Department of Mechanical Engineering, University of Connecticut, Storrs, Connecticut 06269-3139, USA.
Am J Bot. 2012 Dec;99(12):1918-29. doi: 10.3732/ajb.1200141. Epub 2012 Nov 28.
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. •
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. •
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. •
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.
为了更深入地了解响应的基本结构，提出了一种新的树木振动数学模型。它可用于评估树木的稳定性或解释实验数据。
为树干及其 N 个分支的运动建立了一个模型。分支的空间分布和初始方向留给用户指定。牛顿分析产生了（N + 1）个非线性、耦合微分方程，求解后描述了树干和每个分支的响应。模型在平衡附近线性化后，找到固有频率和振动模态形状。然后使用模态分析获得响应（即实际时间历史）的封闭形式表达式。还找到了数值解；这些用于验证解析解，并作为考虑大振幅振动的一种手段。
描述了一种新的基于物理的模型。对于小运动，树的响应可以由各个模态形状和频率构成。还演示了线性理论的局限性以及当树干/分支幅度较大时可以获得的数值解。
本文提出的模型将关键物理纳入到描述树木振动的模型中。它还揭示了振动响应的基本结构，即解的模态性质。演示并讨论了线性解的局限性。
