# A Numerical Approach to Estimate Natural Frequency of Trees with Variable Properties

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{i}is the maximum horizontal displacement, m

_{i}is the mass, η

_{i}is the bending moment in the stem and θ

_{i}is the angular displacement of the stem for the ith vertical section.

_{n}are the solutions of the beam equation and depend on the vibration mode. However, the cross-section of a tree stem is not constant along its length, so it is more realistic to represent it as a tapered cantilever beam. The following equation developed by Blevins [30] can be used to find the natural frequency (f

_{n}) of slender and tapered cantilever beam elements:

_{0}is the basal area moment of inertia, A

_{0}is the area of the cross-section at the base of the cantilever beam (stem), L is the length of the tree stem, E is the modulus of elasticity and ρ is the material density. The unitless parameter, λ, changes due to variations in the physical properties of the tree (such as shape, mass distribution, vertical or horizontal orientation, type of basal support, mode of bending, taper of the beam and so on). Values for trees can be estimated through targeted data collection for a specific species. For example, Milne [28] estimated the average value of λ to be 2.08 (standard deviation = 0.06) for Sitka spruce (Picea sitchensis (Bong.) Carr.) trees.

_{p}is an indicator variable which has a value of 1.0 if the genus is Pinus and 0.0 otherwise. A more recent and comprehensive analysis of tree natural frequency data by Jackson et al. [32] shows that the fundamental natural frequency of conifer trees and broadleaved species with monopodial architecture can be predicted using the equation for the vibration of a tapered cantilever beam. However, for some broadleaved species with more complex architecture (i.e., a more decurrent form), this approximation to a tapered cantilever beam no longer holds.

## 2. Methods

#### 2.1. Theoretical Basis for Vibration Analysis

_{0}-D

_{t})/D

_{0}] and D

_{0}and D

_{t}are the diameters of the tree at the base and top of the tree stem, respectively.

_{0}, E

_{0}, ρ

_{0}and A

_{0}are the area moment of inertia, Young’s modulus, density and cross-section area at the base of the stem, respectively.

#### 2.2. Calculation of Natural Frequency Using Numerical Analysis

#### 2.2.1. Effect of Base Fixity

_{0}I

_{0}. The value of K’ will approach infinity for an equivalent clamped (fixed) case (θ$\to $0) and zero for a pinned condition (where M (base bending moment)$\to $ 0). Validation of the models was conducted by comparison with the previous analyses of Blevins [30].

#### 2.2.2. Effect of Longitudinal Stiffness Variation

_{b}) to the top of the stem (E

_{t}), with E

_{t}/E

_{b}= 0, 0.25, 0.5, 0.75 and 1, (ii) elastic modulus increasing linearly from the bottom (E

_{b}) to one-third of the stem height (E

_{m}), followed by a linear decrease to the top of the stem (Et), with ϕ = 1/3, E

_{t}/E

_{b}= 0.25 and E

_{b}/E

_{m}= 0.25, 0.35, 0.5, 0.75 and 1 and (iii) E

_{t}/E

_{b}= 0.25 and E

_{b}/E

_{m}= 0.5 and varying the position of the gradient change (point m) with L/3, L/2, 2L/3 and 5L/6. In each case, the density (ρ) has been held constant, thus the value of E/ρ changes in a similar manner to that described in Equation (18).

#### 2.2.3. Effect of Radial Stiffness Variation

_{o}= A

_{c}= 50% of the total area of the base cross-section). The stiffness of each ring was then varied according to the ratios: E

_{c}/E

_{o}of 10

^{−3}, 0.1, 0.25, 0.5, 1 and 1.5. The same density was assumed for both the outer-wood and core-wood. To calculate the values of first natural frequency parameter (λ), outer-wood properties at the base were used.

