JH36-400機械壓力機機身部分及其上橫梁加工工藝的設計【說明書+CAD】

JH36-400機械壓力機機身部分及其上橫梁加工工藝的設計【說明書+CAD】,說明書+CAD,JH36-400機械壓力機機身部分及其上橫梁加工工藝的設計【說明書+CAD】,jh36,機械,壓力機,機身,部分,部份,及其,橫梁,加工,工藝,設計,說明書,仿單,cad 譯文 滑動塊轉動曲柄機構的設計 第一部分:多階段動作產生 摘要 設計滑動塊曲柄機構到完成多階段運動產生應用代表性地完成可通過可調節(jié)的平面的四桿運動加速器,這一方法是被提出來了的。這個方法的好處有兩點：第一,多階段的規(guī)定剛體位置是可完成的利用一個機構同較少數(shù)活動部分,它用的活動部分比那平面的四桿機構要少。第二，在這階段滑動塊曲柄運動加速器可以完成階段的規(guī)定剛體位置不需任何人工的或自動調整的它的運動副。一滑塊路徑啟動曲柄運動加速器到完成剛體位置的二階段是設計者利用第7項命令多項式去聯(lián)接那那可調節(jié)的平面四桿運動產生器推桿連接的運動副。這個多項式產生平滑平穩(wěn)徑向位移、速度、加速度和帶有轉角輪廓的邊界條件,這些是可以被呈現(xiàn)的。在本研究中例子問題是考慮一個二階段的運動副轉動平面四桿的機構裝置的調整。 2004 Elsevier 公司版權所有。 1.介紹 平面的四連桿機構廣泛的被被用于機械系統(tǒng)和裝置。由于平面四桿機構的平面運動學,平面的類型和連接軸方向,它可以是實際的設計并且實現(xiàn)這些機構（與大部分四桿空間機構相比較）。 除此之外, 平面的四連桿機構有一廣范系列的圖解式和解析設計和分析方法。 機構分析中產生的問題要求一個剛體是通過一系列規(guī)定位置而被控制的.圖.1顯示了四連桿機構可用于生產這個運動通過制造那剛體作為它的耦合器連接的一部分。圖.2顯示了一部裝配機器的三個位置的運動產生.理想耦合器的運動只能由個別的離散的精確位置近似表示。由于一連接點只有一有限數(shù)的有效的尺寸,設計師可以只規(guī)定一有限數(shù)的精確點。一個四連桿機構可以滿足直到五個規(guī)定位置由那運動產生問題。然而,一個可調節(jié)的四連桿機構可以滿足超過五個給定的位置用這一樣的硬件。一四桿機構的運動副可以用二種不同的方法來調整:可調曲柄／推桿長度（圖.3）和安裝曲柄／推桿聯(lián)桿調整（圖.4）。那可調節(jié)的傳動機構可以供應解決一般平面運動（圖.5）兩個階段的方法。 如果在調整之后,一四連桿機構在第一階段是被設計能達到達位置1，2和3,同相地、這同樣的接合在第二階段可以到達三個新的位置4,5和6。兩個階段的運動可以利用一樣部件通過校準一個或多個接合叁數(shù)來完成,接合可以在這些位置精確地產生運動并且近似表示在其他的位置的運動。連接器的真實運動是精密位置被用愈較多,對理想的運動也愈靠近。 圖.1 平面四桿機構 圖.2 平面四桿卸栽機構 圖.3 可調節(jié)長度的曲柄機構 圖.4 固定長度的曲柄機構 關于運動的產生在可調節(jié)的傳動機構的區(qū)域內,在已出版的作品里[1-19] 略微被限制。上述的工作包括包括 Ahmad和 Waldron的工作[1],他們發(fā)展一方法關于綜合處理一四連桿機構同可調傳動裝置安裝。他們解決二個階段的問題用一最大量總數(shù)的五個位置。Tao和Krishnamoorthy[2]發(fā)明了繪圖的合成程序用尖頭產生可變耦合器彎曲。 圖.5 規(guī)定剛體位置的兩階段 Mcgovern和Sandor[ 3,4]提出了綜合處理可調節(jié)的機構的功能和路徑生成利用合成物耦合器的方法。Funabashietal.[５]介紹一般方法到設計平面,球體和空間機構哪個可以校準的調整輸入/輸出關系。Shoup６設計可調節(jié)的存在于空間的滑動塊曲柄機構被當作可變的換置使用泵。 Cheun-chom 和 Kota[7]介紹了一般的方法關于合成可調節(jié)的機構利用可調節(jié)的二數(shù)。Wilhelm８呈現(xiàn)了為可調節(jié)的四桿機構的二相運動產生問題的合成方法。Wangand Sodhi[9] 呈現(xiàn)了解決為那在每二時期中的二個階段的恰當?shù)囊苿鱼q鏈的三個位置的問題。Russell和Sodhi[10,11] 最近有耐心的介紹這些方法為綜合處理可調節(jié)的空間的機構對于多階段運動產生,空間的RRSS機構可以是綜合處理到完成階段的規(guī)定精確的剛體位置。最近Chang[12] 呈現(xiàn)了可調整四桿機構用指定的切線速度產生圓形的弧。 如果存在過任何性能有關限制到那可調節(jié)的平面的四桿機構, 人工控制或自動控制是被要求完成所有的規(guī)定階段在多階段的申請。人工控制可能是耗費時間的—尤其是如果那調整過程處于被涉及到的收上位置及機件控制被經常地運用。實現(xiàn)自動化調整能力可能使機制不實用從財務的立場來說-尤其當操作和維護開支被考慮的時候。 對于一可調節(jié)的平面的四桿運動加速器它包含移動副和連接長度一起控制推桿連接而曲柄連接只能用移動副控制,一等效的滑動塊曲柄運動加速器可以被設計成能完成多階段的規(guī)定剛體的位置。這種方法的好處是規(guī)定剛體位置的多個階段是可利用一機構與較少數(shù)活動件就能實現(xiàn)的,它與那平面的四連桿機構和那滑動塊曲柄運動加速器相比較只用少數(shù)活動件就可以完成階段的規(guī)定剛體位置而不需要任何實際的或自動操作控制的它的移動副在這些階段中。 在這個一工作中,一種方法設計偏置曲柄運動加速器實現(xiàn)一般地多階段運動產生點樣可利用可調節(jié)的平面的四桿運動加速器來完成是已經被提出來了的。一滑塊路徑啟動曲柄運動加速器到完成剛體位置的二階段是設計者利用第7項命令多項式去連接那那可調節(jié)的平面四桿運動產生器推桿連接的移動副。推桿連接的移動副的徑向位移、速度、加速度和參數(shù)也被規(guī)定利用這個多項式的界限條件的情況 2.剛體規(guī)則和多階段運動鏈鎖反應 存在于這個工作中的滑塊曲柄運動加速器設計法可適應事實上任何多階段運動鏈鎖反應可利用的方法,那方法含有移動副的控制與安裝和可分別地調整曲柄和推桿長度。作者[10,11]發(fā)展了他們整個運動階段鏈鎖反應在這一個研究中被利用的方法。 那平面的四桿運動加速器在圖圖.6是圖解說明了的。在本研究中、連線a0-a1的是表示曲柄而連線b0-b1表示搖桿。平面的四連桿機構的桿a0-a1和桿b0-b1必須滿足那固定長條件因為它的安裝和移動副的連接軸要保持平行。給一固定支點b0和一移動的鉸鏈b1它們的長度條件等價于公式(1)當用合成法合成平面的四連桿機構的曲柄和從動件時[20,21]必須被滿足。 等式（1）可以被重新寫成等式（3）。在等式（3）里，變量R表示曲柄或從動件連桿的長度。這一個工作的一個目的是設計一個等效的滑動塊-曲柄運動加速器作為一可調節(jié)的平面的四桿運動機構。雖然平面的四連桿機構中的曲柄和從動件連桿兩者的運動鉸鏈是可調整的,但只有動件連桿的長度可被調整（非那曲柄連桿)。 通過做這些,這個等效滑動塊曲柄運動機構是被設計成將會有一個固定曲柄連桿長度和一滑動塊路徑這就相當于從動件連桿的調解。 圖.6 平面四桿運動加速器及它剛體上的p、q、r點 （3） 方程（2）是一剛體位移矩陣,它是存在于空間的剛體位移矩陣[20,21]的矩陣與逆矩陣之乘積。為一剛體在適當?shù)奈恢谩癷”和那之后的位置“j”制定坐標,矩陣 [ Dij]是一個變換矩陣要求變換坐標從位置“i”到位置“j”變量 p,q和r在等式方程（2）中表示那剛體在二維空間的位置。雖然這一位置的二維空間位置是通常被描述為單個點和一位移角（例如: p和 ),作家選擇描述剛體使用三個點作為計算的目的。