线性偏摆复合型压电微动平台

线性偏摆复合型压电微动平台 Linear yaw compound piezoelectric micro-motion platform first-author 王　耿（1984-），男，河南南阳人，博士，副教授，硕士生导师，2013年于中国科学院大学获得博士学位，主要研究方向为基于压电陶瓷的微/纳米定位技术、微纳米定位平台的精密跟踪控制。E-mail：wgmouse@163.com http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425014&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425019&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425016&type= 22.01332855 32.00400543 王　耿 （1984-），男，河南南阳人，博士，副教授，硕士生导师，2013年于中国科学院大学获得博士学位，主要研究方向为基于压电陶瓷的微/纳米定位技术、微纳米定位平台的精密跟踪控制。E-mail： wgmouse@163.com wgmouse@163.com corresp E-mail： yonggang_yan@hpu.edu.cn E-mail： yonggang_yan@hpu.edu.cn 闫勇刚（1977-），男，河南孟州人，博士，副教授，硕士生导师，2015年于天津大学获得博士学位，主要研究方向为激光干涉测量技术、微纳测量技术等。E-mail：yonggang_yan@hpu.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425022&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425005&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425023&type= 22.01332855 32.00399780 闫勇刚 （1977-），男，河南孟州人，博士，副教授，硕士生导师，2015年于天津大学获得博士学位，主要研究方向为激光干涉测量技术、微纳测量技术等。E-mail： yonggang_yan@hpu.edu.cn yonggang_yan@hpu.edu.cn 河南理工大学 机械与动力工程学院，河南 焦作 454000 河南理工大学 机械与动力工程学院，河南 焦作 454000 School of Mechanical and Power Engineering， Henan Polytechnic University， Jiaozuo 454000， China School of Mechanical and Power Engineering， Henan Polytechnic University， Jiaozuo 454000， China 为了研究微/纳米技术在光学、精密加工和精确定位等领域的应用，提出了一种线性偏摆复合型压电微动平台，以实现线性运动和偏摆运动。对平台的组成和工作原理进行阐述，建立两种运动的数学模型，并进行运动性能仿真。根据拟定的尺寸加工出样机，对样机的运动能力和频响特性进行测试。最后，通过比例积分微分（Proportional Integral Derivative， PID）反馈实现对平台的跟踪控制。仿真和实验结果表明：该平台具有良好的运动精度和解耦性，一阶固有频率可以达到333.8 Hz，能够实现24.924 μm的线性运动和1.330 mrad的角度摆动，实现了微/纳米级别的运动，在精密光学、精确定位等领域具有广阔的应用前景。 To extend the applications of micro/nano technology in the fields of optics， precision machining， and precise positioning， this study introduces the design， analysis， and testing process for a two-degree-of-freedom micro-movement platform to achieve linear and yaw motion. First， the main composition and working principle of the designed platform are introduced. Subsequently， the mathematical models of the two motions are established and simulated to assess motion performance. A prototype is fabricated considering the size of the piezoelectric platform， and an experimental platform is established to test the motion capability and frequency response. Finally， the precise control of the platform is realized via proportional integral derivative method. The simulation and experimental results demonstrate that the platform achieves good motion accuracy and decoupling performance， with the first natural frequency reaching 333.8 Hz. Moreover， the platform can achieve 24.924 μm linear motion and 1.330 mrad angular swing. The designed structure can serve as reference for the design of micro-motion platform and the research of high-precision motion. 微动平台 线性运动 偏摆运动 柔性机构 数学模型 micro-motion platform linear motion yaw motion flexible mechanism mathematical model 1 引　言 1 微纳米尺度的精密运动对工业与科学的发展至关重要，在原子力显微镜 ［ 1 1 ］ 、微加工装配 ［ 2 2 ］ 、光学元件对准 ［ 3 3 - 4 4 3-4 ］ 、精密印刷 ［ 5 5 ］ 和生物医疗 ［ 6 6 ］ 等领域有着广泛的应用。人们通常采用柔性机构来实现微纳米精密运动，与传统机械关节相比，柔性机构具有制造方便，无侧隙和摩擦，可重复精度高，结构紧凑及无需润滑等优点 ［ 7 7 - 8 8 7-8 ］ 。 微纳米精密运动可以是线性的 ［ 9 9 ］ 或偏摆的 ［ 10 10 ］ ，也可以是一维的 ［ 11 11 ］ 或多自由度的。其中，线性-偏摆相结合的精密运动在许多领域有重要作用。这种运动平台可以同时解决光学中距离和角度的校准问题；在快刀加工中，通过改变实际切削角度，能够明显改善切削力、切削热、加工表面质量以及刀具寿命。 柔性机构在运动行程、运动精度的改善上效果显著 ［ 12 12 - 13 13 12-13 ］ 。Kang等 ［ 14 14 ］ 设计了一款由音圈电机驱动的六自由运动平台，可以实现0.5 mm的线性运动和5 mrad的角度摆动。Clark等 ［ 15 15 ］ 设计了一种基于压电陶瓷驱动的柔性铰链的双自由度微位移平台，可以实现平移以及偏摆，在偏摆上的工作行程有535.8 μrad，且解耦性能良好，一阶模态达到了795 Hz。Kunhai等 ［ 16 16 ］ 设计了一种六自由度精密定位系统，可以实现8.2 μm×10.5 μm×13.0 μm×225 μrad×105 μrad×97 μrad内的运动。然而，上述线性-偏摆微动平台的固有频率普遍较小，其主要原因在于，一方面要实现多种功能；另一方面偏摆运动要有扭矩和偏摆中心，需要多结构间相互作用，使得平台尺寸大、结构复杂。此外，压电材料的固有特性使得以它作为动力源的微动平台存在行程小的问题。 压电陶瓷具有频响高、分辨率高、易于控制和体积小等优点 ［ 17 17 ］ 。本文提出了一种基于压电驱动的双自由度运动平台，在高带宽、大行程的前提下实现了线性运动与偏摆运动的结合。 2 工作原理 1 微动平台主要由底座、上层平台、下层平台、肋板和压电陶瓷组成，上、下平台通过销钉紧密连接，如 图1 图1 所示。 上层平台（ 图2 图2 （a））通过安装孔固定，压电陶瓷安装在压电驱动器中。平行四边形铰链变形，带动中间整体产生线性位移（ 图2 图2 （b））；蛇形铰链变形，运动平台偏摆一定角度（ 图2 图2 （c））。为增大行程，在上层平台安装肋板，将蛇形铰链的左右对称处连接，使它在线性运动时不产生摆动，减小能量损耗。偏摆运动时，由于蛇形铰链的运动和变形一致，因此肋板不对偏摆角度产生影响。 下层平台（ 图3 图3 （a））通过安装孔固定，由压电驱动器驱动，两个压电驱动器输出直线运动，经中心对称结构生成扭矩。当上层的运动平台产生线性位移，会引起薄壁梁的变形（ 图3 图3 （b））；当产生偏摆运动，下层扭矩通过销钉带动上层平台扭转一定角度（ 图3 图3 （c））。其中，压电驱动器的参数与上层平台一致，以便于机械结构的设计和压电陶瓷的选型。 3 理论分析 1 3.1　线性运动的理论模型 2 线性运动时，平台可简化为一个弹簧系统，如 图4 图4 所示。弹簧1，2，3，4分别代表复合平行四边形铰链、薄壁梁、蛇形铰链和压电驱动器。有输入时，弹簧4一边向上位移，一边压缩，设其初始长度为 <math id="M1"><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425048&type= 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3.72533321 为理论输入位移， <math id="M4"><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425063&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425038&type= 3.72533321 3.72533321 为实际输出位移， <math id="M5"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425065&type= 2.45533323 2.79399991 为压缩后的长度。 设弹簧4受压力 <math id="M6"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425070&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425069&type= 1.10066664 3.64066648 作用，弹簧刚度为 <math id="M7"><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425113&type= 16.67933464 3.47133350 ，为横截面的惯性矩。 整理可得： <math display="block" id="M20"><mi>k</mi><mo>=</mo><mfrac><mrow><mi>f</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><mi>I</mi></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425124&type= 22.43666649 8.21266651 . 