中國地質(zhì)大學(xué)長城學(xué)院 本科畢業(yè)設(shè)計(jì)外文資料翻譯 系 別： 工程技術(shù)系 專 業(yè)： 機(jī)械設(shè)計(jì)制造及其自動(dòng)化 姓 名： 殷玉磊 學(xué) 號(hào)： 05211521 2015年 4 月 1 日 外文資料翻譯譯文 通過鍵合圖對剪刀舉升原理做動(dòng)態(tài)分析 Md Toufiqul Islam, Cheng Yin, Shengqi Jian, and Luc Rolland 摘要 本文描繪了基于鍵合圖建模框架 的剪刀舉升機(jī)理（即四桿并聯(lián)機(jī)構(gòu)）下普通多體系統(tǒng)動(dòng)力的實(shí)現(xiàn)。剪刀舉升機(jī)理是高低汽車工業(yè)的首選，系統(tǒng)有一個(gè)自由度，有一些源于經(jīng)典力學(xué)剛體動(dòng)力學(xué)方程的程序 (例如 Classic Newton-DAlembert, Newton-Euler, Lagrange, Hamilton, kanes to name a few)，但是這些對于大型復(fù)雜系統(tǒng)計(jì)算量特別大，從而容易出錯(cuò)，這里我們將生成機(jī)理的多體動(dòng)力學(xué)模型鍵合圖，因?yàn)樗鼮椴话魏我蚬P(guān)系沖突和控制工程的閉環(huán)運(yùn)動(dòng)模型提供了靈活性。本文中的機(jī)理是建模和模擬，以便評估一些特定應(yīng)用需求，如動(dòng)態(tài)和位置精度。提出的機(jī)理的多體動(dòng)力模型提供一個(gè)準(zhǔn)確快捷的方法來交接分析動(dòng)態(tài)機(jī)理，這里沒有采用剪刀舉升機(jī)構(gòu)。仿真給了一個(gè)關(guān)于處于線性位移機(jī)理的不同鏈接長度的電動(dòng)機(jī)轉(zhuǎn)矩大小的清晰想法。 1、 介紹 剪刀舉升類型升降平臺(tái)的主要用途是有無人工的負(fù)載垂直運(yùn)動(dòng)，它們被廣泛用于組裝工作 (如飛機(jī)零部件裝配 ,發(fā)動(dòng)機(jī)部件組裝 )，維護(hù)結(jié)構(gòu)也可用于內(nèi)部材料運(yùn)輸系統(tǒng)。如果正確安裝在汽車或卡車，即適應(yīng)不同的用途，剪刀舉升機(jī)構(gòu)可以移動(dòng) 1。大型復(fù)雜的機(jī)械系統(tǒng)越來越重要，為了剪刀舉升機(jī)理的特有性和功能性，動(dòng)態(tài)仿真系統(tǒng) 非常重要，因?yàn)樗麄兊闹饕δ苁翘岣邉趧?dòng)者的工具和負(fù)載所需的高度 ,同時(shí)允許操作員控制電梯的運(yùn)動(dòng)和位置。因此剪刀舉升機(jī)理的適當(dāng)設(shè)計(jì)、制造和維護(hù)，提高了勞動(dòng)者的工作效率和安全系數(shù) 2。不幸的是，剪刀舉升操作中也會(huì)發(fā)生致命和非致命事故事故 3,4。因此對于調(diào)查系統(tǒng)的動(dòng)態(tài)行為，適當(dāng)?shù)膭?dòng)態(tài)模型是很有必要的。這種機(jī)理有四桿機(jī)構(gòu)和曲柄滑塊機(jī)構(gòu)組成，圖 1顯示了這兩種機(jī)構(gòu)，圖 2顯示了完整的剪刀舉升機(jī)理。這些鏈接形成了一個(gè)平行運(yùn)動(dòng)機(jī)理并有轉(zhuǎn)動(dòng)關(guān)節(jié)鏈接。每個(gè)四個(gè)鏈接形成一個(gè)所有鏈接長度一樣的菱形結(jié)構(gòu)，驅(qū)動(dòng)機(jī)構(gòu)與位于地面移動(dòng)平臺(tái) 的底部連接，基于設(shè)計(jì)準(zhǔn)則，驅(qū)動(dòng)機(jī)構(gòu)可以是電動(dòng)、液壓、氣動(dòng)，系統(tǒng)可以以不同的方式連接它們，連接它們的一些受歡迎方法用來連接彎管接頭的中心，或者作為曲柄滑塊固定在底座上，或者連接到執(zhí)行機(jī)構(gòu)（如液壓、氣動(dòng)）。這里我們把直流電機(jī)作為驅(qū)動(dòng)機(jī)構(gòu)，它位于底部與曲柄滑塊連接，第二節(jié)中介紹關(guān)于多體動(dòng)力學(xué)的問題，第三節(jié)分析機(jī)構(gòu)運(yùn)動(dòng)，第四節(jié)介紹系統(tǒng)動(dòng)態(tài)模型的開發(fā)過程，第五節(jié)提出了仿真結(jié)果，第六節(jié)進(jìn)行模型驗(yàn)證，第七節(jié)討論模型，最后，第八節(jié)得出結(jié)論。 圖 1.四桿機(jī)構(gòu)和曲柄滑塊機(jī)構(gòu) 圖 2.2個(gè)剪刀舉升機(jī)構(gòu)的基本框架 2、 背景研究 1603年，縮放儀由 Christopher Scheiner發(fā)明，被視為四桿機(jī)構(gòu)的第一個(gè)例子 5，后來， James Watt提出四桿機(jī)械裝置，可以做近似直線運(yùn)動(dòng) 6。一些機(jī)構(gòu)用來產(chǎn)生直線輸出運(yùn)動(dòng)，其輸入元素在一天旋轉(zhuǎn)、震蕩、移動(dòng)的直線上。由四連桿機(jī)構(gòu)組成的剪刀舉升裝置是 Larson等人于 1966年創(chuàng)建的 7，這種機(jī)械裝置與可擴(kuò)展的負(fù)載升降機(jī)構(gòu)相關(guān)，更特別的是，這涉及到包括相對立的剪刀機(jī)構(gòu)，每一個(gè)剪刀機(jī)構(gòu)包含一對完全連接的剪刀手臂，伴隨著升降相對關(guān)鍵的運(yùn)動(dòng)。與當(dāng)代設(shè)計(jì)相比，可以提供相對大量鏈接長度相對較 短向上擴(kuò)展，由于是并行裝置，重量非常小剛性更好。