Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T20:54:11.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniqueness in the determination of loads in multi-span beams and plates

Published online by Cambridge University Press:  10 January 2018

ALEXANDRE KAWANO  and
ANTONINO MORASSI
ALEXANDRE KAWANO
Affiliation:
Escola Politecnica, University of Sao Paulo, Sao Paulo, Brazil email: [email protected]
ANTONINO MORASSI
Affiliation:
Polytechnic Department of Engineering and Architecture, Università degli Studi di Udine, Udine, Italy email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Most of the results available on the inverse problem of determining loads acting on elastic beams or plates under transverse vibration refer to single beam or single plate. In this paper, we consider the determination of sources in multi-span systems obtained by connecting either two Euler–Bernoulli elastic beams or two rectangular Kirchhoff–Love elastic plates. The material of the structure is assumed to be homogeneous and isotropic. The transverse load is of the form g(t)f(x), where g(t) is a known function of time and f(x) is the unknown term depending on the position variable x. Under slight a priori assumptions, we prove a uniqueness result for f(x) in terms of observations of the dynamic response taken at interior points of the structure in an arbitrary small interval of time. A numerical implementation of the method is included to show the possible application of the results in the practical identification of the source term.

Keywords

Mechanics of deformable bodiesEuler–Bernoulli equationKirchhoff–Love plate equationInverse problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 1 , February 2019 , pp. 176 - 195
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

A. Kawano acknowledges the support provided by CNPq Proc. 304972/2013-4 and Fapesp Proc. 2017/06452-1; 2017/07189-2. A. Morassi gratefully acknowledges the financial support of the National Research Project PRIN 2015TTJN95 “Identification and monitoring of complex structural systems”.

References

[1] Clough, R. W. & Penzien, J (1995) Dynamics of Structures, 3rd ed., Computers & Structures, Inc., Berkeley, CA.Google Scholar
[2] Hughes, O. F. (1995) Ship Structural Design, 3rd ed., John Wiley & Sons, Jersey City, New Jersey.Google Scholar
[3] Hasanov, A. & Kawano, A. (2016) Identification of an unknown spatial load distribution in a vibrating Euler–Bernoulli beam from limited measured data. Inverse Prob. 32 (5), 055004.Google Scholar
[4] Yamamoto, M. (1996) Determination of forces in vibrations of beams and plates by pointwise and line observations. J. Inverse Ill Posed Prob. 4 (5), 437–457.Google Scholar
[5] Kawano, A. (2015) Uniqueness results in the identification of distributed sources over Germain-Lagrange plates by boundary measurements. Appl. Anal. 95 (9), 2004–2016.Google Scholar
[6] Nicaise, S. & Zaïr, O. (2004) Determination of point sources in vibrating beams by boundary measurements: Identifiability, stability, and reconstruction results. Electron. J. Differ. Equ. 2004 (20), 1–17.Google Scholar
[7] Zaïr, O. (2013) Determination of point sources in vibrating plate by boundary measurements. Appl. Anal. 92 (10), 2061–2075.Google Scholar
[8] Wang, Y. H. (2007) Global uniqueness and stability for an inverse plate problem. J. Optim. Theor. Appl. 132 (1), 161–173.Google Scholar
[9] Alves, C., Silvestre, A. L., Takahashi, T. & Tucsnak, M. (2009) Solving inverse source problems using observability. Applications to the Euler–Bernoulli plate equation. SIAM J. Control Optim. 48 (3), 1632–1659.Google Scholar
[10] Nicaise, S. & Zaïr, O. (2003) Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees. Revista Matemática Complutense 16 (1), 151–178.Google Scholar
[11] Kawano, A. (2013) Uniqueness in the determination of vibration sources in rectangular Germain-Lagrange plates using displacement measurements over line segments with arbitrary small length. Inverse Prob. 29 (8), 085002.Google Scholar
[12] Timoshenko, S. P. & Woinowsky-Krieger, S. (1970) Theory of Plates and Shells, 2nd ed., McGraw-Hill, Singapore.Google Scholar
[13] Kahane, J.-P. (1957) Sur les fonctions moyenne-périodiques bornées. Annales de l'Institut Fourier 7, 293–314.Google Scholar
[14] Olevskii, A. & Ulanovskii, A. (2009) Interpolation in Bernstein and PaleyWiener spaces. J. Funct. Anal. 256 (10), 3257–3278.Google Scholar
[15] Kawano, A. & Zine, A. (2011) Uniqueness and nonuniqueness results for a certain class of almost periodic distributions. SIAM J. Math. Anal. 43 (1), 135–152.Google Scholar