第五章 結論與建議
5.1 結論
本論文乃應用Huang(2002)之成果，利用 Ritz 法探討含有奇異點之斜形 板的振動行為。本論文所研究分析之案例主要有二：懸臂梯形斜板以及懸 臂平行四邊形斜板。茲從本研究之結果於下作ㄧ完整的結論：
a. 從斜形板之收斂性分析，可以發現多項式允許函數之項數愈多，其數值 亦能愈逼近收斂値。其中多項式允許函數在沒有引用角函數時，所得到 之數值解或許會收斂，然而卻不是一個精確値；相對而言，加入角函數 可以加速自然振動頻率收斂之速度。
b. 在 Ritz 法中若只增加滿足傳統邊界條件之多項式函數為允許函數 (admissible function)，而不引入角函數，數值一樣會收斂。然而隨著傾 斜角度之增加，奇異性亦隨之增高；將造成為求得收斂解，須不斷增加 多 項 式 之 項 數 ， 然 此 舉 會 因 引 入 之 項 數 過 多 造 成 數 值 上 的 困 難(ill conditioning)而無法求解。
因此，引入角函數以加速其收斂速度為必要之行為。
c. 觀察懸臂斜形板面外之無因次化頻率，其値將會隨著斜角β 角度之增加
而增加；彎矩與剪力奇異性亦隨之增強。當a /b減小時，勁度則隨之增 加，造成頻率之加大。而當懸臂斜形板之形狀由梯形越接近平行四邊形 時，頻率亦隨之增加。
d. 不論是懸臂平行四邊形厚板或是懸臂梯形厚板、斜角之大小，在其所有
振態中，面內模態之頻率皆比面外模態之頻率下降幅度為大，此點說明 了角函數在面內之振態效果較佳，收斂性亦較好。
5.2 建議
本論文利用 Ritz 法分析含有奇異點之懸臂平行四邊形厚板及懸臂梯形
厚板的自然振動行為。求解上加入角函數，讓 Ritz 法於分析不論由邊界條
件或幾何形狀所產生的應力奇異問題能更為精確。而利用 Ritz 法並引入角
函數，此一作法亦可提供予後人用於研究其它不同邊界條件下具應力奇異 點之厚板自然振動與力學行為。
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表2.1 面外之線性齊次方程式之係數
續表 2.1 面外之線性齊次方程式之係數
2
表2.2 面內之線性齊次方程式之係數
自由端(θ =α)邊界條件 線性齊次方程式之係數
=0
θ
Nr
式(2.35a)
α λ
λ sin( 1)
11 =E0 m m +
b ，
α λ
λ cos( 1)
12 =−E0 m m+
b ，
α μ λ
λ sin( 1)
2 ) 1 )(
1 (
1 0
13 − + −
= m E m
b ，
α μ λ
λ cos( 1)
2 ) 1 )(
1
( 1 0
14 − + −
−
= m E m
b ，
=0 Nθ
式(2.35b)
α λ λ
υ 1) cos( 1)
( 0
21 = − mE m +
b ，
α λ λ
υ 1) sin( 1)
( 0
22 = − mE m +
b ，
α λ
μˆ1 0 cos( 1)
23 =− ⋅E ⋅ m −
b ，
α λ
μˆ1 0 sin( 1)
24 =− ⋅E ⋅ m −
b ，
表4.1 懸臂平行四邊形厚板
面外無因次化頻率(ωa2 ρh/D)之收斂性分析 (h/b=0.1, a/b=1, c/b=1,β =45o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.2 懸臂平行四邊形厚板
面外無因次化頻率(ωa2 ρh/D)之收斂性分析 (h/b=0.1, a/b=1, c/b=1, β =60o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.3 懸臂平行四邊形厚板
面外無因次化頻率(ωa2 ρh/D)之收斂性分析 (h/b=0.1, a/b=1, c/b=1, β =75o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
－: 病態矩陣(ill-conditioning)
表4.4 懸臂平行四邊形厚板
面外無因次化頻率(ωa2 ρh/D)之收斂性分析 (h/b=0.2, a/b=1, c/b=1,β =45o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.5 懸臂平行四邊形厚板
面外無因次化頻率(ωa2 ρh/D)之收斂性分析 (h/b=0.2, a/b=1, c/b=1, β =60o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.6 懸臂平行四邊形厚板
面外無因次化頻率(ωa2 ρh/D)之收斂性分析 (h/b=0.2, a/b=1, c/b=1, β =75o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.7 懸臂平行四邊形厚板
面內無因次化頻率(ωa ρ/E )之收斂性分析 (h/b=0.1, a/b=1, c/b=1,β =45o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.8 懸臂平行四邊形厚板
面內無因次化頻率(ωa ρ/E )之收斂性分析 (h/b=0.1, a/b=1, c/b=1, β =60o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.9 懸臂平行四邊形厚板
