# Assessment of the Response of a Scots Pine Tree to Effective Wind Loading

^{*}

## Abstract

**:**

## 1. Introduction

_{2}balance can be explained partly by storm damage to forest [6]. The winter storm Lothar [7], which hit Europe on 26 December 1999, led to a 30% decline in the European net biome production [6]. It is not only in forests that attempts have been made to reduce damage. In cities great efforts have also been undertaken to maximize tree stability against severe wind loads to maintain the safety of people and property [8].

_{2}sink associated with European forests [7,46].

## 2. Materials and Methods

#### 2.1. Workflow

#### 2.2. Airflow and Stem Orientation Measurements

^{TM}Motion Tracking

^{TM}, InvenSense, USA). The motion tracking devices (D1 to D7) are combinations of 3-axis gyroscopes and accelerometers. While the gyroscopes measured the wind-induced rotational motion as angular velocity (°/s), the accelerometer measured the wind-induced acceleration ($g$) along the stem. To monitor the wind-induced response behavior of the stem, D1 to D7 were mounted at heights ${h}_{1}$ = 0.1 m, ${h}_{2}$ = 2.0 m, ${h}_{3}$ = 4.0 m, ${h}_{4}$ = 6.0 m, ${h}_{5}$ = 8.0 m, ${h}_{6}$ = 10.0 m, and ${h}_{7}$ = 12.0 m to it. The orientation measurements were carried out at several heights to capture the multimodal vibration behavior of the stem. In an earlier study [34], it was found that the Scots pine trees at the measurement site have at least four measurable modes of vibration.

#### 2.3. Analysis of Wind-Induced Tree Response

^{2}/s

^{2}, ${\widehat{M}}_{\mathrm{i},2}:$1.0 m

^{2}/s

^{2}≤ ${\overline{M}}_{\mathrm{i},\mathrm{j}}$ < 2.0 m

^{2}/s

^{2}, ${\widehat{M}}_{\mathrm{i},3}:$2.0 m

^{2}/s

^{2}≤ ${\overline{M}}_{\mathrm{i},\mathrm{j}}$ < 3.0 m

^{2}/s

^{2}and ${\widehat{M}}_{\mathrm{i},4}:{\overline{M}}_{\mathrm{i},\mathrm{j}}$ ≥ 3.0 m

^{2}/s

^{2}).

#### 2.4. Bi-Orthogonal Decomposition

#### 2.5. Singular Spectrum Analysis

#### 2.6. Dynamic Time Warping

#### 2.7. Turbulence Factor Calculation

## 3. Results and Discussion

#### 3.1. Mean Momentum Flux-Induced Tree Response

#### 3.2. Bi-Orthogonal Decomposition

^{2}/s

^{2}, the first component was the only significant BOD component. In all intervals where ${\overline{M}}_{7,\mathrm{j}}$ ≤ 0.85 m

^{2}/s

^{2}, two significant BOD components were detected. The presence of only one significant BOD component, even at very low wind loads, indicates that the kinetic energy transferred to the tree was mostly converted into elastic energy that drove the sway of the stem in one mode. Small amounts of kinetic energy contained in the airflow at canopy top were enough to induce movement in the stem down to ${h}_{2}$. Damping processes such as multiple mass damping in the crown and friction with neighboring trees that can be assigned to the second BOD component [34], were only of minor importance for overall tree motion damping.

#### 3.3. Singular Spectrum Analysis

^{2}/s

^{2}occurs, $VS{1}_{7,\mathrm{j}}$ is about 25%, with ${\overline{M}}_{7,\mathrm{j}}$ = 0.59 m

^{2}/s

^{2}being the estimated change point from a two-phase regression model [66]. The change point clearly indicates the change in the tree response behavior to wind excitation. The change points are also drawn as dashed blue lines in Figure 6a,c,e to emphasize once again that the tree response behavior to wind loading is very quickly dominated by sway in the non-oscillatory SSA mode after the blue line has been crossed. At higher wind loads, $VS{1}_{7,\mathrm{j}}$, $VS{1}_{5,\mathrm{j}},$ and $VS{1}_{3,\mathrm{j}}$ steeply increase and reach values up to 75%. This increase documents the rapid and strong growth of the importance of $NO{S}_{\mathrm{i},\mathrm{j}}$ for the total tree response in the upper parts of the stem.

