化工攪拌器的設(shè)計

化工攪拌器的設(shè)計,化工攪拌器的設(shè)計,化工,攪拌器,設(shè)計 目 錄 1．任務(wù)書………………………………………………………1 2．開題報告……………………………………………………2 3．指導(dǎo)教師評閱表……………………………………………4 4．主審教師評審表……………………………………………5 5．畢業(yè)設(shè)計(論文)答辯評審與總成績評定表………………6 6．畢業(yè)設(shè)計說明書……………………………………………7 7．文獻綜述……………………………………………………45 8．文獻翻譯……………………………………………………52 9．光盤 10．設(shè)計圖紙或?qū)嶒灁?shù)據(jù)記錄 黃河科技學(xué)院畢業(yè)設(shè)計（文獻綜述） 第 6 頁 攪拌器的研究與分析 摘要： 攪拌機式攪拌設(shè)備的心臟。在攪拌機設(shè)計及使用過程中，合理的選取攪拌機的結(jié)構(gòu)，運動和工作參數(shù)，直接關(guān)系到混泥土等材料的攪拌質(zhì)量和攪拌效率。論文對攪拌臂的排列、攪拌葉片的安裝角、拌筒長寬比、攪拌機轉(zhuǎn)速和攪拌時間等主要參數(shù)的選取進行分析與實驗研究。通過歸納，給出了雙臥軸攪拌機的主要參數(shù)，包括攪拌臂排列、葉片安裝角、拌筒長寬比、攪拌線速度等；給出了評價攪拌機參數(shù)合理與否的準(zhǔn)則；給出了攪拌臂排列的基本原則。 關(guān)鍵詞：拌臂排列，葉片安裝角，拌筒長寬比，攪拌線速度 1 攪拌機的簡介 通常攪拌裝置由作為原動機的馬達(電動、風(fēng)動或液壓)，減速機與其輸出軸相連的攪拌抽，和安裝在攪拌軸上的葉輪組成 減速機體通過一個支架或底板與攪拌容器相連。當(dāng)容器內(nèi)部有壓力時，攪拌軸穿過底板進入容器時應(yīng)有一個密封裝置，常用填料密封或機械密封。通常馬達與密封均外購，研究的重點是葉輪。葉輪的攪拌作用表現(xiàn)為“泵送”和 渦流”，即產(chǎn)生流體速度和流體剪切，前者導(dǎo)至全容器中的回流，介質(zhì)易位，防止固體的沉淀并產(chǎn)生對換熱熱管束 (如果有)的沖刷；剪切是一種大回流中的微混合，可以打碎氣泡或不可溶的液滴，造成“均勻”。 氣體和低黏度液體混合機械的特點是結(jié)構(gòu)簡單，且無轉(zhuǎn)動部件，維護檢修量小，能耗低。這類混合機械又分為氣流攪拌、管道混合、射流混合和強制循環(huán)混合等四種。 中、高黏度液體和膏狀物的混合機械，一般具有強的剪切作用；熱塑性的物料混合機主要用于熱塑性物料(如橡膠和塑料)與添加劑混合；粉狀、粒狀固體物料混合機械多為間歇操作，也包括兼有混合和研磨作用的機械，如輪輾機等。 混合時要求所有參與混合的物料均勻分布?；旌系某潭确譃槔硐牖旌?、隨機混合和完全不相混三種狀態(tài)。各種物料在混合機械中的混合程度，取決于待混物料的比例、物理狀態(tài)和特性，以及所用混合機械的類型和混合操作持續(xù)的時間等因素。 液體的混合主要靠機械攪拌器、氣流和待混液體的射流等，使待混物料受到攪動，以達到均勻混合。攪動引起部分液體流動，流動液體又推動其周圍的液體，結(jié)果在溶器內(nèi)形成循環(huán)液流，由此產(chǎn)生的液體之間的擴散稱為主體對流擴散。 當(dāng)攪動引起的液體流動速度很高時，在高速液流與周圍低速液流之間的界面上出現(xiàn)剪切作用，從而產(chǎn)生大量的局部性漩渦。這些漩渦迅速向四周擴散，又把更多的液體卷進漩渦中來，在小范圍內(nèi)形成的紊亂對流擴散稱為渦流擴散。 機械攪拌器的運動部件在旋轉(zhuǎn)時也會對液體產(chǎn)生剪切作用，液體在流經(jīng)器壁和安裝在容器內(nèi)的各種固定構(gòu)件時，也要受到剪切作用，這些剪切作用都會引起許多局部渦流擴散。 攪拌引起的主體對流擴散和渦流擴散，增加了不同液體間分子擴散的表面積減少了擴散距離，從而縮短了分子擴散的時間。若待混液體的粘度不高，可以在不長的攪拌時間內(nèi)達到隨機混合的狀態(tài)；若粘度較高，則需較長的混合時間。 對于密度、成分不同、互不相溶的液體，攪拌產(chǎn)生的剪切作用和強烈的湍動將密度大的液體撕碎成小液滴并使其均勻地分散到主液體中。攪拌產(chǎn)生的液體流動速度必須大于液滴的沉降速度。 少量不溶解的粉狀固體與液體的混合機理，與密度成分不同，互不相溶的液體的混合機理相同，只是攪拌不能改變粉狀固體的粒度。若混合前固體顆粒不能使其沉降速度小于液體的流動速度，無論采用何種攪拌方式都形不成均勻的懸浮液。 不同膏狀物的混合主要是將待混物料反復(fù)分割并使其受到壓、輾、擠等動作所產(chǎn)生的強剪切作用，隨后又經(jīng)反復(fù)合并、捏合，最后達到所要求的混合程度。這種混合很難達到理想混合，僅能達到隨機混合。粉狀固體與少量液體混合后為膏狀物，其混合機理與膏狀物料混合的機理相同。 不同的熱塑性物料以及熱塑性物料與少量粉狀固體的混合，需要依靠強剪切作用，反復(fù)地揉搓和捏合，才能達到隨機混合。 2 攪拌機的發(fā)展史及現(xiàn)狀 攪拌混合設(shè)備是一種應(yīng)用廣泛、品種繁多的流體機械產(chǎn)品，適用于化工、冶金、醫(yī)藥、食品和飼料等領(lǐng)域。攪拌操作是工業(yè)反應(yīng)過程的重要環(huán)節(jié)，它的原理涉及流體力學(xué)、傳熱、傳質(zhì)及化學(xué)反應(yīng)等多種過程，而攪拌器是為了使攪拌介質(zhì)獲得適宜的流動場而向其輸入機械能量的裝置。因此攪拌器也叫做Mixer，或叫做Agitator,Stirrer。廣義的攪拌還包括將固體微粒分散懸浮在溶液里面或?qū)⑷芤鹤兂删鶆虻娜榛?，因此它包括分散器和均質(zhì)機。某些攪拌器能產(chǎn)生極大的剪切力，以獲得細化的粒子比膠體磨大10倍以上的亞微米懸浮體，因此，可用于制造色拉醬、美容乳之類的精細食品和化學(xué)品。石化工業(yè)常用于聚氯乙烯合金、順丁橡膠合釜、反應(yīng)釜、汽提釜等統(tǒng)稱為攪拌容器（Agitatored Vessels，或Stirred Vessels）。 近年來，攪拌器和攪拌容器獲得飛速發(fā)展的同時，正面臨著滿足合理利用資源、節(jié)能降耗和對環(huán)境保護要求的嚴(yán)峻挑戰(zhàn)。攪拌器和攪拌容器在服從裝置規(guī)模經(jīng)濟化和品種多樣化的同時，正日趨大型化。日立制作所自1949年生產(chǎn)攪拌反應(yīng)釜以來已為聚氯乙烯、對苯二甲酸、苯乙烯單體、聚丙烯等裝置生產(chǎn)了攪拌反應(yīng)釜近4000臺，容器的最大容量達576m ，最大直徑達7620 mm，圓筒部分最大長度達 44380 mm，設(shè)計壓力最大 28 MPa，設(shè)計溫度最高 530 cI二，電機最大功率達 1100 kW?；诠?jié)能的要求，開發(fā)出變頻調(diào)速電機、小剪切阻力槳葉、以新型密封代替機械密封和填料密封，以磁力驅(qū)動代替機械傳動。基于降低產(chǎn)品總體成本、減少維修保養(yǎng)成本和提高設(shè)備平均維修間隔時間的要求，大大提高了設(shè)備運行壽命?；跐M足衛(wèi)生和降低清洗和殺菌成本的要求，實現(xiàn)了CIP(就地清洗 )和 SIP(就地殺菌)，提高了自動化水平，避免了人與產(chǎn)品的接觸，減少了人工操作和待機時間，大大提高了產(chǎn)品的衛(wèi)生水平。 3 攪拌過程及攪拌槳葉的分類 攪拌技術(shù)觀點看，流體攪拌可分為五種基本攪拌應(yīng)用，而每一種攪拌應(yīng)用又可根據(jù)物理過程和化學(xué)過程分為兩種類型。因此，總共有十種基本的攪拌應(yīng)用。每一種基本攪拌應(yīng)用都有各自的攪拌特點，過程要求和放大設(shè)計準(zhǔn)則。實際應(yīng)用時，每種攪拌應(yīng)用往往會有幾種基本攪拌應(yīng)用組成，如絮凝攪拌過程由液液混合和固體懸浮兩個基本攪拌應(yīng)用組成。 攪拌機主要有電機、減速裝置、攪拌軸和槳葉等組成。攪拌槳葉的形式多種多樣但無論何種槳葉形式，攪拌機在操作時，其軸功率消耗都產(chǎn)生兩部分作用，一部分是槳葉產(chǎn)生的排液量，另一部分是槳葉產(chǎn)生的壓頭。槳葉產(chǎn)生的壓頭又可分成兩部分，即靜壓頭和剪切力；攪拌機槳葉在操作時，必須克服靜壓頭，而剪切力使得物料分散、混合。因此，根據(jù)槳葉產(chǎn)生排液量，克服靜壓頭和產(chǎn)生剪切力能力的大小，可將所有槳葉分成三種基本類型，即流動型、壓頭型和剪切型。每一種槳葉在提供某種基本作用的同時（如流動型槳葉的基本作用是產(chǎn)生排液量），也提供另外兩種作用（產(chǎn)生剪切和克服靜壓頭）。 ? 根據(jù)不同的攪拌工程對攪拌要求的不同，選擇一種合理的槳葉形式，使得攪拌槳葉提供的排液量，靜壓頭和剪切之匹配能最大限度地滿足攪拌過程的攪拌要求。如固體懸浮及互容液體的混合，要求槳葉能提供大排液量、低剪切。而氣一液分散，要求槳葉能同時提供剪切、排液量和靜壓。 ? 攪拌槳葉的分類，也可以按照槳葉對流體作用所產(chǎn)生的流動型態(tài)來分，可將槳葉分成兩種類型-軸流式槳葉及徑流式槳葉。