A數(shù)學(xué)建模制動(dòng)器試驗(yàn)臺(tái)的控制方法分析

2011高教社杯全國(guó)大學(xué)生數(shù)學(xué)建模競(jìng)賽承諾書(shū)我們仔細(xì)閱讀了中國(guó)大學(xué)生數(shù)學(xué)建模競(jìng)賽的競(jìng)賽規(guī)則.我們完全明白，在競(jìng)賽開(kāi)始后參賽隊(duì)員不能以任何方式（包括電話、電子郵件、網(wǎng)上咨詢(xún)等）與隊(duì)外的任何人（包括指導(dǎo)教師）研究、討論與賽題有關(guān)的問(wèn)題。我們知道，抄襲別人的成果是違反競(jìng)賽規(guī)則的,如果引用別人的成果或其他公開(kāi)的資料（包括網(wǎng)上查到的資料），必須按照規(guī)定的參考文獻(xiàn)的表述方式在正文引用處和參考文獻(xiàn)中明確列出。我們鄭重承諾，嚴(yán)格遵守競(jìng)賽規(guī)則，以保證競(jìng)賽的公正、公平性。如有違反競(jìng)賽規(guī)則的行為，我們將受到嚴(yán)肅處理。我們參賽選擇的題號(hào)是（從A/B/C/D中選擇一項(xiàng)填寫(xiě)）： 我們的參賽報(bào)名號(hào)為（如果賽區(qū)設(shè)置報(bào)名號(hào)的話）：所屬學(xué)校（請(qǐng)?zhí)顚?xiě)完整的全名）：參賽隊(duì)員(打印并簽名)：1.2.3.指導(dǎo)教師或指導(dǎo)教師組負(fù)責(zé)人(打印并簽名)：日期：年月日賽區(qū)評(píng)閱編號(hào)（由賽區(qū)組委會(huì)評(píng)閱前進(jìn)行編號(hào)）：2011高教社杯全國(guó)大學(xué)生數(shù)學(xué)建模競(jìng)賽編號(hào)專(zhuān)用頁(yè)賽區(qū)評(píng)閱編號(hào)（由賽區(qū)組委會(huì)評(píng)閱前進(jìn)行編號(hào)）：賽區(qū)評(píng)閱記錄（可供賽區(qū)評(píng)閱時(shí)使用）：評(píng)閱人評(píng)分備注全國(guó)統(tǒng)一編號(hào)（由賽區(qū)組委會(huì)送交全國(guó)前編號(hào)）：全國(guó)評(píng)閱編號(hào)（由全國(guó)組委會(huì)評(píng)閱前進(jìn)行編號(hào)）：制動(dòng)器試驗(yàn)臺(tái)的控制方法分析摘要制動(dòng)器試驗(yàn)臺(tái)用于模擬理想的路試環(huán)境下汽車(chē)的制動(dòng)過(guò)程，從而檢驗(yàn)制動(dòng)器的性能。試驗(yàn)臺(tái)上用飛輪組的轉(zhuǎn)動(dòng)慣量模擬路試中汽車(chē)的慣性，使得飛輪組的轉(zhuǎn)動(dòng)動(dòng)能與汽車(chē)的平動(dòng)動(dòng)能相當(dāng)。對(duì)于前兩問(wèn)，運(yùn)用力學(xué)知識(shí)，根據(jù)質(zhì)量守恒方程，圓盤(pán)的轉(zhuǎn)動(dòng)慣量計(jì)算公式經(jīng)過(guò)計(jì)算即可得到結(jié)果；對(duì)于第三問(wèn)，通過(guò)題目中所給定的驅(qū)動(dòng)電流的算法。列出微分方程,由理論力學(xué)公式可以知道，，所以可以推出電流與可觀測(cè)量之間的關(guān)系為，即為建立的數(shù)學(xué)模型，將問(wèn)題1和問(wèn)題2的條件帶入，可計(jì)算出驅(qū)動(dòng)電流的大小。 對(duì)于第四問(wèn)，需要分別求出路試時(shí)的制動(dòng)器和相對(duì)應(yīng)的實(shí)驗(yàn)臺(tái)上制動(dòng)器在制動(dòng)過(guò)程中消耗的能量，以求他們的能量差值和相對(duì)誤差。通過(guò)計(jì)算差值和相對(duì)誤差能確定其精確度。對(duì)于第五問(wèn)，可以根據(jù)某時(shí)刻測(cè)得的制動(dòng)扭矩QUOTE控制下一個(gè)時(shí)間段的驅(qū)動(dòng)電流。