這種機(jī)制的另一個(gè)優(yōu)點(diǎn)是 ,它是相對自由的復(fù)雜的連鎖系統(tǒng) ,但仍有高能力的擴(kuò)展與短臂。 A 多體動(dòng)力學(xué) 在 6中討論奇點(diǎn)配置和菱形的運(yùn)動(dòng)機(jī)構(gòu)，在 8中探討了基于 Matlab仿真描繪系統(tǒng)運(yùn)動(dòng)分析的另一種方法。盡管有兩篇文獻(xiàn)提到了動(dòng)態(tài)剪刀舉升機(jī)構(gòu)的處理，文獻(xiàn)數(shù)量有限，但是已經(jīng)詳盡研究了多體動(dòng)力學(xué)建模與仿真的一般問題。尤其是多體動(dòng)力分析由于像 Newton, DAlembert, Euler and Lagrange這樣的先驅(qū)有了豐富的歷史的發(fā)展。在 910中可以發(fā) 現(xiàn)關(guān)于多體動(dòng)力學(xué)的文學(xué)綜述。首先，基于牛頓力學(xué)研究多體動(dòng)力學(xué)系統(tǒng)，歐拉后來用框架來研究剛體運(yùn)動(dòng)，他還利用 freebody原理建模約束和關(guān)節(jié)，到目前為止牛頓歐拉方程適用于多體動(dòng)力學(xué)的研究。與歐拉同時(shí)代的拉格朗日還建立了約束力學(xué)系統(tǒng)的系統(tǒng)分析，變分原理應(yīng)用于系統(tǒng)的總動(dòng)能和勢能考慮其運(yùn)動(dòng)學(xué)約束和相應(yīng)的廣義坐標(biāo)的拉格朗日方程結(jié)果的動(dòng)態(tài)分析是非常有用的多體系統(tǒng)，進(jìn)化的多體系統(tǒng)動(dòng)力學(xué)理論和經(jīng)典力學(xué)和剛體系統(tǒng)的應(yīng)用可以歸因于有效的數(shù)值方法的發(fā)展解決高度非線性方程組產(chǎn)生的動(dòng)態(tài)系統(tǒng) 9。在這種演變過程中，研究人員對建模形式 做出了貢獻(xiàn)，根據(jù) 9，這些形式可分為兩類 ,即數(shù)字和符號(hào)。在 10中 Sinha等人把建模的方法分為兩組，第一組與由基于圖論的系統(tǒng)建模生成的組件相關(guān)，線性圖理論用于分析多體動(dòng)力學(xué)，例子包括 1112。除了線性圖，結(jié)合圖也被用于多體動(dòng)力學(xué)模型，如 131415。在 16中，Diaz等人描述了 3D立體建模的線性圖表方法，他們還直接推論線性圖表反映了系統(tǒng)的拓?fù)浣Y(jié)構(gòu)；因此，它更容易讓非專業(yè)人士創(chuàng)建系統(tǒng)描述。第二組是基于模塊化、面向?qū)ο蟮哪Ｐ?,可以分層次組合成完整的系統(tǒng)，例如 1718。在 19中 Antic等人描繪了結(jié)合圖的優(yōu)點(diǎn)，他們提到理論上它可以描繪子系統(tǒng)和形式主義之間的層次結(jié)構(gòu)和聯(lián)系。這提供了計(jì)算機(jī)建模和仿真的支持。 B 計(jì)算多體動(dòng)力學(xué) 對于建模機(jī)理，多體系統(tǒng)社區(qū)開發(fā)了許多軟件工具；然而，它們在模型描述、力學(xué)基本原理的選擇和拓?fù)浣Y(jié)構(gòu)上廣泛不同，以至于不存在統(tǒng)一的描述模型 9。在20Gillespie等人提出了一種計(jì)算多體動(dòng)力學(xué)的全面審查。一些商業(yè)軟件包可用于數(shù)值解決多體動(dòng)力學(xué)問題，其中一些基于計(jì)算機(jī)輔助工程 (CAE)；例如包括 ADAMS21 DADS 22 和 MESA VERDE 23。 Dymola和 Modelica是兩個(gè)大型系統(tǒng)建模的面向?qū)ο蟮慕ＵZ言 16。 20-Sim軟件是一個(gè)鍵合圖模型多體動(dòng)力學(xué)系統(tǒng)非常有效的和高效的工具 24。它提供了允許創(chuàng)建模型快速和直觀的工具，模型可以通過使用方程 ,建立塊圖、物理組件和結(jié)合圖來建立，它提供了各種建立不同模型的工具箱，模擬和分析他們的性能。該軟件還包含 3 d機(jī)械工具箱為多體建模提供靈活性。 圖 3.動(dòng)力學(xué)分析 3、 機(jī)理的運(yùn)動(dòng)分析 在 7中描述了基于剪刀舉升機(jī)理的短暫的液壓執(zhí)行機(jī)構(gòu)的運(yùn)動(dòng)分析。在多階段剪刀舉升機(jī)理中動(dòng)鏈接形成一個(gè) 菱形的配置， Dr. Rolland做了菱形結(jié)構(gòu)配置和重復(fù)菱形結(jié)構(gòu)的詳細(xì)運(yùn)動(dòng)學(xué)分析 5。剪刀舉升的運(yùn)動(dòng)分析可以通過觀察圖 3 的循環(huán)。從圖 3 中 錯(cuò)誤 !未找到引用源。 102 循環(huán)形成一個(gè)等腰三角形，所以 錯(cuò)誤 !未找到引用源。 。輸入為 s 輸出為 h。應(yīng)用勾股定理： 把（ 4）對時(shí)間求導(dǎo)得到速度。 由 (1)我們可以得到反向關(guān)系： 把（ 6）對時(shí)間求導(dǎo)得到： 把 h對 求導(dǎo)得： 圖 3中 H=2h，圖 2中總高度是 4h，把（ 5）和 (7)對時(shí)間求導(dǎo)得到加速度。 4、 動(dòng)力學(xué)機(jī)理 Dong等人調(diào)查剪刀的動(dòng)態(tài)穩(wěn)定性提升 機(jī)制，并基于動(dòng)態(tài)特性的仿真研發(fā)了集總參數(shù)模型 2。這是一個(gè)基于實(shí)際工作的方法而且在整個(gè)系統(tǒng)上沒有提出動(dòng)態(tài)模型，因此本文提出一種動(dòng)態(tài)模型的完整系統(tǒng)，有吸引力的和快速的所需的時(shí)間和精力相比 ,經(jīng)典的動(dòng)力學(xué)建模方法。 圖 4.