Precision-Motion-Control(book).rar_motion control_tan kok kio

<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/625039a86caf596192fdbe78/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625039a86caf596192fdbe78/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1 h2 y2 ff1 fs0 fc0 sc0 ls1 ws0">Precision Tracking Motion Control</div><div class="t m0 x1 h3 y3 ff2 fs1 fc0 sc0 ls2 ws0">In this chapters, two precision actuation systems using piezoelectric actuators</div><div class="t m0 x1 h3 y4 ff2 fs1 fc0 sc0 ls3 ws0">and permanent magnet linear motors will be presented. Common conﬁgura-</div><div class="t m0 x1 h3 y5 ff2 fs1 fc0 sc0 ls5 ws0">tions of these systems, their mathematical model as well as control schemes</div><div class="t m0 x1 h3 y6 ff2 fs1 fc0 sc0 ls6 ws0">will be included.</div><div class="t m0 x1 h4 y7 ff1 fs2 fc0 sc0 ls7 ws0">2.1 Piezoelectric Actuators</div><div class="t m0 x1 h3 y8 ff2 fs1 fc0 sc0 ls8 ws0">In the area of micro- and nano-scale systems, the piezoelectric actuator (PA)</div><div class="t m0 x1 h3 y9 ff2 fs1 fc0 sc0 ls9 ws0">has become an increasingly popular candidate as a precise actuator, due to</div><div class="t m0 x1 h3 ya ff2 fs1 fc0 sc0 lsa ws0">its ability to achieve high precision and its versatility to be implemented in a</div><div class="t m0 x1 h3 yb ff2 fs1 fc0 sc0 lsb ws0">wide range of applications. More speciﬁcally, the PA can provide very precise</div><div class="t m0 x1 h3 yc ff2 fs1 fc0 sc0 lsc ws0">positioning (of the order of nanometers) and generate high forces (up to few</div><div class="t m0 x1 h3 yd ff2 fs1 fc0 sc0 lsd ws0">thousands newton). The increasingly widespread industrial applications of the</div><div class="t m0 x1 h3 ye ff2 fs1 fc0 sc0 lse ws0">PA in various optical ﬁbre alignment, mask alignment, and medical micro-</div><div class="t m0 x1 h3 yf ff2 fs1 fc0 sc0 lsf ws0">manipulation systems are self-evident testimonies of the eﬀectiveness of the</div><div class="t m0 x1 h3 y10 ff2 fs1 fc0 sc0 ls10 ws0">PA in these application domains.</div><div class="t m0 x2 h3 y11 ff2 fs1 fc0 sc0 ls6 ws0">In spite of the beneﬁts of a PA in these application domains, there are</div><div class="t m0 x1 h3 y12 ff2 fs1 fc0 sc0 lsa ws0">challenges in the design and control of these devices. In the following sections,</div><div class="t m0 x1 h3 y13 ff2 fs1 fc0 sc0 ls11 ws0">issues pertaining to PA design and control will be discussed in detail.</div><div class="t m0 x1 h3 y14 ff3 fs1 fc0 sc0 ls12 ws0">2.1.1 Types of Piezoelectric Actuator Conﬁguration</div><div class="t m0 x1 h3 y15 ff2 fs1 fc0 sc0 lsd ws0">The PA oﬀers unique and compelling advantages in nanometer resolution and</div><div class="t m0 x1 h3 y16 ff2 fs1 fc0 sc0 lsb ws0">high-speed applications. To derive maximum performance from PAs, a variety</div><div class="t m0 x1 h3 y17 ff2 fs1 fc0 sc0 ls13 ws0">of conﬁguration can be designed to adapt to various requirements.</div><div class="t m0 x1 h3 y18 ff3 fs1 fc0 sc0 lse ws0">Stack Design</div><div class="t m0 x1 h3 y19 ff2 fs1 fc0 sc0 ls14 ws0">The most common design for PAs is a stack of ceramic layers (see Figure</div><div class="t m0 x1 h3 y1a ff2 fs1 fc0 sc0 ls15 ws0">2.1). Such devices are capable of achieving high displacements and holding</div></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,-173.234022,-270.244158]}'></div></div> </body> </html>