如果用戶喜歡描述那剛體利用傳統(tǒng)的的標記,這個位移矩陣在方程等式（2）將被替換為簡單的平面剛體位移矩陣[20,21]。因為有四個變量（b0x、b0y、b1x,and b1y）,一個五個剛體位置的最大值可以被確定,不需要任意的選擇一參數(shù)作為其中的一階段（看表1）。 點p、q和r 將不會全部的落在各剛體位置的同一直性上.拿這個預防措施防止那些在剛體位移矩陣（方程等式（2））中的排變成成比例項的。有比例項的排,這些矩陣不能被倒置的。在表1里、給出了為可調節(jié)的平面的四桿運動機構規(guī)定的剛體位置的最大極限數(shù)目轉動曲柄和從動件連桿的固定和移動的鉸鏈的數(shù)目確定了剛體位置的最大極限數(shù)目。 這個例子問題在這個工作中，一個等效滑動塊曲柄是被設計成能完成一二相移動鉸鏈控制請求為一可調節(jié)的平面的四桿運動加速器。 在這二相中,可調整的移動鉸鏈例子的問題在這一個工作中,需要的未知數(shù)是 a0,a1,a1n,b0,b1和b1n，未知數(shù)a0 和 b0表示平面的四桿機構繁榮固定支點。未知數(shù)a1,a1n,b1和b1n表示那移動鉸鏈在平面四桿機構中的1階段和２階段。由于這些未知數(shù)的中間每一個有二組成物,則總共由12個變量來確定。 表1 可調的平面四桿機構的規(guī)定剛體位置和各階段的變化 方程等式 (4)-(8), 是用來計算六個未知者中的五個在a0, a1和a1n 。這個變量a0x 和連桿長度R是確定了的。 方程等式 (9)-(13), 是用來計算六個未知者中的五個在b0, b1和b1n 。這個變量b0x 和連桿長度R1和R2是確定了的。 3.軌道鏈鎖反應級 在前一單元描述了那多階段運動鏈鎖反應級方法之后,用戶可以用合成法合成一個平面的四桿運動加速器和確定移動副的路徑。這個從動件連桿的運動副的軌道必須以一種方式被連接的,這種方式以允許平滑的變位速度，加速度和 變換在確定運動副的軌道之間。突然的或不連續(xù)的變化將最終導致滑動塊轉動曲柄機構的過度磨損。這等效滑動塊曲柄運動加速器的滑動塊路徑將由該從動件連桿的運動副的路徑和連接他們的軌道組成。 在一可調節(jié)的四連桿機構的操作期間由于在一個特別的階段,這轉動曲柄和從動件連桿的運動副的半徑位置是固定的及這個運動副半徑的速度,加速度和轉角是零。這同樣的適用在可調節(jié)的平面的四連桿機構的運動副,固定連桿長度調整的期間 ,當運動副和連桿長度調整被考慮的時候,這運動副的徑向位置，速度,加速度和轉角進行從這連桿參數(shù)在前階段到這后階段連桿參數(shù)的變化。如果轉變曲線的產生是因為這從動件連桿,及這個曲線圖是分段的連接到這個從動件的移動副的曲線圖上的,這就相當于這個階段前后的變化, 一個單一的滑動塊軌道的形成說明在這些階段之間的變化（或從動件連桿移動副的調整）。 一7次順序多項式[22,23]是要求確定這可調整的平面的四桿運動加速器中從動件的運動副的徑向位置，速度，加速度和轉角在這些階段的變化。 這徑向變位，速度，加速度和轉角邊界條件關于這個多項式是 在這一個工作中, R0是從動件連桿在階段一的長度（連桿b0 - b1)而Rf是從動件連桿在階段二的長度（連桿b0 -b1n）。這些約束確定了一線性集的八個方程等式與八個數(shù),它們的解答式是 4. 例問題 帶有固定轉動曲柄的可調整的平面四桿運動加速器的兩個階段的運動副的調節(jié)和從動件從動件長度在這一個斷面是被例證了的。在表2里是列出關于七個規(guī)定剛體位置的點p, q和r在X Y 座標系中的座標。 表2可調整的平面四桿運動加速器規(guī)定剛體的位置 等式方程（4）-（8）用來計算六個未知者中的五個在a0、a1和a1n中。這可變的a0x和連桿長度R1是確定了的（a0x = 0而R1 = 1）利用下列初始值: 這平面的四連桿機構解答表示為 圖.7 可調整的平面四桿運動加速器和相應規(guī)定剛體位置 圖.8 用合成法合成可調整的平面的四桿運動加速器的運動副的軌跡 等式方程.（9）-（13）是用來計算在 b 0 ， b 1 和 b 1n 中六個未知數(shù)中的五個。這可變的b0x和連接長度R1而R2是確定了的（b0x = 1.5、R1 = 1.5、R2 = 1.3）。利用下列初始估計： 這平面的四連桿機構解答表示為: 利用這已計算了的值和運動副的參數(shù),可調整的平面四桿機構運動加速器的結果在圖.7里被說明.在本研究中一等效滑動塊曲柄加速器是被設計成平面的四桿運動加速器的。 用合成法合成可調整的平面的四桿運動加速器在階段一和階段二的開始和結束位置是被說明的在圖.8。 由于這曲柄連桿由一固定的長度的連桿的運動副來控制, 這個 圖.9 等效滑動塊曲柄加速器和剛體的初始位置 滑塊徑向位移 曲柄的角位移 圖.10 合成法合成滑動塊曲柄運動加速器中滑動塊相對曲柄轉角徑向位移 連桿的運動副的全部位置（a1經過a4而a1n經過[ D57] a1n）是位于同一條圓弧上的。而從動件連桿的運動副的位置（b1穿過b4和b1n穿過[ D57] b1n）落在兩條不同的弧上（一個為一個階段）。為了完成等效滑動塊曲柄運動加速器的滑動塊軌道,等式方程（14）是用來計算一連接從動件運動副如在圖.8.的路徑。使用方程（14）和這規(guī)定邊界條件,滑動塊軌道在圖.9是可以被設計的?；瑒訅K軌道產生這徑向變位，速度，加速度和轉角輪廓這在圖.10–13中被說明了。 在表 3 列出是等效平面滑動塊轉動曲柄機構的七個規(guī)定剛體位置點p ,q 和r在X Y 座標系中的值。為了達到位置2,3 和4在表3中,連桿a0–a1需繞X軸分別旋轉到130,125和120。為了要在表 3 中達成位置 5,6 和 7,連桿a0-a1需繞X軸分別旋轉到100,95和90。在這兩個階段,曲柄轉角最初 135 是相對 X軸和剛體點坐標在這個轉動曲柄的位置是表格3中位置 1 中是坐標。 滑塊的徑向速度 曲柄的角位移 圖.11 合成法合成滑動塊曲柄運動加速器中滑動塊相對曲柄轉角徑向速度 滑塊的徑向速度 曲柄的角位移 圖.12 合成法合成滑動塊曲柄運動加速器中滑動塊相對曲柄轉角徑向加速度 曲柄的角位移 圖.12 合成法合成滑動塊曲柄運動加速器中滑動塊相對曲柄轉角徑向加速度 表3 圖.10–13說明了等效滑動塊曲柄運動機構徑向這徑向變位，速度，加速度和轉角(相對于轉動曲柄位移角）在階段1到階段2這其中的變化。這徑向速度，加速度和轉角的邊界條件（等式方程( 16– ( 18)和等式方程( 20) ( 22))是指定到零的,這是為了產生的速度，加速度和轉角輪廓與那外在變化的輪廓是相連的。這徑向位移輪廓邊界條件(等式方程。( 15)和( 19))是表示階段1和階段2從動件連桿的長度( R0 = 1.5和Rf = 1.3)，這也是為了形成的位移與那在外面變化的輪廓是連續(xù)的。 5.討論 由于被綜合的機械裝置所需要，在這一個工作中被呈現(xiàn)的滑件路徑設計方法因只有固定的和運動副對可調整平面的四桿機械裝置對大部份的現(xiàn)有動作是可適用方法。雖然一個二個階段的移動副問題在這一個工作被例證,規(guī)定的剛體位置的另外時期能被插入。 通過計算那平面四桿機構的坐標和移動副分別為附加的階段,和每個另外的階段另外的轉變路徑 (等式方程 (14)-(30)) 等效滑動塊曲柄運動加速器可以被設計成能達到這個附加的階段。 雖然二維的空間的剛體的位置普遍被描述被一點和一個變位角 (p 和 h 舉例來說),作家選擇描述剛體使用三個點作為計算的目的。如果使用者偏愛描述使用的剛體傳統(tǒng)的標記,位移矩陣在方程(2) 將會替換為這傳統(tǒng)的平面剛體被放置成矩陣[20,21]。 計算機輔助設計軟件將規(guī)定在這一個工作和數(shù)學軟件的機械裝置叁數(shù)用來計算機械裝置。這一個軟件使那能夠作成表規(guī)定和有計劃的叁數(shù)被四位有效數(shù)來表示。 