本文已以馮卡門大撓曲理論為基礎，完成預應力下含壓電層不對稱夾固圓型板件 之非線性大撓曲問題探討，並針對其簡化式之線性與原始之非線性問題分別求取其解 析解與數值解。其中線性問題係藉由修正貝索函數與貝索函數的相關定義並配合夾固 式邊界條件來求解，而非線性問題則運用有限差分與迭代技巧以求得其數值解。在各 式參數條件下，所得幾何與結構回應之參數化探討，則發現:
(1)
擬單層疊層板件在極低電壓下，不論是線性解析或非線性數值解，各式預拉伸力 或不同面壓條件下之結果均與Sheplak & Dugundji[13]所提純機械負荷下單層圓 型板件之解相當脗合，即本文所提之方法已獲得驗證。(2)
在各式預拉伸力條件下之結果顯示，在特定電壓下，線性與非線性之解幾乎重 疊，唯於低預拉伸力(k=1)時，隨著電壓逐漸提高，則其差異漸趨明顯，而在較 高預拉伸力下，無論電壓大小如何，板件幾乎完全不受影響。是故線性解於低電 壓情況下仍有其相當準確性，且可適用於中高預拉伸力範圍，但低預拉伸力與較 高電壓下，則須仰賴非線性解來修正。(3)
不同材質疊層板件在低電壓下會有明顯差異，此乃由於無因次定義之故，以致板 件所承受的面壓大小將有所不同。(4)
在低預拉伸力與低面壓狀態下，壓電效應才有略微之影響，且隨著電壓提高影響 更趨顯著。唯提高機械負荷後，壓電效應之影響則被機械效應所涵蓋。(5)
在特定電壓下，板件之預拉伸力與面壓均會影響板件之各式幾何回應，但若兩者 同時存在時，也會互相影響並降低彼此之效應。由於微感測元件多為不對稱疊層材質之形式，且機電藕合效應也是今後發展主要 趨勢，然而在電流轉換過程中，會產生溫度的變化並使得板件產生熱殘留效應。
因此，不對稱疊層受壓電與熱效應合併影響之問題，亦是未來值得探討之主題。
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楊氏係數E( Gpa) 蒲松比ν 各層厚度(μm) [Si/Poly-Si](L
a
) E1
=165、E2
=170 ν1
=0.27、ν2
=0.22 h1
=8、
h2
=10 [SiO2
/ Poly-Si](Lb
) E1
=75、E2
=170 ν1
=0.17、ν2
=0.22 h1
=8、
h2
=10表一、含壓電疊層板疊層材質與厚度尺寸表
(a)
(b)
圖1、(a)預拉力下含壓電不對稱疊層圓型板承受面壓示意圖(b)不對稱疊層板自由體圖
0.1 1 10 100
k
0.0001 0.001 0.01 0.1 1
W
0/P
Sheplak & Dugundji(98') Linear Solution
0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0Sheplak and Dugundji('98) V = 1
k = 1 k = 5 k = 10 k = 20 k = 50
Linear Solution
(a)中心撓度 (b)斜率
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0Sheplak & Dugundji('98) V = 1
k = 1 k = 5 k = 10 k = 20 k = 50
Linear Solution
0 0.2 0.4 0.6 0.8 1
-2 -1 0 1 2 3
/P
Sheplak & Dugundji('98) V = 1
k = 1 k = 5 k = 10 k = 20 k = 50
Linear Solution
(c)撓度 (d)曲率
0 0.2 0.4 0.6 0.8 1
-1200 -800 -400 0 400 800
r/p
0Flowing Saini et al.(2000') Linear Solution(La)
k=1 k=5 k=10 k=20 k=30k=50
(e)壓力敏性
圖2、低電壓擬單層線性解析解與 Sheplak & Dugundji[13] 、Saini et al.[58]比較圖
0.1 1 10 100
k
0.0001 0.001 0.01 0.1 1
W
0/P
V = 1 [Si/Poly-Si]
[SiO2/ Poly-Si]
0.1 1 10 100
k
0.0001 0.001 0.01 0.1 1
W
0/P
V = 5 [Si/Poly-Si]
[SiO2/ Poly-Si]
(a)V=1 (b)V=5
0.1 1 10 100
k
0.0001 0.001 0.01 0.1 1
W
0/P
V = 10 [Si/Poly-Si]
[SiO2/ Poly-Si]
0.1 1 10 100
k
0.0001 0.001 0.01 0.1 1 10
W
0/P
V = 20 [Si/Poly-Si]
[SiO2/ Poly-Si]
(c)V=10 (d)V=20
圖3、不同疊層材質板件(L
a
與Lb
)中心撓度線性解對預拉伸參數關係圖0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0V = 1 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0V = 5 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0V = 10 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0V = 20 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0V = 30 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
(e)V=30
圖4、不同疊層材質板件(L
a
與Lb
)無因次斜率線性解之徑向分佈圖0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=1 V=1 V=5 V=10 V=20 V=30
0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=5 V=1 V=5 V=10 V=20 V=30
(a)k=1 (b)k=5
0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=10 V=1 V=5 V=10 V=20 V=30
0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=20 V=1 V=5 V=10 V=20 V=30
(c)k=10 (d)k=20
0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=50 V=1 V=5 V=10 V=20 V=30
(e)k=50
圖5、L
a
疊層不同預拉伸力與各式電壓下無因次斜率線性解之徑向分佈圖0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=1 V=1 V=5 V=10 V=20 V=30
0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=5 V=1 V=5 V=10 V=20 V=30
(a)k=1 (b)k=5
0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=10 V=1 V=5 V=10 V=20 V=30
0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=20 V=1 V=5 V=10 V=20 V=30
(c)k=10 (d)k=20
0 5 10 15 20 25
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=50 V=1 V=5 V=10 V=20 V=30
(e)k=50
圖6、L
b
疊層不同預拉伸力與各式電壓下無因次斜率線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=1(V=1) E1/E2=0.2 E1/E2=0.5 E1/E2=0.9
0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=5(V=1) E1/E2=0.2 E1/E2=0.5 E1/E2=0.9
(a)k=1 (b)k=5
0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=10(V=1) E1/E2=0.2 E1/E2=0.5 E1/E2=0.9
