未來展望

第五章結論與展望

5.2 未來展望

在垂直置放蜂鳴片的模擬中，發現無論是液面高度或液體厚度都會對共振頻率造成向低頻漂移的影響，實際狀況是否如同模擬般變化，是一可研究的問題，又高度及厚度是否有交互影響？且在不同厚度的狀態，隨著液面高度的增加，共振頻率的變化不若以不同液面高度的觀察，並非成比例性的下降，若是將蜂鳴片轉做感測器使用，於感測的準確度而言，便顯得十分重要。垂直置放蜂鳴片的實驗中存在頻率響應曲線尖點不明顯的情況，初步懷疑是反射波的影響，若能將此影響排除，可以取得較佳的實驗數值，對於蜂鳴片垂直置放的狀態，

會有較清楚的了解，及衍生的應用。

因為初始假設液體重量所造成的預應力變化不大，所以在模擬的過程中給予固定拉力7N，結果可以看到前段的共振頻率已高過實驗值，後段的共振頻率卻是低於實驗值，可以想見液體的多寡所造成的預應力影響有一定程度，模擬徑向的拉力應該隨液面增加而增加，又是否成線性增加？又若將蜂鳴片垂直置放，水壓對於蜂鳴片存在有線性增加的預應力關係，相對於將蜂鳴片水平置放，預應力為平均分佈，

應於模擬中考慮以增加準確性。

研究的標的物為單一尺寸的壓電蜂鳴片，液體也僅使用水當作耦合工具，研究的結果及趨勢是否可以應用在不同尺寸的壓電蜂鳴片，

或採用其他液體與蜂鳴片耦合，如油或酒精等等。

本研究證明液體耦合的狀態下，壓電蜂鳴片的共振頻率會隨著液體的增加而逐漸下降，除了製作成致動器外，亦有成為水位感測器的可能，日後可將其實現，可達到降低成本、增加效率。

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表 2.1 PZT-5A 的材料常數

壓電常數 (cl/m²) 介電常數 (10^-12 F/m) e31 e33 e15 ε11^T ε₃₃^T

-5.4 15.8 12.3 916 830

表 2.2 黃銅的材料常數

楊氏係數E (GPa) 蒲松比ν 密度ρ (kg/m³)

100 0.34 8,500

表2.3 水的材料常數

體模數 κ (GPa) 密度ρ (kg/m³) 黏滯係數 η (Pa·s)

2.2 998 0.001 密度

(kg/m³) 剛性係數 (10¹⁰N/m²)

ρ c₁₁ c₁₂ c₁₃ c₃₃ c₄₄ c₆₆

7,650 12.1 7.54 7.52 11.1 2.11 2.26

表 2.4 PZT-5A 前 4 模態頻率值 Mesh size

(mm) 5 2 1 0.8 0.5 0.4 0.3 Element # 128 480 1,792 2,880 7,168 11,360 20,520

Node # 235 819 2,915 4,683 11,459 18,163 32,619 Mode Resonant frequency (Hz)

W01 3,299 2,790 2,721 2,708 2,699 2,696 2,694 W₁₁ 8,498 6,380 6,084 6,029 5,988 5,975 5,967 W21 16,691 10,922 10,032 9,883 9,776 9,743 9,723 W₀₂ 18,675 12,592 11,812 11,651 11,524 11,484 11,459

表2.5 PZT-4 的材料常數

壓電常數 (cl/m²) 介電常數 (10^-12 F/m) e₃₁ e₃₃ e₁₅ ε₁₁^T ε₃₃^T

-5.2 15.1 12.7 730 635 密度

(kg/m³) 剛性係數 (10¹⁰N/m²)

ρ c11 c12 c13 c33 c44 c66

7,500 13.9 7.78 7.43 11.5 2.56 3.06

表 2.6 PZT-5H 的材料常數

壓電常數 (cl/m²) 介電常數 (10^-12 F/m) e31 e33 e15 ε11^T ε₃₃^T

-6.5 23.3 17.0 1,700 1,470

表 2.7 PZT-7A 的材料常數

壓電常數 (cl/m²) 介電常數 (10^-12 F/m) e31 e33 e15 ε11^T ε₃₃^T

-2.1 9.5 9.2 400 235 密度

(kg/m³) 剛性係數 (10¹⁰N/m²)

ρ c₁₁ c₁₂ c₁₃ c₃₃ c₄₄ c₆₆

7,600 12.6 7.95 8.41 11.7 2.30 2.35

密度

(kg/m³) 剛性係數 (10¹⁰N/m²)

ρ c₁₁ c₁₂ c₁₃ c₃₃ c₄₄ c₆₆

7,650 14.8 7.62 7.42 13.1 2.53 3.60

表2.8 PZT-8 的材料常數

壓電常數 (cl/m²) 介電常數 (10^-12 F/m) e31 e33 e15 ε11^T ε₃₃^T

-4.0 13.8 10.4 898 582

表2.9 不同壓電材料前 4 模態頻率值 Resonant frequency (Hz)

Mode PZT-4 PZT-5A PZT-5H PZT-7A PZT-8 實際值 W01 3,416 3,192 3,125 3,519 3,422 3,451 W₁₁ 7,587 7,176 7,082 7,738 7,607 7,625 W21 12,536 11,934 12,734 12,734 12,579 12,300 W₀₂ 14,757 14,042 14,982 14,982 14,842 14,787

密度

(kg/m³) 剛性係數 (10¹⁰N/m²)

ρ c₁₁ c₁₂ c₁₃ c₃₃ c₄₄ c₆₆

7,600 13.7 6.97 7.16 12.4 3.14 3.37

表2.10 碟型蜂鳴片的前 4 模態共振頻率

表2.11 水平置放蜂鳴片不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W01 W11 W21 W02

