李偉 楊曉東 張偉
摘要: 研究了一種末端帶有檢測(cè)質(zhì)量塊的單軸靜電懸臂梁式微陀螺儀。檢測(cè)質(zhì)量塊受到兩個(gè)固定電極的耦合作用,這兩個(gè)電極均連接直流電壓以使質(zhì)量塊產(chǎn)生較大的靜態(tài)變形。在驅(qū)動(dòng)方向上的電極還受到交流電壓的作用,驅(qū)動(dòng)質(zhì)量塊產(chǎn)生主振動(dòng)。當(dāng)有旋轉(zhuǎn)發(fā)生時(shí),在垂直于主振動(dòng)的敏感方向上質(zhì)量塊會(huì)受到科氏力而產(chǎn)生1個(gè)二次振動(dòng),通過(guò)測(cè)得二次振動(dòng)的幅值大小便可以測(cè)得角速度。首先,依據(jù)Hamilton原理建立了振梁微陀螺儀的控制方程,研究了旋轉(zhuǎn)懸臂梁的兩個(gè)橫向耦合振動(dòng)。其次,分析了多種參數(shù)對(duì)微陀螺儀靜態(tài)變形的影響,并求得了系統(tǒng)前2階固有頻率。研究發(fā)現(xiàn)不同參數(shù)對(duì)系統(tǒng)固有頻率的影響規(guī)律,并討論了系統(tǒng)驅(qū)動(dòng)和敏感方向上的動(dòng)力學(xué)放大效應(yīng)及其校正曲線。
關(guān)鍵詞: 振動(dòng)微陀螺儀; 懸臂梁; 動(dòng)態(tài)分析; 固有頻率; 校正曲線
中圖分類號(hào): O326;TB123 ?文獻(xiàn)標(biāo)志碼: A ?文章編號(hào): 1004-4523(2020)04-0742-08
DOI:10.16385/j.cnki.issn.1004-4523.2020.04.012
引 言
近年來(lái),微機(jī)械系統(tǒng)在制造成本、批量生產(chǎn)、重量、尺寸、耐久性、能耗和與集成電路兼容方面的優(yōu)良性能使得它在測(cè)試和制造新設(shè)備方面具有廣闊的發(fā)展前景[1]。微機(jī)械設(shè)備如微型泵、微鏡、麥克風(fēng)等微型諧振器,隨機(jī)存儲(chǔ)器、微型機(jī)器人、超靈敏傳感器、微陀螺儀,在設(shè)備通信中的高頻操作和快速切換網(wǎng)絡(luò)方面具有很多種類的應(yīng)用[2]。其中微陀螺儀廣泛的存在于工程系統(tǒng)中,如相機(jī)、飛行器、汽車和衛(wèi)星,用于跟蹤它們的方位并且控制它們的路徑。由于微陀螺儀復(fù)雜的動(dòng)力學(xué)特性和極小的檢測(cè)信號(hào)使得其成為微機(jī)械加工中最具挑戰(zhàn)性的器件之一[3]。
微型陀螺儀具有多種類型的驅(qū)動(dòng)和檢測(cè)原理,如靜電或壓電等方式。振梁微陀螺儀是基于振動(dòng)結(jié)構(gòu)兩種模式之間的能量交換而工作的。通過(guò)研究?jī)啥撕?jiǎn)支旋轉(zhuǎn)梁,Yang和Fang[4]建立了壓電振梁陀螺儀的運(yùn)動(dòng)方程,研究了不同的幾何和物理參數(shù)對(duì)電壓敏感性的影響。利用Hamilton原理對(duì)末端帶有質(zhì)量塊的懸臂梁動(dòng)力學(xué)建模,Bhadbhade等[5]提出了一種新的振動(dòng)-扭轉(zhuǎn)型振梁陀螺儀并研究了其陀螺效應(yīng)。為建立微陀螺的頻率方程,Esmaeili等[6]提出了一個(gè)通用的建??蚣?,該框架被模型化為受到一般基座激勵(lì)下末端有質(zhì)量塊的懸臂梁結(jié)構(gòu)。利用此頻差法的思想,Ghommem和Abdelkefi[7]對(duì)納米晶材料頻差陀螺儀進(jìn)行了性能分析。Ghayesh等[8]針對(duì)靜電振動(dòng)微陀螺儀應(yīng)用修正偶應(yīng)力理論,研究了微陀螺儀尺寸效應(yīng)相關(guān)的動(dòng)力學(xué)性能。
在靜電微陀螺儀實(shí)際工作中,當(dāng)微梁末端質(zhì)量塊兩端的固定電極上加載的電壓等于或者大于其臨界電壓時(shí),臨界靜電力可將質(zhì)量塊在很短的時(shí)間內(nèi)吸合,其中恢復(fù)力不能抵抗電容力導(dǎo)致電極相互坍塌,從而產(chǎn)生吸合失穩(wěn)。針對(duì)靜電驅(qū)動(dòng)微懸臂梁由于電場(chǎng)、偏轉(zhuǎn)梁的幾何和慣性等引起的系統(tǒng)非線性,Chaterjee和Pohit[9]建立了有較大間隙且與地面分離的微懸臂梁的綜合模型并對(duì)其進(jìn)行了靜態(tài)分析??紤]靜電力和幾何非線性的影響,Mojahedi等[10-11]研究了靜電微陀螺儀受到靜電驅(qū)動(dòng)和分子間力(范德華力和卡西米爾力)對(duì)系統(tǒng)非線性運(yùn)動(dòng)的靜態(tài)吸合失穩(wěn)和動(dòng)態(tài)特性的影響。利用多尺度方法,Lajimi等[12]以及Ghommem等[13]分別研究了靜電力引起系統(tǒng)非線性時(shí)的動(dòng)力學(xué)和頻率響應(yīng)特性。