#### 2.2.4. Effect of Longitudinal Density Variation

_{b}) to the top of the stem (ρ

_{t}), with ρ

_{t}/ρ

_{b}= 0, 0.25, 0.5, 0.75,1 and 1.25, (ii) density decreasing linearly from the bottom (ρ

_{b}) to half of the stem height (ρ

_{m}), followed by a linear increase to the top of the stem (ρ

_{t}), with ϕ = 1/2, ρ

_{t}/ρ

_{b}= 0. 5 and ρ

_{m}/ρ

_{b}= 0.25, 0.5, 0.75, 1 and 1.25 and (iii) ρ

_{t}/ρ

_{b}= 0.5 and ρ

_{m}/ρ

_{b}= 0.25 and varying the position of the gradient change (point m) with L/3, L/2, 2L/3 and 5L/6. In each case, the stiffness (E) has been held constant, as shown in Equation (19).

#### 2.2.5. Effect of Radial Density Variation

_{o}= A

_{c}= 50% of the total area of the base cross-section). The density for each ring was then varied according to the ratios: ρ

_{c}/ρ

_{o}of 0, 0.25, 0.5, 1 and 1.25. The same elastic modulus was assumed for both the core-wood and outer-wood. To calculate the values of first natural frequency parameter (λ), outer-wood properties at the base were used.

#### 2.2.6. Effect of Crown Mass

_{c}) at different locations along the stem (see Figure 3). (i) In the first case, the additional crown mass was applied to the top of the stem, (ii) in the second, the same crown-to-stem mass ratios were applied to the upper two-thirds (by volume) of the stem and (iii) in the third case, they were applied to the upper half (by volume) of the stem. In all cases, we define different crown-to-stem mass ratios (W

_{c}/W

_{s}) at these particular locations.

#### 2.3. Estimating Natural Frequency for Coniferous and Broadleaved Trees

_{n}) were collated from the literature on field studies of 703 broadleaf and conifer trees growing at different sites and under different conditions, spanning height and DBH ranges of 5.9–35.4 m and 6.9–82.8 cm, respectively [11,12,15,28,31,48,49,50,51,52,53]. These field data were used to validate the tapered cantilever beam model and provide further fundamental understanding of the sway behaviour of conifer and broadleaf trees. Results from these studies were used to estimate the natural frequencies and damping ratios of trees with the presence or absence of foliage, and for full and de-branched crowns, enabling the effects of the crown architecture to also be assessed. The field sway data covers thirteen different species: (a) conifers—Sitka spruce (Picea sitchensis), Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco), Norway spruce (Picea abies), white spruce (Picea glauca (Moench) Voss.), Lodgepole pine (Pinus contorta Dougl.), Scots pine (Pinus sylvestris L.), Corsican pine (Pinus nigra Arnold) and red pine (Pinus resinosa Ait.), and (b) broadleaves—red maple (Acer rubrum L.), sugar maple (Acer saccharum Marsh.), shagbark hickory (Carya ovata (Mill) K. Koch), red oak (Quercus rubra L.) and lime (Tilia europaea L.). Data on the stiffness (E), density (ρ), taper (α) and crown/stem mass ratio (W

_{c}/W

_{s}) were found for each species in the literature and are summarised in Table 1 (see the Electronic Supplementary Material, Tables S1 and S2). While these properties can all vary substantially within a species due to genetic and environmental factors, we only had species-level average values as data for the individual trees were not available. Instead, we assumed that these properties were normally distributed with a standard deviation that was equal to 5% of the mean value. We then ran 100 simulations of the model to test the sensitivity to variations in these properties. The data were split into four groups for analysis: (1) conifers with full crowns, (2) broadleaves with full crowns, (3) debranched conifers and (4) leafless broadleaves. The natural frequency parameter (λ) was assumed to be 1.5, except for the case of conifers with their branches removed, when it was assumed to have a value of 2.5. Ordinary least squared regression was used to investigate the relationship between tree fundamental frequency and DBH/L

^{2}. This relationship was compared among the four analysis groups described above. Due to the limited overlap in values of DBH/L

^{2}for several of the broadleaved species, it was not possible to test whether this relationship differed among species. The level of agreement between the predictions from the tapered cantilever beam model and the field observations of fundamental natural frequency for these groups was assessed using Lin’s concordance correlation coefficient [54,55].