面內無因次化頻率(ωa ρ/E )之收斂性分析 (h/b=0.1, a/b=1, c/b=1, β =75o)
(I,J) in Equations (3.22a)-(3.22e) No.of corner
functions
－: 病態矩陣(ill-conditioning)
表4.10 懸臂平行四邊形厚板
面內無因次化頻率(ωa ρ/E )之收斂性分析 (h/b=0.2, a/b=1, c/b=1,β =45o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.11 懸臂平行四邊形厚板
面內無因次化頻率(ωa ρ/E )之收斂性分析 (h/b=0.2, a/b=1, c/b=1, β =60o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.12 懸臂平行四邊形厚板
面內無因次化頻率(ωa ρ/E )之收斂性分析 (h/b=0.2, a/b=1, c/b=1, β =75o)
(I,J) in Equations (3.22a)-(3.22e) Mode No.of corner
表4.13 懸臂平形四邊形厚板無因次化頻率(ωa2 ρh/D)之收斂性分析
(I,J) in Equations (3.22a)-(3.22e) Mode h/b β
(10,10) (11,11) (12,12) (13,13) (14,14) (15,15) (16,16) 0.1
表4.14 懸臂平行四邊形厚板(c/b=1)面內之無因次化頻率(ωa ρ/E )
Mode Number a/b h/b β
1 2 3 4 5 30 2.728 4.588 6.040 6.364 7.034
45 2.556 4.568 6.376 6.936 7.918 60 2.040 4.628 6.272 8.220 8.920 0.1
75 1.177 3.208 5.316 6.788 8.344 30 2.728 4.588 6.040 6.364 7.036 45 2.556 4.568 6.376 6.936 7.916 60 2.040 4.628 6.072 8.220 8.920 0.5
0.2
75 1.177 3.208 5.316 6.788 8.344 30 2.200 4.904 6.296 13.67 18.19 45 2.149 4.690 6.428 9.200 10.60 60 1.862 4.432 6.097 8.879 11.82 0.1
75 1.150 3.143 5.616 6.802 9.514 30 2.200 4.904 6.296 13.67 18.19 45 2.150 4.690 6.428 9.200 10.60 60 1.862 4.480 6.096 8.878 11.82 1
0.2
75 1.150 3.142 5.616 6.802 9.514 30 1.365 4.851 5.918 10.77 14.94 45 1.242 4.379 4.637 9.915 14.27 60 1.034 4.166 5.931 8.403 12.75 0.1
75 0.604 2.702 5.205 6.215 8.800 30 1.364 4.851 5.918 10.77 14.94 45 1.242 4.379 4.637 9.913 14.26 60 1.006 4.166 5.931 8.403 12.75 2
0.2
75 0.604 2.702 5.207 6.213 8.800
表 4.15 懸臂梯形厚板(c/b=0.25)面內之無因次化頻率(ωa ρ/E )
Mode Number a/b h/b β
1 2 3 4 5 30 2.522 5.100 6.528 9.420 10.11
45 2.168 4.936 6.496 9.504 10.66 60 1.674 4.608 6.160 8.784 11.91 0.1
75 0.979 3.095 5.720 6.698 8.062 30 2.521 5.100 6.528 9.420 10.11 45 2.168 4.936 6.496 9.504 10.66 60 1.672 4.608 6.160 8.784 11.91 0.5
0.2
75 0.979 3.095 5.720 6.698 8.062 30 1.969 5.168 6.324 10.22 13.27 45 1.611 4.776 6.524 9.578 13.49 60 1.179 3.984 6.018 8.246 9.611 0.1
75 0.649 2.436 5.148 6.374 8.418 30 1.969 5.168 6.324 10.22 13.27 45 1.611 4.776 6.524 9.578 13.49 60 1.170 3.984 6.018 8.246 9.610 1
0.2
75 0.648 2.436 5.148 6.374 8.418 30 1.304 4.306 6.160 8.810 13.71 45 1.044 3.663 6.055 7.860 12.54 60 0.742 2.767 5.840 6.500 10.47 0.1
