Multiple resonance damping or how do trees escape dangerously large oscillations?

To further understand the mechanics of trees under dynamic loads, we recorded damped oscillations of a Douglas fir (Pseudotsuga menziesii) tree and of its stem without branches. Eigenfrequencies of the branches were calculated and compared to the oscillation frequency of the intact tree. The term eigenfrequency is used here to characterize the calculated resonance frequency of a branch fixed at the proximal end to a solid support. All large branches had nearly the same frequency as the tree. This property is a prerequisite for the distribution of mechanical energy between stem and branches and leads to an enhanced efficiency of damping. We propose that trees constitute systems of coupled oscillators tuned to allow optimal energy dissipation.     

The theory of tree bole and branch form

Summary Working from the general postulate that natural selection of plant form operates so as to maximize the survival potential of a species, this paper examines the hypothesis that the mechanical support of tree foliage must approach optimality in the use of wood, i.e., that tree stems and branches will have optimal form with respect to the amount of support tissue. Mathematical models of bole and branch form are presented, based on the proposition that either wind or gravity are the primary limiting factors for tree size and shape. Predictions of trunk and branch diameter as a function of tree size were tested with dimensional measurements ofPopulus tremuloides. The individual stems were selected from close-grown stands of differing ages. For small and intermediate trees, trunk diameter is such that stems have only 1.6 times as much wood as the minimum required to keep the tree from buckling under its own weight due to elastic instability. Branch diameters are shown to be close to the minimum required to maintain the spatial position of growing branches, as well as withstand wind forces. This minimal branch cost not only reduces the load which the stem must support against elastic instability, but allows the crown to flex in high winds. The flexing, in turn, reduces the drag force exerted by the wind on the trunk. Thus, the hypothesis that the observed tree form is an optimal design cannot be rejected on the basis of these results. Additional studies are planned with respect to optimal foliage distribution.     

Dynamic analysis of olive trees in intensive orchards under forced vibration

The mechanical harvesting of fruit for oil production in an intensive olive tree orchard is generally accomplished by applying vibration to the tree’s trunk. This vibration is consequently transmitted to the branches, causing the fruit to detach. Although this practice is commonly used, the effects on tree behavior under forced vibration are not firmly established. Dynamic analysis was performed on 17 olive trees (Olea europaea L.) growing in an intensively-managed orchard using modal testing techniques. Modal parameter identification was focused inside the range excitation frequency used by the most commonly available trunk shakers on the market. The olive trees featuring a low morphological variability and modal parameters were obtained for a representative olive tree. The first two modes of vibration of the main tree frame were identified with damping ratios of 26.9 and 17.1% and natural frequencies of 20.2 and 37.7 Hz, respectively. A third mode of vibration of less importance was found at a higher frequency. Therefore, many local modes of vibration were detected near these natural frequencies, primarily located on secondary branches. During the testing, the olive trees behaved like a damped harmonic oscillator with predominantly mass damping in these modes.     

The cause of the prevalent directions of the spiral grain patterns in conifers

 A new theory is presented on the cause of the prevalent directions of the spiral grain patterns found in conifers. The hypothesis is based upon the assumption that spiral grain has a function, i.e. that it represents a growth strategy to ensure survival of the trees. The mechanical function of the tree trunk is placed in focus, that is the ability of the trees to withstand external mechanical loads, mainly from wind. Spiral grain is an optimized growth feature when the trees are exposed to combined bending and torsion. Torsion occurs when the crown is asymmetric in the plane perpendicular to the wind direction. Systematic crown asymmetry, with heavier crowns on the south side, was confirmed by measuring the crown projections on 253 sparsely grown pines; 76.7% of the trees had longer branches on the south than on the north side, and the average length difference was 40.8 cm. By studying wind maps it was seen that most of the coniferous forests have prevailing westerly winds, which, when combined with the crown asymmetry, leads to a prevailing torque. Right-handed spiral grain in the outermost layers of mature trees is proposed to be a strategy to withstand this torque, i.e. to avoid stem breakage. Received: 30 June 1997 / Accepted: 17 November 1997     

The effect of wind exposure on the tree aerial architecture and biomechanics of Sitka spruce (Picea sitchensis, Pinaceae)