對(duì)于第六問(wèn)，由理論推導(dǎo)可知轉(zhuǎn)速與時(shí)間t呈線性關(guān)系，可將原始數(shù)據(jù)分為兩段，分別計(jì)算在理想狀況下路試消耗的能量，再根據(jù)試驗(yàn)臺(tái)是數(shù)據(jù)作比較，得到其誤差。關(guān)鍵詞：制動(dòng)模擬，微分方程，能量誤差，轉(zhuǎn)速補(bǔ)償問(wèn)題的提出在汽車(chē)設(shè)計(jì)階段中，通常利用制動(dòng)器試驗(yàn)臺(tái)代替路試來(lái)檢驗(yàn)制動(dòng)器的性能。模擬試驗(yàn)的原則是試驗(yàn)臺(tái)上制動(dòng)器的制動(dòng)過(guò)程與路試車(chē)輛上制動(dòng)器的制動(dòng)過(guò)程盡可能一致?，F(xiàn)假設(shè)輪胎與地面無(wú)滑動(dòng)，假設(shè)試驗(yàn)臺(tái)采用的電動(dòng)機(jī)的驅(qū)動(dòng)電流與其產(chǎn)生的扭矩成正比；且試驗(yàn)臺(tái)工作時(shí)主軸的瞬時(shí)轉(zhuǎn)速與瞬時(shí)扭矩是可觀測(cè)的離散量。由于制動(dòng)器性能的復(fù)雜性，電動(dòng)機(jī)驅(qū)動(dòng)電流與時(shí)間之間的精確關(guān)系是很難得到的。我們把把整個(gè)制動(dòng)時(shí)間離散化為許多小的時(shí)間段，然后根據(jù)前面時(shí)間段觀測(cè)到的瞬時(shí)轉(zhuǎn)速與瞬時(shí)扭矩，設(shè)計(jì)出本時(shí)段驅(qū)動(dòng)電流的值，這個(gè)過(guò)程逐次進(jìn)行，直至完成制動(dòng)。要解決的問(wèn)題如下：設(shè)車(chē)輛單個(gè)前輪的滾動(dòng)半徑為0.286m，制動(dòng)時(shí)承受的載荷為6230N，求得等效的轉(zhuǎn)動(dòng)慣量。飛輪組由3個(gè)外直徑1m、內(nèi)直徑0.2m的環(huán)形鋼制飛輪組成，厚度分別為0.0392m、0.0784m、0.1568m，鋼材密度為7810kg/m3，基礎(chǔ)慣量為10kg·m2，問(wèn)可以組成哪些機(jī)械慣量？設(shè)電動(dòng)機(jī)能補(bǔ)償?shù)哪芰肯鄳?yīng)的慣量的范圍為[-30,30]kg·m2，對(duì)于問(wèn)題1中得到的等效的轉(zhuǎn)動(dòng)慣量，解釋需要用電動(dòng)機(jī)補(bǔ)償慣量的大小。3.建立電動(dòng)機(jī)驅(qū)動(dòng)電流依賴(lài)于可觀測(cè)量的數(shù)學(xué)模型。在問(wèn)題1和問(wèn)題2的條件下，假設(shè)制動(dòng)減速度為常數(shù)，初始速度為50km/h，制動(dòng)5.0秒后車(chē)速為零，計(jì)算驅(qū)動(dòng)電流。4.對(duì)于與所設(shè)計(jì)的路試等效的轉(zhuǎn)動(dòng)慣量為48kg·m2，機(jī)械慣量為35kg·m2，主軸初轉(zhuǎn)速為514轉(zhuǎn)/分鐘，末轉(zhuǎn)速為257轉(zhuǎn)/分鐘，時(shí)間步長(zhǎng)為10ms的情況，根據(jù)附表給出的數(shù)據(jù)，對(duì)該方法執(zhí)行的結(jié)果進(jìn)行評(píng)價(jià)。5.按照第3問(wèn)導(dǎo)出的數(shù)學(xué)模型，給出根據(jù)前一個(gè)時(shí)間段觀測(cè)到的瞬時(shí)轉(zhuǎn)速與/或瞬時(shí)扭矩，設(shè)計(jì)本時(shí)間段電流值的計(jì)算機(jī)控制方法，并對(duì)該方法進(jìn)行評(píng)價(jià)。6.評(píng)價(jià)第5問(wèn)給出的控制方法的不足之處。且重新設(shè)計(jì)一個(gè)盡量完善的計(jì)算機(jī)控制方法，并作評(píng)價(jià)。二、問(wèn)題的分析本文主要研究的是制動(dòng)器試驗(yàn)臺(tái)上的控制方法分析，通過(guò)把車(chē)輛的路試過(guò)程模擬到試驗(yàn)臺(tái)上，將制動(dòng)器的機(jī)械問(wèn)題定量化，從而更深入的研究制動(dòng)器的控制方法。