<div id="pf2" class="pf w2 h0" data-page-no="2"><div class="pc pc2 w2 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625039a86caf596192fdbe78/bg2.jpg"><div class="t m0 x1 h5 y1b ff4 fs3 fc0 sc0 ls16 ws0">12 2 Precision Tracking Motion Control</div><div class="t m0 x1 h3 y1c ff2 fs1 fc0 sc0 ls4 ws0">forces. Standard designs which can withstand pressures of up to 100kN are</div><div class="t m0 x1 h3 y1d ff2 fs1 fc0 sc0 ls17 ws0">available in commercial products (e.g., products from Physik Instrumente),</div><div class="t m0 x1 h3 y1e ff2 fs1 fc0 sc0 ls13 ws0">and preloaded actuators can also be operated in a push-pull mode.</div><div class="t m0 x3 h5 y1f ff6 fs3 fc0 sc0 ls1a ws0">Fig. 2.1. Construction of a stack actuator</div><div class="t m0 x1 h3 y20 ff3 fs1 fc0 sc0 ls1c ws0">Laminar Design</div><div class="t m0 x1 h3 y21 ff2 fs1 fc0 sc0 lsd ws0">This design uses thin laminated ceramic sheets (see Figure 2.2). When a volt-</div><div class="t m0 x1 h3 y22 ff2 fs1 fc0 sc0 ls2 ws0">age is applied to the device, the actuator sheet contracts. The displacement</div><div class="t m0 x1 h3 y23 ff2 fs1 fc0 sc0 ls0 ws0">in the device is caused by the contraction in the material being perpendicular</div><div class="t m0 x1 h3 y24 ff2 fs1 fc0 sc0 lsd ws0">to the direction of polarization and electric ﬁeld application. The maximum</div><div class="t m0 x1 h3 y25 ff2 fs1 fc0 sc0 ls14 ws0">travel of the laminar actuators is a function of the length of the sheets, while</div><div class="t m0 x1 h3 y26 ff2 fs1 fc0 sc0 ls9 ws0">the number of sheets arranged in parallel will determine the stiﬀness and force</div><div class="t m0 x1 h3 y27 ff2 fs1 fc0 sc0 ls1d ws0">generation of the ceramic element. Laminar actuators are easily integrated in</div><div class="t m0 x1 h3 y28 ff2 fs1 fc0 sc0 ls1e ws0">conventional composite layers.</div><div class="t m0 x1 h3 y29 ff3 fs1 fc0 sc0 ls1f ws0">Tubular Design</div><div class="t m0 x1 h3 y2a ff2 fs1 fc0 sc0 ls1d ws0">The monolithic ceramic tube is yet another form of piezo actuator. Figure</div><div class="t m0 x1 h3 y2b ff2 fs1 fc0 sc0 ls20 ws0">2.3 shows a design structure. The surface of a tube is partitioned into four</div><div class="t m0 x1 h3 y2c ff2 fs1 fc0 sc0 ls11 ws0">regions and they are connected along with one end of the tube to electrodes.</div><div class="t m0 x1 h3 y2d ff2 fs1 fc0 sc0 ls21 ws0">Thus, it becomes possible to apply voltages to the tube to initiate motion in</div><div class="t m0 x1 h3 y2e ff2 fs1 fc0 sc0 ls11 ws0">various directions. For example, when an electric voltage is applied between</div><div class="t m0 x1 h3 y2f ff2 fs1 fc0 sc0 lsc ws0">the outer and inner diameter of a thin-walled tube, the tube contracts axially</div><div class="t m0 x1 h3 y30 ff2 fs1 fc0 sc0 ls22 ws0">and radially. A variety of chemical and materials processing applications use</div><div class="t m0 x1 h3 y31 ff2 fs1 fc0 sc0 ls23 ws0">ceramic tubes. Ceramic tubes are also used to fabricate electrical parts for high</div><div class="t m0 x1 h3 y32 ff2 fs1 fc0 sc0 ls9 ws0">voltage or power applications such as insulators, igniters or heating elements.