【機(jī)械類畢業(yè)論文中英文對(duì)照文獻(xiàn)翻譯】柔性臂架自行式起重機(jī)傾翻載荷,機(jī)械類畢業(yè)論文中英文對(duì)照文獻(xiàn)翻譯,機(jī)械類,畢業(yè)論文,中英文,對(duì)照,對(duì)比,比照,文獻(xiàn),翻譯,柔性,自行,起重機(jī),載荷 柔性臂架自行式起重機(jī)傾翻載荷 柔性臂架自行式起重機(jī)傾翻載荷 S. KILICASLAN, T. BALKAN AND S. K. IDER 土耳其安卡拉 中東科技大學(xué)機(jī)械工程學(xué)院 16531 摘 要 在這一項(xiàng)研究中，自行式起重機(jī)的特性是利用基于柔性多體動(dòng)力學(xué)理論建模分析得到的，用以確定起重機(jī)不產(chǎn)生傾翻危險(xiǎn)的起重量。只有起重機(jī)的臂架被假設(shè)成柔性的，因?yàn)楸奂苁俏ㄒ灰粋€(gè)在吊載過(guò)程中撓度較大的部件。用數(shù)學(xué)方程描述起重機(jī)的剛性部件與柔性部件之間的相互作用與耦合，并且開發(fā)了用于進(jìn)行動(dòng)力學(xué)分析的應(yīng)用軟件。變幅油缸的推力因臂架的仰角不同而不同，臂架運(yùn)動(dòng)的同時(shí)，變幅油缸的推力被計(jì)算出來(lái)，同時(shí)繪制出載荷曲線，并且將結(jié)果與用一臺(tái)10噸的自行式工程起重機(jī)實(shí)驗(yàn)得到的數(shù)據(jù)進(jìn)行比較。 1.緒論 作為機(jī)械系統(tǒng)的起重機(jī)一般被看成包含柔性部件的閉環(huán)的機(jī)械裝置。在以臂架的角度位置作為自變量來(lái)確定起重機(jī)額定起重量的時(shí)候，就需要用到動(dòng)力學(xué)的解決方法。 目前只有極少數(shù)關(guān)于自行式起重機(jī)的動(dòng)力學(xué)分析控制的研究，并且這些研究中的大部分都沒有考慮部件的撓性。Sato和Sakawa建立了一個(gè)動(dòng)力學(xué)模型，用于控制一個(gè)同時(shí)進(jìn)行三種運(yùn)動(dòng)（回轉(zhuǎn)運(yùn)動(dòng)、起升運(yùn)動(dòng)、變幅運(yùn)動(dòng)）的柔性回轉(zhuǎn)起重機(jī)[1]。只有主臂與副臂間的連接被假設(shè)成為柔性的，這樣假設(shè)的目的是能夠使載荷得到恰當(dāng)?shù)霓D(zhuǎn)化，這樣在轉(zhuǎn)化結(jié)束時(shí)，載荷幅值的擺動(dòng)能夠以最快的速度衰減。這個(gè)能夠?qū)ζ鹬貦C(jī)進(jìn)行實(shí)際載荷和額定載荷對(duì)比顯示并加以限制的系統(tǒng)在巴爾干半島上得到應(yīng)用，這個(gè)基于微型計(jì)算機(jī)的控制系統(tǒng)是通過(guò)油壓和臂架仰角來(lái)確定當(dāng)前吊鉤的實(shí)際載荷的。 在這篇文章中，起重機(jī)的特性是通過(guò)柔性多體動(dòng)力學(xué)來(lái)分析得到的。給出了柔性多體動(dòng)力學(xué)的動(dòng)力學(xué)和運(yùn)動(dòng)學(xué)方程，同時(shí)開發(fā)了對(duì)起重機(jī)進(jìn)行動(dòng)力學(xué)分析的應(yīng)用軟件。在多體動(dòng)力學(xué)分析時(shí)，系統(tǒng)的剛性聯(lián)接和柔性運(yùn)動(dòng)通過(guò)運(yùn)用絕對(duì)的耦合和形參變量用公式來(lái)表達(dá)[3, 4]。然后，部件間的連接和指定的運(yùn)動(dòng)用約束方程來(lái)描述。柔性體是通過(guò)有限元方法來(lái)模擬的，形參變量是通過(guò)模型轉(zhuǎn)化得到的，用以代替彈性變量。 變幅油缸的推力因臂架的角度位置不同而不同，臂架運(yùn)動(dòng)的同時(shí)，變幅油缸的推力被計(jì)算出來(lái)，通過(guò)開發(fā)的應(yīng)用軟件來(lái)圖示彈性效果。同時(shí)繪制出臂架不同仰角的載荷曲線，并且將結(jié)果與起重機(jī)制造廠家提供的數(shù)據(jù)進(jìn)行比較。 2.起重機(jī)建模 我們用一臺(tái)930型自行式起重機(jī)的傾翻載荷控制的實(shí)驗(yàn)數(shù)據(jù)用于開發(fā)的軟件的計(jì)算，并且用了這臺(tái)起重機(jī)的結(jié)構(gòu)和參數(shù)[5]。但這個(gè)分析方法只要經(jīng)過(guò)簡(jiǎn)單的修改就可以用于類似的起重機(jī)的分析。 重物吊在吊鉤上，臂架起升，因?yàn)橹匚锲鹕^(guò)高非常危險(xiǎn)，重物的高度是通過(guò)測(cè)量起升繩的長(zhǎng)度來(lái)控制的。在重物起升、下降和運(yùn)輸時(shí)，起重機(jī)是不能回轉(zhuǎn)的，這是由于一些限制條件，比如重物非常大或非常重、空間問(wèn)題等。 在臂架的每一個(gè)角度位置都對(duì)應(yīng)著一個(gè)最大載荷，超過(guò)這個(gè)載荷后起重機(jī)很有可能傾翻。因?yàn)楸奂艿难鼋侵挥性谥匚锲鹕蛳陆禃r(shí)才會(huì)發(fā)生改變，而且恰巧這又是個(gè)平面運(yùn)動(dòng)，所以建模和分析計(jì)算都是在二維空間進(jìn)行的。 圖1.試驗(yàn)用起重機(jī)示意圖 圖2.試驗(yàn)用起重機(jī)動(dòng)力學(xué)模型 單位：mm；A0G 2000;GC 17500;A0A 5823;A0D 5850;AD 565;BD 3455;OB0 2350;OA0 805 圖1表示實(shí)驗(yàn)用起重機(jī)的示意圖，圖2表示它的動(dòng)力學(xué)模型，模型共分成五部分。 橫截面、材料特性及各部分的尺寸是通過(guò)技術(shù)數(shù)據(jù)文件和直接對(duì)實(shí)驗(yàn)起重機(jī)進(jìn)行測(cè)量得到的。部件1（臂架）的橫截面是個(gè)空心的多邊形，壁厚為t，如圖3所示。這個(gè)橫截面的尺寸線性地從A0增大到G，又線性地從G減小到C，截面A0、G和C的尺寸如圖3所示。部件2是個(gè)圓柱形的桿，直徑25mm。部件3是個(gè)液壓活塞桿，直徑180mm壁厚20mm。部件4是變幅油缸筒，內(nèi)徑230mm，外徑246mm， 長(zhǎng)3440mm。部件1的彈性模量和密度分別為200GPa和5750kg/m3。其它部件的密度均認(rèn)為是7850kg/m3。 圖3.部件1（臂架）橫截面.（單位：mm） 當(dāng)考慮了各部件的尺寸（長(zhǎng)度和橫截面）和彈性模量，就可以只認(rèn)為部件1（臂架）是彈性的，這樣的話其他部件均被假設(shè)成是剛性的。 在對(duì)起重機(jī)進(jìn)行分析時(shí)要考慮以下假設(shè)： 1.液壓油的質(zhì)量包含在液壓缸（部件4）的質(zhì)量中，液壓缸質(zhì)量的改變是由于考慮了缸內(nèi)的液壓油質(zhì)量的改變； 2.液壓油被認(rèn)為是不可壓縮的； 3.吊重被看成是集中質(zhì)量，并且通過(guò)一根被看成是剛性桿的繩子連接到臂架頭部。這根繩子在平面內(nèi)可以繞C點(diǎn)自由旋轉(zhuǎn)。只要這個(gè)桿相對(duì)于垂直位置的擺動(dòng)很小并且這個(gè)桿仍然處于張緊狀態(tài)時(shí)這個(gè)假設(shè)就是正確的。在正常操作速度和吊重下這些條件都符合； 4.臂架結(jié)構(gòu)的阻尼比用Rayleigh衰減法來(lái)計(jì)算確定； 5.吊重與地面間的距離假設(shè)保持不變，這是通過(guò)在起重機(jī)臂架升降過(guò)程中改變繩長(zhǎng)來(lái)實(shí)現(xiàn)的。 3.動(dòng)力學(xué)方程 令表示一個(gè)部件的結(jié)構(gòu)，部件k的變形通過(guò)這個(gè)系數(shù)來(lái)定義，表示一個(gè)確定的結(jié)構(gòu)。