鏈接示意圖 A、 每個(gè)鏈接的鍵合圖模型 對于一個(gè)單波束的鍵合圖模型身體被認(rèn)為是質(zhì)量和轉(zhuǎn)動(dòng)慣量，外部力量應(yīng)用于 A、B端口，當(dāng)多體系統(tǒng)中的所有身體包含三個(gè)慣性坐標(biāo) (x,y,)，制定會(huì)變得更容易， B點(diǎn)速度相對 G點(diǎn)的速度關(guān)系為： G點(diǎn)是鏈接的重力中心，設(shè) G點(diǎn)到 B點(diǎn)的距離為 r，則方程將是： 根據(jù) 以上方程， B點(diǎn)的 x、 y軸的分量為： （ 14） G點(diǎn)相對 A點(diǎn)的速度為： A點(diǎn)在 x、 y軸的速度矢量和表示為： 為使 A、 B固定，我們只需在附加力的鍵合圖上應(yīng)用零流源或近零， MTF用于鍵合圖的速度限制。圖 4顯示了梁的鍵合圖，其中長度、質(zhì)量和慣性參數(shù)被視為總體參數(shù)。 圖 5.鏈接鍵合圖 B、 寄生剛度和阻尼 為了建模每個(gè)機(jī)械聯(lián)合，要考慮寄生剛度和阻尼。剛度和阻尼是許多機(jī)械系統(tǒng)設(shè)計(jì)的重要標(biāo)準(zhǔn)。剛性聯(lián)軸器彈簧用于使用機(jī)械連接來消除系統(tǒng)的微分因果關(guān)系。我們可以用寄生剛度和 /或電阻元素 移除能量儲(chǔ)存元素之間的依賴關(guān)系 25，圖 6. 顯示了寄生剛度和阻尼結(jié)構(gòu)應(yīng)用于鍵合圖的設(shè)計(jì)。 C、 電機(jī)建模 在提供的直流電機(jī)中，電壓通過串行連接的電感和電阻去向電樞，然后電樞提供電動(dòng)勢作為機(jī)械輸出。為了在鍵合圖中建模直流電機(jī)，阻力和回轉(zhuǎn)器元素用于表示上述標(biāo)準(zhǔn)。電機(jī)軸是通過電感和電阻元件建模，圖 7顯示了直流電機(jī)的鍵合圖模型。 D、 控制機(jī)制 作為單自由度系統(tǒng)， PID控制器已被選中， PID將在當(dāng)前平臺(tái)位置和期望平臺(tái)位置相比較，基于兩個(gè)高度之間的差異控制電動(dòng)機(jī)的輸出，圖 8顯示控制機(jī)制的示意圖。 圖 6. 寄生剛度和阻尼 圖 7. 直流電機(jī)鍵合圖 PID 分別代表比例、積分和導(dǎo)數(shù)， PID 參數(shù)調(diào)優(yōu)實(shí)現(xiàn)了反復(fù)試驗(yàn)優(yōu)化方法，這主要基于猜測和檢測。在這種方法中，主要貢獻(xiàn)是比例作用 ,它可以通過積分和微分作用。控制機(jī)制是有限制的，它將把輸出限制在制定范圍內(nèi)，以便不切實(shí)際的的值驅(qū)動(dòng)電機(jī)，限制塊后面是一個(gè)調(diào)制源。有兩種可用的鍵合圖工作源，一個(gè)是固定的 ,另一個(gè)是可變的，對于 PID控制器提供的電機(jī)不同輸入，這里我們使用不同的變量工作源。電機(jī)之后的變壓器把旋轉(zhuǎn)運(yùn)動(dòng)變?yōu)橹本€運(yùn)動(dòng)。 圖 8.控制機(jī)制 5、 仿真 表 1.仿真結(jié)果 圖 9.菱形階梯剪刀 舉升平臺(tái)的鍵合圖 A、 向上運(yùn)動(dòng)的仿真（期望高度為 6m，初始高度為 5m） 圖 10.平臺(tái)從初始高度到期望高度 圖 11.原動(dòng)件初始運(yùn)動(dòng)需求 圖 12.原動(dòng)件速率 圖 13.加速效應(yīng) B、 向下運(yùn)動(dòng)仿真（期望高度為 4m，初始高度為 5m） 圖 14. 平臺(tái)從初始高度到期望高度 圖 15. 原動(dòng)件初始運(yùn)動(dòng)需求 圖 16. 原動(dòng)件速率 圖 17.加速效應(yīng) 6、 仿真結(jié)果 對于第一次仿真平臺(tái)從初始高度 5m移動(dòng)到 6m（圖 10），根據(jù)圖 2的剪刀升升機(jī)制基礎(chǔ)上的驅(qū)動(dòng)鏈接應(yīng)該從其原始位置向后的方向在 D增加平臺(tái)的高度從 5米到 6米 (圖11)。圖 12顯示了驅(qū)動(dòng)鏈的速率。衍生物的速度在圖 13中給出了加速度。對于第二次仿真平臺(tái)所需高度被設(shè)定為 4m，比初始高度 5m低（圖 14）。根據(jù)機(jī)制基礎(chǔ)上的驅(qū)動(dòng)鏈接應(yīng)該前進(jìn)的方向在點(diǎn) D降低平臺(tái)的高度和平臺(tái)高度應(yīng)該從 5米減少到 4米 (圖 15)。圖16顯示了驅(qū)動(dòng)鏈的速率。圖 17顯示了所需的向下運(yùn)動(dòng)最終效應(yīng)加速度。現(xiàn)在我們可以說從模擬結(jié)果曲線清晰地描述該模型的功能。 7、 討論 本文提出一種充分剪刀舉升控制機(jī)制模型，對比結(jié)果不幸的是沒有得出剪刀舉升機(jī)制的產(chǎn)物。該模型可以描述整個(gè)動(dòng)力學(xué) (位移、速度、加速度等 ),系統(tǒng)的輸入 是電動(dòng)機(jī)轉(zhuǎn)矩，模型的另一個(gè)重要方面是包含控制器的機(jī)制，對于動(dòng)態(tài)系統(tǒng)最傳統(tǒng)的方法可以不包括控制，但鍵合圖提供了包括電機(jī)動(dòng)力學(xué)和控制重要的優(yōu)勢，這實(shí)際上是允許完整的自動(dòng)動(dòng)態(tài)系統(tǒng)仿真，鍵合圖建模框架的另一個(gè)關(guān)鍵優(yōu)勢是快速的分析機(jī)制，剪刀舉升機(jī)制在速度和加速度上響應(yīng)不快，但是我建議在產(chǎn)業(yè)中進(jìn)行快速直線運(yùn)動(dòng)。