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Oscillation frequencies of tapered plant stems¹

Institute for Biology III, University of Freiburg, Germany

Received for publication March 29, 2001. Accepted for publication July 31, 2001.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS COMPUTATIONS THEORETICAL CONSIDERATIONS APPROXIMATIONS RESULTS AND DISCUSSION LITERATURE CITED

 
Free oscillations of upright plant stems, or in technical terms, slender tapered rods with one end free, can be described by considering the equilibrium between bending moments in the form of a differential equation with appropriate boundary conditions. For stems with apical loads, where the mass of the stem is negligible, Mathematica 4.0 returns solutions for tapering modes = 0, 0.5, and 1. For other values of , including cases where the modulus of elasticity varies over the length of the stem, approximations leading to an upper and a lower estimate of the frequency of oscillation can be derived. For the limiting case of = 0, the differential equation is identical with Greenhill's equation for the stability against Euler buckling of a top-loaded slender pole. For stems without top loads, Mathematica 4.0 returns solutions only for two limiting cases, zero gravity (realized approximately for oscillations in a horizontal orientation of the stem) and for = 0 (Greenhill's equation). Approximations can be derived for all other cases. As an example, the oscillation of an Arundo donax plant stem is described.

Key Words: Arundo donax • differential equations • free oscillation • taper • top load

	INTRODUCTION

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Knowledge of the resonance frequency particularly in the fundamental mode of oscillation is necessary to assess a plant's stability against dynamic wind loads (Kerzenmacher and Gardiner, 1998 ). In addition measurement of the frequency of oscillations of plant stems or slender rods offers an efficient and potentially nondestructive way to determine mechanical properties such as the storage modulus and the loss modulus of elasticity (Young, 1989 ; Vincent, 1990 ; Zebrowski, 1991 ; Niklas, 1992 ; Speck and Spatz, 2000 ). Niklas and Moon (1988) and Niklas (1989, 1990, 1991, 1997 ) have recorded multiple resonance spectra and evaluated the modulus of elasticity for a variety of plant stems. Zebrowski (1999) and Spatz and Zebrowski (2001) measured the natural frequency of free vibrations of upright cereal plant shoots with an apical load. They described oscillations in the vertical and in the horizontal orientation of the stems by an equilibrium of bending moments, being different for the two orientations. Solutions of the corresponding differential equations were presented for untapered columns with negligible self mass.

The present approach generalizes this to plant stems with or without apical loads, with different tapering modes and with changes of the modulus of elasticity and of density from the base to the apex as usually observed in plant stems. Geometric input parameters are the total length, the cross-sectional area and the second moment of area of the stem at the base. The results, therefore, relate to any cross section if, at least for cross sections with a symmetry less than threefold, the plane of bending is specified.

Several examples of the solution of the differential equation are presented and approximations for others are outlined. Intermediate values can be interpolated or calculated directly by the Mathematica 4.0 program given. The accuracy of the approach is tested against the oscillation frequency of a tapered plastic rod. Its applicability for plant stems is exemplified by quantitative analyses of video recordings of an Arundo donax stem.

	MATERIALS AND METHODS

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Arundo donax stems in a wind-sheltered area of the Botanical Garden, Freiburg, Germany, were bent and released by hand. The oscillations were recorded in side view with a video camera. A canvas of 3 x 4 m was used as background for the videos and provided additional wind shelter. The stems carried small markers of high contrast tape at eight positions along the stem. Their displacement from the resting position was analyzed frame by frame with the software SIMI°Motion 4.6 (SIMI GmbH, Unterschleissheim, Germany).

	COMPUTATIONS

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Calculations were carried out with the powerful program Mathematica 4.0 (Wolfram Research, Champaign, Illinois, USA). Values for A, B, G, and H (Eq. 19) are found by nesting intervals. Finding the least positive singularities requires small distortions of the boundary conditions as described by Spatz (2000) . All computations were carried to at least five significant digits. Mathematica 4.0 does not solve the integrals (Eqs. 12, 16, and 19) directly; instead, a table of values is provided and interpolated by polynomials. These can be integrated.

