JXB Advance Access originally published online on April 18, 2005
Journal of Experimental Botany 2005 56(416):1563-1573; doi:10.1093/jxb/eri151
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RESEARCH PAPER |
A mechanical analysis of the relationship between free oscillations of Pinus pinaster Ait. saplings and their aerial architecture
1EPHYSE INRA, BP 81, F-33883 Villenave d'Ornon Cedex, France
2CIRAD, LRBB, Domaine de L'hermitage, 69 route d'Arcachon, F-33612 Cestas Cedex, France
* To whom correspondence should be addressed. Fax: +33 5 57 12 24 20. E-mail: sellier{at}lrbb.u-bordeaux.fr
Received 24 November 2004; Accepted 1 March 2005
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Abstract |
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The aim of this study was to investigate the influence of aerial architecture on the dynamic characteristics of young maritime pines (Pinus pinaster Ait.) using a mechanistic approach. For this purpose, three 4-year-old saplings with prominent differences in their branching patterns were submitted to free oscillation tests. The tests were carried out with different methods and directions of mechanical loading in order to initiate the movement of each sapling. The oscillations of the different architectural elements, i.e. stem and branches of different topological order, were measured with inclinometers and strain gauges fixed to saplings. Successive pruning of the architectural elements was carried out to evaluate their relative influence on the dynamic characteristics of the trees. The aerial systems were digitized before the mechanical tests in order to use 3D visualization techniques and to make architectural analyses of the crown structure. Two distinct modes of deformation were detected during free oscillations. The natural swaying frequency ranged from 0.6–0.8 Hz for the saplings tested at the same period of the year. The frequency variations were partly explained by the morphological differences of the experimental subjects. The motions of the axes were found to depend on their topology, i.e. the movement of the axes of a given branching order was forced by the movement of their respective bearing axis. The axes of third branching order had a significant and negative effect on the damping of the natural deformation mode. Results point out the major role played by foliage, qualitatively and quantitatively, on the damping of tree motions and on coupling the motions of the crown components.
Key words: Aerial architecture, damping, oscillations, tree biomechanics
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Introduction |
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Tree stability in high winds is a major concern for forest managers, since strong gales may cause major damage to the stands and thus severe economic losses. Assessing tree stability involves studying the properties of atmospheric flows over canopies (Raupach, 1988
) as well as understanding the mechanical behaviour of trees until failure (Coutts, 1986; Skatter and Kucera, 2000). Most of the damage to the stands results from dynamic loading applied to the tree (Gardiner, 1995). Wind speed has a time-varying pattern by nature, especially inside the stands where turbulence in airflows is high. Inner stand flows are characterized by low mean windspeed and the occurrence of major gusts of high velocity (Finnigan and Brunet, 1995). The drag forces applied to trees also vary with time, since they derive from wind speeds. These time variations induce dynamic effects in the behaviour of the structure (Oliver and Mayhead, 1974). A static approach does not take into account these dynamic effects and, for a given wind speed, underestimates the mechanical stresses that can be responsible for branch and stem breakage and the bending moment transmitted to the root/soil plate that can result in tree uprooting (Papesch, 1974). Dynamic experiments have already been performed to quantify the displacement gain between static and dynamic approaches (Blackburn et al., 1988; Peltola et al., 1993).
