【機械類畢業(yè)論文中英文對照文獻翻譯】柔性臂架自行式起重機傾翻載荷
【機械類畢業(yè)論文中英文對照文獻翻譯】柔性臂架自行式起重機傾翻載荷,機械類畢業(yè)論文中英文對照文獻翻譯,機械類,畢業(yè)論文,中英文,對照,對比,比照,文獻,翻譯,柔性,自行,起重機,載荷
柔性臂架自行式起重機傾翻載荷
柔性臂架自行式起重機傾翻載荷
S. KILICASLAN, T. BALKAN AND S. K. IDER
土耳其安卡拉 中東科技大學(xué)機械工程學(xué)院 16531
摘 要
在這一項研究中，自行式起重機的特性是利用基于柔性多體動力學(xué)理論建模分析得到的，用以確定起重機不產(chǎn)生傾翻危險的起重量。只有起重機的臂架被假設(shè)成柔性的，因為臂架是唯一一個在吊載過程中撓度較大的部件。用數(shù)學(xué)方程描述起重機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)了用于進行動力學(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，臂架運動的同時，變幅油缸的推力被計算出來，同時繪制出載荷曲線，并且將結(jié)果與用一臺10噸的自行式工程起重機實驗得到的數(shù)據(jù)進行比較。
1.緒論
作為機械系統(tǒng)的起重機一般被看成包含柔性部件的閉環(huán)的機械裝置。在以臂架的角度位置作為自變量來確定起重機額定起重量的時候，就需要用到動力學(xué)的解決方法。
目前只有極少數(shù)關(guān)于自行式起重機的動力學(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建立了一個動力學(xué)模型，用于控制一個同時進行三種運動（回轉(zhuǎn)運動、起升運動、變幅運動）的柔性回轉(zhuǎn)起重機[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，載荷幅值的擺動能夠以最快的速度衰減。這個能夠?qū)ζ鹬貦C進行實際載荷和額定載荷對比顯示并加以限制的系統(tǒng)在巴爾干半島上得到應(yīng)用，這個基于微型計算機的控制系統(tǒng)是通過油壓和臂架仰角來確定當(dāng)前吊鉤的實際載荷的。
在這篇文章中，起重機的特性是通過柔性多體動力學(xué)來分析得到的。給出了柔性多體動力學(xué)的動力學(xué)和運動學(xué)方程，同時開發(fā)了對起重機進行動力學(xué)分析的應(yīng)用軟件。在多體動力學(xué)分析時，系統(tǒng)的剛性聯(lián)接和柔性運動通過運用絕對的耦合和形參變量用公式來表達[3, 4]。然后，部件間的連接和指定的運動用約束方程來描述。柔性體是通過有限元方法來模擬的，形參變量是通過模型轉(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，臂架運動的同時，變幅油缸的推力被計算出來，通過開發(fā)的應(yīng)用軟件來圖示彈性效果。同時繪制出臂架不同仰角的載荷曲線，并且將結(jié)果與起重機制造廠家提供的數(shù)據(jù)進行比較。
2.起重機建模
我們用一臺930型自行式起重機的傾翻載荷控制的實驗數(shù)據(jù)用于開發(fā)的軟件的計算，并且用了這臺起重機的結(jié)構(gòu)和參數(shù)[5]。但這個分析方法只要經(jīng)過簡單的修改就可以用于類似的起重機的分析。
重物吊在吊鉤上，臂架起升，因為重物起升過高非常危險，重物的高度是通過測量起升繩的長度來控制的。在重物起升、下降和運輸時，起重機是不能回轉(zhuǎn)的，這是由于一些限制條件，比如重物非常大或非常重、空間問題等。
在臂架的每一個角度位置都對應(yīng)著一個最大載荷，超過這個載荷后起重機很有可能傾翻。因為臂架的仰角只有在重物起升或下降時才會發(fā)生改變，而且恰巧這又是個平面運動，所以建模和分析計算都是在二維空間進行的。
圖1.試驗用起重機示意圖
圖2.試驗用起重機動力學(xué)模型
單位：mm；A0G 2000;GC 17500;A0A 5823;A0D 5850;AD 565;BD 3455;OB0 2350;OA0 805
圖1表示實驗用起重機的示意圖，圖2表示它的動力學(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技術(shù)數(shù)據(jù)文件和直接對實驗起重機進行測量得到的。部件1（臂架）的橫截面是個空心的多邊形，壁厚為t，如圖3所示。這個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是個圓柱形的桿，直徑25mm。部件3是個液壓活塞桿，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（長度和橫截面）和彈性模量，就可以只認為部件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。
在對起重機進行分析時要考慮以下假設(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.液壓油被認為是不可壓縮的；