0 3,513 7,726 12,715 14,961 1 3,071 6,748 11,070 13,009 2 2,772 6,113 10,058 11,829 3 2,564 5,711 9,482 11,167 4 2,412 5,443 9,139 10,762 5 2,295 5,254 8,928 10,494 10 1,922 4,832 8,616 9,890 15 1,692 4,734 8,589 9,633 20 1,528 4,712 8,587 9,469 30 1,237 4,558 8,388 9,047

共振模態 W₀₁ W₁₁ W₂₁ W₀₂

共振頻率 (Hz) 3,519 7,738 12,734 14,982

表 2.12 垂直置放蜂鳴片不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W01 W11 W21 W02

3 3,461 7,539 7,676 12,447 6 3,408 7,054 7,600 11,544 9 3,151 6,321 7,122 10,868 12 2,700 6,063 6,235 10,444 15 2,290 5,504 5,799 9,711 18 1,920 4,923 5,155 8,903 21 1,784 4,803 4,965 8,754

表3.1 水平置放蜂鳴片量測結果一 Resonant frequency (Hz) 累積注水量

(mm³) W01 W11 W21 W02

0 3,443 7,600 12,425 14,787 500 2,875 6,806 11,531 13,493 1,000 2,850 6,256 10,062 11,875 1,500 2,675 5,831 9,587 11,306 2,000 2,475 5,587 9,343 10,956 2,500 2,375 5,412 9,193 10,681 3,000 2,300 5,287 9,118 10,537 3,500 2,225 5,162 9,043 10,387 4,000 2,150 5,112 9,018 10,312 4,500 2,100 5,087 8,993 10,237 5,000 2,025 5,062 8,968 10,187 5,500 1,975 5,037 8,968 10,112 6,000 1,925 5,037 8,968 10,062 6,500 1,900 5,012 8,943 10,037 7,000 1,856 5,012 8,943 9,987 7,500 1,806 4,987 8,943 9,962 8,000 1,781 4,987 8,943 9,912 8,500 1,756 4,987 8,943 9,887 9,000 1,731 4,987 8,943 9,862 9,500 1,681 4,987 8,943 9,837 10,000 1,656 4,987 8,943 9,812 10,500 1,631 4,987 8,943 9,787 11,000 1,631 4,987 8,943 9,762 11,500 1,606 4,987 8,943 9,737 12,000 1,581 4,987 8,943 9,712 12,500 1,556 4,987 8,943 9,687 13,000 1,531 4,987 8,943 9,662

表3.2 水平置放蜂鳴片量測結果二 Resonant frequency (Hz) 累積注水量

(mm³) W01 W11 W21 W02

0 3,493 7,650 12,400 14,762 500 2,875 6,706 11,456 13,518 1,000 2,875 6,281 10,062 11,900 1,500 2,675 5,856 9,562 11,331 2,000 2,475 5,587 9,293 10,956 2,500 2,400 5,412 9,118 10,731 3,000 2,300 5,287 9,018 10,562 3,500 2,225 5,187 8,943 10,437 4,000 2,150 5,137 8,918 10,337 4,500 2,100 5,087 8,893 10,237 5,000 2,050 5,062 8,893 10,187 5,500 2,000 5,062 8,893 10,137 6,000 1,950 5,037 8,868 10,062 6,500 1,900 5,037 8,868 10,037 7,000 1,856 5,037 8,868 9,987 7,500 1,831 5,037 8,868 9,962 8,000 1,781 5,037 8,868 9,937 8,500 1,756 5,037 8,868 9,887 9,000 1,731 5,037 8,868 9,862 9,500 1,706 5,037 8,868 9,837 10,000 1,681 5,037 8,868 9,812 10,500 1,656 5,037 8,868 9,812 11,000 1,631 5,037 8,868 9,762 11,500 1,606 5,037 8,868 9,762 12,000 1,581 5,037 8,868 9,712 12,500 1,581 5,037 8,868 9,687 13,000 1,556 5,037 8,868 9,662

表3.3 水平置放蜂鳴片量測結果三 Resonant frequency (Hz) 累積注水量

(mm³) W01 W11 W21 W02

0 3,418 7,625 12,075 14,812 500 3,168 6,875 11,056 13,393 1,000 2,850 6,106 9,962 11,975 1,500 2,650 5,856 9,462 11,381 2,000 2,525 5,437 9,093 11,006 2,500 2,400 5,262 8,943 10,781 3,000 2,325 5,187 8,868 10,537 3,500 2,225 5,112 8,818 10,387 4,000 2,175 5,037 8,768 10,337 4,500 2,100 5,012 8,743 10,237 5,000 2,050 4,937 8,743 10,112 5,500 2,000 4,912 8,743 10,062 6,000 1,950 4,912 8,743 9,987 6,500 1,925 4,912 8,743 9,962 7,000 1,900 4,887 8,718 9,912 7,500 1,806 4,887 8,718 9,862 8,000 1,781 4,887 8,743 9,787 8,500 1,756 4,887 8,718 9,787 9,000 1,731 4,887 8,743 9,737 9,500 1,706 4,887 8,743 9,687 10,000 1,681 4,887 8,718 9,637 10,500 1,656 4,887 8,743 9,562 11,000 1,631 4,887 8,743 9,587 11,500 1,606 4,887 8,718 9,562 12,000 1,581 4,887 8,743 9,537 12,500 1,556 4,887 8,718 9,512 13,000 1,531 4,887 8,743 9,512