令表示結(jié)構(gòu)的原點(diǎn)的位置，表示部件k的角速度。 利用有限單元法，部件k的i單元上的任一點(diǎn)P的變形位移向量為： （1） 其中是變形的單元形函數(shù)矩陣，是單元間聯(lián)系的坐標(biāo)變換矩陣，單元節(jié)點(diǎn)位移向量。 點(diǎn)P的速度可表示為： （2） 其中是中點(diǎn)Q到P的變形后的位置向量，是的變形協(xié)調(diào)矩陣，Tk是從到的坐標(biāo)變換矩陣，，是用于減小彈性變形的模型轉(zhuǎn)換變量，是模型形變向量。方程（2）可寫成： （3） 其中是影響系數(shù)矩陣，是部件k的速度向量。 連接各個(gè)部件的系統(tǒng)N的連接處和角度指示在速度水平上用運(yùn)動(dòng)學(xué)約束方程表示為： Cy=g （4） 其中C是雅客比約束矩陣，y是系統(tǒng)的速度向量，由下式確定： yT=[y(1)T…y(N)T] （5） Kane方程用于確定系統(tǒng)的運(yùn)動(dòng)方程： My+CTλ=Q+Fs+Fd+F （6） 其中λ是約束反力向量，M是質(zhì)量矩陣，Q、Fs、Fd及F分別是Coriolis力向量、彈性力向量、阻尼力向量和實(shí)際力向量，分別為： ，,, , （7） 質(zhì)量矩陣Mk和部件k的Coriolis向量Qk為 （8） （9） 其中Ek是部件k中的有限單元的數(shù)量，Vki是單元的體積，是它的密度。 Fsk和Fdk可按下式給出： ， （10） 其中Kk是部件k的結(jié)構(gòu)剛度矩陣，Dk是阻尼矩陣。在仿真時(shí)，結(jié)構(gòu)的質(zhì)量和用于組成Dk的結(jié)構(gòu)的剛度復(fù)數(shù)有2%的衰減。 在平面系統(tǒng)中，減小到，其中是個(gè)標(biāo)量。變成，其中是根據(jù)變換而來(lái)的。 當(dāng)公式（8）和（9）中由空間決定的變量分離后，就能獲得[3,4]不隨時(shí)間改變的矩陣。 （11） ，，，，， （12） （13） （14） （15） ， （16，17） （18） （19） 臂架是靠駕駛員控制的液壓缸來(lái)驅(qū)動(dòng)的，一般來(lái)說(shuō)，運(yùn)動(dòng)的整個(gè)過(guò)程中液壓缸以恒定的速度運(yùn)動(dòng)，所以臂架和活塞的振動(dòng)都能控制在一個(gè)很小的水平上。為了避免沖擊載荷的產(chǎn)生，活塞的開始運(yùn)動(dòng)時(shí)速度從0增大到以及最終停止時(shí)從減小到0，速度的變化假設(shè)成隨時(shí)間呈擺線形變化。這個(gè)理想的速度曲線如圖4所示，并可用下列方程描述： （20） 如果部件1和2的絞點(diǎn)在不同的位置，這個(gè)系統(tǒng)將變成一個(gè)不能動(dòng)的結(jié)構(gòu)。系統(tǒng)之所以能夠運(yùn)動(dòng)是因?yàn)槎呓g點(diǎn)位置相同。所以，部件1和2的約束方程是線性相關(guān)的。由于這個(gè)原因，其中的一個(gè)約束方程可以分解以減少線性損耗。 圖4.擺線形加速度的速度曲線 臂架仰角（度） 活塞反力（kN） 圖5.不同臂架仰角的活塞反力（吊重32.4kN，起升時(shí)間30s） 臂架仰角（度） 活塞反力（kN） 圖6.不同臂架仰角的活塞反力（吊重32.4kN，起升時(shí)間10s） 臂架仰角（度） 橫向位移（m） 圖7.不同臂架仰角的節(jié)點(diǎn)3、8、13的橫向位移（載荷32.4kN，起升時(shí)間30s） 橫向位移（m） 臂架仰角（度） 圖8.不同臂架仰角的節(jié)點(diǎn)3、8、13的橫向位移（載荷32.4kN，起升時(shí)間30s） 圖9.（a）節(jié)點(diǎn)13的橫向位移的時(shí)間響應(yīng) （b）節(jié)點(diǎn)13的橫向位移的快速傅氏變換算法 4.起重機(jī)特性的計(jì)算機(jī)仿真及與實(shí)驗(yàn)數(shù)據(jù)的對(duì)比 用于分析實(shí)驗(yàn)用起重機(jī)的應(yīng)用軟件已經(jīng)開發(fā)完畢。在這個(gè)軟件中，部件1（臂架）的其中任一個(gè)有限單元的形函數(shù)可以代換其他任何一個(gè)。 Balkan已經(jīng)對(duì)臂架的工作范圍內(nèi)，臂架運(yùn)動(dòng)的30s過(guò)程做了實(shí)驗(yàn)[2],選取這個(gè)速度是為了減小彈性變形的作用。在研究中測(cè)量了液壓系統(tǒng)的壓力和臂架的仰角位置。由于臂架的振動(dòng)而引起的油壓的振動(dòng)已經(jīng)通過(guò)控制系統(tǒng)過(guò)濾掉，所以在測(cè)量的數(shù)據(jù)中是看不到的。實(shí)驗(yàn)起重機(jī)在起升過(guò)程中的吊重為32.4kN，液壓系統(tǒng)油壓的變化是在臂架運(yùn)動(dòng)的30s過(guò)程中測(cè)定的。所以在臂架起升的30s過(guò)程中液壓活塞的反力變化與臂架仰角的變化有關(guān)，所以可以根據(jù)吊重32.4kN來(lái)計(jì)算臂架處于不同仰角位置時(shí)的變幅油缸活塞反力。 臂架在起升的30s過(guò)程中不同仰角位置的變幅油缸活塞反力通過(guò)利用計(jì)算機(jī)代碼模擬出來(lái)，并且在圖5中給出。 臂架運(yùn)動(dòng)的30s過(guò)程的實(shí)驗(yàn)結(jié)果也同時(shí)在圖5中給出。這些數(shù)據(jù)不包括活塞加速及減速過(guò)程。并且，由于臂架振動(dòng)帶來(lái)的影響已經(jīng)被濾除，所以在圖中是不能看到的。從圖中可以看出，臂架運(yùn)動(dòng)過(guò)程中仿真的結(jié)果與實(shí)驗(yàn)數(shù)據(jù)非常接近。 吊重32.4kN，起升時(shí)間10s時(shí)臂架不同位置的變幅活塞反力也通過(guò)計(jì)算機(jī)程序計(jì)算出來(lái)了，為的是模擬更有意義的彈性效果，如圖6所示。 在仿真時(shí)，臂架被離散成12個(gè)單元。其中兩個(gè)在A0G之間，臂架橫截面積從A0到G線性地增加。另外的十個(gè)在GC之間，臂架的橫截面積從G到C線性地減小。部件1的阻尼是通過(guò)在最初兩種模式基礎(chǔ)上依次減小2%來(lái)近似計(jì)算的。在模擬起升時(shí)間為30s時(shí)，假設(shè)臂架起升最初1.5s為加速過(guò)程，最后1.5s為減速過(guò)程。在臂架起升時(shí)間為10s的情況下，加速和減速的時(shí)間分別假設(shè)為1s。 吊重為32.4kN，臂架起升時(shí)間為30s和10s的兩個(gè)工況時(shí)，節(jié)點(diǎn)3（節(jié)點(diǎn)3在臂架節(jié)點(diǎn)A0和A之間），節(jié)點(diǎn)8（位于臂架節(jié)點(diǎn)A和C之間），及節(jié)點(diǎn)13（對(duì)應(yīng)于臂架尖端節(jié)點(diǎn)C）的橫向位移都根據(jù)臂架的不同仰角位置計(jì)算出來(lái)了，分別表達(dá)在圖7和圖8中。因?yàn)楣?jié)點(diǎn)3的橫向位移的數(shù)量級(jí)為10-5m，所以這個(gè)節(jié)點(diǎn)的橫向位移在圖中是看不到的。 從圖5-8可以看出活塞反力的幅度和平均值及節(jié)點(diǎn)的橫向位移在起升時(shí)間為10s工況時(shí)的值要大于起升時(shí)間為30s的工況。因此，臂架的彈性效果可以清晰地看出來(lái)。 在所有的仿真過(guò)程中，當(dāng)臂架在起升過(guò)程中，變幅活塞的反力像期望的一樣隨之減小。在變幅活塞加速運(yùn)動(dòng)過(guò)程時(shí)，活塞反力、橫向位移量及它們的振動(dòng)幅度都比勻速運(yùn)動(dòng)期間的要大。