在理想的路試中，以恒定的力踏下制動(dòng)踏板，汽車(chē)以恒定的加速度減速。但是在模擬實(shí)驗(yàn)中，由于飛輪組無(wú)法精確模擬實(shí)際狀況，必須由驅(qū)動(dòng)電流補(bǔ)償不足的轉(zhuǎn)動(dòng)慣量，驅(qū)動(dòng)電流產(chǎn)生驅(qū)動(dòng)扭矩，使得實(shí)驗(yàn)室中制動(dòng)扭矩做功模擬路試時(shí)的制動(dòng)扭矩做功，并且試驗(yàn)室中輪軸轉(zhuǎn)速模擬路試時(shí)的實(shí)際轉(zhuǎn)速。在模擬試驗(yàn)中，任意時(shí)間段內(nèi)制動(dòng)扭矩做功應(yīng)該等于路試的理想情況下制動(dòng)扭矩做的功，據(jù)此可以建立微分方程模型。第一問(wèn)：通過(guò)題目中所給出的等效轉(zhuǎn)動(dòng)慣量的概念，與物理學(xué)知識(shí)列出能量守恒定理的方程，，通過(guò)題目給出的已知條件，帶入數(shù)據(jù)分別計(jì)算出

，根據(jù)等式求解方程，便能得到等效轉(zhuǎn)動(dòng)慣量的值。第二問(wèn)：由理論力學(xué)的知識(shí)，我們可以知道圓盤(pán)轉(zhuǎn)動(dòng)慣量的定義式為：故只要求出圓環(huán)的質(zhì)量即可。圓環(huán)的質(zhì)量等于外徑所構(gòu)成大圓盤(pán)的質(zhì)量減去內(nèi)徑所構(gòu)成小圓盤(pán)的質(zhì)量。通過(guò)EXCEL計(jì)算，能求出飛輪慣量。我們發(fā)現(xiàn)機(jī)械慣量有8種情況，將其一一計(jì)算出來(lái)，通過(guò)第一問(wèn)所求出的等效轉(zhuǎn)動(dòng)慣量，選擇出合適的機(jī)械慣量，通過(guò)題目所給出的補(bǔ)償能量相應(yīng)的慣量的范圍為[-30,30]，可以計(jì)算出電動(dòng)機(jī)需補(bǔ)償多大的慣量。第三問(wèn)：通過(guò)題目中所給定的驅(qū)動(dòng)電流的算法。列出微分方程,由理論力學(xué)公式可以知道，，所以可以推出電流與可觀測(cè)量之間的關(guān)系為，即為建立的數(shù)學(xué)模型，將問(wèn)題1和問(wèn)題2的條件帶入，可計(jì)算出驅(qū)動(dòng)電流的大小。第四問(wèn)：根據(jù)題目中所給定的評(píng)價(jià)控制方法優(yōu)劣的一個(gè)重要數(shù)量指標(biāo)是能量誤差的大小，所以我們通過(guò)能量誤差指標(biāo)來(lái)檢驗(yàn)該種控制方法的優(yōu)劣。因此，需要分別求出路試時(shí)的制動(dòng)器和相對(duì)應(yīng)的實(shí)驗(yàn)臺(tái)上制動(dòng)器在制動(dòng)過(guò)程中消耗的能量，以求他們的能量差值和相對(duì)誤差。通過(guò)計(jì)算差值和相對(duì)誤差能確定其精確度。第五問(wèn)：由于題目要求在給出前一個(gè)時(shí)間段觀測(cè)到的瞬時(shí)轉(zhuǎn)速和瞬時(shí)扭矩，根據(jù)動(dòng)量矩定理和微分方程模型，能得到本時(shí)間段電流值。第六問(wèn)：根據(jù)轉(zhuǎn)速與時(shí)間的線性關(guān)系，將原始數(shù)據(jù)分為兩個(gè)階段進(jìn)行理想路試的能耗計(jì)算，在得到其誤差。合理假設(shè)假設(shè)汽車(chē)路試時(shí)地面的滾動(dòng)摩擦力很小，在我們討論的范圍內(nèi)可以忽略不計(jì)；2.假設(shè)車(chē)前輪以及模擬的前輪的飛輪和轉(zhuǎn)軸都是剛體；3.假設(shè)補(bǔ)償能量通過(guò)提高主軸轉(zhuǎn)速來(lái)補(bǔ)償；4.假設(shè)汽車(chē)在路試時(shí)受到恒定的制動(dòng)力，因而其在理想路試的狀態(tài)下作勻減速運(yùn)動(dòng)；5.忽略試驗(yàn)數(shù)據(jù)的獲得過(guò)程的主觀誤差和儀器誤差；6.