所謂軸流式槳葉，是指槳葉的主要排液方向與攪拌軸平行，螺旋推進式槳葉即是一種典型的軸流式槳葉；所謂徑流式槳葉，是指槳葉的主要排液方向與攪拌軸垂直。 帶有“Sabre"形狀葉片的攪拌槳，攪拌能耗量小，產(chǎn)生的流動為主導(dǎo)軸向型，確保非常有效。帶有450傾斜平板葉片的軸向攪拌槳，對中小體積的攪拌最為經(jīng)濟。這種攪拌槳葉產(chǎn)生的流動為主導(dǎo)軸向型帶徑?向流，產(chǎn)生剪切擾動。在不粘的介質(zhì)中這種攪拌槳葉對大多數(shù)應(yīng)用均非常理想，特別是那些需要高速低能耗的場合。例如： 被用于進行懸浮或熱交換。傾斜的槳葉低速運轉(zhuǎn)，產(chǎn)生較高的擾動。這種基本攪拌槳葉通常對一些簡單攪拌應(yīng)用有效。 螺旋推進式型槳葉，對小體積的攪拌最為經(jīng)濟。在無粘性的介質(zhì)中，適合于氣-液交換及熱交換。用于固體、混合物、乳液的傳統(tǒng)槳葉，產(chǎn)生中等水平產(chǎn)生徑向流，具高抗動性和高能耗，專用于特殊應(yīng)用。由于重量原因，這種槳葉僅用小直徑，經(jīng)常用高速運行（電機直接驅(qū)動）。 4 攪拌機的分類 攪拌機是以混合、揉和方式調(diào)整物料稠度的一種機械設(shè)備。攪拌機在工業(yè)生產(chǎn)中，特別是在建筑、水泥等領(lǐng)域有著非常重要的應(yīng)用。攪拌機按照的分類方式很多，下分多個種類，以下是常見的攪拌機劃分方法與攪拌機種類。 4.1 攪拌機的作業(yè)方式分類 攪拌機按照作業(yè)方式上的差別，可以分為循環(huán)作業(yè)式攪拌機和連續(xù)作業(yè)式攪拌機兩種。 循環(huán)作業(yè)式攪拌機是以周期循環(huán)方式，順序完成供料、攪拌和卸料三道工序，對于物料用量的控制較為精準(zhǔn)，物料攪拌的效果較好。目前，在實際生產(chǎn)中應(yīng)用的攪拌機多屬于循環(huán)作業(yè)式攪拌機。 連續(xù)作業(yè)式攪拌機對物料的處理，同樣經(jīng)過供料、攪拌和卸料三道工序，但是這三道工序是在攪拌機附屬的筒體內(nèi)連續(xù)完成的。連續(xù)作業(yè)式攪拌機對物料的配比控制能力較差、也不易掌握物料攪拌的時間，但連續(xù)作業(yè)式攪拌機的生產(chǎn)能力較高、生產(chǎn)量較大，適合物料處理效果要求低的攪拌工作。 4.2 攪拌機的攪拌方式分類 攪拌機按照攪拌方式上的差別，可以分為自落式攪拌機和強制式攪拌機兩種。 自落式攪拌機是攪拌鼓轉(zhuǎn)動而攪拌鼓內(nèi)的葉片相對靜止。自落式攪拌機工作時，攪拌鼓會旋轉(zhuǎn)帶動混合物料，葉片將混合物料提升到一定高度后，物料會在自身的重力作用下灑落，完成攪拌的過程。 強制式攪拌機是攪拌鼓保持靜止而葉片強制攪拌。強制式攪拌機工作時葉片會在轉(zhuǎn)軸的帶動下轉(zhuǎn)動，強制攪拌混合物料。強制式攪拌機的攪拌質(zhì)量好、攪拌效率高，但是葉片磨損速度很快，且需要很大的動力輸出。 4.3 攪拌機的裝置方式分類 攪拌機按照裝置方式分為固定式攪拌機和移動式攪拌機兩種。固定式攪拌機安裝在固定基座上，整機無法移動，生產(chǎn)效率高，多適用于攪拌樓或攪拌站使用。移動式攪拌機安裝在汽車上，易于移動、機動性好，多適用于各種小型工程。 4.4 攪拌機的容量大小分類 攪拌機的設(shè)計容量范圍很大，從50L到3000L都有。小型攪拌機的出料容量為50到250L，而中型攪拌機的出料容量就上升至300到500L，大型攪拌機的容量則高達1000到3000L。 4.5 攪拌機的內(nèi)部構(gòu)造分類 槳式攪拌器 有平槳式和斜槳式兩種。平槳式攪拌器由兩片平直槳葉構(gòu)成。槳葉直徑與高度之比為 4～10,圓周速度為1.5～3m/s，所產(chǎn)生的徑向液流速度較小。斜槳式攪拌器的兩葉相反折轉(zhuǎn)45°或60°，因而產(chǎn)生軸向液流。槳式攪拌器結(jié)構(gòu)簡單，常用于低粘度液體的混合以及固體微粒的溶解和懸浮。 5 攪拌機的應(yīng)用范圍 新型攪拌器系換代產(chǎn)品，是化工和建材行業(yè)攪拌設(shè)備無可替代的產(chǎn)物，實現(xiàn)了正確“攪和拌”的問世，從而淘汰其它攪拌設(shè)備所以承但的重任。它以其超常規(guī)的構(gòu)思和精銳的技術(shù)含量，合理的設(shè)計水準(zhǔn)，填補了國際空白。其廣泛用于油漆、涂料、染料、制革、醫(yī)藥、飲料、粘膠劑、食品、洗滌品、化妝品及各種固態(tài)物體等。有取之不盡的財富。對物體分散、乳化、均質(zhì)、調(diào)色等較之傳統(tǒng)攪拌機的攪拌效果更加理想、直觀、是攪拌行業(yè)的一次革命。另一方面，我們和一些發(fā)達國家還存在一定的距離，這就需要我們汲取和借鑒國外的先進技術(shù)，使我們的產(chǎn)品更加完美。 參考文獻 [1] 黎明，化工行業(yè)標(biāo)準(zhǔn)-攪拌器[M].北京：化學(xué)工業(yè)出版社,2008-10. [2] 陳志平，攪拌與混合設(shè)備設(shè)計選用手冊[M]. 北京：化學(xué)工業(yè)出版社. [3] 李國剛，固體廢物實驗與監(jiān)測分析方法[M]. 北京：化學(xué)工業(yè)出版社. [4] 成大先, 機械設(shè)計手冊[M]（第五版）（第1卷）. 北京：化學(xué)工業(yè)出版社. [5] 初志，吳巖石等編.化工容器技術(shù)問答--化工設(shè)備技術(shù)問答叢書[M]. 北京：化學(xué)工業(yè)出版社. [6] 譚天恩，竇梅，周明華.化工原理[M] (上)(普通高等教育十五國家級規(guī)劃教材). 北京： 化學(xué)工業(yè)出版社,2006年08月 [7] 曲文海，朱有庭.?于浦義化工設(shè)備設(shè)計手冊[M](上 下). 北京：化學(xué)工業(yè)出版社. [8] 倫世儀.生化工程[M]（第二版）.化學(xué)工業(yè)出版社. [9] 湯善甫，朱思明主編.化工設(shè)備機械基礎(chǔ)[M]（第二版）.上海：華東理工大學(xué)出版社，2004.12. [10]Yang D Y, Jung D W, Song I S, etal. Comparative Investigation into Implicit, explicit, and iterative implicit/explicit schemes for the simulation of sheet-metal forming processes[J].Journal of Materials Processing Technology.1995(50):39-53 [11]Ronda J, Mercer C D, Bothma A S, etal. Simulation of square-cup deep-drawing with various friction and material models[J].Journal of Materials Processing Technology. 1995(50):92-104. [12] Zhoua D .Wagonera R H. Development and application of sheet-forming simulation [J].Jounral of Materials Processing technology.1995(50)1-16 畢業(yè)設(shè)計 文獻翻譯 院（系）名稱 工學(xué)院機械系 專業(yè)名稱 機械設(shè)計制造及其自動化 學(xué)生姓名 指導(dǎo)教師 2012年 03月 27日 畢業(yè)設(shè)計 文獻綜述 院（系）名稱 工學(xué)院機械系 專業(yè)名稱 機械設(shè)計制造及其自動化 學(xué)生姓名 指導(dǎo)教師 2010年 03月 27日 黃河科技學(xué)院畢業(yè)設(shè)計(文獻翻譯) 第 7 頁 基于機床混合模型的參數(shù)曲線高速插補速度極值分析 塞巴斯蒂安四蒂馬爾，日達噸法魯克 美國加州大學(xué)戴維斯分校，機械系和航空工程系，美國 加州95616 2005年7月7日收稿， 2006年3月23修訂 ,2006年4月10日發(fā)表 摘 要 算法是隨著估算進給速度的曲率的變化而發(fā)展的，這確保了一個3軸的最低運動時間，數(shù)控機床受固定軸加速度范圍和驅(qū)動電機輸出扭矩特性的軸速度約束。對于由一個多項式參數(shù)曲線指定一個路徑，最優(yōu)時間的進給速度確定一個分段曲線函數(shù)的參數(shù)解析與細分，對應(yīng)限制一個軸的加速度飽和常數(shù)。進給速度之間的始發(fā)點段，可通過數(shù)值計算的解決方法。對于細分固定加速度的（平方）的最佳進給速度是合理的曲線參數(shù)。對于速度依賴加速度范圍，最佳進給速度在一種新的超越函數(shù)，其值及封閉的形式表達可有效地計算使用，實時控制一個特殊的算法。最佳進給速度推導(dǎo)出一個實時插補算法，可以直接從驅(qū)動器的解析路徑描述機器。