6.結論 為滑件曲柄機構的一個設計方法達成多階段動作鏈鎖反應級請求的完成通過平面的四連桿機構帶有可調整的運動副是被呈現(xiàn)在這項研究中的。這種方法的好處是那使用它,規(guī)定剛體使用一機構與較少數(shù)活動部分可完成的多階段位置,它用到的活動部分部分與平面的四連桿機構相比較是少的。這一個方法的另一種利益是使用它,滑塊曲柄機構能被設計達成規(guī)定剛體的各階段不需任何的人工的或自動操作控制它的運動副在這些階段中。一滑塊路徑啟動曲柄運動加速器到完成剛體位置的二階段是設計者利用第7項命令多項式去連接那那可調節(jié)的平面四桿運動產生器推桿連接的移動副。例子問題在本研究中認為一二階段的移動副調整可調整平面的四桿機械裝置。 33 On the design of slider-crank mechanisms. Part I: method is twofold. First, multiple phases of prescribed rigid body positions are achievable using a mech- * Corresponding author. Tel.: +1 973 596 3362; fax: +1 973 642 4282. E-mail address: sodhiadm.njit.edu (R.S. Sodhi). Mechanism and Machine Theory 40 (2005) 285299 Mechanism and Machine Theory 0094-114X/$ - see front matter C211 2004 Elsevier Ltd. All rights reserved. anism with fewer moving parts than the planar four-bar mechanism. Second, the slider-crank motion gen- erator can achieve phases of prescribed rigid body positions without any physical or automated adjustments of its moving pivots between phases. A slider path that enables the slider-crank motion gen- erator to achieve two phases of prescribed rigid body positions is designed by using 7th order polynomials to connect the moving pivot paths of the follower link of the adjustable planar four-bar motion generator. This polynomial generates smooth radial displacement, velocity, acceleration and jerk proles with bound- ary conditions that can be prescribed. The example problem in this work considers a two-phase moving pivot adjustment of a planar four-bar mechanism. C211 2004 Elsevier Ltd. All rights reserved. multi-phase motion generation Kevin Russell a , Raj S. Sodhi b, * a Armaments Engineering and Technology Center, US Army Research, Development and Engineering Center, Picatinny Arsenal, NJ 07806-5000, USA b Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA Received 24 February 2003; received in revised form 12 July 2004; accepted 12 July 2004 Available online 28 September 2004 Abstract A method for designing slider-crank mechanisms to achieve multi-phase motion generation applications typically accomplished by adjustable planar four-bar motion generators is presented. The benet of this doi:10.1016/j.mechmachtheory.2004.07.009 1. Introduction Planarfour-barmechanismsarewidelyusedinmechanicalsystemsanddevices.Duetothepla- nar kinematics, joint type and joint axis orientations of the planar four-bar mechanism, it can be practical to design and implement these mechanisms (compared to most four-bar spatial mecha- nisms). In addition, an extensive array of graphical and analytical design and analysis methods exists for planar four-bar mechanisms. Motion generation problems in mechanism synthesis require that a rigid body be guided through a series of prescribed positions. The four-bar linkage shown in Fig. 1 can be used to pro- duce this motion by making the rigid body as a part of its coupler link. Fig. 2 shows motion gen- erationforthethreepositionsinanassemblymachine.Anidealmotionofthecouplercanonlybe approximated by several discrete precision positions. Since a linkage has only a nite number of signicant dimensions, the designer may only prescribe a nite number of precision points. A four-bar linkage can satisfy up to ve prescribed positions for the motion generation problem. However, an adjustable four-bar linkage can satisfy more than ve given positions