</div></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,-162.262385,-270.244158]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625039a86caf596192fdbe78/bg3.jpg"><div class="t m0 x4 h5 y1b ff4 fs3 fc0 sc0 ls24 ws0">2.1 Piezoelectric Actuators 13</div><div class="t m0 x5 h5 y33 ff6 fs3 fc0 sc0 ls1a ws0">Fig. 2.2. Laminar design</div><div class="t m0 x6 h5 y34 ff6 fs3 fc0 sc0 ls1a ws0">Fig. 2.3. Tub e design</div><div class="t m0 x1 h3 y35 ff3 fs1 fc0 sc0 ls27 ws0">Partial-Rotation Design</div><div class="t m0 x1 h3 y36 ff2 fs1 fc0 sc0 ls28 ws0">The design employs a piezoelectric cylinder from LPZT (lithium-lead-zirconium-</div><div class="t m0 x1 h3 y37 ff2 fs1 fc0 sc0 lsd ws0">titanate) ceramic with radial polarization. The sketch of the design is pre-</div><div class="t m0 x1 h3 y38 ff2 fs1 fc0 sc0 ls2 ws0">sented in Figure 2.4, in which a piezoelectric cylinder (2) is mounted on a</div><div class="t m0 x1 h3 y39 ff2 fs1 fc0 sc0 ls29 ws0">rigid base (1). The movement at the free end of the piezoelectric cylinder is</div><div class="t m0 x1 h3 y3a ff2 fs1 fc0 sc0 ls14 ws0">transmitted to a friction pad (3). A spring (4) provides a normal force against</div><div class="t m0 x1 h3 y3b ff2 fs1 fc0 sc0 ls2a ws0">the friction pad to generate friction force. A stopper (5) and a rod (6) serve</div><div class="t m0 x1 h3 y3c ff2 fs1 fc0 sc0 ls2b ws0">to act as the axis of rotation.</div><div class="t m0 x2 h3 y3d ff2 fs1 fc0 sc0 ls2c ws0">The actuator is designed based on the indirect mode of actuation, therefore</div><div class="t m0 x1 h3 y3e ff2 fs1 fc0 sc0 ls21 ws0">reducing the eﬀect of hysteresis in the actuator output. The outer surface of</div><div class="t m0 x1 h3 y3f ff2 fs1 fc0 sc0 ls0 ws0">the piezoelectric cylinder is divided into several sections. In the design, four</div><div class="t m0 x1 h3 y40 ff2 fs1 fc0 sc0 ls6 ws0">sections are used, although the number of sections can be increased if a ﬁner</div><div class="t m0 x1 h3 y41 ff2 fs1 fc0 sc0 lse ws0">resolution is desired. The inner surface is a common electrical contact point.</div><div class="t m0 x1 h3 y42 ff2 fs1 fc0 sc0 lse ws0">With this construction, the cylinder will expand unevenly when a voltage</div></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,-173.234022,-270.244158]}'></div></div>

<div id="pf4" class="pf w2 h0" data-page-no="4"><div class="pc pc4 w2 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625039a86caf596192fdbe78/bg4.jpg"><div class="t m0 x1 h5 y1b ff4 fs3 fc0 sc0 ls16 ws0">14 2 Precision Tracking Motion Control</div><div class="t m0 x7 h5 y43 ff6 fs3 fc0 sc0 ls1a ws0">Fig. 2.4. Construction of partially-rotating actuator</div><div class="t m0 x1 h3 y44 ff2 fs1 fc0 sc0 ls10 ws0">is applied across a particular section. By applying a voltage orderly across</div><div class="t m0 x1 h3 y45 ff2 fs1 fc0 sc0 lsc ws0">adjacent sections in a controlled manner (i.e., V</div><div class="t m0 x8 h6 y46 ff8 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x9 h3 y47 ff2 fs1 fc0 sc0 ls0 ws0">, V</div><div class="t m0 xa h6 y46 ff8 