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**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) LFI – Low Frequency Interface Good morning, ladies and gentlemen. Inserire la prima slide My name is Valerio Villa, I teach mechanics at the Faculty of Engineering of the University of Brescia. You will not see my name between the authors of this memory: in fact I work in the same research group as the authors, but I am not involved in this research. In this time in our University there is plenty of didactic work, due both to examinations time and switching from old to new graduation rules. As a consequence the authors were not allowed to come here, and I was charged with this presentation. Due to my lack of experience with the subject, I am afraid you will not enjoy it as much as I have enjoyed touring around this very nice town. ICEM th International Conference on Experimental Mechanics 29 August - 2 September 2004, Politecnico di Bari, Bari, Italy

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**Uncertainty estimation: statistical problem **

Foreword 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) Anyhow I’ll do my best to be clear and to answer your questions, if they will not be too difficult. The research I am presenting was caused by a problem occurred to a firm located nearby Milan and operating in the aerospace area, the Laben (Alenia group). They are involved in the development of an instrument … inserire e commentare la seconda slide (testo) e la terza presentata come figura di riferimento della precedente Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**The Low Frequency Instrument**

Introduction 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. Anyhow I’ll do my best to be clear and to answer your questions, if they will not be too difficult. The research I am presenting was caused by a problem occurred to a firm located nearby Milan and operating in the aerospace area, the Laben (Alenia group). They are involved in the development of an instrument … inserire e commentare la seconda slide (testo) e la terza presentata come figura di riferimento della precedente Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**The Low Frequency Instrument**

Overall Sight BEU FPU Waveguides structures Support 1.5 m Anyhow I’ll do my best to be clear and to answer your questions, if they will not be too difficult. The research I am presenting was caused by a problem occurred to a firm located nearby Milan and operating in the aerospace area, the Laben (Alenia group). They are involved in the development of an instrument … inserire e commentare la seconda slide (testo) e la terza presentata come figura di riferimento della precedente Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**The Low Frequency Instrument**

Overall Sight Horn LFI HFI Main Frame Waveguides Support Structure BEU Bipodes Anyhow I’ll do my best to be clear and to answer your questions, if they will not be too difficult. The research I am presenting was caused by a problem occurred to a firm located nearby Milan and operating in the aerospace area, the Laben (Alenia group). They are involved in the development of an instrument … inserire e commentare la seconda slide (testo) e la terza presentata come figura di riferimento della precedente Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**The Low Frequency Instrument**

Focal Plane Unit FPU Bipodes Interfaces to satellite 30 GHz feed horns 70 GHz feed horns 44 GHz HFI Anyhow I’ll do my best to be clear and to answer your questions, if they will not be too difficult. The research I am presenting was caused by a problem occurred to a firm located nearby Milan and operating in the aerospace area, the Laben (Alenia group). They are involved in the development of an instrument … inserire e commentare la seconda slide (testo) e la terza presentata come figura di riferimento della precedente Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**The Low Frequency Instrument**

Focal Plane Unit FPU Feed Horns (1) Main Frame Top (2) Main Frame (3) Bipodes (4) Interfaces to satellite Anyhow I’ll do my best to be clear and to answer your questions, if they will not be too difficult. The research I am presenting was caused by a problem occurred to a firm located nearby Milan and operating in the aerospace area, the Laben (Alenia group). They are involved in the development of an instrument … inserire e commentare la seconda slide (testo) e la terza presentata come figura di riferimento della precedente Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Geometrical Approach 1 - Feed Horn 2 - Frame top (OMT for 70 GHz FH)**

Assembly Modelling 1 - Feed Horn 2 - Frame top (OMT for 70 GHz FH) 3 - Main Frame 4 - Exapode x y z 4 3 2 xA4 yA4 zA4 xA3 yA3 zA3 xA2 yA2 zA2 xA1 yA1 zA1 xL2 yL2 zL2 xL1 yL1 zL1 xL3 yL3 zL3 xL4 yL4 zL4 1 PC PC – Phase Center (a theoretical Point) basta commentarla, eventualmente tornando alla 5 per mostrare le relazioni tra i blocchi logici e le parti meccaniche. Basically the blue axes are referred to the supporting structure, while the red ones are refferred to the supported item. In case of perfect coupling the red and the blue axes are superimposed. As a consequence of the coupling errors the red axes are displaced from the blue ones. In other terms, at any interface level the blue axes are integral with the preceding item, while the red ones are integral with the following one. Local reference frame Coupling reference frame Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Dimensional tolerances Geometrical tolerances Fit tolerances **