with the same Coupler 286 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 Fig. 1. Planar four-bar mechanism. Fig. 2. Planar four-bar loading mechanism. K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 287 hardware. The moving pivots of a four-bar linkage can beadjusted in two dierent ways: with adjustable crank/follower lengths (Fig. 3) and with xed crank/follower link adjustments (Fig. 4). The adjustable linkages can provide solution for two phases of general plane motion (Fig. 5). If a four-bar linkage is designed to reach positions 1, 2 and 3 in phase 1, after the adjustments, the Fig. 3. Adjustable crank length. Fig. 4. Fixed crank length. PHASE ONE PHASE TWO 4 1 2 3 5 6 Fig. 5. Two phases of prescribed rigid body positions. same linkage can reach three new positions 4, 5 and 6 in the second phase 2. Both phases of mo- tioncanbeaccomplishedusingthesamehardwarebyadjustingoneormoreofthelinkageparam- eters. The linkage can create the motion precisely at these positions and will approximate the motion at other positions. The more precision positions are used, the closer to the ideal motion is the actual motion of the coupler. In the area of adjustable linkages for motion generation, published work is somewhat limited 119. Previous work includes the work of Ahmad and Waldron 1 who developed a technique for synthesizing a four-bar linkage with adjustable driven xed pivot. They solved two-phase problemswith a maximumtotal numberofvepositions. Tao andKrishnamoorthy 2 developed graphical synthesis procedures to generate variable coupler curves with cusps. McGovern and Sandor 3,4 presented methods to synthesize adjustable mechanisms for function and path gen- eration using complex variables. Funabashi et al. 5 presented general methods to design planar, sphericalandspatialmechanismswhichcanadjustinputoutputrelationships.Shoup6designed adjustable spatial slider-crank mechanism to be used as a variable displacement pump. Cheun- chom and Kota 7 have presented general methods for the synthesis of adjustable mechanisms using adjustable dyads. Wilhelm 8 developed synthesis techniques for two-phase motion gener- ation problems of adjustable four-bar linkages. Wang and Sodhi 9 developed solutions for the two-phase adjustable moving pivot problems with three positions in each of the two phases. Rus- sell and Sodhi 10,11 recently presented methods for synthesizing adjustable three-dimensional mechanisms for multi-phase motion generation with tolerances. Using these methods, spatial RRSS mechanisms can be synthesized to achieve phases of prescribed precise rigid body positions and rigid body positions with tolerances. Recently Chang 12 presented synthesis of adjustable four-bar mechanisms generating circular arcs with specied tangential velocities. If there is any performance-related limitation to the adjustable planar four-bar mechanism, it is that manual or automated adjustments are required to achieve all of the prescribed phases in multi-phase applications. Manual adjustments can be time consumingespecially if the adjust- mentprocedureis involvedandthemechanismadjustmentsmustbeperformedfrequently.Imple- mentingautomatedadjustmentcapabilitiesmaymakethemechanismimpracticalfromanancial standpointespecially when operations and maintenance expenditures are considered. For an adjustable planar four-bar