在臂架勻速起升的過(guò)程中，活塞反力的大小、橫向位移及各自的振動(dòng)幅度都平穩(wěn)地減小。在臂架減速起升的過(guò)程中，活塞反力和橫向位移減小，但振動(dòng)的幅度卻增加了。但在減速的過(guò)程中，活塞反力數(shù)值及幅值、橫向位移的變化卻比加速過(guò)程的小。 從仿真中還可以看出存在兩種類型的振動(dòng)。一種是由臂架的振動(dòng)產(chǎn)生的，周期較小，另一種由于載荷的振動(dòng)產(chǎn)生的，周期較大。吊重32.4kN，起升時(shí)間為10s工況時(shí)，臂架尖端（節(jié)點(diǎn)13）的橫向振動(dòng)的時(shí)間響應(yīng)如圖9（a）所示。圖9（a）中數(shù)據(jù)快速傅氏變換算法數(shù)值曲線在圖9（b）中給出。 頻率小的振動(dòng)是由于系統(tǒng)的激勵(lì)產(chǎn)生的，它的大小變化符合這個(gè)規(guī)律。當(dāng)臂架向上運(yùn)動(dòng)時(shí)，它朝垂直的位置變化，從而引起臂架的橫向偏轉(zhuǎn)量減小。由于載荷振動(dòng)而產(chǎn)生的振動(dòng)的頻率也降到這個(gè)頻率范圍。頻率大于1.5Hz的振動(dòng)是由于臂架以它的自然頻率振動(dòng)。自然頻率隨時(shí)間變化是多體系統(tǒng)的一個(gè)特性，并且這引起了我們?cè)趫D9（a）中看到的尖銳的信號(hào)。 5.起升能力的仿真 起重機(jī)傾翻的模擬是在閉環(huán)的條件下來(lái)完成的，其目的是為了能夠看到在起重機(jī)臂架向上及向下運(yùn)動(dòng)的過(guò)程中究竟何時(shí)發(fā)生傾翻。當(dāng)起重機(jī)的一個(gè)支腿受到地面的支反力變成零時(shí)，傾翻便發(fā)生了。利用起重機(jī)底盤的自由體受力圖，如圖10所示，傾翻時(shí)可用方程（21）表示： （21） 式中，、和、是起重機(jī)底盤對(duì)臂架及變幅油缸支反作用力的分力，、是地面對(duì)起重機(jī)兩個(gè)支腿的反作用力，是起重機(jī)底盤的重力。 圖10. 起重機(jī)底盤的自由體受力圖.單位：m.A5.50；C3.37；A12.57；A24.60；B11.77；B22.25 圖11.起升能力曲線. 30s仿真；10s仿真；廠家的數(shù)據(jù) 當(dāng)小于或等于零時(shí)，就滿足了傾翻的條件。對(duì)于臂架起升時(shí)間分別為10s和30s的兩種工況，對(duì)于不同吊重，對(duì)應(yīng)的力變成零的臂架位置是通過(guò)開發(fā)的應(yīng)用軟件計(jì)算出來(lái)的。實(shí)驗(yàn)用起重機(jī)的仿真結(jié)果在圖11中給出了。試驗(yàn)用起重機(jī)生產(chǎn)廠家提供的額定起重量也同時(shí)列于圖11中。其中的幅度R是這樣定義的，起重機(jī)在傾翻方向上由回轉(zhuǎn)中心到臂架端點(diǎn)，即載荷所在位置的水平距離，通過(guò)式計(jì)算得出。 從圖11中可以看出，當(dāng)臂架運(yùn)動(dòng)時(shí)間減少時(shí)，相同幅度時(shí)對(duì)應(yīng)的許用起重量隨之減少了。雖然在算得許用起重量的時(shí)候沒有包含任何類似臂架運(yùn)動(dòng)時(shí)間的信息，但與臂架運(yùn)動(dòng)時(shí)間為30s時(shí)的許用起重量的曲線非常相似。除此之外，制造廠家還注釋說(shuō)這些許用載荷數(shù)據(jù)應(yīng)該以一個(gè)安全系數(shù)來(lái)用，安全系數(shù)取為1.5。 6.結(jié)論 在研究中，起重機(jī)的起重能力是通過(guò)柔性多體動(dòng)力學(xué)分析得到的。為了達(dá)到這個(gè)分析目的，我們開發(fā)了一個(gè)能夠?qū)ζ鹬貦C(jī)進(jìn)行動(dòng)力學(xué)分析的應(yīng)用軟件。系統(tǒng)的剛性連接和彈性運(yùn)動(dòng)通過(guò)運(yùn)用絕對(duì)的耦合和形參變量用公式來(lái)表達(dá)[3, 4]。然后，部件間的鉸接和特定的運(yùn)動(dòng)是通過(guò)約束方程來(lái)表達(dá)的。柔性體是通過(guò)有限元方法來(lái)模擬的，形參變量是通過(guò)模型轉(zhuǎn)化得到的，用以代替彈性變量。并且利用計(jì)算機(jī)程序?qū)Φ踔貫?2.4kN，臂架起升運(yùn)動(dòng)時(shí)間分別為30s和10s兩種工況的變幅油缸活塞反力進(jìn)行了仿真，運(yùn)動(dòng)速度的加速和減速時(shí)的加速度都是圓滑的擺線形。并用臂架運(yùn)動(dòng)時(shí)間為30s的仿真結(jié)果與實(shí)驗(yàn)的結(jié)果進(jìn)行了對(duì)比。臂架運(yùn)動(dòng)時(shí)間分別為30s和10s兩種工況下，節(jié)點(diǎn)3、節(jié)點(diǎn)8和節(jié)點(diǎn)13在臂架處于不同仰角位置的橫向位移都計(jì)算出來(lái)了。最后，形成了臂架起升時(shí)間分別為10s和30s的兩種工況的載荷曲線，并將其與起重機(jī)生產(chǎn)廠家提供的數(shù)據(jù)進(jìn)行對(duì)比。 從分析中可以看出臂架運(yùn)動(dòng)時(shí)間將很顯著地影響起重機(jī)的動(dòng)力學(xué)特性。在活塞運(yùn)動(dòng)速度較低時(shí)（比如臂架起升時(shí)間為30s），臂架的柔性起的作用很小，所以臂架在這時(shí)可以看成是剛性體。然而，當(dāng)活塞運(yùn)動(dòng)速度提高時(shí)（比如臂架起升時(shí)間為10s），臂架的柔性起顯著的作用。在臂架運(yùn)動(dòng)時(shí)間為10s時(shí)，從活塞反力和起升能力的仿真結(jié)果中可以看出臂架的柔性起的作用。 另外需要注意的是，這里計(jì)算出來(lái)的載荷曲線是在活塞運(yùn)動(dòng)速度曲線為擺線形的情況下得出的，這是為了近似地模擬有經(jīng)驗(yàn)的駕駛員操作時(shí)的速度特性。其他形狀的速度曲線對(duì)應(yīng)的載荷曲線也同樣可以通過(guò)開發(fā)的計(jì)算機(jī)程序計(jì)算生成。變幅活塞突然加速和減速時(shí)將會(huì)顯著的降低起升能力，這是由于這將會(huì)產(chǎn)生較大的橫向位移和慣性沖擊載荷。 參考文獻(xiàn)：略 - 13 - Journal of Sound and Vibration (1999) 223(4), 645±657 Article No. jsvi.1999.2154, available online at http://www.idealibrary.com on TIPPING LOADS OF MOBILE CRANES WITH FLEXIBLE BOOMS S. KILIC?ASLAN, T. BALKAN AND S. K. IDER Department of Mechanical Engineering, Middle East Technical University, 06531 Ankara, Turkey (Received 23 July 1997, and in ?nal form 4 January 1999) In this study the characteristics of a mobile crane are obtained by using a ˉexible multibody dynamics approach, for the determination of safe loads to prevent tipping of a mobile crane. Only the boom of the crane is assumed to be ˉexible since it