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Dynamic Modelling of a Controlled Orthotropic Plate: Analytic and Data-Driven Approaches in the Frequency Domain

Alexander Zuyev, Francesco Pellicano, Antonio Zippo, Giovanni Iarriccio

TL;DR

This work addresses frequency-domain modelling of vibrations in an orthotropic Kirchhoff plate with a laminated piezosensor under lumped actuation. It develops an analytic modal framework using separation of variables and a Krylov-function expansion, projecting onto a Galerkin basis to obtain a finite-dimensional system and a closed-form transfer function $H_{\mathcal N}(s)$ in terms of modal data and $\lambda_n$. Experimental validation on a three-layer graphite-epoxy plate demonstrates good agreement between measured modal frequencies and the model predictions, and the model-based transfer function reproduces the observed frequency response over the studied range. The work lays groundwork for data-driven reduced-order modeling and future application of the Loewner framework for identification in practical settings.

Abstract

This paper is devoted to the mathematical modelling of a vibrating orthotropic plate equipped with a laminated piezosensor, under the influence of a lumped force actuation. We employ the Kirchhoff plate theory to derive the corresponding partial differential equation, assuming free boundary conditions. Analytical solutions for this boundary value problem are explored in the form of series expansions, using products of Krylov functions. Utilizing Galerkin's method, this mathematical model is transformed into an infinite-dimensional control system characterized by modal coordinates. The transfer function of such a system is explicitly evaluated in the single-input single-output case. The computation of coefficients for finite-dimensional approximate systems is formalized in an algorithm with an arbitrary number of degrees of freedom. Our numerical study confirms that the modeled input-output behavior shows acceptable agreement over the given frequency range.

Dynamic Modelling of a Controlled Orthotropic Plate: Analytic and Data-Driven Approaches in the Frequency Domain

TL;DR

This work addresses frequency-domain modelling of vibrations in an orthotropic Kirchhoff plate with a laminated piezosensor under lumped actuation. It develops an analytic modal framework using separation of variables and a Krylov-function expansion, projecting onto a Galerkin basis to obtain a finite-dimensional system and a closed-form transfer function in terms of modal data and . Experimental validation on a three-layer graphite-epoxy plate demonstrates good agreement between measured modal frequencies and the model predictions, and the model-based transfer function reproduces the observed frequency response over the studied range. The work lays groundwork for data-driven reduced-order modeling and future application of the Loewner framework for identification in practical settings.

Abstract

This paper is devoted to the mathematical modelling of a vibrating orthotropic plate equipped with a laminated piezosensor, under the influence of a lumped force actuation. We employ the Kirchhoff plate theory to derive the corresponding partial differential equation, assuming free boundary conditions. Analytical solutions for this boundary value problem are explored in the form of series expansions, using products of Krylov functions. Utilizing Galerkin's method, this mathematical model is transformed into an infinite-dimensional control system characterized by modal coordinates. The transfer function of such a system is explicitly evaluated in the single-input single-output case. The computation of coefficients for finite-dimensional approximate systems is formalized in an algorithm with an arbitrary number of degrees of freedom. Our numerical study confirms that the modeled input-output behavior shows acceptable agreement over the given frequency range.
Paper Structure (6 sections, 1 theorem, 36 equations, 2 figures, 1 table)

This paper contains 6 sections, 1 theorem, 36 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Controlled Kirchhoff plate model
  3. Separation of variables and spectral problem
  4. Transfer function
  5. Experimental results and numerical simulations
  6. Conclusion

Key Result

Proposition 1

The solutions to boundary value problem BVP satisfy the following properties.

Figures (2)

  • Figure 1: Experimental transfer function.
  • Figure 2: Theoretical transfer function for Galerkin's approximation.

Theorems & Definitions (1)

  • Proposition 1