Other experiments (Gardiner, 1994; Guitard and Castera, 1995; Lohou et al., 2003) have been designed to characterize wind-induced oscillations of trees and to study energy transfer between the airflow and trees. Trees absorb energy from the airflow they are submitted to through their aerial system. Absorption of energy from turbulent winds mainly occurs in a frequency range closely related to the swaying frequency of trees (Mayer, 1987). The energy available for the tree at these frequencies is that most likely to result in stem breakage or uprooting (Wood, 1995; Moore, 2002). Another intrinsic characteristic of the dynamic behaviour of the tree is its damping ratio. As the damping ratio expresses the efficiency of the tree to dissipate motion energy, it appears necessary to consider it as an indicator of the tree stability to wind, in addition to the swaying frequency. The damping process results from inner and outer friction mechanisms activated during the structural motions. In the case of trees, several sources of energy dissipation are active simultaneously. Milne (1991) has quantified the participation of three known sources of damping in Picea sitchensis: first, the tree crown clashes with neighbouring trees (accounting for 50% of total damping), second, the aerodynamic drag of the foliage (40%) and, last, wood viscosity (10%). Although contact with neighbours may be a major damping source for trees growing in dense stands, it was not taken into account in this study which is centred on individual trees. Other sources are expected to be involved in the damping of stem motions. They are the friction between soil and root elements (England et al., 2000) and the swaying of branches (Scannell, 1984). However, as far as is known to the authors, the influence of branches on damping has not been qualified experimentally. The effect of branch oscillations on the apparent damping of the stem is due to the energy transfer occurring between the different dynamic subsystems, i.e. the different axes. On the other hand, there is no loss of energy originating from these movements if considering the balance at the whole structure level. In addition, Moore (2002) stated that relatively small variations in mechanical properties of the branches can greatly influence the dynamic behaviour of the entire structure. Such features highlight the impact of architecture on the tree aerodynamics. Evidence of the influence of tree architecture on wind firmness has also been shown by Fourcaud et al. (1999) who carried out mechanical studies on two rubber tree (Hevea brasiliensis) clones that had similar wood properties but dissimilar crown structures (Cilas et al., 2004).
In the present study, it was investigated how the aerial system of young pines can affect their dynamic characteristics. For this purpose, a mechanistic approach was used that relied on the theory of structure dynamics. Experiments were specifically designed to assess and describe the underlying mechanisms involved in tree oscillations.
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Materials and methods |
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Tree material
Site and subjects:
All experiments described in this section were performed on three even-aged Maritime pines (Pinus pinaster Ait.) that were located at INRA Pierroton in South-West of France at 44°73' N of latitude, 0°68' W of longitude and 53 m of altitude. The saplings, named (S1), (S2), and (S3), were grown in two separate forest blocks. The first block, of circular pattern, had been subjected to artificial wind loading during previous experiments (Berthier and Stokes, 2005
). Saplings (S1) and (S2) were located in the first row of this block in front of a clearing. The second block was the control block and was only subjected to natural wind. Sapling (S3) belonged to the second block and, unlike (S1) and (S2), had no close neighbours. The wind experiments had ceased 1 year before these experiments began. In July 2002, each subject was 4 years old. Neighbouring trees were pulled away or cut before the experiments. The general morphology of saplings (Table 1) was described with the following parameters: d0,13, the diameter of the stem measured at 0.13 m from the basis; h, the total height of the sapling; m, the total biomass of the sapling; mcrown/mstem, the ratio between the crown mass and the stem mass; zG/h, the relative height of the centre of mass of the whole sapling; crown eccentricity, the distance between the centres of mass of the crown and of the stem projected on the horizontal plane.
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Aerial architecture:
These saplings were chosen as the experimental subjects because of the prominent differences in their aerial system. The architecture of each sapling was digitized before any destructive testing. For this purpose, the 3D digitizing device (Polhemus, USA, 3space FASTRAK) was used. The architectural measurements included the spatial coordinates, the diameter, and the branching order of the axis. They were done for each vegetative axis at singular locations: at the base and the top of the axis, at the base and the top of foliated zones, if any, and finally at additional points needed to describe the curvature of the axis.
Data issued from the measurements were coded into the MTG format file (Godin and Caraglio, 1998) for each sapling by using the DIPLAMI Software (Sinoquet et al., 1997). To sum up, in this format, each point of measurement is described by its position in crown topology and is given attributes (e.g. spatial coordinates and diameter). The visualization and the analyses of the architectural data contained in the MTG files were performed with the software AMAPmod (Godin et al., 1997). The digitized saplings are shown in Fig. 1.
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Free oscillation tests
Initiation of tree motions:
Free oscillations of saplings were obtained by three different initial displacements using a nylon rope whose mass (0.02 kg) was negligible compared with that of the saplings. In the first case, the displacement was applied to the stem at one-third of the tree height. The stem was pulled once and released. In the second case, the displacement was repeated three times, also at one-third of the tree height, by synchronizing the pulls with the tree movement. In this case of applied displacement, the initial velocity was not equal to zero at the start of free oscillations. In the third and last case, the displacement was applied once and at the top of the stem. Moreover, the stem was initially bent in two different directions X and Y (Fig. 2) for every case of displacement.
