Tree Physiology 26, 799–806
© 2006 Heron Publishing—Victoria, Canada
A finite element model for investigating effects of aerial architecture on
tree oscillations
DAMIEN SELLIER,1,2 THIERRY FOURCAUD1,3,4 and PATRICK LAC1
1
UMR LRBB, Domaine de l’Hermitage, 69 route d’Arcachon, F-33612 Cestas Cedex, France
2
UR EPHYSE INRA, B81 F-33883 Villenave d’Ornon Cedex, France
3
UMR AMAP, CIRAD, TA40/PS2, Boulevard de la Lironde, F-34398 Montpellier Cedex 5, France
4
Corresponding author ([email protected])
Received June 10, 2005; accepted September 30, 2005; published online March 1, 2006
Summary A finite element model was developed to study
the influence of aerial architecture on the structural dynamics
of trees. The model combines a complete description of the
axes of the aerial architecture of the plant with numerical techniques suitable for dynamic nonlinear analyses. Trees were
modeled on the basis of morphological measurements that
were previously made on three 4-year-old Pinus pinaster Ait.
saplings originating from even-aged stands. Calculated and
measured oscillations were compared to evaluate model behavior. The computations allowed the characteristics of the
fundamental mode of vibration to be estimated with satisfactory accuracy. Inclusion of a topological description of the aerial system in a mechanical model provided insight into the
effect of tree architecture on tree dynamic behavior. Simplifications of the branching pattern in the model led to overestimations of the natural swaying frequency of saplings by 10 to
20%. Inadequate values of stem and root anchorage stiffness
resulted in errors of 10 to 20%. Modeling results indicated that
aerodynamic drag of needles is responsible for 80% of the
damping in the studied trees. Additionally, damping of stem
movement is reduced by one half when branch oscillations are
not considered. It appears that the efficiency of the dissipative
mechanisms depends directly on crown topology.
Keywords: damping, dynamics, numerical analysis, oscillation frequencies, tree biomechanics.
Introduction
A knowledge of structural dynamics is crucial for the analysis
of both applied and fundamental aspects of tree biomechanics.
Trees can be subjected to a shockwave following impacts
caused by rockfall in mountainous terrain (Dorren and Berger
2006). Additionally, trees are subjected to highly fluctuating
wind speeds within canopies where turbulence is known to be
mechanically active (Finnigan 2000). Dynamic analysis is
needed to evaluate tree resistance in such cases. Moreover, dynamic loads are involved in thigmomorphogenesis (Jaffe et al.
2002), a phenomenon that can involve modification of biomass allocation in plants (Niklas 1998, Fournier et al. 2005).
Because wind-induced movement is the primary abiotic
cause of tree damage, several dynamic models have been developed to investigate the mechanics of individual trees (Wood
1995). Studies of plant oscillations by theoretical analysis
(Papesch 1974, Spatz and Speck 2002) or numerical analysis
(Kerzenmacher and Gardiner 1998) have all focused on the
stem of the plant while simplifying the rest of the aerial system. In contrast, Moore (2002) developed a dynamic model
based on the Finite Element Method (FEM) where branches
connected to the stem are described as flexible and oscillating
elements and hence branch properties modify the dynamics of
the whole structure. These numerical studies have shown that
the architecture of the aerial system is a key component of tree
stability because it is the interface between the atmospheric
flow and the element that is critical for wind-firmness. When
subjected to wind, branches, twigs and leaves bend (Speck
2003), twist or even break (Niklas 2000) thereby reducing the
exposed area as wind speed increases.
We describe a dynamic model that has its foundation in a
FEM-based plant architectural model that was developed to
perform static structural analyses of tree architecture (Fourcaud et al. 2003b). Maritime pine (Pinus pinaster Ait.) saplings were used to test the model and for the numerical analyses. The morphology and mechanical characteristics of these
saplings have been already described by Sellier and Fourcaud
(2005).
Because of the complexity and variability of aerial architecture among trees, we used numerical techniques to perform
dynamic analyses of systems with geometric nonlinearity
(Fourcaud and Lac 2003). In a first step, the capability of the
model to simulate the free oscillations of the saplings was
evaluated based on two criteria of particular interest: (1) tree
oscillatory frequencies were calculated because the resonance
phenomenon, which is potentially destructive, can occur when
the frequency of gust arrival is close to the frequency of tree
sways; and (2) the damping ratio of tree structure was calculated because this criterion expresses the efficiency of movement dissipation. Damping has several sources in the case of
isolated trees. Most of it is linked to friction processes such as
internal friction of woody material (Spatz et al. 2004) or
800
SELLIER, FOURCAUD AND LAC
aerodynamic drag of foliage (Milne 1991). Oscillations of
branches can inhibit stem oscillations (Speck and Spatz 2004).
