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研究生：	郭仕堯 Kuo, Shih-Yao
論文名稱：	複合夾心板之氣熱彈分析 Aerothermoelastic Analysis of Composite Sandwich Plates
指導教授：	蕭樂群 Shiau, Le-Chung
學位類別：	博士 Doctor
系所名稱：	工學院 - 航空太空工程學系 Department of Aeronautics & Astronautics
論文出版年：	2002
畢業學年度：	90
語文別：	英文
論文頁數：	125
中文關鍵詞：	線性顫振、非線性顫振、自由振動、熱應力後挫屈、熱應力挫屈、氣熱彈、複合夾心板
外文關鍵詞：	nonlinear flutter, linear flutter, free vibration, thermal buckling, composite sandwich plate, thermal postbuckling, aerothermoelastic
相關次數：	點閱：118 下載：4
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72個自由度高精確度的高階三角形板元素用以探討複合夾心板之氣熱彈特性，Von Karman幾何大變形假設和半穩定空氣動力學理論作為分析的基礎，複合夾心板是由複合層板面板與蜂巢結構心材所組成，夾心板之側向位移由彎矩撓曲及剪力變形疊加而得。研究範疇包括複合夾心板之熱應力挫屈、熱應力後挫屈；熱應力挫屈後複合夾心板之自由振動、線性顫振與非線性顫振等課題，深入研究複合夾心板的幾何形狀、纖維角度、邊界條件、材料係數、疊層順序與溫度分佈等變數。當複合夾心板承受溫度影響時可能發生熱應力挫屈現象，研究發現挫屈模態與複合夾心板之纖維角度和展弦比有關。溫度高於臨界挫屈溫度時，複合夾心板進入熱應力後挫屈階段，幾何大變形引入非線性勁度矩陣，此時非線性平衡方程式可由Newton-Raphson解法求解。結果顯示複合夾心板挫屈模態可能發生轉移，轉移後之模態取決於纖維角度與展弦比。溫度上升時與下降時，熱應力後挫屈之變形路徑可能不同。為了分析溫度高於臨界挫屈溫度時，複合夾心板之自由振動行為，複合夾心板之總位移分別以靜態與動態位移量之和表示，並假設其動態位移量遠小於靜態位移。先以Newton-Raphson解法求取靜態挫屈變形，再於頻率域求得自然頻率與振動模態。結果顯示振動模態的排列順序為溫度的函數，最低模態可能由高模態主宰。挫屈模態發生轉移時，複合夾心板之勁度突然改變，振動模態也會劇烈變化。溫度梯度增強複合夾心板之勁度，沒有振動模態頻率會陡降為零。挫屈模態發生轉移時，振動模態也只會緩慢變化。線性顫振可由頻率域求解，一般而言，顫振耦合發生於(1,1)與(2,1)模態之間。對於較狹長和較厚之格板，顫振耦合發生於(n,1)與(n+1,1)模態之間。非線性顫振問題於時間域求得結果，Newmark數值積分法配合靜態壓縮與模態縮減用以節省計算時間。研究發現溫度上升不一定降低顫振邊界，反而可能因為挫屈模態發生轉移，複合夾心板形成浪板形狀，提供額外勁度，造成顫振邊界上升。溫度梯度使得複合夾心板在溫度低於臨界挫屈溫度時便開始變形，並且明顯穩定複合夾心板之動態行為。五種不同行為─平坦、挫屈、有限循環運動、週期性運動、散漫運動均可有效預測，並有深入剖析。空氣動力與空氣摩擦生熱交互作用所引起的非線性行為有廣泛探討與充分瞭解，本文提供首篇先進複合夾心板氣熱彈行為的完整研究。

