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Ecology |
Institut für Biologie III, University of Freiburg, D-79104 Freiburg, Germany; Forstliche Versuchs- und Forschungsanstalt Baden-Württemberg, D-79100 Freiburg, Germany
Received for publication December 11, 2006. Accepted for publication August 7, 2007.
ABSTRACT
To further understand the mechanics of trees under dynamic loads, we recorded damped oscillations of a Douglas fir (Pseudotsuga menziesii) tree and of its stem without branches. Eigenfrequencies of the branches were calculated and compared to the oscillation frequency of the intact tree. The term eigenfrequency is used here to characterize the calculated resonance frequency of a branch fixed at the proximal end to a solid support. All large branches had nearly the same frequency as the tree. This property is a prerequisite for the distribution of mechanical energy between stem and branches and leads to an enhanced efficiency of damping. We propose that trees constitute systems of coupled oscillators tuned to allow optimal energy dissipation.
Key Words: biomechanics • damping • Douglas fir • oscillations • resonance • theory of bending vibrations
Upon dynamic wind loads (Scannell, 1984
; Schindler and Mayer, 2003
), trees may undergo large sways and even fail in a resonance catastrophe. Oscillation damping is essential for the survival of a tree. Here we describe how "structural damping" (Niklas, 1992
) accounts for a rapid decay of oscillations in a Douglas fir (Pseudotsuga menziesii). Such damping greatly reduces the danger of tree failure in gusty winds.
Oscillation and oscillation damping of trees have widely been studied (Mayhead, 1973
; Scannell, 1984
; Milne, 1991
; Peltola et al., 1993
; Brüchert et al., 2003
; James, 2003
; Moore and Maguire, 2005
; James et al., 2006
; Spatz et al., 2006
) and theoretically analyzed (Kerzenmacher and Gardiner, 1998
; Brüchert et al., 2003
). Reviews with additional citations are available (Mayer, 1987
; Moore and Maguire, 2004
). "Pull and release" experiments led to a description of eigenfrequencies and damping ratios in relation to the architecture of plants (Speck and Spatz, 2004
; Sellier and Fourcaud, 2005
; Brüchert and Gardiner, 2006
). Finite element analysis provided a theoretical description of these oscillations (Moore and Maguire, 2006
; Sellier et al., 2006
).
When friction among different trees or among different branches and dissipative mechanisms in the root–soil system are set aside, there are two principal sources of damping: aerodynamic damping and viscous damping within the material. In complex structures like trees, processes can enhance damping. It can often be observed that in gusty winds branches do not sway in line with the stem but instead move relatively independently. In this case, energy is distributed between the stem and branches, where it is dissipated more effectively than when the structure is too stiff to allow relative movement between stem and branches. The phenomenon is referred to as structural damping (Niklas, 1992
) and has also been observed in plants such as the giant reed Arundo donax (Speck and Spatz, 2004
) and Stipa gigantea (Spatz et al., 2004
). Structural damping can be effected by loose coupling to a mass (termed mass damping; James, 2003
) or by the distribution and dissipation of mechanical energy through resonance phenomena within the tree (James et al., 2006
; Spatz et al., 2006
). We use the term multiple resonance damping for the latter process.
To contribute to a better understanding of the physics of tree mechanics under dynamic loads, we focused on the physics of oscillation damping in a complex system such as a tree. We demonstrate that the principle of multiple resonance damping is valid in a young Douglas fir.
THEORETICAL CONSIDERATIONS
Eigenfrequencies
The fourth order differential equation for free, undamped bending oscillations of an upright slender rod or stem without an apical load can be formulated as an equilibrium of distributed loads:
|
| (1) |
is the density of the stem at a particular height, and 
is the circular frequency.
