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(American Journal of Botany. 2004;91:789-796.)
© 2004 Botanical Society of America, Inc.

Biomechanics

Damped oscillations of the giant reed Arundo donax (Poaceae)1

Olga Speck2 and Hanns-Christof Spatz

Competence Network Biomimetics, Plant Biomechanics Group Freiburg, Fakultät für Biologie, Albert-Ludwigs-Universität Freiburg, Schänzlestr. 1, D-79104 Freiburg, Germany

Received for publication September 19, 2003. Accepted for publication January 30, 2004.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 LITERATURE CITED
 
The slender upright culms of the giant reed (Arundo donax L.) are often exposed to dynamic wind loads causing significant swaying. The giant reed has slightly tapered hollow stems (4–6 m high) with flat leaves and an extensive underground rhizomatous system with solid branches bearing adventitious roots. Quantitative analyses of videorecordings prove that A. donax responds to dynamic deflections of the stem with damped harmonic bending oscillations. The logarithmic decrement can be used to calculate the relative damping, as a measure of the plant's capacity to dissipate vibrational energy. Plants with leaves have a significantly higher damping compared to plants without leaves. A comparison of the relative damping of plants with and without leaves shows that this finding is only partly due to aerodynamic resistance of the leaves. Structural damping also contributes considerably to the overall damping of the foliate A. donax stem. By stepwise removal of the underground plant organs the influence of rhizome, roots, and soil on the vibrational behavior was determined. The data indicate that underground plant organs as well as leaf sheaths covering the nodes have no significant influence on damping.

Key Words: aerodynamic damping • Arundo donax • biomechanics • material damping • oscillation • plants • structural damping


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 LITERATURE CITED
 
If the dense stands of the giant reed (Arundo donax) are subjected to dynamic wind loads, the slender culms respond with bending vibrations and pronounced damping. Four ways of dissipating the mechanical energy can be distinguished: (1) friction between neighboring plants in a dense stand, (2) aerodynamic resistance of the leaves and, to a lesser extent, of the stem (aerodynamic damping), (3) damping as a result of the movement of lateral or terminal organs such as leaves relative to the stem (structural damping), and (4) damping generated within the tissues of the plant (material damping) (cf. Niklas, 1992 ). In natural habitats A. donax grows in dense communities, and friction between neighboring plants may contribute significantly to damping of oscillations (cf. Farquhar and Eggleton, 2000 ; Doaré et al., 2003a , b ). However, especially at the edge of stands, where often the highest wind loads are found, the plants behave at least partly as individuals (Speck, 2003 ). In this study, we only investigated the oscillation and damping behavior of individual plants. By stepwise removal of plant organs, the influence of leaves, leaf sheath, rhizome, roots, and soil on the vibrational and damping behavior was determined.

Other than in the engineering sciences the term "structural damping" is used in the biological literature (Niklas, 1992 ; Brüchert et al., 2003 ) to describe damping that results from the oscillation of lateral or terminal organs with different frequencies (Fournier et al., 1993 ) or phases from those of the main part of the stem. This leads to energy transfer to the periphery where it is dissipated most efficiently via aerodynamic or material damping.

Measurement of oscillation frequencies of plant stems offers an efficient way to determine mechanical properties related to energy storage and dissipation such as storage modulus and loss modulus (Young, 1989 ; Vincent, 1990 ; Niklas, 1992 ; Speck and Spatz, 2000 ). Zebrowski (1999) and Spatz and Zebrowski (2001) measured the natural frequency of free vibrations of upright cereal plant shoots with an apical load. Spatz and Speck (2002) have generalized this approach for tapered plant stems with and without top loads, including cases in which the modulus of elasticity varies over the length of the stem.

For an integral understanding of the damping behavior of individual A. donax plants, the entire system, consisting of aerial and underground plant organs, has to be taken into consideration. The aim of this paper concerns three main aspects: the quantitative analyses of a plant's oscillation behavior, an estimate of the plant's capacity to absorb vibrational energy due to aerodynamic, structural, and material damping, and the influence of various plant organs on damping the oscillations.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 APPENDIX
 LITERATURE CITED
 
Plant material
The giant reed (A. donax L.) belongs to the family of Poaceae. It is distributed from the Mediterranean region to China and Japan (Walters et al., 1984 ). Hedgerows of A. donax plants are extensively planted in southern Europe to serve as wind shelters, because the slender swaying culms have excellent wind resistance (Tutin et al., 1980 ).