表3.4 垂直置放蜂鳴片量測結果一 Resonant frequency (Hz) 累積注水量

(mm³) W01 W11 W21 W02

0 3,468 7,950 12,400 14,762 500 3,468 7,600 11,875 14,368 1,000 3,468 7,600 11,875 14,412 1,500 3,468 7,600 11,875 14,437 2,000 3,468 7,275 11,875 14,318 2,500 3,443 7,175 11,681 14,068 3,000 3,443 7,075 10,262 12,225 3,500 3,318 6,856 9,887 11,656 4,000 3,293 6,781 9,412 11,531 4,500 3,218 6,556 9,787 11,356 5,000 3,143 6,181 9,762 11,156 5,500 3,050 6,081 9,762 11,081 6,000 3,000 5,906 9,762 10,831 6,500 2,875 5,881 9,737 10,681 7,000 2,775 5,881 9,662 10,537 7,500 2,625 5,731 9,662 10,537 8,000 2,600 5,681 9,662 10,512 8,500 2,525 5,612 9,687 10,512 9,000 2,475 5,587 9,712 10,512 9,500 2,400 5,362 9,662 10,487 10,000 2,350 5,212 9,637 10,462 10,500 2,350 5,212 9,612 10,462 11,000 2,150 5,162 9,587 10,462 11,500 2,150 5,162 9,587 10,462 12,000 2,125 5,087 9,387 10,412 12,500 1,875 4,987 9,318 10,437 13,000 1,806 4,887 9,118 10,437

表3.5 垂直置放蜂鳴片量測結果二 Resonant frequency (Hz) 累積注水量

(mm³) W01 W11 W21 W02

0 3,518 7,700 12,225 14,712 500 3,518 7,600 11,900 14,387 1,000 3,493 7,475 12,050 14,368 1,500 3,418 7,500 11,856 14,368 2,000 3,493 7,350 11,756 14,043 2,500 3,493 7,350 11,706 14,068 3,000 3,418 7,525 10,387 12,900 3,500 3,368 7,475 9,812 12,825 4,000 3,318 7,325 9,512 12,100 4,500 3,293 7,250 9,462 11,481 5,000 3,243 7,275 9,462 11,531 5,500 3,025 7,475 9,712 11,456 6,000 2,975 7,200 9,562 11,506 6,500 2,900 6,900 9,537 11,531 7,000 2,825 6,756 9,512 12,300 7,500 2,725 6,706 9,512 11,481 8,000 2,650 6,681 9,487 11,481 8,500 2,550 5,612 8,718 11,481 9,000 2,475 5,612 8,793 11,181 9,500 2,425 5,512 8,768 11,131 10,000 2,375 5,437 8,768 11,106 10,500 2,375 5,362 8,818 11,131 11,000 2,150 5,162 8,693 11,031 11,500 2,150 5,137 8,693 11,081 12,000 2,075 5,037 8,643 11,006 12,500 2,025 5,012 8,643 10,587 13,000 1,950 4,837 8,618 10,462

表3.6 垂直置放蜂鳴片量測結果三 Resonant frequency (Hz) 累積注水量

(mm³) W01 W11 W21 W02

0 3,543 7,750 12,400 14,862 500 3,468 7,525 12,275 14,662 1,000 3,468 7,525 12,300 14,687 1,500 3,443 7,425 12,200 14,587 2,000 3,443 7,400 12,225 14,412 2,500 3,443 7,250 11,925 14,512 3,000 3,418 6,856 11,731 13,643 3,500 3,393 6,681 11,356 13,243 4,000 3,393 6,681 11,031 12,650 4,500 3,268 6,631 10,756 12,125 5,000 3,218 6,506 10,437 12,150 5,500 3,100 6,631 10,437 11,706 6,000 3,050 6,606 10,162 11,656 6,500 2,900 6,506 10,087 11,556 7,000 2,850 6,456 9,737 11,331 7,500 2,750 6,056 9,687 11,056 8,000 2,675 5,981 9,462 11,031 8,500 2,600 5,831 9,268 11,006 9,000 2,500 5,781 8,468 10,731 9,500 2,475 5,612 8,268 11,056 10,000 2,400 5,981 7,975 10,756 10,500 2,375 5,906 8,418 10,437

11,000 2,175 5,806 8,168 9,762 11,500 2,125 5,756 8,075 9,737 12,000 2,100 5,656 7,975 9,662 12,500 1,950 5,387 7,875 9,587 13,000 1,950 5,412 7,950 9,637

表3.7 水平置放蜂鳴片阻尼因子 ζ_r

W₀₁ W₁₁

液面高度

h (mm) 測試一測試二測試三測試一測試二測試三 0 0.00829 0.01137 0.01960 0.00821 0.00762 0.01824 1.105 0.00686 0.00825 0.00841 0.01464 0.00921 0.01667 2.210 0.00882 0.00873 0.00955 0.01649 0.01367 0.02386 3.316 0.01253 0.01588 0.01551 0.02163 0.01974 0.01615 4.421 0.01294 0.01022 0.01011 0.01607 0.02048 0.01494 5.526 0.00841 0.00875 0.00854 0.01667 0.02314 0.01687 6.631 0.00885 0.00849 0.00953 0.02204 0.02653 0.02168 7.737 0.00827 0.00875 0.00943 0.02983 0.02888 0.02257 8.842 0.01009 0.00786 0.00939 0.02742 0.01803 0.02181 9.947 0.00913 0.00875 0.01018 0.02756 0.01720 0.02125 11.052 0.00787 0.00877 0.00937 0.02573 0.01781 0.02231 12.158 0.00860 0.00960 0.00949 0.02538 0.01878 0.02437 13.263 0.00816 0.00927 0.00912 0.02475 0.01932 0.02847 14.368 0.00908 0.00883 0.00875 0.02483 0.01976 0.03065 15.473 0.00874 0.00879 0.00929 0.02502 0.02003 0.03146 16.579 0.01069 0.00919 0.00964 0.02498 0.02038 0.03220 17.684 0.00886 0.00960 0.00884 0.02405 0.02057 0.03242 18.789 0.00958 0.00829 0.00986 0.02349 0.02010 0.03234 19.894 0.00883 0.00797 0.01102 0.02338 0.02067 0.03255 21.000 0.00815 0.00793 0.01172 0.02242 0.02107 0.03254 22.105 0.00974 0.01009 0.01102 0.02344 0.02083 0.03210 23.210 0.00975 0.01110 0.01083 0.02285 0.02074 0.03236 24.315 0.01099 0.00860 0.01216 0.02361 0.02061 0.03234 25.421 0.01052 0.01033 0.01153 0.02316 0.02104 0.03265 26.526 0.01071 0.01105 0.01114 0.02332 0.02077 0.03283 27.631 0.01121 0.01055 0.01418 0.02329 0.02087 0.03314 28.736 0.01238 0.01096 0.01267 0.02307 0.02103 0.03272