根据结构的对称性以及平行四边形铰链的复合结构， <math id="M21"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">24</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425125&type= 17.01799965 9.05933285 ， <math id="M22"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">24</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425131&type= 17.01799965 9.05933285 。 3 . 1 . 2　 <math id="M23"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425902&type= 2.79399991 3.72533321 的计算 3 蛇形铰链部分对线性和偏摆有直接影响且结构最复杂。因其结构对称，取其右下结构（ 图6 图6 （a））分析，为保证分析精度，采用有限单元法。假设肋板与上层平台位于同一水平面，将结构简化为15个梁单元，如 图6 图6 （b）所示。 取第 <math id="M24"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425160&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425159&type= 1.10066664 2.79399991 个单元分析（ 图6 图6 （c））有： <math display="block" id="M25"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover></mrow><mrow><mi>x</mi><mi>p</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover></mrow><mrow><mi>x</mi><mi>q</mi></mrow></msub><mo>=</mo><mo>-</mo><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfenced><mo>=</mo><mo>-</mo><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover></mrow><mrow><mi>y</mi><mi>p</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover></mrow><mrow><mi>y</mi><mi>q</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>M</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>M</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425162&type= 72.47466278 63.41533279 ， 其中： <math id="M26"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425166&type= 2.62466669 2.79399991 是杨氏模量， <math id="M27"><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425138&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425169&type= 2.87866688 2.79399991 ， <math id="M28"><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425141&type= 1.26999998 2.79399991 为梁的横截面积和最小惯性矩， <math id="M29"><mover accent="true"><mi>ξ</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425146&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425170&type= 1.60866666 4.14866638 为梁相对坐标下的物理量，下标 p 和 q 表示梁单元的两端， <math id="M30"><msub><mrow><mi>F</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425176&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425173&type= 3.72533321 3.72533321 ， <math id="M31"><msub><mrow><mi>F</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425177&type= 3.64066648 3.72533321 和 <math id="M32"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425184&type= 3.38666677 2.87866688 代表杆端的轴力、剪力和弯矩， <math id="M33"><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425187&type= 1.86266661 2.79399991 ， <math id="M34"><mi>v</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425202&type= 1.86266661 2.79399991 和 <math id="M35"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425210&type= 1.77800000 2.79399991 分别为杆端的轴向位移、横向位移和转角位移。将 式（6） 式（6） 整合成矩阵形式有： <math display="block" id="M36"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover></mrow><mrow><mi>x</mi><mi>p</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover></mrow><mrow><mi>y</mi><mi>p</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>M</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover></mrow><mrow><mi>x</mi><mi>q</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>F</mi><mo>¯</mo></mover></mrow><mrow><mi>y</mi><mi>q</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>M</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></mtd><mtd><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></mtd><mtd><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd><mtd><mfrac><mrow><mn mathvariant="normal">4</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd><mtd><mfrac><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></mtd><mtd><mo>-</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac></mtd><mtd><mo>-</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd><mtd><mfrac><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd><mtd><mfrac><mrow><mn mathvariant="normal">4</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>p</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>u</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>v</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>θ</mi><mo>¯</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425229&type= 107.35732269 65.70133209 . 可简写为： <math display="block" id="M37"><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi mathvariant="bold-italic">k</mi><mi mathvariant="bold-italic">δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425233&type= 11.43000031 2.87866688 ， 式中 <math id="M38"><mi mathvariant="bold-italic">F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425219&type= 2.62466669 2.79399991 ， <math id="M39"><mi mathvariant="bold-italic">k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425242&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425253&type= 1.86266661 2.79399991 和 <math id="M40"><mi mathvariant="bold-italic">δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425244&type= 2.20133328 2.79399991 分别代表局部坐标系下的广义力、单位刚度矩阵和广义位移。 