Structural dynamics and resonance in plants with nonlinear stiffness

ARTICLE IN PRESS

Journal of Theoretical Biology 234 (2005) 511–524 www.elsevier.com/locate/yjtbi

Structural dynamics and resonance in plants with nonlinear stiffness Laura A. Miller Department of Biology, Box 90338, Duke University, Durham, NC 27708, USA Received 7 January 2004; received in revised form 7 December 2004; accepted 8 December 2004 Available online 2 February 2005

Abstract Although most biomaterials are characterized by strong stiffness nonlinearities, the majority of studies of plant biomechanics and structural dynamics focus on the linear elastic range of their behavior. In this paper, the effects of hardening (elastic modulus increases with strain) and softening (elastic modulus decreases with strain) nonlinearities on the structural dynamics of plant stems are investigated. A number of recent studies suggest that trees, crops, and other plants often uproot or snap when they are forced by gusting winds or waves at their natural frequency. This can be attributed to the fact that the deflections of the plant, and hence mechanical stresses along the stem and root system, are greatest during resonance. To better understand the effect of nonlinear stiffness on the resonant behavior of plants, plant stems have been modeled here as forced Duffing oscillators with softening or hardening nonlinearities. The results of this study suggest that the resonant behavior of plants with nonlinear stiffness is substantially different from that predicted by linear models of plant structural dynamics. Parameter values were considered over a range relevant to most plants. The maximum amplitudes of deflection of the plant stem were calculated numerically for forcing frequencies ranging from zero to twice the natural frequency. For hardening nonlinearities, the resonant behavior was ‘pushed’ to higher frequencies, and the maximum deflection amplitudes were lower than for the linear case. For softening nonlinearities, the resonant behavior was pushed to lower frequencies, and the maximum deflection amplitudes were higher than for the linear case. These nonlinearities could be beneficial or detrimental to the stability of the plant, depending on the environment. Damping had the effect of drastically decreasing deflection amplitudes and reducing the effect of the nonlinearities. r 2005 Elsevier Ltd. All rights reserved. Keywords: Resonance; Biomechanics; Biomaterials; Plants; Nonlinear dynamics

1. Introduction Several investigations in recent years have suggested that trees, crops and other plants often uproot or snap when they are forced by wind gusts at their natural frequency in transverse vibration (Baker, 1995; Blackburn and Petty, 1988; Kerzenmacher and Gardiner, 1998). Other studies on benthic organisms also suggest that the dynamic forces produced by waves are significant to the stability of the organisms (Denny et

Corresponding author. Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112-0090, USA. Tel.: +1 801 585 1637. E-mail address: [email protected] (L.A. Miller).

0022-5193/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2004.12.004

al., 1998; Gaylord et al., 2001; Utter and Denny, 1996). This has led to the development of a variety of mathematical models to study resonance behavior in plant stems based on mass and flexural stiffness distributions (Bru¨chert et al., 2003; Denny et al., 1998; Finnigan and Mulhearn, 1978; James, 2003; Spatz and Speck, 2002; Spatz and Zebrowski, 2001). Most of these models make the simplifying assumption that the force required to bend the plant varies linearly with deflection. Most biomaterials, however, exhibit strong nonlinear stiffness properties, particularly at the large deformations when failure occurs. This paper compares the resonance behavior of linear and nonlinear plant stems modeled as Duffing oscillators. The solutions of these systems were calculated numerically using parameter values relevant to plants.

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L.A. Miller / Journal of Theoretical Biology 234 (2005) 511–524

1.1. Previous research Several studies have characterized the large amplitude deflections in plants stems that are produced by turbulent or gusty winds (Finnigan, 1979; Flesch and Grant, 1992; Mayer, 1987). By measuring dynamic wind speeds and the resulting deflections of the plants, these studies have shown that a particular range of gust frequencies cause large amplitude deflections in plant stems or shoots. These frequencies are close to the natural frequency of the plant stem (i.e. the frequency of unforced vibration when the stem is deflected from its resting position and released). Finnigan (1979) found that oscillations in wheat stalks were largest during wind gusts at the plants’ natural frequencies in the range of about 0.5–1 Hz. Flesch and Grant (1991) also found that vibrations in corn stalks were greatest when forced at their natural frequencies of about 1–2 Hz. Other studies have shown that vibrations in trees are greatest during successive wind gusts of 0.4–1 Hz (Blackburn and Petty, 1988; Holbo et al., 1980; Oliver and Mayhead, 1974) which is approximately the range of natural frequencies measured in other trees (Baker, 1997; Kerzenmacher and Gardiner, 1998; Milne, 1991). Since the dynamics of swaying plant stems appear to be so important to their stability, a number of mathematical models have been proposed to predict stem dynamics and natural frequencies. Most of these models are based either on spring–mass–damper systems described by a second-order ordinary differential equation (Denny et al., 1998; James, 2003; Milne, 1991) or as continuous models of cantilever beams described by a fourth order partial differential equation (Finnigan and Mulhearn, 1978; Bru¨chert et al., 2003; Mayer, 1987; Spatz and Speck, 2002). Such models have been used to predict natural frequencies and dynamic behavior with relatively good agreement (Baker, 1995; Milne, 1991). These models, however, best characterize the dynamics of the plants in their linear elastic range since resistance to bending is either given as a linear term, or the biomaterial is assumed to be Hookean. Furthermore, most tests of the models do not consider large deformations of the plant during destructive winds or waves when nonlinearities are likely to be significant. 1.2. Nonlinear stiffness in biological beams At large deformations, the elastic modulus of most biomaterials is characterized by some form of nonlinearity. These materials can be classified as hard springs (elastic modulus increases with deflection) and soft springs (elastic modulus decreases with deflection), or some combination of the two. A survey of many biomaterials with examples of softening and hardening