3.吊重被看成是集中質(zhì)量，并且通過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點自由旋轉(zhuǎn)。只要這個桿相對于垂直位置的擺動很小并且這個桿仍然處于張緊狀態(tài)時這個假設(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衰減法來計算確定；
5.吊重與地面間的距離假設(shè)保持不變，這是通過在起重機臂架升降過程中改變繩長來實現(xiàn)的。
3.動力學(xué)方程
令表示一個部件的結(jié)構(gòu)，部件k的變形通過這個系數(shù)來定義，表示一個確定的結(jié)構(gòu)。令表示結(jié)構(gòu)的原點的位置，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)點位移向量。
點P的速度可表示為：
（2）
其中是中點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。
連接各個部件的系統(tǒng)N的連接處和角度指示在速度水平上用運動學(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)的運動方程：
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，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，其中是個標(biāo)量。變成，其中是根據(jù)變換而來的。
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驅(qū)動的，一般來說，運動的整個過程中液壓缸以恒定的速度運動，所以臂架和活塞的振動都能控制在一個很小的水平上。為了避免沖擊載荷的產(chǎn)生，活塞的開始運動時速度從0增大到以及最終停止時從減小到0，速度的變化假設(shè)成隨時間呈擺線形變化。這個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的絞點在不同的位置，這個系統(tǒng)將變成一個不能動的結(jié)構(gòu)。系統(tǒng)之所以能夠運動是因為二者絞點位置相同。所以，部件1和2的約束方程是線性相關(guān)的。由于這個原因，其中的一個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，起升時間10s）
臂架仰角（度）
橫向位移（m）
圖7.不同臂架仰角的節(jié)點3、8、13的橫向位移（載荷32.4kN，起升時間30s）
橫向位移（m）
臂架仰角（度）
圖8.不同臂架仰角的節(jié)點3、8、13的橫向位移（載荷32.4kN，起升時間30s）
圖9.（a）節(jié)點13的橫向位移的時間響應(yīng)
（b）節(jié)點13的橫向位移的快速傅氏變換算法
4.起重機特性的計算機仿真及與實驗數(shù)據(jù)的對比
用于分析實驗用起重機的應(yīng)用軟件已經(jīng)開發(fā)完畢。在這個軟件中，部件1（臂架）的其中任一個有限單元的形函數(shù)可以代換其他任何一個。
Balkan已經(jīng)對臂架的工作范圍內(nèi)，臂架運動的30s過程做了實驗[2],選取這個速度是為了減小彈性變形的作用。在研究中測量了液壓系統(tǒng)的壓力和臂架的仰角位置。由于臂架的振動而引起的油壓的振動已經(jīng)通過控制系統(tǒng)過濾掉，所以在測量的數(shù)據(jù)中是看不到的。實驗起重機在起升過程中的吊重為32.4kN，液壓系統(tǒng)油壓的變化是在臂架運動的30s過程中測定的。所以在臂架起升的30s過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來計算臂架處于不同仰角位置時的變幅油缸活塞反力。
臂架在起升的30s過程中不同仰角位置的變幅油缸活塞反力通過利用計算機代碼模擬出來，并且在圖5中給出。
臂架運動的30s過程的實驗結(jié)果也同時在圖5中給出。這些數(shù)據(jù)不包括活塞加速及減速過程。并且，由于臂架振動帶來的影響已經(jīng)被濾除，所以在圖中是不能看到的。從圖中可以看出，臂架運動過程中仿真的結(jié)果與實驗數(shù)據(jù)非常接近。
吊重32.4kN，起升時間10s時臂架不同位置的變幅活塞反力也通過計算機程序計算出來了，為的是模擬更有意義的彈性效果，如圖6所示。
在仿真時，臂架被離散成12個單元。其中兩個在A0G之間，臂架橫截面積從A0到G線性地增加。另外的十個在GC之間，臂架的橫截面積從G到C線性地減小。部件1的阻尼是通過在最初兩種模式基礎(chǔ)上依次減小2%來近似計算的。在模擬起升時間為30s時，假設(shè)臂架起升最初1.5s為加速過程，最后1.5s為減速過程。在臂架起升時間為10s的情況下，加速和減速的時間分別假設(shè)為1s。
吊重為32.4kN，臂架起升時間為30s和10s的兩個工況時，節(jié)點3（節(jié)點3在臂架節(jié)點A0和A之間），節(jié)點8（位于臂架節(jié)點A和C之間），及節(jié)點13（對應(yīng)于臂架尖端節(jié)點C）的橫向位移都根據(jù)臂架的不同仰角位置計算出來了，分別表達在圖7和圖8中。因為節(jié)點3的橫向位移的數(shù)量級為10-5m，所以這個節(jié)點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)點的橫向位移在起升時間為10s工況時的值要大于起升時間為30s的工況。因此，臂架的彈性效果可以清晰地看出來。