于是一个梁单元的整体刚度矩阵有： <math display="block" id="M41"><mi mathvariant="bold-italic">k</mi><mo>=</mo><mi mathvariant="bold-italic">T</mi><mi mathvariant="bold-italic">k</mi><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425260&type= 13.80066681 2.87866688 ， 其中 <math id="M42"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425266&type= 2.62466669 2.79399991 为整体坐标系和局部坐标系的转换矩阵，即： <math display="block" id="M43"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>α</mi></mtd><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>α</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>α</mi></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>α</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>α</mi></mtd><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>α</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>α</mi></mtd><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>α</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425272&type= 68.15666962 28.53266525 . 依次对每个梁单元进行上述计算，再整合即可得到整体刚度矩阵。由于计算量大，通过计算机辅助完成 <math id="M44"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425276&type= 2.79399991 3.72533321 的计算。 3 . 1 . 3　 <math id="M45"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425864&type= 2.79399991 3.72533321 与 <math id="M46"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425377&type= 1.86266661 3.64066648 的计算 3 对于压电执行器，考虑到结构的对称性，取左上四分之一进行分析，如 图7 图7 所示。 当压电执行器只受压电陶瓷驱动时，对它进行受力分析， AB 和 AC 段都会有变形，引入边界约束条件，有： <math display="block" id="M47"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>△</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>l</mi></mrow><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>l</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi></mrow><mrow><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mo>△</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">3</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>m</mi><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>x</mi></mrow></msub><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>m</mi><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>m</mi><mi>l</mi></mrow><mrow><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi><mi>C</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>x</mi></mrow></msub><msubsup><mrow><mi>l</mi></mrow><mrow><mi>A</mi><mi>C</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">3</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mi>C</mi></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>l</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi><msubsup><mrow><mi>l</mi></mrow><mrow><mi>A</mi><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi><mi>C</mi></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425305&type= 69.00333405 42.33333206 ， 其中下标 <math id="M48"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425321&type= 1.86266661 2.79399991 ， <math id="M49"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425325&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425310&type= 1.10066664 2.79399991 分别表示轴向和切向。 整理可得放大比： <math display="block" id="M50"><mi>η</mi><mo>=</mo><mn mathvariant="normal">2</mn><mfrac><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>y</mi></mrow></msub></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>x</mi></mrow></msub></mrow></mfrac><mo>=</mo><mn mathvariant="normal">2</mn><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi><mo>-</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi><mi>C</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425326&type= 57.74266815 8.89000034 . 同理，当只有 <math id="M51"><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425345&type= 2.53999996 2.79399991 端受到一个竖直向下的力 <math id="M52"><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425336&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425348&type= 2.28600001 4.57200003 时，可得其刚度为： <math display="block" id="M53"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mi>l</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">12</mn><msub><mrow><mi>I</mi></mrow><mrow><mi>l</mi></mrow></msub><mi>l</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><msup><mrow><mi mathvariant="normal">n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></msub><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><msup><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>θ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425353&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425350&type= 45.72000122 9.05933285 . 将 <math id="M54"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425881&type= 2.79399991 3.72533321 ~ <math id="M55"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425864&type= 2.79399991 3.72533321 ， <math id="M56"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425377&type= 1.86266661 3.64066648 带入 式（3） 式（3） 即可得到线性运动中输入与输出的关系。 3.2　偏摆运动的理论模型 2 偏摆运动同样可以简化为弹簧系统（ 图8 图8 （a）所示），弹簧4，5，6，7分别代表压电执行器、传递杆、薄壁梁和蛇形铰链， <math id="M57"><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425383&type= 2.37066650 3.72533321 为弹簧 <math id="M58"><mi>j</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425386&type= 1.10066664 3.64066648 的刚度。