nonlinearities can be found in Vincent (1990) and Wainwright et al. (1982). Gaylord and Denny (1997) also describe how Hookean materials behave nonlinearly under the large deformations which are characteristic of almost all plant stems. A number of plant and animal biomaterials have the property of hardening nonlinear stiffness, which gives their stress–strain curve a typical ‘J-shape’. In animals, collagenous biomaterials represent a nice example of a hard spring. The stretching of elastin contributes to the low modulus region, and the stretching of the collagen contributes to the high modulus region. Examples of plant biomaterials that behave like hard springs for a certain range of strain include balsa wood (Easterling et al., 1982) and spruce wood (Widehammar, 2004) under radial and tangential compression, and the stipes of the giant kelp Macrocystis pyrifera (Utter and Denny, 1996). A variety of biomaterials also behave like soft springs, due to factors such as an elastic modulus that decreases with deflection or the behavior of the soil–root system. Examples of ‘soft’ biomaterials include balsa wood in axial compression (Easterling et al., 1982), the stipes of the kelp Nereicytis leutkeana (Denny and Hale, 2003), sclerenchyma tissue of Aristolochia macrophylla and Arabidopsis (Spatz et al., 1999, Ko¨hler and Spatz, 2002), and the strengthening tissues of Equisetum hyemale in tension (Speck et al., 1998). Yield might also be described as a softening nonlinearity and could be significant to the stability of the plant because this nonlinear range is ultimately responsible for structural failure (Ko¨hler and Spatz, 2002). The dynamics of such nonlinearities, however, are rarely studied in a biological context. Several models have been proposed to explain the nonlinear behavior of biomaterials under large deformations. These models are based on the arrangement of the microfibril arrays within the biomaterials and how the fiber orientations change under deformation. Vincent (1990) suggests a general model for obtaining the Jshaped stress–strain curve observed for many biomaterials. Basically, fibers are oriented at an angle relative to the direction of strain. As the material is stretched, the fibers progressively orient towards the direction of extension, increasing the elastic modulus. Spatz et al. (1999) suggest a model based on microfibril reorientation and slippage that explains how biomaterials might behave like soft springs. Fibers of higher elastic moduli are embedded in a weaker and more pliant matrix. In the initial linear elastic range, the biomaterial acts like a homogeneous material with the fibers and matrix undergoing the same strain. Above the yield point, the microfibers begin to slide past each other when the matrix undergoes plastic deformation (Bodig and Jayne, 1993). This leads to a lower elastic modulus at higher strains.

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1.3. Theory of linear and nonlinear spring– mass– damper systems The simplest model that has been used to describe plant stem structural dynamics and resonance is a simple linear ordinary differential equation (ODE) describing a spring–mass–damper system. An excellent description of this model, its derivation, and solutions has been given in a biological context by Denny (1988). The equation of motion for this system may be written as follows: mx00 ðtÞ þ cx0 ðtÞ þ k1 xðtÞ ¼ F ðtÞ,

(1)

where m is the mass, c is the damping coefficient, k1 is proportional to the stiffness of the spring, and F(t) is the applied force as a function of time (see Fig. 1). x(t) gives the displacement of the stem as a function of time. This displacement may be described as the horizontal displacement or angular displacement of a point on the stem. The choice of x(t) determines the values of the corresponding constants c and k1. In terms of plant structural dynamics, the first term of the equation describes the effective mass of the plant times its acceleration, the second term describes the friction in the system due to aerodynamic, material and structural damping of the plant, the third term describes the resistance of the trunk or stem to bending, and the forcing term describes the effective wind force on the plant. In reality, the damping term is probably not linearly proportional to velocity. This is a rough approximation, and more complicated terms could be added in subsequent models. Note also that this equation gives motion at one point. The motion of the point it describes depends upon the calculation of the

Fig. 1. Diagram of a spring–mass–damper system (left) and inverted pendulum (right). In the spring–mass–damper system, the mass of the system is given as m, the spring constant is given as k1, and the damping coefficient is given as c. The horizontal displacement from rest is x(t), and the periodic force applied to the system as a function of time is given as F(t). A plant stem or other cantilever beam can be modeled as a spring–mass–damper system by setting the effective mass, effective spring constant, and damping coefficient at a point on the stem equal to m, k1 and c, respectively. x(t) then describes the displacement of a point on the stem from equilibrium, and F(t) describes the effective force applied to the stem at the same point. The same ODE can also be used to model an inverted pendulum. In this case, x(t) describes angular displacement, m describes the effective mass at the top of the beam, k describes the resistance to bending, and c is a damping coefficient. The choice of x(t) (horizontal or angular displacement) determines the values of m, k1, and c.

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effective mass, effective stiffness, and effective force (which should all be estimated at that same point). Discussions on how to calculate these ‘effective’ parameters at different points along a beam or stem can be found in Denny (1988) and Wilson (1984). The phenomenon of resonance can be described mathematically by solving for the dimensionless response amplitude of the system for various forcing frequencies. Assume that the system is forced by a harmonic loading function of frequency o and amplitude po: F ðtÞ ¼ po sinðotÞ.

(2)

The displacement as a function of time may be expressed as:
 po xðtÞ ¼ HðoÞ sinðotÞ, (3) k1 where po is the maximum applied force and HðoÞ is a dimensionless function that depends on the forcing frequency. Substituting Eqs. (2) and (3) into Eq. (1) and solving for HðoÞ results in the following: 8" 9
2 #2
2 =1=2 < o o HðoÞ ¼ 1 þ 4z2 , (4) : oo oo ; rffiffiffiffiffi k1 , oo ¼ m z¼

c c ¼ pffiffiffiffiffiffiffiffiffi , 2moo 2 k1 m

(5) (6)

where oo is the natural frequency and z is the damping ratio (Fig. 2). Soft and hard nonlinear structures have often been modeled as Duffing oscillators (Clough and Penzien, 1975; Wilson, 1984; Thompson and Steward, 1986). The Duffing oscillator was originally introduced to model the large amplitude vibration modes of a steel beam subjected to periodic forces (Duffing, 1918). The Duffing oscillator has since become one of the standard models for studying nonlinear forced systems, particularly for beams subjected to external forces. Since most biomaterials have nonlinear stiffness, modeling biological structures as Duffing oscillators could also be beneficial in gaining a better understanding of resonance in plants. The justification for this choice of model is based on history and simplicity: this is a well-studied nonlinear model that is relatively simple, lends itself to some analytical work, and yet has very complex behavior. To model a system as a Duffing oscillator, a cubic stiffness term, k3 x3 ðtÞ; is added to the left-hand side of Eq. (1). mx00 ðtÞ þ cx0 ðtÞ þ k1 xðtÞ þ k3 x3 ðtÞ ¼ F ðtÞ,