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 xb h3 y47 ff2 fs1 fc0 sc0 ls0 ws0">, V</div><div class="t m0 xc h6 y46 ff8 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 xd h3 y47 ff2 fs1 fc0 sc0 ls0 ws0">, V</div><div class="t m0 xe h6 y46 ff8 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 xf h3 y47 ff2 fs1 fc0 sc0 ls2e ws0">,andsoon),</div><div class="t m0 x1 h3 y48 ff2 fs1 fc0 sc0 lsb ws0">an incomplete rotating wave can be produced at the end of the piezoelectric</div><div class="t m0 x1 h3 y49 ff2 fs1 fc0 sc0 lsc ws0">cylinder, ready to be transmitted to the friction disc mounted on top of the</div><div class="t m0 x1 h3 y4a ff2 fs1 fc0 sc0 lsd ws0">cylinder. To reverse the direction of the rotation, a negative voltage can be</div><div class="t m0 x1 h3 y4b ff2 fs1 fc0 sc0 lsd ws0">applied across the sections. By controlling the duration of the positive and</div><div class="t m0 x1 h3 y4c ff2 fs1 fc0 sc0 ls2f ws0">negative voltage applied, the sweeping angle of the motor can be controlled.</div><div class="t m0 x1 h3 y4d ff3 fs1 fc0 sc0 ls14 ws0">2.1.2 Mathematical Model</div><div class="t m0 x1 h3 y4e ff2 fs1 fc0 sc0 ls6 ws0">In this section, the dynamics of a linear and partially-rotating PA will be dis-</div><div class="t m0 x1 h3 y4f ff2 fs1 fc0 sc0 ls2c ws0">cussed. The piezo design presented can be used for linear or partial-rotation</div><div class="t m0 x1 h3 y50 ff2 fs1 fc0 sc0 lsa ws0">application. The dynamic model is derived from the physical characteristics</div><div class="t m0 x1 h3 y51 ff2 fs1 fc0 sc0 ls31 ws0">of the systems, including its piezoelectric eﬀect and the structure and con-</div><div class="t m0 x1 h3 y52 ff2 fs1 fc0 sc0 ls32 ws0">struction of the piezoelectric element.</div><div class="t m0 x1 h3 y53 ff3 fs1 fc0 sc0 ls17 ws0">Linear PA System</div><div class="t m0 x1 h3 y54 ff2 fs1 fc0 sc0 ls11 ws0">The linear system considered here is based on a piezoelectric stack. The analy-</div><div class="t m0 x1 h3 y55 ff2 fs1 fc0 sc0 ls31 ws0">sis may begin from the constitutive equations of piezoelectricity, followed by</div><div class="t m0 x1 h3 y56 ff2 fs1 fc0 sc0 ls2 ws0">the generation of force, and then the movement of the object.</div><div class="t m0 x2 h3 y57 ff2 fs1 fc0 sc0 ls31 ws0">The eﬀect of piezoelectricity is taken into account by its constitutive equa-</div><div class="t m0 x1 h3 y58 ff2 fs1 fc0 sc0 ls33 ws0">tionsasfollows:</div></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,-162.262385,-270.244158]}'></div></div>

<div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625039a86caf596192fdbe78/bg5.jpg"><div class="t m0 x4 h5 y1b ff4 fs3 fc0 sc0 ls24 ws0">2.1 Piezoelectric Actuators 15</div><div class="t m0 x10 h3 y1c ff7 fs1 fc0 sc0 ls0 ws0">σ</div><div class="t m0 x11 h6 y59 ff9 fs4 fc0 sc0 ls34 ws0">ij</div><div class="t m0 x12 h3 y1c ff2 fs1 fc0 sc0 ls0 ws0">= c</div><div class="t m0 x13 h6 y5a ff9 fs4 fc0 sc0 ls0 ws0">E</div><div class="t m0 x13 h6 y5b ff9 fs4 fc0 sc0 ls34 ws0">ijkl</div><div class="t m0 x14 h3 y1c ff7 fs1 fc0 sc0 ls0 ws0">s</div><div class="t m0 x15 h6 y59 ff9 fs4 fc0 sc0 ls35 ws0">kl</div><div class="t m0 x16 h3 y1c ffa fs1 fc0 sc0 ls0 ws0">− e</div><div class="t m0 x17 h6 y59 ff9 fs4 fc0 