^{2}/s

^{2}, more than 20% of the variance contained in the ${O}_{7,\mathrm{j}}$ time series was assigned to stem motion in the fundamental mode (Figure 7a). With increasing wind load, $VS{2}_{7,\mathrm{j}}$ rapidly drops to just over 10%. Again, the value of ${\overline{M}}_{7,\mathrm{j}}$ < 0.70 m

^{2}/s

^{2}, where the distribution of signal energy on the response components changes, corresponds to the change point of the results from a two-phase regression model (Figure 7b). The dashed blue line in Figure 7a reinforces the impression on the wind load-dependent drop of $VS{2}_{7,\mathrm{j}}$ further. Similar changes as in $VS{2}_{7,\mathrm{j}}$ can also be found in $VS{2}_{5,\mathrm{j}}$ (Figure 7c) and $VS{2}_{3,\mathrm{j}}$ (Figure 7e). At the stem base, $VS{2}_{1,\mathrm{j}}$ is almost independent of ${\overline{M}}_{1,\mathrm{j}}$ (Figure 7g). It varies at about 40% over the course of the day with decreasing tendency at higher wind loading.

#### 3.4. Dynamic Time Warping

^{2}/s

^{2}and $NO{S}_{7,68}$ reached values of 12° (Figure 8a). It is evident that the temporal variation of $NO{S}_{7,68}$ does not always match the temporal variation of low-pass filtered ${M}_{7,68}$ although there are only a few meters distance between the sample tree and the airflow measurement at the canopy top.

^{2}/s

^{2})) in dependence of low-pass filtered ${M}_{7,68}$.

^{2}/s

^{2}) at ${h}_{1}$ to 0.32 °/(m

^{2}/s

^{2}) at ${h}_{7}$. In the upper parts of the trunk, the dispersion of the $S{l}_{\mathrm{i},\mathrm{j}}$-values also increases, which expresses the range of possible stem reactions to the effective wind loads. In contrast to previous approaches, this type of analysis provides probabilistic estimates of the wind load-dependent effective tree response instead of providing fixed $Sl$ values.

#### 3.5. Turbulence Factor

^{2}/s

^{2}(Figure 11). The median of ${T}_{1,\mathrm{j}}$ = 6.5 is significantly (95% confidence) higher than at all other heights. The medians of ${T}_{2,\mathrm{j}}$ = 5.4 to ${T}_{7,\mathrm{j}}$ = 4.8 vary in a narrow range with no significant differences as indicated by the overlap of the notches in the boxes. That ${T}_{\mathrm{i},\mathrm{j}}$ is not a constant value can be deduced from the displayed interquartile ranges. At ${h}_{1}$, ${T}_{1,\mathrm{j}}$ shows the greatest variability. There, it varies by 2.1 between ${T}_{1,\mathrm{j}}$ = 5.8 and ${T}_{1,\mathrm{j}}$ = 7.9. From ${h}_{2}$ to ${h}_{7}$, the interquartile ranges span 1.3 to 1.7. Since ${T}_{\mathrm{i},\mathrm{j}}$ is not a constant along the stem and is dependent on the wind load, it would therefore not be appropriate to use ${T}_{\mathrm{i},\mathrm{j}}$ as a fixed value in systems for storm damage analysis, but to implement it probabilistically.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Acronyms | |