This paper reports on the effect of wind loading below damaging strength on tree mechanical and physical properties. In a wind-exposed Sitka spruce stand in western Scotland, 60 trees at four different levels of wind exposure (10 m, 30 m, 50 m, 90 m from edge) were characterized for stem and crown size and shape and mechanical properties, including structural Young's modulus (E(struct)), natural frequency, and damping ratio. E(struct) increased from the stand edge to the mid-forest, but with a large inter-tree variation. Swaying frequency and damping ratio of the trees also increased with distance from edge. Wind-exposed edge trees grew shorter, but more tapered with an overall lower E(struct), allowing for greater flexural stiffness at the stem base due to the larger diameter and for higher flexibility in the crown region of the stem. The trees at the middle of the stand compensated for their increased slenderness with a higher E(struct). Thus, for the different requirements for wind-firmness at stand edge and mid-forest, an adapted combination of tree form and mechanical properties allows the best withstanding of wind loads. The results show the requirement to understand the different strategies of trees to adapt to environmental constraints and the heterogeneity of their growth reactions in response to these strategies.     

The effect of crown architecture on dynamic amplification factor of an open-grown sugar maple (Acer saccharum L.)

Tree failure may cause significant economic and societal disruptions in urban environments. A better understanding of the relationship between branches and stem as they affect the dynamic response of decurrent trees under wind loading is needed to reduce the risk of tree failure. Finite element (FE) models were used to identify the parameters that primarily impact tree response. A base model was developed using data from a sugar maple (Acer saccharum L.) located in Belchertown, MA, USA, from which parametric models were subsequently developed. Confidence in the base model was gained by comparing the natural frequency of this tree with experimental results. Results from a parametric study incorporating changes in eight different tree parameters (stem diameter, slenderness ratio of branches, number of branches, damping ratio, branch attachment heights, branch attachment angles, branch azimuth angles, and elastic modulus) are then presented to help identify critical model properties that affect the dynamic amplification factor (Rd) of the tree. A single parameter was varied in each model while keeping others unchanged from the base model. Parameters with the greatest effect on Rd included stem diameter, number and slenderness of branches in the crown, elastic modulus of stem and branches, and damping ratio. Thus, it may be possible to use pruning to alter crown architecture to reduce the risk of tree failure.     

The periodic motion of lodgepole pine trees as affected by collisions with neighbors

The periodic sways of a group of ten Pinus contorta var. latifolia (lodgepole pine) trees with slender stems from the Two Creeks site (TW) and ten stout trees from the Chickadee site (CH) both in Alberta, Canada were quantified. Tree displacement at TW was measured during periods of consistent wind direction with three mean wind speeds (1.9, 4.6, and 5.4 m/s) and for two mean wind speeds at CH (5.0 and 7.9 m/s). Spectral analysis of sway displacement data showed a decrease in the frequency with wind speed for trees at TW, but remained unchanged for trees at CH. Significant correlations between tree sway frequency and amplitude during high winds at TW indicate a loss of sway energy concomitant with the occurrence of high collision intensity. These observations support the hypothesis that inter-crown collisions have an important influence on the sway frequency of trees and should be incorporated into efforts to model their sway dynamics. We also present a theoretical collision-damped sway model which supports our empirical findings.     

On oblique growth of trees under the action of winds

It is shown that an inclined regular growth of tree trunks under the action of prevailing winds may be explained as a response of the plant to the mechanical action of the wind, in conformity with the Schwendener theory of shape of plants based on the concept of maximum strength. A possible mechanism of such response is suggested. The condition of maximum strength determines the inclination of the tree trunk on the basis of purely mechanical considerations. A numerical example computed for a palm agrees with observational data.     

Analysis and simulation of dynamic response behavior of Scots pine trees to wind loading

This paper presents an empirical approach for the decomposition, simulation, and reconstruction of wind-induced stem displacement of plantation-grown Scots pine trees. Results from singular spectrum analysis (SSA) allow a low-dimensional characterization of the complex and complicated tree motion patterns in response to non-destructive wind excitation. Since motion of the sample trees was dominated by sway in the first mode, the application of SSA on time series of sample trees’ stem displacement yielded characteristic and distinguishable non-oscillatory trend components, quasi-oscillatory sway, and noise, of which only the non-oscillatory components were correlated directly with wind characteristics. Although sway in the range of the dominant damped fundamental frequency dominated the measured stem displacement signals, it was almost decoupled from near-surface airflow. The ability to discriminate SSA-components is demonstrated based on correlation and spectral analysis. These SSA-components, as well as wind speed measured in the canopy space of the Scots pine forest, were used to train neural networks, which could then reasonably simulate tree response to wind excitation.     

