A physics-based link model for tree vibrations.

PREMISE OF THE STUDY: A new mathematical model for the vibration of trees is presented for developing a more thorough understanding of the underlying structure of the response. It may be used, for example, to assess the stability of a tree or to interpret experimental data. • METHODS: A model is developed for the motion of the trunk and its N number of branches. The spatial distribution and initial orientation of the branches are left for the user to prescribe. A Newtonian analysis yields (N + 1) nonlinear, coupled differential equations that, when solved, describe the response of the trunk and each branch. After the model is linearized near equilibrium, the natural frequencies and vibration mode shapes are found. Closed-form expressions for the response (i.e., the actual time histories) are then obtained using modal analysis. Numerical solutions are also found; these are used to validate the analytical solutions and to serve as a means for considering large amplitude vibrations. • KEY RESULTS: A new physics-based model is described. For small motion, the tree response may be constructed from the individual mode shapes and frequencies. Also demonstrated are the limitations of the linear theory as well as numerical solutions that can be obtained when trunk/branch amplitudes are large. • CONCLUSIONS: The model presented here incorporates critical physics into a model that describes tree vibrations. It also sheds light on the underlying structure of the vibration response in terms of the modal nature of the solution. Limitations to the linear solutions are demonstrated and discussed.

Reducing stem bending increases the height growth of tall pines.

The hypothesis was tested that upper limits to height growth in trees are the result of the increasing bending moment of trees as they grow in height. The increasing bending moment of tall trees demands increased radial growth at the expense of height growth to maintain mechanical stability. In this study, the bending moment of large lodgepole pine (Pinus contorta Dougl. Ex Loud. var. latifolia Engelm.) was reduced by tethering trees at 10 m height to counter the wind load. Average bending moment of tethered trees was reduced to 38% of control trees. Six years of tethering resulted in a 40% increase in height growth relative to the period before tethering. By contrast, control trees showed decreased height growth in the period after tethering treatment. Average radial growth along the bole, relative to height growth, was reduced in tethered trees. This strongly suggests that mechanical constraints play a crucial role in limiting the height growth of tall trees. Analysis of bending moment and basal area increment at both 10 m and 1.3 m showed that the amount of wood added to the stem was closely related to the bending moment produced at these heights, in both control and tethered trees. The tethering treatment also resulted in an increase in the proportion of latewood at the tethering height, relative to 1.3 m height. For untethered control trees, the ratio of bending stresses at 10 m versus 1.3 m height was close to 1 in both 1998 and 2003, suggesting a uniform stress distribution along the outer surface of the bole.

Comparison of the structural and physical properties of human hair eumelanin following enzymatic or acid/base extraction.

Eumelanin was isolated from a sample of black, Indonesian human hair using three different published procedures: two different acid/base extractions and an enzymatic extraction. The morphology and spectroscopic properties of the isolated pigments differ significantly. The acid/base procedures both yield an amorphous material, while enzymatic extraction yields ellipsoidal melanosomes. Amino acid analysis shows that there is protein associated with the isolated pigments, accounting for 52, 40 and 14% of the total mass for the two acid/base extractions and the enzymatic extraction, respectively. The amino acid compositions do not correlate with those of keratin or tyrosinase. Metal elemental analysis shows that the acid/base extraction removes a majority of many metal ions bound to the pigment. Chemical degradation analysis by KMnO4/H+ and H2O2/OH- indicates significant differences between the pigments isolated by acid/base and enzymatic extraction. After correction for the protein mass in the two pigments, the lower yields of both pyrrole-2,3,5-tricarboxylic acid and pyrrole-2,3-dicarboxylic acid, eumelanin degradation products, indicate acid/base extraction modifies the chemical structure of the melanin, consistent with the result of Soluene solubilization assay. While the optical absorption spectra of the bulk pigments are similar, the spectra of the molecular weight less than 1000 mass fractions differ significantly. The data clearly indicate that pigment obtained from human hair by acid/base extraction contains significant protein, exhibits destruction of the melanosome, and possesses altered molecular structure. The acid/base extracted hair melanin is not representative of the natural material and is a poor model system for studying the physical and biological properties of melanins. The enzymatically extracted hair melanin, on the contrary, retains the morphology of intact melanosomes and is an excellent source of human melanin.