<abstract xmlns="http://www.w3.org/1999/xhtml">Two main shortcomings of common formulations, encountered in the literature concerning the linear problems of structural dynamics are revealed: the implicit, not discussed, postulation, of the use of Kelvin – Voigt constitutive laws (which is often infirmed by experience) and the calculation difficulties involved by the attempts to use other constitutive laws. In order to overcome these two categories of shortcomings, the use of the bilateral Laplace – Carson transformation is adopted. Instead of the dependence on time, <italic>t</italic>, of a certain function <italic>f</italic> (<italic>t</italic>), the dependence of its <italic>image f#</italic> (<italic>p</italic>) on the complex parameter <italic>p</italic> = <italic>χ</italic> + i<italic>ω</italic> (<italic>ω</italic>: circular frequency) will occur. This leads to the formulation of associated non-classical eigenvalue problems. The basic relations satisfied by the eigenvalues <bold><italic>λ</italic></bold><italic>r</italic>#(<italic>p</italic>) and the eigenvectors <italic>v</italic><italic>r</italic>#(<italic>p</italic>) of dynamic systems are examined (among other, the property of orthogonality of eigenvectors is replaced by the property of <italic>pseudo-orthogonality</italic>). The case of points <italic>p</italic> = <italic>p’</italic>, where multiple eigenvalues occur and where, as a rule, chains of principal vectors are to be considered, is discussed. An illustrative case, concerning a non-classical eigenvalue problem, is presented. Plots of variation along the <italic>ω</italic> axis, for the real and imaginary components of eigenvalues and eigenvectors, are presented. A brief final discussion closes the paper.</abstract>

Two main shortcomings of common formulations, encountered in the literature concerning the linear problems of structural dynamics are revealed: the implicit, not discussed, postulation, of the use of Kelvin – Voigt constitutive laws (which is often infirmed by experience) and the calculation difficulties involved by the attempts to use other constitutive laws. In order to overcome these two categories of shortcomings, the use of the bilateral Laplace – Carson transformation is adopted. Instead of the dependence on time, t, of a certain function f (t), the dependence of its image f# (p) on the complex parameter p = χ + iω (ω: circular frequency) will occur. This leads to the formulation of associated non-classical eigenvalue problems. The basic relations satisfied by the eigenvalues λr#(p) and the eigenvectors vr#(p) of dynamic systems are examined (among other, the property of orthogonality of eigenvectors is replaced by the property of pseudo-orthogonality). The case of points p = p’, where multiple eigenvalues occur and where, as a rule, chains of principal vectors are to be considered, is discussed. An illustrative case, concerning a non-classical eigenvalue problem, is presented. Plots of variation along the ω axis, for the real and imaginary components of eigenvalues and eigenvectors, are presented. A brief final discussion closes the paper.

An Approach to Some Non-Classical Eigenvalue Problems of Structural Dynamics

Dr., M., Academy of Technical Sciences of Romania

Dr., senior researcher, URBAN – INCERC, Romania.

Modelling in Civil Environmental Engineering

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"An Approach to Some Non-Classical Eigenvalue Problems of Structural Dynamics"}

Two main shortcomings of common formulations, encountered in the literature concerning the linear problems of structural dynamics are revealed: the implicit,...

An Approach to Some Non-Classical Eigenvalue Problems of Structural Dynamics

Published Online: Jan 22, 2016

Page range: 21 - 32

DOI: https://doi.org/10.1515/mmce-2015-0017

Keywords
Non-classical eigenvalue problems, Laplace – Carson transformation, pseudo-orthogonality, multiple eigenvalues, singular eigenvectors

© 2015 Horea Sandi et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

An Approach to Some Non-Classical Eigenvalue Problems of Structural Dynamics

Published Online: Jan 22, 2016

Page range: 21 - 32

DOI: https://doi.org/10.1515/mmce-2015-0017

KeywordsNon-classical eigenvalue problems, Laplace – Carson transformation, pseudo-orthogonality, multiple eigenvalues, singular eigenvectors

© 2015 Horea Sandi et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Non-classical eigenvalue problems, Laplace – Carson transformation, pseudo-orthogonality, multiple eigenvalues, singular eigenvectors