Crown structure and wood properties: Influence on tree sway and response to high winds

Wind can alter plant growth and cause extensive, irreversible damage in forested areas. To better understand how to mitigate the effects of wind action, we investigated the sensitivity of tree aerodynamic behavior to the material and geometrical factors characterizing the aerial system. The mechanical response of a 35-yr-old maritime pine (Pinus pinaster, Pinaceae) submitted to static and dynamic wind loads is simulated with a finite element model. The branching structure is represented in three dimensions. Factor effects are evaluated using a fractional experimental design. Results show that material properties play only a limited role in tree dynamics. In contrast, small morphological variations can produce extreme behaviors such as either very little or nearly critical dissipation of stem oscillations. Slender trees are shown to be relatively more vulnerable to stem breakage than uprooting. Dynamic loading leads to deflections and forces up to 20% higher near the base of the tree than those calculated for a static loading of similar magnitude. Effects of branch geometry on dynamic amplification are substantial yet not linear. The flexibility of the aerial system is found to be critical to reducing the resistance to the airflow and thus to minimizing the risk of failure.     

A scaling law for the effects of architecture and allometry on tree vibration modes suggests a biological tuning to modal compartmentalization

Wind is a major ecological factor for plants and a major economical factor for forestry. Mechanical analyses have revealed that the multimodal dynamic behavior of trees is central to wind-tree interactions. Moreover, the trunk and branches influence dynamic modes, both in frequency and location. Because of the complexity of tree architecture, finite element models (FEMs) have been used to analyze such dynamics. However, these models require detailed geometric and architectural data and are tree-specific-two major restraints for their use in most ecological or biological studies. In this work, closed-form scaling laws for modal characteristics were derived from the dimensional analysis of idealized fractal trees that sketched the major architectural and allometrical regularities of real trees. These scaling laws were compared to three-dimensional FEM modal analyses of two completely digitized trees with maximal architectural contrast. Despite their simplifying hypotheses, the models explained most of the spatiotemporal characteristics of modes that involved the trunk and branches, especially for sympodial trees. These scaling laws reduce the tree to (1) a fundamental frequency and (2) one architectural and three biometrical parameters. They also give quantitative insights into the possible biological control of wind excitability of trees through architecture and allometries.     

The mechanical stability of the world’s tallest broadleaf trees

The factors that limit the maximum height of trees, whether ecophysiological or mechanical, are the subject of longstanding debate. Here, we examine the role of mechanical stability in limiting tree height and focus on trees from the tallest tropical forests on Earth, in Sabah, Malaysian Borneo, including the recently discovered tallest tropical tree, a 100.8 m Shorea faguetiana named Menara. We use terrestrial laser scans, in situ strain gauge data and finite element simulations, to map the architecture of tall tropical trees and monitor their response to wind loading. We demonstrate that a tree's risk of breaking due to gravity or self‐weight decreases with tree height and is much more strongly affected by tree architecture than by material properties. In contrast, wind damage risk increases with tree height despite the larger diameters of tall trees, resulting in a U‐shaped curve of mechanical risk with tree height. Our results suggest that the relative rarity of extreme wind speeds in north Borneo may be the reason it is home to the tallest trees in the tropics. Abstract in MALAY is available with online material.     

A physics-based link model for tree vibrations

• Premise of the study: A new mathematical model for the vibration of trees is presented for developing a more thorough understanding of the underlying structure of the response. It may be used, for example, to assess the stability of a tree or to interpret experimental data. • Methods: A model is developed for the motion of the trunk and its N number of branches. The spatial distribution and initial orientation of the branches are left for the user to prescribe. A Newtonian analysis yields (N + 1) nonlinear, coupled differential equations that, when solved, describe the response of the trunk and each branch. After the model is linearized near equilibrium, the natural frequencies and vibration mode shapes are found. Closed-form expressions for the response (i.e., the actual time histories) are then obtained using modal analysis. Numerical solutions are also found; these are used to validate the analytical solutions and to serve as a means for considering large amplitude vibrations. • Key results: A new physics-based model is described. For small motion, the tree response may be constructed from the individual mode shapes and frequencies. Also demonstrated are the limitations of the linear theory as well as numerical solutions that can be obtained when trunk/branch amplitudes are large. • Conclusions: The model presented here incorporates critical physics into a model that describes tree vibrations. It also sheds light on the underlying structure of the vibration response in terms of the modal nature of the solution. Limitations to the linear solutions are demonstrated and discussed.     

