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(American Journal of Botany. 2008;95:1523-1537.) doi: 10.3732/ajb.0800161 © 2008 Botanical Society of America, Inc.

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A scaling law for the effects of architecture and allometry on tree vibration modes suggests a biological tuning to modal compartmentalization¹

² Department of Mechanics, LadHyX, Ecole Polytechnique-CNRS, 91128 Palaiseau, France ³ UMR547 PIAF, INRA, Univ Blaise Pascal, F-63100 Clermont Ferrand

ABSTRACT

Wind is a major ecological factor for plants and a major economical factor for forestry. Mechanical analyses have revealed that the multimodal dynamic behavior of trees is central to wind–tree interactions. Moreover, the trunk and branches influence dynamic modes, both in frequency and location. Because of the complexity of tree architecture, finite element models (FEMs) have been used to analyze such dynamics. However, these models require detailed geometric and architectural data and are tree-specific—two major restraints for their use in most ecological or biological studies. In this work, closed-form scaling laws for modal characteristics were derived from the dimensional analysis of idealized fractal trees that sketched the major architectural and allometrical regularities of real trees. These scaling laws were compared to three-dimensional FEM modal analyses of two completely digitized trees with maximal architectural contrast. Despite their simplifying hypotheses, the models explained most of the spatiotemporal characteristics of modes that involved the trunk and branches, especially for sympodial trees. These scaling laws reduce the tree to (1) a fundamental frequency and (2) one architectural and three biometrical parameters. They also give quantitative insights into the possible biological control of wind excitability of trees through architecture and allometries.

Key Words: allometry • biomechanics • dimensional analysis • dynamics • frequency • model • scaling • tree architecture • vibrational properties • wind

Received for publication 7 May 2008. Accepted for publication 9 September 2008.

FOOTNOTES

¹ The authors dedicate this article to the late Dr. H. Sinoquet, who pioneered the use of 3D digitizing and virtual plants for physical issues related to plant biology and who left us too early. They thank Drs. H. Sinoquet and D. Sellier for access to the digitized tree databases and N. Dones for teaching the basics of MTG handling and of tree architecture. Thanks also to Dr. B. Roman for pointing out the similarities in the frequency ranges of very distinct trees and the two anonymous reviewers and Dr. B. E. Hazen for providing helpful suggestions to revise the manuscript and improve the English. This work was supported by ANR grant ANR-06-BLAN-0210-02 "Chêne-Roseau".

⁴ Author for correspondence (e-mail: moulia{at}clermont.inra.fr)

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