In a second phase, the model was used to quantify the influence of crown architecture on the dynamic characteristics of
the studied saplings.
We applied the model to three 4-year-old Pinus pinaster saplings in an even-aged stand as described by Sellier and Fourcaud (2005). The three saplings are referred to as S1, S2 and
S3 hereafter. The main morphological features of the saplings
are given in Table 1. Model input data were derived from morphological measurements and mechanical experiments performed in situ on the saplings.
tree height. The second mode was not active when movement
was induced with other initial conditions. The nature of deformation involved in this second mode was not determined
experimentally.
Additional mechanical tests were performed with sapling
S1. A horizontal force was statically applied to the tree stem in
two orthogonal directions, 1 and 2, successively. The force induced a bending moment that caused curvature of the stem and
a tilt of the root system through a small angle. Values of stem
deflection and basal tilt angle for a given applied force depend
on stem bending stiffness and root-anchorage flexibility. Deviation angle from rest position was measured with inclinometers at the base of the stem and at half of tree height (Model
6900461, Sensorex, France) to estimate the elastic properties
of the stem and of the root anchorage, as described by Neild
and Wood (1999).
Mechanical measurements
Geometrical modeling
The dynamic characteristics of the saplings were measured by
free sway tests (Sellier and Fourcaud 2005). Motion was initiated by pulling the saplings out of their static equilibrium position with a rope and then releasing them from the induced
strain. Frequency and damping ratios were derived from the
oscillations following the initial release. Frequency was calculated as the inverse of the mean time lag between successive
maxima of angular displacements in stem oscillations. The
damping ratio was calculated by an exponential regression between the amplitude and the number of cycles. Sapling S1 was
subjected to dynamic tests before and after successive prunings, i.e., with the entire crown, without needles, without
third-order axes A3 and finally with the single stem. The
branching order was defined as the topological distance of an
axis in the crown: the stem is the axis of first order (A1),
branches inserted on the stem are second-order axes (A2) and
axes inserted on A2 are third-order axes (A3). For each sapling, the fundamental mode of vibration was fully characterized in terms of frequency and damping ratio, and described as
a bending mode. A second mode of vibration of higher frequency was observed in situ. This mode was observed only
when movement was initiated by pulling the stem at a third of
Finite Element (FE) modeling and analyses were carried out
with ABAQUS software (ABAQUS Inc., Providence, RI).
Vegetative axes of the plant aerial system were modeled
with three-dimensional beam elements (Fourcaud et al.
2003a). Timoshenko’s beam theory was applied because
stems and branches are slender bodies with small transverse
dimensions compared with the longitudinal dimension (Timoshenko 1976). The beam elements were chosen with an associated linear interpolation field because of the large number of
elements used to describe the whole branching system. Element cross sections were assumed circular and constant over
the element length. This limitation increased the number of elements necessary to model each axis so that the axis tapering
mode was reasonably described. The connection between an
axis and its bearing axis was modeled as being rigid.
A common approach in modeling tree dynamics involves
aggregating branches on the stem at their point of insertion (Guitard and Castéra 1995, Kerzenmacher and Gardiner
1998). This simplification is made because dynamic calculations are time-consuming; however, it may introduce prediction errors. For this reason, we compared alternate modeling
approaches. In addition to the first modeling method where
branches were modeled with beam elements (model M1),
branches were also modeled by point masses aggregated on
the stem (model M2) and by cylindrical rigid bodies
(model M3). Unless stated otherwise, dynamic calculations
were carried out with model M1.
Needles present on vegetative axes are not explicitly described in the FE model. They are represented by modifying
the mass and dissipative properties of each supporting element. The root system was modeled by a U-joint connector element. This connector allowed angular displacement around
two orthogonal and horizontal directions while the remaining
degrees of freedom were blocked. The mechanical behavior of
such a connector is described in the Material modeling section.
Architectural data, i.e., spatial coordinates, diameter and
topological relationships of aerial axes, were used to create
FE meshes. These data were taken from Multi-scale Tree
Materials and methods
Tree material
Table 1. Measured morphological characteristics of the saplings (S1,
S2 and S3). The value within brackets for S3 corresponds to the height
of the stem. For the other two saplings, heights of the tree and the stem
was identical. An asterisk indicates that values were estimated. Abbreviations: d0.13 = stem diameter at 0.13 m from ground; h = tree
height; m = tree mass; mcrown /m = relative crown biomass; and z G /h =
relative height of the center of mass (from Sellier and Fourcaud 2005).