The giant reed commonly sends tall, hollow stems to a height of 4–6 m from an extensive, persistent underground rhizomatous system. The branched pachymorph rhizome is solid, bears scale leaves, produces extensive adventitious roots at the nodes, and terminates distally in a vertical culm (Bell, 1991 ). The hollow stems are divided into nodes and internodes, while glabrous leaf sheaths cover the nodes. With a maximum outer diameter at the base of about 3.5 cm, the slenderness ratio of the stems, calculated as length/ basal diameter is very high (to 100–200). Its leaves attain, depending on the height above ground, a length to 86 cm and a width to 7.4 cm at the leaf base. The alternate leaves are regularly spaced in two opposite ranks on the same plane (Everett, 1980 ).

Arundo donax was obtained from outdoor cultivations in the Botanical Garden of the University of Freiburg (Germany). A total of 12 plants in the first year of growth, already vegetatively entirely differentiated, were used. The tested plants were all clonal and propagated vegetatively from a single stock. They grew in nutrient-rich, sandy-loamy soil. Total lengths, total mass, internodal lengths, outer radius, wall thickness, and density of the culms were measured. In addition, size and mass of every single leaf were determined for each plant.

Experimental procedure
Experiments were carried out when no wind was noticeable. The oscillation was started by the release of the stem after it was displaced by hand. For two-dimensional analysis, the swaying stems were recorded from the side (Fig. 1) by video equipment (Sharp, Osaka, Japan, Model VL-C750S) at a frame rate of 25 or 50 frames/s. The electronic shutter of the camera was set at 1/1000 s to guarantee distortion-free images. A blue canvas of 3 x 4 m was used as background for the videorecordings and provided additional wind shelter. Points on the stems at different levels above ground were marked with high contrast tape.



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  		Fig. 1. Schematic drawing of the experimental setup for oscillation analyses with videorecordings

 
Image analysis
Video footage was transferred from the external video device onto the hard disk of the computer with the software FASTCap (Fast Multimedia, Washington, D.C., USA, 1999, version 2.1.411). After the analog/digital transformation, the videorecordings were presented in AVI (Audio Video Interlaced) format. The video footage (in AVI format) of the motion of the culms was quantitatively analyzed using the image analysis software SIMI°Motion (SIMI GmbH, Unterschleissheim, Germany). Both a coordinate system and time information were necessary. The coordinate system consisted of a "pass point system," in which the object space was calibrated, and a "zero point system," which ensured the precise congruence of the serial pictures. The time information was given by the exact run specification of the video camera (25 or 50 frames per second [fps]). The zoom was adjusted to ensure optimal spatial resolving power. The temporal resolving power was directly proportional to the frame rate of 25 or 50 fps (Baumann, 1982 ). For a sufficiently exact calculation of the oscillation frequency, the scanning rate had to exceed at least five-fold the oscillating frequency of the object; otherwise, the frequencies simulated might be too low (Waller and Schmidt, 1989 ). The defined-object points were digitized with the mouse and cursor in each video picture. These screen coordinates and the calibration coordinates were merged into a new data set of coordinates. These coordinates enabled the analysis of the oscillation and damping behavior of the object and the determination of the relevant parameters for subsequent simulation. The displacement of up to eight marker points along the swaying object was examined.

Alternative experimental procedure
To analyze more specimens than possible with the time-consuming video analysis with the potential to analyze 10–20 repetitions of every experiment, oscillation frequencies and damping were measured at only one level above ground (40% height of the plant stem) with a device (Fig. 2) originally designed to measure the damped oscillations of a torsion pendulum (Köhler et al., 2000 ). The object at rest was placed between two laser beams that were directed towards two slits on a detector unit. Upon swaying, the object interrupts one or the other of the light beams, resulting in a drop of the output voltage of the detector. The output is transferred to a computer via an analog/digital converter at a sampling interval of 2 ms. For every cycle, four interruption signals are recorded. The time between the interruptions of alternate laser beams provides a measure of the velocity of the object at the height of the optical recording. This is an indirect measure of the amplitude, as the velocity is the time derivative of the amplitude. Damping of the oscillation is observed as the decrease of the amplitude with time over several cycles.



