含電流變材料與磁流變材料的三明治樑之振動與動態穩定性分析

Vibration and Dynamic Stability of Sandwich Beams with Electrorheological and Magnetorheological Materials

葉日豐(Zi-Fong Yeh)

指導教授 ： 施延欣

中原大學/工學院/機械工程研究所/博士(2005年)

https://doi.org/10.6840/cycu200500504

摘要

中文摘要

本文係針對兩端簡支撐之黏彈性樑、內含電流變材料的三明治樑和內含磁流變材料的三明治樑等三種樑結構，就自然振動頻率、損失因子、及受軸向週期性負荷作用下之動態穩定性等議題進行研究。在黏彈性樑中，所使用的黏彈性材料之彈性模數是複數型態且為結構振動頻率的函數，分析中，使用歐拉-伯努利(Eluer-Bernoulli)樑理論導出黏彈性樑的運動方程式，再經過Galerkin方法，將黏彈性樑的運動方程式轉換成具複數係數的Mathieu方程式，此等複數係數同時為結構振動頻率的函數，文中提出一複數式增量調合平衡法以適用複數係數的Mathieu方程式，同時結合半區間數值方法，以分析得黏彈性樑動態穩定性區域，同時就樑的長度及靜力對臨界負荷之比等因素對動態穩定性區域的影響進行研究。另外，在內含電流變材料的三明治樑方面，則係於三明治樑的中間層內填充電流變材料，此電流變材料的剪彈性模數是複數型態且為電場強度的函數；至於內含磁流變材料的三明治樑，則是在三明治樑的中間層中填充磁流變材料，而其剪彈性模數亦為複數型態，但其為磁場強度的函數；文中使用Mead與Markus三明治樑理論，推導出這兩種三明治樑的運動方程式，運用Galerkin方法，分別將內含電流變材料的三明治樑和內含磁流變材料的三明治樑的運動方程式簡化為一含複數係數的Mathieu方程式，而其中的係數則分別為電場強度和磁場強度的函數，文中係使用複數式增量調合平衡法，以分別求得內含電流變材料的三明治樑和內含磁流變材料的三明治樑的動態穩定性區域，同時分別就電場強度、磁場強度、三明治樑中間層厚度、樑的長度及靜力對臨界負荷之比等因素對動態穩定性區域的影響進行探討。在上述運用Galerkin方法時，將可分別求得每一種樑之自然振動頻率與損失因子的計算式，文中同時就每一種樑的此種振動特性進行分析與討論。

關鍵字

三明治樑 ； 磁流變 ； 電流變 ； 黏彈性樑 ； 增量調合平衡法、動態穩定性

並列摘要

Abstract

This dissertation investigates the vibration and dynamic stability of a simply supported viscoelastic beam, a sandwich beam with an electrorheological core and a sandwich beam with a magnetorheological core. All of beams subject to an axial harmonic load. The dynamic stability of a viscoelastic beam with a complex elastic modulus that depends on the vibrating frequency, was considered. The governing equation of motion is theoretically derived using Euler-Bernoulli theory. Applying Galerkin’s method to simplify the governing equation of motion into a complex Mathieu equation with frequency-dependent coefficients. Then, the boundaries of region of dynamic stability are determined by coupling the numerical binary search procedure and the complex incremental harmonic balance method, both of which are developed in this dissertation. The effects of beam length and static load parameter factor are discussed. The electrorheological beams and magnetorheological beams were obtained by sandwiching electrorheological material and magnetorheological material between two elastic face plates, respectively. The complex shear modulus of electrorheological material is a function of the applied electric field. The complex shear modulus of magnetorheological material is a function of the applied magnetic field. The theoretical model is developed from Mead & Markus sandwich beam theory. Galerkin’s method is used to simplify the governing equation of motion to the complex Mathieu equation. The complex incremental harmonic balance method is employed to determine the dynamic stability of the sandwich beam with an electrorheological core and the sandwich beam with a magnetorheological core. The influences of the electric field in the case of the sandwich beam with electrorheological core, the magnetic field in the case of the sandwich beam with the magnetorheological core, the core thickness, the beam length and the static load parameter factor on the dynamic stability are addressed. The formulae for natural frequency and loss factor of these simply supported beams can be obtained in Galerkin’s procedure. These vibration characteristics of these beams are also elucidated.

並列關鍵字

dynamic stability ； electrorheological ； magnetorheological ； incremental harmonic balance method ； viscoelastic beam ； sandwich beam

參考文獻

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