Biomechanical constraints on tree architecture

Key message

Mechanical properties of wood constrain most conifers to an excurrent form and limit the width of tree crowns. Development of support tissue alters allometric relations during ontogeny.

Abstract

Biomechanical constraints on tree architecture are explored. Torque on a tree branch is a multiplicative function of mass and moment arm. As such, the need for support rises faster than branch length, which leads to increased taper as branch size increases. This violates assumptions of models, such as the pipe-model theory, for large trees and causes changing allometry with tree size or exposure. Thus, assumptions about optimal design for light capture, self-similarity, or optimal hydraulic architecture need to be modified to account for mechanical constraints and costs. In particular, it is argued that mechanical limitations of compression wood in conifers prevent members of this taxon from developing large branches. With decurrent form ruled out (for larger species), only a conical or excurrent form can develop. Wind is shown to be a major mortality risk for trees. Adaptations for wind include dynamic responses of wood properties and height. It is argued that an adaptation to wind could be the development of an open crown in larger trees to let the wind penetrate, thereby reducing wind-throw risk. It is thus argued that crown shape and branching may result not just from optimal light capture considerations but also from adaptation to and response to wind as well as from mechanical constraints. Results have implications for allometric theory, life history theory, and simulations of tree architecture.

     

A mathematical model to describe the dynamic response of a spruce tree to the wind

 A mathematical, computer-based, dynamic sway model of a Sitka spruce (Picea sitchensis) tree was developed and tested against measurements of the movement of a tree within a forest. The model tree was divided into segments each with a stiffness, mass and damping parameter. Equations were formulated to describe the response of every segment which together form a system of coupled differential equations. These were solved with the aid of matrices and from the resulting modes, the transfer function of the tree was found and used to calculate the movement of the tree in the wind. Comparison of the modelled movement of a tree in response to the measured wind speed above a forest canopy gave good agreement with the measured movement of the top of the tree but less satisfactory agreement close to the base. The comparison also pointed to the complexity of tree response to the wind and inadequacies in the model. In particular, the branches need to be treated as coupled cantilevers attached to the stem rather than simply as masses lumped together. Received: 18 February 1997 / Accepted: 16 December 1997     

Measure of simultaneous tree sways and estimation of crown interactions among a group of trees

We present a technique to measure the simultaneous sway of a group of trees and reconstruct the frequency of crown collisions and sway dynamics of individual or groups of trees. We placed a biaxial clinometer (tiltmeter) at the live crown base in each of ten neighboring 15-m-tall lodgepole pine trees in Alberta, Canada. Tree bole rotation at tiltmeter mount height was recorded during windy conditions at a rate of 10 times/s for the cluster of trees. Rotation angles were used in a bole curve calculation to estimate tree displacement in 2-dimensional (x, y) space. Collision reconstruction was done in Arc/Info by assigning asymmetrical crown area dimensions (polygons) to calculated bole displacement for each tree. Reconstruction of each time step measured any overlaps between crown polygon areas. Crown polygon overlaps estimated in this manner allowed assessment of collision frequencies, area of crown overlap during collisions, and identification of the tree(s) that a subject tree contacted. Collision statistics are only given for trees interior to the sensored cluster (n=3). For 15.0 min of data with an average wind speed of 4.5 m/s and a maximum of 10.0 m/s there was an average of 65 collisions/min for each tree, and an average collision overlap area of 24%. This frequency and depth of collisions supports the notion that wind-induced crown interaction inhibits lateral shoot extension and is an important mechanism for the development of crown asymmetry and crown shyness. Insight into dynamic tree sway behavior and crown interactions will allow estimation and cultivation of a forest stand structure that is more resistant to damage from wind. The techniques of recording multiple simultaneous bole sway and their reconstruction are applicable to a broad range of wind-forest interaction research.     