Characteristic
S1
S2
S3
d 0,13 (cm)
h (m)
m (kg)
mcrown /m
z G /h
5.1
2.57
6.6
0.58
0.42
4.3
2.29
4.64
0.59
0.43
5.7
1.76 (0.93)
6.67*
0.75*
0.42
TREE PHYSIOLOGY VOLUME 26, 2006
EFFECT OF AERIAL ARCHITECTURE ON TREE OSCILLATIONS
Graph files (MTG; Godin and Caraglio 1998) that were created by digitizing aerial architecture (Sellier and Fourcaud
2005). Data interfacing between MTG files and ABAQUS input files was performed by the mean of scripts developed in
Python language.
Convergence tests were made to determine the number of
elements needed to model wood axes. Based on a study undertaken with a direct time integration solver (see Dynamic analyses section), we calculated that the fundamental frequency of a
cylindrical axis converged towards the theoretical solution
with a minimum of eight elements to model the axis. Hence,
we modeled A2 with eight elements. Sixteen elements were
used for stem A1 to describe the tapering mode with improved
accuracy. To model A3, we reduced the number of elements to
four, because large numbers of small discrete elements in the
model led to prohibitive computational times (Courant condition, cf. Gerardin and Rixen 1997). Finally, the FE mesh was
built according to the predefined location of the tree whorls.
The size of modeled structures ranged from 2600 degrees of
freedom for S1 to 5100 degrees of freedom for S3. Element
nodes have six degrees of freedom, one degrees of freedom for
each displacement component (three transverse and three angular).
Material modeling
Some material properties of the structural components were
needed for the dynamic analyses. Mass, damping and stiffness
of the beam elements were required to solve the equations of
movement:
Mu&& + Cu& + Ku = F (t )
(1)
where M, C, K are the inertia, damping and stiffness matrices
of the system, respectively, F(t) is the time-dependent load
&& u& and u are the acceleration, speed and
column vector and u,
displacement column vectors, respectively. In a first approach,
the material properties were kept constant within a given
branching order for each sapling.
Wood density is an input parameter of the model. For a
given branching order and for each sapling, the wood density
value was calculated by dividing the measured mass of wood
components by the measured volume of these components.
Wood density ranged from 1100 to 1350 kg m – 3. The volume
of each sapling was measured when the saplings were being
digitized. Because digitizing is a discrete process that resulted
in partially underestimating the real volume of vegetative
axes, it led to wood density values that were higher than expected for green wood material. The method nevertheless ensured that the biomass of the modeled tree was equal to that of
the measured tree. The density of finite elements bearing needles was modified to take into account the needle mass considered to be uniformly distributed along the element. The mean
linear mass for needles was 0.24 kg m –1 for stems, 0.14 kg m – 1
for A2 and 0.07 kg m –1 for A3.
Wood mechanical behavior was modeled as an elastic material. This assumption is commonly used in static modeling at
801
the tree scale (Gaffrey and Kniemeyer 2002, Fourcaud et al.
2003a) and appeared sufficient to fulfil the objective of this
study. Orthotropic features of wood behavior were neglected
because of the use of beam theory. Hence, the material characteristics used to model the stiffness of wood axes were the longitudinal modulus of elasticity (E), and the Poisson’s ratio (ν).
The value of ν (0.38) was derived as described by Guitard and
Fournier (1994). An elastic mechanical behavior was used to
model root anchorage. It was assumed that rotations applied to
the soil–root plate were small enough to avoid nonlinear behavior (e.g., plastic strain or damage). The U-joint connector
was characterized by a bending stiffness, kS. For sapling S1,
values of E and kS were fitted by replicating static tests with the
FE model so that measured and calculated rotations were
equal (cf. Table 2). Values of E and kS averaged over test directions 1 and 2 were chosen, namely E = 1.12 GPa and kS =
342 Nm rad – 1, because the model does not model anisotropy
of bending stiffness. The modulus of elasticity for wood axes
with a branching order superior to one was not measured and
was therefore assumed to be equal to the modulus of elasticity
of the stem. Elastic properties of stem and root anchorage were
not measured in saplings S2 and S3: values of E and kS obtained for sapling S1 were initially used for the other saplings.
Rayleigh’s hypothesis was used to model material damping.
This hypothesis assumes that the damping matrix C is proportional to the inertia and stiffness matrices, M and K, respectively, according to:
C = α M + βK
(2)
This hypothesis is appropriate because the branch damping
ratio is partly correlated with the mass of needles present on
the branch (Moore 2002). The β coefficient in the damping expression was neglected because wind-induced movement is
usually limited to the first modes of vibration (Mayer 1987).
The stiffness-dependent part of damping is effective only for
high frequency modes. However, β was set at 0.001 rather than
a null value to prevent the occurrence of numeric noise at high
frequencies. Two values of the α coefficient were used in the
model, i.e., αW for wood and αN for needles. Energy dissipation through the soil–root plate was modeled by a dashpot attached to the connector with a dissipative parameter cS. The
parameters were calibrated so that the mean damping ratios
measured for sapling S1 were equal to the calculated
ξ msr
0
damping ratios ξ cal
0 in test simulations:
ξ cal
= ξ msr
0
0
(3)
Table 2. Measured values of the stem longitudinal modulus of elasticity (E) and the bending stiffness of anchorage (kS), resulting from
static tests carried out on sapling S1.
