4 James et al.: Tree Biomechanics Literature Review: Dynamics Figure 1. Dynamic models using a spring-mass-damper system representing: (a) a tree as a single mass (Miller 2005), and (b) as multiple masses with a trunk and branches (James et al. 2006). represent the trunk and branches (James et al. 2006; Theckes et al. 2011; Murphy and Rudnicki 2012). A simple spring-mass-damper system (Figure 1a) is described by a second-order differential equation: [1] mẍ + cẋ + kx = f(t) where c, m, and k are damping coefficient, mass, and stiffness, respectively; x, ẋ, and ẍ are the displace- ment, velocity, and acceleration, respectively; and f(t) is the wind-induced time varying (dynamic) force. Equation 1 describes the motion of a single degree of freedom system (SDOF), and if used for analysis of a tree (Figure 1a), approximates the tree to a single oscillating mass (m) with a stiffness (k) and a damping (c) (Miller 2005). A more complex mass model representing branches as oscillating masses attached to a main trunk (Figure 1b) extends this concept to consider branches as oscillating masses attached to the main trunk (James et al 2006). The oscillating lumped-mass model has been used for trees (Milne 1991; Baker and Bell 1992; Peltola and Kellomaki 1993; Guitard and Castera 1995; Pel- tola 1996a; Baker 1997; Kerzenmacher and Gardiner 1998; Saunderson et al. 1999; Flesch and Wilson 1999b; England et al. 2000; Miller 2005; James et al. 2006; Jonsson et al. 2007; James 2010; Thekes et al. 2011; Murphy and Rudnicki 2012). Analyses of the mass-spring-damper model of a tree may include a spectral analysis approach using Fourier transformations and transfer functions based on a SDOF model that is oſten not explicitly stated (Peltola 1996b; Rudnicki et al. 2008; Schindler 2008). A simple model of a tree (Figure 1a) has a dynamic response whose response amplitude is fre- quency dependent. Depending on the frequency of ©2014 International Society of Arboriculture sway, the dynamic response is domi- nated by stiffness, damping, or inertia (Balachandran and Magrab 2004). At low frequencies, the response is domi- nated by stiffness. As the frequency of the applied force increases, the dynamic response increases until it equals the natural frequency of the system. At this point resonance occurs and there is an amplification of the sway, which depends on the damping, and is known as the damping-dominated region. As frequencies increase further, the rapid force impulses do not cause the mass to move because of its inertia; this is known as the inertial region. In the damping-dominated region, at fre- quencies close to the natural frequency, the amplification of sway response has been described for trees as a DAF (Sellier and Four- caud 2009; Ciſtci et al. 2013) (James 2010). The DAF applied to trees by Sellier and Four- caud (2009) was defined as the ratio of the maxi- mum displacement under turbulent wind to the displacement caused by the static, instantaneous wind force. DAF was calculated at breast height and at the base of the live crown of a 35-year-old maritime pine (Pinus pinaster Ait.), with values between 0.98 and 1.19. These values seem low due to the DAF being based on displacements, which at breast height would always be small. Ciſtci (2012) used FEM to investigate the effect of branches on DAF of a large sugar maple (Acer saccharum L.), also finding that changes to tree geometry induced greater changes in DAF. How- ever, recent studies have indicated that different growth forms in woody plants show distinct onto- genetic trends in mechanical properties (Dahle and Grabosky 2010b; Speck and Burgert 2011), so material properties cannot be ignored in dynamic analyses (Moore and Maguire 2008; Ciſtci 2012). The DRF (James 2010) was defined as the ratio of maximum base moment to mean base moment. It varied among species; more flexible trees (Cupressus sempervirens L., Wash-
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