寬頻激振下基於非線性振動控制(SSD)及壓電振能擷取(SSH)技術的數學建模

Abstract

在環保意識抬頭下，更換和廢棄電池的處理成為一個棘手問題。因此，替代能源 （太陽能、熱能、動能等）在近年來廣受人們的重視。其中又以振動能較不受環境因素影響以及在振幅和頻寬上較易做調整而廣受注目。在多種的振動換能器中，壓電型換能器利用其特性使機械振動能直接轉換電能輸出。壓電材料具有高功率密度及容易整合的特性，因此壓電振能擷取的研究在國際上被廣為討論。 法國學者 Richard 在1998年，提出了非線性介面電路應用於振動控制並於2005年擴展為振能擷取介面電路。非線性介面電路中包含了一個同步切換開關，此開關在振動達到極值時關閉，使得壓電元件的跨壓與振動速度間的相位差減到最低並提升振動能的轉換，因而提升振動抑制的效果。經過壓電材料轉換後的能量可透過介面電路及外加負載做擷取及利用，因此非線性技術應用於振動能的擷取使擷取效率大為提升。過去的研究多針對於單一頻率振動源做討論，然而，在實際應用中，振動能的來源大多為寬頻且隨機，而此振動來源使得非線性技術的效能評估更加困難。 本文的目的在於建立一個可應用於寬頻振動源的數學模型。透過此模型，非線性技術的效能能夠直接被評估。運用頻域自採樣和自疊頻的概念，此模型建立於頻率域，因此相對於傳統的時域遞回分析，本文提供了一個可直接表示系統響應以及包含多頻訊息的數學模型。此外，本模型考慮了開關切換延遲所帶來的效果並且可被應用於開關頻率不同於兩倍振動子自然頻率的狀況下。本文將數學模型的建立分成兩部分，首先對振動控制技術做討論，然後擴展到能量擷取技術。在每一部份又對位移輸入及力（加速度）輸入分別進行討論。在數學模型建立後，本文也對多個常見振動形式做理論分析並將運用理論分析（針對不同振動形式對數學模型做簡化）及寬頻數學模型的模擬結果與運用時域分析的結果做比較。利用本數學模型，在單頻激力輸入下的模擬結果與文獻中的實驗數據結果相符。 利用寬頻數學模型及理論分析所得到的系統響應與時域遞回分析的結果相當符合，因此驗證了本文所提出的寬頻數學模型的有效性。此外，在理論分析的過程中也對幾種頻域分析時會有的現象，譬如Gibbs現象，海森堡的測不準原理和時域混疊進行了討論。此外，在模擬結果中顯示，開關切換的延遲若在特定範圍內對振能的擷取影響不大。

The decrease in consumption of electronic components has allowed the growth of mobile wireless applications and with this rapid growth, the replacement and disposal of battery have become a problem. Therefore, alternative power sources from ambient environment recently grasped people's interest. The basic idea of energy harvesting consists of converting a given source to a more useful form of energy. Among the numerous available energy sources, piezoelectric materials that convert vibrational energy into electrical energy received much attention as low-level mechanical vibrations are available in many environments and as piezoelectric transducers allow the direct conversion of vibrations into electricity. In addition, piezoelectric materials feature high power density and have promising integration potentials. Recently, nonlinear techniques were proposed for vibration control and extended as an energy conversion interface in order to increase the efficiency of power harvesting using piezoelectric materials. The nonlinear interfaces consists in a switch device connected with the piezoelectric element. The basic concept of using nonlinear techniques is to turn on the switch when the vibration gets its maximum and minimum values and so the voltage would be inverted to hold the magnitude. The switching frequency depends on the electromechanical structural response and the input excitation. The nonlinear process therefore induces an additional piecewise voltage which could be seen as a dry friction effect on the system and leads to a vibration damping process. In a electromechanical point of view, the induced damping effect is explained by the increase of the converted energy by the piezoelectric element. The converted energy could be harvested and accordingly, the nonlinear technique has also been used in energy harvesting system, showing bright performance in monochromatic excitation. However, in practical application, the excitation would be broadband and random rather than single frequency and so the performance when using nonlinear techniques would be more complicated to evaluate. The purpose of this work is to construct a broadband modeling using the concept of frequency-domain self-sampling and self-aliasing and permit containing more frequency information. The broadband modeling is firstly discussed with vibration control techniques as an introductory section and then extended to energy harvesting techniques. The modeling is separated into two parts: displacement input and force excitation. With this broadband modeling, the systematic performance could be described directly instead of the classic recursive time-domain analysis considering a switching delay and with a switching frequency which could be other than $2omega_0$. The broadband modeling is also analyzed with several well-known excitation cases as theoretical analysis and the simulation results based on theoretical analysis and broadband modeling are then compared with the time-domain resolution (classic time-domain analysis) to show its effectiveness. The prediction of harvested power under a monochromatic force excitation is also compared with the experimental results in previous literature. Simulation results from theoretical analysis and numerical calculation based on broadband modeling match well with the time-domain resolution considering different excitations and the broadband modeling is validated to be effective. In addition, through the result of theoretical analysis, several phenomena due to the frequency-domain analysis like Gibbs phenomenon, Heisenberg's uncertainty principle and time-domain aliasing are discussed. From simulation results, the effect of switching delay is shown to be limited within a specific range of switching delay.