基于有限元比亞迪F3制動器的設(shè)計【鼓式制動器】【說明書+CAD+PROE】

基于有限元比亞迪F3制動器的設(shè)計【鼓式制動器】【說明書+CAD+PROE】,鼓式制動器,說明書+CAD+PROE,基于有限元比亞迪F3制動器的設(shè)計【鼓式制動器】【說明書+CAD+PROE】,基于,有限元,f3,制動器,設(shè)計,說明書,仿單,cad,proe SY-025-BY-1 畢業(yè)設(shè)計（論文）題目審定表 指導(dǎo)教師姓名 王永梅 職稱 講師 從事 專業(yè) 車輛工程 是否外聘 □是■否 題目名稱 基于有限元比亞迪F3制動器的設(shè)計 課題適用專業(yè) 車輛工程 課題類型 Z 課題簡介：（主要內(nèi)容、意義、現(xiàn)有條件、預(yù)期成果及表現(xiàn)形式。） 一、主要內(nèi)容：確定鼓式制動器的基本參數(shù)，對制動器的制動鼓、蹄片和支撐的幾何尺寸進行及強度校核，利用Pro/E軟件建立制動器三維模型裝配圖，利用Ansys軟件對摩擦襯片有限元分析。 二、意義：汽車制動性能是確保車輛行駛的主、被動安全性和提升車輛行駛動力性決定因素之一。鼓式制動器是應(yīng)用非常廣泛的一種制動器，尤其優(yōu)良的制動效果及簡單的結(jié)構(gòu)形式。應(yīng)用Pro/E軟件建立鼓式制動器主要零件的實體模型，并完成虛擬裝配，利用Ansys軟件對制動器摩擦襯片等主要零件進行有限元分析，為鼓式制動器的設(shè)計與研究提供了一種方法，可縮短制動器的研發(fā)周期，降低產(chǎn)品設(shè)計成本，并為以后進一步優(yōu)化設(shè)計、制造及運動分析奠定了基礎(chǔ)。 三、現(xiàn)有條件：大量有關(guān)鼓式制動器的報刊和書籍，汽車底盤實驗室。 四、預(yù)期成果：設(shè)計一款結(jié)構(gòu)優(yōu)化的鼓式制動器。 五、表現(xiàn)形式：鼓式制動器CAD裝配圖及零件圖一套，鼓式制動器三維裝配圖、分解圖和零件圖，1.5萬字說明書一份。 指導(dǎo)教師簽字： 年 月 日 教 研 室 意 見 1 選題與專業(yè)培養(yǎng)目標的符合度 □好 □較好 □一般 □較差 2 對學(xué)生能力培養(yǎng)及全面訓(xùn)練的程度 □好 □較好 □一般 □較差 3 選題與生產(chǎn)、科研、實驗室建設(shè)等實際的結(jié)合程度 □好 □較好 □一般 □較差 4 論文選題的理論意義或?qū)嶋H價值 □好 □較好 □一般 □較差 5 課題預(yù)計工作量 □較大 □適中 □較小 6 課題預(yù)計難易程度 □較難 □一般 □較易 教研室主任簽字： 年 月 日 系（部）教學(xué)指導(dǎo)委員會意見： 負責(zé)人簽字： 年 月 日 注：課題類型填寫 W.科研項目；X.生產(chǎn)（社會）實際；Y.實驗室建設(shè)；Z.其它。 中文翻譯 應(yīng)用熱工 三維制動器瞬態(tài)溫度場的緊急制動 為了準確掌握在葫蘆的緊急制動蹄片的溫度場的變化規(guī)律，制動時，三維（3- D）的瞬態(tài)溫度場的理論模型，根據(jù)熱傳導(dǎo)，能量轉(zhuǎn)換和分布規(guī)律的理論，以及礦山提升機運行的緊急情況制動。一種溫度場的解析推導(dǎo)了采用積分變換法。此外，溫度模擬實驗場進行了溫度場和溫度梯度和內(nèi)部的變化規(guī)律獲得。同時，通過模擬葫蘆的緊急制動條件下，實驗測量制動蹄的溫度，同時進行。結(jié)果發(fā)現(xiàn)，通過比較模擬結(jié)果與實驗數(shù)據(jù)，即三維瞬態(tài)溫度場模型的制動蹄片是有效和實用，和分析解決方案解決了積分變換方法是正確的。 1、 簡介 提升機的緊急制動是一個轉(zhuǎn)變過程機械能轉(zhuǎn)化為對制動摩擦熱能量。該礦山提升機緊急制動過程中具有以下特點高速，重載，而這種情況更糟糕的是比剎車條件的車輛，火車等[1-3,6,10,11]。以前對剎車片的溫度場的重點工作[1-4,10,12,13]。特別是，由于制動蹄是固定的過程中緊急制動，所以有更強烈的溫度上升制動器蹄片。制動蹄片是一種復(fù)合材料，以及溫度上升，從摩擦產(chǎn)生的熱能是最重要的因素影響制動器蹄片摩擦磨損性能同制動安全性能[5-10]。因此，有必要調(diào)查關(guān)于制動器蹄片的溫度場來調(diào)查剎車片的。 制動器蹄片的溫度場目前的理論模型基于一維或二。 Afferrante[11]建立了一個二維（2- D）的多層模型來估計瞬態(tài)演化在多盤離合器溫度擾動和在操作過程中剎車。納吉[12]建立了一維數(shù)學(xué)模型來描述一個制動熱行為系統(tǒng)。 Yevtushenko和Ivanyk[13]推導(dǎo)了瞬態(tài)溫度場的一軸對稱熱傳導(dǎo)問題2三維坐標。這是困難的這些模式，以反映制動器蹄片真實溫度場的三維幾何圖形。 解決的方法剎車片的三維瞬態(tài)溫度場集中有限元法[1-3,14-17]，近似集成的方法[4,18]，格林函數(shù)法[12]和Laplace變換方法[9,13]等，前三者方法是數(shù)值求解方法和低是相對的準確性。例如，有限元方法可以解決復(fù)雜熱傳導(dǎo)問題，但計算精度解決方案是比較低，這是影響網(wǎng)密度，步長等。雖然拉普拉斯變換解決方法是分析方法，它是難以解決的方程復(fù)雜邊界的熱傳導(dǎo)。