一維多層性材料之機械振動系統研究及其在光學黏度計之應用

Abstract

本文主要探討一維震動機制系統(one-dimensional mechanical systems with piecewise constant properties)的分析與應用。而振動分析的第一步驟通常是利用數值解或分析解找到自然頻率(natural frequency)下之振盪頻譜。對一個有固定材料參數的振盪系統如同樑(beam)、繩(string)和懸臂樑(cantilever beam)是比較簡單處理的。然而，很多實際的應用包含了不連續的非固定的材料參數。一個複合系統(composed system)來說，數值解常常被用判定其模態。但藉由數值計算並無法正確得到各個參數(彈性係數、密度、斷面長度、斷面材料等)對複合系統的影響。為了求得各參數對該複合系統的影響，分析解是有必要的。文假設一維系統與N段固定參數(N-piecewise constant properties)，並導入一「有效傳遞速度」(effective propagation velocity)求得其分析解。為了測試本理論在第二模態(N=2)的可靠度，利用散射光圖樣辦識的光學實驗方法得到，第一模態頻率的實驗數據與分析解所得的解相同。為了延伸此分析解，吾人設計一實驗如下：將一繩(fiber)之下半部分浸入液體中並振動該繩，並視液體為阻尼後量測該繩的振盪機制。無論是最大振幅(maximum vibrational amplitude)或是能量帶寬的變異(power bandwidth variation)，分析解所得之值與實驗數據是吻合的，而由該設計實驗所推出分析解也列在本論文中。

This thesis deals with an analysis and application of the vibrating one-dimensional mechanical systems with piecewise constant properties. It is a well known fact that the fist step in a vibration analysis of any mechanical system is to find spectrum of their natural frequencies. It can be done either numerically or analytically. For simple vibrating systems like beam, string or cantilever with constant mechanical properties the analytical determination of the normal modes is known and is preferable one. However many real applications involve not a constant but a piecewise constant and, factually discontinuous properties. For composed systems the numerical determination of the normal modes is usually performed. We have found here that the normal modes for one–dimensional mechanical problems with piecewise constant properties can be obtained by solving an appropriate transcendental equation. This equation of course contains all information about given mechanical system (elastic moduli, density, cross-sectional area of material, thickness of each layer with constant mechanical properties, etc.). In this thesis transcendental equation is derived for case of one-dimensional systems with N - piecewise constant properties. The physical interpretation of normal modes for this kind of mechanical systems by introducing the “effective” propagation velocity has been put forward. An optical method utilizing a forward light scattering pattern has been carried out to test the validity of the theory for N = 2. Theoretical predictions of the first mode frequency and experimental data agree precisely. Moreover the extension of the analysis and experiments for a case of the partially submerged fiber in fluid with taking into account of a viscous damping has been performed. From obtained results a simple way for viscosity extraction from either the maximum vibrational amplitude or the power bandwidth variation has been suggested and tested experimentally. Besides an explicit formula for the achievable accuracy of the viscosity sensing is derived here as well.