杆上 a ， b 的变形可以看作是 <math id="M59"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425468&type= 2.28600001 4.57200003 和 <math id="M60"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425478&type= 2.28600001 4.57200003 的共同作用引起的（如 图8 图8 （b）所示）， a ， b 点的位移为： <math display="block" id="M61"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>f</mi><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>f</mi><mn mathvariant="normal">2</mn><mi>a</mi></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425397&type= 57.82733154 8.80533314 ， <math display="block" id="M62"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>f</mi><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>f</mi><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfrac><mo>+</mo><mi>β</mi><msub><mrow><mi>r</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425412&type= 65.95532990 8.80533314 ， 其中： <math id="M63"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425872&type= 3.80999994 3.72533321 ， <math id="M64"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425618&type= 3.80999994 3.72533321 ， <math id="M65"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425421&type= 3.80999994 3.72533321 ， <math id="M66"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425878&type= 3.80999994 3.72533321 表示梁上 <math id="M67"><mi>a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425430&type= 1.86266661 2.79399991 ， <math id="M68"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425694&type= 1.86266661 2.79399991 点关于 <math id="M69"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425468&type= 2.28600001 4.57200003 ， <math id="M70"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425478&type= 2.28600001 4.57200003 的刚度，即 <math id="M71"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>f</mi><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>/</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425481&type= 20.65866661 4.82600021 ， <math id="M72"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>f</mi><mn mathvariant="normal">2</mn><mi>a</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>/</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425487&type= 20.65866661 4.82600021 ， <math id="M73"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>f</mi><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>/</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425492&type= 20.65866661 4.82600021 ， <math id="M74"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>X</mi></mrow><mrow><mi>f</mi><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>/</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425512&type= 20.65866661 4.82600021 。结合 式（14） 式（14） 和 式（15） 式（15） ，有： <math display="block" id="M75"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub></mrow></mfrac></mrow></mfenced><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>b</mi></mrow></msub><mi>β</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425523&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425517&type= 44.28066635 10.75266743 ， <math display="block" id="M76"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>J</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425521&type= 12.61533356 7.95866632 ， 其中 <math id="M77"><mi>J</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>a</mi></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425529&type= 72.47466278 10.75266743 。 偏摆角度的输出为： <math display="block" id="M78"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>J</mi><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425556&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425552&type= 23.36800003 8.80533314 . 3 . 2 . 1　 <math id="M79"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425558&type= 2.79399991 3.72533321 ， <math id="M80"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425575&type= 2.79399991 3.72533321 的计算 3 根据杆的拉、压变形 <math id="M81"><mi>δ</mi><mo>=</mo><mfrac><mrow><mi>F</mi><mi>L</mi></mrow><mrow><mi>E</mi><mi>A</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425582&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425565&type= 12.95400047 7.95866632 ，有： <math display="block" id="M82"><msub><mrow><mi>k</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>λ</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo>/</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425586&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425584&type= 21.16666603 4.14866638 ， 其中 <math id="M83"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425601&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425588&type= 1.77800000 2.79399991 分别为5，6。 3 . 2 . 2　 <math id="M84"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425602&type= 3.80999994 3.72533321 ， <math id="M85"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425877&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425876&type= 3.80999994 3.72533321 ， <math id="M86"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425618&type= 3.80999994 3.72533321 ， <math id="M87"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425628&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425612&type= 3.80999994 3.72533321 的计算 3 如 图9 图9 所示，由挠度公式可知： <math display="block" id="M88"><mi>ω</mi><mo>=</mo><mfrac><mrow><mi>F</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>b</mi></mrow></msub><mi>x</mi></mrow><mrow><mn mathvariant="normal">6</mn><mi>E</mi><mi>I</mi><mi>l</mi></mrow></mfrac><mo stretchy="false">(</mo><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425625&type= 38.94666672 7.95866632 <math