	THEORETICAL CONSIDERATIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS COMPUTATIONS THEORETICAL CONSIDERATIONS APPROXIMATIONS RESULTS AND DISCUSSION LITERATURE CITED

 
Top load, negligible mass of the stem
Vertical orientation of the stem
The differential equation for free vibrations of an upright slender rod or stem with an apical load (Fig. 1) can be formulated as an equilibrium of bending moments. Neglecting the mass of the stem itself we obtain for deflections within the linear elastic range

View larger version (16K): [in this window] [in a new window]   Fig. 1. Scheme illustrating a slender tapered plant stem with an apical load, oscillating around an upright resting position

(1)

For plant stems it is reasonable to assume that EI varies smoothly and monotonously over its length. We can approximate

With = y/y_T, Differential Eq. 1 can be rewritten in the form

For A = 0 (i.e., = 0) Eq. 3a–e becomes identical to Greenhill's equation for the stability of a top-loaded slender rod against Euler buckling (Greenhill, 1881 ; Spatz, 2000 ).

The remaining problem is to find numerical values for combinations of A and B, which characterize the fundamental frequency of free vibration. The equivalent condition for any given value of B is the solution of Differential Eq. 3a with the boundary conditions (Eq. 3d), which fullfills (z_T) = 1.

Figure 2 is an example of a solution of Differential Eq. 3 as returned by Mathematica 4.0. The output gives the function (z). The plot (z) against z displays the actual bending line. Figure 3 shows how the acceleration term A and therefore according to Eq. 3c depends on the gravitational term B for those values of the tapering mode for which a solution is available. The lines through the data points are fitted by a second-order polynomial

View larger version (14K): [in this window] [in a new window]   Fig. 2. Formulation of the Differential Eq. 3a–e and its solution by Mathematica 4.0. This is given in form of a complex function. The imaginary part is of the order of 10–15. The graph displays the actual bending line, with z = 1 at the base and 0 at the very tip of the stem. The correct value for A is found as the solution that fulfils T = 1. The correct value for B is found as singularity of the differential equation. The singularity lies in the interval where the solution switches from positive to negative values

View larger version (14K): [in this window] [in a new window]   Fig. 3. Solutions of Differential Eq. 3a–e. The "acceleration term" A is plotted against the "gravitational term" B for tapering modes = 0, 0.5, 1, ß = 0, and zT = 0.05. The lines through the data points are fitted by second order polynomials (see text). The data for = 1 are plotted on a different scale in Fig. 9

View larger version (17K): [in this window] [in a new window]   Fig. 4. The "acceleration term" A plotted against the "gravitational term" B for = 0.5, ß = 0, and different positions zT of the top load. The data points are fitted by second-order polynomials

Eq. 6a–d is equivalent to Eq. 3a–e for B = 0, i.e., in principle for zero gravity. It should be noted though that Eq. 4 is only valid if the stem in its resting position can be considered nearly horizontal over its entire length. This requires a very stiff rod, such that the downward deflection is small. From cantilever theory it follows that this can be expressed by

Exact solutions in form of linear combinations of elementary functions are obtained for Differential Eq. 6a–d, i.e., for oscillations of a top-loaded plant stem or rod oriented horizontally. Figure 5 shows how the acceleration term A and therefore according to Eq. 6b depends on the tapering mode and the mode of dependence of the modulus of elasticity ß.

View larger version (17K): [in this window] [in a new window]   Fig. 5. Solutions of Differential Eq. 6a–d for oscillations with horizontal orientation of the stem are available for almost all combinations of and ß. The lines through the data points are fitted by third-order polynomials

2

View larger version (15K): [in this window] [in a new window]   Fig. 6. Scheme illustrating a slender tapered plant stem without a top load oscillating around an upright resting position

For H = 0 (i.e., = 0) Eq. 8a–e becomes identical to Greenhill's equation for the stability of a slender pole without top load (Greenhill, 1881; Spatz, 2000 ).

