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Università degli Studi di Brescia Dipartimento 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 th International Conference on Experimental Mechanics 29 August - 2 September 2004, Politecnico di Bari, Bari, Italy

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 2/22 Foreword Statistical method for the prediction of tolerancing effects in design A. 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)

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 3/22 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 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 Instrument Introduction Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 4/22 The Low Frequency Instrument Overall Sight Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan BEU FPU Waveguides Support structures 1.5 m

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 5/22 The Low Frequency Instrument Overall Sight Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan Horn LFI HFI Main Frame Waveguides Support Structure Waveguides BEU Bipodes

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 6/22 The Low Frequency Instrument Focal Plane Unit FPU Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan Bipodes Interfaces to satellite 30 GHz feed horns 70 GHz feed horns 44 GHz feed horns HFI

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 7/22 The Low Frequency Instrument Focal Plane Unit FPU Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan Feed Horns (1) Main Frame Top (2) Main Frame (3) Bipodes (4) Interfaces to satellite

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 8/22 x y z 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 Geometrical Approach Assembly Modelling Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 9/22 Geometrical Approach Uncertainty Contributions Dimensional tolerances Geometrical tolerances Fit tolerances Thermal distortions* Changes of position and orientation of each local reference frame Fit modalities Changes of position and orientation of each local coupling reference frame *thermal effects are here disregarded Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 10/22 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 Geometrical Approach Homogeneous Matrixes Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 11/22 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 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 Geometrical Approach Homogeneous Matrixes Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 12/22 x y z 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 Geometrical Approach Homogeneous Matrixes Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 13/22 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 x x Tolerancing Specifications Effects Reference frames definition Drawings are courtesy of Alenia S.p.A. - LABEN Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 14/22 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 Effects Reference Frames Definition Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 15/22 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 Effects Uncertainty Contributions x y z P x y z P P z y x x Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 16/22 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 Effects Uncertainty Contributions Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 17/22 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 Analysis Monte Carlo Simulation Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 18/22 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 V PC1 vector. The three found rotations and the the three found translations give the FH space location. Numerical Statistical Analysis Monte Carlo Simulation Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 19/22 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 Statistical Analysis Monte Carlo Simulation Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 20/22 Qualitative Results A FH d.o.f. Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 21/22 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 Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan Method

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Università degli Studi di Brescia Dipartimento di Ingegneria Meccanica 22/22... thank you Authors are grateful to: ALENIA S.p.A. - LABEN (Vimodrone, Italy) CENTROTECNICA S.a.S. (Milano, Italy) Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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