從實驗結(jié)果的執(zhí)行情況看，時間最優(yōu)的3軸數(shù)控由于采用開放式構(gòu)架的軟件驅(qū)動進給速度控制器給出。該算法是一種顯著的改善[蒂馬爾支持SD，法魯克逆轉(zhuǎn)錄，史密斯給付，博亞杰夫建議。算法的時間最優(yōu)控制沿著彎曲的數(shù)控機床刀具路徑。機器人集成制造2005; 21:37-53]，因為除了電壓限制運動排除了沿直線或接近直線路徑段任意高速的可能性。 2006愛思唯爾版權(quán)所有。 關(guān)鍵詞：3軸加工，進給速度的函數(shù)，加速度的極值，時間最優(yōu)路徑遍歷，噪音控制 1 簡介 時間最優(yōu)控制在以往的研究領(lǐng)域，機器人技術(shù)[1-7]和數(shù)控加工[8-10]關(guān)注與一個指定的路徑最短時間穿越了一系統(tǒng)具有已知的動態(tài)和在指定的范圍運動的執(zhí)行機構(gòu)。該方案解決這些問題的一個典型招致控制“噪音”戰(zhàn)略，其中至少有一個輸出系統(tǒng)飽和執(zhí)行器在每個瞬間整個路徑遍歷。這些研究通常假定驅(qū)動器常與對稱力極限（獨立驅(qū)動的速度和方向）而且一般不解決問題的速度，超過該范圍執(zhí)行器可以發(fā)揮最大的力量。 固定場直流電動機是最常見的定位在機器人及數(shù)控加工輪廓的應(yīng)用[11]。由于他們的扭矩輸出是成正比對電樞電流，恒轉(zhuǎn)矩對稱限制反映了馬達的最大電流容量電樞繞組。保持恒轉(zhuǎn)矩輸出不斷變化的有關(guān)電樞電壓反電動勢的（正比于電機轉(zhuǎn)速）否則控制電樞電流供應(yīng)[10]。 除了電樞，電流限制應(yīng)用電樞電壓可能會受到限制的問題引起的電機特性或電樞電源。 這樣電壓限制限制了生產(chǎn)的運動能力最大輸出扭矩，速度有限的范圍內(nèi)。超出此范圍，最大適用電樞電壓不電樞電流是限制因子電機扭矩輸出，速度依賴造成最大力矩電機的增加呈線性下降速度[10]。 在3軸加工中，最大電流容量一軸驅(qū)動電機施加一個恒定的加速度限制在軸速度降低，最大電壓容量規(guī)定在較高軸速度依賴加速度極限速度。從目前有限的過渡到電機軸的操作發(fā)生在過渡速度。在下面的速度過渡的速度，最高軸加速度保持不變。在速度大于過渡的速度，最大軸加速度線性軸的速度下降，在下降到零軸空載速度。 為了保證時間的最優(yōu)路徑遍歷符合這兩個驅(qū)動器電流和電壓的限制，算法必須考慮到這兩個常數(shù)和在每臺機器軸加速度限制。這本文推廣了以前的研究結(jié)果[9]用人唯一不變的加速度式（1假設(shè)高速任意核算結(jié)果，如果路徑中包含擴展線性段），并介紹了新算法現(xiàn)實的時間來計算最優(yōu)進給速度為笛卡爾與驅(qū)動電機軸數(shù)控機床同時受電壓和電流限制。列入的加速式招致重大，定性以較早的算法在許多方面的變化[9]，其中包括一套可行的進給速度和加速組合的速度限制曲線（可變編碼）;可能的切換不同類型點;以及進給速度的極值函數(shù)的形式相平面軌跡。然而，對于笛卡爾數(shù)控與軸獨立驅(qū)動的機器，它仍然是可能的以獲取基本上封閉形式解的進給速度，由于計算能力的根源某些多項式方程。 我們首先回顧了第2個DC電機運行并在第3軸加速度范圍。我們介紹了最低時的遍歷問題常和速度依賴軸彎曲的路徑加速度限制在第4節(jié)，我們得出進給速度恒和速度的表達式依賴極值加速度軌跡。飼料加速度限制，可變長編碼，和進給速度破發(fā)點，然后對第5-7分別進行討論。經(jīng)過討論的進給速度計算在第8和實時數(shù)控插補算法在第9，我們目前的細節(jié)進給速度計算和機實施效果。在第10條的幾個例子。最后，第11節(jié)總結(jié)我們的結(jié)果并提出了一些結(jié)論說這番話的。 2 直流電動機轉(zhuǎn)矩限制 為加深對軸的性質(zhì)背景，適當(dāng)?shù)闹苯鞘郊铀俣葦?shù)控機床，我們開始與一固定場區(qū)的簡要概述了通常用于驅(qū)動小型至中型電機。銑床（見其更完整的細節(jié)操作 [10]）。該方程管運作電機是也就是說，電機的輸出轉(zhuǎn)矩T是成正比的，電樞電流I，反電動勢是成正比。 電機角速度，電樞和應(yīng)用電壓V等于反電動勢和總結(jié)的壓降電樞電阻R的KT和柯相稱因素，所謂的扭矩常數(shù)和反電動勢常數(shù)，是內(nèi)在的物理一個給定的電機性能的影響。從這些表現(xiàn)形式，你可以很容易地推導(dǎo)出電動機轉(zhuǎn)矩轉(zhuǎn)速的關(guān)系 在給付是失速扭矩,和 無負(fù)載速度。所以，電機轉(zhuǎn)矩降低,線性電動機的速度增加，從時 到時。 參見[12]更完整的細節(jié)。 在發(fā)動機啟動和低速時，反電動勢E是小相比，施加電壓V，以及限流設(shè)備是用來限制電流I為（大約）常數(shù)的最大值，以防止伊利姆電樞繞組的損壞。因此，電機轉(zhuǎn)矩輸出保持恒定在整個低轉(zhuǎn)速范圍的操作。 隨著馬達的加快，電樞電壓應(yīng)用最終達到最大電機或電源供應(yīng)器 額定電壓。這發(fā)生在過渡的速度，定義 對于速度高于催產(chǎn)素大，電樞電壓（而不是比目前的）是在電機轉(zhuǎn)矩限制因素輸出。在電壓限制，扭矩T線性下降隨著電機轉(zhuǎn)速澳，下降至零，空載轉(zhuǎn)速的實現(xiàn)。 圖（1）描述了電機的制約電流和電壓范圍，和在為積極和消極的馬達速度。該約束定義兩個平行帶，其交集形成定義可行的制度直流電動機運行。所有受理的組合電動機的扭矩和速度，按照給定的電樞電流和電壓范圍，在這個謊言。 對超出的部分延伸無負(fù)載在每個方向符合再生電機，制動其中意味著外部扭矩申請。由于沒有這樣的扭矩可在驅(qū)動器中的數(shù)控機床馬達，可行的扭矩范圍/速度降低狀態(tài)來表示空載速度最高電機轉(zhuǎn)速，高產(chǎn)的六面平行四邊形，如圖1所示。 這六個面平行四邊形定義了三個不同的直流馬達轉(zhuǎn)速范圍，具有鮮明的最低和每最大扭矩限制，即： 3 軸加速度限制 在高速加工[8,13,14]慣性力可能稱霸切削力，摩擦等，尤其是工具路徑的高曲率。會計軸慣性，軸的速度和加速度是成比例的力矩電機和電機速度分別?？紤]，也就是說，x軸。如果它是有效質(zhì)量的Mx和驅(qū)動，由驅(qū)動電機通過彈性模量Kx（即滾珠絲桿，線性軸速度是關(guān)系到汽車的角相應(yīng)的軸加速度以電動機轉(zhuǎn)矩T是ax=KxT/Mx。注意到進給速度可被視為一個數(shù)量級v和載體由單位路徑切線的特定方向，我們有和電機轉(zhuǎn)速為 因此，上面導(dǎo)出的轉(zhuǎn)矩限制相當(dāng)于X軸加速度限制 其中VT是軸過渡的速度，V0的是軸空載速度，我們定義通過對速度的依賴加速度限制，軸速度VX始終保持在區(qū)間 軸轉(zhuǎn)速范圍內(nèi) ,最低軸加速度和最高限額都是固定的，因此，這被稱為制度的不斷限制在X軸。軸速度范圍，為其中一個加速度是固定的二是依靠速度，被稱為混合為X軸的限制制度。在制度不變的限制，加速范圍可寫為。對于混合限制制度，加速范圍可能表現(xiàn)在表格 在路徑遍歷，每個軸在一個月內(nèi)運作，其加速度限制制度獨立于其他軸，每一個都可能加速極限之間切換，按照制度與工具的變化路徑幾何形狀和進給速度。因此，有四個加速度限制制度的可能組合，其中的x，y軸，Z軸（見表1）。對于一個平面曲線，涉及的僅有的兩個機軸運動，有三個可能的組合：常量/恒，恒/混合，和混合/混合。每個組合的加速度極限，除了要具體分析計算的時間最優(yōu)進給速度。 4 時間最優(yōu)的進給速度 考慮到學(xué)位曲線描述的路徑 與對照點。如果指弧長沿曲線測量，我們定義參數(shù)速度 切線的單位和（主軸）和正常向量曲率（4）定義 與此相反，與我們可以寫 現(xiàn)在假設(shè)我們遍歷與進給速度（速度的曲線）指定由該函數(shù)。由于衍生金融工具方面時間t和參數(shù)x，我們以點表示和素數(shù)，分別為，由有關(guān) 速度和加速度向量由每個點給出由 切向分量的消失如果V 是常數(shù)，而正常（向心力）組件的如果消失K=0。其時的進給速度（衍生的加速度）給出的角度來看， 我們希望盡量減少沿線rexT遍歷時間，開始和結(jié)束休息時，受限制的加速度表格（3）和其他類似用語機軸。這些要求可以在以下方面措辭以下優(yōu)化問題使得 其中指的是笛卡爾每一個組成部分， 正如在第3節(jié)，軸加速度的形式是 4.1 恒定加速度軌跡 從關(guān)系和 我們可以寫 對于給定的曲線的X軸 組件（說）一個定義為加速的 在我們寫，因為它是方便工作對進給速度平方（見[9詳情]）。 在一個不斷加速階段極值加速度限制，其中一個組成部分，是加速等于加上或減去相應(yīng)的約束，一條件是產(chǎn)生一個為q的線性微分方程如果x是加快軸，這個方程承認(rèn)為（平方）進給速度，即封閉形式解。定義為 其中積分常數(shù)C是取決于指定的一個已知點，對軌跡：關(guān)于進一步解決（10）的方法詳情中可以 在[9] 找到。 4.2 加速度極值軌跡 考慮到當(dāng)x軸（假定）執(zhí)行一個加速極值，通過定義加速度極值約束決定進給速度v形式。