(7)

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If the beam or stem has nonlinear stiffness, this equation may now be written as follows:

mg 3 x ðtÞ ¼ F ðtÞ. mx00 ðtÞ þ cx0 ðtÞ þ ðk1  mgÞxðtÞ þ k3 þ 6 (10)

8

Response Amplitude

ζ = 0.07 6 ζ = 0.1 4 ζ = 0.2 2

0 0

0.5 1 1.5 Forcing Frequency / Natural Frequency

2

Fig. 2. Harmonic response functions of a lightly damped, linear spring–mass system. Dimensionless response amplitude is plotted against the forcing frequency divided by the natural frequency. In this graph, the values of the damping ratio z are varied from 0.2 to 0.07. A periodic force is applied to the system at frequencies that vary from zero (steady force) to twice the natural frequency of the system. The dimensionless response amplitude describes the maximum distance deflected from the equilibrium position divided by the distance deflected by a static force of the same magnitude. In other words, the response amplitude equals one when the forcing frequency is zero. Resonance occurs when the forcing frequency is close to the natural frequency of the system.

where k3 is the cubic stiffness coefficient. k3 40 models a hardening nonlinearity, and k3 o0 models a softening nonlinearity. The addition of this nonlinear cubic term is what distinguishes this model from previously proposed models of plant stem dynamics. An argument can be made that the Duffing oscillator is the simplest appropriate nonlinear model available. A linear term and a nonlinear term are needed to describe the initial linear-elastic region of the stress–strain curve and the subsequent nonlinearities under large deformations. To get the proper behavior of the system in both directions of deflection, a second-order term would need to be signed ðk2 xðtÞjxðtÞjÞ; making analytical work difficult (this is true of any term whose power is not an odd integer). This model can also be used as an approximation to an inverted pendulum (Marion and Thorton, 2004). Let x(t) describe the angular displacement of the stem, k1 describe the resistance to bending, and m be the effective mass at the top of the stem or beam. The equation of motion is as follows: mx00 ðtÞ þ cx0 ðtÞ þ k1 xðtÞ  mg sinðxðtÞÞ ¼ F ðtÞ.

(8)

Taking the first two non-zero terms of the Taylor series expansion of sinðxÞ about x ¼ 0 yields the following: mg 3 mx00 ðtÞ þ cx0 ðtÞ þ k1 xðtÞ  mgxðtÞ þ x ðtÞ ¼ F ðtÞ. 3! (9)

In this case, (k1mg) and (k3+mg/6) are the coefficients of the ‘resistance to deflection’ terms. In a manner similar to the calculation of the harmonic response function for the linear system, response amplitudes may be approximated for an undamped Duffing oscillator using perturbation methods (Hayashi, 1985; Wilson, 1984). Analytical approximations for a damped Duffing oscillator, however, are not available and must be determined numerically. The undamped response function is given by the following equation:
 4 k 1 o2 4 po 3 a  1 a ¼ 0, (11) 2 3 k3 oo 3 k3 where a is the maximum amplitude of vibration, po is the maximum applied force, k1 is the linear stiffness constant, k3 is the cubic stiffness constant, oo is the linear natural frequency, and o is the forcing frequency. The response amplitudes for this system can be found by solving for the roots of the cubic polynomial in Eq. (11). The number of real roots may be determined with use of results given by Rosenbach et al. (1958). Consider the following polynomial with real coefficients a and b: a3 þ aa þ b ¼ 0.

(12)

Let R be defined as follows: R¼

1 3 1 2 a þ b. 27 4

(13)

Then there is one real root if R40; there are at least two equal real roots if R is zero, and there are three real roots if Ro0: The roots of Eq. (11) were approximated for different values of o=oo and are plotted as response amplitudes in Fig. 3. The nonlinear stiffness has the effect of ‘‘pushing over’’ the resonance peak when compared to the linear case. The stronger the nonlinearity, the greater the distortion would be. Note that in the linear case, the response amplitude ‘blows up’ when the forcing frequency equals the natural frequency, but the response amplitude of the nonlinear system is finite for finite forcing frequencies. Damping the system would have the effect of ‘capping’ the curves at an amplitude about equal to the maximum amplitude seen in the equivalently damped linear case when k35k1. The effect of damping for strongly nonlinear systems, however, must be determined numerically. For a certain range of forcing frequencies, the response amplitudes become multi-valued. This corresponds to forcing frequencies that yield more than one real root of Eq. (11). In the damped case, the response

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8

8

6

k3 = 0.025

6 Amplitude

Amplitude

k3 = −0.025 k3 = −0.05

4

k3 = 0.05

4

k3 = −0.1 2

k3 = 0.1

2

0

0 0

0.5

1 ω / ωo

1.5

2

0

0.5

1 ω / ωo

1.5

2

Fig. 3. Response curves for undamped soft (left) and hard (right) spring-mass systems. In this case, k1 ¼ 1; m ¼ 0:1; po ¼ 1; and k3 is varied from 0.1 to 0.1. The nonlinearities have the effect of ‘‘pushing over’’ the resonance peak when compared to the linear system shown in Fig. 2. The stronger the nonlinearity, the greater the distortion from the linear case. Damping the system would have the effect of ‘capping’ the curves at an amplitude about equal to the maximum amplitude seen in the equivalently damped linear case where k3 5k1 : For strong nonlinearities, the effects of damping must be calculated numerically. Note that for certain forcing frequencies the response curve is multi-valued.