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In September 2002, the saplings were tested with the first and second cases of initial displacement described above. Tests were repeated four times with the same direction and the same initial displacement. Tests on sapling (S3) were repeated only twice in each configuration due to the sudden occurrence of wind. In December 2002, sapling (S2) was tested with the first and third cases of initial displacement described above. Free oscillation tests were repeated twice in each configuration. Further in the paper, sapling (S2) is referred to as (S2.s) for the September experiments and as (S2.d) for the December experiments. The architecture of (S2.d) was not digitized. Only the diameter of its stem was measured in December (Fig. 3). One-way ANOVA tests were performed on the sampled values obtained for each sapling to evaluate if there were any significant differences induced by the initial direction or the case of displacement.
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Progressive pruning:
Two methods of pruning were used to investigate the relative influence of the crown elements on the dynamics of saplings. Both methods were composed of successive phases. The first method was applied to sapling (S1). After the first pruning phase, needles were removed. After the second phase, 3°A (3rd order axes, namely the branches inserted in branches) were removed. After the third and final phase, 2°A (2nd order axes, namely the branches inserted in the stem) were removed. The second pruning method, which was carried out on (S2.d), consisted of removing successively the axes by branching order without removing the foliage initially.
Free oscillation experiments were performed at each intermediate step of pruning by following the methodology described previously for the entire saplings. The foliage and each branching order were weighed after pruning. For sapling (S3), the mass was estimated on the basis of its volume (obtained through digitizing) and of wood and foliage density calculated for the other two saplings.
Measurement of stem movement:
The movement of the stem was measured with three inclinometers (Sensorex, France, model 6900461) fastened to the stem of each sapling at 0.1 m above the stem base, at one-third and at one-half of the tree height, respectively. The mass of the inclinometers, namely 0.2 kg each, was not negligible compared with the total mass of the saplings. Their additional mass altered the dynamics of the saplings studied because of equation (5). However, care was taken not to fasten them in the highest and most slender part of the stem in order to minimize their influence. The inclinometers are 2-way sensors which record rotations with the vertical direction in two orthogonal vertical planes. Their sampling frequency is limited to 10 Hz, which was expected to be sufficient to measure the oscillation frequency of saplings according to the previous studies carried out on young trees (Milne, 1990; Guitard and Castera, 1995). Data were recorded with a personal computer equipped with a data logging device (National Instruments, USA, NI 6023E). Each free oscillation test lasted for 30 s. Unless specified otherwise, the rotations recorded by the inclinometers in both orthogonal directions were averaged.
Measurement of branch motions:
In order to measure the oscillations of individual branches, sapling (S1) was equipped with plastic-backed reversible strain gauges (Kyowa, Japan, KFG-10-120-C-11, 100 mm length, 120 Ohm resistance). The gauges were connected in quarter bridge mode to two signal conditioning cards (Sensorex, France, 2-way conditioning unit, 9350 series) to transform electric signals into micro-deformations. No significant variations of temperature were supposed to occur during such short runs of measurement (30 s). Therefore, the use of half bridge mode for temperature compensation did not appear justified in these experiments. The gauges were glued with liquid Loctite 401 multi-usage glue to the wood after having removed the bark locally at 5 cm from the point of origin of the branch.
A pair of branches was equipped with two gauges on each branch, allowing the measurement of the bending strains in both the vertical and horizontal planes. The branches were oriented in diametrically opposed directions and inserted in the same whorl on the stem. Two other pairs of diametrically opposed branches were equipped with one gauge on each branch, allowing recording of the bending strain in the vertical plane only. Spatial orientation of both pairs was similar and closely matched the X direction of initial displacement. The branches of the first pair originated in a whorl located in the lower half of the stem. By contrast, the second pair of branches was inserted in a whorl located in the upper half. Finally, a branch (2°A) and its three daughter branches (3°A) were equipped with one gauge each to measure the bending strain in the vertical plane. Altogether, 12 gauges were used on sapling (S1).