通過以上描述，就是在這種情況下推導(dǎo)出進給速度的微分方程 在我們不斷介紹方程（11）是一階變系數(shù)非線性微分方程。這對來說，可以專門寫作 Robotics and Computer-Integrated Manufac paths form Previous studies of time-optimal control in the fields of the speed and direction of actuation), and generally do not actuators can exert their maximum force. Fixed-field DC motors are common to most positioning armature voltage may be subject to limits arising from the motor characteristics or armature power supply. Such voltage limits confine the ability of the motor to produce ARTICLE IN PRESS the maximum output torque to a finite range of speeds. Beyond this range, maximum applied armature voltage not armature currentis the factor limiting the motor 0736-5845/$-see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2006.07.002 C3 Corresponding author. E-mail addresses: sdtimarucdavis.edu (S.D. Timar), faroukialgol.engr.ucdavis.edu (R.T. Farouki). robotics 17 and CNC machining 810 were concerned with the minimum-time traversal of a prescribed path by a system with known dynamics and specified bounds on the motive-force capacity of its actuators. The solutions to such problems characteristically incur a bang-bang control strategy, in which the output of at least one system actuator is saturated at each instant throughout the path traversal. These studies typically assume actuators with constant and symmetric force limits (independent of and contouring applications in robotics and CNC machin- ing 11. Since their torque output is directly proportional to the armature current, the constant symmetric torque limits reflect the maximum current capacity of the motor armature windings. Constant torque output is maintained by continuously varying the armature voltage in relation to the back EMF (proportional to the motor speed) or otherwise controlling the armature current supply 10. In addition to the armature current limits, the applied CNC machine subject to both fixed and speed-dependent axis acceleration bounds arising from the output-torque characteristics of the axis drive motors. For a path specified by a polynomial parametric curve, the time-optimal feedrate is determined as a piecewise-analytic function of the curve parameter, with segments that correspond to saturation of the acceleration along one axis under constant or speed- dependent limits. Break points between the feedrate segments may be computed by numerical root-solving methods. For segments that correspond to fixed acceleration bounds, the (squared) optimal feedrate is rational in the curve parameter. For speed-dependent acceleration bounds, the optimal feedrate admits a closed-form expression in terms of a novel transcendental function whose values may be efficiently computed, for use in real-time control, by a special algorithm. The optimal feedrate admits a real-time interpolator algorithm, that can drive the machine directly from the analytic path description. Experimental results from an implementation of the time-optimal feedrate on a 3-axis CNC mill driven by an open-architecture software controller are presented. The algorithm is a significant improvement over that proposed in Timar SD, Farouki RT, Smith TS, Boyadjieff CL. Algorithms for time-optimal control of CNC machines along curved tool paths. Robotics Comput Integrated Manufacturing 2005;21:3753, since the addition of motor voltage constraints precludes the possibility of arbitrarily high speeds along linear or near-linear path segments. r 2006 Elsevier Ltd. All rights reserved. Keywords: 3-Axis machining; Feedrate functions; Acceleration constraints; Time-optimal path traversal; Bang-bang control; Real-time interpolators 1. Introduction address the question of the range of speeds over which the Algorithms are developed to compute the feedrate variation along a curved path, that ensures minimum traversal time for a 3-axis Time-optimal traversal of curved under both constant and speed-dependent Sebastian D. Timar, Department of Mechanical and Aeronautical