流體致振壓電能量擷取系統之數值開發與風洞實驗

Numerical Calculation and Experimental Measurement of Vortex-Induced Vibration for Piezoelectric Energy Harvester

洪國勛(Guo-Hsun Hung)

指導教授 ： 黃育熙

國立臺灣大學/工學院/機械工程學研究所/碩士(2022年)

https://doi.org/10.6342/NTU202201530

摘要

本論文利用平板疊加法理論與晶格波茲曼法的流場模擬，探討壓電陶瓷雙晶片於單邊固定之邊界下受渦流激振之振動特性，並以有限元素法模擬與實驗量測進行驗證。平板理論解析首先以線性壓電理論結合力量與力矩平衡條件式，將壓電片三層的結構等效為單層矩形平板之參數，使用疊加法將平板拆成四個結構進行疊加，其特性能使四個邊界條件自動滿足，而剩下的四個邊界條件則利用正交函數展開得到壓電平板在單邊固定邊界下的共振頻率與模態振形，同時與有限元素法之計算結果進行比較，驗證理論分析之準確性。
使用晶格波茲曼法進行二維圓柱繞流之升力係數驗證，進一步以二維與三維模型模擬圓柱複合平板置於流場所產生的渦街現象，計算流體對平板的激振力，接著利用平板模態之正交特性代入流體作用力求出暫態面外位移與電壓響應。以有限元素法進行單向耦合及雙向耦合的振動分析，並與晶格波茲曼法結合平板理論的結果進行比較，從模擬結果發現單向耦合僅能觀察出平板的第一共振頻率，雙向耦合能觀察到平板第一共振頻率以及流場激發的耦合頻率，晶格波茲曼法結合平板理論也能同時觀察到流體頻率與平板第一共振頻率，且平板振動的位移結果與雙向耦合較相似。而使用晶格波茲曼法可配合圖形處理器(GPU)進行加速運算，其計算速度可比使用前加快2至3倍，並且進行流固耦合模擬的時間與有限元素法軟體相比具有優勢。
實驗部分以電子斑點干涉術針對壓電平板進行即時動態量測，紀錄模態振形與共振頻率，結合穩態之雷射都卜勒振動儀量測結果進行比較，共振頻率與模態振形皆與理論、有限元素法符合。風洞實驗首先以量測的管路阻力曲線與廠商提供之風機性能曲線進行風機的更換，提高風洞流速至14 m/s。以熱線風速計量測圓柱複合壓電平板在不同流速下造成的渦街頻率變化，利用雷射都卜勒振動儀及示波器同時量測置於風洞的壓電平板面外位移與電壓，與本論文開發之晶格波茲曼法結合平板理論所計算之面外位移與電壓具有良好的對應性。

關鍵字

壓電平板 ； 疊加法 ； 自然頻率 ； 模態振形 ； 晶格波茲曼法 ； 流固耦合 ； 單向耦合 ； 雙向耦合 ； 渦流致振 ； 能量擷取系統 ； 風洞量測

並列摘要

This research analyzed the out-of-plate vibration and voltage of piezoelectric ceramic bimorphs in the cantilevered boundary condition, which is based on theoretical analysis of thin plate theory, flow field simulation with lattice Boltzmann method (LBM), and experimental measurements. The vibration of the plate is derived from four structures by the superposition method to satisfy the plate's boundary conditions. The resonant frequency and mode shape of the piezoelectric plate under the cantilevered boundary condition are obtained by theoretical analysis and verified with the finite element method (FEM) results.
LBM numerically calculates the vortex-induced vibration through a cylinder. The phenomenon in vortex street produced by the cylindrical composite plate in the flow field is generated by two-dimensional and three-dimensional LBM models. The results of lift coefficient and Strouhal number in LBM are verified with the FEM results from fluid-structure interaction by one-way and two-way coupling. It is presented from the FEM results that the one-way coupling can only obtain the first resonant frequency from vortex street in correspondence with the LBM. The two-way coupling can be obtained from both the plate's first resonant frequency and the flow field's excitation frequency. In consideration of the stiffness of the piezoelectric plate, the accuracy of vortex-street frequency is good enough in one-way coupling interaction to calculate the fluid speed generated in the correspondent natural frequency of the plate. The lift extracted from LBM is also input into the transient response of the piezoelectric orthogonally plate, to obtain the vibrating displacement and electrical voltage. In the study, LBM was developed in a corporation with the graphics processing unit (GPU) to accelerate the calculation, and the computing speed is 2 to 3 times faster than before architecture. The calculation time for LBM combined with the thin plate theory is superior to the FEM.
The experimental techniques are used to verify the results from the calculation. The electronic speckle pattern interferometry (ESPI) is used to perform real-time dynamic measurement on the piezoelectric plate and record the mode shape and resonant frequency. The result is compared with the measurement of the laser Doppler vibrometer (LDV), which indicate that the resonant frequency and mode shape are consistent with the theoretical solution and the FEM calculation. In the wind tunnel, the fan was replaced according to the measurement of the pipeline resistance curve and fan performance curve. The result of the replacement shows that the flow rate in the wind tunnel was increased to 14 m/s in the achievement of vortex-induced vibration for a piezoelectric plate. The hot wire anemometer measures the vortex-street frequency of the cylindrical composite piezoelectric plate at various flow rates. The vibrating displacement obtained by LDV is used simultaneously to measure the correspondent voltage of the piezoelectric plate in the wind tunnel. It is concluded that the best efficiency in electromechanical coupling transformation can be calculated by LBM combined with thin plate theory for piezoelectric energy harvester in vortex-induced vibration. The results from theoretical analysis and numerical calculation have good agreement with the experimental measurement.

並列關鍵字

piezoelectric plate ； superposition method ； natural frequency ； mode shape ； lattice Boltzmann method (LBM) ； fluid-structure interaction ； one-way coupled system ； two-way-coupled system ； vortex-induced vibration ； energy harvesting system ； wind tunnel

參考文獻

[1]W. Ritz, “Theorie der Transversalschwingungen einer quadratischen Platte mit freien Randern.”, Annalen der Physik, vol. 333(4), pp. 737-786, 1909.

[2]D. Young, “Vibration of rectangular plates by the Ritz method.”, Journal of Applied Mechanics-Transactions of the ASME, vol. 17(4), pp. 448-453,1950.

[3]G. B. Warburtion, “The vibration of rectangular plates.” Proceedings of the Institution of Mechanical Engineers, vol. 168(1), pp. 371-384,1954.

[4]A. W. Leissa, “Vibration of plates, NASA SP-160.” US Washington. W. Leissa. J, Sound and Vibration, vol. 56, pp. 313, 1978.

[5]D. J. Gorman, “Free vibration analysis of cantilever plates by the method of superposition”, Journal of Sound and Vibration, vol. 49, pp.453-467, 1976.