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A Semi-analytical Approach to Approximate Chattering Time of Rocking Structures

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Mathematical Modeling in Cultural Heritage (MACH 2021)

Part of the book series: Springer INdAM Series ((SINDAMS,volume 55))

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Abstract

This paper investigates the seismic behaviour of a freestanding rigid rocking block when subjected to low amplitude sinusoidal ground excitations. An important scenario in such cases is the complete chattering the block might exhibit. Complete chattering occurs when the block undergoes a theoretically infinite sequence of decaying impacts that converge to the state of persistent (continuous) contact in finite time, even under a nonzero ground excitation. This study proposes a semi-analytical approach that approximates the (finite) time required for this to happen, i.e. chattering time. Specifically, this paper provides a detailed description of the semi-analytical scheme and shows the influence of the amplitude of the ground acceleration on the approximation of the chattering time. Importantly, the proposed scheme is based on the realisation that, during chattering, and after a sufficiently large number of impacts, the ratio of the time-intervals of every two consecutive impacts becomes constant and equal to the square of the coefficient of restitution. The proposed semi-analytical approach efficiently approximates chattering time providing a state-of-the-art mathematical formulation of the chattering phenomenon for the rocking problem.

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Author information

Authors and Affiliations

  1. Department of Civil Engineering, ISISE, University of Minho, Guimarães, Portugal

    Anastasios I. Giouvanidis & Paulo B. Lourenço

  2. Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

    Elias G. Dimitrakopoulos

Authors
  1. Anastasios I. Giouvanidis
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  2. Elias G. Dimitrakopoulos
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  3. Paulo B. Lourenço
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Corresponding author

Correspondence to Anastasios I. Giouvanidis .

Editor information

Editors and Affiliations

  1. Istituto per le Applicazioni del Calcolo "Mauro Picone" (IAC), Consiglio Nazionale delle Ricerche (CNR), Roma, Italy

    Gabriella Bretti

  2. Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy

    Cecilia Cavaterra

  3. Dipartimento di Architettura, Università degli Studi di Sassari, Alghero, Sassari, Italy

    Margherita Solci

  4. Institute of Applied Mathematics and Information Technologies (IMATI), Consiglio Nazionale delle Ricerche (CNR), Genova, Italy

    Michela Spagnuolo

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Giouvanidis, A.I., Dimitrakopoulos, E.G., Lourenço, P.B. (2023). A Semi-analytical Approach to Approximate Chattering Time of Rocking Structures. In: Bretti, G., Cavaterra, C., Solci, M., Spagnuolo, M. (eds) Mathematical Modeling in Cultural Heritage. MACH 2021. Springer INdAM Series, vol 55. Springer, Singapore. https://doi.org/10.1007/978-981-99-3679-3_8

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