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 Basta commentarla. Basically, from the red axes 3) to the blue axes 3) the coupling errors are relevant, while from the blue axes 3) to the red axes 2) the thermo-structural deformations are relevant. The coupling errors are expected to be in the range of the dimensional and geometric tolerances, while the thermostructural deformations can be evaluated with usual FEM procedures. Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Geometrical Approach Homogeneous Matrixes 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 xB yB zB OB xA yA zA OA P To describe and connect the various error contribution a matrix based method has been used. Such method has been developed and tested by our university fellows of the robotic department, to whom we are indebted. As per our information it is original, but I am not acquainted enough with the mathematical aspect to go deeper into it. Were anybody interested, I may supply the address of my proper colleague. Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Geometrical Approach 4 3 2 1 PC ML1 MA1 ML2 MA2 ML3 MA3 ML4 MA4**

Homogeneous Matrixes x y z 4 3 2 1 xL2 yL2 zL2 xL1 yL1 zL1 xL3 yL3 zL3 xL4 yL4 zL4 xA4 yA4 zA4 xA3 yA3 zA3 xA2 yA2 zA2 xA1 yA1 zA1 PC ML1 MA1 ML2 MA2 ML3 MA3 ML4 MA4 Position vector of the PC in the L1 ref. frame 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 ... A set of n. 8 matrices is listed, fit to describe all the error causes in the single feed horn supporting chain Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Geometrical Approach 4 3 2 1 PC ML1 MA1 ML2 MA2 ML3 MA3 ML4 MA4**

Homogeneous Matrixes x y z 4 3 2 1 xL2 yL2 zL2 xL1 yL1 zL1 xL3 yL3 zL3 xL4 yL4 zL4 xA4 yA4 zA4 xA3 yA3 zA3 xA2 yA2 zA2 xA1 yA1 zA1 PC ML1 MA1 ML2 MA2 ML3 MA3 ML4 MA4 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 What is most interesting: operating properly the error matrices, a new matrix is obteined which describes the integrated effect of all the considered errors: this procedure really simulates the error propagation along the structural chain Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Tolerancing Specifications Effects**

Reference frames definition FH 44 GHz x y z P Drawings are courtesy of Alenia S.p.A. - LABEN 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 slides 13 alla fine – basta commentarle seguendo quello che c’è scritto. Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Tolerancing Specifications Effects**

Reference Frames Definition 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 slides 13 alla fine – basta commentarle seguendo quello che c’è scritto. Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Tolerancing Specifications Effects**

Uncertainty Contributions 1 3 2 x y z P translation along x of the L1 origin rotations around A1 z & y axis 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) slides 13 alla fine – basta commentarle seguendo quello che c’è scritto Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Tolerancing Specifications Effects**

Uncertainty Contributions 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. slides 13 alla fine – basta commentarle seguendo quello che c’è scritto Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Numerical Statistical Analysis**

Monte Carlo 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 (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. slides 13 alla fine – basta commentarle seguendo quello che c’è scritto Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Numerical Statistical Analysis**

Monte Carlo Simulation 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. slides 13 alla fine – basta commentarle seguendo quello che c’è scritto Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Numerical Statistical Analysis**

Monte Carlo Simulation 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. slides 13 alla fine – basta commentarle seguendo quello che c’è scritto Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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Qualitative Results A FH d.o.f. slides 13 alla fine – basta commentarle seguendo quello che c’è scritto Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**Conclusions and Discussion**

Method 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 Anyhow I’ll do my best to be clear and to answer your questions, if they will not be too difficult. The research I am presenting was caused by a problem occurred to a firm located nearby Milan and operating in the aerospace area, the Laben (Alenia group). They are involved in the development of an instrument … inserire e commentare la seconda slide (testo) e la terza presentata come figura di riferimento della precedente Statistical method for the prediction of tolerancing effects in design A. Magalini, D. Vetturi, L. Pagan

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**... 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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