因此，所謂的解析解積分變換方法通過[19]，因為它是解決問題的合適非均質(zhì)瞬態(tài)熱傳導(dǎo)。為了掌握制動器蹄片的溫度變化規(guī)律在葫蘆的緊急制動領(lǐng)域，提高安全可靠性制動，一個3- D的制動器蹄片瞬態(tài)溫度場研究了在積分變換方法的基礎(chǔ)上，和有效性證明了數(shù)值模擬和實驗研究。 2、 理論分析 2、1理論模式 圖1顯示了葫蘆的制動摩擦副示意圖。為了分析制動器蹄片的三維溫度場，圓柱坐標（r，，z）是通過結(jié)構(gòu)來描述幾何如圖所示。 2，其中R是剎車點之間的距離和制動盤的旋轉(zhuǎn)軸; 為圓心角;這三者之間的制動蹄摩擦點和表面的距離。至于幾何結(jié)構(gòu)參數(shù)和圖2所示。它看到，顯然，這是制動器蹄片的溫度T是函數(shù)的圓柱坐標（r，，z）和時間（t）。根據(jù)熱理論傳導(dǎo)，三維瞬態(tài)熱傳導(dǎo)微分方程是獲得如下： （1） 其中a是熱擴散，；是熱導(dǎo)率；為密度；是比熱容量。 2.2、邊界條件 2.2.1、熱流量及其分布系數(shù) 這是在緊急制動產(chǎn)生的摩擦熱難要在短時間內(nèi)發(fā)出，因此它幾乎完全吸收剎車對。由于制動器蹄片是固定的，摩擦溫度多面大幅上升，這最終會影響其摩擦學(xué)更嚴重的行為。為了掌握真實該制動器蹄片溫度場在緊急制動時，熱流量及其分布系數(shù)摩擦表面必須確定準確。根據(jù)操作緊急制動，條件假設(shè)制動速度光盤隨時間呈線性，熱流量，得到公式 （2） 其中q為熱摩擦表面流動; P是比壓之間的制動對; 的和是最初的線性和角速度在制動盤; l是剎車副之間的摩擦系數(shù); 是整個制動時間，k是熱分布流系數(shù)。假設(shè)摩擦熱量轉(zhuǎn)移到完全制動運動鞋和制動盤，分布的熱流量系數(shù)根據(jù)得到的一維熱傳導(dǎo)分析。圖。 3顯示了聯(lián)系兩個半平面示意圖。在一維瞬態(tài)熱傳導(dǎo)的條件，對摩擦表面（z = 0處）的溫度上升，得到公式 （3） 其中q為在平面吸收一半熱流。和熱流量是從Eq獲得的 （4） 假設(shè)兩個半飛機具有相同的溫度上升，對摩擦表面，然后在熱流量比進入兩個半平面可表示為 其中下標S和D意味著制動器蹄片和制動盤，分別。根據(jù)Eq。（5），分配系數(shù)熱流根據(jù)這個公式獲得進入制動器蹄片。 2.2.2、在邊界系數(shù)對流換熱 至于側(cè)面和頂面制動器蹄片，得到他們的對流換熱系數(shù)，分別按自然對流換熱邊界條件直立板和橫板 圖1-制動摩擦副示意圖 圖2、三維幾何模型的制動器蹄片。 圖3、兩個半平面示意圖 其中下標L和U代表側(cè)面和頂部表面，h分別為對流換熱系數(shù)在邊界上，DT是之間的溫差邊界和環(huán)境，L是較短維邊界。 2.2.3、初始和邊界條件 制動器蹄片之間的接觸和制動盤表面受到不斷熱流在緊急制動過程qs的。 制動蹄片的邊界都用空氣的自然對流。邊界和初始條件可以表示為 其中是制動器蹄片在t=0的初始溫度。 2.3。積分變換求解方法 積分變換的方法有兩個解決問題的步驟。首先，只有作出適當(dāng)?shù)姆e分變換空間 變量，熱傳導(dǎo)原方程可以簡化由于考慮到時間與常微分方程變量t然后，通過采取逆變換關(guān)于解常微分方程的解析解在關(guān)于空間和時間變量溫度場可以得到的。積分變換方法應(yīng)用于求解方程。 （1）邊界條件方程。（8）。用積分變換有關(guān)空間變量（r，，z）的反過來，他們可能會偏微分方程是''消滅“。編寫公式來表示的運作采取逆變換與積分變換方面到Z，這些被定義為 其中是的積分變化，是特征函數(shù)。 提交Eq，獲得以下方程： 以同樣的方式，逆變換與積分變換關(guān)于和r分別定義 最后，根據(jù)上面的積分變換，方程1）（8）可以簡化為如下： 解決方案可以獲得通過解式。(16)。以反變換關(guān)于根據(jù)Eqs。(九)、(12)和(14),的解析制動器的三維瞬態(tài)溫度場分布 3.仿真和實驗 圖4顯示了一半的制動器剖面樣品。線c、d的中心線,底線的橫截面上的分別。樣品的尺寸是:一個= 137.5 mm,b = = 1 / 6毫米,半162.5 rad,l = 6毫米。閘瓦的材料和盤式制動器是石棉和16Mn,分別。他們的參數(shù)和條件的緊急制動見表1。 假設(shè)摩擦系數(shù)和制動襯墊比壓在緊急制動過程是不變的?；谝陨戏治瞿Ｐ?模擬閘瓦的三維溫度場進行與到…= 7.23 s。溫度的變化規(guī)律 圖4 把剖面的一半剎車蹄的樣品 表1剎車副的基本參數(shù)和緊急制動條件 與內(nèi)部溫度梯度場進行了分析。什么是顯示在無花果里都是片面的。5 - 9的仿真結(jié)果相符合。 什么是顯示在圖5是閘瓦的三維溫度場當(dāng)時間7.23 s。它被認為是從圖5的最高溫度是396.534閘瓦制動,其K后最低溫度和熱是能量293歐幾里得主要集中 圖5 三維溫度場的剎車蹄(t = 7.23 s) 圖6 溫度的改變對摩擦表面與時間t 圖7 溫度的改變對線d用時間t 圖8 溫度梯度的變化與時間線c t 圖9 溫度的改變不同深度隨時間的線c t 層上的摩擦表面的熱影響層(命名),既體現(xiàn)了熱diffusibility閘瓦的很差。