Theory of free oscillations: equations of movement:
Free oscillations of the system occur after an initial displacement has been imposed and released. On release of the structure, the system oscillates until it reaches its equilibrium state again. The movement of a given system occurs only through its modes of displacement when external forces are absent. The movement associated with each mode can be expressed by the motion equation for a single degree-of-freedom system (Clough and Penzien, 1975):
 | (1) |
The response in displacement for one mode is given by the solution of equation (1), namely:
 | (2) |
is the time independent amplitude of the oscillations and 
is the phase shift. Their expressions are not detailed here (cf. Clough and Penzien, 1975). 
and d are the angular frequency and dampened angular frequency of the oscillations and are expressed by the relationship (equation 3).
 | (3) |
is the damping ratio, defined as the ratio between the damping c and the critical damping of the system as defined in equation (4).
 | (4) |
The global response of the system can be expressed as a linear combination of the response of each activated mode. The number and the nature of the deformation modes included in the response depend on the time variations and the geometry of the load as well as on the mass and stiffness properties of the structure. Few deformation modes are active in the case of trees in free oscillations or in wind-induced oscillations. Only the natural swaying mode which is the fundamental (of lowest frequency) bending mode is usually active, although a second mode of higher frequency can sometimes also participate in tree oscillations (Mayer, 1987).
Determination of the dynamic characteristics of the structure:
The swaying frequency, f, and damping ratio, , of the modes were obtained by fitting the theoretical response given by equation (2) with the experimental response of the system in free oscillations. Experimentally, f was derived from the modal response curve as being the inverse of the mean time between two successive amplitude peaks. Because of equation (3), and d are equivalent for low dampened systems, i.e. <0.2.
Theoretically, f is expressed as follows:
 | (5) |
). However, systems that are composed of numerous axes are usually solved by numerical approaches because of the complex coupling between the different axes (Fournier et al., 1993).
Experimentally, was determined by fitting a decreasing exponential law on the successive displacement peaks. Theoretically, the damping of motions results from complex physical processes and is sometimes expressed by the mean of empirical relationships such as for damping induced by aerodynamic drag:
 | (6) |
air is the air density, Cd is the drag coefficient, and A the area of the element exposed to the fluid. However, other sources of damping such as damping induced by branch oscillations cannot be expressed by such relationships. An experimental approach is commonly the only way to determine the damping of real systems properly.
Signal processing:
When several modes were superimposed in the experimental response, the time series of displacement associated with each mode was determined through Continuous Wavelet Transform (CWT). Wavelet transform (Mallat, 1999) consists in representing a signal in the time-scale domain. Contrary to Fourier transform (frequency domain), the timing and the duration of an event is not lost after the transform is applied. A complex Morlet wavelet was used in this study's analysis. It was chosen because it has a better resolution in frequency than time compared with a Gaussian wavelet (Hangan et al., 2001). Furthermore, using a complex wavelet made it possible to uncouple phase and amplitude of the responses. Representation of the energy density of a given signal in the time-scale domain is called a scalogram. Scalograms were used in this study to show the deformation modes that were active in the free oscillations of saplings.
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Results |
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Aerial architecture of saplings
Distributions of mass and geometry by branching order are summarized in Table 2. The mass of sapling (S2) was 30% lower than that of sapling (S1) whereas its height was about 10% lower. The mean basal diameter of its branches was, according to the branching order, 11–17% lower than that of the branches of sapling (S1). Despite these morphological differences, their branching patterns shared several similarities. The relative amount of biomass allocated to the crown (60%) and foliage (45%) was equal for both saplings. Their crowns were eccentric in the same proportions, namely 13 cm from the stem, and in comparable directions. During the growth of sapling (S2) however, the 3°A development was larger than that of sapling (S1). It is shown by the total length of its axes (23 m versus 11 m), their number (114 versus 46), and their relative biomass (19% versus 13%).
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Sapling (S3) was completely different from an architectural point of view. A noteworthy discrepancy was that the apex of its primary axis had died prematurely, resulting in a fork composed of two 2°A of equivalent dimensions. The total height of the sapling was 32% lower than that of sapling (S1) whereas the total mass of those saplings was very close. The crown biomass of sapling (S3) represented 75% of its total mass. As said previously, it was only 60% for the other saplings. Sapling (S3) was the most branched of the subjects with 181 branches and a length of axes equal to 58.6 m altogether. It also appeared to be the stiffest with significantly higher basal diameters for each branching order than the other saplings (as a reminder, the bending stiffness of a cylindrical axis depends positively on d4 whereas its mass increases with d2). The crown of sapling (S3) was eccentric, although the direction of eccentricity was completely different from what was observed for (S1) and (S2).