sc0 ls35 ws0">kij</div><div class="t m0 x18 h3 y1c ff7 fs1 fc0 sc0 ls0 ws0">V</div><div class="t m0 x19 h6 y59 ff9 fs4 fc0 sc0 ls0 ws0">k</div><div class="t m0 x1a h3 y1c ff7 fs1 fc0 sc0 ls0 ws0">, (2.1)</div><div class="t m0 x1b h3 y5c ff7 fs1 fc0 sc0 ls0 ws0">D</div><div class="t m0 x1c h6 y5d ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x12 h3 y5e ff2 fs1 fc0 sc0 ls0 ws0">= e</div><div class="t m0 x13 h6 y5d ff9 fs4 fc0 sc0 ls34 ws0">ikl</div><div class="t m0 x1d h3 y5e ff7 fs1 fc0 sc0 ls0 ws0">s</div><div class="t m0 x1e h6 y5d ff9 fs4 fc0 sc0 ls35 ws0">kl</div><div class="t m0 x1f h3 y5e ff2 fs1 fc0 sc0 ls0 ws0">+ </div><div class="t m0 x20 h6 y5f ff9 fs4 fc0 sc0 ls0 ws0">S</div><div class="t m0 x20 h6 y60 ff9 fs4 fc0 sc0 ls34 ws0">ij</div><div class="t m0 x21 h3 y5e ff7 fs1 fc0 sc0 ls0 ws0">V</div><div class="t m0 x22 h6 y5d ff9 fs4 fc0 sc0 ls0 ws0">k</div><div class="t m0 x23 h3 y5e ff7 fs1 fc0 sc0 ls0 ws0">, (2.2)</div><div class="t m0 x1 h3 y61 ff2 fs1 fc0 sc0 ls8 ws0">where σ is stress vector, s is strain vector, V is electric ﬁeld vector, D is electric</div><div class="t m0 x1 h3 y62 ff2 fs1 fc0 sc0 ls39 ws0">displacement vector, c is elastic stiﬀness constant matrix, e is piezoelectric</div><div class="t m0 x1 h3 y63 ff2 fs1 fc0 sc0 ls21 ws0">constant matrix, and  is permittivity constant matrix. Superscripts E and S</div><div class="t m0 x1 h3 y64 ff2 fs1 fc0 sc0 ls2 ws0">denote constant electric ﬁeld and constant strain, respectively.</div><div class="t m0 x2 h3 y65 ff2 fs1 fc0 sc0 lsf ws0">The driving force of the linear PA is generated by the piezoelectric stack</div><div class="t m0 x1 h3 y66 ff2 fs1 fc0 sc0 ls4 ws0">cylinder, where each layer exerts a force of F</div><div class="t m0 x24 h6 y67 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x25 h3 y68 ff2 fs1 fc0 sc0 ls28 ws0">. Taking into account the hys-</div><div class="t m0 x1 h3 y69 ff2 fs1 fc0 sc0 ls1f ws0">teresis of the piezoelectric stack, the driving force can be expressed according</div><div class="t m0 x1 h3 y6a ff2 fs1 fc0 sc0 ls3c ws0">to Goldfarb and Celanovic (1997) as follows:</div><div class="t m0 x26 h3 y6b ff7 fs1 fc0 sc0 ls0 ws0">F</div><div class="t m0 x27 h6 y6c ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x28 h3 y6d ff2 fs1 fc0 sc0 ls0 ws0">=</div><div class="t m0 x29 h7 y6e ffb fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x2a h3 y6f ff7 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x5 h6 y70 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x1b h3 y71 ff2 fs1 fc0 sc0 ls0 ws0">(x − x</div><div class="t m0 x2b h6 y70 ff9 fs4 fc0 sc0 ls0 ws0">b</div><div class="t m0 x2c h8 y72 ffc fs5 fc0 sc0 ls0 ws0">i</div><div class="t m0 x2d h3 y71 ff2 fs1 fc0 sc0 ls0 ws0">) if |k</div><div class="t m0 x2e h6 y70 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x2f h3 y71 ff2 fs1 fc0 sc0 ls0 ws0">(x − x</div><div class="t m0 x30 h6 y70 ff9 fs4 fc0 sc0 ls0 ws0">b</div><div class="t m0 x31 h8 y72 ffc fs5 fc0 sc0 ls0 ws0">i</div><div class="t m0 x32 h3 y71 ff2 fs1 fc0 sc0 ls0 ws0">)| <f</div><div class="t m0 x33 h6 y70 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x34 h3 y73 ff7 fs1 fc0 sc0 ls0 ws0">f</div><div class="t m0 x35 h6 