Eigenfrequencies can be calculated by solving this differential equation with appropriate boundary conditions, implying inter alia that the objects are embedded at their base (Spatz and Speck, 2002
). This approach is possible even for tapered objects with a gradient of the modulus of elasticity and of its density along its length. For horizontally oriented objects (in our case the branches), exact solutions are available. For vertically oriented objects (the tree or the stem), close approximations can be obtained (Spatz and Speck, 2002
).
For the formulation of the differential equation, a coordinate z = x/L along the length L is defined with z = 1 at the base and z = 0 at the tip. The taper is described as the diameter d = dBz
, with dB the diameter at the base and the tapering mode. Analogous power functions describe the modulus of elasticity E as E = EBzß and the density as = Bz
.
The differential equation for undamped oscillations can then be written as
|
| (2) |
|
| (3) |
|
| (4) |
/4 (dB/2)4. For horizontal objects, the gravitational term G = 0. With the help of the powerful program Mathematica 4.0 (Wolfram Research, Champaign, Illinois, USA), we can find numerical values for combinations of G and H, which characterize the fundamental frequency of free vibrations.
Damping due to aerodynamic resistance can be calculated under the assumption (to be disproved later) that the branches oscillate in line with the stem, with the same frequency, amplitude, and phase (Speck and Spatz, 2004
). The loss of mechanical energy in every half cycle is computed, and thus the decline of the amplitude of displacement with time can be estimated.
|
| (5) |
), the change of the modulus of elasticity (ß), and the change of the density () along the stem (Spatz and Speck, 2002
). For the tree studied, N4 = 0.00897.
If the loss of energy is only due to viscous damping within the material of the stem and to damping from the aerodynamic drag of the branches (and usually to a much smaller extent of the stem), the overall loss of energy can be approximated in discrete steps of half cycles by
|
| (6) |
|
| (7) |
The energy loss due to aerodynamic drag depends on the third power of the amplitude, while the energy loss due to viscous damping depends on the square of the amplitude. Therefore, the relative damping from these two sources is only additive for small changes of the amplitude, which necessitates a stepwise procedure to calculate the overall damping.
For small velocities, the aerodynamic drag force FD is proportional to the square of the velocity v:
|
| (8) |
Air the density of air, and A the sail area. For velocities higher than those encountered here, CD and A can no longer be considered as constant.
The energy loss can be obtained by integration over a half cycle:
|
| (9) |
in Eq. 9 with 
The calculation of the amplitude yn(hi) at height hi requires the information about the bending line. If the tapering mode (), the change in the modulus of elasticity (ß), and the change in the density () along the stem are known, the bending line can be obtained as the solution of the differential equation for bending oscillations (Eq. 2).
MATERIALS AND METHODS
Morphology
In a forest near Freiburg, Germany, a 12-yr-old, 5.03-m-high Douglas fir with a basal diameter of 7.0 cm was studied, and the following characteristics were measured: the height of the tree, the length of all 90 primary branches and 28 primary sprigs with length 
0.2 m, the diameter of branches and sprigs at the base, their orientation in space, and the height above ground of their origin. After oscillations were recorded (see Recording of oscillations section), the tree was felled. Stem and branches, placed in plastic bags with wet towels to prevent drying, were carried to the laboratory, stored at ambient temperature, weighed, and photographed in two orientations. The taper of the stem was obtained from the relation between diameter and height. Also the taper of 15 primary branches was determined.
Mass distribution
One day after felling, the mass distribution along a 1.54-m-long primary branch, including secondary and all higher-order branches, was determined quantitatively by weighing sections of 5 cm length. The resulting distribution was confirmed later for 24 branches of different length from a neighboring tree.
Moduli of elasticity along the length of the stem and of 14 primary branches of different basal diameter were measured in three-point-bending experiments using an Instron universal testing machine (IST GmbH, Darmstadt, Germany). All measurements were carried out within 1 wk after felling.
