D. Cambiaghi Planck LFI Alignment Analysis Approach SLIDE 22DESIGN 2004 – DUBROVNIK... thank you