y74 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x11 h3 y73 ff7 fs1 fc0 sc0 ls1f ws0">sgn(˙x) else</div><div class="t m0 x36 h3 y75 ff7 fs1 fc0 sc0 ls0 ws0">, (2.3)</div><div class="t m0 x37 h3 y76 ff7 fs1 fc0 sc0 ls0 ws0">F</div><div class="t m0 x38 h6 y77 ff9 fs4 fc0 sc0 ls0 ws0">D</div><div class="t m0 x28 h3 y78 ff2 fs1 fc0 sc0 ls0 ws0">=</div><div class="t m0 x39 h6 y79 ff9 fs4 fc0 sc0 ls0 ws0">n</div><div class="t m0 x29 h7 y7a ffb fs1 fc0 sc0 ls0 ws0"></div><div class="t m0 x3a h6 y7b ff9 fs4 fc0 sc0 ls0 ws0">i=1</div><div class="t m0 x10 h3 y78 ff7 fs1 fc0 sc0 ls0 ws0">F</div><div class="t m0 x3b h6 y77 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x1c h3 y78 ff7 fs1 fc0 sc0 ls0 ws0">, (2.4)</div><div class="t m0 x1 h3 y7c ff2 fs1 fc0 sc0 ls8 ws0">where k is the stiﬀness, x is the position, x</div><div class="t m0 x3c h6 y7d ff9 fs4 fc0 sc0 ls0 ws0">b</div><div class="t m0 x3d h3 y7e ff2 fs1 fc0 sc0 lsd ws0">is the position of the block, f is</div><div class="t m0 x1 h3 y7f ff2 fs1 fc0 sc0 ls42 ws0">the breakaway force, and index (.)</div><div class="t m0 x2c h6 y80 ff9 fs4 fc0 sc0 ls0 ws0">i</div><div class="t m0 x3e h3 y81 ff2 fs1 fc0 sc0 ls43 ws0">denotes each layer of the stack.</div><div class="t m0 x1 h3 y82 ff3 fs1 fc0 sc0 ls44 ws0">Partially-Rotating PA System</div><div class="t m0 x1 h3 y83 ff2 fs1 fc0 sc0 lsb ws0">In this section, the dynamics of a partially-rotating system is discussed. The</div><div class="t m0 x1 h3 y84 ff2 fs1 fc0 sc0 ls38 ws0">analysis includes the piezoelectric eﬀect and external eﬀect from the spring.</div><div class="t m0 x1 h3 y85 ff2 fs1 fc0 sc0 ls0 ws0">The piezoelectric eﬀect is analyzed in a similar manner to that used for the</div><div class="t m0 x1 h3 y86 ff2 fs1 fc0 sc0 ls1d ws0">linear system as explained in Equations (2.1) and (2.2).</div><div class="t m0 x2 h3 y87 ff2 fs1 fc0 sc0 lsa ws0">Such a partial-rotation system has been used for piercing of eggs in intra-</div><div class="t m0 x1 h3 y88 ff2 fs1 fc0 sc0 ls9 ws0">cytoplasmic sperm injection. Detailed dynamic of the system is presented as</div><div class="t m0 x1 h3 y89 ff2 fs1 fc0 sc0 ls21 ws0">follows.</div><div class="t m0 x2 h3 y8a ff2 fs1 fc0 sc0 ls12 ws0">Because the motor is pressed by a spring force, an axial loading is present</div><div class="t m0 x1 h3 y8b ff2 fs1 fc0 sc0 lsa ws0">in the cylinder. The cylinder is regarded as a tube.</div><div class="t m0 x2 h3 y8c ff2 fs1 fc0 sc0 ls6 ws0">The following force equation applies:</div><div class="t m0 x3f h3 y8d ff7 fs1 fc0 sc0 ls0 ws0">F</div><div class="t m0 x13 h6 y8e ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x40 h3 y8f ff2 fs1 fc0 sc0 ls0 ws0">= k</div><div class="t m0 x41 h6 y8e ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x42 h3 y8f ff7 fs1 fc0 sc0 ls0 ws0">y</div><div class="t m0 x43 h6 y8e ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x44 h3 y8f ff7 fs1 fc0 sc0 ls0 ws0">, (2.5)</div><div class="t m0 x1 h3 y90 ff2 fs1 fc0 sc0 ls8 ws0">where F</div><div class="t m0 x45 h6 y91 ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x46 h3 y92 ff2 fs1 fc0 sc0 ls9 ws0">is the spring force, k</div><div class="t m0 x47 h6 y91 ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x48 