8

1

Amplitude

6

2

4

2 3

0

0

0.5

1

1.5 ω / ωo

2

2.5

3

Fig. 4. Response curve for a hard spring–mass system. In this case, k1 ¼ 1; m ¼ 0:1; po ¼ 1; and k3 ¼ 0:05: The first dotted curve approximates the shape of the curve for a moderately damped system, and the second dotted curve approximates the shape of the curve for a lightly damped system. For a range of o=oo greater than 1.65 (denoted by the star), the solution becomes multi-valued if damping is sufficiently light. The amplitude chosen by the system depends upon the initial conditions. Rapid changes from one branch to the other are called jumps. For this range of o=oo ; the system can become chaotic.

amplitudes become multi-valued if the damping is sufficiently small (Fig. 4). In the multi-valued range of the graph, the response amplitudes can take on any value of the branches. The value taken depends upon the initial conditions. The star in Fig. 4 denotes the bifurcation point (when the system becomes multivalued). Erratic jumps in the response amplitudes are observed in the multi-valued range of forcing frequencies for lightly damped systems. Fig. 5 shows the

response amplitudes for soft and hard nonlinear systems at different forcing amplitudes (po). Changes in po and o=oo can produce erratic changes in response amplitudes called jumps, which lead to chaos (Moon and Holmes, 1979). Further discussions on jumps in Duffing oscillators can be found in Nayfeh and Mook (1979) and Wu et al. (2001). In this paper, response amplitudes under periodic forcing will be calculated for hard and soft nonlinear spring–mass–damper systems for a range of parameter values relevant to plant stems. These results will be compared to the linear model in order to better understand the effects of nonlinear stiffness on resonance in plant stems and to gain insight into how failure might occur.

2. Methods In order to cover a large space of biologically relevant parameters, Eq. (7) was rewritten in dimensionless form. Dividing Eq. (7) by k1 and substituting Eqs. (5) and (6) into the result yields
2 1 2z 0 k3 p x00 ðtÞ þ x ðtÞ þ xðtÞ þ x3 ðtÞ ¼ o F ðotÞ. oo oo k1 k1 (14) The following change of variables is made: x X¼ , Lmax T ¼ oo t,

(15)

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6

6

5

5 po = 1.0

4

po = 0.5 3

Amplitude

Amplitude

4

po = 0.25

3

2

2

1

1

0

0

0.5

1 ω / ωo

1.5

2

po = 1.0 po = 0.5 po = 0.25

0 0

0.5

1 ω / ωo

1.5

2

Fig. 5. Response curves for undamped soft (left) and hard (right) spring–mass systems. In this case, k1 ¼ 1; m ¼ 0:1; k3 ¼ 0:05; and the forcing amplitude po is varied from 0.25 to 1.0. The entire harmonic response curve increases as po is increased. In the damped case, each curve would be ‘capped’ at an amplitude that is about equal to the maximum amplitude for an equivalently damped linear system if k3 5k1 : In a damped case, the maximum amplitude of deflection would increase as po increases. If the damping of the system is sufficiently light or the forcing amplitude is sufficiently large, slight changes in forcing frequency and magnitude lead to jumps and chaos.

d dt d 1 d ¼ ¼ , dT dT dt oo dt

3 2.5



2 2 d2 d 1 d 1 d ¼ , ¼ 2 dT oo dt oo dt2 dT where Lmax is a characteristic maximum deflection. In this case, Lmax is the deflection at which failure occurs. This is a rough approximation since the deformation resulting in failure most likely depends upon the rate of strain, soil conditions and other factors. If x represents the deflection from equilibrium, then X ¼ x=Lmaxx represents the actual deflection divided by the deflection at which failure occurs. So a deflection of X ¼ 1 results in failure. Making these substitutions gives the following:
 K3 3 po o 00 0 X ðTÞ þ 2zX ðTÞ þ X ðTÞ þ X ðTÞ ¼ F T , oo K1 K1 (16) K 1 ¼ k1 Lmax ; K 3 ¼ k3 ðLmax Þ3 ;

(17)

where K 3 =K 1 will be referred to as the nonlinear ratio, and po =K 1 will be called the normalized forcing amplitude. To simulate a periodic gusting wind, the forcing term is set to the following:


 o 1 o 1 þ sin F T ¼ T . (18) oo 2 oo This function provides a ‘gusting’ force that does not reverse direction over time. This forcing function is an approximation of the dynamic forces plants might face

Force / K1

2 1.5 1 0.5 0

0

0.2

0.4 0.6 L / Lmax

0.8

1

Fig. 6. Dimensionless force displacement curves for nonlinear springs (K 3 =K 1 ¼ 1; 0.5, 0, 1, 2). These values of the nonlinear ratio correspond to the lines graphed from bottom to top. L is the distance a mass is displaced from rest as a result of a steady applied force of magnitude po/K1. Lmax is a characteristic displacement. In this case, Lmax is assumed to be the deflection at which failure occurs. K1 equals k1 Lmax ; where k1 is the linear spring constant. K 3 equals k3 ðLmax Þ3 ; where k3 is the cubic spring constant. In the case of a soft spring, the force displacement curve is constructed to plateau at its maximum rather than decrease (see Methods).

in the environment. In reality, wind gusts are not sinusoidal and forcing amplitudes are stochastic. Values of the nonlinear ratio were varied from 1 (softening nonlinearity) to 2 (hardening nonlinearity). In the soft spring case, however, the force displacement curve was designed to plateau at its maximum value rather than decrease (Fig. 6). In reality, force displacement curves for biomaterials can be more complex than

