離散Aldo模型在Barber條件下的熱彈穩定性

Abstract

熱彈問題在許多領域上有重要的應用。事實上，有許多工業問題都起因於接觸表面上的熱彈不穩定性。熱彈問題與接觸表面的熱阻有很大的關係。傳統上，熱阻被定義為接觸壓力的不連續函數：接觸壓力不為零時熱阻為零，接觸壓力為零時熱阻為無窮大。在傳統的熱阻定義下，熱彈問題可能沒有平衡解。為了解決這個問題，Barber提出了一個新的熱阻模型：熱阻會隨著接觸表面的壓力或間隙而連續地改變。這個熱阻條件被稱為Barber熱阻條件或Barber條件。為了說明Barber熱阻的特性，Barber分析了一個簡化的粗糙面模型─Aldo模型。Aldo模型包含了一個由兩個彈性柱體支撐的剛體。經由分析這個粗糙面模型，Barber獲得了一些接觸面間熱彈特性的通則。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。此外，為了專注於柱體的力學特性的影響而不涉及複雜的數學運算，將連續Aldo模型進一步的簡化為離散模型。首先探討單一柱體的受力-變形關係，接著推導雙柱體模型的統御方程式。雙柱體的分歧分析奠基於單一柱體的受力-變形關係。平衡解的穩定性決定於擾動線性系統的特徵值，並由數值積分的法方來驗證。最後，我們比較Aldo模型在Barber條件下的穩定特性，我們的結果指出，柱體的力學特性會對Aldo模型的穩定性有顯著的影響。

Thermoelasticity has important applications in various fields. In fact, many industrial problems are resulted from thermoelastic instability at the contact interface. Thermoelasticity depends heavily on the thermal resistance at the interface. Traditionally, the thermal resistance is defined as a discontinuous function of the contact pressure: the value is zero for nonzero contact pressure and jumps to infinity as the contact pressure drops to zero. With the traditional thermal resistance, a static thermoelastic problem may not have a solution. To overcome this difficulty, Barber proposed a new model of thermal resistance: the thermal resistance changes continuously with the pressure or gap between two interacting surfaces. This new thermal resistance is referred to as Barber’s thermal condition or Barber’s condition. To show the characteristics of the new thermal resistance model, Barber analyzed a simplified surface model, the Aldo model, which consists of a rigid plate supported by two elastic rods. With this simple surface model, Barber obtained some general properties regarding the thermoelastic interactions between two contact surfaces 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. Moreover, to focus on the effects of the mechanical properties of the rods without involving complicated mathematical calculations, the continuous Aldo model is further simplified to a discrete model. We first investigate the load-deflection relationship of a single rod model. Then, we derive the governing equations of the two-rod Aldo system. Bifurcation analysis of the two-rod model is conducted on the basis of the load-deflection relationship of a single rod model. 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 compared the stability characteristics of the Aldo model with those of Barber’s. Our results indicate that mechanical properties of the rods significantly influence the stability of the Aldo model.