is the only element that has considerable deˉections in applications. The coupled rigid and elastic motions of the crane are formulated and software is developed in order to carry out the dynamic analysis. The variation of piston force with respect to boom angular position for dierent boom motion times are simulated, load curves are generated and the results are compared with the experimental results obtained from a 10 t mobile crane. # 1999 Academic Press 1. INTRODUCTION Cranes as mechanical systems are in general closed-chain mechanisms with ˉexible members. In the problem of determination of safe loads which as a function of the boom angular position, the solution of the dynamic equations is necessary. There are few studies related to the dynamics and control of mobile cranes for various applications. In almost all of these studies the body ˉexibility is not taken into consideration. A dynamic model for the control of a ˉexible rotary crane which carries out three kinds of motion (rotation, load hoisting and boom hoisting) simultaneously is derived by Sato and Sakawa [1]. Only the joint between the boom and the jib is assumed to be ˉexible. The goal is to transfer a load to a desired place in such a way that at the end of the transfer the swing of the load decays as quickly as possible. The application of a hook load and safe load indicator and limiter for mobile cranes is presented by Balkan where the microprocessor-based control system for the determination of current hook load is based on oil pressure and boom angle [2]. In this paper, mobile crane characteristics are determined by using ˉexible multibody analysis. Kinematics and equations of motion of the ˉexible multibody system are derived. Software has been developed to carry out dynamic analysis of the crane. In the ˉexible dynamic analysis, the coupled rigid and elastic motion of the system is formulated by using absolute co-ordinates 0022±460X/99/240645 13 $30.00/0  # 1999 Academic Press 646  S. KILIC?ASLAN ET AL. and modal variables [3, 4]. Then, joint connections and prescribed motions are imposed as constraint equations. The ˉexible body is modelled by the ?nite element method and modal variables are used as the elastic variables by utilizing modal transformation. The variations of the piston force with respect to the boom angular positions are analyzed for different boom motion times to illustrate the effect of ˉexibility by using the developed software. Load curves are generated for various boom motion times and compared to those of the manufacturer. 2. MODELLING OF THE CRANE Since there is experimental work on the tipping load control of a COLES Mobile 930 crane, for the application of the developed software, the structure of the above mentioned crane and its parameters are used [5]. However, the method of analysis can easily be applied to similar types of cranes with simple modi?cations. In general, mobile cranes are operated under blocked conditions by the vertical jacks. The load is attached to the hook and the boom is hoisted. Since the excessive raising of the load is dangerous, the height of the load is controlled by lengthening the rope. During hoisting, lowering and transportation of the load, the crane is not rotated, due to some restrictions such as very huge and/or heavy loads, space problems, etc. In every angular position of the boom, there is a maximum load above which tipping might probably occur. Since the angular position of the boom changes only during the up and down motion of the load, which is