Factors Affecting the Compliance and Sway Properties of Tree Branches Used by the Sumatran Orangutan (Pongo abelii)

The tropical arboreal environment is a mechanically complex and varied habitat. Arboreal inhabitants must adapt to changes in the compliance and stability of supports when moving around trees. Because the orangutan is the largest habitual arboreal inhabitant, it is unusually susceptible to branch compliance and stability and therefore represents a unique animal model to help investigate how animals cope with the mechanical heterogeneity of the tropical canopy. The aim of this study was to investigate how changes in compliance and time of oscillation of branches are related to easily observable traits of arboreal supports. This should help predict how supports react mechanically to the weight and mass of a moving orangutan, and suggest how orangutans themselves predict branch properties. We measured the compliance and time of oscillation of branches from 11 tree species frequented by orangutans in the rainforest of Sumatra. Branches were pulled at several points along their length using a force balance at the end of a stiff rope, and the local diameter of the branch and the distance to its base and tip were measured. Compliance was negatively associated with both local diameter and length to the tip of the branch, and positively, if weakly, associated with length from the trunk. However, branch diameter not only predicted compliance best, but would also be easiest for an orangutan to observe. In contrast, oscillation times of branches were largely unaffected by local diameter, and only significantly increased at diameters below 2 cm. The results of this study validate previous field research, which related locomotory modes to local branch diameter, while suggesting how arboreal animals themselves sense their mechanical environment.     

Mechanical stimuli regulate the allocation of biomass in trees: demonstration with young Prunus avium trees

Background and Aims

Plastic tree-shelters are increasingly used to protect tree seedlings against browsing animals and herbicide drifts. The biomass allocation in young seedlings of deciduous trees is highly disturbed by common plastic tree-shelters, resulting in poor root systems and reduced diameter growth of the trunk. The shelters have been improved by creating chimney-effect ventilation with holes drilled at the bottom, resulting in stimulated trunk diameter growth, but the root deficit has remained unchanged. An experiment was set up to elucidate the mechanisms behind the poor root growth of sheltered Prunus avium trees.

Methods

Tree seedlings were grown either in natural windy conditions or in tree-shelters. Mechanical wind stimuli were suppressed in ten unsheltered trees by staking. Mechanical stimuli (bending) of the stem were applied in ten sheltered trees using an original mechanical device.

Key Results

Sheltered trees suffered from poor root growth, but sheltered bent trees largely recovered, showing that mechano-sensing is an important mechanism governing C allocation and the shoot–root balance. The use of a few artificial mechanical stimuli increased the biomass allocation towards the roots, as did natural wind sway. It was demonstrated that there was an acclimation of plants to the imposed strain.

Conclusions

This study suggests that if mechanical stimuli are used to control plant growth, they should be applied at low frequency in order to be most effective. The impact on the functional equilibrium hypothesis that is used in many tree growth models is discussed. The consequence of the lack of mechanical stimuli should be incorporated in tree growth models when applied to environments protected from the wind (e.g. greenhouses, dense forests).Key words: Prunus avium, growth, mechanical stress, bending, biomass, shoot/root ratio, wind, shelter     

Size- and Age-dependent Variation in the Properties of Sap- and Heartwood in Black Locust (Robinia pseudoacaciaL.)

Dissection and mechanical bending experiments showed that the cross-sectional area and elastic moduli of sap- and heartwood varied within the trunk and branches as a function of the distance from the top of a 43-year-old black locust tree (Robinia pseudoacaciaL.). Wood in branches less than 1 m from the top of the tree consisted entirely of sapwood; the majority of the wood from more basipetal (and older) parts of the tree was heartwood. The Young's elastic moduli of sap- and heartwood increased towards the base of the trunk, and, on average, the modulus of the sapwood was 35%less than that of the heartwood. Younger, more distal tree limbs, therefore, were more flexible than older portions of the same tree. Simple bending experiments showed that the flexural rigidity of young limbs was governed by the location, physical properties, and the relative quantities of the two types of wood. The rigidity of limbs increased toward the base of the tree, and was dominated by sapwood in young limbs and by heartwood in the oldest parts of the tree. These trends predict that the younger, distal limbs of this tree can more easily deflect and bend in the wind, thereby reducing drag and the total bending moment on the tree trunk, while older limbs and the trunk are sufficiently rigid to support static self-loadings. Further study, however, is required to determine whether the trends reported here apply to all trees of this species and to trees of different species.