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  		Fig. 2. Schematic drawing of the experimental setup for oscillation analyses with laser detector equipment

 
Influence of plant organs on oscillation damping
To determine the influence of plant organs on the oscillation and damping behavior of A. donax, plant organs were removed sequentially. Five experiments were carried out with each individual plant. Oscillations were recorded of the foliated plant (1), and the plant with leaves cut off (2) both rooted in the soil. The same plant was dug out and the rhizome without roots (3) was embedded in a plastic trough with a base of at least 0.6 x 0.36 m filled with 60 kg cement. To avoid any extra movement, the trough was guyed with four steel cables to the ground. Subsequently, (4) the stem without rhizome was embedded in cement as before. To afford a sure hold, the hollow stem was fixed with a mounting system consisting of an H-shaped aluminum plate with a thorn that rises into the cavity of the basal-most internodes of the stem, two half shells, and two hose clamps fixing the stem base to that thorn (Fig. 3). For the final experiment, the leaf sheaths were removed from the embedded plant stem (5). Relative damping of A. donax stems found under test protocol 5 was set to 100%, because this experiment was carried out for all plants. One of the plants was videorecorded; oscillation frequency and damping for the five remaining plants were measured as described under "alternative experimental procedure."



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  		Fig. 3. Equipment to ensure a sure hold of stems without rhizome. An H-shaped aluminum plate with a thorn (1) that was stuck into the pith cavity of the basalmost internodes of an Arundo donax stem is embedded in cement. The stem base is further fixed with two aluminum half shells (2) and two hose clamps (3)

 
Data evaluation
Quantitative analyses of oscillations
As will be shown in the results section, the bending oscillations represent damped harmonic oscillations, which can be described by the following equation:

(1)
where y is the displacement, y0 is the displacement at t = 0, {delta} is the decay constant, {omega} is the angular frequency, and t is the time.

For beams bending vibrations with small amplitudes can be described by second-order differential equations (Kamke, 1983 ). In its general form, the differential equation describing the equilibrium of bending moments for a plant stem without a top load but finite self-mass results in (Spatz and Speck, 2002 ; cf. their eq. 8c):

(2)
where L is the length of the stem, EB is the modulus of elasticity of the stem wall at the stem base, IB is the axial second moment of area at the stem base, {rho}B is the density at the stem base, AB is the cross-sectional area of the stem base, and {lambda}1 is the eigenvalue for the fundamental frequency. The eigenvalue {lambda}1 depends not only on the boundary conditions, but also on the tapering mode ({alpha}), the variation of the modulus of elasticity along the stem (ß), and on the variation of density along the stem ({gamma}) (cf. eqs. 3–6) (Speck and Spatz, 2000 ; Spatz and Speck, 2002 ; Brüchert et al., 2003 ). Both the shape of the bending line and the eigenvalue for any set of values for these parameters were obtained using Mathematica 4.0 (Wolfram Research, Champaign, Illinois, USA).

Definition of parameters
The variations of the properties I, E, and {rho} along the stem can be modeled by eqs. 3–6, where D is the absolute distance from the apex, and z the relative distance from the apex (z = D/L) is a dimensionless coordinate with z = 1 at the stem base and z = 0 at the apex of the stem:

{abot-91-06-15-e3}

where I(z) is the axial second moment of area, E(z) is the modulus of elasticity, EI(z) is the flexural stiffness, and {rho}(z) is the density.

Relative damping
The complex modulus of elasticity E can be resolved into its components: storage modulus (E') and loss modulus (E''). As the logarithmic decrement {Delta} is known as the reduction of the amplitude per oscillation, the relative damping (E''/E') can be calculated from the formula (Vincent, 1990 ; compare also eq. 2 in the Appendix):

(7)

Aerodynamic damping
An upper estimate of the loss of energy due to aerodynamic drag of the leaves can be calculated (see Appendix), taking into account that for small oscillation velocities of A. donax (<1.5 m/s), the drag force is proportional to the velocity squared (Speck, 2003 ).