actually a planar motion, modelling and analysis are carried out in two dimensions. Figure 1. Schematic representation of the test crane. MOBILE CRANES  (5)  C  n(5)  647 n(5) 1 n(2) n2 G  A  n1  Body 1 D Body 3  Body 5 1 2 n(1) Body 2 n(1) 1 2 n(2) n(3) (3) 1 (1) (2)  n(3) 1 B  Body 4  a A0 2 n(4) (4) O n(4) 1 2 B0 Figure 2. Kinematic model of the test crane. Dimensions in mm: A0G 2000; GC 17 500; A0A 5823; A0D 5850; AD 565; BD 3455; OB02350; OA0805. Schematic representation of the test crane is shown in Figure 1 and the kinematic model of the test crane which can be represented by ?ve bodies is shown in Figure 2. Cross-section, material properties and dimensions of each body are obtained from the technical data sheet and measured directly from the test crane. The cross-section of Body 1 is a hollow polygon of thickness, t, as shown in Figure 3. The cross-section dimensions increase from A0to G and decrease from G to C linearly, and the dimensions at sections A0, G and C are shown in Figure 3. Body 2 is a cylindrical rod 25 mm in diameter and Body 3 is a piston 180 mm in e0 c0 A0G C f0  t=20 e0453 c0200 f0120 160 160 440 207 120 120 Figure 3. Cross-section of Body 1. Dimensions in mm. 648  S. KILIC?ASLAN ET AL. diameter with spool thickness of 20 mm. Body 4 is a cylinder with inner diameter of 230 mm, outer diameter of 246 mm and length of 3440 mm. Additionally, modulus of elasticity and mass density of the Body 1 are taken as 200 GPa and 5750 kg/m3, respectively. Mass density of other bodies are taken as 7850 kg/m3. When dimensions (lengths and cross-sections) and elastic properties of the bodies of the crane are considered, it is suf?cient to take only the Body 1 (the boom) as ˉexible. In this case, other bodies are assumed to be rigid. The following assumptions are considered in the analysis of the crane. 1. The mass of the hydraulic oil is included in the mass of the cylinder (Body 4). Varying mass of the cylinder due to varying amounts of hydraulic oil inside it is taken into consideration. 2. Hydraulic oil is assumed incompressible. 3. The hook load is considered as a point mass and connected to the end of the boom with a rope which is taken as a rigid rod. This rope is free for planar rotation about point C. This assumption is valid as long as the oscillations of the rod about the vertical position are small and the rod remains in tension. These conditions are satis?ed for normal operation speeds and hook loads. 4. The structural damping of the boom is taken into account by assuming Rayleigh damping. 5. The distance between the load and the base is assumed to be kept constant by varying the length of the rope during the up and down motion of the crane. 3. DYNAMIC EQUATIONS Let nk represent a body reference frame relative to which the deformation of Body k is de?ned and n represent a ?xed frame. Let xk represent the position of the origin Q of nk in n, and ok be the angular velocity of Body k. Using the ?nite element method, the deformation displacement vector uki of an arbitrary point P in element i of Body k is where fki is the element shape function matrix transformed to nk, Bki is the element connectivity Boolean matrix and ak is the vector of body nodal variables. The velocity of P is written as where qki is the position vector from Q to P in nk including deformation, ~qki is the skew symmetric matrix of qki, Tk is the co-ordinate transformation from nk to n, o" k TkiTok, wk is modal transformation used to reduce the elastic degrees of freedom, and Zk is the vector of body