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what is considered here. Many biomaterials contain regions of both hardening and softening nonlinearities, as well as stronger nonlinearities. The range of parameters considered, however, should provide some insight into the behavior of more complicated biomaterials. To consider a wide range of biologically relevant values, damping ratios were varied from 0.05 to 0.50, and a value of 0.15 was used as a typical case. Damping ratios have been measured experimentally in several plant stems and trunks. Finnigan and Mulhearn (1978) measured a damping ratio of about 0.17 in wheat stalks. Baker (1995) and Flesch and Grant (1992) measured damping ratios of about 0.15 in trees and corn stalks, respectively. Using deflection data given by Spatz and Speck (2002), an Arundodonax plant without leaves has a damping ratio of about 0.2. Bru¨chert et al. (2003) outlined four main factors contributing to the damping of plant stems: (1) aerodynamic damping, (2) viscoelastic damping within the stem and root system, (3) interference or friction with surrounding neighbors, and (4) structural or mass damping due to oscillations of different parts of the plant at different frequencies. All of these factors could conceivably vary substantially in different environmental and experimental conditions. If the wind gusts with the motion of the plant, aerodynamic damping should be reduced. Interference from neighbors can vary greatly depending upon the plant’s surroundings (dense stand vs. a single plant). Structural damping should vary depending upon whether or not the species is branched. Although more work is needed to determine how much the damping ratio can vary in plant stems, the range of values considered here should cover most scenarios. The normalized forcing amplitude (po =K 1 ) was set to 0.25, 0.33, and 0.66 unless otherwise stated. These values were chosen to give maximum deflections close to the characteristic maximum deflection ðL Lmax Þ for three cases (K 3 =K 1 ¼ 1; 0, and 2). If Lmax is chosen to represent the displacement of the stem at which failure is likely to occur, then these values of po =K 1 represent the forcing amplitudes likely to cause failure for the corresponding cases. The forcing frequency divided by the natural frequency (o=oo ) was varied from 0 to 2. This range of frequencies models the displacement of the stem resulting from a steady force (o=oo ¼ 0) through displacements resulting from an applied force oscillating at twice the stem’s natural frequency (o=oo ¼ 2). These values likely characterize the entire range of resonance behavior relevant to plants. An explicit Runge–Kutta solver available on Matlab 5.3, the MathWorks, Inc., was used to numerically solve Eq. (16). Simulations were run for 40 periods of oscillation. For each set of parameters, ‘forward’ and ‘backward’ scans were performed. This technique was

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used to ensure that the upper portion of the response curve was simulated since the solution of the Duffing equation is dependent upon the initial conditions (Tufillaro et al., 1992). The forward and backward scans were performed as follows: during the first 10 oscillations, the forcing frequency was set to a low frequency (o ¼ 0:1oo ) for the forward scan or a high frequency (o ¼ 2oo ) for the backward scan. During the next 30 oscillations, the forcing frequency was set to the forcing frequency considered for the simulation. The maximum deflection amplitude was taken over the last 20 periods of oscillation. These maximum amplitudes were plotted as functions of the forcing frequency, damping ratio, and nonlinear ratio.

3. Results 3.1. Hardening nonlinearities In order to compare the harmonic response curve of a linear and hard nonlinear spring–mass–damper system, simulations were performed for three values of the nonlinear ratio (K 3 =K 1 ¼ 0; 1; 2) and five damping ratios (z ¼ 0:1; 0:2; 0:3; 0:4; 0:5). Fig. 7A shows the response amplitudes for these nonlinear ratios with the normalized forcing amplitude set to 0.33. This value of the forcing amplitude corresponds to the normalized force likely to cause failure in the linear case. For K 3 =K 1 ¼ 1 and 2, the nonlinear response is ‘pushed’ to the right in comparison to the linear case. The maximum amplitudes of deflection are also smaller and occur at higher forcing frequencies. These effects are greater for the more nonlinear case and diminish as the damping ratio increases. To consider the harmonic response at forcing amplitudes likely to cause failure in the strongly nonlinear case (K 3 =K 1 ¼ 2), simulations were also performed at a normalized forcing amplitude of 0.66 as shown in Fig. 7B. The differences between the linear and nonlinear cases become more apparent at this higher forcing amplitude. The maximum amplitude of deflection in the strongly nonlinear case (K 3 =K 1 ¼ 2) is about half that of the linear case for a damping ratio of 0.10. This maximum also occurs at a forcing frequency that is more than 1.6 times higher than the linear case for a damping ratio of 0.10. The maximum amplitude of deflection in the moderately nonlinear case (K 3 =K 1 ¼ 1) is about 70% of the maximum amplitude of deflection in the linear case. This maximum occurs at a forcing frequency about 1.4 times greater than the linear case for a damping ratio of 0.10. The differences between the linear and nonlinear cases diminish, however, as the damping ratio is increased. To compare the shape of the response amplitude curve for various values of the normalized forcing

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K3 / K1 = 0

Deflection / Max Deflection

1 0.8

K3 / K1 = 1

1

po/K1 = 0.33

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0 0

1

2

0 0

(A) Deflection / Max Deflection

2

2

2

2

1.5

1.5

1.5

1

1

0.5

0.5

0

1

2

0

1

2

0

1

2

1

0.5 0

0

1

2

1.5 Deflection / Max Deflection

1 ω / ωo

po/K1 = 0.66

0

0

(B)

1 ω / ωo

2

1.5

1.5

1

1

1

0.5

0.5

0.5

ζ = 0.15

0 (C)

