246 Walker and Dahle: Likelihood of Failure of Trees Along Utility Rights-of-Way dynamically, is a simplification of the actual dynamic process of windthrow, since inertial forces only develop at the mass points (James et al. 2014). Even so, this method has been used to develop spring-mass- damper models for trees as a single mass or as a com- plex system of coupled masses that represent the trunk and branches (Milne 1991; Miller 2005; James et al. 2014). The uniformly distributed mass method considers a tree as a beam or column, with its mass uniformly distributed along its length. A fourth-order partial dif- ferential equation has been used to study the oscilla- tions and damping of woody and nonwoody plants (Gardiner et al. 2000; Spatz 2000; Moore and Magu- ire 2008; James et al. 2014; James et al. 2018). The FEM combines features of both the lumped- mass and uniformly distributed mass procedures (Sell- ier et al. 2006; Moore and Maguire 2008; Theckes et al. 2011; Ciftci et al. 2014a; James et al. 2014). FEM divides a structure, in this case a tree, into an appro- priate number of elements: beams, whose sizes may vary, and the ends of which, nodes, become the gen- eralized coordinate points. An advantage of FEM is that complex wind-loading scenarios can be modeled (James et al. 2014). Yet, FEM’s reliability is limited by its requirements of multiple accurate, empirical measurements peculiar to the individual tree and its loading conditions (James et al. 2014). All models used for dynamic analysis of trees make assumptions and may not accurately represent the complex dynamics of trees (Moore and Maguire 2004). Models must account for the damping and dynamic contribution of branches (de Langre 2008; Rodriguez et al. 2008; James et al. 2014; James et al. 2018). Additionally, trees require multi-degrees of freedom, or multimodal analysis, to model dynamic interactions between the branches and trunk, and lit- erature is lacking on how these interactions take place (Sellier et al. 2006; de Langre 2008; Rodriguez et al. 2008; James et al. 2014). Damping dissipates energy and thus reduces the amplitude of oscillation through the frictional forces of aerodynamic drag and collisions as well as inter- nal, viscoelastic forces (Milne 1991; James et al. 2006; James et al. 2014). Damping forces are consid- ered velocity dependent and are most effective around the natural frequency, while having little effect at lower and higher frequencies where the inertia of a tree’s mass is the dominant effect (James et al. 2014). ©2022 International Society of Arboriculture Furthermore, damping is usually not well understood in vibrating structures or in nature (Clough and Pen- zien 1993; James et al. 2014). The effect of damping may be nonlinear, thus it may potentially result in a higher level of complexity than seen in most dynamic models to this point (James et al. 2014). Multimodal response in branched structures occurs when several coupled masses (branches) oscillate in a complex manner, with in-phase and out-of-phase responses such that several modal swap responses are possible (Rodriguez et al. 2008; James et al. 2014). Furthermore, where multimodal response occurs, a damping effect known as “mass damping” may also occur (James et al. 2014). Mass damping was described by Den Hartog (1956) and has been defined for trees (James et al. 2006). Mass damping occurs when the branches sway together or against each other, in-phase and out-of-phase, respectively (de Langre 2008; Theckes et al. 2011; James et al. 2014). Mass damping allows for the dissipation of forces exerted by wind on tree crowns in a nondestructive fashion. Additionally, trees may also dissipate wind energy through a mechanism called “multiple resonance damping” (Spatz et al. 2007), “multiple mass damp- ing” (James et al. 2006), or “branch damping” (Spatz and Theckes 2013; James et al. 2014). Gardiner et al. (2008) published a review of pre- dictive, mechanistic models of wind damage to for- ests. These models attempt to capture the physical processes involved in tree uprooting or failure typi- cally through a 2-step process. The initial stage is to calculate the above-canopy “critical wind speed” (CWS) required to cause windthrow within a forest (Gardiner et al. 2008). The second stage is to use some assessment of the local wind climatology to cal- culate the probability of such a wind speed occurring at the geographic location of the trees (Gardiner et al. 2008). They termed this probability of damage the “risk of damage” (Gardiner et al. 2008). The approaches used to calculate the CWS and the local wind climate may vary between the different predictive models (Gardiner et al. 2008). These predictive mechanistic models attempt to approximate the CWS of trees based on the antici- pated wind-related forces and the counteracting and combined resistive forces of their roots and stem (Gardiner et al. 2008). When predicting the CWS, the resistance to overturning is based upon correlations between the bending moment required to cause
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