Rasekh和Khadem[14]提出了一種在模式匹配條件下具有高工作頻率的振梁陀螺儀,但是,Lajimi等[13]將此模型稱之為梁-質(zhì)量型陀螺儀,Lajimi等通過(guò)引入剛體的轉(zhuǎn)動(dòng)慣量得出了更為準(zhǔn)確的梁-剛體型陀螺儀模型,并用有限元法做了驗(yàn)證。基于靜電驅(qū)動(dòng)和電阻變化檢測(cè)原理,Ghommem和Abdelkefi[16]設(shè)計(jì)了一種新型微陀螺儀并用仿真結(jié)果證明了新傳感技術(shù)的可行性。通過(guò)引入懸臂梁末端質(zhì)量塊的偏心率,Lajimi等[17-18]研究了非線性微陀螺儀的參數(shù)性能和機(jī)械熱噪聲。
本文依據(jù)Hamilton原理對(duì)靜電振梁陀螺儀建立了動(dòng)力學(xué)方程,主要研究無(wú)量綱參數(shù)αv,w對(duì)微陀螺儀的靜態(tài)和動(dòng)態(tài)性能。參數(shù)αv,w是由方程無(wú)量綱化得到的,跟梁的長(zhǎng)度和電容器的面積成正比,跟梁的抗彎剛度和質(zhì)量塊與電容器的間隙距離成反比,如改變梁的長(zhǎng)度或抗彎剛度即可改變無(wú)量綱參數(shù)αv,w的值,由此可統(tǒng)一研究無(wú)量綱參數(shù)αv,w對(duì)微陀螺的性能影響。本文研究了無(wú)量綱參數(shù)對(duì)微陀螺靜態(tài)特性的影響,發(fā)現(xiàn)轉(zhuǎn)速、轉(zhuǎn)動(dòng)慣量和梁末端質(zhì)量塊對(duì)系統(tǒng)靜態(tài)特性沒(méi)有顯著影響,但隨著無(wú)量綱參數(shù)αv,w的增大,系統(tǒng)吸合失穩(wěn)越來(lái)越小。接著分析了參數(shù)αv,w對(duì)系統(tǒng)1階和2階固有頻率的影響。最后研究了驅(qū)動(dòng)電壓VAC和參數(shù)αv,w對(duì)系統(tǒng)敏感和驅(qū)動(dòng)方向上的動(dòng)力學(xué)放大和校正曲線的影響。
1 微陀螺儀建模
3.3 校正曲線
在參數(shù)VDC=4, αv,w=0.01694, ω=1.3, c =0.065和Mr=1作用下,圖8展示系統(tǒng)在敏感和驅(qū)動(dòng)方向上的校正曲線隨驅(qū)動(dòng)電壓VAC幅值的變化。當(dāng)其他參數(shù)不變時(shí),隨著VAC的增大,在圖8(a)中可發(fā)現(xiàn)敏感方向上的最大位移vM是增大的,并且,當(dāng)轉(zhuǎn)速Ω增大時(shí),vM是隨著轉(zhuǎn)速呈線性正比的,由此可測(cè)量出振動(dòng)微陀螺的轉(zhuǎn)速。當(dāng)其他參數(shù)不變時(shí),隨著VAC幅值的增大,在圖8(b)中發(fā)現(xiàn)驅(qū)動(dòng)方向上的最大位移wM是增大的,但是,當(dāng)轉(zhuǎn)速Ω增大時(shí),wM是不變的。VAC幅值越大,vM越大,系統(tǒng)的敏感度越高。
在參數(shù)VDC=4, VAC=0.1, ω=1.3, c =0.065和Mr=1作用下,圖9描述系統(tǒng)在敏感和驅(qū)動(dòng)方向上的校正曲線隨參數(shù)αv,w的變化。當(dāng)其他參數(shù)不變時(shí),隨著αv,w的增大,在圖9中發(fā)現(xiàn)vM和wM是增大的,但當(dāng)Ω增大時(shí),wM是不變的,vM是隨著Ω呈線性正比的,由此可測(cè)量出振動(dòng)微陀螺的轉(zhuǎn)速。當(dāng)無(wú)量綱參數(shù)αv,w=0.01894時(shí),敏感方向上的最大位移vM相比αv,w=0.01694和αv,w=0.01494高很多,αv,w越大,系統(tǒng)能達(dá)到的振動(dòng)幅值越大,靈敏度越高,跟3.2節(jié)得到了一致的結(jié)論。
4 結(jié) 論
本文介紹了一種單軸振梁微陀螺儀,利用Hamilton原理對(duì)懸臂梁陀螺進(jìn)行了建模,同時(shí)得到了系統(tǒng)的運(yùn)動(dòng)方程和邊界條件。通過(guò)設(shè)解的形式并代入邊界條件對(duì)方程進(jìn)行了求解,分析了系統(tǒng)的靜態(tài)變形并得到了系統(tǒng)的前2階固有頻率,以及研究了靜電微陀螺儀的動(dòng)力學(xué)放大效應(yīng)和校正曲線。以下是得到的幾點(diǎn)結(jié)論。