Engineerin Received 7 July 2005; received in revised Abstract turing 23 (2007) 563579 by Cartesian CNC machines axis acceleration bounds Rida T. Farouki C3 g, University of California, Davis, CA 95616, USA 23 March 2006; accepted 10 April 2006 ARTICLE IN PRESS torque output, resulting in a speed-dependent maximum torque that decreases linearly with increasing motor speed 10. In 3-axis machining, the maximum current capacity of an axis drive motor imposes a constant acceleration limit at lower axis speeds, and the maximum voltage capacity imposes a speed-dependent acceleration limit at higher axis speeds. The transition from current-limited to voltage- limited operation of the motor occurs at the axis transition speed. At speeds below the transition speed, the maximum axis acceleration remains constant. At speeds greater than the transition speed, the maximum axis acceleration decreases linearly with the axis speed, dropping to zero at the axis no-load speed. To guarantee that time-optimal path traversals conform to both actuator current and voltage limits, algorithms must account for the regimes of both constant and speed- dependent acceleration limits on each machine axis. This paper generalizes the results of a previous study 9 employing only constant acceleration bounds (an assump- tion that incurs arbitrarily high speeds if the path contains extended linear segments), and introduces new algorithms to compute realistic time-optimal feedrates for Cartesian CNC machines with axis drive motors subject to both current and voltage limits. The inclusion of speed- dependent acceleration bounds incurs significant, qualita- tive changes to many aspects of the earlier algorithm in 9including the set of feasible feedrate and feed acceleration combinationsv; a; the nature of the velocity limit curve (VLC); the different types of possible switching points; and the form of the feedrate function for extremal phase-plane trajectories. Nevertheless, for Cartesian CNC machines with independently driven axes, it is still possible to obtain an essentially closed-form solution for the time- optimal feedrate, given the ability to compute the roots of certain polynomial equations. We begin by reviewing DC motor operation in Section 2 and the axis acceleration bounds in Section 3. We introduce the problem of minimum-time traversal of curved paths with constant and speed-dependent axis acceleration limits in Section 4, and we derive feedrate expressions for constant and speed-dependent extremal acceleration trajectories. Feed acceleration limits, the VLC, and feedrate break points are then addressed in Sections 57, respectively. Following a discussion of the feedrate computation in Section 8, and the real-time CNC inter- polator algorithm in Section 9, we present details of feedrate computation and machine implementation results for several examples in Section 10. Finally, Section 11 summarizes our results and makes some concluding remarks. 2. DC motor torque limits As background for understanding the nature of the axis S.D. Timar, R.T. Farouki / Robotics and Computer-In564 acceleration bounds appropriate to Cartesian CNC ma- chines, we begin with a brief overview of the fixed-field DC motors that are commonly used to drive small-to-medium milling machines (see 10 for more complete details of their operation). The equations governing the operation of fixed- field motors are T K T I; EK E o; V EIR, i.e., the motor output torque T is proportional to the armature current I, the back EMF E is proportional to the motor angular speed o, and the applied armature voltage V is equal to the sum of the back EMF and the voltage drop across the armature resistance R. The proportionality factors K T and K E , called the torque constant and back EMF constant, are intrinsic physical properties of a given motor. From these expressions, one can easily derive the motor torquespeed relation T T s 1C0 o o 0 C18C19 , (1) where T s K T V=R is the stall torque, and o 0 V=K E is the no-load