Recording of oscillations
All field measurements were carried out on a dry and calm day in June 2004 after a longer period of dry weather. This ensured that the tree was well rooted in the ground. Neighboring trees were bound back to prevent contact of the swaying tree with their branches. Extensometers (Wessolly, Stuttgart, Germany) were attached in two orientations at a height of 2.26 m above ground. This way large enough amplitudes of the oscillation could be recorded without undue influence of the extra mass of 320 g of the extensometers.. Recording equipment was developed at the Institute for Forest Utilization, University of Freiburg. A rope attached to the tree 2.63 m above ground, some distance above the extensometers, was used to pull the tree out of its resting position by 0.3 to 0.4 m measured at the height of the extensometers. After release, the output of the extensometers was recorded for several seconds with a sampling interval of 0.1 s. A calibration gave the absolute values of the displacements.
After oscillations of the intact tree were recorded, all primary branches and sprigs and the top 80 cm of the stem were cut off, and the oscillation of the remaining stem was recorded.
Data evaluation
The output of the extensometers was transferred to Excel and simulated by the function
|
| (10) |
and the characteristic decay time 
. Damping ratios are expressed as the ratios of the loss modulus to the elastic modulus
|
| (11) |
RESULTS
Morphometric data
The taper of the stem was nearly linear and the cross-section nearly circular. The length and diameter of all branches are depicted in Fig. 1 as a function of the height of their origin above ground.
|
= 0.716, r2 = 0.96.
|
equiv = 239 x height + 2481, r2 = 0.0665.
Figure 3 shows the mass distribution along the axes of a 1.26-m-long branch. The distribution is typical for a total of 25 branches with length between 0.23 m and 1.37 m. From these data, the power function of the equivalent density (the effective mass per volume) along the primary axis can be calculated with an exponent = –0.457 x length – 0.848, r2 = 0.564 for branches over 0.2 m long.
|
; Blackburn, 1997
; Brüchert, 1998
). The top part of the Douglas fir has a much lower stiffness than the main part of the stem. The data can be approximated by a power function with an exponent ß = 0.8. While this appears to be a very rough approximation, the calculations are not very sensitive to the value of ß. A variation of ß within the limits 0.6 ß 1.0 leads only to a variation of calculated within 6%.
|
|
, an indirect confirmation of the value for ß. It should be noted, however, that this assumption may not be justified for older trees (Spatz and Brüchert, 2000
).
|
= 3.78 ± 0.09 s–1, N = 12, agreed fairly well with the computed value of calculated = 4.47 s–1.
|
= 11.4 ± 0.5 s–1, N = 9, of the debranched stem can be accounted for by the reduced length of the stem and the loss of the mass of the branches. The computed value was calculated = 12.26 s–1.
|
also observed high damping ratios for some of the unpruned Douglas fir trees in their study. Because under the conditions of our measurements damping in the root–soil system should be minimal (Speck and Spatz, 2004
), this value should amount to the sum of aerodynamic damping and viscous damping in the green wood.
Under the assumption (to be disproved!) that the branches do move in line and in phase with the stem, we can derive a theoretical estimate for aerodynamic damping of the tree (Speck and Spatz, 2004
). For this purpose, the sail area (A) of each of the 90 branches and 28 sprigs was estimated from the photographs. The total A was 2.37 m2 or 1.79 m2 when corrected for shading within every whorl. By recording the amplitude of the oscillation and computing the relative amplitude as a function of height, we determined that aerodynamic damping was approximately as effective as damping within the wood of the stem (Fig. 7). Using a drag coefficient 0.5 CD 1 (Niklas, 1992
; Wood, 1995
; Spatz and Brüchert, 2000
), we obtained an upper estimate of 0.035 E''/E' 0.05. This low value results from the fact that the amplitudes of the oscillation are small, particularly for the lower part of the tree, which has large branches (Fig. 1).
