### 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 language | English |
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Title of host publication | 7th European Conference on Structural Dynamics, EURODYN 2008 |

Editors | M. J. Brennan |

Publisher | University of Southampton, Institute of Sound Vibration and Research |

ISBN (Electronic) | 9780854328826 |

Publication status | Published - 2008 |

Externally published | Yes |

Event | 7th European Conference on Structural Dynamics - University of Southampton, Southampton, United Kingdom Duration: 7 Jul 2008 → 9 Jul 2008 |

### Conference

Conference | 7th European Conference on Structural Dynamics |
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Abbreviated title | EURODYN 2008 |

Country | United Kingdom |

City | Southampton |

Period | 7/07/08 → 9/07/08 |

### Fingerprint

### Cite this

*7th European Conference on Structural Dynamics, EURODYN 2008*University of Southampton, Institute of Sound Vibration and Research.

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*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Bernoulli's transformation of the response of an elastic body and damping

AU - Nikitas, Vasilios

AU - Nikitas, Alexandros

AU - Nikitas, Nikolaos

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Bernoulli's transformation

KW - Damping

KW - Elastic response

KW - Hooke's law

UR - http://www.scopus.com/inward/record.url?scp=84959925488&partnerID=8YFLogxK

M3 - Conference contribution

BT - 7th European Conference on Structural Dynamics, EURODYN 2008

A2 - Brennan, M. J.

PB - University of Southampton, Institute of Sound Vibration and Research

ER -