Simulating the dynamic behavior of Douglas-fir trees under applied loads by the finite element method

The finite element method of structural analysis was used to model the dynamic behavior of three 20-year-old Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) trees subjected to applied loading. Detailed measurements of stem and branch geometry were made for each tree, enabling the first-order branches of each tree to be represented as individual cantilever beams attached to the stem. Three values for branch modulus of elasticity (E) were assumed: 4, 5 and 6 GPa. For two trees with relatively large crown masses (175 and 250 kg), significantly improved estimates of natural frequency were obtained when the branches were modeled as separate cantilever beams rather than as a series of discrete masses attached to the stem. Closest agreement with the results from field sway tests was found when branch E was 4 GPa. Oscillations of individual branches contributed to the damping of tree oscillations--a phenomenon known as structural damping--with the contribution increasing as branch E decreased. When branch E was 4 GPa, the phase difference between the oscillation of the stem and that of some branches was almost 180 degrees. We applied a series of forces separately to the stem and branches of each tree and determined the mechanical transfer function for each loading case. These transfer functions were similar to the theoretical transfer function for a damped harmonic oscillator, but showed a smaller tree response at higher loading frequencies, particularly when branch E was 4 GPa. Branch structural properties, particularly modulus of elasticity, appear to be important in defining overall tree behavior under applied loading.