為了靈便的溫度變化規(guī)律的摩擦表面,在緊急制動過程的摩擦表面的變化的溫度與時間t進行了模擬。什么是在圖6中顯示,揭示了摩擦表面的溫度,然后增加首先減小的趨勢。這是因為,高速度的盤式制動器是在開始的時候,結(jié)果造成大heat-flow而對流換熱系數(shù)低邊界上的那一刻,所以溫度增加;后期的制動的heatflow量減少的速度,而對流換熱系數(shù)高,由于溫差較大的差異,從而導(dǎo)致減少邊界溫度。無花果。6、7,反映了溫度變化規(guī)律進行了徑向尺寸:在外面的溫度高于閘瓦里面,并且外面的溫度變化較大。 圖8論證了溫度梯度的變化規(guī)律的方向沿z。最高溫度梯度的摩擦層是由3.739 105 K / m與方向會急驟下降沿z。最低價值只是4.597 1011 K / m。在開始的時候,溫度梯度的熱影響層是最高,而溫度接近周圍的溫度。象剎車的推移,溫度梯度漸次降低,直到最后。圖9所示的是變化的溫度不同深度隨時間的線c t。溫度會急驟下降隨著z,、邊界條件等影響有窩內(nèi)部溫度。溫度增高但z P0.0006米。一旦z是由0.002米,制動過程中溫度的差別小于3 k .這表明,熱能集中在熱影響層,其厚度是關(guān)于0.002米。 為了證明的解析模型,實驗進行了摩擦試驗機,如圖10。實驗原理如下:當(dāng)剎車開始,兩種制動蹄制動圓盤也要被推遲到一定壓力p和溫度點e在摩擦表面熱電偶測量。因為試樣厚度太厚,而且摩擦試驗機的結(jié)構(gòu)是有限的,很難固定熱電偶在剎車蹄。因此,熱電偶是固定的直接對盤式制動器是封閉,點e列圖。10。圖11顯示的溫度變化規(guī)律的兩種情況下點在e的緊急制動。 從圖11,觀察點e增加時的溫度,在第一,然后減少,最高溫度低于,通過仿真實驗數(shù)據(jù)也落后。在圖11a,模擬溫度達到最大427.14凱西在3.6 s而來的實驗數(shù)據(jù)和最大435.65凱西在3.8秒。在圖11b,仿真結(jié)果達到最大469.55凱西在4.5 s而來到479.68實驗數(shù)據(jù)K在5秒。它被認為是從圖11,通過實驗測量溫度低于仿真結(jié)果,在第一,然后它相反的。這是因為熱電偶本身的能量吸收熱量閘瓦在開始,然后將其釋放到剎車蹄當(dāng)溫度下降。對比實驗數(shù)據(jù)和仿真結(jié)果表明,仿真結(jié)果表明,兩者吻合較好,誤差的實驗,他們的最高溫度是1.99% 圖10 圖解的摩擦測試儀。 圖11a 溫度的變化規(guī)律與時刻t的e點(p = 1.38 = 0 - 1兆帕,證明米/秒)。 圖11b 溫度的變化規(guī)律與時刻t的e點(p = 1.5895%兆帕,證明=長1 - 2.5米/秒)。 和2.16%,分別。這表明,解析解的三維瞬態(tài)溫度場是正確的。 4.結(jié)論 (1)的理論模型建立了三維瞬態(tài)溫度場的理論根據(jù)熱傳導(dǎo)及緊急制動條件的礦山提升機。這個積分變換方法應(yīng)用于解決的理論模型,并對溫度場的解析解,推導(dǎo)出。這表明,積分變換方法是有效解決這一問題的三維瞬態(tài)溫度場。 (2)基于解析解的理論模型,并采用數(shù)值分析模擬溫度分布的變化規(guī)律下緊急制動狀態(tài)。仿真結(jié)果表明:摩擦表面溫度的增加降低;首先,然后在開始的溫度梯度的熱影響層的最高,其次是溫度增加迅速,正如制動過程正在進行中,溫度梯度溫度的增加呈減少趨勢;窩;邊界條件影響了內(nèi)部溫度上升;熱能量都集中在熱影響層,其厚度約2毫米。 (3) 實驗數(shù)據(jù)與仿真結(jié)果吻合良好,誤差對他們的最高溫度是大約2%,這證明了積分變換方法的正確性求解理論模型的三維瞬態(tài)溫度場。解析模型能夠反映出的變化規(guī)律閘瓦的三維瞬態(tài)溫度場在緊急剎車。 出處 本項目是支持的重點工程,中國教育部(批準號:)資助107054)和程序為新世紀優(yōu)秀人才(批準號:)資助的大學(xué)。NCET-04-0488)。 參考 [1] y .楊、江康鈺周、數(shù)值模擬研究的三維熱應(yīng)力場與復(fù)雜邊界問題,《工程熱27(3)(2007)487-489。 [2] l . j .的歌,Z.Y.李郭的研究;(3)快速有限元的仿真模型,對車輛制動熱分析系統(tǒng)仿真學(xué)報,17(12)(2005)。2877 2869-2872。 [3] 邱智賢高,X.Z.林、瞬態(tài)溫度場分析剎車在引入非完全軸對稱三維模型、期刊的材料加工技術(shù)129(1 - 3)(2002)513-517。 [4] 應(yīng)用文獻的理發(fā)師,李,并沒有變法防守的瞬態(tài)熱彈性接觸問題解的速度膨脹法,穿265(3勝4敗)(2008)402-410。 [5] 張亞蘭,Z.C. 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Thuresson、穩(wěn)定的滑動contact-comparison銷釘,并且建立有限元模型,穿261(7 - 8)(2006)896-904。 [17] H.S.氣、陳護升的一天,調(diào)查的閥瓣/襯墊摩擦界面溫度,穿262制動(5 - 6)(2007)505-513。 [18] D.P.劉、懸,近似計算方法,梅摩擦摩擦溫度在絡(luò)筒機上襯砌雜志》、《中國礦業(yè)大學(xué)及技術(shù)26(1)(1997)70 - 72。