display="block" id="M89"><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>a</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425634&type= 18.96533203 4.48733330 ， <math display="block" id="M90"><mi>ω</mi><mo>=</mo><mfrac><mrow><mi>F</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">6</mn><mi>E</mi><mi>I</mi><mi>l</mi></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><mi>l</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>-</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>a</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mi>x</mi><mo>-</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425650&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425637&type= 66.12466431 9.99066734 <math display="block" id="M91"><mo stretchy="false">(</mo><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>l</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425642&type= 17.94933319 3.55599999 . 于是有： <math display="block" id="M92"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><mi>I</mi><mi>l</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><mi>l</mi><mo>-</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><mi>I</mi><mi>l</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>a</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><mi>I</mi><mi>l</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><mi>I</mi><mi>l</mi></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425680&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425663&type= 59.68999863 44.95800018 ， 其中： l 1 ， l 2 分别对应 l a ， l b ， <math id="M93"><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>l</mi><mo>-</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425667&type= 22.26733208 3.72533321 。 为方便计算，这里给出在 <math id="M94"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425685&type= 2.28600001 4.57200003 ， <math id="M95"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425704&type= 2.28600001 4.57200003 作用下 <math id="M96"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425710&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425694&type= 1.86266661 2.79399991 点的偏转角度（本机构 <math id="M97"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425698&type= 2.28600001 4.57200003 作用于中点）： <math display="block" id="M98"><mi>ϕ</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">6</mn><mi>E</mi><mi>I</mi><mi>l</mi></mrow></mfrac><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mfrac><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">3</mn><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425717&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425700&type= 64.34667206 10.58333302 . 3 . 2 . 3　 <math id="M99"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425720&type= 2.79399991 3.72533321 的计算 3 当运动平台摆动时，上层主要是蛇形铰链产生的变形，取其右下结构进行分析，如 图10 图10 所示，偏摆主要由四根细长梁变形产生。 对梁进行受力分析，由于梁的端部转角为0，于是有： <math display="block" id="M100"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi>f</mi><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">3</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>m</mi><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>=</mo><mi>d</mi></mtd></mtr><mtr><mtd><mfrac><mrow><mi>f</mi><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>m</mi><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">4</mn><mi>f</mi><mi>r</mi><mo>+</mo><mn mathvariant="normal">4</mn><mi>m</mi><mo>=</mo><mi>M</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425727&type= 28.95599937 26.16200066 ， 其中： <math id="M101"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425838&type= 3.38666677 2.87866688 为下层平台传递的扭矩， <math id="M102"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425763&type= 2.37066650 3.72533321 为梁的长度，即 <math id="M103"><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>R</mi><mo>-</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425754&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425753&type= 16.17133331 3.72533321 ， <math id="M104"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425756&type= 2.62466669 2.79399991 为材料的杨氏模量， <math id="M105"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425771&type= 2.37066650 3.72533321 为梁的惯性矩。 图10 图10 中， <math id="M106"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>d</mi><mo>/</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425761&type= 13.20800018 4.14866638 ， <math id="M107"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>d</mi><mo>/</mo><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425786&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425784&type= 22.18266678 4.14866638 ， <math id="M108"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>d</mi><mo>/</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425788&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425787&type= 20.99733353 4.14866638 ， <math id="M109"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">d</mi><mo>/</mo><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425789&type= 21.84399986 4.14866638 。联立可得： <math display="block" id="M110"><mi>M</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi><mi>r</mi></mrow><mrow><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi><mi>r</mi></mrow><mrow><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425792&type= 49.69933319 9.05933285 . 运动平台偏摆时，下层结构会吸收一部分能量，主要集中在杆 ab 和薄壁梁的变形上，如 图11 图11 所示。 <math id="M111"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425811&type= 3.64066648 3.72533321 的力臂很小，生成的扭矩可忽略不计。 中间运动平台偏摆时，会引起薄壁梁的位移和偏转，于是有： <math display="block" id="M112"><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">3</mn><mi>E</mi><mi>I</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>E</mi><mi>I</mi></mrow></mfrac><mo>=</mo><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow></mfenced><msub><mrow><mi>r</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>+</mo><mi>ϕ</mi><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425829&type= 56.04933929 8.21266651 ， <math display="block" id="M113"><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>l</mi></mrow><mrow><mi>E</mi><mi>I</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><mi>E</mi><mi>I</mi></mrow></mfrac><mo>=</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi></mrow></msub><mo>-</mo><mi>ϕ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425815&type= 35.13666534 8.21266651 ， 得： <math display="block" id="M114"><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">6</mn><mi>E</mi><mi>I</mi><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow></mfenced><msub><mrow><mi>r</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">4</mn><mi>E</mi><mi>I</mi><mi>l</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><mi>E</mi><mi>I</mi><mi>ϕ</mi><mi>l</mi></mrow><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425836&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425834&type= 70.01933289 9.31333351 . 当传递到中间运动平台 <math id="M115"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425838&type= 3.38666677 2.87866688 扭矩时： <math display="block" id="M116"><mi>M</mi><mo>=</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>-</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425819&type= 26.58533096 4.57200003 . 由 式（25） 式（25） 和 式（28） 式（28） 可得，当平台输出 <math id="M117"><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425846&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425844&type= 4.65666676 3.72533321 时有： <math display="block" id="M118"><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi><mi>r</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn mathvariant="normal">2</mn><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>-</mo><msub><mrow><mi>M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425848&type= 44.11133575 9.05933285 . 联立 式（16） 式（16） 、 式（19） 式（19） 、式（22）~ 式（23） 式（23） 和式（29）~ 式（30） 式（30） 可得： <math display="block" id="M119"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">O</mi><mi mathvariant="normal">P</mi></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mfrac><mrow><mn mathvariant="normal">8</mn><mi>E</mi><mi>I</mi></mrow><mrow><mi>l</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">12</mn><mi>E</mi><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi><mi>r</mi></mrow><mrow><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>E</mi><mi>I</mi><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow><mrow><mi>l</mi></mrow></mfrac></mrow><mrow><mn mathvariant="normal">2</mn><msub><mrow><mi>r</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>E</mi><mi>I</mi><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow><mrow><mi>l</mi></mrow></mfrac><mfenced separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425852&type= 66.04000092 20.48933220 ， 其中 <math id="M120"><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">6</mn><mi>E</mi><mi>I</mi><mi>l</mi></mrow></mfrac><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mo>+</mo><mfrac><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">3</mn><msup><mrow><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425862&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425859&type= 63.75400162 10.58333302 。 将 <math id="M121"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425866&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425864&type= 2.79399991 3.72533321 ~ <math id="M122"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425892&type= 2.79399991 3.72533321 ， <math id="M123"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425884&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425872&type= 3.80999994 3.72533321 ， <math id="M124"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425885&type= 3.80999994 3.72533321 ， <math id="M125"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425877&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425876&type= 3.80999994 3.72533321 和 <math id="M126"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425878&type= 3.80999994 3.72533321 代入式（16）~ 式（18） 式（18） ，即可得到偏摆运动中输入与输出的关系。 分析模型可知：增大线性位移，应减小 <math id="M127"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425890&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425881&type= 2.79399991 3.72533321 ， <math id="M128"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425898&type= 2.79399991 3.72533321 ，增大 <math id="M129"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425902&type= 2.79399991 3.72533321 ，即要求四边形铰链和细长梁的径向刚度尽可能小，蛇形铰链细梁的轴向刚度尽可能大，细梁间夹角尽可能小；增大偏摆，应减小 <math id="M130"><msub><mrow><mi>k</mi></mrow><mrow><mn mathvariant="normal">7</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425892&type= 2.79399991 3.72533321 ，即要求蛇形铰链16根细梁的径向刚度尽可能小。