Bernoulli's transformation of the response of an elastic body and damping

Vasilios Nikitas, Alexandros Nikitas, Nikolaos Nikitas

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Bernoulli's transformation and the related separation of variables method or modal analysis as classically applied to the partial differential equation of motion of an elastic continuum will always conclude an undamped response. However, this conclusion lacks reliability, since the underlying analysis assumes either integralwise differentiability (i.e. differentiation and integration signs are interchangeable) or termwise differentiability (i.e. the derivative of an infinite series of terms equals the sum of the derivatives of the terms) for Bernoulli's transformation, which not only is arbitrary but also is responsible for the undamped response. This paper using Bernoulli's transformation examines an elastic uniform column ruled by the generalized Hooke's law and subjected to axial surface tractions at its free end or a free axial vibration, and shows that the above differentiability assumptions underlying classical analysis are equivalent and actually constitute a limitation to the class of the response functions. Only on this limitation, damping appears to be inconsistent with the elastic column response. Removing the limitation through nontermwise differentiability of Bernoulli's transformation results in a damped response of the elastic column, which indicates that damping actually complies with the generalized Hooke's law as applied to elastic continua.

Original languageEnglish
Title of host publication7th European Conference on Structural Dynamics, EURODYN 2008
EditorsM. J. Brennan
PublisherUniversity of Southampton, Institute of Sound Vibration and Research
ISBN (Electronic)9780854328826
Publication statusPublished - 2008
Externally publishedYes
Event7th European Conference on Structural Dynamics - University of Southampton, Southampton, United Kingdom
Duration: 7 Jul 20089 Jul 2008

Conference

Conference7th European Conference on Structural Dynamics
Abbreviated titleEURODYN 2008
CountryUnited Kingdom
CitySouthampton
Period7/07/089/07/08

Fingerprint

Damping
Derivatives
Modal analysis
Partial differential equations
Equations of motion

Cite this

Nikitas, V., Nikitas, A., & Nikitas, N. (2008). Bernoulli's transformation of the response of an elastic body and damping. In M. J. Brennan (Ed.), 7th European Conference on Structural Dynamics, EURODYN 2008 University of Southampton, Institute of Sound Vibration and Research.
Nikitas, Vasilios ; Nikitas, Alexandros ; Nikitas, Nikolaos. / Bernoulli's transformation of the response of an elastic body and damping. 7th European Conference on Structural Dynamics, EURODYN 2008. editor / M. J. Brennan. University of Southampton, Institute of Sound Vibration and Research, 2008.
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Nikitas, V, Nikitas, A & Nikitas, N 2008, Bernoulli's transformation of the response of an elastic body and damping. in MJ Brennan (ed.), 7th European Conference on Structural Dynamics, EURODYN 2008. University of Southampton, Institute of Sound Vibration and Research, 7th European Conference on Structural Dynamics, Southampton, United Kingdom, 7/07/08.

Bernoulli's transformation of the response of an elastic body and damping. / Nikitas, Vasilios; Nikitas, Alexandros; Nikitas, Nikolaos.

7th European Conference on Structural Dynamics, EURODYN 2008. ed. / M. J. Brennan. University of Southampton, Institute of Sound Vibration and Research, 2008.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Nikitas V, Nikitas A, Nikitas N. Bernoulli's transformation of the response of an elastic body and damping. In Brennan MJ, editor, 7th European Conference on Structural Dynamics, EURODYN 2008. University of Southampton, Institute of Sound Vibration and Research. 2008