在產(chǎn)業(yè)里機(jī)制可用于構(gòu)建高性能線性致動(dòng)器，這里鍵合圖建模比方程建模更具優(yōu)勢，因?yàn)樗褂昧藞D形的方法，并能快速簡易實(shí)現(xiàn)，鍵合圖利用二維圖紙的關(guān)系可表達(dá)方程編程相比之下更自然 26。在模擬中不同參數(shù)組合可以用來模擬觀 察反應(yīng)，從仿真輸出來看，可以出發(fā)一些優(yōu)化一些想法，如仿真可用于電動(dòng)機(jī)轉(zhuǎn)矩的優(yōu)化剪刀提升機(jī)制 ,由于鍵合圖仿真非常迅速，幾個(gè)參數(shù)組合可以在很短的時(shí)間內(nèi)模擬，該模型的缺點(diǎn)之一是軟件不可用。為優(yōu)化PID控制器在這里我們使用了猜測和檢查方法，但我們可以實(shí)現(xiàn)任何其他 PID調(diào)優(yōu)技術(shù)，在 20-sim PID塊包含溫順常數(shù)，它影響差異化的行為，理解一個(gè) PID回路是基于線性控制 , 它可能不會(huì)給剪刀舉升平臺(tái)上由非線性方程管理的最好結(jié)果。 8、 結(jié)論 在這項(xiàng)研究中剪刀舉升機(jī)制的動(dòng)態(tài)行為建模和模擬使用商業(yè)軟件包（ 20sim），模型的函數(shù)性 由所需輸出條件的應(yīng)用程序來證明。使用控制機(jī)制的一個(gè)合適的控制器 (PID控制器 )，仿真結(jié)果表明 ,該設(shè)計(jì)幾乎可以模擬一個(gè)實(shí)時(shí)應(yīng)用程序和仿真數(shù)據(jù)可以有效地用于優(yōu)化設(shè)計(jì)，工業(yè)優(yōu)化專業(yè)軟件 (例如設(shè)計(jì)專家 ,一款統(tǒng)計(jì)軟件等 )都可以使用。 20 sim軟件包鍵合圖建模和解決提供了良好的靈活性。盡管程序需要一些時(shí)間學(xué)習(xí)適度陡峭的學(xué)習(xí)曲線，自動(dòng)化的模型生成功能可以在開發(fā)階段的多體系統(tǒng)提供很大的幫助。 外文原文 題目： Dynamic Analysis of Scissor Lift Mechanism through Bond Graph Modeling Md Toufiqul Islam, Cheng Yin, Shengqi Jian, and Luc Rolland AbstractThis paper describes the implementation of general multibody system dynamics on Scissor lift Mechanism (i.e. four bar parallel mechanism) within a bond graph modeling framework. Scissor lifting mechanism is the first choice for automobiles and industries for elevation work. The system has a one degree of freedom. There are several procedures for deriving dynamic equations of rigid bodies in classical mechanics (i.e. Classic Newton-DAlembert, Newton- Euler, Lagrange, Hamilton, kanes to name a few). But these are labor-intensive for large and complicated systems thereby error prone. Here the multibody dynamics model of the mechanism is developed in bond graph formalism because it offers flexibility for modeling of closed loop kinematic systems without any causal conflicts and control laws can be included. In this work, the mechanism is modeled and simulated in order to evaluate several application-specific requirements such as dynamics, position accuracy etc. The proposed multibody dynamics model of the mechanism offers an accurate and fast method to analyze the dynamics of the mechanism knowing that there is no such work available for scissor lifts. The simulation gives a clear idea about motor torque sizing for different link lengths of the mechanism over a linear displacement. I. INTRODUCTION The main use of scissor lift type elevating platforms is vertical transportation of load with or without human. They are widely used for the assembly works (e.g. aircraft parts assembly, motor parts assembly), the maintenance of constructions or they can be used in the inner material transportation