The remaining problem is to find numerical values for H and G, which characterize the fundamental frequency of free vibration. The equivalent condition is the least positive singularity in the solution of the differential equation with the above boundary conditions. Mathematica 4.0 solves Differential Eq. 8a–e only for G = 0 or H = 0 (Fig. 7). Approximations for G and H 0 can be obtained as outlined below.

View larger version (19K): [in this window] [in a new window]   Fig. 7. Formulation of Differential Eq. 8a–e and its solution by Mathematica 4.0 in terms of hypergeometric functions. The correct values for G with H = 0 and H with G = 0 are found as singularities of the differential equation. As in Fig. 2, the graph displays the actual bending line, with z = 1 at the base and 0 at the tip of the stem

View larger version (23K): [in this window] [in a new window]   Fig. 8. Solutions of Differential Eq. 8a–e for G = 0, i.e., for oscillations with horizontal orientation of the stem, are available for almost all combinations of , ß, and . The lines through the data points are fitted by fifth-order polynomials (Table 1 )

View this table: [in this window] [in a new window]   Table 1. Within deviations of <0.5% the data shown in Fig. 8 can be approximated by H0 = c0 + c1 + c2 2 + c3 3 + c4 4 + c5 5. The table lists values for c0 to c5 for some values of ß and typically found in plant stems

	APPROXIMATIONS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS COMPUTATIONS THEORETICAL CONSIDERATIONS APPROXIMATIONS RESULTS AND DISCUSSION LITERATURE CITED

For all other cases approximations can be obtained by considering the potential energy during oscillations (Timoshenko and Young, 1948 ).

The strain energy of bending the stem is

With = y/y_T, Eq. 3a and 3b and the abbreviations

(12)

The evaluation of the numerical terms K₁ and K₂ requires the knowledge of the bending line y(z). Timoshenko and Young (1948) used a cosine function. A better approximation, especially for strongly tapered columns, uses the bending lines for the two limiting cases: (1) For A 0, (z) is chosen as solution of Eq. 3a–e with A = 0, i.e., Greenhill's equation for Euler buckling; (2) For B 0, (z) is chosen as solution of Differential Eq. 3a–e with B = 0, i.e., zero gravity or horizontal orientation provided that the downward deflection in its resting position is small.

This is illustrated for = 1 in Fig. 9. Applying Eq. 10 and choosing a solution of Eq. 3a–e for A = 0 leads to a straight line through (0, A₀) and (B*, 0), and choosing a solution of equation Eq. 3a–e for B = 0 leads to a straight line through (0, A*) and (B₀, 0), where A₀ = A(B = 0) and B₀ = B(A = 0) are exact solutions.

(6)

View larger version (17K): [in this window] [in a new window]   Fig. 9. The "acceleration term" A plotted against the "gravitational term" B for = 1, ß = 0, and zT = 0.05. Squares represent solutions of Differential Eq. 3a–e. The graph illustrates the procedure as described in the Approximations section. A straight line through A0 and B0 gives a lower estimate, and the lines from A0 or B0 to the intercept of the two straight lines defined by Eq. 12 provide an upper estimate. The approximations become increasingly better as becomes smaller (compare Fig. 3 )

View this table: [in this window] [in a new window]   Table 2. Characteristic values A0, A*, B0 and B* for upper and lower estimates of oscillation frequencies as a function of the tapering mode for ß = 0 and zT = 0.05. A* and B* are defined as intercepts in Fig. 9. Values for ß 0 can be interpolated by replacing with + ß/4

View larger version (17K): [in this window] [in a new window]   Fig. 10. Values for A0, A*, B0, B* for = 0.25, zT = 0.05 as function of the mode of positional dependence of the modulus of elasticity ß

The strain energy of bending the stem can be calculated analogically to Eq. 9

During the oscillation each part of the stem experiences a vertical displacement. The corresponding potential energy is

The bending line y(z) for H 0 is approximated as the solution of Differential Eq. 8a–e for H = 0, i.e., Greenhill's equation. The bending line y(z) for B 0 is approximated as the solution of Differential Eq. 8a–e for B = 0, i.e., zero gravity.