The ratio between maximal widths across the X direction and the Y direction of loading was 1.16, 0.86, and 1.10, respectively, for the crowns of (S1), (S2), and (S3). The ratio values were calculated in order to estimate the elliptic shape of the crowns as viewed from above. The relative height of the centre of mass was constant at about 42% of total height between the experimental subjects, despite the featured differences of their aerial system.
Dynamic characteristics of the whole saplings
Modes of deformation:
Only one mode of deformation was observed in the free oscillations when the displacement was applied successively three times at the first third of tree height or when it was applied once at the tree top. However, an additional mode of deformation was active when the movement was initiated by displacing the sapling stem once and at one-third of tree height. The oscillations associated with the second mode were initially of the same magnitude as those of the first mode, but decreased more quickly. The second mode was indeed a short-lived event which occurred only during the very beginning of free oscillations. It was observed on sapling (S3) that oscillations occurring at an intermediate frequency between the two modes was mainly transferred towards the first mode (Fig. 4).
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The participation, or the non-participation, of the different deformation modes did not depend on the direction in which the initial displacement was applied. Moreover, all the saplings behaved in a similar way for the same cases of initial displacement.
Oscillatory frequencies:
The oscillation frequency of saplings (S2.s) and (S3) in the first deformation mode, f0, were very close with 0.76 Hz and 0.8 Hz, respectively. For sapling (S1), f0 was significantly lower at 0.6 Hz. When tested in December, sapling (S2), namely (S2.d), happened to sway at an increased value of f0 (1.03 Hz). As the frequencies of this mode were the lowest that were measured, it is referred to as the fundamental mode. The values of f0 depended neither on the case of initial displacement nor on its direction. The values of f1 (frequency of the second mode) are given as an estimate, since it was not possible to determine them accurately on the basis of time series sampled at 10 Hz. The values of f0 and f1 are given in Table 3 concurrently with the damping ratios.
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Damping of tree motions:
Only the damping ratios associated with the first mode of oscillations are presented in this section as this mode did not depend on the initial displacement and since it was dominant in the time-displacement response. Sapling (S1) presented a mean damping ratio of 0.098. In other terms, the amplitude of its rotations was divided by two at each oscillatory cycle. The stem movement was attenuated less efficiently for saplings (S2.s) and (S3), with respective damping ratios of 0.063 and 0.062. If compared to (S2.s), i.e. the same sapling three months earlier, the ratio of (S2.d) decreased to 0.051.
The case of initial displacement led to no significant differences between the damping ratios. However, the direction in which the sapling was initially pulled had a major influence. The difference between the damping ratios for X and Y directions, 
and 
was significant (P <0.05) for saplings (S2.s) and (S2.d) and very significant (P <0.001) for sapling (S1). However, no significant dependence on the initial direction was found for the remaining sapling, (S3). The significance tests were performed on two samples of eight values each for (S2.s) and (S1) and two samples of four values for (S2.d) and (S3).
Relative influence of the different architectural elements
The role of needles:
According to the results obtained for sapling (S1) shown in Table 4, the damping that was induced by the presence of needles in the crown was the major source of damping. After the removal of foliage, decreased from 0.118 to 0.038 and from 0.078 to 0.044. After the removal of needles during the successive pruning performed on (S1), the difference between and was no more significant during the oscillations performed later. On the other hand, the difference between and remained significant (P <0.05) through the pruning phases carried out on sapling (S2.d). However, contrary to sapling (S1), its remaining axes still had needles at each intermediate phase.
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From the strain measurements carried out on the branches of sapling (S1), it was observed that once the foliage had been removed, a deformation mode appeared in the oscillations of branches in addition to the first mode. Whereas the first deformation mode was shared by both the stem and branches, the other was active only in branches. The frequency of this last mode was different between branches and ranged from 2.5 Hz to 4.17 Hz. In other terms, it happened to be a distinct mode for each branch. This mode was not active when the foliage was still present and the branches oscillated only according to the same modes of deformation than those of the stem.