(在中國)。 [19] 《孫子兵法》Z.Y.摩擦學(xué)行為研究對提升機的制動蹄機盤式制動器(論文)、中國礦業(yè)大學(xué),2007年,pp.103-109技術(shù)(中文)。(4)。 14 field Chen Hoist the heat the temperature C211 2008 Elsevier Ltd. All rights reserved. is a process energy hoist situation on 13,6,10,11 is fixed tion of temperature perturbations in multi-disk clutches and brakes during operation. Naji 12 established one-dimensional mathematical model to describe the thermal behavior of a brake system. Yevtushenko and Ivanyk 13 deduced the transient tem- perature field for an axi-symmetrical heat conductivity problem with 2-D coordinates. It is difficult for these models to reflect the real temperature field of brake shoe with 3-D geometry. 2. Theoretical analysis 2.1. Theoretical model Fig. 1 shows the schematic of hoists braking friction pair. In or- der to analyze brake shoes 3-D temperature field, the cylindrical coordinates (r,u,z) is adopted to describe the geometric structure shown in Fig. 2, where r is the distance between a point of brake shoe and the rotation axis of brake disc; u is the central angle; z * Corresponding author. Tel.: +86 13805209649; fax: +86 516 83590708. Applied Thermal Engineering 29 (2009) 932937 Contents lists available E-mail address: (Y.-x. Peng). emergency braking, so there is more intense temperature rise in brake shoe. The brake shoe is kind of composite material, and the temperature rise resulting from frictional heat energy is the most important factor affecting tribological behavior of brake shoe and the braking safety performance 510. Therefore, it is necessary to investigate the brake shoes temperature field with respect to investigating brake pads. Current theoretical models of brake shoes temperature field are based on one dimension or two. Afferrante 11 built a two-dimen- sional (2-D) multilayered model to estimate the transient evolu- method is an analytic solution method, it is difficult to solve the equation of heat conduction with complicated boundaries. There- fore, the analytic solution called integral-transform method is adopted 19, because it is suitable for solving the problem of non-homogeneous transient heat conduction. In order to master the change rules of brake shoes temperature fieldduringhoistsemergencybrakingandimprovethesafereliabil- ity of braking, a 3-D transient temperature field of the brake shoe was studied based on integral-transform method, and the validity is proved by numerical simulation and experimental research. 