75 0.395 1.541 3.591 6.155 6.685 30 1.304 4.306 6.162 8.810 13.71 45 1.044 3.663 6.056 7.860 12.54 60 0.742 2.767 5.842 6.499 10.47 2
0.2
75 0.395 1.541 3.591 6.154 6.684
表4.16 懸臂梯形厚板(c/b=0.5)面內之無因次化頻率(ωa ρ/E )
Mode Number a/b h/b β
1 2 3 4 5 30 2.662 4.996 6.288 8.566 8.694
45 2.424 6.416 8.952 9.272 9.376 60 1.972 4.568 6.706 6.110 8.842 0.1
75 1.170 3.200 5.711 6.830 9.590 30 2.662 4.996 6.288 8.564 8.696 45 2.430 6.416 8.952 9.272 9.376 60 1.971 4.568 6.706 6.110 8.842 0.5
0.2
75 1.174 3.200 5.711 6.830 9.590 30 2.045 6.444 10.48 12.68 14.34 45 1.776 4.898 6.402 9.854 12.980 60 1.386 4.550 6.122 8.664 12.610 0.1
75 0.808 2.908 5.452 6.518 9.107 30 2.046 6.444 10.48 12.68 14.34 45 1.776 4.898 6.402 9.854 12.98 60 1.386 4.550 6.122 8.664 12.61 1
0.2
75 0.808 2.908 5.452 6.518 9.106 30 1.298 5.855 9.880 14.96 15.370 45 1.079 4.220 5.808 9.095 14.15 60 0.796 3.395 5.807 7.610 12.13 0.1
75 0.438 2.000 4.628 6.035 7.710 30 1.298 5.853 9.882 14.95 15.37 45 1.079 4.220 5.808 9.097 14.15 60 0.796 3.395 5.807 7.608 12.13 2
0.2
75 0.438 2.000 4.628 6.037 7.708
表 4.17 懸臂梯形厚板(c/b=0.75)面內之無因次化頻率(ωa ρ/E )
Mode Number a/b h/b β
1 2 3 4 5 30 2.550 4.796 6.100 7.308 7.798
45 2.530 6.384 7.996 8.312 9.816 60 2.032 4.612 6.084 8.704 9.484 0.1
75 1.163 3.172 6.820 8.931 10.60 30 2.204 4.796 6.100 7.308 7.796 45 2.530 6.384 7.996 8.312 9.816 60 2.032 4.612 6.084 8.704 9.484 0.5
0.2
75 1.163 3.172 6.820 8.931 10.60 30 2.141 5.004 6.432 10.06 11.17 45 1.979 4.814 6.490 9.598 11.69 60 1.646 4.466 6.132 8.906 12.39 0.1
75 1.012 3.004 5.534 6.790 9.479 30 2.140 5.004 6.432 10.06 11.17 45 1.979 4.814 6.490 9.598 11.69 60 1.646 4.466 6.132 8.906 12.39 1
0.2
75 1.012 3.004 5.534 6.790 9.480 30 1.328 4.859 5.830 10.52 15.19 45 1.153 4.538 5.845 9.760 14.52 60 0.888 3.888 5.800 8.245 12.51 0.1
75 0.509 2.420 5.735 6.020 14.33 30 1.328 4.859 5.832 10.52 15.18 45 1.152 4.538 5.845 9.761 14.52 60 0.888 3.888 5.802 8.243 12.51 2
0.2
75 0.509 2.420 5.742 6.018 14.33
表4.18 懸臂平行四邊形厚板(c/b=1)面外之無因次化頻率(ωa2 ρh/D)
Mode Number a/b h/b β
1 2 3 4 5 30 3.845 5.709 9.883 17.14 19.87 45 4.319 7.460 11.55 18.65 21.75 60 4.828 12.16 16.04 21.64 25.19 0.1
75 5.451 19.89 23.89 35.35 40.47 30 3.459 4.908 8.094 13.13 13.80 45 3.805 6.266 9.261 13.87 14.31 60 4.205 9.323 12.16 13.54 16.45 0.5