system. The scissor lift structure can be mobile if it is mounted on the vehicle or the right truck, i.e. adaptable to different purposes 1. There is an increasing importance of modeling and simulation of complex and large mechanical systems. For scissor lifting mechanism proper and functional dynamic simulation of the system has a great importance as their primary function is to elevate worker tools and load to a desired height while allowing the operator to control the movement and position of the lift. Therefore proper designing, manufacturing and maintenance of scissor lifting mechanism not only increase productivity but also workers safety 2. Unfortunately, fatal accidents and non-fatal incidents have also happened during scissor lift operations 3,4.Therefore a proper dynamic model is very necessary to investigate the dynamic behavior of the system. This mechanism is comprised of two very well-known mechanisms which are four bar mechanism and slider crank mechanism. Fig.1 shows the mentioned two mechanisms and Fig.2 shows the complete system model of the scissor lift mechanism. The links form a parallel kinematic mechanism and they are connected by revolute joints. Each four link form a rhombus like structure and where length of each link is same. There is a driving mechanism connected with the lower end of the moving link which is located at the ground platform. The driving mechanism can be electric, hydraulic or pneumatic based on design criteria. They can be connected to the systems in different ways. Some very popular ways of connecting them are to connect at the center of corner joint or connect it on the base as slider crank or connected to any actuator link (e.g. hydraulic, pneumatic etc.). But here we considered one DC motor as the driving mechanism. It is located at the base and connected with the slider crank. A brief literature review relating to this multibody dynamics problem is presented in Section II. The kinematics of the mechanism is analyzed in Section III. The development process of the dynamic model of the system is offered in Section IV. The result from the simulation is presented in Section V. The model was verified in Section VI. Discussion about the model is presented in the Section VII. Finally, Section VIII offers the concluding remarks. Fig. 1. Four bar and Slider crank mechanism Fig. 2. Basic Construction of 2 stage Scissor Lift Mechanism II. BACKGROUND STUDY In 1603, the Pantograph was invented by Christopher Scheiner, which may be regarded as the first example of the four-bar linkage 5. Later James Watt proposed a four-bar mechanism which can generate roughly