Damping in the debranched stem (E''/E' = 0.042 ± 0.007, N = 8) was much less than in the intact tree. Because aerodynamic drag is negligible for the debranched stem, this value represents the viscous damping in the wood of the stem. If we again assume that the branches move in line and in phase with the stem, this value should also represent the viscous friction in the wood of the tree with all its branches. Clearly, the two damping mechanisms considered do not sum up to the damping of the tree. According to the principle of proof via contradiction, the relative movement of the branches with respect to the stem, and similarly the relative movement of the rather floppy top part of the tree with respect to main part the stem, must strongly influence the damping of the tree: structural damping must account for the largest part of the tree's capability to reduce the amplitude of oscillations.
Structural damping can even be observed in a simple physical model made of steel wire and pieces of cork and consisting of a main axis and two side arms (Spatz et al., 2006
) As long as the frequency of the entire structure was sufficiently different from their eigenfrequencies, the side arms vibrated in line with the main axis. However, when the frequency was close to the eigenfrequencies of the side arms quite complex movements were observed; the side arms oscillated with phases that differed from each other and from the main axis. In this mode of damping, energy is transferred from the main axis to the side arms, and damping is almost twice as high as in the case where the frequencies are sufficiently different.
Calculation of the eigenfrequency of the branches
In finite element calculations of virtual trees, Moore and Maguire (2006)
showed that overall damping of a tree would depend on the modulus of elasticity of its branches and thus on their eigenfrequencies. Here we propose that structural damping in the Douglas fir tree is due to multiple resonance damping. For multiple resonance damping, the eigenfrequencies of the branches should be close to the natural frequency of the entire tree. Figure 8 shows that this is indeed the case for all branches longer than 0.5 m.
|
Pull and release tests can be used to describe the physical principles of oscillation damping in a complex system like a tree. Our observations reveal that the remarkably high damping in the tree studied is mainly due to a distribution of mechanical energy between stem, primary side branches, and higher order branches, where the energy is dissipated more effectively. Analogous concepts have been developed by Scannell (1984)
and James et al. (2006)
.
The mechanism of multiple resonance damping is well known in the engineering sciences (Thomson and Dahleh, 1998
), in particular where damping elements are installed in high-rise buildings (Holmes, 2001
) as in the Taipeh Tower. Coupled damped oscillators can exchange energy provided that the resonance spectra have some overlap (Fig. 9). Their half-width depends on the degree of damping.
|
In a natural situation, the wind acts primarily on the branches, which then exchange mechanical energy with the stem. A good alternative to pull and release tests is the observation of the movements of a tree in the wind and a Fourier analysis of the data (Gardiner, 1995
; James et al., 2006
). It can be observed that in turbulent wind, branches will undergo large relative movements, while the stem is less affected. Energy exchange is not only a question of amplitudes and frequencies, but also of phase relations. Under certain conditions, the excitation from two or more branches can partly cancel each other.
The answer to the question of how trees escape dangerously large oscillations of the stem is that the tree reacts to dynamic wind loads as a system of coupled damped oscillators. The tree's capacity to distribute energy over the entire structure leads to a higher damping ratio and additionally to less strain on the stem as compared to a structure with much stiffer side branches. Consequently, the danger of failure of the tree in gusty winds is greatly reduced.
This contribution focuses on the physics of oscillation damping in a complex system like a tree. Our measurements, limited to one Douglas fir tree, can only demonstrate that trees can utilize multiple resonance damping. We do not imply that all trees or even all Douglas firs can benefit from this principle. However, a similar relation between eigenfrequencies of a tree and its branches has been reported by Fournier et al. (1993)
for Pinus pinaster. Further studies are necessary to test whether multiple resonance damping also applies to other conifers as well as to deciduous tree species.
FOOTNOTES
1 The authors are indebted to the Institute for Forest Utilization, University of Freiburg, for lending their equipment to record oscillations. They also thank two unknown reviewers for many helpful suggestions to improve the manuscript. 
4 Author for correspondence (spatz{at}biologie.uni-freiburg.de
)
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