Finally, without foliage, the main plane of oscillations switched direction several times during a swaying test (Fig. 5). With foliage, however, the main plane of oscillations was constant over time and stayed oriented in the initial direction of release.
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The role of vegetal axes:
The stem and the branches of the whole sapling (S1) oscillated at the same frequency of 0.6 Hz, i.e. according to the first mode. The second mode, characterized by a frequency of 2.14 Hz, that was observed in the stem movement, was also observed in the oscillations of several tested branches when the first case of initial displacement was applied. However, only the first mode was active in the oscillations of every branch. No correlations were found between the position of the branches in the crown and the possible presence of the second mode in their oscillations.
For the branches where it was measured, no significant differences were found between the frequency of vertical and lateral oscillations. The diametrically opposed branches oscillated vertically in phase opposition. By contrast, the branches that belonged to the same vertical plane and on the same side of the stem oscillated with no phase shift (Fig. 6). Moreover, the branches oriented in the direction of the movement bent upwards, whereas branches on the opposite side bent downwards. When the stem reached its maximal amplitude and began to sway in the opposite direction, the direction of branch oscillations changed with a phase shift with the stem.
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With or without bearing foliage, the three instrumented 3°A and their mother branch always oscillated at similar frequencies (Fig. 7). Moreover, the 3°A swayed in phase with the movement of the parent branch.
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Finally, the mean damping ratio increased in a significant way consecutively to the removal of every 3°A for saplings (S1) and (S2.d) (Table 4). In the case of (S2.d),
0 increased from 0.051 to 0.059. In the case of (S1), 0 increased from 0.041 to 0.052, namely by more than 25%, which is noteworthy due to the fact that the 3°A were not bearing needles and represented less than 1% of the sapling biomass before being pruned. |
Discussion |
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When submitted to a dynamic load, the trees behaved as a system of coupled and dampened oscillators. In the case of free oscillations, the nature of the coupling between the different axes directly depended on their topological position in the aerial system. Therefore, the motions of given axes were forced by the motions of their respective mother axis. As a consequence, the stem immediately appeared to govern the oscillations of the whole system. Its modes of displacement were transmitted to the 2°A, and, subsequently, from the 2°A to the 3°A. This purely topological effect, which resulted from applying the mechanical load on the 1°A, was further reinforced by the fact that, for any dynamic system, the predominant elements in the response are the heaviest ones. For trees, the mass of individual axes usually decreases as their branching order increases.
Foliage significantly affected the efficiency of the coupling of motions in the aerial system of the plant. First, when the needles were present, the vertical oscillations of the 2°A were correlated with their location in the crown. After needle removal, this was not observed anymore. Furthermore, without foliage, the movement of the 2°A was composed of both forced swaying, induced by the stem oscillations, and free swaying. The hypothesis that the 2°A oscillated partly at their own natural frequency (or more accurately, the natural frequency of the subsystem composed of the 2°A and its borne axes) relies on the fact that these oscillations were different for each 2°A and not present in the stem movement, therefore defining a singular mode for each 2°A. Free oscillation tests carried out independently on each 2°A would be necessary to support this hypothesis fully. The foliage, nevertheless, played a significant role by making the tree structure to oscillate as a whole.
The deformation mode of lowest frequency that was observed in every test was the fundamental mode of bending of the structures. Its frequency corresponded approximately to the oscillation frequency of the stem alone, reduced by the additional mass of the branches. The existence of the second identified mode relied heavily on the case of initial displacement. This mode could either be the first mode of stem torsion of the stem or the first harmonic of bending of the stem. Due to the bending nature of the initial loading, it can be assumed to be a bending mode and, therefore, the first harmonic. Nonetheless, the nature of this mode is not discussed further here as the saplings were not equipped to record the torsion strain in the stem.
The frequency of the oscillations in the first mode, i.e. the natural swaying frequency of the tree, varied between the saplings tested. Since saplings (S1) and (S3) had very similar total biomasses and relative heights of mass centre, the fact that sapling (S3) had a higher natural swaying frequency than sapling (S1) is explained by their difference in slenderness only. The natural swaying frequency of sapling (S3) is higher due to its smaller size and its higher basal diameter. For sapling (S2.s), the comparison with (S1) and (S3) is less evident, since its total biomass and the geometrical characteristics of its stem were very different. It is noteworthy that its natural swaying frequency was closer to that of the sapling that had fewer similarities in its crown pattern.