## 3. Results

#### 3.1. Effect of Base Fixity

#### 3.2. Effect of Longitudinal Stiffness Variation

#### 3.3. Effect of Radial Stiffness Variation

^{−3}is a lower bound for the behaviour. Beyond this point, the behaviour of the stem approaches that of a hollow section. For the stiffness ratio of E

_{c}/E

_{o}= 1.5, an increase in the natural frequency of up to 3% was found.

#### 3.4. Effect of Longitudinal Density Variation

#### 3.5. Effect of Radial Density Variation

#### 3.6. Effect of Crown Mass

_{c}/W

_{s}= 0), taper has a noticeable effect on the natural frequency parameter. However, as the ratio of crown mass to stem mass increases, the natural frequency becomes invariant to changes in taper, particularly when the crown mass is added at 2L/3 and L/2 (Figure 9b,c). This suggests that the crown weight is dominating the taper effect to some extent. However, adding a crown mass to the very top of a tapered column reduces the natural frequency while increased taper ratio in the absence of a crown mass increases natural frequency (Figure 9a).

#### 3.7. Estimating Natural Frequency for Coniferous and Broadleaved Trees

^{2}and comparatively low natural frequency). The fundamental natural frequencies predicted by the model proposed in this paper were in general agreement with field observations for conifers with full crown (concordance correlation coefficient = 0.79) (Figure 11). The model overpredicted the natural frequencies of coniferous trees with their branches removed and had much poorer performance for broadleaved trees, particularly those with full crown (concordance correlation coefficient < 0.3). This latter result was mostly driven by the dataset for lime trees where there was a weak relationship between tree natural frequency and DBH/L

^{2}. With the exception of these data, all the simulations produced results that were in general agreement with field observations.

## 4. Discussion

^{2}[11,32]. Interestingly, the presence of a crown seems to reduce the natural frequency and improves the fit of the model compared with the situation where the model is applied to a stem with the crown removed. Broadleaves tend to have a more complex geometry, with multiple close natural frequencies and are therefore more dependent on the branch/crown architecture and presence of leaves [17,32,53]. For the broadleaves with crowns, the model fit is generally quite poor, and as would be anticipated, this improves with the removal of the leaf mass. As occurs with excurrent trees, the natural frequency of decurrent trees also reduces with the removal of the leaf mass from the crown, although this change is relatively small compared to that for excurrent trees. These findings are supported by field studies [8,11,15,28], where pruning and seasonal changes have more profound effects on excurrent trees. The change in natural frequency with pruning have mostly been associated with mass change, but numerical studies by Kerzenmacher and Gardiner [19] showed that mass change alone could not completely account for the change in natural frequency, as trees went from a fully-branched state to a de-branched state. They hypothesized that branch swaying also had an effect on overall dynamic behaviour of trees.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**a**) Schematic of longitudinal stiffness variation scenarios (i) to (iii), (

**b**) schematic of radial stiffness variation scenarios and (

**c**) schematic of longitudinal density variation scenarios (i) to (iii).

**Figure 4.**(

**a**) Comparison of theoretical and numerical first natural frequency parameter ratio with relative rotational base stiffness, and (

**b**) taper ratio versus first natural frequency parameter ratio with varying rotational base stiffness.

**Figure 5.**(

**a**) Taper ratio versus first natural frequency parameter ratio with longitudinal stiffness variation linearly from the bottom to the top end of the tree (first scenario), (

**b**) taper ratio versus first natural frequency parameter ratio with longitudinal stiffness variation linearly from the bottom to one-third of the tree height, followed by a linear variation till the top end of the tree (second scenario) and (

**c**) taper ratio versus first natural frequency parameter ratio with longitudinal stiffness variation linearly from the bottom to L’, followed by a linear variation till the top end of the tree (third scenario).

**Figure 6.**Taper ratio versus first natural frequency parameter ratio with radial stiffness variation.

**Figure 7.**(

**a**) taper ratio versus first natural frequency parameter ratio with longitudinal density variation linearly from the bottom to the top end of the tree (first scenario), (

**b**) taper ratio versus first natural frequency parameter ratio with longitudinal density variation linearly from the bottom to the half of the tree height, followed by a linear variation till the top end of the tree (second scenario) and (

**c**) taper ratio versus first natural frequency parameter ratio with longitudinal density variation linearly from the bottom to L’, followed by a linear variation till the top end of the tree (third scenario).