0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=20(V=1) E1/E2=0.2 E1/E2=0.5 E1/E2=0.9
(c)k=10 (d)k=20
0 0.2 0.4 0.6 0.8 1
-2 -1.6 -1.2 -0.8 -0.4 0
/ W
0k=50(V=1) E1/E2=0.2 E1/E2=0.5 E1/E2=0.9
(e)k=50
圖7、低電壓(V=1)及各式預拉伸力下不同模數比板件之斜率線性解
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0V = 1 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0V = 5 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0V = 10 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0V = 20 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0V = 30 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10 k = 20 k = 50
(e)V=30
圖8、不同疊層材質板件(L
a
與Lb
)無因次撓度線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0k=1(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0k=5(La) V=1 V=5 V=10 V=20 V=30
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0k=10(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0k=20(La) V=1 V=5 V=10 V=20 V=30
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/ W
0k=50(La) V=1 V=5 V=10 V=20 V=30
(e)V=30
圖9、L
a
板件不同預拉伸力下無因次撓度線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/W
0k=1(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/W
0k=5(Lb) V=1 V=5 V=10 V=20 V=30
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/W
0k=10(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/W
0k=20(Lb) V=1 V=5 V=10 V=20 V=30
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W/W
0k=50(Lb) V=1 V=5 V=10 V=20 V=30
(e)V=30
圖10、L
b
板件不同預拉伸力下無因次撓度線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
V = 1 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10
k = 20 k = 50
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
V = 5 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10
k = 20 k = 50
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
V = 10 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10
k = 20 k = 50
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
V = 20 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1
k = 5 k = 10
k = 20 k = 50
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
V = 30 [Si/Poly-Si]
[SiO2/ Poly-Si]
k = 1 k = 5 k = 10
k = 20 k = 50
(e)V=30
圖11、不同疊層材質板件(L
a
與Lb
)無因次曲率線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
k=1(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
k=5(La) V=1 V=5 V=10 V=20 V=30
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
k=10(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
k=20(La) V=1 V=5 V=10 V=20 V=30
(c)V=10 (d)V=20
圖12、L
a
板件不同預拉身力下無因次曲率線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
k=1(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
k=5(Lb) V=1 V=5 V=10 V=20 V=30
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
k=10(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
k=20(Lb) V=1 V=5 V=10 V=20 V=30
(c)V=10 (d)V=20
圖13、L
b
板件不同預拉身力下無因次曲率線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-3 -2 -1 0 1 2
V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-1.2 -0.8 -0.4 0 0.4 0.8
W
V=1 V=5 V=10 V=20 V=30
圖14、純斜率與撓度線性解之徑向分佈圖
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0V=1 La Lb k=1
k=5 k=10
k=20 k=50k=30
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0V=5 La Lb k=1
k=5 k=10
k=20k=30k=50
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0V=10 La Lb
k=1
k=5 k=10
k=20k=30 k=50
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0V=20 La Lb
k=1 k=5
k=10
k=20k=30 k=50
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0V=30 La Lb k=1
k=5 k=10
k=20k=30 k=50
(e)V=30
圖15、L
a
與Lb
疊層不同之預拉伸力、電壓下壓力敏性線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0k=1 V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0k=5 V=1 V=5 V=10 V=20 V=30
(a)k=1 (b)k=5