speed. Hence, the motor torque decreases linearly with increasing motor speed, from T T s at o 0toT 0atoo 0 .See12 for more complete details. At motor start-up and low speeds, the back EMF E is small compared to the applied voltage V, and a current- limiting device is used to constrain the current I to an (approximately) constant maximum value I lim to prevent damage to the armature windings. Hence, the motor torque output remains constant at T lim K T I lim throughout the low-speed range of operation. As the motor speeds up, the applied armature voltage eventually reaches the maximum motor or power supply voltage rating, V lim . This occurs at the transition speed, defined by o t V lim C0I lim R K E . (2) For speeds greater than o t , the armature voltage (rather than the current) is the limiting factor on the motor torque output. At the voltage limit, the torque T decreases linearly with increasing motor speed o, dropping to zero when the no-load speed o 0 is attained. Fig. 1 depicts the motor constraints imposed by the current and voltage limits, I lim and V lim ,intheo; Tplane for both positive and negative motor speeds. The constraints define two parallel strips, whose intersection forms a paralellogram that defines the feasible regime of DC motor operation. All admissible combinations of motor torque and speed, consistent with the given armature current and voltage limits, lie within this paralellogram. The portions of the paralellogram extending beyond the no-load speed in each direction (ooC0o 0 and o4o 0 ) correspond to regenerative braking of the motor, which implies application of an external torque. Since no such tegrated Manufacturing 23 (2007) 563579 torque is available in the context of CNC machine drive motors, the range of feasible torque/speed states is reduced speed-dependent acceleration limits, the axis speed v x always remains in the intervalC0v 0 ;v 0 C138. Within the axis speed range v x 2C0v t ;v t , the mini- mum and maximum axis acceleration limits are both fixed, and hence this is referred to as the constant limits regime for ARTICLE IN PRESS drive abc constant constant constant mixed constant constant mixed mixed constant mixed mixed mixed to indicate the no-load speed as the maximum motor speed, yielding the six-sided parallelogram shown in Fig. 1. The six-sided parallelogram defines three distinct DC motor speed ranges, each with distinct minimum and maximum torque limits, namely: C0T lim o 0 o o 0 C0o t pTp T lim for C0o 0 popC0o t , C0T lim pTp T lim for C0o t popo t , C0T lim pTp T lim o 0 C0o o 0 C0o t for o t popo 0 . 3. Axis acceleration limits In high-speed machining 8,13,14 inertial forces may dominate cutting forces, friction, etc., especially for tool T Fig. 1. Left: the maximum current and voltage limits impose constant and speed-de (shaded) of feasible motor torque/speed values. Right: since the motors that of feasible torque/speed values is truncated to form a six-sided parallelogram. S.D. Timar, R.T. Farouki / Robotics and Computer-In paths of high curvature. Accounting for the axis inertias, the axis speeds and accelerations are proportional to the motor speeds and motor torques, respectively. Consider, say, the x-axis. If it has effective mass M x and is actuated by a drive motor through a ball screw of modulus K x (i.e., the linear axis velocity v x is related to the motor angular speed o by v x o=K x ), the axis acceleration correspond- ing to motor torque T is a x K x T=M x . Noting that the feedrate may be regarded as a vector of magnitude v and direction given by the unit path tangent tt x ; t y ; t z ,we have v x t x v and the motor rotational speed is oK x t x v. Hence, the torque limits derived above are equivalent to the x-axis acceleration limits C0 A x v 0 v x v 0 C0v t pa x pA x for C0v 0 pv x pC0v t , C0 A x pa x pA x for C0v t pv x pv t , C0 A x pa x pA x v 0 C0v x v 0 C0v t for v t pv x pv 0 , 3 where v t is the axis transition speed, v 0 is the axis no-load speed, and we define A x