modal variables. Equation (2) can be expressed as MOBILE CRANES  649 The joint connections and prescribed motions in the system of N interconnected bodies are represented by kinematic constraint equations expressed at velocity level as where y is the system generalized speed vector given by  C is the constraint Jacobian matrix which can be formed by the velocity inˉuence coef?cient matrices and g indicates the prescribed velocities. Kane's equations are used to determine the equations of motion of the system as  where l is the vector of constraint forces, M is the generalized mass matrix, Q, Fs, Fd and F are vectors of Coriolis forces, elastic forces, damping forces and applied forces, respectively and 650  v0 v(t) 0  S. KILIC?ASLAN ET AL. t1t2  t3 Figure 4. Velocity pro?le with cycloidal acceleration and deceleration. where Kk is the structural stiffness matrix and Dk is the structural Rayleigh damping matrix of Body k. In the simulations, the weights of the structural mass and stiffness matrices used in forming Dk correspond to a 2% damping ratio. When the space dependent terms in equations (8) and (9) are separated, a set of time invariant matrices are obtained [3, 4]. These mass properties are evaluated once in advance. Equation (6) and the derivative of equation (4) represent linear equations for the accelerations y and the constraint forces l. The accelerations obtained from these equations are numerically integrated by using a variable step, variable order predictor±corrector algorithm to obtain the time history of the generalized speeds and generalized co-ordinates. The boundary conditions used for the description of the deformation of Body 1 are that for axial deformation A0is ?xed, and for bending A0is hinged and A is ?xed. The ?rst axial mode, the ?rst bending mode of part A0A and the ?rst two bending modes of part AC of Body 1 are taken as the modal co-ordinates since the higher modes are observed negligible. Therefore the generalized speed vector of the system is MOBILE CRANES  651  The boom is driven by a hydraulic actuator which is controlled by the operator. In general, throughout the motion, the hydraulic actuator is driven with constant velocity v0so that the boom and piston oscillations are kept to a minimum level. Moreover, to avoid impact loading, the actuator velocity is increased from zero to v0at the beginning of the motion and decreased from v0 to zero at the end of the motion which can be assumed cycloidal in time. This desired velocity pro?le is shown in Figure 4 and can be expressed as follows. If the pivots of Bodies 1 and 2 were at different points, the system would be a structure. The system is moveable owing to the special dimension obtained due to the concurrency of the pivots. Thus, the constraint equations written for Body 1 and Body 2 are linearly dependent. For this reason, one of the constraint equations is dropped to remove the linear dependency. 652  600 400 200 0  20  S. KILIC?ASLAN ET AL. Experimental Simulation 40  60  80 Boom angular position (degree) Figure 5. Piston force with respect to boom angular position (32á4 kN hook load and 30 s boom upward motion). 