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0k=10 V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0k=20 V=1 V=5 V=10 V=20 V=30
(c)k=10 (d)k=20
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0k=30 V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-8000 -4000 0 4000
r/p
0k=50 V=1 V=5 V=10 V=20 V=30
(e)k=30 (f)k=50
圖16、L
a
疊層不同預拉伸力與電壓下壓力敏性線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0k=1 V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0k=5 V=1 V=5 V=10 V=20 V=30
(a)k=1 (b)k=5
0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0k=10 V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0k=20 V=1 V=5 V=10 V=20 V=30
(c)k=10 (d)k=20
0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0k=30 V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0k=50 V=1 V=5 V=10 V=20 V=30
(e)k=30 (f)k=50
圖17、L
b
疊層不同預拉伸力與電壓下壓力敏性線性解之徑向分佈圖0.1 1 10 100
k
0.1 1 1x101 1x102 1x103 1x104 1x105
P
maxV=1 La Lb
Sheplak & Dugundji
Fit 1: Spline smoothing
0.1 1 10 100
k
0.1 1 1x101 1x102 1x103 1x104 1x105
P
maxV=5 La Lb
Sheplak & Dugundji
(a)V=1 (b)V=5
0.1 1 10 100
k
0.1 1 1x101 1x102 1x103 1x104 1x105
P
maxV=10 La Lb
Sheplak & Dugundji
0.1 1 10 100
k
0.1 1 1x101 1x102 1x103 1x104 1x105
P
maxV=20 La Lb
Sheplak & Dugundji
(c)V=10 (d)V=20
0.1 1 10 100
k
0.1 1 1x101 1x102 1x103 1x104 1x105
P
maxLa V=1 V=5 V=10 V=20
0.1 1 10 100
k
0.1 1 1x101 1x102 1x103 1x104 1x105
P
maxLb V=1 V=5 V=10 V=20
(e)L
a
(f)Lb
圖18、不同電壓下預拉伸參數對線性行為最大面壓圖
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0V=1 La Lb
Shplak &Dugundji
k=1k=5 k=10k=20 k=50k=100
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0V=5 La Lb
k=1k=5 k=10k=20 k=50k=100
(a)V=1 (b)V=5
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0V=10 La Lb
k=1k=5 k=10k=20 k=50k=100
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0V=20 La Lb
k=1k=5 k=10k=20 k=50k=100
(c)V=10 (d)V=20
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0V=30 La Lb
k=1k=5 k=10k=20 k=100k=50
(e)V=30
圖19、L
a
和Lb
板件不同面壓與電壓下中心撓度與面壓之關係圖0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=1(La) V=1 V=5 V=10 V=20 V=30
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=5(La) V=1 V=5 V=10 V=20 V=30
(a)k=1 (b)k=5
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=10(La) V=1 V=5 V=10 V=20 V=30
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=20(La) V=1 V=5 V=10 V=20 V=30
(c)k=10 (d)k=20
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=50(La) V=1 V=5 V=10 V=20 V=30
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=100(La) V=1 V=5 V=10 V=20 V=30
(e)k=50 (f)k=100 圖20、L
a
板件不同面壓與電壓下中心撓度與面壓之關係圖0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=1(Lb) V=1 V=5 V=10 V=20 V=30
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=5(Lb) V=1 V=5 V=10 V=20 V=30
(a)k=1 (b)k=5
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=10(Lb) V=1 V=5 V=10 V=20 V=30
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=20(Lb) V=1 V=5 V=10 V=20 V=30
(c)k=10 (d)k=20
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=50(Lb) V=1 V=5 V=10 V=20 V=30
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100
W
0k=100(Lb) V=1 V=5 V=10 V=20 V=30
(e)k=50 (f)k=100 圖21、L
b
板件不同面壓與電壓下中心撓度與面壓之關係圖0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
V=1 La
Lb
Sheplak & Dugundji
k=1k=5 k=10k=20 k=50k=100
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
V=5 La
Lb
k=1k=5 k=10k=20 k=50k=100
(a)V=1 (b)V=5
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
V=10 La
Lb
k=1k=5 k=10k=20 k=50k=100
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
V=20 La
Lb
k=1k=5 k=10k=20 k=50k=100
(c)V=10 (d)V=20
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
V=30 La
Lb
k=1k=5 k=10k=20 k=50k=100
(e)V=30
圖22、L
a
和Lb
板件不同面壓與電壓下中心曲率非線性解與面壓之關係圖0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=1(La) V=1 V=5 V=10 V=20 V=50