考虑结构紧凑性、运动稳定性和加工条件，取平台中平行四边形铰链、蛇形铰链和细长梁的宽度为0.8 mm，蛇形铰链的细梁间夹角为15°，由此计算理论模型得：平台可以实现25.284 μm和1.450 mrad的运动。 4 仿真分析 1 运用ANSYS Workbench 2019 R3对机构进行仿真分析。由于结构中细小尺寸较多，为细化网格，使用四面体网格划分，使用patch conforming算法。节点数量超过26 000个，元素的数量超过11 000个，模型的每个截面至少跨越了5个元素。上、下层平台材料选用7075铝合金，其余选用45钢，以减小不必要的变形，在各安装孔处设置固定约束。 4.1　数学模型验证 2 根据预选的压电陶瓷规格，给定几组可实现的位移，记录中间运动平台的运动，然后将相同的输入带进理论模型，得出的数据整理后与仿真结果对比，如 图12 图12 所示。线性运动的理论数据最大误差在1%左右，偏摆运动的最大误差在8.5%左右。 值得注意的是，线性运动的理论数值小于仿真结果，而偏摆的理论数值却大于仿真结果。其原因主要在于：在压电执行器的计算中，为方便计算简化了模型， 式（12） 式（12） 中的 <math id="M131"><msub><mrow><mi>d</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425914&type= 3.38666677 3.72533321 减小，导致线性模型的最终数据小于仿真结果；而偏摆理论模型中，尽管也涉及到 <math id="M132"><msub><mrow><mi>d</mi></mrow><mrow><mi>x</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425920&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425918&type= 3.38666677 3.72533321 减小的问题，但模型中简化了 ab 杆和薄壁梁，将梁的两端柔性铰链的扭转刚度省略，当作转动副考虑，其次忽略了 <math id="M133"><msub><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425896&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=28425922&type= 3.64066648 3.72533321 的作用，以及上层平台中除蛇形铰链外其余结构的微小变形。虽然运动结果有偏差，仍在可允许的范围内。 4.2　运动精度与耦合性分析 2 通过ANSYS仿真分析得到平台的线性度误差为0.030 6 μm/μm，即平台每产生1 μm的线性运动时会在其正交方向上产生约0.030 6 μm的偏移。平台的定轴精度为0.139 7 μm/mrad，即平台每产生1 mrad的偏摆角度时会在其圆心产生0.139 7 μm的偏移。 由于线性运动和偏摆运动相耦合，当两种运动同时产生时，行程会有所下降，结果如 图13 图13 所示。位移（偏摆）最多减小0.035 μm （0.003 mrad），所引起的误差在一定范围内可通过增大驱动电压来补偿。 5 实　验 1 本文加工出的微动平台样机如 图14 图14 所示，整体尺寸（不包含底座）为88 mm×83.8 mm×18 mm。实验平台如 图15 图15 所示，主要包括微动平台、驱动电源（MDT694B，Thorlabs）、调理电路电源（APS3005D，深圳市安泰信科技有限公司）、dSPACE、电容式传感器（E-501，PI公司）、压电陶瓷（PSt150/5×5/20L，哈尔滨芯明天科技有限公司）、信号调理电路（西安骏腾电子科技有限公司）和应变式传感器。 首先，由计算机产生一定波形的信号，经dSPACE的D/A模块将数字信号转化为模拟信号，模拟信号经驱动电源放大后作用在压电陶瓷上，使之产生一定的位移和力，驱动平台做线性和偏摆运动。集成在平台上的应变传感器将采集到的电压信号经信号调理电路放大后传送给dSPACE，最后dSPACE将采集到的信号传送给计算机进行后期处理。 5.1　测　量 2 实验采用应变式传感器对运动进行测量，因为电容式传感器安装不方便且存在校准误差。应变式传感器利用4个相同的应变片搭建惠斯通电桥。V+、V0供电，S+、S - 测试信号，2个应变片的粘贴位置如 图16 图16 所示。图示位置应变易检测，而且不会因为另一种运动受到影响，另外两个应变片粘贴在无应变区域。做线性运动时， 图16 图16 （a）的 R 2 和 R 1 位置分别产生拉、压应变，引起 R 1 ， R 2 的反向变化， R 0 不变，S+，S - 之间产生电压差。做偏摆运动时， 图16 图16 （b）的 R 1 和 R 2 位置均产生拉应变，引起 R 1 ， R 2 电阻的变化， R 0 不变，S+，S - 之间产生电压差。下层平台有两组电桥是为了验证两个压电陶瓷能否正常工作，在后续的实验数据测试中只记录其中一组数据。 实验前，精准安装电容传感器，对应变传感器进行标定。具体方法如下：给定压电陶瓷驱动电压，分别记录电容传感器和应变传感器的电压信号，确定比例系数为11.656 mV/μm（313.758 mV/mrad），即每输出1 μm（1 mrad）时，应变传感器输出11.656 mV（313.758 mV）的电压信号。再改变驱动电压，对比例系数进行验证，结果如 图17 图17 所示。通过测试，平台可以在105 V下实现24.924 μm的线性运动，与理论模型的误差为1.428%；112.5 V下实现1.330 mrad的角度摆动，与理论模型的误差为9.022%。由此证明了理论模型与仿真模型的正确性。 5.2　频率响应测试 2 平台有两个自由度的运动， 图18 图18 为对应的第一和第六阶模态效果图。给上、下层压电陶瓷分别施加1~700 Hz，1~400 Hz的正弦扫频信号，电压幅值为15 V，得到的频率响应如 图19 图19 所示，微动平台的一阶固有频率在333.8 Hz左右。当偏摆运动时，265 Hz左右的扫频段也有小峰值出现，这是由下层的两个压电陶瓷的性能和参数不完全一致引起的局部固有频率。 5.3　开闭环测试 2 为测试结构性能，分别对平台的线性和偏摆运动做了开/闭环测试。其中，开环测试依次给定线性（偏摆）90 V，1 Hz的正弦波、方波和三角波，产生了21.364 μm（1.063 mrad）的运动，结果如 图20 图20 所示，其迟滞效果如 图21 图21 （a）所示。闭环采用PID控制，同样给定90 V的电压，其结果如 图21 图21 （b）所示，迟滞效应有效降低，在低频下对平台的线性和偏摆运动实现了良好的跟踪控制。 1;2;3; 1;2; 6 结　论 1 本文提出了一种二自由度微动平台，可以同时实现直线运动与偏摆运动，建立了精确的数学模型，并通过仿真和实验验证了该平台的运动性能。结果表明，该平台能够实现24.924 μm的线性运动和1.330 mrad的角度偏摆，通过PID实现了低频下的跟踪控制，从而为微动平台的设计和高精密运动研究提供参考。未来将主要围绕共同驱动和高频控制两方面展开研究。 参考文献 BYKOV V A ， NOVIKOV Y A ， RAKOV A V ， et al . Defining the parameters of a cantilever tip AFM by reference structure ［J］. Ultramicroscopy ， 2003 ， 96 （ 2 ）： 175 - 180 . doi: 10.1016/s0304-3991(03)00005-6 http://dx.doi.org/10.1016/s0304-3991(03)00005-6 Defining the parameters of a cantilever tip AFM by reference structure WANG S M ， YE Z Z ， YEH C C ， et al . New design of a low-cost micro machine tool with the high-precision variable-resolution mechanism ［J］. Optical and Quantum Electronics ， 2016 ， 48 （ 12 ）： 565 . doi: 10.1007/s11082-016-0836-z http://dx.doi.org/10.1007/s11082-016-0836-z New design of a low-cost micro machine tool with the high-precision variable-resolution mechanism WANG G ， YAO X M ， CUI J J ， et al . A novel piezoelectric hysteresis modeling method combining LSTM and NARX neural networks ［J］. Modern Physics Letters B ， 2020 ， 34 （ 28 ）： 2050306 . doi: 10.1142/s0217984920503066 http://dx.doi.org/10.1142/s0217984920503066 A novel piezoelectric hysteresis modeling method combining LSTM and NARX neural networks 陈国真 ， 徐斯强 ， 刘品宽 ， 等 . 大行程快速反射镜的结构设计及带宽特性 ［J］. 光学 精密工程 ， 2020 ， 28 （ 1 ）： 90 - 101 . doi: 10.3788/ope.20202801.0090 http://dx.doi.org/10.3788/ope.20202801.0090 大行程快速反射镜的结构设计及带宽特性 CHEN G ZH ， XU S Q ， LIU P K ， et al . Structural design and bandwidth characteristic of a fast steering mirror with large travel range ［J］. Opt. Precision Eng. ， 2020 ， 28 （ 1 ）： 90 - 101 . （in Chinese） . doi: 10.3788/ope.20202801.0090 http://dx.doi.org/10.3788/ope.20202801.0090 Structural design and bandwidth characteristic of a fast steering mirror with large travel range ZHAO L ， WANG H ， DUAN W B ， et al . Design and analysis of a bi-axial centralized butterfly flexure hinge for fast steering mirrors ［J］. Journal of Astronomical Telescopes， Instruments， and Systems ， 2020 ， 6 ： 048003 . doi: 10.1117/1.jatis.6.4.048003 http://dx.doi.org/10.1117/1.jatis.6.4.048003 Design and analysis of a bi-axial centralized butterfly flexure hinge for fast steering mirrors WANG P Y ， XU Q S . Design and testing of a flexure-based constant-force stage for biological cell micromanipulation ［J］. IEEE Transactions on Automation Science and Engineering ， 2018 ， 15 （ 3 ）： 1114 - 1126 . doi: 10.1109/tase.2017.2733553 http://dx.doi.org/10.1109/tase.2017.2733553 Design and testing of a flexure-based constant-force stage for biological cell micromanipulation LUO Y Q ， LIU W Y . Analysis of the displacement of distributed compliant parallel-guiding mechanism considering parasitic rotation and deflection on the guiding plate ［J］. Mechanism and Machine Theory ， 2014 ， 80 ： 151 - 165 . doi: 10.1016/j.mechmachtheory.2014.06.005 http://dx.doi.org/10.1016/j.mechmachtheory.2014.06.005 Analysis of the displacement of distributed compliant parallel-guiding mechanism considering parasitic rotation and deflection on the guiding plate 张旭 ， 赖磊捷 ， 李朋志 ， 等 . 