在所有的仿真過程中，當(dāng)臂架在起升過程中，變幅活塞的反力像期望的一樣隨之減小。在變幅活塞加速運動過程時，活塞反力、橫向位移量及它們的振動幅度都比勻速運動期間的要大。在臂架勻速起升的過程中，活塞反力的大小、橫向位移及各自的振動幅度都平穩(wěn)地減小。在臂架減速起升的過程中，活塞反力和橫向位移減小，但振動的幅度卻增加了。但在減速的過程中，活塞反力數(shù)值及幅值、橫向位移的變化卻比加速過程的小。
從仿真中還可以看出存在兩種類型的振動。一種是由臂架的振動產(chǎn)生的，周期較小，另一種由于載荷的振動產(chǎn)生的，周期較大。吊重32.4kN，起升時間為10s工況時，臂架尖端（節(jié)點13）的橫向振動的時間響應(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。
頻率小的振動是由于系統(tǒng)的激勵產(chǎn)生的，它的大小變化符合這個規(guī)律。當(dāng)臂架向上運動時，它朝垂直的位置變化，從而引起臂架的橫向偏轉(zhuǎn)量減小。由于載荷振動而產(chǎn)生的振動的頻率也降到這個頻率范圍。頻率大于1.5Hz的振動是由于臂架以它的自然頻率振動。自然頻率隨時間變化是多體系統(tǒng)的一個特性，并且這引起了我們在圖9（a）中看到的尖銳的信號。
5.起升能力的仿真
起重機傾翻的模擬是在閉環(huán)的條件下來完成的，其目的是為了能夠看到在起重機臂架向上及向下運動的過程中究竟何時發(fā)生傾翻。當(dāng)起重機的一個支腿受到地面的支反力變成零時，傾翻便發(fā)生了。利用起重機底盤的自由體受力圖，如圖10所示，傾翻時可用方程（21）表示：
（21）
式中，、和、是起重機底盤對臂架及變幅油缸支反作用力的分力，、是地面對起重機兩個支腿的反作用力，是起重機底盤的重力。
圖10. 起重機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，就滿足了傾翻的條件。對于臂架起升時間分別為10s和30s的兩種工況，對于不同吊重，對應(yīng)的力變成零的臂架位置是通過開發(fā)的應(yīng)用軟件計算出來的。實驗用起重機的仿真結(jié)果在圖11中給出了。試驗用起重機生產(chǎn)廠家提供的額定起重量也同時列于圖11中。其中的幅度R是這樣定義的，起重機在傾翻方向上由回轉(zhuǎn)中心到臂架端點，即載荷所在位置的水平距離，通過式計算得出。
從圖11中可以看出，當(dāng)臂架運動時間減少時，相同幅度時對應(yīng)的許用起重量隨之減少了。雖然在算得許用起重量的時候沒有包含任何類似臂架運動時間的信息，但與臂架運動時間為30s時的許用起重量的曲線非常相似。除此之外，制造廠家還注釋說這些許用載荷數(shù)據(jù)應(yīng)該以一個安全系數(shù)來用，安全系數(shù)取為1.5。
6.結(jié)論
在研究中，起重機的起重能力是通過柔性多體動力學(xué)分析得到的。為了達到這個分析目的，我們開發(fā)了一個能夠?qū)ζ鹬貦C進行動力學(xué)分析的應(yīng)用軟件。系統(tǒng)的剛性連接和彈性運動通過運用絕對的耦合和形參變量用公式來表達[3, 4]。然后，部件間的鉸接和特定的運動是通過約束方程來表達的。柔性體是通過有限元方法來模擬的，形參變量是通過模型轉(zhuǎn)化得到的，用以代替彈性變量。并且利用計算機程序?qū)Φ踔貫?2.4kN，臂架起升運動時間分別為30s和10s兩種工況的變幅油缸活塞反力進行了仿真，運動速度的加速和減速時的加速度都是圓滑的擺線形。并用臂架運動時間為30s的仿真結(jié)果與實驗的結(jié)果進行了對比。臂架運動時間分別為30s和10s兩種工況下，節(jié)點3、節(jié)點8和節(jié)點13在臂架處于不同仰角位置的橫向位移都計算出來了。最后，形成了臂架起升時間分別為10s和30s的兩種工況的載荷曲線，并將其與起重機生產(chǎn)廠家提供的數(shù)據(jù)進行對比。
從分析中可以看出臂架運動時間將很顯著地影響起重機的動力學(xué)特性。在活塞運動速度較低時（比如臂架起升時間為30s），臂架的柔性起的作用很小，所以臂架在這時可以看成是剛性體。然而，當(dāng)活塞運動速度提高時（比如臂架起升時間為10s），臂架的柔性起顯著的作用。在臂架運動時間為10s時，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，這里計算出來的載荷曲線是在活塞運動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)驗的駕駛員操作時的速度特性。其他形狀的速度曲線對應(yīng)的載荷曲線也同樣可以通過開發(fā)的計算機程序計算生成。變幅活塞突然加速和減速時將會顯著的降低起升能力，這是由于這將會產(chǎn)生較大的橫向位移和慣性沖擊載荷。
參考文獻：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
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