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=5(La) V=1 V=5 V=10 V=20 V=50
(a)k=1 (b)k=5
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=10(La) V=1 V=5 V=10 V=20 V=50
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=20(La) V=1 V=5 V=10 V=20 V=50
(c)k=10 (d)k=20
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=50(La) V=1 V=5 V=10 V=20 V=50
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=100(La) V=1 V=5 V=10 V=20 V=50
(e)k=50 (f)k=100
圖23、L
a
板件不同面壓與電壓下中心曲率非線性解與面壓之關係圖0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=1(Lb) V=1 V=5 V=10 V=20 V=50
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=5(Lb) V=1 V=5 V=10 V=20 V=50
(a)k=1 (b)k=5
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=10(Lb) V=1 V=5 V=10 V=20 V=50
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=20(Lb) V=1 V=5 V=10 V=20 V=50
(c)k=10 (d)k=20
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=50(Lb) V=1 V=5 V=10 V=20 V=50
0.1 1 1x101 1x102 1x103 1x104 1x105 1x106
P
0.01 0.1 1 10 100 1000
(0)
k=100(Lb) V=1 V=5 V=10 V=20 V=50
(e)k=50 (f)k=100
圖24、L
b
板件不同面壓與電壓下中心曲率非線性解與面壓之關係圖0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0La
V=1 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
Sheplak & Dugundji
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0La
V=5 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0La
V=10 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0La
V=20 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0La
V=30 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
(e)V=30
圖25、L
a
和Lb
板件不同面壓與電壓下無因次斜率非線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=1(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=10(La) V=1 V=5 V=10 V=20 V=30
(a)P=1 (b) P=10
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=100(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=1000(La) V=1 V=5 V=10 V=20 V=30
(c) P=100 (d) P=1000
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=10000(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=100000(La) V=1 V=5 V=10 V=20 V=30
(e) P=10000 (f) P=100000
圖26、L
a
板件不同面壓與電壓下無因次斜率非線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=1(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=10(Lb) V=1 V=5 V=10 V=20 V=30
(a) P=1 (b) P=10
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=100(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=1000(Lb) V=1 V=5 V=10 V=20 V=30
(c) P=100 (d) P=1000
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=10000(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
-2.5 -2 -1.5 -1 -0.5 0
/ W
0P=100000(Lb) V=1 V=5 V=10 V=20 V=30
(e) P=10000 (f) P=100000
圖27、L
b
板件不同面壓與電壓下無因次斜率非線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0La
V=1 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
Sheplak & Dugundji
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0La
V=5 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0La
V=10 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0La
V=20 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0La
V=30 P=1 P=10 P=100 P=1000 P=10000 P=100000
Lb
(e)V=30
圖28、L
a
和Lb
板件不同面壓與電壓下無因次撓度非線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=1(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=10(La) V=1 V=5 V=10 V=20 V=30
(a) P=1 (a) P=10
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=100(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=1000(La) V=1 V=5 V=10 V=20 V=30
(c) P=100 (d) P=1000
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=10000(La) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=100000(La) V=1 V=5 V=10 V=20 V=30
(e) P=10000 (f) P=100000
圖29、L
a
板件不同面壓與電壓下無因次撓度非線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=1(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=10(Lb) V=1 V=5 V=10 V=20 V=30
