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Technical Briefs

The Multimodal Dynamics of a Walnut Tree: Experiments and Models

[+] Author and Article Information
M. Rodriguez

Department of Mechanics, LadHyX, Ecole Polytechnique, CNRS, 91128 Palaiseau, France; INRA, UMR 547 PIAF, 63100 Clermont-Fd, France

S. Ploquin

INRA, UMR 547 PIAF, 63100 Clermont-Fd, France; Université Blaise Pascal, UMR 547 PIAF, 63100 Clermont Fd, France

B. Moulia

INRA, UMR 547 PIAF, 63100 Clermont-Fd, France; Université Blaise Pascal, UMR 547 PIAF, 63100 Clermont Fd, Francemoulia@clermont.inra.fr

E. de Langre1

Department of Mechanics, LadHyX, Ecole Polytechnique, CNRS, 91128 Palaiseau, Francedelangre@ladhyx.polytechnique.fr

1

Corresponding author.

J. Appl. Mech 79(4), 044505 (May 11, 2012) (5 pages) doi:10.1115/1.4005553 History: Received November 25, 2010; Revised September 08, 2011; Posted January 30, 2012; Published May 11, 2012; Online May 11, 2012

The dynamics of a walnut tree is investigated in order to better understand the mechanical interaction of trees with wind. Experimental data on the vibrational modes of the tree are obtained, using pull and release tests. These results are then used to validate a previous analytical approach that predicts the organization of modal frequencies as a function of two allometry parameters describing the branched tree geometry. In addition to these experimental results, vibration modes are also obtained from a finite element computation using a detailed digitization of the tree geometry. The comparison between experiments, computation, and the simple analytical approach confirm the specific organization of modes of such branched trees, with a high modal density and a spatial localization. Then the possible biological importance of this organization and the potential biomimetic applications are outlined.

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Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Idealized branched tree. Top: a branched tree and self-similar subsets. Center: computed modes [14]. Bottom: schematic representation of the mode shapes corresponding to groups of different orders (redrawn from [14]).

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Figure 2

Experimental setup. Left: the young walnut seen from x-direction. Right: positions of the excitation (point E) and location of sensors (point S1, S2, and S3).

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Figure 3

Displacements at sensors S1, S2, and S3 during a pull and release test

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Figure 4

Normalized PSD of responses for each sensor: S1 (top), S2 (center), and S3 (bottom). Peaks are identified with arrows, secondary peaks with dashed arrows, and labeled according to the frequency range. Higher modes are only seen in the displacement of sensors in the branches.

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Figure 5

Digitized geometry of the walnut tree used for the computation of the allometry parameters λ and β in Sec. 3 and for the finite element computations in Sec. 4

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Figure 6

Normalized frequencies of modes of the walnut tree, as a function of the group of modes. (▪) Experimental data from Table 1. (—) Prediction using Eq. 1 with λ and β derived from the actual tree geometry. The gray area corresponds to the 90% confidence interval on the geometrical parameters.

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Figure 7

Normalized computed eigenfrequencies of the walnut. Frequencies are grouped according to the main localization of modal deformations.

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Figure 8

Frequency ratio of modes of different groups from experiments (▪), finite-element computations, (□) and from the prediction using the allometry parameters (—)

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