假設(shè)主軸單位步長(zhǎng)內(nèi)角加速度速度為定值，步長(zhǎng)間角加速度可以越變。四、定義符號(hào)說(shuō)明m 模擬前輪的質(zhì)量G 載荷的重力g 重力加速度D 飛輪的外直徑d 飛輪的內(nèi)直徑k 比例系數(shù)r 模擬前輪的半徑v 模擬前輪的速度 鋼材密度 不同環(huán)形飛輪的厚度（j=1,2,3） 等效的轉(zhuǎn)動(dòng)慣量 機(jī)械轉(zhuǎn)動(dòng)慣量 扭矩 理想路試初始能量 理想路試結(jié)束能量 能量消耗之差 相對(duì)誤差 制動(dòng)車(chē)輪的角速度 每個(gè)步長(zhǎng)的角速度 制動(dòng)車(chē)輪的角加速度 車(chē)輪的加速度 驅(qū)動(dòng)電流 飛輪轉(zhuǎn)速五、模型的建立、求解1，對(duì)問(wèn)題一的解答：

汽車(chē)平動(dòng)所具有的動(dòng)能為；飛輪組系統(tǒng)轉(zhuǎn)動(dòng)時(shí)所具有的動(dòng)能為；兩者應(yīng)該相等，所以=QUOTE；由QUOTE得等效慣量為：代入數(shù)據(jù)，G=6230N, g=9.8,r=0.286m得 =51.99522，對(duì)問(wèn)題二的解答：由理論力學(xué)的知識(shí)，我們可以知道圓盤(pán)轉(zhuǎn)動(dòng)慣量的定義式為：故只要求出圓盤(pán)的質(zhì)量即可。題中的圓環(huán)可以看作由外徑構(gòu)成的大圓盤(pán)去掉由內(nèi)徑構(gòu)成的小圓盤(pán)。圓環(huán)的轉(zhuǎn)動(dòng)慣量等于外徑所構(gòu)成大圓盤(pán)的轉(zhuǎn)動(dòng)慣量減去內(nèi)徑所構(gòu)成小圓盤(pán)的轉(zhuǎn)動(dòng)慣量。用EXCLE求解得到下表：表1：飛輪組的轉(zhuǎn)動(dòng)慣量規(guī)格123質(zhì)量238.047476.093952.187轉(zhuǎn)動(dòng)慣量30.0460.09120.18為了計(jì)算的方便，轉(zhuǎn)動(dòng)慣量取整數(shù)。當(dāng)=0.0392m時(shí)，J=30當(dāng)=0.0784m時(shí)，J=60當(dāng)=0.1568m時(shí)，J=120考慮基礎(chǔ)慣量為10，所以可以組成的機(jī)械慣量有：10，40，70，100，130，160，190，220，共八種情況。方案一：組合出40的機(jī)械慣量，并用電動(dòng)機(jī)補(bǔ)償12的慣量。方案二：組合出70的機(jī)械慣量，并用電動(dòng)機(jī)補(bǔ)償-18的慣量。兩個(gè)方案電動(dòng)機(jī)補(bǔ)償都在[-30,30]之間主要誤差是由于電流補(bǔ)償?shù)哪遣糠郑杂呻妱?dòng)機(jī)補(bǔ)償?shù)膽T量的大小越小越好！對(duì)于問(wèn)題一中得到的轉(zhuǎn)動(dòng)慣量52，選擇40的機(jī)械慣量最為接近，也最為合理，用電動(dòng)機(jī)補(bǔ)償12的轉(zhuǎn)動(dòng)慣量。3，對(duì)問(wèn)題三的解答：通過(guò)題目中所給定的驅(qū)動(dòng)電流的算法。列出微分方程,由理論力學(xué)公式可以知道，，所以可以推出電流與可觀測(cè)量之間的關(guān)系為，即為建立的數(shù)學(xué)模型。其中，I表示電流；T表示扭矩；A表示加速度；表示角加速度。將問(wèn)題1和問(wèn)題2的條件帶入，可計(jì)算出驅(qū)動(dòng)電流的大小。得到驅(qū)動(dòng)電流4，對(duì)問(wèn)題四的解答：根據(jù)題目中所給定的評(píng)價(jià)控制方法優(yōu)劣的一個(gè)重要數(shù)量指標(biāo)是能量誤差的大小。我們通過(guò)能量誤差指標(biāo)來(lái)檢驗(yàn)該種控制方法的優(yōu)劣，因此，需要分別求出路試時(shí)的制動(dòng)器和相對(duì)應(yīng)的實(shí)驗(yàn)臺(tái)上制動(dòng)器在制動(dòng)過(guò)程中消耗的能量，以求他們的能量差值。