Direction 1
Direction 2
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E (GPa)
kS (Nm rad – 1)
1.05
1.20
335
349
802
SELLIER, FOURCAUD AND LAC
where ξ 0 is the damping ratio of the stem in the fundamental
mode of deformation. This ratio expresses the ability of the
structure to dissipate movement efficiently. The measured
value of ξ 0 varied with crown pruning. The first step of the calibration process was to determine pairs of values (αW and cS)
that satisfied Equation 3 for the S1 stem, i.e., when the sapling
branches were removed. A parametric study was performed
for this purpose. As a second step, oscillations were simulated
with each of the previous pairs (αW and cS) for the sapling
composed of the stem and the A2 branches. This second series
of simulations allowed a new unique (αW and cS) pair to be obtained that satisfied Equation 3. Finally, the value of the missing parameter αN was obtained by performing a simulation for
the entire sapling S1, i.e., with all its branches and their needles.
Dynamic analyses
Different numerical methods were used to determine frequencies and damping ratios of the modeled saplings. Calculation
of oscillatory modes and frequencies relied on the modal
method (Gerardin and Rixen 1997), which consists of solving
the eigenvalue system associated with the equations of motion.
During the process of mode extraction, no mechanical loads
were taken into account, allowing the eigenmodes to be studied independently of their possible activation in sapling motion. The first 250 modes were extracted for each sapling.
Participation of a given mode was computed relative to structural displacements. Participation in a transverse displacement
was calculated as the ratio between the modal effective mass
and the total mass of the structure. Participation in an angular
displacement was calculated as the ratio between the effective
modal inertia and the corresponding moment of inertia.
Oscillations of modeled saplings were calculated with direct time integration techniques based on the Newmark algorithm (cf. Gerardin and Rixen 1997). Calculation of movement
could have been achieved with modal techniques but this approach would have been limited to linear dynamics. Time integration techniques were chosen because the model was initially designed to predict wind-induced tree motions. Such
motions can include geometric nonlinearity such as crown
streamlining induced by wind. The initiation of tree swaying
in numerical simulations corresponded to the initial displacement applied during the field experiments, i.e., constant loading was applied on the stem at a third of tree height for 5 s and
then released.
Results
Oscillatory frequencies
Natural swaying frequency Deformed shapes as well as patterns of mechanical stresses are shown in Figure 1 for saplings
in the fundamental mode of vibration. For the three saplings,
participation of this mode in stem bending was always greater
than 50% of the modal response. Comparison between measured values and values calculated with model M1 showed a
close agreement for the frequency of the fundamental mode
( f0). For sapling S1, the predictive error on f0 did not exceed 5%
irrespective of the degree of crown pruning (Table 3). The maximum error (4.5%) was obtained when the needles were removed from the tree. The error was less than 1% for the stem
alone. Calculations underestimated f0 except when the sapling
comprised stem and A2.
The frequency value for the entire structure varied significantly according to the modeling approach. The value of f0 was
0.58 Hz for sapling S1 when calculated with model M1. The
predictive error was 3.5% when compared to the measured frequency. By aggregating branch masses on the stem (model
M2), the calculated frequency was 0.72 Hz. Thus, model M2
overestimated the f0 value measured during field experiments
by about 20%. Calculated f0 of the whole structure was 0.63 Hz
when the branches were modeled as rigid bodies (model M3).
This value corresponded to a predictive error of 6.5% when
compared with measured value and 10% when compared with
the value obtained with flexible branches by model M1.
Numerical analyses performed on saplings S2 and S3 relied
on model M1 and on the elastic characteristics of sapling S1.
Calculated frequencies for their fundamental mode of vibra-
Figure 1. Deformed shape and mechanical Mises stresses of the three
modeled saplings in their fundamental
mode (bending) of vibration. Mises
stresses give an average representation
of overall stress components calculated
according to the local referential.
TREE PHYSIOLOGY VOLUME 26, 2006
EFFECT OF AERIAL ARCHITECTURE ON TREE OSCILLATIONS
Table 3. Measured and calculated frequencies of the fundamental
mode of vibration for sapling S1 as a function of crown pruning.
Crown state
f0 (Hz)
Table 4. Frequency of the second mode as observed experimentally
versus frequency of the second bending mode as calculated for the
three saplings (S1, S2 and S3).