By considering the total transverse displacement of a sandwich plate be the sum of the displacement due to bending of the plate and that due to shear deformation of the core, a 72 degree-of-freedom high precision high order triangular plate element is developed for aerothermoelastic analysis of composite sandwich plates. The Von Karman large deflection assumptions and quasi-steady supersonic aerodynamic theory are used for the aerothermoelastic analysis of composite sandwich panel consisting of two laminated face sheets and an orthotropic honeycomb core. The scope of this thesis includes thermal buckling and thermal postbuckling of composite sandwich plates; free vibration, linear flutter, and nonlinear flutter of thermally buckled composite sandwich plates. The effects of plate geometry, fiber orientation, boundary conditions, material properties, stacking sequence, and temperature distribution are discussed in details. The thermal buckling analysis has been represented as the standard eigenvalue problem. Numerical results show that the buckling mode that sandwich plate will buckle into is dependent on the fiber orientation in the plate and aspect ratio of the plate. In the thermal postbuckling analysis of composite sandwich plates. The nonlinear system equation is solved by Newton-Raphson method. Numerical results show that the buckle pattern that sandwich plate with angle-ply laminates faces will change is dependent on the fiber orientation in the plate and aspect ratio of the plate. The path of cooling down process is different from that of heating up process. The buckle pattern change will occur at lower temperature during cooling down process while it will occur at higher temperature in the heating up process. In the free vibration analysis of thermally buckled composite sandwich plates, the total response of the plate is assumed to be the sum of displacement due to static deformation and that due to dynamic deformation. The nonlinear governing equations can be written into two sets of equations. The static deflection of the thermally buckled plate is first solved by Newton-Raphson method. Then, another set of linear equation may determine the dynamic behavior of the thermally buckled plate in frequency domain. By examining the buckling and free vibration modes of the plate, a clear picture of buckle pattern change and lowest vibration mode shift is presented. Numerical results show that the order of the vibration mode is a function of the temperature rise. The higher vibration mode with multiple waves may become the fundamental vibration mode. Due to this buckle pattern change, the stiffness of the plate is also suddenly changed that in turn makes the natural frequency of the plate jump. The higher vibration mode may become the fundamental vibration mode after buckle pattern change. The temperature gradient induces thermal moments and makes the plate stiffer. None of the natural frequencies will drop to zero as the temperature is increased and lowest vibration mode shift induced by buckle pattern change will occur slowly. In the flutter analysis of thermally buckled composite sandwich plates, Newton-Raphson method and Newmark method are used to solve the nonlinear problem in frequency and time domains, respectively. Static condensation and modal reduction are adopted for saving computing time. In general, the coalescence pair usually is between natural modes (1,1) and (2,1). For longer or thicker plates, the flutter mode shape is the combination of (n,1) and (n+1,1) flutter modes. Numerical results show that the elevated temperature may not only decrease flutter boundary in general, but also increase flutter boundary due to the buckling pattern suddenly changes in some special cases. The corrugated deflection pattern will offer additional geometrical nonlinear stiffness. The critical aerodynamic pressure of the buckled plate with high temperature rise may be higher than the one without thermal effect. The temperature gradient makes the plates deflect under the critical buckling temperature and stabilized the dynamic behavior of the thermally buckled composite sandwich plates significantly. All five types of panel behavior such as flat, buckled, limit-cycle, periodic, and chaotic motions can be identified successively. The nonlinear behavior of composite sandwich plates subjected to aerodynamic loading and aerodynamic heating is extensively explored and reasonably understood. This dissertation provides aerothermoelastic analysis of composite sandwich plate that is still lacking. It is the first study devoted to the complete aerothermoelastic analysis of advanced composite sandwich plate.

CONTENTS

ABSTRACT...........................................................................................................................ii
CONTENTS......................................................................................................................xviii
LIST OF TABLES..............................................................................................................xxi
LIST OF FIGURES............................................................................................................xxii
NOMENCLATURE...........................................................................................................xxv

Chapter                                                              page
I.	INTRODUCTION............................................................................................................1
II.	FORMULATION OF FINITE ELEMENT....................................................................14
   2.1 Formulation of Composite Sandwich Plates............................................................14
   2.2 Strain Energy, Kinematic Energy, and Work...........................................................17
   2.3 Triangular Plate Element...........….………………….............................................18
   2.4 Element Stiffness, Mass, Aerodynamic Matrices and Thermal Loading Vector….21
III.	THERMAL BUCKLING AND POSTBUCKLING ANALYSIS OF COMPOSITE SANDWICH PLATES..................................................................................................24
   3.1 Equilibrium Equation...............................................................................................24
   3.2 Solution Procedures for Thermal Postbuckling Analysis.........................................26
   3.3 Numerical Results....................................................................................................27
      3.3.1 Thermal Buckling............................................................................................27
           [(0/90)2/core]s Sandwich Plate.......................................................................28
           [(±q)2/core]s Sandwich Plate..........................................................................29
      3.3.2 Thermal Postbuckling......................................................................................31
           [(0/90)2/core]s Sandwich Plate.......................................................................31
           [(±q)2/core]s Sandwich Plate..........................................................................33
Temperature Gradient Effect..........................................................................36
IV.	FREE VIBRATION ANALYSIS OF THERMALLY BUCKLED COMPOSITE SANDWICH PLATES..................................................................................................38
   4.1 Equation of Motion..................................................................................................38
   4.2 Solution Procedures..................................................................................................40
   4.3 Numerical Results....................................................................................................41
      [(0/90)2/core]s Sandwich Plate..................................................................... ...........42
      [(±q)2/core]s Sandwich Plate....................................................................................43
      Temperature Gradient Effect....................................................................................45
V.	FLUTTER ANALYSIS OF THERMALLY BUCKLED COMPOSITE SANDWICH PLATES.........................................................................................................................47
   5.1 Equation of Motion..................................................................................................47
   5.2 Solution Procedures..................................................................................................49
      5.2.1 Frequency Domain Formulation for Linear Flutter.........................................49
      5.2.2 Time Domain Modal-Formulation for Nonlinear Flutter................................51 
   5.3 Numerical Results....................................................................................................55
      5.3.1 Linear Panel Flutter.........................................................................................55
      5.3.2 Nonlinear Panel Flutter ..................................................................................58     
VI.	CONCLUSIONS…………………………………........................................................63
REFERENCES.....................................................................................................................66
APPENDIX..........................................................................................................................74
TABLES...............................................................................................................................77
FIGURES.............................................................................................................................79
PUBLICATION LIST........................................................................................................123
VITA...................................................................................................................................124