以下兩圖分別實(shí)驗(yàn)是中扭矩和轉(zhuǎn)速與時(shí)間的折線圖。圖一：扭矩隨時(shí)間的變化圖從圖中可以看出，制動(dòng)扭矩在0~1秒內(nèi)呈現(xiàn)遞增特性，而在1秒之后在一個(gè)恒定值附近小范圍的波動(dòng)?？梢钥闯觯@是控制系統(tǒng)的調(diào)節(jié)作用。圖二：轉(zhuǎn)速隨時(shí)間的變化圖轉(zhuǎn)速曲線在剛開(kāi)始一段時(shí)間內(nèi)變動(dòng)并不規(guī)律，在1秒附近有一次突然地減小，而1秒之后則基本符合線性規(guī)律。實(shí)驗(yàn)臺(tái)上制動(dòng)器在制動(dòng)過(guò)程中消耗的能量由力學(xué)知識(shí)可知道處理后得到結(jié)果如下（部分，全部結(jié)果見(jiàn)附錄1）扭矩(N.m)轉(zhuǎn)速(rpm)時(shí)間(s)角速度(每秒)單位時(shí)間內(nèi)功（J）總功(J)40514.33053.860511650.53860511721.544204749291.942240513.790.0153.803962980.5380396321.521585240513.240.0253.746367120.53746367121.4985469..................288.75257.174.6726.930779420.26930779477.7626256我們可以看到表格中的E表示在實(shí)驗(yàn)臺(tái)上制動(dòng)器在制動(dòng)過(guò)程中消耗的能量大小為我們計(jì)算路試時(shí)的能量時(shí)，利用公式,初始時(shí)和末時(shí)的能量之差即為制動(dòng)器在制動(dòng)過(guò)程中消耗的能量。所以路試時(shí)消耗的總能量路試和試驗(yàn)臺(tái)上制動(dòng)器在制動(dòng)過(guò)程中消耗的能量之差,即為衡量該控制方法優(yōu)劣的標(biāo)準(zhǔn)。其相對(duì)誤差是：通過(guò)查閱資料了解到誤差在5%附近可以說(shuō)明模型是較為精確的。5，對(duì)問(wèn)題五的解答：假設(shè)t時(shí)刻驅(qū)動(dòng)扭矩為L(zhǎng)(t)，主軸的角速度為QUOTE，經(jīng)過(guò)時(shí)間增加d，時(shí)間內(nèi)制動(dòng)扭矩做功為就是在試驗(yàn)中制動(dòng)扭矩做的功。模擬等效系統(tǒng)的動(dòng)能增量為：就是在理想路試上制動(dòng)扭矩所做的功。為了模擬等效系統(tǒng)，兩者應(yīng)該相等：得制動(dòng)扭矩為QUOTE，再根據(jù)動(dòng)量矩定理，制動(dòng)力矩為：公式：QUOTE已知，于是微分方程模型為：根據(jù)微分方程(1)，得：QUOTE所以，得到控制電流為：QUOTE于是，可以根據(jù)某時(shí)刻測(cè)得的制動(dòng)扭矩QUOTE控制下一個(gè)時(shí)間段的驅(qū)動(dòng)電流。6，對(duì)問(wèn)題六的解答：從第四問(wèn)中可以看出，實(shí)際試驗(yàn)過(guò)程中常常會(huì)有不確定因素出現(xiàn)而導(dǎo)致實(shí)際轉(zhuǎn)速偏離理論值，而上題中控制方法沒(méi)有考慮隨機(jī)因素的影響，所以還是存在不足。下面討論對(duì)于這種方法的改進(jìn)。從第四問(wèn)中可以看出，實(shí)際試驗(yàn)過(guò)程中常常會(huì)有不確定因素出現(xiàn)而導(dǎo)致實(shí)際轉(zhuǎn)速偏離理論值，而上題中控制方法沒(méi)有考慮隨機(jī)因素的影響，所以還是存在不足。下面討論對(duì)于這種方法的改進(jìn)。假設(shè)某次試驗(yàn)的主軸初始角速度為QUOTE，根據(jù)（1），可得到角速度的理論值為：QUOTE但實(shí)際試驗(yàn)中由于不確定因素的影響，測(cè)得的QUOTE，于是要通過(guò)調(diào)整驅(qū)動(dòng)扭矩L(t)補(bǔ)償做功偏差，使得下一個(gè)時(shí)間段內(nèi)制動(dòng)扭矩做功的偏差等于驅(qū)動(dòng)扭矩補(bǔ)償?shù)哪芰浚航獾么思礊樾拚蟮尿?qū)動(dòng)扭矩控制表達(dá)式。