Subject
Complete
Without needles
Without A3
Stem only
Measured
Calculated
0.60
1.13
1.09
1.39
0.58
1.08
1.10
1.38
tion showed a significant inconsistency when compared with
measured values (Figure 2). The relative error on f0 was about
10% for both trees. The f0 value was underestimated for sapling S2, whereas it was overestimated for sapling S3. Errors on
the frequency calculation were reduced to less than 1% by
modifying the values of parameters E and kS for saplings S2
and S3 by +30% and –15%, respectively.
Secondary modes of vibration For each sapling, frequencies
of the vibration modes that have a significant participation
(> 5%) in the modal response were compared with frequencies
of the second mode recorded during in situ tests. Among them,
the second bending mode of the stem had the closest frequency
to the measured values (Table 4). The maximal relative error on
f1 was found for sapling S1 with an underestimation of about
10%. Errors in f1 of less than 5% were obtained for saplings S2
and S3 after calibrating E and kS. Error values were partly the
result of inaccuracies in the modeling hypotheses; however, the
803
S1
S2
S3
f1 (Hz)
Measured
Calculated
2.14
2.61
2.98
1.90
2.59
3.04
field measurements were inaccurate, in part, because of the
high ratio between measured frequencies, ranging from 2 to
3 Hz, and the sampling frequency, equal to 10 Hz.
No single torsion mode was identified through numerical
analysis. On the contrary, several vibration modes involved the
bending of branch groups, which induces torsion in the stem.
The participation of each mode to rotation around the z-axis
(vertical) ranged from 5 to 15%. Torsion results are presented
for indicative purposes only because spiral grain is not considered in the model.
Damping of natural oscillations
The study of damping was limited to the fundamental mode of
vibration because this mode governed tree sways during field
experiments and because it did not depend on the initial conditions. The main focus was to identify the individual roles of
crown components on the overall damping of the saplings.
Evaluation of the dissipative parameters When both damping matrix coefficients αN and αW (Equation 2) and the dissipative parameter cS, which is related to root anchorage, were
fitted for sapling S1, we obtained αN = 0.68; αW = 0.32 and cS =
5.72 Pa s. These values were also used for material modeling of
saplings S2 and S3 to simulate free oscillations.
Results showed good agreement between measured and calculated damping ratios of the stem for the fundamental mode,
ξ 0 (cf. Table 5). The relative error on ξ 0 was low—it did not exceed 5%. The maximum error was found for sapling S3 where
wood density and biomass distribution of needles were approximated. Use of a direct measurement of biomass distribution gave a numerical error in ξ 0 close to zero for sapling S2.
Anisotropic damping, i.e., difference in damping ratio according to orthogonal directions 1 and 2, was observed for sap-
Table 5. Predicted values of the mean damping ratio in the fundamental mode of vibration (ξ 0) for saplings S2 and S3 versus the measured
values. The asterisk (*) indicates the target value used to calibrate the
model.
Subject
Figure 2. Calculated frequency of the fundamental vibration mode
( f0) for saplings S2 and S3 obtained before and after adjustment of
wood modulus of elasticity (E) and anchorage bending stiffness (ks).
S1
S2
S3
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ξ 0 (%)
Measured
Calculated
9.8
6.4
6.2
9.8*
6.4
5.9
804
SELLIER, FOURCAUD AND LAC
ling S1 during field experiments. Values of ξ 0 were 11.8 and
7.8% in both directions of sway initiation, respectively. This
anisotropy was not reproduced by the numerical simulations
where values of ξ 0 were equal to 9.8% independent of the test
direction. The absence of damping anisotropy in calculated oscillations was also observed in the other saplings; however, the
numerical and experimental damping values were equivalent
with respect to the mean ξ 0 value.
Influence of crown components on damping Numerical analyses were performed on the whole sapling S1 to evaluate the
relative participation of dissipative sources on tree sway damping. For this purpose, simulations were performed with all
dissipative parameters set to zero except one. Results showed
that needles were responsible for about 80% of the overall
damping (Figure 3). The oscillation damping resulting from
wood viscosity and the soil/root system were equivalent, each
one accounting for about 10% of the overall damping.
The role of branch oscillations on the damping of sapling S1
was assessed through simulations performed with different
and highly contrasting branch stiffnesses. A numerical ξ 0
value of 9.8% was obtained with model M1 with flexible
branches compared with a value of 4.6% when using model
M3 with rigid branches.
Field experiments on sapling S1 revealed a negative influence of A3 on the damping of stem sways. After removal of
needles, ξ 0 equalled 4.2% when A3 were still present within
the crown. The ξ 0 value increased to 5.2% after removal of A3.
When performed on sapling S1 without foliage, calculations
gave a ξ 0 value of 4.7% with the A3 axes and 5.2% without.
Hence, the model was able to simulate the increase in damping
caused by the removal of A3, although it underestimated the
decrease compared with the experimental data.
