以希爾伯特轉換最佳化多方向傳遞波驅動的二維壓電平板致動器

Abstract

本研究宗旨在於開發以行進波驅動之二維壓電平板致動器系統，並控制其在平面上可達成多方向之運動。在結構設計上，本研究使用壓電材料與不鏽鋼薄板之複合結構，使用PZT作為致動器，並以不鏽鋼薄板經由板金加工後來模擬簡支端的邊界條件，在此邊界條件下有利於數學模型的建立，並可求得簡單的解析解，來完成數值模擬的部分。從結構的設計上，本研究僅使用一個壓電致動器就可以達成在多方向的運動控制，具有製造成本低與可控性高的特點。在驅動方法上，本研究採用整數倍頻驅動方法，並非直接驅動在共振頻率上，而是透過靠近共振頻率的兩個呈整數倍關係的頻率疊加，使得產生的行進波形具有週期性，也提升其穩定性。此一驅動方法也藉由有限元素模擬驗證其可行性。由於驅動頻率和結構共振頻率間存在著相位差，因此本研究也透過實驗量測的結果，利用等效電路(equivalent circuit)與波德圖(Bode plot)計算並補償其相位差值。 本研究引用了希爾伯特轉換理論進行最佳化參數設計，並對產生的行進波進行定量分析，評估行進波的效率；在數值模擬的部分透過調控兩輸入訊號間之電壓比值與相位差值，探討輸入的電壓比值與相位差值對於產生行進波的影響，分別在結構之x方向與y方向上產生穩定的行進波；在實驗上以雷射測振儀也驗證了理論與數值模擬的正確性，最後透過不同的輸入電壓大小與荷重實驗，量測系統之運動速度，來分析其驅動效率。本研究開發之二維壓電平板致動器在x方向上，輸入電壓比為72V：108V下，運動速度最高可達3.45 mm/s，而其可負載至最大荷重為26.2g，加上整體結構的重量共為43.6g；而在y方向上，輸入電壓比為64.3V：115.7V下，運動速度最高可達5.50 mm/s，其可負載至最大荷重為30.2g，加上整體結構的重量共為47.6g，驗證了此二維壓電平板致動器的效能。

The aim of this work is to develop a planar type two-dimensional piezoelectric actuator system driven by traveling waves and to control its multi-direction planar motion. This piezoelectric actuator is composed of one 45mm by 31.8mm by 0.2mm PZT plate and a 50mm by 40mm by 10mm stainless steel box made of 0.5 mm thickness shim. The PZT was served as an actuator mounted on top of a bent stainless steel on four edges of a rectangular plate to simulate simply supported boundary condition on all edges. To use a single PZT sheet to drive the piezoelectric actuator moving multi-directionally, a two-dimensional analytical model was developed, and numerical and finite element analyses are used to assist the design for optimization. To achieve multi-directional motorization, a two-integer-frequency two-mode (TIF-TM) method was adopted in this study. By setting the driving frequencies of the activated modes to be an integer multiplier, we could stabilize the period of the generated traveling waves and thus increase the propelling efficiency. In order to verify the feasiblilty of TIF-TM method, FEM simulation was applied. Since the driving frequencies deviated from the corresponding resonant frequency, phase compensation calculated by Bode plot was added to minimize the phase lag. It is verified that one piezoelectric actuator is sufficient to create multi-dimensional locomotion, which is superior to previously reported methods that all need more than one piezoelectric sheet. This design has advantages of low manufacturing cost and high controllability. Furthermore, optimization was approached using Hilbert transform, where the voltage ratio and phase difference of the driving signal are optimized. The contribution of driving voltage ratio and phase difference to generating a traveling wave in both x- and y-directions are analyzed and discussed by using numerical sumulation. It is also experimentally verified by measuring the vibrating profile of the planar type two-dimensional piezoelectric actuator system. Last but not least, the velocity test was done under different input voltage and preloads. It is verified that the moving speed can reach 3.45 mm/s in x-direcion under the condition of 72V：108V voltage ratio, and its maximum loading is 43.6g. On the other hand, the maximum moving speed in y-direction is 5.50 mm/s, and its maximun loading is 47.6g.