(1)不論如何變化系統(tǒng)參數(shù),當(dāng)變形達(dá)到間隙間距的33%時(shí),該系統(tǒng)就會(huì)吸合失穩(wěn)。
(2)在不同的αv,w影響下,當(dāng)系統(tǒng)發(fā)生吸合失穩(wěn)的時(shí)候,系統(tǒng)1階固有頻率減小直到零,系統(tǒng)2階固有頻率并不等于零并依然存在。
(3)當(dāng)梁的長(zhǎng)度增大或梁的抗彎剛度減小或電容器的面積增大或質(zhì)量塊與電容器的間隙距離減小時(shí),可使參數(shù)αv,w增大,相同的電壓驅(qū)動(dòng)系統(tǒng)產(chǎn)生更大的變形,系統(tǒng)能達(dá)到的振動(dòng)幅值越大,靈敏度越高。
(4)隨著驅(qū)動(dòng)電壓VAC幅值和無(wú)量綱參數(shù)αv,w的增大,系統(tǒng)在驅(qū)動(dòng)和敏感方向上的最大位移wM和vM是增大的,但是,當(dāng)轉(zhuǎn)速Ω增大時(shí),wM是不變的,而vM是隨著轉(zhuǎn)速呈線性正比的,由此即可測(cè)量出振動(dòng)微陀螺的轉(zhuǎn)速。
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Abstract: This paper introduces a single-axis electrostatic micro-gyroscope made of a cantilever beam and a proof mass fixed at the free end of the beam. The proof mass is under the coupled action of two fixed electrodes, both of which are connected to DC voltage to produce larger static deformation. The electrode in the driving direction is also subjected to AC voltage, which drives the proof mass to produce the primary vibration. Due to the rotation, a second-order vibration is generated by the Coriolis force in the sense direction which is perpendicular to the primary vibration. Therefore, the angular speed can be measured by detecting the amplitude of the second-order vibration. First, the governing equations of vibrating beam micro-gyroscope are deduced by Hamilton principle and the two transverse coupled vibrations of the rotating cantilever beam are investigated. Next, the influence of multiple parameters on the static deformation of micro-gyroscope is analyzed and the first two natural frequencies of the system are obtained. The effects of different parameters on the natural frequencies of the system are presented, and the dynamic amplification effect and its calibration curve in the drive and sense directions of the system are discussed.
Key words: vibrating micro-gyroscope; cantilever beam; dynamic analysis; natural frequency; calibration curve