Discussion
The frequency of natural sways of an existing tree was calculated by the FEM with a satisfactory accuracy with only measured geometrical and material properties as model input. The
Figure 3. Relative participation of the dissipative sources (needles,
wood and soil/root) in the damping ratio of the stem of sapling S1.
predictive error in frequency never exceeded 4.5% through the
different pruning phases of sapling S1, even though the modulus of elasticity of the branches was only approximated. However, the error in frequency calculation was more important
(10%) for saplings S2 and S3 as a result of the use of the elastic
characteristics of sapling S1, indicating that stem and root anchorage stiffness has a marked influence on the frequency of
tree natural sways. Thus, the errors introduced as a result of incorrectly estimating stiffness indicate that inter-tree variability
in stem and root anchorage stiffness parameters cannot be
neglected in dynamic modeling exercises.
A common modeling approach that involves crown simplification by neglecting branch geometry and in considering
only branch masses aggregated on the stem leads to an important error when calculating f0. This error arises because of misrepresentation of both stiffness and inertia matrices as a result
of crown simplification. The relative error is twice as large as
the error caused by ignoring inter-tree variability in stem and
anchorage elastic characteristics. Modeling branches with
rigid bodies decreases the error in frequency calculation when
compared with the crown simplification approach, because the
error is then caused by misrepresenting the stiffness matrix
only. Moreover, the resulting calculation errors are in good
agreement with those obtained by Fournier et al. (1993) who
found errors ranging from 10 to 15% between rigid and flexible branch modeling in adult Pinus pinaster and Hevea brasiliensis (Willd. ex A. Juss.) Müll. Arg. Thus, a more realistic
model of aerial architecture consistently predicts the swaying
frequency on the basis of measured structural properties only,
whereas models that rely on crown simplifications require adjusted parameters to predict the fundamental frequency of a
given tree with an equivalent accuracy.
Numerical analysis made it possible to identify the nature of
the second vibration mode that was observed during field experiments. This mode is found to be the second bending mode
of the stem. In broader terms, the model may act as a prospective tool to study the modal response of trees. Although
tree sways are mainly governed by the fundamental vibration
mode, analysis of higher frequency modes is of particular interest when several modes are likely to be activated by dynamic mechanical loads, e.g., rock impact or wind turbulence.
We found little to no variability of the dissipative parameters
among the studied trees. Nevertheless, the quality of damping
ratio prediction depends on accurate descriptions of the spatial
distribution of needles and wood mass within the aerial system. Overall, the modeled trees presented a damping behavior
quantitatively similar to the measured trees.
We showed that damping of stem movement is governed by
the friction of needles in surrounding air. On average, the needles comprised 45% of total aerial biomass in the studied saplings. Therefore, the dominance of aerodynamic damping over
other dissipative sources may be related to the juvenile state of
these trees because relative biomass of needles decreases with
age. Our results are nonetheless consistent with those obtained
by Milne (1991) on 26-year-old Picea sitchensis (Bong.) Carrière trees showing that 80% of the total damping was due to
aerodynamic drag in the absence of neighbouring trees. Wood
TREE PHYSIOLOGY VOLUME 26, 2006
EFFECT OF AERIAL ARCHITECTURE ON TREE OSCILLATIONS
viscous behavior and the soil–root interface appeared as minor dissipative sources in our study. Although a minor effect of
soil and roots on stem damping has been predicted (Speck and
Spatz 2004), the role of the soil–root plate on damping may
vary greatly according to soil type and water content.
The failure of the numerical analysis to identify damping
anisotropy may be associated with our restrictive working hypotheses on aerodynamic friction of needles, which is the primary source of damping in the studied structures. In our
model, the aerodynamic drag scales up with mass rather than
with needle frontal area. In future studies, an improved physical model for movement dissipation induced by air friction
should allow the difference between measured and calculated damping to decrease. However, the results of the current
model are consistent with experimental data, allowing consideration of the general behavior of damping.
The damping ratio of sapling S1 decreased by about 50% as
a result of the complete stiffening of its branches. This has several possible causes. One is linked to the particular behavior of
A2 during plant oscillations as measured in field experiments
(Sellier and Fourcaud 2005). When the stem oscillates, the
front A2 bends upwards while the rear A2 bends downwards.
The A2 movement, thus opposes stem bending, forming a
mechanism that hampers stem oscillations. However, the main
cause of branch-induced damping when branches are flexible
is that the strain energy initially contained in the stem is transferred to the entire branch system. This results in an increased
branch motion velocity that subsequently causes an overall
increase in damping because damping scales increase with
displacement velocity. It explains why ξ 0 was so low when
branches were modeled as rigid bodies because such damping mechanisms were totally inhibited. Damping induced by
branch oscillations is coupled to the other dissipative processes affecting branches such as aerodynamic friction or material viscosity. In the absence of these processes, energy will
be conserved within the aerial system and energy transfer
within crown elements will not by itself cause any dissipation.