h3 y92 ff2 fs1 fc0 sc0 ls2b ws0">is the spring constant, and y</div><div class="t m0 xe h6 y91 ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x49 h3 y92 ff2 fs1 fc0 sc0 ls17 ws0">is the spring</div><div class="t m0 x1 h3 y93 ff2 fs1 fc0 sc0 lsa ws0">displacement. The compression force is within control since the spring dis-</div><div class="t m0 x1 h3 y94 ff2 fs1 fc0 sc0 ls10 ws0">placement, y</div><div class="t m0 x4a h6 y95 ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x4b h3 y96 ff2 fs1 fc0 sc0 ls9 ws0">, is adjustable. The spring constant depends on the type of the</div><div class="t m0 x1 h3 y97 ff2 fs1 fc0 sc0 ls29 ws0">spring.</div><div class="t m0 x2 h3 y98 ff2 fs1 fc0 sc0 ls22 ws0">The spring force causes the cylinder to contract axially due to stress:</div><div class="t m0 x4c h3 y99 ff7 fs1 fc0 sc0 ls0 ws0">σ</div><div class="t m0 x4d h6 y9a ff9 fs4 fc0 sc0 ls45 ws0">ax</div><div class="t m0 x4e h3 y9b ff2 fs1 fc0 sc0 ls0 ws0">=</div><div class="t m0 x4f h3 y9c ff7 fs1 fc0 sc0 ls0 ws0">F</div><div class="t m0 x50 h6 y9d ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x51 h3 y9e ff7 fs1 fc0 sc0 ls0 ws0">A</div><div class="t m0 x48 h6 y9f ff9 fs4 fc0 sc0 ls0 ws0">a</div><div class="t m0 x2c h3 y9e ff7 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x1d h3 ya0 ff2 fs1 fc0 sc0 ls0 ws0">=</div><div class="t m0 x3c h3 ya1 ff7 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x4 h6 y9d ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x52 h3 ya2 ff7 fs1 fc0 sc0 ls0 ws0">y</div><div class="t m0 x18 h6 y9d ff9 fs4 fc0 sc0 ls0 ws0">s</div><div class="t m0 x53 h6 ya3 ff8 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x53 h6 ya4 ff8 fs4 fc0 sc0 ls0 ws0">4</div><div class="t m0 x54 h3 ya5 ff7 fs1 fc0 sc0 ls0 ws0">π (d</div><div class="t m0 x21 h6 ya6 ff8 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x21 h6 ya7 ff8 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x55 h3 ya5 ffa fs1 fc0 sc0 ls0 ws0">− d</div><div class="t m0 x56 h6 ya6 ff8 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x56 h6 ya7 ff8 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x31 h3 ya5 ff2 fs1 fc0 sc0 ls0 ws0">)</div><div class="t m0 x57 h3 ya8 ff7 fs1 fc0 sc0 ls0 ws0">, (2.6)</div><div class="t m0 x1 h3 ya9 ff2 fs1 fc0 sc0 ls8 ws0">where σ</div><div class="t m0 x45 h6 yaa ff9 fs4 fc0 sc0 ls45 ws0">ax</div><div class="t m0 x58 h3 yab ff2 fs1 fc0 sc0 ls9 ws0">is the axial stress (stress due to axial force), A</div><div class="t m0 x59 h6 yaa ff9 fs4 fc0 sc0 ls45 ws0">ax</div><div class="t m0 x5a h3 yab ff2 fs1 fc0 sc0 ls46 ws0">is the area of</div><div class="t m0 x1 h3 yac ff2 fs1 fc0 sc0 lsa ws0">the cylinder, and d</div><div class="t m0 x5b h6 yad ff8 fs4 fc0 sc0 ls0 ws0">2</div><div class="t m0 x28 h3 yae ff2 fs1 fc0 sc0 ls3b ws0">and d</div><div class="t m0 x34 h6 yad ff8 fs4 fc0 sc0 ls0 ws0">1</div><div class="t m0 x6 h3 yae ff2 fs1 fc0 sc0 ls9 ws0">are the outer and inner diameter of the cylinder,</div><div class="t m0 x1 h3 yaf ff2 fs1 fc0 sc0 ls2c ws0">respectively; and the strain:</div></div><div class="pi" data-data='{"ctm":[2.037103,0.000000,0.000000,2.037103,-173.234022,-270.244158]}'></div></div>