motion generator that incorporates both moving pivot and link length adjustments for the follower link and only moving pivot adjustments for the crank link, an equivalent slider-crank motion generator can be designed to achieve multiple phases of prescribed rigid body positions. The benets of the method are that multiple phases of prescribed rigid body positions are achievable using a mechanism with fewer moving parts than the planar four-bar mechanism and the slider-crank motion generator can achieve phases of prescribed rigid body positions without any physical or automated adjustments of its moving pivots between phases. In this work, a method to design slider-crank motion generators to achieve multi-phase motion generationapplicationstypicallyaccomplishedbyadjustableplanarfour-barmotiongeneratorsis presented. A slider path that enables the slider-crank motion generator to achieve two phases of rigidbodypositionsisdesignedbyusing7thorderpolynomialstoconnectthemovingpivotpaths of the follower link of the adjustable planar four-bar motion generator. The radial displacement, velocity, acceleration and jerk parameters of the moving pivot of the follower link are also pre- 288 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 scribed using the boundary conditions of these polynomials. 2. Rigid body guidance and multi-phase motion generation The slider-crank motion generator design method presented in this work is adaptable to virtu- ally any multi-phase motion generation method available that incorporates moving pivot adjust- ments with xed and adjustable crank and follower lengths respectively. The authors 10,11 developed the multi-phase motion generation method utilized in this work. The planar four-bar motion generator is illustrated in Fig. 6. In this work, linka 0 a 1 is the des- ignated crank link and linkb 0 b 1 is the designated follower link. Links a 0 a 1 andb 0 b 1 of the pla- narfour-barmechanismmustsatisfytheconstantlengthconditiononlysinceitsxedandmoving pivot joint axes remain parallel. Given a xed pivot b 0 and a moving pivot b 1 the constant length condition in Eq. (1) 20,21 must be satised whensynthesizing the crank and follower links of the planar four-bar mechanism. b j C0b 0 T b j C0b 0 b 1 C0b 0 T b 1 C0b 0 j 2;3; .; n 1 where b 0 b 0 x ; b 0y ;1 b 1 b 1x ; b 1y ;1 b j D ij C138b 1 Eq. (1) follower K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 289 j a j j p Y b p X r j j i 0 1 a 0 a 1 b b q r q i i One objective of this work is to design an equivalent slider-crank motion generator for an adjustable planar four-bar motion generator. Although the moving pivots of both the crank 111 111 can be rewritten as Eq. (3). In Eq. (3), the variable R represents the length of the crank or link. and D ij C138 p jx q jx r jx p jy q jy r jy 2 6 4 3 7 5 p ix q ix r ix p iy q iy r iy 2 6 4 3 7 5 C01 2 Fig. 6. The planar four-bar motion generator with rigid body points p, q and r. and follower link of the planar four-bar mechanism are adjustable, only the length of the follower link will be adjusted (not the crank link). By doing this, the equivalent slider-crank motion gen- erator to be designed will have a xed crank link length and a slider path that accounts for the adjustment of the follower link. b j C0b 0 T b j C0b 0 R 2 j 2;3; .n 3 Eq.