K3 / K1 = 2

1

0

1

2

0

0

1 ω / ωo

2

0

Fig. 7. Maximum response amplitudes for linear and nonlinear hard spring–mass–damper systems. Normalized deflections (L/Lmax) are plotted against the forcing frequency divided by the linear natural frequency (o=oo ). The nonlinear ratio ðK 3 =K 1 Þ is set to 0, 1, and 2. (A) The normalized forcing amplitude (po/K1) is set to 0.33. This value corresponds to the normalized force like to cause failure in the linear case when z ¼ 0:1: The damping ratio (z) is set to 0.1, 0.2, 0.3, 0.4, and 0.5 (corresponding to the curves graphed from top to bottom in each plot). For each damping ratio, the nonlinear response is ‘pushed’ to the right in comparison to the linear case. The maximum amplitudes of deflection are also smaller and occur at higher forcing frequencies. These differences diminish as the damping ratio increases. (B) Same as A, but the normalized forcing amplitude (po/K1) is now set to 0.66. This corresponds to the normalized force likely to cause failure in the strongly nonlinear case (K 3 =K 1 ¼ 2) when z ¼ 0:1: The effect of the hardening nonlinearity on the response curve becomes greater at this higher forcing amplitude in each case. The nonlinearities further distort the harmonic response curve to higher frequencies and lower amplitudes when compared to A. (C) The damping ratio (z) is set to 0.15, and the normalized forcing amplitude (po =K 1 ) is set to 0.132, 0.264, 0.396, 0.528, and 0.66 (corresponding to the lines drawn from bottom to top of each graph). The effect of the nonlinearity on the response amplitude increases as the forcing amplitude increases. When po =K 1 ¼ 0:132; the effect of the nonlinearity is minimal.

amplitude, simulations were performed for three cases (K 3 =K 1 ¼ 0; 1, and 2) at five forcing amplitudes (po =K 1 ¼ 0:132; 0.264, 0.396, 0.528, 0.66) in Fig. 7C. The damping ratio in each case was set to z ¼ 0:15: As po =K 1 is increased, the entire response amplitude curve increases. The effect of the nonlinearity also increases as the forcing amplitude increases. Resonance behavior is pushed to higher frequencies under larger forces, even

for the same value of K3/K1. This result implies that nonlinear systems would appear linear during small deflections, but the behavior could drastically change during large amplitude deflections. The response amplitudes for a range of hardening nonlinearities at various forcing frequencies are shown in Fig. 8A. Two forcing amplitudes were considered. po =K 1 ¼ 0:33 models the force that would likely cause

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p0/K1 = 0.33

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Fig. 8. Contour plots of the normalized maximum deflection (L/Lmax). (A) L/Lmax as a function of the forcing frequency divided by the natural frequency (o=oo ) and the nonlinear ratio (K 3 =K 1 ). The normalized forcing amplitude (po =K 1 ) is set to 0.33 (left) and 0.66 (right). The damping ratio is set to 0.15. The nonlinear ratio is varied from 0 (linear spring) to 2 (nonlinear hard spring). When K 3 =K 1 0; maximum deflections occur when the forcing frequency is near the natural frequency. As K3/K1 increases, the resonance behavior shifts to higher frequencies, and the maximum amplitude of deflection over all frequencies decreases. The larger forcing amplitude results in larger maximum deflections and augments the effect of the hardening nonlinearity. (B) Maximum L/Lmax as a function of the nonlinear ratio (K 3 =K 1 ) and the damping ratio (z). Maximum L/Lmax was taken as the maximum deflection over all forcing frequencies ð0pop2oo Þ for each nonlinear ratio and damping ratio pair. The normalized forcing amplitude is set to 0.33 (left) and 0.66 (right). K3/K1 is varied from 0 (linear spring) to 2 (nonlinear hardening spring). The damping ratio is varied from 0.05 to 0.25. Maximum deflections occur when K 3 =K 1 0 and the damping ratio is small (z 0:05). Increasing the damping ratio and/or increasing the nonlinear ratio decreases the maximum amplitude of deflection. As K3/K1 increases, the resonance behavior shifts to higher frequencies.

failure in the linear case (K 3 =K 1 ¼ 0), and po =K 1 ¼ 0:66 models the normalized force likely to cause failure in the strongly nonlinear case (K 3 =K 1 ¼ 2). Simulations were performed for nonlinear ratios ranging from 0 to 2 at a damping ratio of 0.15. When K 3 =K 1 0; maximum deflections occur when the forcing frequency is near the natural frequency. As K 3 =K 1 increases, the resonance behavior shifts to higher frequencies, and the maximum amplitude of deflection over all frequencies decreases. The maximum response amplitudes over a range of forcing frequencies for various nonlinear ratios and damping ratios are shown in Fig. 8B. The same normalized forcing amplitudes were considered (po =K 1 ¼ 0:33 and 0.66). Simulations were performed for nonlinear ratios ranging from 0 to 2 and damping ratios ranging from 0.05 to 0.25. The maximum response amplitude for each nonlinear ratio and damping ratio pair was taken as the maximum of the response amplitude curve over forcing frequencies ranging from 0 to 2oo : Maximum deflections occur

when K 3 =K 1 0 and the damping ratio is small (z 0:05). Increasing the damping ratio or increasing the nonlinear ratio decreases the maximum amplitude of deflection. As K 3 =K 1 increases, the resonance behavior shifts to higher frequencies. This behavior could be beneficial to the plant if large successive gusts at higher frequencies are less common. 3.2. Softening nonlinearities To compare the resonance behavior of a linear and soft nonlinear spring-mass-damper systems, simulations were performed for three values of the nonlinear ratio (K 3 =K 1 ¼ 1; 0.5, 0) and five damping ratios ðz ¼ 0:1; 0:2; 0:3; 0:4; 0:5Þ: Fig. 9A shows the response amplitudes for these nonlinear ratios with the normalized forcing amplitude set to 0.33. This value of the normalized forcing amplitude corresponds to the normalized force likely to cause failure in the linear case. For K 3 =K 1 ¼ 0:5 and 1, the nonlinear response is