a straight line motion 6. Some mechanisms are designed to produce straight-line output motion from an input element which rotates, oscillates or moves also in a straight line. However, the scissor lifting mechanism an extrapolation from the four bar linkage was patented by Larson et al. in 1966 7. This mechanism relates to extensible lift mechanism for elevating a load. More particularly, this relates to such mechanism including an opposed pair of scissor mechanisms, each of which includes a pair of scissor arms pivotally connected together, where relative pivotal movement of the arms accompanies extension of the lift mechanism. In comparison to the contemporary designs, this mechanism can provide relatively large amount of upward extension with a relatively short link length. Weight is very small and rigidity is better due to parallel mechanism. Another big advantage of this mechanism is that it is relatively free of complicated linkage systems, but still capable of high extension with a short arm. A. Multibody Dynamics The singularity configurations and the kinematics of the rhombus part of the mechanism is discussed in 6. Another approach to describe the kinematic analysis of the system based on MATLAB simulation is discussed in 8. Though the mentioned two literatures dealing with the dynamics of the scissor lift mechanism are very limited, the general problem of modeling and simulation of multibody dynamics has been exhaustively studied. Especially analytic multibody dynamics has a rich history of development owing to pioneers like Newton, DAlembert, Euler and Lagrange. Detailed literature reviews on multibody dynamics can be found in 9, 10. Primarily, the study on the dynamics of multibody systems is based on Newtonian mechanics. Euler later contributed the framework to study the motion of rigid bodies. He also used the freebody principle to model constraints and joints. To date the Newton-Euler equations facilitates the study of multibody Dynamics. Eulers coetaneous Lagrange also established a systematic analysis of constrained mechanical systems. The variational principle applied to the total kinetic and potential energy of the system considering its kinematic constrains and the corresponding generalized coordinates result in the Lagrangian equations that are very useful to the dynamic analysis of multibody systems. Evolution of multibody system dynamics from the theory and applications of classical mechanics and rigid body systems can be attributed to the development of efficient numerical means to solve the highly nonlinear equations resulting from the dynamics of the system 9. In