0.2
75 4.696 8.487 14.58 15.50 22.79 30 3.857 8.888 23.30 24.31 37.14 45 4.381 10.56 24.84 28.34 45.08 60 5.032 14.87 27.16 38.43 44.32 0.1
75 5.734 22.80 39.36 55.35 70.72 30 3.724 8.094 19.72 21.44 30.80 45 4.179 9.567 21.06 23.70 36.60 60 4.733 13.20 22.16 29.29 41.49 1
0.2
75 5.363 19.64 23.75 36.04 42.97 30 3.689 15.24 24.27 44.88 68.96 45 4.011 17.16 28.43 47.37 78.80 60 4.422 20.12 36.03 54.06 90.05 0.1
75 4.868 24.69 58.51 62.05 111.70 30 3.646 14.30 22.76 41.47 61.52 45 3.946 16.11 25.85 43.79 69.05 60 4.323 19.43 30.82 49.54 78.32 2
0.2
75 4.740 23.30 40.01 56.48 96.43
表 4.19 懸臂梯形厚板(c/b=0.25)面外之無因次化頻率(ωa2 ρh/D)
Mode Number a/b h/b β
1 2 3 4 5 30 3.991 10.09 19.33 20.00 28.89 45 4.257 11.73 21.11 22.52 31.45 60 4.677 15.32 23.04 29.59 38.80 0.1
75 5.148 19.94 22.77 38.53 43.36 30 3.576 7.975 13.33 14.80 20.52 45 3.766 8.904 13.68 16.11 21.78 60 4.063 10.06 13.72 18.95 21.96 0.5
0.2
75 4.480 7.721 14.29 16.76 22.68 30 4.326 16.51 25.23 37.47 57.00 45 4.486 18.10 28.20 41.25 59.08 60 4.746 20.90 34.31 49.02 68.66 0.1
75 5.075 23.62 44.60 56.11 92.02 30 4.156 14.50 20.35 30.67 44.87 45 4.279 15.77 21.47 32.62 46.21 60 4.496 17.98 22.39 37.25 48.05 1
0.2
75 4.802 18.06 20.69 37.12 44.87 30 4.572 21.41 37.85 54.54 85.07 45 4.648 21.93 44.24 56.77 92.28 60 4.799 23.03 55.27 62.98 105.19 0.1
75 4.995 24.32 61.59 85.65 114.82 30 4.508 20.53 32.38 49.66 71.97 45 4.570 20.99 36.32 50.87 77.27 60 4.698 21.94 41.85 53.11 86.55 2
0.2
75 4.882 23.11 39.47 56.19 82.67
表4.20 懸臂梯形厚板(c/b=0.5)面外之無因次化頻率(ωa2 ρh/D)
Mode Number a/b h/b β
1 2 3 4 5 30 3.872 7.872 16.17 19.88 25.19 45 4.292 9.328 18.31 21.79 27.46 60 4.800 13.12 21.72 26.16 33.58 0.1
75 5.412 19.75 23.76 35.92 42.84 30 3.477 6.472 12.39 13.80 18.31 45 3.772 7.503 11.93 13.80 19.31 60 4.156 9.696 13.52 15.27 20.82 0.5
0.2
75 4.687 8.484 14.65 15.37 22.83 30 4.010 13.38 23.46 32.32 53.69 45 4.293 15.12 26.45 35.33 55.41 60 4.693 18.87 31.00 44.44 61.22 0.1
75 5.161 23.47 42.69 55.31 88.70 30 3.863 11.83 19.60 25.54 43.20 45 4.100 13.22 21.17 29.37 44.34 60 4.438 16.11 22.34 35.08 48.17 1
0.2
75 4.846 19.23 21.89 38.87 44.01 30 4.063 20.30 29.19 51.70 75.89 45 4.214 21.18 34.89 53.68 86.40 60 4.434 22.84 46.34 58.19 103.3 0.1