(a) P=1 (a) P=10
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=100(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=1000(Lb) V=1 V=5 V=10 V=20 V=30
(c) P=100 (d) P=1000
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=10000(Lb) V=1 V=5 V=10 V=20 V=30
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
W /W
0P=100000(Lb) V=1 V=5 V=10 V=20 V=30
(e) P=10000 (f) P=100000
圖30、L
b
板件不同面壓與電壓下無因次撓度非線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
Sheplak & Dugundji(98') V=1(Volt)
P=1 P=10 P=100 P=1000 P=10000 P=100000
La Lb
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
V=5(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La Lb
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
V=10(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La Lb
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
V=20(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La Lb
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
-4 0 4
/P
V=30(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La Lb
(e)V=30
圖31、L
a
與Lb
板件不同面壓與電壓下無因次曲率非線性解之徑向分佈圖0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
r/S
0Sheplak & Dugundji V=1(Volt)
P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
r/S
0V=5(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
r/S
0V=10(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
r/S
0V=20(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
r/S
0V=30(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
(e)V=30
圖32、L
a
與Lb
板件不同面壓與電壓下無因次徑向力合之分佈圖0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0Sheplak & Dugundji V=1(Volt)(La)
P=1 P=10 P=100 P=1000 P=10000 P=100000
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=5(Volt)(La) P=1 P=10 P=100 P=1000 P=10000 P=100000
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=10(Volt)(La) P=1 P=10 P=100 P=1000 P=10000 P=100000
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=20(Volt)(La) P=1 P=10 P=100 P=1000 P=10000 P=100000
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=30(Volt)(La) P=1 P=10 P=100 P=1000 P=10000 P=100000
(e)V=30
圖33、L
a
板件不同面壓與電壓下壓環向力合之徑向分佈圖0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=1(Volt)(Lb) P=1 P=10 P=100 P=1000 P=10000 P=100000
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=5(Volt)(Lb) P=1 P=10 P=100 P=1000 P=10000 P=100000
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=10(Volt)(Lb) P=1 P=10 P=100 P=1000 P=10000 P=100000
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=20(Volt)(Lb) P=1 P=10 P=100 P=1000 P=10000 P=100000
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=30(Volt)(Lb) P=1 P=10 P=100 P=1000 P=10000 P=100000
(e)V=30
圖34、L
b
板件不同面壓與電壓下壓環向力合之徑向分佈圖0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0Sheplak & Dugundji V=1(Volt)
P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=5(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=10(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=20(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
(c)V=10 (d)V=20
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
S
/S
0V=30(Volt) P=1 P=10 P=100 P=1000 P=10000 P=100000
La
Lb
(e)V=30
圖35、L
a
與Lb
板件不同面壓與電壓下壓環向力合之徑向分佈圖0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0Flowing Saini et al.(2000') Linear Solution(La)
k=1 k=5k=10 k=20
k=30k=50
Nonlinear Solution(La)
0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0V=5 Linear Solution(La)
k=1 k=10 k=5
k=20
k=50k=30
Nonlinear Solution(La)
(a)V=1 (b)V=5
0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0V=10 Linear Solution(La)
k=1 k=5k=10 k=20
k=50k=30
Nonlinear Solution(La)
0 0.2 0.4 0.6 0.8 1
-4000 -2000 0 2000 4000
r/p
0V=20
Linear Solution(La) k=1
k=5k=10 k=20
k=50k=30
Nonlinear Solution(La)
(c)V=10 (d)V=20
圖36、各式電壓下之 L