BOD | bi-orthogonal decomposition |

CWS | critical wind speed |

D1-D7 | seven MEMS motion-tracking devices installed at ${h}_{1}$ to ${h}_{7}$ |

MEMS | micro electro-mechanical systems |

S1-S5 | five ultrasonic anemometers installed on the meteorological measurement tower |

S3 | ultrasonic anemometer installed close to canopy top |

SSA | singular spectrum analysis |

Symbols | |

$d$ | Euclidean distance |

${f}_{0}$ | fundamental sway frequency of the stem (Hz) |

$g$ | gravity acceleration (m/s^{2}) |

$G$ | gust factor |

${h}_{\mathrm{i}}$ | mounting heights ($\mathrm{i}$ = 1, …, 7) of D1-D7 along the stem of the Scots pine tree (m) |

$L$ | embedding dimension |

${M}_{\mathrm{i},\mathrm{j}}$ | momentum flux in height i and interval j (m^{2}/s^{2}) |

${\overline{M}}_{\mathrm{i},\mathrm{j}}$ | interval mean of ${M}_{\mathrm{i},\mathrm{j}}$ (m^{2}/s^{2}) |

${\widehat{M}}_{\mathrm{i},\mathrm{k}}$ | mean of ${\overline{M}}_{\mathrm{i},\mathrm{j}}$ assigned to four classes (m^{2}/s^{2}) |

$NO{S}_{\mathrm{i},\mathrm{j}}$ | non-oscillatory sway component determined with SSA in height i and interval j (°) |

$NO{S}_{{50}_{\mathrm{i},\mathrm{j}}}$ | 50th percentile (median) of $NO{S}_{\mathrm{i},\mathrm{j}}$ (°) |

$NO{S}_{{98}_{\mathrm{i},\mathrm{j}}}$ | 98th percentile of $NO{S}_{\mathrm{i},\mathrm{j}}$ (°) |

${O}_{\mathrm{i},\mathrm{j}}$ | orientation in height i and interval j (°) |

${\overline{O}}_{\mathrm{i},\mathrm{j}}$ | interval mean of ${O}_{\mathrm{i},\mathrm{j}}$ (°) |

${\widehat{O}}_{\mathrm{i},\mathrm{k}}$ | mean of ${\overline{O}}_{\mathrm{i},\mathrm{j}}$ assigned to four ${\widehat{M}}_{\mathrm{i},\mathrm{k}}$ classes (°) |

$O{S}_{\mathrm{i},\mathrm{j}}$ | oscillatory sway component determined with SSA in height i and interval j (°) |

${R}^{2}$ | coefficient of determination |

$S{l}_{\mathrm{i},\mathrm{j}}$ | slope of regression line calculated between $NO{S}_{\mathrm{i},\mathrm{j}}$ and low-pass filtered ${M}_{\mathrm{i},\mathrm{j}}$ (°/(m^{2}/s^{2})) |

${T}_{\mathrm{i},\mathrm{j}}$ | turbulence factor calculated in height i and interval j |

${u}_{\mathrm{i},\mathrm{j}}$ | wind vector component in east-west direction in height i and interval j (m/s) |

${u}_{\mathrm{i},\mathrm{j}}^{\u2019}$ | fluctuation of ${u}_{\mathrm{i},\mathrm{j}}$ (m/s) |

${v}_{\mathrm{i},\mathrm{j}}$ | wind vector component in north-south direction in height i and interval j (m/s) |

${v}_{\mathrm{i},\mathrm{j}}^{\u2019}$ | fluctuation of ${v}_{\mathrm{i},\mathrm{j}}$ (m/s) |

$V{B}_{\mathrm{j}}$ | variance explained by BOD components in interval j (%) |

$V{S}_{\mathrm{i},\mathrm{j}}$ | variance explained by SSA components in height i and interval j (%) |

$VS{1}_{\mathrm{i},\mathrm{j}}$ | variance explained by $NO{S}_{\mathrm{i},\mathrm{j}}$ (%) |