(2)isarigidbodydisplacementmatrix.Itisaderivativeofthespatialrigidbodydisplacement matrix 20,21. Given the coordinates for a rigid body in position i and the subsequent j, ma- trixD ij is the transformationmatrixrequiredtotransformcoordinatesfrom positioni toposi- tion space. singlepointandadisplacementangle(pand h forexample),theauthorschosetodescribetherigid Prescribed Number 290 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 Number of unknowns Number of free choices 15 4 0 28 6 311 8 m rigid body position and phase variations for the adjustable planar four-bar mechanism of phases Maximum number of rigid body positions Crank or follower links body using three points for computational purposes. If the user prefers to describe the rigid body using conventional notation, the displacement matrix in Eq. (2) will be replaced with the conven- tional plane rigid body displacement matrix 20,21. Since there are four variables (b 0 x ,b 0y ,b 1x and b 1y ), a maximum of ve rigid body positions can be prescribed, with no arbitrary choice of parameter for one phase (see Table 1). Points p, q and r should not all lie on the same line in each rigid body position. Taking this precaution prevents the rows in the rigid body displacement matrix (Eq. (2) from becoming pro- portional. With proportional rows, this matrix cannot be inverted. In Table 1, the maximum numbers of prescribed rigid body positions for the adjustable planar four-bar motion generator for several phases are given. The number of xed and moving pivot coordinates for the crank and follower links determine the maximum number of rigid body posi- tions. In the example problem in this work, an equivalent slider crank is designed to achieve a two-phase moving pivot adjustment application for an adjustable planar four-bar motion generator. In the two-phase, adjustable moving pivot example problem in this work, the required un- knowns are a 0 , a 1 , a 1n , b 0 , b 1 and b 1n . The unknowns a 0 and b 0 represent the xed pivots of the planar four-bar mechanism. The unknowns a 1 , a 1n , b 1 and b 1n represent the moving pivots in phase 1 and phase 2 of the planar four-bar mechanism. Since each of these unknowns has two components, there are a total of 12 variables to determine. a 0 a 0 x ; a 0y a 1 a 1x ; a 1y a 1n a 1nx ; a 1ny b 0 b 0 x ; b 0y b 1 b 1x ; b 1y b 1n b 1nx ; b 1ny Table 1 j. Variablesp,qandrinEq. (2)representthepositionoftherigidbodyintwo-dimensional Althoughthepositionofarigidbodyintwo-dimensionalspaceiscommonlydescribedbya 5+3(m C0 1) 2 + 2m 0 13 1 0 13 1 0 1 D 14 C138b 1 C0b 0 T D 14 C138b 1 C0b 0 C0R 2 0 11 3. Trajectory After willallow moving on the will be them. During positions ities, pivot, ing pivot K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 291 jerks of the moving pivots undergo a transition from the link parameters in the former phase to the link parameters inthe latter phase. Iftransition curvesare generatedfor the follower link, and these smoothdisplacement,velocity,accelerationandjerktransitionsbetweenthedetermined pivot paths. Abrupt or discontinuous transitions will ultimately result in excessive wear slider-crank mechanism. The slider path of the equivalent slider-crank motion generator comprised of the moving pivot paths of the follower link and the trajectories that connect the operation of an adjustable four-bar mechanism within a particular