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Deflection / Max Deflection

K3 / K1 = 0 2

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Fig. 9. Maximum response amplitudes for linear and soft spring-mass-damper systems. Normalized deflections (L/Lmax) are plotted against the forcing frequency divided by the linear natural frequency (o=oo ). The nonlinear ratio (K3/K1) is set to -1, -0.5 and 0. (A) The normalized forcing amplitude (po =K 1 ) is set to 0.33. This value corresponds to the force like to cause failure in the linear case when z ¼ 0:1: The damping ratio (z) is set to 0.1, 0.2, 0.3, 0.4, and 0.5 (corresponding to the curves graphed from top to bottom in each plot). For each damping ratio, the nonlinear response is ‘pushed’ to the left in comparison to the linear case. The maximum amplitudes of deflection are also larger and occur at lower forcing frequencies. These differences diminish as the damping ratio increases. (B) Same as A, but the normalized forcing amplitude (po =K 1 ) is now set to 0.25. This corresponds to the normalized force likely to cause failure in the nonlinear case (K 3 =K 1 ¼ 1) when z ¼ 0:1: The effect of the softening nonlinearity on the response curve is lower at this smaller forcing amplitude. (C) The damping ratio (z) is set to 0.15, and the normalized forcing amplitude (po =K 1 ) is set to 0.05, 0.1, 0.15, 0.2, and 0.25 (corresponding to the curves graphed from bottom to top of each plot). The effect of the nonlinearity on the response amplitude decreases as the forcing amplitude decreases. When po =K 1 ¼ 0:05; the effect of the nonlinearity is minimal.

‘pushed’ to the left in comparison to the linear case. Note that this is the opposite effect of the hardening case shown in Fig. 7A. The maximum amplitudes of deflection are larger in the nonlinear case and occur at lower forcing frequencies. These differences diminish as the damping ratio increases. To consider the harmonic response at forcing amplitudes likely to cause failure in the strongly nonlinear case (K 3 =K 1 ¼ 1), simulations were also performed at a normalized forcing amplitude of 0.25 as shown in Fig. 9B. The effect of the nonlinearity is diminished in the case of the lower forcing amplitude.

The maximum amplitude of deflection in the strongly nonlinear case (K 3 =K 1 ¼ 1) is about 1.4 times greater than the linear case when the damping ratio is set to 0.10. This maximum amplitude of deflection over all frequencies occurs at a forcing frequency that is about 65% of the linear natural frequency when the damping ratio equals 0.10. To compare the shape of the response amplitude curve for various values of the normalized forcing amplitude, simulations were performed for three cases (K 3 =K 1 ¼ 1; 0.5, 0) at five forcing amplitudes (po =K 1 ¼ 0:05; 0.1, 0.15, 0.2, 0.25) in Fig. 9C. The

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Fig. 10. Contour plots of the normalized maximum deflection (L/Lmax). (A) L/Lmax as a function of the forcing frequency divided by the natural frequency (o=oo ) and the nonlinear ratio (K3/K1). The normalized forcing amplitude (po =K 1 ) is set to 0.33 (left) and 0.25 (right). The damping ratio is set to 0.15. The nonlinear ratio (K3/K1) is varied from 1 to 0. When K 3 =K 1 0; maximum deflections occur when the forcing frequency is near the natural frequency. As K3/K1 decreases, the resonance behavior shifts to lower frequencies, and the maximum amplitude of deflection over all frequencies increases. The larger forcing amplitude results in larger maximum deflections and augment the effect of the softening nonlinearity. (B) Maximum L/Lmax as a function of the nonlinear ratio (K3/K1) and the damping ratio (z). Maximum L/Lmax was taken as the maximum deflection over all forcing frequencies ð0pop2oo Þ for each nonlinear ratio and damping ratio pair. The normalized forcing amplitude is set to 0.33 (left) and 0.25 (right). K3/K1 is varied from 1 to 0. The damping ratio is varied from 0.05 to 0.25. Maximum deflections occur when K 3 =K 1 1 and the damping ratio is small (z 0:05). Decreasing the damping ratio and/or decreasing the nonlinear ratio increases the maximum amplitude of deflection. As K3/K1 decreases, the resonance behavior shifts to lower frequencies. Finally, the far left portions of the contour plots appear ‘spotted’ because these are chaotic regions of the graph. During some simulations, substantially larger deflections occurred due to chaos.

damping ratio in each case was set to 0. As po =K 1 increases, the entire response curve also increases. The effect of the nonlinearity increases as the forcing amplitude increases. Resonance behavior is pushed to lower frequencies under larger applied forces. This implies that the results of dynamic tests at small deflections could differ greatly from the dynamics at larger amplitudes in a nonlinear system. The response amplitudes for a range of softening nonlinearities over a range forcing frequencies are shown in Fig. 10A. Two normalized forcing amplitudes were considered. po =K 1 ¼ 0:33 models the normalized force likely to cause failure in the linear case. po =K 1 ¼ 0:25 models the normalized force likely to cause failure in the strongly nonlinear case (K 3 =K 1 ¼ 1). Simulations were performed for nonlinear ratios ranging from 0 to 1 at a damping ratio of 0.15. For K 3 =K 1 0; maximum deflections occur when the forcing frequency is near the natural frequency. As K 3 =K 1 decreases, the resonance behavior shifts to lower frequencies. Furthermore, the maximum amplitude of deflection

over all frequencies increases as K 3 =K 1 becomes more negative. The maximum response amplitudes over a range of forcing frequencies for various softening nonlinearities and damping ratios are shown in Fig. 10B. The same normalized forcing amplitudes were considered (po =K 1 ¼ 0:25 and 0.33). Simulations were performed for nonlinear ratios ranging from 1 to 0 and damping ratios ranging from 0.05 to 0.25. The maximum response amplitude for each nonlinear ratio and damping ratio pair was taken as the maximum of the response amplitude curve for forcing frequencies ranging from 0 to 2oo : Larger deflections occur as K3/K1 becomes more negative. As the damping ratio decreases, the maximum deflection rapidly increases. In addition, the resonance behavior of the system shifts to lower frequencies as K3/K1 becomes more negative. This is significant to the stability of the plant because the range of resonance behavior shifts to lower frequencies as the system becomes more nonlinear. This could increase the probability that the plant experiences wind gusts that