4. COMPUTER SIMULATION OF THE CRANE CHARACTERISTICS AND COMPARISON WITH THE EXPERIMENTAL RESULTS Software has been developed for the analysis of the test crane. In this software, one can take any number of ?nite elements and modal variables for Body 1. Experimental studies have been carried out by Balkan for the working range of the boom in which the boom was moved in 30 s [2]. This speed was selected in order to minimize the effect of ˉexibility. In that study, the pressure in the hydraulic actuator and the angular positions of the boom were measured. The oscillations in the pressure resulting from the boom oscillations are ?ltered out 800 400 0 20 40 60 80 Boom angular position (degree) Figure 6. Piston force with respect to boom angular position (32á4 kN hook load and 10 s boom upward motion). Piston force (kN) Piston force (kN) 0.00 –0.08 –0.16  MOBILE CRANES Node 3 8 12  653 20 40 60 80 Boom angular position (degree) Figure 7. Transverse deˉections of nodes 3, 8 and 13 with respect to boom angular position (32á4 kN hook load and 30 s boom upward motion). in the control system, hence they are not seen in the measured data. The test crane was moved with a 32á4 kN hook load in the upward direction, and the variations of the pressures in the hydraulic actuator with respect to the boom angular positions are obtained for the 30 s motion of the boom. Therefore, the variations of the piston force with respect to the boom angular positions for the 30 s boom upward motion can be calculated for the 32á4 kN hook load. The variations of the piston force with respect to the boom angular positions for the 32á4 kN hook load are simulated for the 30 s boom upward motion by using the computer code and given in Figure 5. Experimental results for the 30 s motion of the boom are also shown in Figure 5. The data do not include piston acceleration and deceleration intervals. Moreover, since the boom oscillations are ?ltered out, they are not seen in the Node 3 0.00 8 –0.08 13 –0.16 20  40  60  80 Boom angular position (degree) Figure 8. Transverse deˉections of nodes 3, 8 and 13 with respect to boom angular position (32á4 kN hook load and 10 s boom upward motion). Transverse deflection (m) Transverse deflection (m) 654  0.00  S. KILIC?ASLAN ET AL. –0.08 –0.16  0  2  4  6  8  10  0  4  8 Time (s) Frequency (Hz) Figure 9. (a) Time response of transverse deˉection of node 13. (b) FFT of transverse deˉec- tion of node 13. ?gure. It is seen from the ?gure that simulation and experimental results for the 30 s boom motion are close to each other. Similarly, the variations of the piston force with respect to the boom angular positions for the 32á4 kN hook load are simulated for the 10 s boom upward motion by using the computer code in order to make the effect of ˉexibility more signi?cant as shown in Figure 6. In the simulations, the boom is discretized by 12 ?nite elements. Two of them are taken on A0G where the cross-sectional area is increasing from A0to G linearly and ten of them are taken on GC where the cross-sectional area is decreasing from G to C linearly. Damping is included for Body 1 by using a 2% damping ratio for the ?rst two modes. It is assumed that the ?rst 1á5 s is used for the acceleration and the last 1á5 s is used for the deceleration of the boom for the 30 s boom motion. In the case of 10 s boom motion, acceleration and deceleration intervals are assumed to be 1 s. A0 FA01B0 FA 02 F B01  FB02  B1A1 FA  A FC A  C A2  B2  B FB Figure 10. Free body diagram of the crane chassis. Dimensions in m: A 5á50; C 3á37; A12á57; A24á60; B11á77; B22á25. Tranverse deflection (m) Arbitrary units 160 120 80 40 0  MOBILE CRANES  655 6 10 Radius (m) 14 18 Load (kN) 656  S. KILIC?ASLAN ET AL. The magnitudes at small frequencies correspond to the trend due to the excitation of the system. As the boom moves upwards, it goes towards the vertical position causing the boom transverse deˉections to decrease. The frequency due to the load oscillations also falls into this frequency range. The frequencies over 1á5 Hz are due to the boom oscillations at its natural frequencies. The variation of the natural frequency with time is a characteristic of multibody systems and results in a chirp signal as seen in Figure 9(a). 