phase, the radial of the moving pivots of the crank and follower links are constant and the radial veloc- accelerations and jerks of these moving pivots are zero. The same holds true during moving constant link length adjustments of the adjustable planar four-bar mechanism. When mov- and linklengthadjustments considered,the radialpositions, velocities, accelerations and tion, the user can synthesize a planar four-bar motion generator and determine the paths of its moving pivots. The moving pivot paths of the follower link must be connected in a manner that 1 D 56 C138b 1n C0b 0 T D 56 C138b 1n C0b 0 C0R 2 1 0 12 D 57 C138b 1n C0b 0 T D 57 C138b 1n C0b 0 C0R 2 1 0 13 generation incorporating the multi-phase motion generation method described in the previous sec- Eqs. (4)(8), wereused to calculateveof the six unknownsina 0 ,a 1 anda 1n . The variablea 0 x and the link length R 1 are specied. D 12 C138a 1 C0a 0 T D 12 C138a 1 C0a 0 C0R 2 1 0 4 D 13 C138a 1 C0a 0 T D 13 C138a 1 C0a 0 C0R 2 1 0 5 D 14 C138a 1 C0a 0 T D 14 C138a 1 C0a 0 C0R 2 1 0 6 D 56 C138a 1n C0a 0 T D 56 C138a 1n C0a 0 C0R 2 1 0 7 D 57 C138a 1n C0a 0 T D 57 C138a 1n C0a 0 C0R 2 1 0 8 Eqs. (9)(13) were used to calculate ve of the six unknowns in b 0 , b 1 and b 1n . The variable b 0 x and the link lengths R 1 and R 2 are specied. D 12 C138b 1 C0b 0 T D 12 C138b 1 C0b 0 C0R 2 1 0 9 D C138b C0b T D C138b C0b C0R 2 0 10 curves are connected piecewise to the moving pivot curves of the follower corresponding to thephases transition A andjerk tion The are Inthis is the eight equations with eight unknowns whose solutions are 292 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 a 2 R 0 2 25 a 3 R v 0 26 a 1 _ R 0 24 a 0 R 0 23 dh 0 _ R 0 16 dR 2 h 0 dh 2 0 R 0 17 dR 3 h 0 dh 3 0 R v 0 18 Rh f R f 19 dRh f dh f _ R f 20 dR 2 h f dh 2 f R f 21 dR 3 h f dh 3 f R v f 22 work,thetermR 0 isthelengthofthefollowerlinkinphaseone(linkb 0 b 1 )andthetermR f length of the follower link in phase two (link b 0 b 1n ). The constraints specify a linear set of Rh 0 R 0 15 dRh 0 beforeandafterthistransition,asinglesliderpathisgeneratedthatwillaccountforthe between phases (or follower link moving pivot adjustment). 7th order polynomial 22,23 is required to specify the radial position, velocity, acceleration parametersofthemovingpivotofthefollowerlinkoftheadjustableplanarfour-barmo- generator during the transition between phases. Rha 0 a 1 h a 2 h 2 a 3 h 3 a 4 h 4 a 5 h 5 a 6 h 6 a 7 h 7 14 radial displacement, velocity, acceleration and jerk boundary conditions for this polynomial 6 a 4 35 h 4 f R f C0 R 0 C0 _ R 0 h f C0 1 2 R 0 h 2 f C0 1 6 R v 0 h 3 f C18C19 C0 15 h 3 f C0 _ R 0 C0 R 0 h f C0 1 2 R v 0 h 2 f C18C19 5 2h 2 f C0 R 0 C0R v 0 h f 1 6h f R v 0 27 a 5 C084 h 5 f R f C0 R 0 C0 _ R 0 h f C0 1 2 R 0 h 2 f C0 1 6 R v 0 h 3 f C18C19 39 h 4 f C0 _ R 0 C0 R 0 h f C0 1 2 R v 0 h 2 f C18C19 C0 7 2h 3 f C0 R 0 C0R v 0 h f 1 2h 2 f R v 0 28 a 6 70 6 R f C0 R 0 C0 _ R 0 h f C0 1 R 0 h 2 f C0 1 R v 0 h 3 f C18C19 34 5 C0 _ R 0 C0 R 0 h f C0 1 R v 0 h 2 f C18C19 h 7 f 2 f 6 f h 6 f 2 f 2 v 1 v 4. Example Two-phase xed the X Table Prescribed Phase Position Position Position Position Phase Position Position Position K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 293 C0 h 5 f C0 R 0 C0R 0 h f 6h 4 f R 0 30 problem moving pivot adjustments of the adjustable planar four-bar motion generator with crank and adjustable follower lengths are exemplied in this section. Listed in Table 2 are - and Y-coordinates of p, q and r for seven prescribed rigid body positions. 