this evolution process, several modeling formalisms have been contributed by the researchers. According to 9, these formalisms can be categorized into two classes; namely, numeric and symbolic. Sinha et al. in 10 classified the modeling approaches into two groups. The first group is relevant to component based systems modeling that is based on graph theory. Linear graph theory has been used in the analysis of multibody dynamics; examples include 11, 12. Besides linear graphs, bond graphs have also been used to model multibody dynamics; e.g. 13, 14, 15. Diaz et al. in 16 describes linear graphs method to model 3D body mechanics. They also comment that linear graphs reflect the topology of the system directly; hence, it is easier for nonspecialists to create system descriptions. The second group is based on modular, object oriented models that can be hierarchically combined into complete systems; e.g., 17,18. Antic et al. in 19 describes the advantages of bond graph. They mentioned it can describe hierarchical structure and connections between subsystems of a system and formalism introduced in the theory. This offers computer support for modeling and simulation. B. Computational Multibody Dynamics For modeling of mechanisms the multibody systems community developed many software tools; however, they differ widely in terms of model description, choice of basic principles of mechanics and topological structure so that a uniform description of models does not exist 9. Gillespie et al. presented a comprehensive review on computational multibody dynamics in 20. Several commercial software packages are available to numerically solve the multibody dynamics problem. Some of these are based on Computer Aided Engineering (CAE); examples include ADAMS 21, DADS 22 and MESA VERDE 23. Dymola and Modelica are two object oriented modeling languages for large system modeling 16. 20-Sim software is a very effective and efficient tool to model multibody dynamic systems for bond graph 24. It provides tools that allow creating models very quickly and intuitively. Models can be created by using equations, block diagrams, physical components and bond graphs. It provides various tool boxes to build different models, to simulate them and analyze their performance. The software also contains 3D mechanics toolbox which provides flexibility for multibody modeling. Fig.3. Figure for kinematic analysis III. KINEMATIC ANALYSIS OF THE MECHANISM Brief kinematic analysis of hydraulic actuator based scissor lift mechanism is described in 7. In the multi stages scissor lift mechanism the moving links forms a rhombus configuration. A