$VS{2}_{\mathrm{i},\mathrm{j}}$ | variance explained by $O{S}_{\mathrm{i},\mathrm{j}}$ (%) |

${w}_{\mathrm{i},\mathrm{j}}$ | vertical wind vector component in height i and interval j (m/s) |

${w}_{\mathrm{i},\mathrm{j}}^{\u2019}$ | fluctuation of ${w}_{\mathrm{i},\mathrm{j}}$ (m/s) |

${x}_{\mathrm{i},\mathrm{j}}$ | orientation component in east-west direction in height i and interval j (°) |

${x}_{\mathrm{i},\mathrm{j}}^{\u2019}$ | fluctuation ${x}_{\mathrm{i},\mathrm{j}}$ (°) |

${y}_{\mathrm{i},\mathrm{j}}$ | orientation component in north-south direction in height i and interval j (°) |

${y}_{\mathrm{i},\mathrm{j}}^{\u2019}$ | fluctuation of ${y}_{\mathrm{i},\mathrm{j}}$ (°) |

Subscripts | |

$\mathrm{i}$ | index for MEMS motion tracking devices |

$ix$ | index for horizontal moves in dynamic time warping |

$iy$ | index for vertical moves in dynamic time warping |

$\mathrm{j}$ | index for irregular, device-specific analysis intervals |

$\mathrm{k}$ | index for mean momentum flux class |

$p$ | index for momentum flux sample in dynamic time warping |

$P$ | maximum of $p$ |

$q$ | index for orientation sample in dynamic time warping |

$Q$ | maximum of $q$ |

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**Figure 1.**Workflow for quantifying the effective wind-induced response of the sample tree to effective dynamic and quasi-static wind loads. The parts of the figure marked in red indicate the steps and data used to quantify the effective tree response to wind loading.

**Figure 3.**Mean orientation (${\overline{O}}_{1,\mathrm{j}}$ to ${\overline{O}}_{7,\mathrm{j}}$) during irregular intervals ($\mathrm{j}$) at measuring heights (

**a**–

**g**) ${h}_{1}$ to ${h}_{7}$ plotted against mean momentum flux at canopy top (${\overline{M}}_{1,\mathrm{j}}$ to ${\overline{M}}_{7,\mathrm{j}}$) on 30 January 2019. ${R}^{2}$ is the coefficient of determination.

**Figure 4.**(

**a**) Mean momentum flux at canopy height (${\overline{M}}_{7,\mathrm{j}}$) and mean orientation at ${h}_{7}$ = 12 m (${\overline{O}}_{7,\mathrm{j}}$) in 156 irregular intervals on 30 January 2019; (

**b**) mean orientation (${\widehat{O}}_{\mathrm{i},\mathrm{k}}$) at ${h}_{1}$ = 0.1 m to ${h}_{7}$ = 12 m in four classes of mean momentum flux calculated at canopy height (${\widehat{M}}_{\mathrm{i},1}:{\overline{M}}_{\mathrm{i},\mathrm{j}}$ < 1.0 m

^{2}/s

^{2}, ${\widehat{M}}_{\mathrm{i},2}:$ 1.0 m

^{2}/s

^{2}≤${\overline{M}}_{\mathrm{i},\mathrm{j}}$ < 2.0 m

^{2}/s

^{2}, ${\widehat{M}}_{\mathrm{i},3}:$ 2.0 m

^{2}/s

^{2}≤ ${\overline{M}}_{\mathrm{i},\mathrm{j}}$ < 3.0 m

^{2}/s

^{2}and ${\widehat{M}}_{\mathrm{i},4}:{\overline{M}}_{\mathrm{i},\mathrm{j}}$ ≥ 3.0 m

^{2}/s

^{2}).

