Barber邊界條件下Aldo模型的熱彈性穩定性分析

Abstract

熱彈問題在許多領域有重要的應用。傳統的熱阻定義（有間隙時熱阻為無限大，接觸時熱阻為零）可能導致靜態熱彈問題的解不存在。為了處理這個問題，Barber提出了隨著接觸面間壓力或間隙而連續變化熱阻。應用這個熱阻觀念，Barber分析了一個極簡化的粗糙表面模型–Aldo模型，並得到一些接觸面間熱彈特性的通則。Aldo模型由兩根柱體阻成，然而，Barber的分析中並未考慮這些柱體的力學特性，例如阻尼、波傳速度等。本論文探討這些特性對Aldo模型熱彈穩定性的影響。為此，首先由單一柱體的受力-變形關係探討Aldo模型平衡解的分岐行為。接著推導統御方程式，由在平衡解附近線性化系統的特徵值判別其穩定性，並和數值積分的結果相驗證。最後探討不同系統參數值下的分歧行為和穩定，並和Barber的結果相比對。我們發現柱體的力學特性會影響Aldo模型的穩定性，Barber的結果相當於柱體具有高阻尼時的情形。

Thermoelasticity has important applications in different fields. Traditionally, the thermal resistance is defined as infinite between non-contact surfaces and zero for contact surfaces. With this kind of thermal resistance, a static thermoelastic problem may not have solutions. To overcome this difficulty, Barber proposed a new thermal resistance concept: the thermal resistance depends continuously on the pressure or gap between two interacting surfaces. Using this new thermal resistance, Barber analyzed a simplified surface model, the Aldo model, and obtained some general properties regarding the thermoelastic interactions between two contact surfaces. The Aldo model is composed of two elastic rods. However, Barber did not consider the effects of important mechanical properties of the rods, e.g. damping and wave velocity. This thesis aims to study the effects of the mechanical properties of the rods on the stability of the Aldo model. To this end, we first investigated the bifurcation of the Aldo model on the basis of the load-deflection curve of a single rod. Then, we derived the nonlinear governing equations of the Aldo system. The stability of an equilibrium solution was determined by the eigenvalues of the assoicated linearized system. The results were verified using numerical integration. Finally, we studied the bifurcation behavior and stability of the Aldo model with the variation of parameters, and compared the results with those of Barber’s. Results of this study indicate that mechanical properties of the rods significantly influence the stability of the Aldo model. Barber’s results correspond to the cases with high damping values.