表3.8 垂直置放蜂鳴片阻尼因子 ζ_r

W₀₁ W₁₁

液面高度

h (mm) 測試一測試二測試三測試一測試二測試三 0 0.00619 0.00718 0.00557 0.00667 0.00670 0.00747 1.898 0.00601 0.00667 0.00573 0.00762 0.00798 0.01322 2.657 0.00623 0.01156 0.00578 0.00951 0.00772 0.01414 4.025 0.00624 0.00616 0.00683 0.00980 0.01070 0.00827 4.917 0.00554 0.00718 0.00652 0.01575 0.00919 0.00791 5.752 0.00607 0.00694 0.00635 0.00959 0.01040 0.01280 6.548 0.00665 0.00838 0.00702 0.01003 0.01181 0.01301 7.314 0.00550 0.02143 0.00910 0.01217 0.01584 0.02447 8.059 0.00763 0.00741 0.00981 0.01254 0.02449 0.02023 8.786 0.01377 0.00645 0.00851 0.02190 0.01196 0.01235 9.501 0.01026 0.01279 0.00890 0.01824 0.01254 0.01426 10.207 0.00965 0.00941 0.00809 0.03156 0.01825 0.01284 10.907 0.01082 0.01212 0.00966 0.02074 0.02023 0.01454 11.603 0.00992 0.01239 0.00864 0.02008 0.02320 0.01767 12.297 0.01127 0.01295 0.00915 0.02128 0.02637 0.03557 12.993 0.01227 0.01398 0.00992 0.02089 0.02834 0.02167 13.692 0.01197 0.01241 0.01069 0.02049 0.01840 0.01677 14.397 0.01279 0.01372 0.00980 0.02007 0.02591 0.02244 15.110 0.01346 0.01481 0.01134 0.01526 0.03409 0.02491 15.836 0.01479 0.01580 0.01547 0.02482 0.03609 0.01846 16.578 0.01679 0.01731 0.01484 0.02422 0.03290 0.01554 17.340 0.01960 0.01785 0.02194 0.02209 0.03038 0.01552 18.131 0.01893 0.01731 0.01649 0.02288 0.03264 0.01910 18.960 0.02047 0.02548 0.01750 0.02490 0.04899 0.02125 19.843 0.01085 0.02893 0.01720 0.03697 0.06956 0.02440 20.808 0.03591 0.03008 0.01876 0.02522 0.06663 0.06238 21.922 0.04155 0.03914 0.01880 0.03112 0.02399 0.06810

表3.9 水平置放蜂鳴片比例阻尼常數

α β

液面高度

h (mm) 測試一測試二測試三測試一測試二測試三 0 248.128 433.936 626.215 0.24x10^-6 0.13x10^-6 0.49x10^-6 1.105 32.326 191.717 56.803 0.66x10^-6 0.33x10^-6 0.76x10^-6 2.210 55.849 113.598 -61.857 0.81x10^-6 0.62x10^-6 1.25x10^-6 3.316 104.277 290.493 345.184 1.12x10^-6 0.86x10^-6 0.62x10^-6 4.421 223.841 44.864 135.067 0.74x10^-6 1.13x10^-6 0.74x10^-6 5.526 38.356 -53.688 40.959 0.95x10^-6 1.41x10^-6 0.95x10^-6 6.631 -24.922 -107.070 4.380 1.35x10^-6 1.69x10^-6 1.30x10^-6 7.737 -156.179 -126.233 -8.995 1.99x10^-6 1.90x10^-6 1.39x10^-6 8.842 -50.257 9.190 6.950 1.77x10^-6 1.11x10^-6 1.35x10^-6 9.947 -73.141 53.053 45.628 1.82x10^-6 1.02x10^-6 1.28x10^-6 11.052 -79.378 49.175 12.467 1.73x10^-6 1.07x10^-6 1.39x10^-6 12.158 -43.563 65.305 -2.615 1.67x10^-6 1.12x10^-6 1.54x10^-6 13.263 -42.573 52.662 -52.676 1.63x10^-6 1.17x10^-6 1.85x10^-6 14.368 -11.057 38.303 -78.139 1.61x10^-6 1.22x10^-6 2.01x10^-6 15.473 -17.476 36.548 -62.117 1.63x10^-6 1.24x10^-6 2.05x10^-6 16.579 40.302 46.861 -53.547 1.56x10^-6 1.25x10^-6 2.09x10^-6 17.684 5.558 58.335 -68.466 1.54x10^-6 1.25x10^-6 2.12x10^-6 18.789 31.012 31.157 -36.754 1.49x10^-6 1.25x10^-6 2.08x10^-6 19.894 16.576 20.963 -4.135 1.49x10^-6 1.30x10^-6 2.06x10^-6 21.000 10.403 18.527 17.780 1.44x10^-6 1.33x10^-6 2.04x10^-6 22.105 42.963 74.236 7.862 1.47x10^-6 1.25x10^-6 2.02x10^-6 23.210 49.193 99.046 4.733 1.43x10^-6 1.22x10^-6 2.04x10^-6 24.315 73.785 43.745 38.554 1.45x10^-6 1.27x10^-6 2.00x10^-6 25.421 67.800 80.644 25.544 1.42x10^-6 1.26x10^-6 2.03x10^-6 26.526 71.702 98.770 17.558 1.43x10^-6 1.22x10^-6 2.06x10^-6 27.631 83.834 87.341 83.573 1.42x10^-6 1.24x10^-6 2.01x10^-6 28.736 110.893 95.642 55.966 1.37x10^-6 1.24x10^-6 2.01x10^-6