K x T lim =M x . By virtue of the T pendent torque limits, respectively, forming a four-sided parallelogram CNC machine axes will not exceed the no-load motor speed, the region Table 1 The four possible combinations of acceleration-limited regimes for a 3-axis CNC machine (here a; b; c denotes any permutation of the axes x; y; z) Axis tegrated Manufacturing 23 (2007) 563579 565 the x-axis. The axis speed ranges v x 2C0v 0 ;C0v t and v x 2v t ;v 0 , for which one acceleration limit is fixed and the other is speed dependent, are called the mixed limits regimes for the x-axis. In the constant limits regime, the acceleration bounds may be written as a x A x , with a x C61. For the mixed limits regime, the acceleration bounds may be expressed in the form A x g x v 0 C0v x v 0 C0v t and C0g x A x , where g x C01 for v x 2C0v 0 ;C0v t i.e., t x o0, and g x 1 for v x 2v t ;v 0 i.e., t x 40. Similar considerations apply to the y- and z-axis. During a path traversal, each axis operates within one of its acceleration limit regimes independently of the other axis, and each may switch between the acceleration limit regimes in accordance with variations in the tool path geometry and feedrate. Consequently, there are four possible combinations of acceleration-limited regimes among the x-, y-, z-axis (see Table 1). For a planar curve, ARTICLE IN involving motion of only two machine axes, there are three possible combinations: constant/constant, constant/mixed, and mixed/mixed. Each combination of acceleration limits incurs a specific analysis to compute the time-optimal feedrate. The case in which all axes are in the constant regime is covered by our earlier study 9, but cases involving one or more of the axes in the mixed regime have not been previously addressed. 4. Time-optimal feedrates Consider a path described by a degree-n Bezier curve rx X n k0 p k n k C18C19 1C0 x nC0k x k ; x20;1C138 (4) with control points p k x k ; y k ; z k , k0; .; n 15.Ifs denotes arc length measured along the curve, we define the parametric speed by sxjr 0 xj ds dx . The unit tangent and (principal) normal vectors and the curvature of (4) are defined by t r 0 s ; n r 0 C2r 00 jr 0 C2r 00 j C2t; k jr 0 C2r 00 j s 3 (5) and, conversely, with s 0 r 0 C1r 00 =s we may write r 0 st; r 00 s 0 ts 2 kn. (6) Now suppose we traverse the curve with feedrate (speed) specified by the function vx. Since derivatives with respect to time t and the parameter xwhich we denote by dots and primes, respectivelyare related by d dt ds dt dx ds d dx v s d dx , the velocity and acceleration vectors at each point are given by v_rvt; ar _vtkv 2 n. (7) The tangential component _vt of a vanishes if vconstant, while the normal (centripetal) component kv 2 n vanishes if k0. The time derivative of the feedrate (the feed acceleration) is given in terms of x as _vvv 0 =s. We wish to minimize the traversal time along rx, starting and ending at rest, subject to acceleration limits of the form (3) and analogous expressions for the other machine axes. These requirements can be phrased in terms of the following optimization problem: min vx T Z 1 0 s v dx (8) such that S.D. Timar, R.T. Farouki / Robotics and Computer-In566 A i;min pa i xpA i;max for x20;1C138, Z1C0 v t v 0 . Eq. (11) is a first-order, non-linear differential equation with variable coefficients. It may be written exclusively in terms of x as 00 0 C18C19 2 where ix; y; z refers to each of the Cartesian components a x ; a y ; a z of a. As noted in Section 3, the axis acceleration bounds A i;min , A i;max are of the form C0A i ;A i or A i g i v 0 C0v i v 0 C0v t ;C0g i A i . 