75 4.690 24.67 63.02 72.33 116.9 30 4.011 19.36 25.88 47.08 65.70 45 4.144 20.19 29.66 49.09 73.30 60 4.340 21.71 35.73 52.87 85.70 2
0.2
75 4.579 23.38 39.56 57.45 95.84
表 4.21 懸臂梯形厚板(c/b=0.75)面外之無因次化頻率(ωa2 ρh/D)
Mode Number a/b h/b β
1 2 3 4 5 30 3.851 6.528 13.62 19.85 26.44 45 4.311 8.076 14.49 21.32 23.51 60 4.781 12.36 18.61 23.37 28.91 0.1
75 4.974 5.820 19.76 27.04 35.35 30 3.462 6.811 10.23 13.69 15.74 45 3.798 6.686 11.36 11.62 14.00 60 4.201 9.407 13.27 14.24 19.00 0.5
0.2
75 4.723 8.476 14.22 14.56 15.50 30 3.901 10.74 23.25 28.51 47.96 45 4.313 12.41 26.10 31.43 50.62 60 4.866 16.43 29.25 41.73 57.03 0.1
75 5.500 22.80 40.89 55.54 81.30 30 3.763 9.649 19.63 24.62 38.94 45 4.118 11.07 21.33 26.62 41.03 60 4.588 14.33 22.92 32.60 45.46 1
0.2
75 5.155 19.45 23.16 37.33 43.57 30 3.820 18.11 25.21 48.53 71.09 45 4.054 19.60 29.91 50.78 82.22 60 4.362 22.19 39.17 56.49 97.36 0.1
75 4.702 24.93 61.90 64.86 114.7 30 3.774 17.02 23.25 44.68 62.76 45 3.988 18.49 26.53 46.68 70.97 60 4.267 20.96 32.13 51.59 82.79 2
0.2
75 4.582 23.54 39.55 57.42 98.58
表4.22 懸臂平行四邊形厚板(c/b=1)之無因次化頻率(ωa2 ρh/D)
Mode Number a/b h/b β
1 2 3 4 5 30 3.845 5.709 9.883 13.64* 17.14
45 4.319 7.460 11.55 12.78* 18.65 60 4.828 10.20* 12.16 16.04 21.64 0.1
75 5.451 5.886* 16.04* 19.89 23.89 30 3.459 4.908 6.819* 8.094 11.47*
45 3.805 6.266 6.389* 9.261 11.42*
60 4.205 5.099* 9.323 11.57* 12.16 0.5
0.2
75 2.708* 4.696 7.302* 8.487 13.29*
30 3.857 8.888 22.00* 23.30 24.31 45 4.381 10.56 21.49* 24.84 28.34 60 5.032 14.87 18.62* 27.16 38.43 0.1
75 5.734 11.50* 22.80 31.43* 39.36 30 3.724 8.094 11.00* 19.72 21.44 45 4.179 9.567 10.75* 21.06 23.45 60 4.733 9.310* 13.20 22.16 22.40*
1
0.2
75 5.363 5.750* 15.71* 19.64 23.75 30 3.689 15.24 24.27 27.29* 44.88 45 4.011 17.16 24.83* 28.43 47.37 60 4.422 20.12 20.67* 36.03 54.06 0.1
75 4.868 12.08* 24.69 54.04* 58.51 30 3.646 13.64* 14.30 22.76 41.47 45 3.946 12.42* 16.11 25.85 43.79 60 4.323 10.06* 19.43 30.82 41.66*
2
0.2
75 4.740 6.039* 23.30 27.02* 40.01
* ：面內(in-plane)模態之頻率
表 4.23 懸臂梯形厚板(c/b=0.25)之無因次化頻率(ωa2 ρh/D)
Mode Number a/b h/b β
1 2 3 4 5 30 3.991 10.09 12.61* 19.33 20.00 45 4.257 10.84* 11.73 21.11 22.52 60 4.677 8.372* 15.32 23.04 23.08*
0.1
75 3.776* 5.148 12.18* 19.94 22.77 30 3.576 6.302* 7.975 12.75* 13.33 45 3.766 5.421* 8.904 12.34* 13.68 60 4.180* 10.06 11.52* 13.72 15.40*