Statistics
For the statistical analyses, we calculated means with standard deviation. Means were compared using either a Student's test or ANOVA (post-hoc test following Scheffé, SPSS, version 10.1; SPSS, Chicago, Illinois, USA; Sachs, 1993 ).


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 APPENDIX
 LITERATURE CITED
 
Morphological and mechanical properties
For the seven tested plants, a double logarithmic plot of the axial second moment of area as a function of the relative distance from the apex yields a slope that gives an average power coefficient of 4{alpha} = 0.75 ± 0.14 (Table 1). The value 4{alpha} + ß is determined from measurements of the bending stiffness as function of the position along the stem. A larger data set from the same clonal stand of A. donax (Spatz et al., 1997 ) gives a value of 4{alpha} + ß = 1.71 ± 0.11. Therefore, the variation of the modulus of elasticity can be described by an average power coefficient ß = 0.96 ± 0.14. The density of the tissues comprising the stem of A. donax without leaves was constant, therefore {gamma} = 0. To compare experimental values from foliated stems with the theoretical calculations, the mass of the leaves was included in an effective density that varies along the stem with {gamma}effective = –0.29 ± 0.03. This is justified for the quantitative analysis of oscillations (Brüchert et al., 2003 ). Knowing the length, the parameters {alpha}, ß, {gamma}, and the oscillation frequency, we can calculate the modulus of elasticity at the base according to eq. 2. For seven plants without leaves, we found an average of EB = 4.79 ± 0.70 GPa as compared to 5.23 ± 1.25 GPa determined by conventional three-point bending tests on an independent set from the same clonal stand of A. donax (H. Beismann, unpublished data).


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  		Table 1. Stem length (L), tapering mode ({alpha}), variation of the elastic modulus along the stem (ß), variation of the effective density along the stem with the mass of the leaves included ({gamma}effective), angular frequency ({omega}), modulus of elasticity at the base of the stem (EB), and relative damping (E"/E') of videorecorded Arundo donax plants with and without leaves. Means and standard deviations of determinations at different levels above ground are given. Due to uncertainty of the effective length of the A. donax plants with leaves, the modulus of elasticity is only shown for plant stems without leaves

 
Quantitative analyses of oscillation
For all test protocols, quantitative analyses of the videorecordings of swaying stems show that A. donax behaves as a damped harmonic oscillator (Fig. 4). With the exception of the first period, the amplitude decreases exponentially with time (Fig. 5) and can be precisely described by eq. 1. According to test protocols 1 and 2, two types of experiments were analyzed: oscillations of the foliate plant and oscillations of the same plant having the leaves cut off. A reduction of the distal mass by removal of the apical leafy stem part and the consequent reduction of stem length from 4.76 ± 0.45 to 4.13 ± 0.45 m caused an increase in the angular frequency of the stem's oscillation from 2.77 ± 0.51 to 3.91 ± 0.55 s–1 (Table 1). (Speck et al., 1998 , 2000; Speck and Fässler, 2000 ).



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  		Fig. 4. Displacement as a function of time of an Arundo donax plant with leaves (A) and without leaves (B) at eight levels aboveground. See figure insert for details

 


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  		Fig. 5. Displacement as a function of time of an Arundo donax plant with leaves (A) and without leaves (B) in 3.6 m above ground. With the exception of the first period, the amplitude of the curves determined from the videorecordings decreases exponentially with time (lines with open boxes). With this exception, the oscillation can be described very precisely by eq. 1 (solid lines). The envelopes (dashed lines) are exponential curves

 
With the help of Mathematica 4.0, relative amplitudes (y/ ymax) of the modeled swaying stems with and without leaves in different levels above ground can be calculated. A plot of the relative amplitudes calculated with Mathematica for the modeled stems and those determined from the video observations of the real stems gives a regression line with R2 = 0.9988 (Fig. 6), which proves that the oscillations can be described adequately as bending vibrations. For the set of seven stems of A. donax that were videorecorded and for which all geometric parameters were known, R2 = 0.9982 ± 0.0009 was found for stems with leaves and R2 = 0.9973 ± 0.0013 for stems without leaves. These coefficients of determination do not include the first period of oscillation (compare Fig. 5).



