# A view on parametric uncertainties of structural and vibro-acoustic systems

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## Abstract

A view on uncertainties of the parameters of structural and vibro-acoustic systems A view on parametric uncertainties of structural and vibro-acoustic systems Christophe Lecomte Institute of Sound and Vibration Research (ISVR) and Southampton Statistical Sciences Research Institute (S3RI) University of Southampton Why is uncertainty important? Prediction Design Detection Control High-frequency mid-frequency Multiscale Linear algebra solvers Inverse problems Eigenproblems Pseudospectrum Orthogonal polynomials Special functions November 17, 2011, @Christophe Lecomte A view on parametric uncertainties 2 Uncertainty Plan of talk • Definition of stochastic systems • Propagation of uncertainties • Bayesian identification of uncertain parameters • Matrix point of view November 17, 2011, @Christophe Lecomte A view on parametric uncertainties 3 Zooming in on linear algebra Consider a structural or vibro-acoustic system • Nominally, 𝐀 𝜔 𝒙 𝜔, 0 = 𝐟 𝜔 = 2𝑓 is a frequency parameter – For example, • 𝐊 + i 𝜔 𝐂 – 𝜔2 𝐌 𝒙 𝜔, 0 = 𝐟 Quadratic function of the parameter 𝜔 • 𝐀𝟎 + 𝜔𝐀𝟏 + 𝜔 2𝐀𝟐 + ⋯ 𝒙 𝜔 = 𝒇 𝜔 Polynomial system • i𝜔𝐈 − 𝐁0 − 𝐁𝑖 𝑚 𝑖=1 e −i𝜔𝜏𝑖 𝒙 𝜔 = 𝒇 Non-polynomial (time-delay) system - Dynamic stiffness matrix, 𝐀 - Force vector, 𝒇 - Response vector, 𝒙 - Stiffness matrix, 𝐊 - Damping matrix, 𝐂 - Mass matrix, 𝐌 - Taylor matrices, 𝐀𝑗 - Delay matrices, 𝐁𝑗 November 17, 2011, @Christophe Lecomte A view on parametric uncertainties 4 Stochastic system • If a disturbance affects the matrix [𝐀 𝜔 − 𝐃 𝜔 ]𝒙 𝜔,𝐃 . = 𝒇 and if D(w) can be expressed in terms of parameters 𝑠1, 𝑠2,… 𝐀 𝜔 – 𝑠𝑗 𝑗 𝐃𝑗(𝜔) 𝒙(𝜔, 𝑠1, 𝑠2, … ) = 𝒇 There are then several parameters, 𝜔, 𝑠1, 𝑠2, … • The system is called stocha

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