Table 3 gives values for G₀, G*, H₀, and H* as function of the tapering mode .

View this table: [in this window] [in a new window]   Table 3. Characteristic values H0, H*, G0, and G* for upper and lower estimates of oscillation frequencies as a function of the tapering mode for ß = 0 and = 0. H* and G* are defined as intercepts analogous to A* and B* (Fig. 9). Solutions of Differential Eq. 8 with H = 0 (Greenhill's equation) are not available for 1.5 (Spatz, 2000)

View larger version (16K): [in this window] [in a new window]   Fig. 11. Approximations of the "acceleration term" H as function of the "gravitational term" G. The values for H0 and G0 (solid symbols) are calculated for ß = 0 and = 0. The numbers near the dashed lines denote the tapering mode

During the oscillation each part of the stem as well as the additional load experiences a vertical displacement

As outlined above the evaluation of the numerical terms N₁, N₂, N₃, and N₄ requires the knowledge of the normalized bending line (z). In general this is only available if three of the terms A, B, G, or H are zero, while the fourth is determined as a solution of Differential Eqs. 3a–e or 8a–e. The numerical terms N₁, N₂, N₃, and N₄ can be computed for the corresponding function (z) being different for the four limiting cases A₀, B₀, G₀, or H₀. Table 4 shows the results of these computations for a given set of input values , ß, , z_T, and z (tip). The values for A*, B*, G*, and H* allow to construct plots as in Fig. 9 to determine upper and lower estimates for the oscillation frequencies or for the stability against Euler buckling. Some special cases deserve mentioning. (a) G = 0; H = 0 describes the case of a stem with a top load but negligible mass of the stem itself as discussed above; (b) A = 0 and B = 0 describes the case of a stem with nonnegligible mass and no top load as discussed above; (c) B = 0; G = 0 describes the oscillation of a stem in the horizontal orientation with non-negligible mass and an additional mass attached (Spatz and Zebrowski, 2001 ). For = 0, ß = 0, = 0, and z_T = 0 and the oscillation frequency being dominated by the additional mass M the upper estimate reads

(11)

identical with the approximation given by Young (1989 ; p. 715). If the mass of the stem is dominating we obtain an upper estimate

(12)

The lower estimate is given by

(13)

(d) A = 0; H = 0 corresponds to = 0, it therefore describes the stability against Euler buckling for a stem with a nonnegligible self mass and an additional load attached (Spatz, 2000 ). (e) and (f) The remaining cases A = 0; G = 0 and B = 0; H = 0 are physically not easily realized.

View this table: [in this window] [in a new window]   Table 4. Values for the approximation of oscillation frequencies for = 0.5, ß = 0, = 0, zT = 0.05, and z(tip) = 0.0009.

	RESULTS AND DISCUSSION

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As a test for the accuracy of the calculations the oscillation frequencies in the upright orientation of a polyvinyl chloride rod manufactured to give a tapering mode of = 0.1415 and a nontapered rod of the same material were recorded. Differential Eq. 8a–e was solved for ß = 0, = 0, and G = 0 and the solution corrected for the reduction of the oscillation frequency due to a finite gravitational term according to Eq. 16. This results in a determination of the modulus of elasticity of 3.40 GPa for the tapered rod and 3.27 GPa for the nontapered rod compared to 3.32 GPa from independent three-point-bending experiments.

As an example of the applicability of the approach to the oscillation of plant stems the results of the analysis of video recordings of an Arundo donax stem are presented in Fig. 12. In this particular example the leaves were removed. The length of the stem is L = 4.33 m. The sample is part of a study of damping of oscillations performed on eight plants under various conditions (to be published elsewhere). The data are compatible with a damped harmonic oscillation, implying that the oscillation did not go beyond the linear elastic range of the material. The frequency of oscillation was = 3.648 ± 0.004 sec^–1. The analysis of the amplitude as function of the height along the plant stem shows that it is a bending oscillation. It is not compatible with a pendulum with a hinge in the transition zone from stem to underground rhizome.

