以希爾伯特轉換成本函數分析之最佳化高模態傳遞波及其在多方向二維壓電平板致動器之應用

Abstract

本研究旨在開發一種以高模態疊加產生行進波以驅動自走式二維壓電平板致動器，驅動特定壓電片使結構激發出不同模態，使其在平面上達成多方向運動之計畫目標。本研究以不鏽鋼薄板作為基板，以四塊尺寸相同之PZT作為致動器，邊界夾具以V形鋁材溝槽製作，用以卡住不鏽鋼薄板，以模擬簡支端的邊界條件，並透過理論模型求得解析解來進行數值模擬以進行最佳化之設計。本研究透過壓電片位置設計，使驅動不同壓電片組合可以產生不同方向之行進波，增加行進波方向的可能性。在驅動方法上，使用高模態疊加行進波，大幅提升速度表現，並可於結構上產生兩個波峰處來推動致動器。本研究有別於以往直接驅動在共振頻上的驅動方式，而是採用整數倍頻驅動方法，在共振頻附近選擇兩個呈整數倍關係的頻率疊加，使行進波具有週期性，提升其穩定度。本研究亦同時利用希爾伯特轉換理論進行驅動參數最佳化，並對其定量方法設計出三種成本函數，在數值模擬的部分透過調變驅動訊號間電壓比及項位差，觀察成本函數與行進波效率之間的關係，接著利用有限元素模擬驗證此分析方法其可行性。在實驗上以雷射測振儀驗證了理論與數值模擬的正確性，最後透過給予不同輸入總電壓及荷重，觀察致動器之行走軌跡及移動速度，藉此分析其驅動效率。由本研究所開發的理論模型及成本函數分析方法所開發之二維壓電平板致動器，在結構本體重量16g荷重下，在+x及－x方向上，驅動頻率選擇1855赫茲及7420赫茲，其最大速度分別可達95.5 mm/s (60Vpp)及102.92 mm/s (60Vpp)；在+y及－y方向上，驅動頻率選擇1775赫茲及7100赫茲，其最大速度分別可達133.34 mm/s (60Vpp)及132.42 mm/s (50Vpp)。最後將致動器置於軌道上，在總電壓為100Vpp無外加負載情況下最大速度可達177.35 mm/s，而在70g荷重時，最高速度可達247.2 mm/s，為之前驅動方法的36倍，載重能力提升約4倍之多，驗證本研究所提出的方法具有較佳之驅動效率。

The aim of this study is to develop a self-propelled two-dimensional piezoelectric actuator driven by traveling waves generated by the combination of high-order bending modes. The actuator structure can be excited to move in different directions by activating four different piezoelectric sheets. The base plate of this actuator is a stainless-steel sheet, and four piezoelectric sheets of identical size are attached to its surface. Aluminum fixtures with V grooves are used to simulate simply support boundary conditions. An analytical solution was developed to describe the traveling waves and assist in the design of the two-dimensional piezoelectric actuator. In addition, the Hilbert transform is used to optimize the driving parameters, and three cost functions are designed to quantify the efficiency of generated traveling waves. Applying the developed analytical solution and the cost functions, traveling waves generated by higher-order modes can be generated. The concept of the two-integer-frequency-two-mode (TIF-TM) method is adopted, and two frequencies near the resonant frequency that are in an integer multiple relationship are selected. It creates a periodic and stable traveling wave in x- or y-direction. Furthermore, the experimental findings demonstrate that the driving efficiency is significantly improved comparing to previously reported methods. For a 16g loading (the weight of the motor), the maximum traveling speed in +x and -x directions are 95.5 mm/s (60Vpp) and 102.92 mm/s (60Vpp). The maximum traveling speed in +y and -y directions are 133.34 mm/s (60Vpp) and 132.42 mm/s (50Vpp). The moving speed can reach 177.35 mm/s without applied load when placing the motor on a track. The moving speed can reach 247.2 mm/s with a 70g load. It is about 36 times faster than previous reported methods. The maximum loading is 180g, and it is about 4 times higher then previously reported methods. These results suggest that the present driving method is more efficient and advanced.