Furthermore, damping caused by branch oscillations is expected to vary significantly among plant species (Brüchert et
al. 2003).
The negative influence of A3 on damping of stem oscillations was observed both in experimental and numerical results.
Underestimation of this effect in the model may be caused by
the small number of elements used to mesh A3. The mechanism of damping induced by branch oscillations is assumed to
be responsible for the influence of A3 on stem damping. If, as
shown in our results, A2 axes inhibit the oscillations of stem
A1, thus causing an increase in ξ 0, the A3 axes may similarly
inhibit the oscillations of A2 axes. Consequently, the A3 axes
would lead to a decrease in ξ 0 because their oscillations reduce
the positive impact of A2 sways on stem damping. This hypothesis, if true, implies that the topological structure of the
plant directly affects its damping characteristics and mechanisms.
In conclusion, the capability of the FEM to predict the dynamic characteristics of trees, as well as the importance of
using a full description of plant aerial axes, has been demon-
805
strated. Numerical results are in good agreement with previous
experimental studies and show that a fine description of the
branching structure is necessary to understand the role of the
aerial architecture in tree oscillations. Further development of
the model is necessary to study the mechanical behavior of
trees submitted to external dynamic loading, e.g., coupling
tree dynamics and drag forces caused by turbulent wind. Our
model could then be used to analyze mechanical energy transfer from the foliage to the stem and the root system.
Dupuy et al. (2005) recently developed a static finite element model that allows the mechanical behavior of the root–
soil plate to be simulated. This model is based on the same
methodology (Fourcaud et al. 2003b) as our model. Coupling
our aboveground model with the root model of Dupuy et al.
(2005) could facilitate studies of the mechanisms involved in
the breakage and uprooting of trees subjected to strong winds.
Such a complete dynamical model could be used to improve
the existing static models devoted to the estimation of wind
damage at the stand level (Ancelin et al. 2004a).
Coupling between tree growth and tree biomechanics has
also been performed in previous works to simulate the morphological adaptation of trees to external loads (Fourcaud et
al. 2003a, Ancelin et al. 2004b). However, these models did
not consider the dynamic responses of plants. The integration
of a dynamic model in a tree growth model could provide additional insight into how plant architecture could be optimally
designed to resist wind forces (Stokes et al. 1995, Telewski
1995).
Acknowledgments
The authors thank Dr. Yves Caraglio (AMAP, CIRAD), Dr. Barry
Gardiner (Forest Research, UK) and Dr. Yves Brunet (EPHYSE,
INRA) for their help. This numerical study is part of the VENFOR
project funded by the GIP-ECOFOR through the ‘Forêt, vent et
risques’ programme.
References
Ancelin, P., B. Courbaud and T. Fourcaud. 2004a. Developing an individual tree based mechanical model to predict wind damage
within forest stands. For. Ecol. Manage. 203:101–121.
Ancelin, P., T. Fourcaud and P. Lac. 2004b. Modeling the biomechanical behavior of growing trees at the forest stand scale. Part I:
development of an incremental transfer matrix method and application on simplified tree structures. Ann. For. Sci. 61: 263–275.
Brüchert, F., O. Speck and H.-C. Spatz. 2003. Oscillations of plants’
stems and their damping: theory and experimentation. Phil. Trans.
Roy. Soc. Lond. B 358:1487–1492.
Dorren, L.K.A. and F. Berger. 2006. Stem breakage of trees and energy dissipation during rockfall impacts. Tree Physiol. 26:63–71.
Dupuy, L., T. Fourcaud and A. Stokes. 2005. A numerical investigation into the influence of soil type and root architecture on tree anchorage. Plant Soil 278:119–134.
Finnigan, J.J. 2000. Turbulence in plant canopies. Annu. Rev. Fluid
Mech. 32:519–571.
Fourcaud, T. and P. Lac. 2003. Numerical modeling of shape regulation and growth stresses in trees. i. An incremental static finite element formulation. Trees 17:23–30.
TREE PHYSIOLOGY ONLINE at http://heronpublishing.com
806
SELLIER, FOURCAUD AND LAC
Fourcaud, T., P. Blaise, P. Lac, P. Castera and P. de Reffye. 2003a. Numerical modeling of shape regulation and growth stresses in trees.
ii. Implementation in the AMAPpara software and simulation of
tree growth. Trees 17:31–39.
Fourcaud, T., L. Dupuy, D. Sellier, P. Ancelin and P. Lac. 2003b. Application of plant architectural models to biomechanics. In 2003
Int. Symp. Plant Growth Modeling, Simulation, Visualization and
Their Applications. Eds. M. Jaeger and B.G. Hu. Tsinghua University Press–Springer-Verlag, Berlin, pp 462–479.