0.5
0.2
75 2.448* 4.480 7.721 7.738* 14.29 30 4.326 16.51 19.69* 25.23 37.47 45 4.486 16.11* 18.10 28.20 41.25 60 4.746 11.79* 20.90 34.31 40.08*
0.1
75 5.075 6.485* 23.62 24.36* 44.60 30 4.156 9.843* 14.50 20.35 25.84*
45 4.279 8.053* 15.77 21.47 23.88*
60 4.496 5.852* 17.98 19.92* 22.39 1
0.2
75 3.242* 4.802 12.18* 18.06 20.69 30 4.572 21.41 26.08* 37.85 54.54 45 4.648 20.87* 21.93 44.24 56.77 60 4.799 14.84* 23.03 55.27 55.33*
0.1
75 4.995 7.893* 24.32 30.82* 61.59 30 4.508 13.04* 20.53 32.38 43.06*
45 4.570 10.44* 20.99 36.32 36.63*
60 4.698 7.421* 21.94 27.67* 41.85 2
0.2
75 3.947* 4.882 15.41* 23.11 35.91*
* ：面內(in-plane)模態之頻率
表4.24 懸臂梯形厚板(c/b=0.5)之無因次化頻率(ωa2 ρh/D)
Mode Number a/b h/b β
1 2 3 4 5 30 3.872 7.872 13.31* 16.17 19.88 45 4.292 9.328 12.12* 18.31 21.79 60 4.800 9.861* 13.12 21.72 22.84*
0.1
75 5.412 5.850* 15.95* 19.75 23.76 30 3.477 6.472 6.655* 12.39 12.49*
45 3.772 6.075* 7.503 11.93 13.80 60 4.156 4.927* 9.696 11.42* 13.52 0.5
0.2
75 2.934* 4.687 8.001* 8.484 14.28 30 4.010 13.38 20.45* 23.46 32.32 45 4.293 15.12 17.76* 26.45 35.33 60 4.693 13.86* 18.87 31.00 44.44 0.1
75 5.161 8.084* 23.47 29.08* 42.69 30 3.863 10.23* 11.83 19.60 25.54 45 4.100 8.882* 13.22 21.17 24.49*
60 4.438 6.928* 16.11 22.34 22.75*
1
0.2
75 4.042* 4.846 14.54* 19.23 21.89 30 4.063 20.30 25.96* 29.19 51.70 45 4.214 21.18 21.58* 34.89 53.68 60 4.434 15.92* 22.84 46.34 58.19 0.1
75 4.690 8.755* 24.67 40.00* 63.02 30 4.011 12.98* 19.36 25.88 47.08 45 4.144 10.79* 20.19 29.66 42.20*
60 4.340 7.962* 21.71 33.95* 35.73 2
0.2
75 4.378* 4.579 20.00* 23.38 39.56
*：面內(in-plane)模態之頻率
表 4.25 懸臂梯形厚板(c/b=0.75)之無因次化頻率(ωa2 ρh/D)
Mode Number a/b h/b β
1 2 3 4 5 30 3.851 6.528 12.75* 13.62 19.85 45 4.311 8.076 12.65* 14.49 21.33*
60 4.781 10.16* 12.36 18.61 23.06*
0.1
75 4.974 5.820 5.965* 16.40* 19.76 30 3.462 5.510* 6.811 10.23 11.99*
45 3.798 6.326* 6.686 11.36 11.62 60 4.201 5.081* 9.407 11.53* 13.27 0.5
0.2
75 2.918* 4.723 7.930* 8.476 14.22 30 3.901 10.74 21.41* 23.25 28.51 45 4.313 12.41 19.79* 26.10 31.43 60 4.866 16.43 16.46* 29.25 41.73 0.1
75 5.500 10.12* 22.80 30.05* 40.89
30 3.763 9.649 10.70* 19.63 24.62 45 4.118 9.896* 11.07 21.33 24.07*
60 4.588 8.231* 14.33 22.33* 22.92 1
0.2