The increase in frequency of sapling (S2) which occurred after the 3 month interval was coherent with the theoretical expression of swaying frequency (cf. equation 5). The secondary growth of the stem that occurred during the same period led to an increased stiffness of the stem. Although late secondary growth was unexpected at this period of the year, it has previously been recorded by Lemoine (1979) for Maritime pines. Nonetheless, more accurate predictions of the swaying frequencies would require numerical modelling (Fournier et al., 1993) to understand the interactions between the stem and the branches.
The variations of the natural swaying frequency recorded through the successive pruning performed on saplings (S1) and (S2.d) were consistent with previous observations made on adult trees. First, a high percentage of the crown biomass has to be pruned before any change can be seen in the swaying frequency of the whole structure (Moore, 2002). Furthermore, the higher the centre of the mass, the lower the natural frequency (Sugden, 1962). Thus, if pruning leads to a noteworthy lowering of the centre of the mass, it also leads to an increase in natural frequency.
According to the results of this study, the foliage was the major source of damping in the structure. It is consistent with previous observations describing the aerodynamic drag of foliage as being the dominant factor when neighbouring trees are absent (Milne, 1991). It also makes sense physically, since aerodynamic damping increases with the squared value of the displacement velocity as shown in equation (6) and since needles are located on the extreme part of the axes where their relative velocity is the highest. The influence of the remaining sources (wood viscosity, soil/root friction, branch oscillations) of damping in structural movement cannot, however, be neglected, since they reached, for example, 40% of total damping in the case of sapling (S1).
The 3°A had a negative effect on the amount of damping: the oscillations of the saplings were at least 15% less dampened with the 3°A than without. Such an effect was unexpected because the structural properties (geometry and mass) of the 3°A were weak compared with the other elements composing the structure. This effect on damping could be explained by the topological location of these axes. This mechanism has still to be investigated. Moreover, the negative influence of the 3°A explains why the motions of saplings (S2.s) and (S3) were less dampened than those of sapling (S1). Indeed, these saplings have favoured the development of 3°A during their growth compared to sapling (S1).
According to the observations of this study, the damping anisotropy directly results from the foliage. After foliage removal from sapling (S1), anisotropy was no longer significant. The drag induced by the foliage directly depends on its frontal area opposed to tree movement. Crown width ratios between the directions of initial displacement allowed the foliage area to be estimated since it was not measured. For saplings (S1), (S2.s), and (S2.d), the initial direction for which damping ratio was the highest was equivalent to the direction where width was higher. Standard deviations resulting from the determination of the damping ratios highlight the need for further studies to confirm the present measurements.
Although the equations of movement take into account the geometry and material properties of the system studied, the results cannot be extended directly to older trees because of the modification of shape and material during growth. The measurement of the tree aerial architecture did not bring a definite way to assess its dynamic characteristics. As stated previously, two trees with an important discrepancy in their crown patterns can have a similar dynamic behaviour. However, it does not mean that architecture is not influential since it was observed that the different elements in the crown were plainly involved in the response to mechanical loading, especially the foliage. This points to the fact that several coupled parameters are used to describe the tree architecture and that their relative influence might compensate for each other. Uncoupling such parameters could be done in the future through numerical modelling by using the data and the results presented in the present study.
When the tree oscillations are wind-induced, movements are transmitted from the foliage to the stem through branches. The resulting load on the tree is the opposite of free oscillations. Which modes of displacements are activated by the wind and how gust energy is transmitted through the crown elements to the stem, are still open questions.
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Acknowledgements |
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This study was funded by the GIP-Ecofor as part of the VENFOR project. Many thanks to Dr Yves Brunet, Professor Barry Gardiner, Dr Patrick Lac, Dr Slobodan Mickovski, and Dr Alexia Stokes for their scientific assistance. The authors would also like to thank Damien Soulier and Frederic Lagane for their invaluable help with many technical aspects.
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References |
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