Startseite Simultaneous determination of mass density and flexural rigidity of the damped Euler–Bernoulli beam from two boundary measured outputs
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Simultaneous determination of mass density and flexural rigidity of the damped Euler–Bernoulli beam from two boundary measured outputs

  • Cristiana Sebu ORCID logo EMAIL logo
Veröffentlicht/Copyright: 25. Oktober 2022
Journal of Inverse and Ill-posed Problems
Aus der Zeitschrift Journal of Inverse and Ill-posed Problems Band 30 Heft 6

Abstract

In this paper, we study the inverse coefficient problem of identifying both the mass density ρ ⁢ ( x ) > 0 and flexural rigidity r ⁢ ( x ) > 0 of a damped Euler–Bernoulli (cantilever) beam governed by the equation ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ) , subject to boundary conditions u ⁢ ( 0 , t ) = u x ⁢ ( 0 , t ) = 0 , u x ⁢ x ⁢ ( ℓ , t ) = 0 , - ( r ⁢ ( x ) ⁢ u x ⁢ x ⁢ ( x , t ) ) x | x = ℓ = g ⁢ ( t ) , from the available measured boundary deflection ν ⁢ ( t ) := u ⁢ ( ℓ , t ) and rotation θ ⁢ ( t ) := u x ⁢ ( ℓ , t ) at the free end of the beam. The distinctive feature of the considered inverse coefficient problem is that not one, but two Neumann-to-Dirichlet operators have to be formally defined. The inverse problem is hence formulated as a system of nonlinear Neumann-to-Dirichlet operator equations with the right-hand sides consisting of the measured outputs. As a natural consequence of this approach, a vector-form Tikhonov functional is introduced whose components are squares of the L 2 -norm differences between predicted and measured outputs. We then prove existence of a quasi-solution of the inverse problem and derive explicit gradient formulae for the Fréchet derivatives of both components of the Tikhonov functional. These results are instrumental to any gradient based algorithms for reconstructing the two unknown coefficients of the considered damped Euler–Bernoulli beam.

Keywords: Simultaneous determination of coefficients; damped Euler–Bernoulli beam; measured boundary data; Neumann-to-Dirichlet mappings; existence of a quasi-solution; Fréchet differentiability; gradient formula
MSC 2010: 74G75; 65M32; 49J20

Acknowledgements

I am grateful to Professor Alemdar Hasanov for introducing me to this problem during his research visit at the University of Malta in January 2022, for his encouragement to work on it, and for his valuable scientific feedback and continued support.

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Received: 2022-05-30
Revised: 2022-06-20
Accepted: 2022-07-04
Published Online: 2022-10-25
Published in Print: 2022-12-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Parameter identification for portfolio optimization with a slow stochastic factor
  3. Simultaneous inversion of a fractional order and a space source term in an anomalous diffusion model
  4. On the inverse gravimetry problem with minimal data
  5. On stable parameter estimation and short-term forecasting with quantified uncertainty with application to COVID-19 transmission
  6. Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems
  7. Convergence analysis of iteratively regularized Gauss–Newton method with frozen derivative in Banach spaces
  8. Identification of an unknown flexural rigidity of a cantilever Euler–Bernoulli beam from measured boundary deflection
  9. Hadamard’s example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation
  10. Ill-posed problems and the conjugate gradient method: Optimal convergence rates in the presence of discretization and modelling errors
  11. Simultaneous determination of mass density and flexural rigidity of the damped Euler–Bernoulli beam from two boundary measured outputs
https://doi.org/10.1515/jiip-2022-0044
Schlagwörter für diesen Artikel
Simultaneous determination of coefficients; damped Euler–Bernoulli beam; measured boundary data; Neumann-to-Dirichlet mappings; existence of a quasi-solution; Fréchet differentiability; gradient formula

Artikel in diesem Heft

  1. Frontmatter
  2. Parameter identification for portfolio optimization with a slow stochastic factor
  3. Simultaneous inversion of a fractional order and a space source term in an anomalous diffusion model
  4. On the inverse gravimetry problem with minimal data
  5. On stable parameter estimation and short-term forecasting with quantified uncertainty with application to COVID-19 transmission
  6. Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems
  7. Convergence analysis of iteratively regularized Gauss–Newton method with frozen derivative in Banach spaces
  8. Identification of an unknown flexural rigidity of a cantilever Euler–Bernoulli beam from measured boundary deflection
  9. Hadamard’s example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation
  10. Ill-posed problems and the conjugate gradient method: Optimal convergence rates in the presence of discretization and modelling errors
  11. Simultaneous determination of mass density and flexural rigidity of the damped Euler–Bernoulli beam from two boundary measured outputs
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