2 rigid body positions for the adjustable planar four-bar motion generator pqr 1 1 C00.5175, 0.9640 C00.2148, 1.5049 0.3551, 1.3103 2 C00.4502, 1.0207 C00.1413, 1.5581 0.4263, 1.3570 3 C00.3786, 1.0720 C00.0645, 1.6064 0.5011, 1.3997 4 C00.3030, 1.1173 0.0152, 1.6492 0.5792, 1.4382 2 5 C00.0583, 1.2042 0.4515, 1.6834 0.9782, 1.3914 6 0.1449, 1.2155 0.5383, 1.6945 1.0648, 1.4023 h f 2 6 h f 2 13 2h 4 f C0 R 0 C0R v 0 h f 1 2h 3 f R v 0 29 a 7 C020 R f C0 R 0 C0 _ R 0 h f C0 1 R 0 h 2 C0 1 R v 0 h 3 C18C19 10 C0 _ R 0 C0 R 0 h f C0 1 R v 0 h 2 C18C19 7 0.2319, 1.2195 0.6249, 1.6988 1.1517, 1.4070 Eqs. (4)(8) were used to calculate ve of the six unknowns in a 0 , a 1 and a 1n . The variable a 0 x and the link length R 1 were specied (a 0 x =0 and R 1 =1). Using the following initial guesses: a 0y 0:1 a 1 C00:5;0:5 a 1n C00:5;0:5 the planar four-bar mechanism solutions converged to a 0y 0:0761 a 1 C00:7049;0:7859 a 1n C00:1739;1:0608: a0 b0 a1 b1 b1n X Y a1n r1 p1 p2 p3 p4 p5 p6 p7 q1 q2 q3 q4 q5 q6 q7 r2 r3 r4 r5 r6r7 Fig. 7. Adjustable planar four-bar motion generator and corresponding prescribed rigid body positions. a b 4 1n 57 1 a 1 b b a 1n 4 D 57 D 1n a 1n b Y 294 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 b 0 X a 0 Fig. 8. Moving pivot paths of the synthesized adjustable planar four-bar motion generator. Eqs. (9)(13) were used to calculate ve of the six unknowns in b 0 , b 1 and b 1n . The variable b 0 x and the links length R 1 and R 2 were specied (b 0 x =1.5, R 1 =1.5 and R 2 =1.3). Using the following initial guesses: b 0y 0:1 b 1 0:6;1:2 b 1n 0:6;1:2 the planar four-bar mechanism solutions converged to b 0y C00:1064 b 1 0:6821;1:1505 b 1n 1:2964;1:1775: Using the calculated xed and moving pivot parameters, the resulting adjustable planar four- bar motion generator is illustrated in Fig. 7. An equivalent slider-crank motion generator was de- signed for the planar four-bar motion generator in this work. In Fig. 8, the starting and ending positions for the synthesized adjustable planar four-bar mo- tion generator in phases one and two are illustrated. Since the crank link underwent a constant 0 a a 1 b 4 b 1 1n b Y X q1 r1 p1 phase 1 phase 2transition Fig. 9. Equivalent slider-crank motion generator and initial rigid body position. K. Russell, R.S. Sodhi / Mechanism and Machine Theory 40 (2005) 285299 295 1.25 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 crank displacement angle rad 1.30 slider radial displacement 1.35 1.40 1.45 1.50 1.55 Fig. 10. Radial slider displacement versus crank angle for synthesized slider-crank motion generator. link length moving pivot adjustment, all of the moving pivot positions (a 1 through a 4 and a 1n through D 57 a 1n ) for this link lie on the same circle. The moving pivot positions (b 1 through b 4 and b 1n through D 57 b 1n ) for the follower link lie on two dierent circles (one for each phase). To complete the slider path of the equivalent slider-crank motion generator (the path between b 1 and b 1n ), Eq. (14) was used calculate a path to link both of the follower moving pivot arcs in Fig. 8. Using Eq. (14) and the prescri