75 5.060* 5.155 15.02* 19.45 23.16 30 3.820 18.11 25.21 26.55* 48.53 45 4.054 19.60 23.05* 29.91 50.78 60 4.362 17.76* 22.19 39.17 56.49 0.1
75 4.702 10.17* 24.93 48.40* 61.90 30 3.774 13.28* 17.02 23.25 44.68 45 3.988 11.52* 18.49 26.53 45.38*
60 4.267 8.882* 20.96 32.13 38.88*
2
0.2
75 4.582 5.085* 23.54 24.20* 39.55
* ：面內(in-plane)模態之頻率
sin ] ) 2 / ( [ cos tan
] sin ) 2 / ( 2 )
2 / [(
2 /
1
2 / 1 2
2
β ξ η
β θ ξ
β η ξ
ξ η β π α
−
= −
−
− +
−
=
+
=
−
b
b b
r η Y
c C
D
ξ
α
θ
β B
A b/2
b X
r
a L
圖 1.1 斜形板示意圖(固定於θ =0處)
y
x v
u θ
u
rv
θ圖3.1 卡氏座標與極座標之轉換關係
Mode β a / b
1 2 3 4 5
0.5
(3.845) (5.709) (9.883) (13.64*) (17.14)
1
(3.857) (8.888) (22.00*) (23.30) (24.31) 30o
2
(3.689) (15.24) (24.27) (27.29*) (44.88)
*：面內(in-plane)之模態
圖 4.1a 懸臂平行四邊形厚板振態圖 (β =30o，c/b=1，h/b=0.1)
Mode β a / b
1 2 3 4 5
0.5
(4.319) (7.460) (11.55) (12.78*) (18.65)
1
(4.381) (10.56) (21.49*) (24.84) (28.34)
2
(4.011) (17.16) (24.83) (28.43) (47.37) 45o
*：面內(in-plane)之模態
圖4.1b 懸臂平行四邊形厚板振態圖(β =45o，c/b=1，h/b=0.1)
Mode β a / b
1 2 3 4 5
0.5
(4.828) (10.20*) (12.16) (16.04) (21.64)
1
(5.032) (14.87) (18.62*) (27.16) (38.43) 60o
2
(4.422) (20.12) (20.67*) (36.03) (54.06)
*：面內(in-plane)之模態
圖 4.1c 懸臂平行四邊形厚板振態圖(β =60o，c/b=1，h/b=0.1)
Mode β a / b
1 2 3 4 5
0.5
(5.451)
(5.886*) (16.04*)
(19.89) (23.89)
1
(5.734) (11.50) (22.80) (31.43*) (39.36)
2
(4.868) (12.08*) (24.69) (54.04*) (58.51) 75o
*：面內(in-plane)之模態
圖4.1d 懸臂平行四邊形厚板振態圖(β =75o，c/b=1，h/b=0.1)
Mode β a / b
1 2 3 4 5
0.5
(3.459) (4.908) (6.819*) (8.094) (11.47*)
1
(3.724) (8.094) (11.00*) (19.72) (21.44)
2
(3.646) (13.64) (14.30) (22.76) (41.47) 30o
*：面內(in-plane)之模態
圖 4.1e 懸臂平行四邊形厚板振態圖(β =30o，c/b=1，h/b=0.2)
Mode β a / b
1 2 3 4 5
0.5
(3.805) (6.266) (6.389*) (9.261) (11.42*)
1
(4.179) (9.567) (10.75*) (21.06) (23.45*)
2
(3.946) (12.42*) (16.11) (25.85) (43.79) 45o
*：面內(in-plane)之模態
圖4.1f 懸臂平行四邊形厚板振態圖(β =45o，c/b=1，h/b=0.2)
Mode β a / b
1 2 3 4 5
0.5
(4.205) (5.099*) (9.323) (11.57*) (12.16)
1
(4.733) (9.310*) (13.20) (22.16*) (22.40) 60o
2
(4.323) (10.06*) (19.43) (30.82) (41.66*)
*：面內(in-plane)之模態
圖4.1g 懸臂平行四邊形厚板振態圖(β =60o，c/b=1，h/b=0.2)
Mode β a / b
1 2 3 4 5
0.5
(2.708*)
(4.696) (7.302*) (8.487) (13.29*)
1
(5.363) (5.750*) (15.71*) (19.64) (23.75)
2
(4.740)
(6.039*) (23.30) (27.02*) (40.01) 75o
*：面內(in-plane)之模態
圖4.1h 懸臂平行四邊形厚板振態圖(β =75o，c/b=1，h/b=0.2)