表 3.10 垂直置放蜂鳴片比例阻尼常數

α β

液面高度

h (mm) 測試一測試二測試三測試一測試二測試三 0 176.476 229.764 121.368 0.20x10^-6 0.18x10^-6 0.26x10^-6 1.898 139.295 167.378 -18.779 0.26x10^-6 0.26x10^-6 0.57x10^-6 2.657 98.756 449.162 -40.220 0.37x10^-6 0.13x10^-6 0.62x10^-6 4.025 88.622 59.650 165.223 0.39x10^-6 0.44x10^-6 0.28x10^-6 4.917 -88.526 159.754 157.249 0.70x10^-6 0.32x10^-6 0.27x10^-6 5.752 82.652 112.651 15.451 0.38x10^-6 0.40x10^-6 0.55x10^-6 6.548 100.992 153.101 31.084 0.40x10^-6 0.45x10^-6 0.59x10^-6 7.314 -20.279 760.650 -188.383 0.58x10^-6 0.33x10^-6 1.27x10^-6 8.059 83.837 -231.250 -25.334 0.54x10^-6 1.25x10^-6 0.98x10^-6 8.786 161.336 52.951 131.292 0.97x10^-6 0.50x10^-6 0.52x10^-6 9.501 100.661 363.446 124.865 0.78x10^-6 0.38x10^-6 0.57x10^-6 10.207 -198.634 56.648 115.578 1.55x10^-6 0.84x10^-6 0.53x10^-6 10.907 87.541 115.698 143.060 0.90x10^-6 0.96x10^-6 0.62x10^-6 11.603 69.344 116.094 33.909 0.89x10^-6 1.01x10^-6 0.84x10^-6 12.297 112.906 83.133 -310.938 0.92x10^-6 1.20x10^-6 1.99x10^-6 12.993 163.095 101.983 6.774 0.88x10^-6 1.29x10^-6 1.13x10^-6 13.692 160.657 202.498 134.046 0.86x10^-6 0.76x10^-6 0.80x10^-6 14.397 197.477 138.407 -9.128 0.83x10^-6 1.17x10^-6 1.23x10^-6 15.110 259.278 -11.793 14.726 0.66x10^-6 1.94x10^-6 1.38x10^-6 15.836 207.629 72.311 295.460 1.05x10^-6 1.76x10^-6 0.77x10^-6 16.578 282.162 171.123 308.412 0.98x10^-6 1.55x10^-6 0.61x10^-6 17.340 401.062 221.582 507.386 0.82x10^-6 1.40x10^-6 0.47x10^-6 18.131 349.303 172.738 295.740 0.89x10^-6 1.54x10^-6 0.83x10^-6 18.960 344.059 257.432 298.627 1.13x10^-6 2.37x10^-6 0.95x10^-6 19.843 -67.876 196.301 248.725 2.24x10^-6 3.28x10^-6 1.18x10^-6 20.808 725.908 252.884 -108.400 0.87x10^-6 3.17x10^-6 3.78x10^-6 21.922 802.940 859.358 -158.722 1.17x10^-6 0.65x10^-6 4.14x10^-6

表3.11 水平置放蜂鳴片不同液面高度之共振頻率 Resonant frequency (Hz)

液面高度

h (mm) W01 W11 W21 W02

0 3513.7 7726.4 12715 14961 2.210 2725.5 6027.5 9947.1 11705 4.421 2362.8 5368.3 9073.8 10669 6.631 2147.7 5055.6 8773.7 10227 9.947 1925.7 4842.4 8644.4 9895.8 13.263 1763.4 4770.3 8636.2 9723.7 16.579 1635.3 4738.2 8624 9583

19.894 1531.4 4725.2 8617.5 9474.2 28.736 1328.1 4720.1 8618.5 9273

表3.12 垂直置放蜂鳴片不同液面高度之共振頻率 Resonant frequency (Hz)

液面高度

h (mm) W₀₁ W₁₁ W₂₁ W₀₂ 0 3465.9 7593.7 12636 14844 2.657 3459.4 7540.6 12461 14655 4.917 3442.8 7321.9 11898 14377 6.548 3388.8 6933.9 11515 14240 8.786 3179.9 6363.8 10928 13517 10.907 2869.5 6116.3 10651 12460 12.993 2550.5 5954.7 10164 11649 15.110 2277.1 5483.6 9688.5 11266 20.808 1912.3 4941 8981.8 10536

表3.13 水平置放蜂鳴片加入阻尼效應之共振頻率 Resonant frequency (Hz)

液面高度

h (mm) W01 W11 W21 W02

0 3513.4 7725.9 12714 14959 2.210 2725.4 6026.6 9943.1 11699 4.421 2362.7 5367.6 9070.7 10664 6.631 2147.6 5054.3 8766.8 10216 9.947 1925.6 4841.4 8638.3 9886.7 13.263 1763.4 4769 8628.6 9712.9 16.579 1635.3 4736.7 8615.5 9571.3 19.894 1531.3 4723.9 8609.2 9463.2 28.736 1328 4718.7 8610.7 9263.4

表3.14 垂直置放蜂鳴片加入阻尼效應之共振頻率 Resonant frequency (Hz)

液面高度

h (mm) W₀₁ W₁₁ W₂₁ W₀₂ 0 3465.8 7593.5 12635 14843 2.657 3459.3 7540.2 12459 14652 4.917 3442.8 7321.5 11896 14374 6.548 3388.7 6933.4 11513 14236 8.786 3179.8 6363.1 10925 13512 10.907 2869.4 6115.4 10647 12453 12.993 2550.3 5953.3 10158 11639 15.110 2277 5482 9680.3 11253 20.808 1911.6 4935.9 8955.7 10495

表3.15 水平置放蜂鳴片加入阻尼效應之共振頻率變化頻率變化 ∆f_r/f_r (%)