View larger version (25K): [in this window] [in a new window]   Fig. 12. Displacement at various heights above ground as function of time for an Arundo donax plant without leaves. The lines through the data points correspond to an equation for a damped harmonic oscillation

Eq. 16 written in the form

	FOOTNOTES

 
¹ Helpful discussions with Drs. M. Neumann and T. Speck are gratefully acknowledged. We thank L. Schweizer for his help with the quantitative video analyses, discussions, and suggestions.

	LITERATURE CITED

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS COMPUTATIONS THEORETICAL CONSIDERATIONS APPROXIMATIONS RESULTS AND DISCUSSION LITERATURE CITED

 
Greenhill G. 1881 Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and the greatest height to which a tree of given proportions can grow. Proceedings of the Cambridge Philosophical Society 4: 65-73

Kerzenmacher T. B. Gardiner 1998 A mathematical model to describe the dynamic response of a spruce tree to the wind. Trees 12: 385-394[CrossRef]

Niklas K. J. 1989 Nodal septa and the rigidity of aerial shoots of Equisetum hyemale. American Journal of Botany 76: 521-531[CrossRef][ISI]

Niklas K. J. 1990 The mechanical significance of clasping leaf sheaths in grasses: evidence from two cultivars of Avena sativa. Annals of Botany 65: 505-512[Abstract/Free Full Text]

Niklas K. J. 1991 Bending stiffness of cylindrical plant organs with a ‘core-rind’ construction: evidence from Juncus effusus leaves. American Journal of Botany 78: 561-568[CrossRef][ISI]

Niklas K. J. 1992 Plant biomechanics: an engineering approach to plant form and function. The University of Chicago Press, Chicago, Illinois, USA

Niklas K. J. 1997 Responses of hollow, septate stems to vibrations: biomechanical evidence that nodes can act mechanically as spring-like joints. Annals of Botany 80: 437-448[Abstract/Free Full Text]

Niklas K. J. F. C. Moon 1988 Flexural stiffness and modulus of elasticity of flower stalks from Allium sativum as measured by multiple resonance frequency spectra. American Journal of Botany 75: 1517-1525[CrossRef][ISI]

Spatz H.-Ch. 2000 Greenhill's formula for the critical Euler buckling length revisited. In H.-Ch. Spatz and T. Speck [eds.], Plant biomechanics. Thieme Verlag, Stuttgart, Germany

Spatz H.-Ch. H. Beismann F. Brüchert A. Emanns T. Speck 1997 Biomechanics of the giant reed Arundo donax. Philosophical Transactions of the Royal Society of London 352: 1-10[CrossRef]

Spatz H.-Ch. J. Zebrowski In press Oscillation frequencies of plant stems with apical loads. Planta 214: 215-219

Speck O. H.-Ch. Spatz 2000 Oscillation and damping behaviour of the giant reed Arundo donax. In H.-Ch. Spatz and T. Speck [eds.], Plant biomechanics. Thieme Verlag, Stuttgart, Germany

Timoshenko S. D. H. Young 1948 Advanced dynamics. McGraw-Hill, New York, New York, USA

Vincent J. F. V. 1990 Structural biomaterials. Princeton University Press, Princeton, New Jersey, USA

Young W. C. 1989 Roark's formulas for stress and strain. McGraw-Hill, New York, New York, USA

Zebrowski J. 1991 The use of free vibrations to measure peduncle stiffness in Triticale. Journal of Experimental Botany 42: 1207-1212[Abstract/Free Full Text]

Zebrowski J. 1999 Dynamic behaviour of inflorescence bearing Triticale and Triticum stems. Planta 207: 410-417[CrossRef][ISI][Medline]

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