LIST OF TABLES

Table                                                                page
2.1  Material properties for the graphite epoxy laminated face sheet.................................77
2.2  Material properties of aluminum honeycomb core......................................................77
3.1  Critical buckling temperature of conventional sandwich plate...................................78

LIST OF FIGURES

Figure                                                               page
2.1	  Geometry of a rectangular honeycomb core sandwich panel.....................................79
2.2	  72 degree of freedom triangular plate element...........................................................80
2.3	  The 8×4 mesh of a rectangular plate with a/b=2.......................................................81
3.1	Critical buckling temperature vs aspect ratio for [(0/90)2/core] s rectangular sandwich plates......................................................................................................82
3.2	Boundary of thermal buckling mode for simply supported sandwich plates....…......83
3.3	Critical buckling temperature vs core thickness for [(0/90)2/core] s square sandwich plates.........................................................................................................................84
3.4	Boundary of thermal buckling mode for simply supported [(±q)2/core] s sandwich plates............................................................................................................................85
3.5	Critical buckling temperature vs aspect ratio for simply supported [(±q)2/core]s sandwich plates............................................................................................................86
3.6	Critical buckling temperature vs fiber orientation for simply supported [(±q)2/core]s sandwich plates............................................................................................................87
3.7	Central deflection vs temperature rise for [(0/90)2/core] s square sandwich plates.....88
3.8	Central deflection vs temperature rise for [(0/90)2/core] s rectangular sandwich plates............................................................................................................................89
3.9	Central line deflection curve for [(0/90)2/core] s sandwich plate (a/b=4)...................90
3.10	Central line deflection curve for [(0/90)2/core] s sandwich plate (a/b=5)...................91
3.11	Central deflection vs temperature for [(±q)2/core] s square sandwich plates...............92
3.12	3-D deflection shape for [(±60) 2/core] s square composite sandwich plate................93
3.13	Boundary of mode shape for [(±q)2/core] s square composite sandwich plates...........94
3.14	Boundary of mode shape for [(±q)2/core] s composite sandwich plates (a/b=2).........95
3.15	Thermal postbuckling behavior of [(±60) 2/core] s square sandwich plate..............…96
3.16	Temperature gradient effect on central deflection of [(±q)2/core]s sandwich plates..97
4.1	Natural frequencies for [(+45/-45/+45) square laminate.........................................…98
4.2	Natural frequencies for [(0/90)2/core]s square composite sandwich plate..............…99
4.3	Natural frequencies for [(0/90)2/core]s composite sandwich plate (a/b=2).............100
4.4	Natural frequencies for [(0/90)2/core]s composite sandwich plate (a/b=3)...........…101
4.5	Boundary of first vibration mode shape for [(±q)2/core]s square sandwich plates....102
4.6	Natural frequencies for [(±60)2/core]s square composite sandwich plate.................103
4.7	Contour plot of first mode for [(±60)2/core]s square composite sandwich plate.......104
4.8	Natural frequencies for [(±75)2/core]s square composite sandwich plate.................105
4.9	Temperature gradient on vibration of thermally buckled [(±60)2/core]s square sandwich plate..............................................................................................…….....106
4.10	Temperature gradient on vibration of thermally buckled [(±75)2/core]s square sandwich plate................................................................................................….......107
5.1	Flutter coalescence plot of [(0/90)2/core]s rectangular sandwich plates....................108
5.2	Critical dynamic pressure vs aspect ratio for [(±q)2/core]s rectangular sandwich plates..........................................................................................................................109
5.3	Critical dynamic pressure vs fiber angle for [(±q)2/core]s rectangular sandwich plates..........................................................................................................................110
5.4	Flutter boundary vs fiber angle for [(±q)2/core]s square sandwich plates.…………111
5.5	Stability boundary of [(±30)2/core]s and [(±45)2/core]s square sandwich plates...…112
5.6	Temperature gradient on flutter boundary of [(±q)2/core]s sandwich plates (DT/DTcr =1).............................................................................................................................113
5.7	Stability boundary of square isotropic plate with uniform temperature....................114
5.8	Temperature gradient on stability boundary of [(±45)2/core]s square sandwich plate...........................................................................................................................115
5.9	Phase plane plot for [(±45)2/core]s square sandwich plate (l=100)..........................116
5.10	Phase plane plot for [(±45)2/core]s square sandwich plate (DT/DTcr=3)...................117
5.11	Response of [(±45)2/core]s square composite sandwich plate (Tt =1.5DT, Tb =0.5DT, l=250).......................................................................................................................118
5.12	Temperature gradient on stability boundary of [(±30)2/core]s square sandwich plate...........................................................................................................................119
5.13	Contour plot of static deflection of [(±30)2/core]s sandwich plate (DT/DTcr=3).......120
5.14	Flutter coalescence plot of [(±30)2/core]s sandwich plate (DT/DTcr=3)....................121
5.15	Temperature gradient on stability boundary of [(0/90)2/core]s square sandwich plate...........................................................................................................................122

                                    

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