**Figure 5.**Explained variance associated with (

**a**) the first ($V{B}_{1}$) and (

**b**) the second ($V{B}_{2}$) bi-orthogonal decomposition (BOD) component resulting from wind-induced stem motion; (

**c**) ratio $V{B}_{1}/V{B}_{2}$; all as a function of mean momentum flux at canopy top (${\overline{M}}_{7,\mathrm{j}}$).

**Figure 6.**(

**a**,

**c**,

**e**,

**g**) Explained variance associated with the non-oscillatory ($NO{S}_{\mathrm{i},\mathrm{j}}$) singular spectrum analysis (SSA) component ($VS{1}_{\mathrm{i},\mathrm{j}}$ with i = 1, 3, 5, 7) as a function of mean momentum flux at canopy top (${\overline{M}}_{\mathrm{i},\mathrm{j}}$) on 30 January 2019. The red lines that are shown in the subplots in the right column of the figure were calculated with (

**b**,

**d**,

**f**) two-phase regression models and (

**h**) a second-order polynomial regression model.

**Figure 7.**(

**a**,

**c**,

**e**,

**g**) Explained variance associated with the oscillatory ($O{S}_{\mathrm{i},\mathrm{j}}$) singular spectrum analysis (SSA) component ($VS{2}_{\mathrm{i},\mathrm{j}}$ with i = 1, 3, 5, 7) in dependence of mean momentum flux at canopy top (${\overline{M}}_{\mathrm{i},\mathrm{j}}$) on 30 January 2019. The red lines that are shown in the subplots in the right column of the figure were calculated with (

**b**,

**d**,

**f**) two-phase regression models and (

**h**) a second-order polynomial regression model.

**Figure 8.**(

**a**) Low-pass filtered momentum flux (${M}_{7,68}$) and non-oscillatory SSA component ($NO{S}_{7,68}$) related to measurements at ${h}_{7}$ = 12 m in the device-specific, irregular interval 68; (

**b**) series of dynamically time warped ${M}_{7,68}$ and $NO{S}_{7,68}$; (

**c**) scatter plot of dynamically time warped ${M}_{7,68}$ and $NO{S}_{7,68}$. The color bar indicates the density of the points. ${R}^{2}$ is the coefficient of determination.

**Figure 9.**Boxplots of device- and interval-specific slope ($S{l}_{\mathrm{i},\mathrm{j}}$) values of the regression line calculated from low-pass filtered momentum flux at canopy top (${M}_{\mathrm{i},\mathrm{j}}$) and non-oscillatory SSA component ($NO{S}_{\mathrm{i},\mathrm{j}}$) for all intervals where ${\overline{M}}_{\mathrm{i},\mathrm{j}}$ > 1.0 m

^{2}/s

^{2}. The whiskers include all values that lie within a distance from the first and third quartiles that is less than 1.5 times the interquartile range.

**Figure 11.**Turbulence factor ${T}_{\mathrm{i},\mathrm{j}}$ at heights ${h}_{1}$ to ${h}_{7}$ for all intervals where mean momentum flux at canopy top ${\overline{M}}_{\mathrm{i},\mathrm{j}}$ > 1.0 m

^{2}/s

^{2}. The red vertical line indicates medians of ${T}_{\mathrm{i},\mathrm{j}}$. The whiskers include all values that lie within a distance from the first and third quartiles that is less than 1.5 times the interquartile range.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Schindler, D.; Kolbe, S. Assessment of the Response of a Scots Pine Tree to Effective Wind Loading. *Forests* **2020**, *11*, 145.
https://doi.org/10.3390/f11020145

**AMA Style**

Schindler D, Kolbe S. Assessment of the Response of a Scots Pine Tree to Effective Wind Loading. *Forests*. 2020; 11(2):145.
https://doi.org/10.3390/f11020145

**Chicago/Turabian Style**

Schindler, Dirk, and Sven Kolbe. 2020. "Assessment of the Response of a Scots Pine Tree to Effective Wind Loading" *Forests* 11, no. 2: 145.
https://doi.org/10.3390/f11020145