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Analytical solutions for vibrating fractal composite rods and beams

机译：分形复合杆和梁振动的解析解

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摘要

Fractals have the potential to describe complex microstructures but presently no solution methodologies exist for the prediction of deformation on transiently deforming fractal structures. This is achieved in this paper with the development of analytical solutions on vibrating composite rods and beams. The fractals considered are necessarily deterministic and relatively simple in form to demonstrate the solution methodology. The solutions are limited to beams and rods constructed from an idealised-composite material consisting of relatively large rigid particles embedded in an infinitely thin pliable matrix. Although, as a result, the fractal composite system is not representative of a realistic physical system the methodologies presented do serve to highlight the practical difficulties in using fractals in structural dynamics. Static loading is restricted to spatially invariant axial forces and bending moments as solutions on a unified state of continuum stress are sought which then serve as initial conditions for the vibratory problem. It is demonstrated that measurable displacement is possible on a fractal structure and that finite measures of total, kinetic and strain energy are simultaneously achievable. The approach involves the use of modal analysis to determine modes at natural frequencies that satisfy boundary conditions. These are combined to provide a free vibration solution on a fractal that satisfies the initial conditions in the form of a fractal displacement field.
机译：分形有可能描述复杂的微观结构，但目前尚无用于预测瞬态变形分形结构变形的解决方法。这是通过开发振动复合杆和梁的解析解决方案而实现的。考虑的分形必须是确定性的，并且在形式上相对简单以证明求解方法。该解决方案仅限于由理想复合材料构成的梁和杆，该理想复合材料由嵌入无限薄的柔韧性基质中的相对较大的刚性颗粒组成。尽管结果表明，分形复合系统不能代表现实的物理系统，但所提出的方法确实可以突显出在结构动力学中使用分形的实际困难。静态载荷被限制在空间不变的轴向力和弯矩上，这是因为寻求连续应力统一状态的解决方案，然后将其作为振动问题的初始条件。结果表明，在分形结构上可以测量位移，并且可以同时获得总，动能和应变能的有限量度。该方法涉及使用模态分析来确定满足边界条件的固有频率下的模。将它们组合在一起，以分形位移场的形式提供满足初始条件的分形上的自由振动解。

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