5. SIMULATION OF THE LIFTING CAPACITY ON THE HOOK Tipping simulation is performed in the blocked condition of the crane to see when tipping occurs as the boom moves in the upward and downward directions. When one of the reaction forces coming from the ground to the vertical jacks becomes zero, tipping occurs. Using the free body diagram of the crane chassis, shown in Figure 10, equation (21) is written for the tipping case as where FA01, FA02 and FB01, FB02 are components of the reaction forces exerted by the boom and the cylinder on the crane chassis; FAand FBare the reaction forces exerted by the ground to the jacks and FCis the body force of the crane chassis. When FAis smaller than or equal to zero, tipping condition occurs. For the 30 s and 10 s boom motions, the boom angular positions where FAbecomes zero are determined by using the developed software for different hook loads. The simulation results for the test crane are given in Figure 11. The allowable load speci?ed by the manufacturer of the test crane is also shown in Figure 11 where radius is de?ned as the horizontal distance from the vertical axis of rotation of the crane to the tip of the boom at the tipping position, calculated as It can be seen from Figure 11 that when the boom motion time is decreased, the allowable load for the same radius decreases. Although there is no information about the conditions such as boom motion time while the allowable load data are being obtained, the plot of the allowable load is very similar to 30 s boom motion time simulation. In addition to this, it is noted by the manufacturer that these allowable load data should be used with a safety factor of 1-5. 6. CONCLUSION In this study, the mobile crane characteristics are determined by using ˉexible multibody analysis. In order to achieve this goal software has been developed which is capable of carrying out dynamic analysis of the crane. The coupled rigid and elastic motions of the system are formulated by using absolute co-ordinates and modal variables [3, 4]. Then, joint connections and prescribed motions are imposed as constraint equations. The ˉexible body is MOBILE CRANES  657 modelled by the ?nite element method and the modal variables are used as the elastic variables by utilizing modal transformation. The variations of piston force with respect to the boom angular positions for 32á4 kN hook load are simulated for both 30 s and 10 s boom upward motions by using the computer code for the velocity pro?le with cycloidal acceleration and deceleration. 30 s boom motion simulations are compared with the experimental results. Moreover, transverse deˉections of node 3, node 8 and node 13 are obtained with respect to the boom angular positions for both 30 s and 10 s boom upward motion. Finally, load curves are generated for the 30 s motion and 10 s motion and compared with those of the manufacturer. It is seen from the analysis that the boom motion time affects the crane dynamics considerably. For lower piston speeds (i.e., 30 s motion of the boom), the effect of ˉexibility is very small. Thus, the boom can be taken as a rigid body. However, when the piston speed is increased (i.e., 10 s motion of the boom), the effect of ˉexib