**Figure 9.**(

**a**) Taper ratio versus first natural frequency parameter ratio with crown mass ratio (first scenario; the additional crown mass was applied to the top of the stem), (

**b**) taper ratio versus first natural frequency parameter ratio with crown mass ratio (second scenario; the additional crown mass was applied to the upper two-thirds (by volume) of the stem) and (

**c**) taper ratio versus first natural frequency parameter ratio with crown mass ratio (third scenario; the additional crown mass was applied to the upper half (by volume) of the stem).

**Figure 10.**Relationship between natural frequency and the ratio of diameter at breast height (DBH) to total tree height squared for selected trees from the published literature.

**Figure 11.**Comparison between predicted natural frequency and observed natural frequency for selected trees from the published literature. The grey points show the range of simulated values and the coloured symbols represent the mean result for each tree. The solid black line indicates a 1:1 relationship.

**Table 1.**Average values of wood properties (modulus of elasticity of green wood, E

_{g}, and density ρ), tree taper (α) and the ratio of crown mass to stem mass (Wc/Ws) for each species used in the analysis to estimate fundamental natural sway frequency. The number of trees (n) for which sway data were obtained from the literature is also given.

Code | Tree Species | n | E_{g} (GPa) | ρ (kg/m^{3}) | E_{g}/ρ | Taper (α) | Wc/Ws |
---|---|---|---|---|---|---|---|

CP | Corsican pine (Pinus nigra) | 57 | 8.70 | 657 | 0.013 | 0.85 | 0.34 |

DF | Douglas-fir (Pseudotsuga menziesii) | 17 | 9.83 | 583 | 0.017 | 0.77 | 0.16 |

LP | Lodgepole pine (Pinus contorta) | 40 | 6.90 | 487 | 0.014 | 0.83 | 0.33 |

NS | Norway spruce (Picea abies) | 32 | 6.23 | 598 | 0.010 | 0.90 | 0.32 |

RP | Red pine (Pinus resinosa) | 300 | 8.80 | 410 | 0.021 | 0.81 | 0.22 |

SP | Scots pine (Pinus sylvestris) | 20 | 7.33 | 700 | 0.010 | 0.90 | 0.29 |

SS | Sitka spruce (Picea sitchensis) | 175 | 7.53 | 447 | 0.017 | 0.84 | 0.50 |

WS | White spruce (Picea glauca) | 6 | 7.40 | 466 | 0.016 | 0.91 | 0.34 |

Averages | 7.77 | 551 | 0.015 | 0.85 | 0.31 | ||

LT | Lime (Tilia europaea) | 18 | 11.7 | 530 | 0.022 | 0.84 | 0.18 |

AR | Red maple (Acer rubrum) | 7 | 9.6 | 524 | 0.018 | 0.67 | 0.22 |

QR | Red oak (Quercus rubra) | 11 | 9.9 | 665 | 0.015 | 0.75 | 0.32 |

CO | Shagbark hickory (Carya ovata) | 5 | 10.8 | 649 | 0.017 | 0.76 | 0.39 |

SM | Sugar maple (Acer saccharum) | 15 | 10.7 | 560 | 0.019 | 0.79 | 0.18 |

Averages | 10.54 | 585.6 | 0.018 | 0.76 | 0.26 |

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**MDPI and ACS Style**

Dargahi, M.; Newson, T.; R. Moore, J. A Numerical Approach to Estimate Natural Frequency of Trees with Variable Properties. *Forests* **2020**, *11*, 915.
https://doi.org/10.3390/f11090915

**AMA Style**

Dargahi M, Newson T, R. Moore J. A Numerical Approach to Estimate Natural Frequency of Trees with Variable Properties. *Forests*. 2020; 11(9):915.
https://doi.org/10.3390/f11090915

**Chicago/Turabian Style**

Dargahi, Mojtaba, Timothy Newson, and John R. Moore. 2020. "A Numerical Approach to Estimate Natural Frequency of Trees with Variable Properties" *Forests* 11, no. 9: 915.
https://doi.org/10.3390/f11090915