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Description of Plates with the Matrix Klein–Gordon Equation.

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Title: Description of Plates with the Matrix Klein–Gordon Equation.
Authors: Kniazeva, K. S.1 (AUTHOR) knyazevaks05@gmail.com, Shelest, E. L.1 (AUTHOR), Shanin, A. V.1 (AUTHOR)
Source: Acoustical Physics. Oct2025, Vol. 71 Issue 5, p675-682. 8p.
Subjects: Klein-Gordon equation, Structural plates, Vibration (Mechanics), Waveguides, Finite element method, Differential equations
Abstract: The matrix Klein–Gordon equation (MKGE) describes waveguides with a discrete structure of their cross section. It may also be considered as a continuous waveguide model constructed by applying the waveguide finite element method. In this paper, the possibility of describing the bending vibrations of a thin plate with MKGE is studied. The problem is not trivial, as MKGE contains second-order time and coordinate derivatives, and bending vibrations are described by an equation with a fourth-order coordinate derivative. In this paper, matrix Klein–Gordon equations of different dimensions are derived for a plate. It is shown that a linear approximation of plate deformation over a cross section leads to overestimated plate stiffness values, but more complicated models give correct values. [ABSTRACT FROM AUTHOR]
Copyright of Acoustical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Description of Plates with the Matrix Klein–Gordon Equation.
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  Data: <searchLink fieldCode="JN" term="%22Acoustical+Physics%22">Acoustical Physics</searchLink>. Oct2025, Vol. 71 Issue 5, p675-682. 8p.
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  Data: <searchLink fieldCode="DE" term="%22Klein-Gordon+equation%22">Klein-Gordon equation</searchLink><br /><searchLink fieldCode="DE" term="%22Structural+plates%22">Structural plates</searchLink><br /><searchLink fieldCode="DE" term="%22Vibration+%28Mechanics%29%22">Vibration (Mechanics)</searchLink><br /><searchLink fieldCode="DE" term="%22Waveguides%22">Waveguides</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+equations%22">Differential equations</searchLink>
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  Data: The matrix Klein–Gordon equation (MKGE) describes waveguides with a discrete structure of their cross section. It may also be considered as a continuous waveguide model constructed by applying the waveguide finite element method. In this paper, the possibility of describing the bending vibrations of a thin plate with MKGE is studied. The problem is not trivial, as MKGE contains second-order time and coordinate derivatives, and bending vibrations are described by an equation with a fourth-order coordinate derivative. In this paper, matrix Klein–Gordon equations of different dimensions are derived for a plate. It is shown that a linear approximation of plate deformation over a cross section leads to overestimated plate stiffness values, but more complicated models give correct values. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Acoustical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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        Type: general
      – SubjectFull: Structural plates
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      – SubjectFull: Vibration (Mechanics)
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      – SubjectFull: Waveguides
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              Text: Oct2025
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