液面高度

h (mm) W01 W11 W21 W02

0 -0.009% -0.006% -0.008% -0.013%

2.210 -0.004% -0.015% -0.040% -0.051%

4.421 -0.004% -0.013% -0.034% -0.047%

6.631 -0.005% -0.026% -0.079% -0.108%

9.947 -0.005% -0.021% -0.071% -0.092%

13.263 0.000% -0.027% -0.088% -0.111%

16.579 0.000% -0.032% -0.099% -0.122%

19.894 -0.007% -0.028% -0.096% -0.116%

28.736 -0.008% -0.030% -0.091% -0.104%

表3.16 垂直置放蜂鳴片加入阻尼效應之共振頻率變化頻率變化∆f_r/ f_r (%)

液面高度

h (mm) W₀₁ W₁₁ W₂₁ W₀₂

0 -0.003% -0.003% -0.008% -0.007%

2.657 -0.003% -0.005% -0.016% -0.020%

4.917 0.000% -0.005% -0.017% -0.021%

6.548 -0.003% -0.007% -0.017% -0.028%

8.786 -0.003% -0.011% -0.027% -0.037%

10.907 -0.003% -0.015% -0.038% -0.056%

12.993 -0.008% -0.024% -0.059% -0.086%

15.110 -0.004% -0.029% -0.085% -0.115%

20.808 -0.037% -0.103% -0.291% -0.389%

表4.1 水平置放節點施力 7N 不同液面高之共振頻率 Resonant frequency (Hz)

液面高度

h (mm) W01 W11 W21 W02

0 3,513 7,726 12,715 14,961 1 3,334 7,163 11,431 13,327 2 3,009 6,486 10,378 12,110 3 2,783 6,055 9,776 11,422 4 2,617 5,766 9,418 11,002 5 2,488 5,561 9,199 10,725 10 2,077 5,109 8,885 10,117 15 1,825 5,006 8,859 9,865 20 1,646 4,983 8,857 9,704 30 1,403 4,977 8,857 9,484

表4.2 垂直置放節點施力 7N 不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W₀₁ W₁₁

3 3,789 8,135

6 3,713 7,782

9 3,422 7,123

12 2,928 6,514

15 2,497 5,999

18 2,188 5,515

21 1,989 5,213

表4.3 厚度 1mm 不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W01 W11 W21 W02

3 3,462 7,614 12,530 14,698 6 3,439 7,502 12,238 14,507 9 3,358 7,308 11,986 14,295 12 3,231 7,132 11,782 13,766 15 3,113 6,948 11,522 13,364 18 2,937 6,568 10,914 12,788 21 3,020 6,635 10,904 12,775

表4.4 厚度 2mm 不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W₀₁ W₁₁ W₂₁ W₀₂

3 3,461 7,610 12,508 14,673 6 3,423 7,420 11,994 14,635 9 3,281 7,079 11,564 13,921 12 3,061 6,782 11,225 13,048 15 2,873 6,491 10,809 12,491 18 2,666 6,022 10,071 11,819 21 2,728 6,025 9,949 11,668

表4.5 厚度 3mm 不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W01 W11 W21 W02

3 3,461 7,609 12,503 14,668 6 3,417 7,383 11,885 14,306 9 3,237 6,951 11,338 13,727 12 2,957 6,574 10,928 12,679 15 2,719 6,221 10,426 12,041 18 2,488 5,700 9,620 11,328 21 2,528 5,653 9,433 11,084

表4.6 厚度 4mm 不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W01 W11 W21 W02

3 3,461 7,609 12,502 14,666 6 3,414 7,365 11,832 14,278 9 3,210 6,874 11,211 13,619 12 2,889 6,442 10,763 12,463 15 2,616 6,052 10,213 11,790 18 2,365 5,502 9,370 11,058 21 2,387 5,417 9,143 10,756

表4.7 厚度 5mm 不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W01 W11 W21 W02

3 3,461 7,608 12,502 14,666 6 3,413 7,355 11,806 14,264 9 3,193 6,825 11,137 13,556 12 2,841 6,354 10,668 12,324 15 2,544 5,704 10,090 11,638 18 2,277 5,372 9,225 10,896 21 2,283 5,261 8,974 10,557

表4.8 厚度 10mm 不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W₀₁ W₁₁ W₂₁ W₀₂

0.003 3,461 7,607 12,497 14,659 0.006 3,411 7,346 11,782 14,250 0.009 3,160 6,746 11,031 13,466 0.012 2,734 6,187 10,542 12,066 0.015 2,368 5,713 9,927 11,364 0.018 2,130 5,255 9,212 10,868 0.021 2,000 4,953 8,737 10,188

表4.9 厚度 15mm 不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W01 W11 W21 W02

3 3,461 7,608 12,502 14,666 6 3,411 7,346 11,782 14,250 9 3,156 6,736 11,020 13,455 12 2,708 6,156 10,530 12,015 15 2,313 5,665 9,914 11,299 18 1,970 5,061 9,003 10,529 21 1,876 4,888 8,712 10,078

表4.10 厚度 20mm 不同液面高之共振頻率 Resonant frequency (Hz) 液面高度

h (mm) W01 W11 W21 W02

3 3,461 7,608 12,502 14,666 6 3,411 7,346 11,782 14,250 9 3,155 6,735 11,019 13,454 12 2,702 6,150 10,528 12,003 15 2,296 5,653 9,912 11,279 18 1,937 5,045 8,999 10,501 21 1,819 4,871 8,706 10,030

表 4.11 不同液面高之共振頻率降低量 Resonant frequency change (Hz) 液面高度

h (mm) W01 W11 W21 W02

1.105 -479 -830 -953 -1319

2.210 -114 -581 -1319 -1552

3.316 -192 -367 -491 -577 4.421 -175 -310 -294 -367 5.526 -100 -175 -159 -241 6.631 -83 -109 -83 -186 7.737 -83 -100 -67 -142 8.842 -67 -58 -33 -75 9.947 -58 -33 -25 -91 11.052 -59 -42 -8 -75 12.158 -50 -17 0 -59 13.263 -50 -8 -9 -66 14.368 -33 -8 -8 -25 15.473 -38 -9 -8 -50 16.579 -56 -8 0 -34