在t時(shí)刻成立：QUOTE于是，建立轉(zhuǎn)速修正模型：QUOTEQUOTE（2）下面我們對(duì)問(wèn)題四給出的數(shù)據(jù)驗(yàn)證這種控制方法確實(shí)能夠減小能量誤差。由修正模型解得修正后的角速度為：對(duì)其離散化，得到：QUOTE而總之，是關(guān)于t的線性函數(shù)，將圖（2）分為兩段，得到兩個(gè)線段，其端點(diǎn)為（0,514.3），（0.95,475）；和（1,465.2），（4.67,257.17）。試驗(yàn)臺(tái)上消耗能量不變，為：理想路試消耗能量為：誤差相對(duì)問(wèn)題四中的相對(duì)能量誤差5.38%減小，只有2.31%。同時(shí)，此控制方法對(duì)于突變情況具有良好的反映能力，說(shuō)明此方法相對(duì)比較完善。六、結(jié)果評(píng)價(jià)1、從圖五可以看出，（1）式的常微分方程模型可以很好的模擬沒(méi)有不確定因素干擾的試驗(yàn)，但出現(xiàn)不確定因素干擾時(shí)，模擬偏差較大。2、問(wèn)題六中建立的修正模型(2)，通過(guò)時(shí)刻監(jiān)測(cè)主軸轉(zhuǎn)速，調(diào)整驅(qū)動(dòng)電流值修正由不確定因素導(dǎo)致的轉(zhuǎn)速偏差，能較好的模擬路試時(shí)制動(dòng)器的做功。參考文獻(xiàn)：[1]姜啟源，謝金星，葉俊數(shù)學(xué)模型（第三版）北京高等教育出版社2003；[2]薛毅，數(shù)學(xué)建模北京工業(yè)大學(xué)出版2004；附件扭矩(N.m)轉(zhuǎn)速(rpm)時(shí)間(s)角速度(每秒)單位時(shí)間內(nèi)功（J）總功(J)40514.33053.860511650.53860511721.5442046649291.9421540513.790.0153.803962980.5380396321.5215851940513.240.0253.746367120.53746367121.4985468541.25513.790.0353.803962980.5380396322.1941347343.75513.790.0453.803962980.5380396323.5392338145513.790.0553.803962980.5380396324.2117833447.5513.240.0653.746367120.53746367125.5295243850513.240.0753.746367120.53746367126.8731835653.75512.690.0853.688771250.53688771328.8577145555512.690.0953.688771250.53688771329.5288241957.5512.150.153.632222580.53632222630.8385279958.75512.150.1153.632222580.53632222631.5089307762.5512.150.1253.632222580.53632222633.5201391262.5512.690.1353.688771250.53688771333.5554820367.5512.150.1453.632222580.53632222636.2017502467.5512.150.1553.632222580.53632222636.2017502472.5511.60.1653.574626720.53574626738.8416043775511.60.1753.574626720.53574626740.1809700481.25511.060.1853.518078050.53518078143.4834384286.25511.60.1953.574626720.53574626746.2081155591.25511.60.253.57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