MIT2_092F09_project1 - 2.093 Term Project Frequency...

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2.093 Term Project Frequency Response of Trees Department of Civil and Environmental Engineering Massachusetts Institute of Technology November 2010
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1 Introduction and Motivation Plants have always been a ubiquitous component of human experience. Plants are a permanent fixture in the natural world, imported or accounted for in the urban setting, and harvested to sustain a population. When one thinks of plants, the mind naturally wanders to the swaying of branches or waves rippling through crop fields. These images are apt, as our most common experiences with plants (the live ones, at least) involve a common component: wind. The response of plants to wind loading is an essential question. Permanent uprooting as well as mechanical failure of trunks, branches, and/or stems causes economic damage, poses a threat to human safety, and can cause the loss of significant portions of agricultural crops [1]. Perhaps sur- prisingly, the other major concern for modeling the interaction of wind and plants comes from the computer graphics industry, where a realistic behavior must be attained in order to fool the eye - and must be done in the computationally cheapest way possible [1]. The current study investigates the response of trees to harmonic wind loading. This response is dictated by three main components: (1) Time-varying excitation load caused by wind-induced drag; (2) The dynamic behavior of the tree, and; (3) damping processes. Wind inputs energy into the system. The dynamic behavior of the tree is dictated by the exchange between kinetic energy (expressed by portions of the tree moving with certain velocities) and elastic strain energy (expressed by the deformation of portions of the tree that have the potential to spring back into their original configuration). Damping removes energy of the system and is responsible for the decay of oscillatory amplitudes. It is caused mainly by turbulence created by the interaction between parts of the tree and air as well as (to a lesser extent) the production of heat through the internal material friction in the tree. Obviously, the dynamic response of a tree depends heavily on its actual geometric structure as well as its physical properties. Most trees have a branched structure, expressing repeating architectures on different scales. One common mode of branching is sympodial branching, which involves a single segment growing until it branches into two lateral segments and no axial segment continuing [2]. This architecture shall be used in the model tree described in the next section. The paper shall proceed to describe a model tree and its material properties and explain the finite element model used. It shall then provide a simplified model that was tested for agreement with a mathematical solution. The paper shall then proceed to explain results obtained from finite element modeling. Natural frequencies for the first 25 modes shall be shown. The evolution of mode shapes shall be illustrated for both increasing frequencies as well as decreasing element size. The undamped response
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