电磁驱动柔顺微定位平台闭环频域逆迭代学习控制 ［J］. 光学 精密工程 ， 2021 ， 29 （ 9 ）： 2149 - 2157 . doi: 10.37188/OPE.20212909.2149 http://dx.doi.org/10.37188/OPE.20212909.2149 电磁驱动柔顺微定位平台闭环频域逆迭代学习控制 ZHANG X ， LAI L J ， LI P ZH ， et al . Closed-loop inverse iterative learning control in frequency-domain for electromagnetic driven compliant micro-positioning platform ［J］. Opt. Precision Eng. ， 2021 ， 29 （ 9 ）： 2149 - 2157 . （in Chinese） . doi: 10.37188/OPE.20212909.2149 http://dx.doi.org/10.37188/OPE.20212909.2149 Closed-loop inverse iterative learning control in frequency-domain for electromagnetic driven compliant micro-positioning platform ZHONG X Q ， LIU C X ， DI Z M ， et al . Optimization and experiment of two-dimensional parallel decoupling image stabilization mechanism ［J］. International Journal of Precision Engineering and Manufacturing ， 2020 ， 21 （ 10 ）： 1965 - 1974 . doi: 10.1007/s12541-020-00393-x http://dx.doi.org/10.1007/s12541-020-00393-x Optimization and experiment of two-dimensional parallel decoupling image stabilization mechanism CLARK L ， SHIRINZADEH B ， ZHONG Y M ， et al . Design and analysis of a compact flexure-based precision pure rotation stage without actuator redundancy ［J］. Mechanism and Machine Theory ， 2016 ， 105 ： 129 - 144 . doi: 10.1016/j.mechmachtheory.2016.06.017 http://dx.doi.org/10.1016/j.mechmachtheory.2016.06.017 Design and analysis of a compact flexure-based precision pure rotation stage without actuator redundancy DAS T K ， SHIRINZADEH B ， GHAFARIAN M ， et al . Design， analysis and experimental investigations of a high precision flexure-based microgripper for micro/nano manipulation ［J］. Mechatronics ， 2020 ， 69 ： 102396 . doi: 10.1016/j.mechatronics.2020.102396 http://dx.doi.org/10.1016/j.mechatronics.2020.102396 Design， analysis and experimental investigations of a high precision flexure-based microgripper for micro/nano manipulation ZHU J X ， HAO G B . Design and test of a compact compliant gripper using the Scott–Russell mechanism ［J］. Archives of Civil and Mechanical Engineering ， 2020 ， 20 （ 3 ）： 81 . doi: 10.1007/s43452-020-00085-3 http://dx.doi.org/10.1007/s43452-020-00085-3 Design and test of a compact compliant gripper using the Scott–Russell mechanism NAN Z J ， XU Q S ， ZHANG Y B ， et al . Force-sensing robotic microinjection system for automated multi-cell injection with consistent quality ［J］. IEEE Access ， 2019 ， 7 ： 55543 - 55553 . doi: 10.1109/access.2019.2913592 http://dx.doi.org/10.1109/access.2019.2913592 Force-sensing robotic microinjection system for automated multi-cell injection with consistent quality KANG S ， LEE M G ， CHOI Y M . Six degrees-of-freedom direct-driven nanopositioning stage using crab-leg flexures ［J］. IEEE/ASME Transactions on Mechatronics ， 2020 ， 25 （ 2 ）： 513 - 525 . doi: 10.1109/tmech.2020.2972301 http://dx.doi.org/10.1109/tmech.2020.2972301 Six degrees-of-freedom direct-driven nanopositioning stage using crab-leg flexures CLARK L ， SHIRINZADEH B ， BHAGAT U ， et al . Development and control of a two DOF linear-angular precision positioning stage ［J］. Mechatronics ， 2015 ， 32 ： 34 - 43 . doi: 10.1016/j.mechatronics.2015.10.001 http://dx.doi.org/10.1016/j.mechatronics.2015.10.001 Development and control of a two DOF linear-angular precision positioning stage CAI K H ， TIAN Y L ， WANG F J ， et al . Design and control of a 6-degree-of-freedom precision positioning system ［J］. Robotics and Computer-Integrated Manufacturing ， 2017 ， 44 ： 77 - 96 . doi: 10.1016/j.rcim.2016.08.005 http://dx.doi.org/10.1016/j.rcim.2016.08.005 Design and control of a 6-degree-of-freedom precision positioning system WU S L ， SHAO Z X ， SU H J ， et al . An energy-based approach for kinetostatic modeling of general compliant mechanisms ［J］. Mechanism and Machine Theory ， 2019 ， 142 ： 103588 . doi: 10.1016/j.mechmachtheory.2019.103588 http://dx.doi.org/10.1016/j.mechmachtheory.2019.103588 An energy-based approach for kinetostatic modeling of general compliant mechanisms 10.37188/OPE.20223009.1058 TH744 TP183 A 1004-924X（2022）09-1058-13 王耿,魏维宁,代军等.线性偏摆复合型压电微动平台[J].光学精密工程,2022,30(09):1058-1070. 王耿,魏维宁,代军等.线性偏摆复合型压电微动平台[J].光学精密工程,2022,30(09):1058-1070. WANG Geng,WEI Weining,DAI Jun,et al.Linear yaw compound piezoelectric micro-motion platform[J].Optics and Precision Engineering,2022,30(09):1058-1070. WANG Geng,WEI Weining,DAI Jun,et al.Linear yaw compound piezoelectric micro-motion platform[J].Optics and Precision Engineering,2022,30(09):1058-1070. 河南省科技攻关项目（No.202102210082，No.212102210323）；河南理工大学杰出青年基金资助项目（No.J2020-5）；河南理工大学博士基金资助项目（No.B2014-037，No.B2017-32）；河南理工大学青年骨干教师基金资助项目（No.2017XQG-04） 2022-01-06 2022-03-15 2022-05-10 2022-05-26 光学精密工程 9 30