Fournier, M., P. Rogier, E. Costes and M. Jaeger. 1993. Modélisation
mécanique des vibrations propres d’un arbre soumis aux vents, en
fonction de sa morphologie. Ann. Sci. For. 50:401–412.
Fournier, M., A. Stokes, C. Coutand, T. Fourcaud and B. Moulia.
2006. Tree biomechanics and growth strategies in the context of
forest functional ecology. In Biomechanics: A Mechanical Approach to the Ecology of Animals and Plants, Eds. A. Herrel,
T. Speck and N. Rowe. CRC Press, LLC, USA, pp 1–34.
Gaffrey, D. and O. Kniemeyer. 2002. The elasto-mechanical behaviour of Douglas fir, its sensitivity to tree specific properties, wind
and snow loads, and implications for stability—a simulation study.
J. For. Sci.48:49–69.
Gerardin, M. and D. Rixen. 1993. Théorie des vibrations—application à la dynamique des structures. Masson, Paris, 420 p.
Godin, C. and Y. Caraglio. 1998. A multiscale model of plant topological structures. J. Theor. Biol. 191:1–46.
Guitard, D. and M. Fournier. 1994. Comportement mécanique du
matériau bois. In Le Bois, Matériau D’Ingénierie. Ed. N. Arbolor.
Arbolor, Nancy, France, pp 91–125.
Guitard, D.G.E. and P. Castéra. 1995. Experimental analysis and mechanical modeling of wind-induced tree sways. In Wind and Trees.
Eds. M.P. Coutts and J. Grace. Cambridge University Press, London, pp 182–194.
Jaffe, M.J., A.C. Leopold and R.C. Staples. 2002. Thigmo responses
in plants and fungi. Am. J. Bot. 89:375–382.
Kerzenmacher, T. and B.A. Gardiner. 1998. A mathematical model to
describe the dynamic response of a spruce tree to the wind. Trees
12:385–394.
Mayer, H. 1987. Wind-induced tree sways. Trees 1:195–206.
Milne, R. 1991. Dynamics of swaying Picea sitchensis. Tree Physiol.
9:383–399.
Moore, J.R. 2002. Mechanical behaviour of coniferous trees subjected to wind loading. Ph.D. Thesis, Oregon State University,
Corvallis, Oregon, 211 p.
Neild, S.A. and C.J. Wood. 1999. Estimating stem and root-anchorage flexibility in trees. Tree Physiol. 19:141–151.
Niklas, K.J. 1998. Effects of vibration on mechanical properties and
biomass allocation pattern of Capsella bursa-pastoris (cruciferae).
Ann. Bot. 82:147–156.
Niklas, K.J. 2000. Computing factors of safety against wind-induced
tree stem damage. J. Exp. Bot. 51:797–806.
Papesch, A.J.G. 1974. A simplified theoretical analysis of the factors
that influence windthrow of trees. In 5th Australasian Conference
on Hydraulics and Fluid Dynamics, University of Canterbury,
Christchurch, New Zealand, pp 235–242.
Sellier, D. and T. Fourcaud. 2005. A mechanical analysis of the relationship between free oscillations of Pinus pinaster Ait. saplings
and their aerial architecture. J. Exp. Bot. 56:1563–1573.
Spatz, H.-C. and O. Speck. 2002. Oscillation frequencies of tapered
plant stems. Am. J. Bot. 89:1–11.
Spatz, H.-C., A. Emmans and O. Speck. 2004. The structural basis of
oscillation damping in plant stems—biomechanics and biomimetics. J. Bionics Engineering 1:149–158.
Speck, O. 2003. Field measurements of wind speed and reconfiguration in Arundo donax (Poaceae) with estimates of drag forces. Am.
J. Bot. 90:1253–1256.
Speck, O. and H.-C. Spatz. 2004. Damped oscillations of the giant
reed Arundo donax (Poaceae). Am. J. Bot. 91:789–796.
Stokes, A., F.H. Fitter and M.P. Coutts. 1995. Responses of young
trees to wind and shading: effects on root architecture. J. Exp. Bot.
46:1139–1146
Telewski, F.W. 1995. Wind-induced physiological and developmental
responses in trees. In Wind and Trees. Eds. M.P. Coutts and
J. Grace. Cambridge University Press, London, pp 237–263.
Timoshenko, S.P. 1976. Strength of materials. 3rd Edn. Krieger Publishing group, Hutington, NY, 456 p.
Wood, C.J. 1995. Understanding wind forces on trees. In Wind and
Trees. Eds. M.P. Coutts and J. Grace. Cambridge University Press,
London, pp 133–164.
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