detailed kinematic analysis of the rhombus structure configuration and repeated rhombus structure is done by Dr. Rolland 5. The kinematic analysis for the scissor lift can be done by looking at the loops of Fig. 3. From fig. 3, = and the loop 102 forming an isosceles triangle. So, l12 = l/2 = l02 (Where l=length of each link). Input is S and output is h. Applying Pythagoras theorem: Differentiating (4) with respect to time gives the velocity. From (1) we can write another inverse relationship: By differentiating (6) with respect to time we get: The height h with respect to would be: Fig. 3 shows H =2h and the total height from Fig. 2 is 4h. Further differentiation of (5) and (7) with respect to time will give the acceleration. IV. DYNAMICS OF THE MECHANISM Dong et al. investigate the dynamic stability of scissor lift mechanism and a lumped parameter model was developed based on the dynamic characteristics resulted from the simulation 2. It was an approach based on practical work and no dynamic model was presented on the whole system. Hence this paper presents a dynamic model of the complete system which is attractive and fast compared to time and effort required by classic dynamics modeling methods. Fig.4. Schematic of a Single Link A. Bond graph Model of Each Link For bond graph modeling of a single beam a body is considered with mass and rotational inertia. External forces are applied at port A and B. Formulation become much easier when all bodies in a multibody system contains three inertial coordinate(x,y,). The velocity of the point B with respect to the point G can be formulated as: Where the point G is the center of the gravity of the link. If the distance from the point G to the point B is r then the equation will be: From the above equation the velocity component along x axis and velocity component along y axis of the point B are: To make A or B fixed we just need to apply zero flow source or approximately zero on parasitic spring in the bond graph. MTF is used in the bond graph to get the velocity constraints. Fig.4 shows the bond graph model of a single beam where length, mass and inertia parameters are considered as global parameter. Fig. 5. Bond graph of a single link B. Parasitic Stiffness and Damping To model each mechanical joint parasitic stiffness and damping are considered. Stiffness and damping are important criteria for many mechanical system designs. Stiff coupling springs are useful to use at mechanical joints to eliminate derivative causality of the system. We can use parasitic stiffness and/or resistive elements to remove dependencies among energy storage eleme