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  		Fig. 6. Relative amplitudes determined from videorecordings of a real Arundo donax plant without leaves plotted against relative amplitudes calculated with Mathematica 4.0 for the same modeled plant as described in the text. The 0 along both axes indicates zero amplitude at the base of the plant, and 1 indicates the maximum amplitude close to the apical tip. The regression line has a coefficient of determination of R2 = 0.9988. The data shown are taken after the first period of oscillation

 
Modes of damping
Because the decrease in amplitude per oscillation was determined from the analyses, the relative damping could be calculated according to eq. 7 (Table 1). Relative damping of plants with leaves and rooted in the soil yields within the first period a value of 0.36 ± 0.04, and after the first period, a value of 0.19 ± 0.05. E''/E' of plants without leaves amounts to 0.14 ± 0.02 within the first period, which is approximately twice as high as the value of 0.07 ± 0.02 found after the initial phase. Figure 7 shows the relative damping observed in seven plants during several periods of the oscillation for test protocols 1 and 2. Two aspects are evident. First, damping is much higher in the initial phase than in subsequent periods (P < 0.001). Second, damping is much higher for leafy plants than for plants without leaves (P < 0.001).



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  		Fig. 7. Values for relative damping (E''/E') of seven Arundo donax plants with (dashed line) and without leaves (solid line) plotted against the period number. The bold lines denote the averages. Two phases can be distinguished: an initial phase within the first period and a second phase for the subsequent periods

 
Starting from material damping within the culm (test protocol 2), an upper approximation shows that damping due to aerodynamic resistance of the leaves should increase the relative damping by at most a factor of 1.9, which does not account for the 2.8-fold difference observed in experiments. Thus, structural damping must play a significant role in leafy plants at least within the three periods of the oscillations observable in our experiments.

Influence of plant organs on oscillation damping
To compare the data of all studied plants, relative damping for plants without leaf sheaths (test protocol 5) was set to 100% (Fig. 8). E''/E' yields 470 ± 111 (n = 3) for plants with leaves (test protocol 1, not shown in Fig. 8), 123 ± 21 (n = 5) for plants without leaves (test protocol 2), 104 ± 10 (n = 5) for plants having the rhizome embedded in cement (test protocol 3), and 106 ± 8 (n = 6) for stems without rhizome embedded in cement (test protocol 4). As discussed, these data do not include the first period. The differences between leafy plants and plants having the leaves cut off confirm the data shown in Fig. 7. On the other hand, the small differences due to step-by-step removal of underground organs (roots and soil, rhizome), and leaf sheaths were not statistically significant. One exception should be mentioned: Plant 1 (dotted columns), videorecorded one year before the other plants, had a 1.5-fold difference between being rooted in the soil and being embedded in cement.



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  		Fig. 8. Relative damping E''/E' of six Arundo donax plants. The following experiments with different test protocols are shown in this figure: (2) plant with leaves cut off, (3) plant dug out and the rhizome (without roots) embedded in cement, (4) the stem (without rhizome) embedded in cement, (5) in addition the leaf sheaths were removed. The relative damping (E''/E') for plant stems without leaf sheaths (5) was set to 100%. Dotted columns of plant 1 are from videorecordings. All other measurements were carried out as described in Alternative experimental design and are means of 10– 20 observations per plant and test protocol. Error bars show their standard deviations. The horizontal line at 100% provides a visual guide

 

   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
APPENDIX
 LITERATURE CITED
 
Video analysis demonstrates that with the exception of the first period, swaying of individual A. donax stems can be described as a damped harmonic bending oscillation. A point force is applied to the stem before release. Therefore, the bending line (Spatz and Speck, 2002 ) differs in the initial phase from the later periods, when the forces during oscillation have to be considered as distributed loads. As will be discussed later, this may be part of the reason for the pronounced damping in the initial phase.

