Mark Rudnicki

A physics-based link model for tree vibrations.

UNLABELLED 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.