17.684 -33 0 8 -50

18.789 -25 0 -8 -25

19.894 -25 0 8 -33

21.000 -34 0 0 -33

22.105 -25 0 -8 -34

23.210 -25 0 8 -33

24.315 -16 0 0 -17

25.421 -25 0 -8 -16

26.526 -25 0 8 -34

27.631 -17 0 -8 -25

28.736 -25 0 8 -16

0 2000 4000 6000 8000 10000 Frequency (Hz)

-60 -40 -20 0 20

Magnitude (dB)

Electricity Vibration

圖1.1 蜂鳴片一的頻率響應曲線[22]

0 2000 4000 6000 8000 10000 Frequency (Hz)

-60 -40 -20 0 20

Magnitude (dB)

vibration electricity Impedance

圖1.2 蜂鳴片二的頻率響應曲線[22]

3dB

3dB 3,881.25Hz

6,887.50Hz

3,821.88Hz 3,965.63Hz

6,762.50Hz 7,096.88Hz W₀₁

W02

0 2000 4000 6000 8000 10000 Frequency (Hz)

-60 -40 -20 0 20

Magnitude (dB)

Electricity Vibration

圖1.3 蜂鳴片三的頻率響應曲線[22]

0 2000 4000 6000 8000 10000 Frequency (Hz)

-80 -60 -40 -20 0

Magnitude (dB)

gap= 0 µm gap= 10 µm gap= 20 µm

圖1.4 流體負載之蜂鳴片一頻率響應曲線[22]

Set material property Build model

Mesh

Set analysis type and method Define loads and constraints Solving

General Postprocessor Launch ANSYS

Solution Preprocessor

Set element type

Solid5 Fluid80

Brass Water

PZT

Mesh size = 0.001m

Static

Modal

Read Results Result Summary

Plot Results

Reduced

圖2.2 SOLID5 的示意圖 X

圖2.4 壓電蜂鳴片結構示意圖

圖2.5 二維截面圖 hPZT

h r

r_PZT

J I G

H F

C B L

壓電材料黃銅

水 D

E M Z

圖2.6 建模完成圖

圖 2.7 網格切割完成圖

電性邊界固定邊界

0 Volt UR＝0

UR＝0 Uθ＝0 UZ＝0

1 2 3 4 Mode

0 4000 8000 12000 16000 20000

R es ona nt F re qu enc y ( H z)

圖2.9 元素尺寸的收斂性探討

1mm

0.3mm 0.4mm 0.5mm 0.8mm

1 2 3 4 Mode

4000 8000 12000 16000

R es ona nt F re qu enc y ( H z)

圖 2.10 不同壓電材料的蜂鳴片共振頻率比較圖

：PZT-4

：PZT-5A

：PZT-5H

：PZT-7A

：PZT-8

：實際值

圖2.11 碟型蜂鳴片 W₀₁共振模態

圖 2.12 碟型蜂鳴片 W11共振模態

圖2.13 碟型蜂鳴片 W₂₁共振模態

圖2.14 碟型蜂鳴片 W02共振模態

圖 2.15 W21模態振型節徑與節圓示意圖

圖2.16 蜂鳴片水平置放於圓形容器底部試驗裝置

注水裝置

節徑節圓

圖2.17 蜂鳴片垂直置放於圓形容器左側

圖2.18 模擬水平置放蜂鳴片之 9 種液面高度

10mm

5mm

5mm 5mm

1mm 1mm 1mm 1mm 1mm

水平置放蜂鳴片

0 10 20 30 Depth h (mm)

0 4000 8000 12000 16000

Resonant Frequency (Hz)

W₀₁ W₁₁ W₂₁ W₀₂

圖2.19 水平置放蜂鳴片不同液面高之共振頻率模擬值

圖 2.21 水平置放蜂鳴片與水耦合之 W11振型圖

圖 2.22 水平置放蜂鳴片與水耦合之 W 振型圖

圖 2.23 水平置放蜂鳴片與水耦合之 W₀₂振型圖

圖2.24 模擬垂直置放蜂鳴片之 7 種液面高度垂直置放蜂鳴片

3mm 3mm 3mm 3mm 3mm 3mm 3mm

圖 2.25 垂直置放蜂鳴片之邊界條件設定圖液體表面

z＝0

0 4 8 12 16 20 24 Depth h (mm)

0 4000 8000 12000 16000

Resonant Frequency (Hz)

W₀₁ W₁₁ W₂₁ W₀₂

圖2.26 垂直置放蜂鳴片不同液面高之共振頻率模擬值

圖 2.27 垂直置放蜂鳴片與水耦合之 W 振型圖

圖 2.28 垂直置放蜂鳴片與水耦合之 W₁₁振型圖

圖 2.30 垂直置放蜂鳴片與水耦合之 W02振型圖

0 4000 8000 12000 16000

Frequency (Hz) -80

-60 -40 -20 0

Magnitude (dB)

：in air

：in water

圖3.2 壓電蜂鳴片夾具示意圖

r_out r_in

r_m

圖 3.3 實驗容器示意圖 r_out r_in

h_s t

r_m

圖3.4 電極塗佈完成

圖3.6 蜂鳴片夾置完成上視圖

3.7 磁性定位分注器

圖3.8 實驗架構

Response Exciting

Input Output

0 4000 8000 12000 16000 20000 Resonant Frequency (Hz)

-80 -60 -40 -20 0

Magnitude (dB)

圖3.10 SigLab 頻率響應圖

0 10 20 30

Depth h (mm) 0

4000 8000 12000 16000

Resonant Frequency (Hz)

W₀₁ W₁₁ W₂₁ W₀₂

圖3.11 水平置放蜂鳴片共振頻率

：試驗一

：試驗二

：試驗三

0 10 20 30 Depth h (mm)

0 4000 8000 12000 16000

Resonant Frequency (Hz)

W₀₁ W₁₁ W₂₁ W₀₂

圖3.12 水平置放蜂鳴片共振頻率平均值

0 4000 8000 12000 16000 20000 Resonant Frequency (Hz)