The influence of various plant organs on different modes of damping, such as aerodynamic, structural, and material damping, can be determined by stepwise removal of plant organs. Arundo donax stems with leaves have a much higher overall damping—composed of all three modes of damping relevant for isolated plants—than stems without leaves. An estimate of damping through aerodynamic resistance of the foliage amounts to an average of 37%. Structural damping contributes an average of 26% to the overall damping. In our case, this can be ascribed mainly to vibrations of the leaves with higher frequency components and/or different phase angles as compared to the stem. Caused by these asynchronies, mechanical energy is transferred to and dissipated in the leaves with higher efficiency than in the stem. Structural damping is particularly noticeable in the initial phase of the oscillations (up to 35%) and in very tall, slender foliated stems. The upper parts of the stem can oscillate differently from the basal part of the stem. Aside from this a considerable part of the overall damping of the leafy giant reed, as calculated from data of plants without leaves, can be attributed to material damping and yields an average of 37%.

In stems without leaves, damping is mainly a property of the stem itself. Because of the small projection area of the stem, aerodynamic damping of the stem alone is only 1/8 of that for the foliate stem, and therefore much smaller than material damping within the stem. In mechanical terms, the tissues comprising the stem can be seen as composite materials. In cross-section of the hollow stem (Spatz et al., 1997 ), the vascular bundles are almost evenly distributed in the lignified parenchyma, only slightly increasing in density toward the stem periphery. From the point of view of stability against static bending loads, it would be more efficient if all stiff sclerenchymatous fibers were located near the periphery. Embedding stiff material in a less stiff matrix will, upon bending, lead to shearing at the interface, which leads to loss of energy and consequently, oscillation damping. This anatomical structure may also explain the use of A. donax stems for reeds of musical instruments in which damping properties at very high frequencies are essential (Obataya and Norimoto, 1999a , b ; Obataya et al., 1999 ).

Viscoelastic behavior is shown explicitly for the rhizome of A. donax (Speck and Spatz, 2003 ). However, comparison of the results found under test protocols 3 and 4 shows that very little energy is dissipated in the rhizome. Removal of rhizome and roots at the same time did not influence damping behavior significantly. The act of securely fixing the plant in cement in such a manner seems to mimic efficiently the behavior of a properly rooted plant in compacted soil. The removal of any parts of the plant located below the point of fixation has no effect upon the behavior of the culm's oscillations above that point. However, in the natural habitat with often water-soaked soil, viscoelastic properties of the rhizome as an interface between stem and roots and soil may play a role in energy dissipation.

Indeed one exception was found. As noted, the relative damping of plant 1 tested one year earlier differs from the other plants under test protocol 2. This can possibly be ascribed to different soil moisture conditions at the time of the measurements. Whether the roots influence the damping behavior or whether this difference depends entirely on the soil conditions cannot be determined with our experimental procedure. Preliminary results suggest that the adventitious roots may largely differ in their mechanical properties. Some adventitious roots consisting of root cortex and central cylinder can be characterized by high critical strains and non-brittle fracture propagation. In other roots, the cortex represents a dried-out sheath, which easily peels off so that only the prominent central cylinder remains, showing pronounced stiffness in tension and brittleness in the fracture. It can be hypothesized that these differences depend on the age or lignification of the roots or on the season when the roots were harvested. The variety of mechanical properties that are important for damping oscillation becomes even larger when also considering the interface where the adventitious roots merge smoothly into the rhizome. Further experiments taking into account soil conditions and mechanical properties of the roots are necessary to understand the role of the interaction of underground plant organs and soil on damping behavior.

As shown in protocol 5 loss of energy from friction between the stem and leaf sheaths covering the nodes plays no significant role. To a minor extent and only for large amplitudes, hollow, septate stems under bending vibrations may be damped when internodes bend and become oval beyond the linear elastic range. In addition, septate nodes may store and absorb strain energy and only partially release this energy for elastic restoration of the stem's original shape (Niklas, 1997 ).