4.1. Constant acceleration trajectories From the relations (5), (7), ss 0 r 0 C1r 00 , and _vvv 0 =s, we may write a vv 0 s 2 r 0 v 2 s 3 sr 00 C0s 0 r 0 . For a given curve rxxx; yx; zx the x-axis component (say) of the acceleration a is defined by a x q 0 2s 2 x 0 q s 3 sx 00 C0s 0 x 0 , (9) where we write qv 2 , since it is convenient to work with the square of the feedrate (see 9 for further details). During an extremal acceleration phase under constant acceleration limits, one component of the acceleration is equal to plus or minus the corresponding bound, a condition that yields a linear differential equation for q. If x is the extremally accelerating axis, this equation admits a closed-form solution for the (squared) feedrate, namely q s x 0 C16C17 2 C2a x A x x, (10) where the integration constant C is determined by specifying a known point x C3 ; qx C3 on the trajectory: Cx 0 x C3 =sx C3 2 qx C3 C02a x A x xx C3 . Further details of the solution method for (10) may be found in 9. 4.2. Speed-dependent acceleration trajectories Consider the determination of the feedrate v when the x- axis (say) executes an extremal acceleration defined by a speed-dependent acceleration bound, of the form described above. The differential equation governing the feedrate under such circumstances is t x _vkn x v 2 A x Zv 0 t x vC0 g x A x Z 0, (11) where we introduce the constant PRESS tegrated Manufacturing 23 (2007) 563579 vv 0 x x 0 C0 s s v 2 A x Zv 0 svC0 g x A x Z s x 0 0. feedrate consistent with the axis constraints, and the range a min x; vpapa max x; v of possible feed accelerations at each feedrate v less than v lim x. In the case of constant acceleration bounds on all axes, the acceleration constraints at each curve point x describe strips in the v 2 ; a plane, bounded by parallel line pairs. The intersection of these strips defines a parallelogram, whose interior constitutes the set of feasiblev 2 ; avalues, and whose right-most vertex defines v lim x. For each feedrate v less than v lim x, the upper parallelogram boundary defines the maximum feed acceleration a max x; v, and the lower parallelogram boundary defines the minimum feed acceleration a min x; v. We refer the reader to 9 for complete details. In the case of mixed acceleration bounds, either the lower or the upper constraint involves both v and v 2 ,as well as a, and is thus not describable by a linear relation in ARTICLE IN PRESS To obtain a closed-form integration of this equation, we note that vv 0 x 00 x 0 C0 s 0 s C18C19 v 2 1 2 s x 0 C16C17 2 d dx x 0 s v C18C19 2 . Hence, since g 2 x 1, we obtain d dx x 0 s v v 0 C18C19 2 2 g x A x Zv 2 0 x 0 1C0g x x 0 s v v 0 C18C19 . Writing ux 0 =sv=v 0 , this gives u du dx g x A x Zv 2 0 x 0 1C0g x u, which is amenable to separation of variables, giving Z udu 1C0g x u g x A x Zv 2 0 Z x 0 dx. Noting again that g 2 x 1, this can be integrated to obtain 1C0g x uC0ln1C0g x u g x A x Zv 2 0 xc, the integration constant c being determined from a known initial condition. We note that g x ug x x 0 =sv=v 0 satisfies 0pg x up1, since 0pv=v 0 p1, C01px 0 =sp1, and g x has the same sign as x 0 =s. Hence, the argument of the logarithm occurring above is between 0 and 1. Now let ck be the transcendental function that is defined implicitly as the solution of the equation ckC0lnckk. (12) By differentiating, we see that dc dk C0 ck 1C0ck , and hence the function ckis monotone decreasing if its range is confined to 0pckp1. The corresponding domain is 1pkp1. Using the function c, we can write the feedrate explicitly in terms of the curve parameter x as vxg x v 0 sx x 0 x 1C0c g x A x Zv 2 0 xxc C18C19C20C21 . We regard ck as a basic transcendental function, of similar stature to the trigonometric or logarithmic func- tions. To use it in the context of real-time motion control, an efficient means to evaluate this function is required. Re-writing (12) in the form ckexpC0kexpck (13) yields the iteration sequence for ckdefined by c r expC0kexpc rC01 ; r1;2; . (14) with a suitable starting approximation c 0 . For 1oko1 S.D. Timar, R.T. Farouki / Robotics and Computer-In and 0ocko1, the derivative of the right-hand side of (13) with respect to c is of magnitude less than 1, and hence To estimate starting values c 0 , we use linear interpolation between a sequence of pre-computed values (see Table 2). 5. Set of possiblev;avalues In the comp