-100 -80 -60 -40 -20 0

Magnitude (dB)

0 4000 8000 12000 16000 20000 Resonant Frequency (Hz)

-100 -80 -60 -40 -20 0

Magnitude (dB)

圖3.14 液面高 8.059mm 垂直置放蜂鳴片頻率響應曲線

0 4000 8000 12000 16000 20000 Resonant Frequency (Hz)

-100 -80 -60 -40 -20 0

Magnitude (dB)

0 5 10 15 20 25 Depth h (mm)

0 4000 8000 12000 16000

Resonant Frequency (Hz)

W₀₁ W₁₁ W₂₁ W₀₂

圖3.16 垂直置放蜂鳴片共振頻率

0 5 10 15 20 25

Depth h (mm) 0

4000 8000 12000 16000

Resonant Frequency (Hz)

W₀₁ W₁₁ W₂₁ W₀₂

圖3.17 垂直置放蜂鳴片共振頻率平均值

：試驗一

：試驗二

：試驗三

圖3.18 SDOF 的頻率響應函數圖

0 2000 4000 6000 8000 10000

Resonant Frequency (Hz)

-80

0 10 20 30 Depth h (mm)

0.004 0.008 0.012 0.016 0.02

Damping factor

圖 3.20 水平置放蜂鳴片 W₀₁模態阻尼因子

0 10 20 30

Depth h (mm) 0

0.01 0.02 0.03 0.04

Damping factor

圖 3.21 水平置放蜂鳴片 W₁₁模態阻尼因子

：試驗一

：試驗二

：試驗三

：平均值

：試驗一

：試驗二

：試驗三

：平均值

0 5 10 15 20 25 Depth h (mm)

0 0.01 0.02 0.03 0.04 0.05

Damping factor

圖 3.22 垂直置放蜂鳴片 W₀₁模態阻尼因子

0 5 10 15 20 25

Depth h (mm) 0

0.02 0.04 0.06 0.08

Damping factor

圖 3.23 垂直置放蜂鳴片 W₁₁模態阻尼因子

：試驗一

：試驗二

：試驗三

：平均值

：試驗一

：試驗二

：試驗三

：平均值

0 10 20 30 Depth h (mm)

-200 0 200 400 600 800

Mass damping α (rad/s)

圖 3.24 水平置放蜂鳴片質量阻尼常數

0 10 20 30

Depth h (mm) 0

4 8 12 16 20 24

S tif fn es s d am pin g β (x 10

-7

s/ ra d)

：試驗一

：試驗二

：試驗三

：平均值

：試驗一

：試驗二

：試驗三

：平均值

0 5 10 15 20 25 Depth h (mm)

-400 0 400 800 1200

Mass damping α (rad/s)

圖 3.26 垂直置放蜂鳴片質量阻尼常數

0 5 10 15 20 25

Depth h (mm) 0

10 20 30 40 50

S tif fn es s d am pin g β (x 10

-7

s/ ra d)

圖 3.27 垂直置放蜂鳴片勁度阻尼常數

：試驗一

：試驗二

：試驗三

：平均值

：試驗一

：試驗二

：試驗三

：平均值

0 10 20 30 Depth h (mm)

-0.0016 -0.0012 -0.0008 -0.0004 0

Resonant frequency change rate (%)

W₀₁ W₁₁

W₂₁

W₀₂

圖3.28 蜂鳴片水平置放時阻尼效應之共振頻率變化

0 5 10 15 20 25

Depth h (mm) -0.004

-0.003 -0.002 -0.001 0

Resonant frequency change rate (%)

W₀₁ W₁₁

W₂₁ W₀₂

圖3.29 蜂鳴片垂直置放時阻尼效應之共振頻率變化

0 10 20 30 Depth h (mm)

0 4000 8000 12000 16000

Resonant Frequency (Hz)

W₀₁ W₁₁ W₂₁ W₀₂

圖 4.1 蜂鳴片水平置放之實驗與模擬比較

：模擬值

：實驗值

電性邊界

固定邊界

0 Volt UX＝0

UY＝0 UZ＝0 預拉力

0 10 20 30 Depth h (mm)

8000 9000 10000 11000 12000 13000

Resonant Frequency (Hz)

圖4.3 預應力模擬結果與實驗值比較

0 10 20 30

Depth h (mm) 0

4000 8000 12000 16000

Resonant Frequency (Hz)

W₀₁ W₁₁ W₂₁ W₀₂

圖4.4 節點施力 7N 模擬結果與實驗值比較

：σr＝66.36 MPa

：σ_r＝74.27 MPa

：σ_r＝84.88 MPa

：實驗值

：σ_r＝74.27 MPa

：實驗值

0 5 10 15 20 25 Depth h (mm)

0 2000 4000 6000 8000

Resonant Frequency (Hz)

W₀₁ W₁₁

圖 4.5 水厚度 30mm 蜂鳴片垂直置放之實驗值與模擬結果

0 5 10 15 20 25

Depth h (mm) 0

2000 4000 6000 8000 10000

Resonant Frequency (Hz)

W₀₁ W₁₁

圖 4.6 水厚度 30mm 蜂鳴片預應力模擬結果與實驗值

在文檔中液體耦合圓形壓電層板之振動分析 (頁 46-125)

第五章 結論與展望

5.2 未來展望

參考文獻

Electricity Vibration

1 2 3 4 Mode

0 4000 8000 12000 16000 20000

R es ona nt F re qu enc y ( H z)

1 2 3 4 Mode

4000 8000 12000 16000

R es ona nt F re qu enc y ( H z)

0 2000 4000 6000 8000 10000

Resonant Frequency (Hz)

-80

0 10 20 30

Depth h (mm) 0

4 8 12 16 20 24

S tif fn es s d am pin g β (x 10

s/ ra d)

0 5 10 15 20 25

Depth h (mm) 0

10 20 30 40 50

S tif fn es s d am pin g β (x 10

s/ ra d)

第五章結論與展望