Oscillation damping is of vital importance for plants. Three different mechanisms are observed in single plant stems: aerodynamic, structural, and material damping. A comparative approach of oscillation damping in A. donax and six other species of four families suggests that embedding stiff material in a less stiff matrix is the basic principle of material damping. Thus, from a mechanical point of view the distribution of vascular bundles in the plant stems is a trade-off between efficient damping and yet high stability against mechanical load.


   APPENDIX
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
LITERATURE CITED
 
Estimate of the contribution of aerodynamic resistance to the damping of oscillations in ARUNDO DONAX.

The displacement (y) of an object under damped-free vibrations can be described by

(A1)
where y0 is the displacement at t = 0, {delta} is the decay constant, {omega} is the angular frequency, and t is the time.

If n denotes the period number such that tn+1 – tn = T = 2{pi}/{omega} the logarithmic decrement (Vincent, 1990 ) can be written as

{abot-91-06-15-ea2}

where E'' is the loss modulus and E' the storage modulus of elasticity.

The energy of oscillation is

(A3)
where yn(Top) is the amplitude at the top of the stem, {rho}B,Stem the density of the stem at its base, AB,Stem the area of the stem's cross-section at its base, L the length of the stem, and N4 a numerical factor depending on the tapering mode of the stem ({alpha}), the change of the modulus of elasticity (ß), and of the density ({gamma}) along the stem. It can be approximated using solutions of the differential equations for bending oscillations (Spatz and Speck, 2002 ).

Because the energy is proportional to the square of the amplitude eq. 2 can be rewritten as

{abot-91-06-15-ea4}

where n + 1/2; denotes the half period.

If the loss of energy is only due to damping within the plant stem's material and damping due to the aerodynamic resistance of the leaves (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

(A5)

Neglecting the aerodynamic resistance of the stem, the energy loss due to the material properties of the plant stem can be obtained from oscillation damping of the stem from which all leaves have been removed.

For small velocities (<1.5 m/s; Speck, 2003 ), the calculated drag force FD is proportional to the square of the velocity. The aerodynamic resistance is given by

(A6)
where CD is the drag coefficient, {rho}Air the density of air, Aleaves the projection surface area of the leaves, and v the velocity of movement.

The energy loss in a half cycle is

{abot-91-06-15-ea7}

Substituting v = dy/dt from eq. 1 results for {delta} << {omega} in

{abot-91-06-15-ea8}

Because the energy loss due to aerodynamic resistance depends on the third power of the amplitude and that of the stem alone on the square of the amplitude, the relative damping due to these two sources is only additive for small changes of the amplitude, which necessitates a stepwise procedure to calculate the overall damping.

For plants like Cyperus alternifolius, in which the leaves are concentrated at the top of the plant, this procedure is straightforward. However, for Arundo donax, with leaves distributed over the entire stem, the aerodynamic resistance is a function of the projection area of each of the leaves and the amplitude at the height of the leaf base. This can be taken into account by replacing Aleavesyn3 in eq. 8 by an effective projection area

{abot-91-06-15-ea9}

The calculation of the amplitude yn (hi) at height i requires the knowledge of the bending line. If the tapering mode ({alpha}) and the change of the modulus of elasticity (ß) and of the density ({gamma}) along the stem are known, this can be obtained as a solution of the differential equation for bending oscillations (Spatz and Speck, 2002 ).


   FOOTNOTES
 
1 The authors wish to thank Prof. T. Speck (Botanical Garden, University of Freiburg, Germany) and Dr. N. P. Rowe (University of Montpellier, France) for valuable help and advice. We wish to thank L. Schweizer (Institute of Sport and Sport Science, University of Freiburg, Germany) for his help with the quantitative video analyses of oscillating stems and helpful discussions and suggestions. We thank A. Emanns (Institute for Biology III, University of Freiburg, Germany) for his gentle and precise assistance in the field experiments in Freiburg. We would like to thank two anonymous referees for extremely helpful comments. This study was in part supported by the DaimlerChrysler AG and Alumni Freiburg. Back

2 E-mail: olga.speck{at}biologie.uni-freiburg.de Back


   LITERATURE CITED
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 LITERATURE CITED
 
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