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cause resonance. Finally, the far left portions of the contour plots appear ‘discontinuous’ because these are chaotic regions of the graph. During some simulations, substantially larger deflections occurred due to chaos. 4. Discussion The results of this paper show that hardening and softening nonlinearities in the elastic modulus of plant stems can have significant influence on their resonance behavior at large amplitudes of deflection. Plant stems and other biological beams that behave like hard springs experience resonance at higher frequencies and at lower amplitudes than what would be predicted by linear theory. Biological beams that act like soft springs experience resonance at lower frequencies and at larger amplitudes than what would be predicted by linear theory. For both types of nonlinearities, resonant behavior is distributed over a larger range of frequencies and when compared to the linear case. This result is supported by several studies that have measured deflections of plant shoots in the field. The results of such studies on corn and wheat stalks suggest that there are no clear resonant peaks when power spectra are made from deflection data of field measurements (Finnigan, 1979; Wilson et al., 1982). Rather, resonant behavior is spread over a wide range of frequencies. Denny et al. (1998) also found that resonant behavior was spread over a large range of frequencies for several wave swept organisms. Plant structures have a unique capability of regularly withstanding substantial resonant loading. On the other hand, man-made structures often fail when forced at their natural frequency. A famous example of this is the failure of the Tacoma Narrows Bridge (Billah and Scanlan, 1991). The changes in resonance behavior due to nonlinearities in the elastic modulus of the plant could be beneficial or detrimental to the plant under resonant loading, depending upon the environment. In the case of hardening nonlinearities, resonant behavior is shifted to higher frequencies. This could be advantageous if higher frequency gusts are relatively infrequent and of lower energy. For the same amplitude driving force, the deflections during resonance are lower than those of the corresponding linear system. Another advantage of the hardening nonlinearity is that it could allow plants to reorient into lower drag configurations, yet behave like stiffer structures during large deflections. In the case of softening nonlinearities, resonant behavior is shifted to lower frequencies. This could be advantageous if low frequency gusts or waves are relatively infrequent. Gaylord and Denny (1997) describe how such a strategy might be used by some kelps, i.e. the direction of the waves change before the stipe is fully extended. In addition, oscillation amplitudes are decreased as the driving frequency increases beyond the

natural frequency of the system. Similar to the hardening case, a ‘soft’ plant might also avoid the effects of resonance by moving resonance away from the driving frequency, if the driving frequency is relatively constant. On the other hand, such behavior could be detrimental in certain environments. If the resonant behavior of the plant stem is spread over a large range of lower frequencies, a plant stem could conceivably be ‘trapped’ in resonance by successive wind gusts or waves in that large frequency range. Damping can also dramatically reduce resonant amplitudes for a given force. Milne (1991) suggested that three main components contribute to the damping of a medium sized tree: (1) friction from branches interacting with neighboring trees, (2) aerodynamic drag due to foliage, and (3) viscoelastic damping in the stem. He estimated the importance of these factors to the overall damping of the tree as the ratio 5:4:1. The two most important factors could vary greatly depending upon the proximity of neighboring trees or plants, and whether or not wind gusts with the motion of the plant. The third factor, viscoelastic damping of the stem, would seem to be the most uniform, but it also has the smallest contribution to the total damping. This suggests that there might be ancestral or developmental constraints that limit the degree to which the stem can be damped. 4.1. Chaotic stems In the presence of stochastic forces, the Duffing oscillator exhibits chaotic motion for a certain range of parameter values. Denny et al. (1997) have discussed chaos and its significance to the stability of kelps. They suggest that chaotic dynamics can lead to large destructive stresses on plants. In the model presented in this paper, chaos will occur when the forcing amplitude is sufficiently large and/or the damping is sufficiently light. This transition to chaos occurs when the response amplitude curve of the nonlinear system becomes multi-valued. The threshold into chaotic behavior is determined by o=oo ; K 3 =K 1 ; po =K 1 ; and z: This could be used define a morphospace of biological structures ‘tuned’ to avoid chaos. If forcing amplitude becomes quite large or damping becomes sufficiently light (i.e. the wind gusts with the motion of the plant, the plant stand becomes less dense, structural damping is reduced as a result of a broken branch, material damping changes with humidity or soil moisture) nonchaotic plants could become chaotic. Perhaps this is one factor that contributes to catastrophic failure, but more work is needed on this complicated subject. 4.2. Further modeling To gain a complete understanding of how plants deal with resonance and why they fail, further work

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describing the behavior of plant stems beyond their linear elastic range is needed. With the use of detailed data on how the elastic modulus varies with deformation, more sophisticated models of plant stem dynamics could be developed. For example, alternative force–displacement relationships (such as a higher-order polynomial, logarithmic, or exponential models) could be measured and tested in a manner similar to that presented in this paper. The ODE model of the Duffing oscillator could also be extended to a continuous partial differential equation model. Such a model could describe deflection, mass, and stiffness as a function of height. The advantages of the model presented in this paper, however, are its simplicity, the relative ease of obtaining some analytical solutions, and the wealth of work available on the Duffing equation. In addition to the nonlinearities of the elastic modulus, other mechanical properties of the plant stem could be significant to the resonant behavior at large amplitudes. For example, asymmetries in the elastic properties or mass distribution along the stem could contribute to twisting motions and torsional strains. As a result, coupled twisting and bending could be significant to the resonant behavior of the plant. Such asymmetries could be incorporated into the model by describing the stem dynamics with coupled bending and twisting equations. Furthermore, the elastic modulus of biomaterials is often dependent upon the rate of strain. This property could also be incorporated into the model by making k1 and k3 velocity dependent.

Acknowledgements I would like to thank Steven Vogel, Hugh Crenshaw, and Bill Kier who provided excellent comments on this work as members of my thesis committee. I would also like to thank James Wilson for insights provided from many conversations on structural dynamics. Finally, I would like to thank two anonymous referees for their detailed and thought provoking comments.

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