1. Introduction The hoists emergency braking mechanical energy into frictional heat emergency braking process of mining of high speed and heavy load, and this ing condition of vehicle, train and so work focused on the brake pads temperature Especially, because the brake shoe 1359-4311/$ - see front matter C211 2008 Elsevier Ltd. All doi:10.1016/j.applthermaleng.2008.04.022 of transforming of brake pair. The has the characteristic is worse than brak- . The previous field 14,10,12,13. during the process of The methods solving brake pads 3-D transient temperature field concentrated on finite element method 13,1417, approx- imate integration method 4,18, Greens function method 12 and Laplace transformation method 9,13, etc. The former three methods are numerical solution methods and are of low relative accuracy. For example, finite element method can solve the com- plicate heat conduction problem, but the accuracy of computa- tional solution is relatively low, which is affected by mesh density, step length and so on. Though the Laplace transformation Integral-transform method Emergency braking with experimental data, that the 3-D transient temperature field model of brake shoe is valid and prac- tical, and analytic solution solved by integral-transform method is correct. Three-dimensional transient temperature emergency braking Zhen-cai Zhu, Yu-xing Peng * , Zhi-yuan Shi, Guo-an College of Mechanical and Electrical Engineering, China University of Mining and Technology, article info Article history: Received 22 November 2007 Accepted 27 April 2008 Available online 6 May 2008 Keywords: Brake shoe Three-dimensional Transient temperature field abstract In order to exactly master braking, the theoretical model according to the theory of operating condition of mining deduced by adopting integral-transform field were carried out and ent were obtained. At the same for measuring brake shoes Applied Thermal journal homepage: www.elsevi rights reserved. of brake shoe during hoists Xuzhou 221116, China change rules of brake shoes temperature field during hoists emergency of three-dimensional (3-D) transient temperature field was established conduction, the law of energy transformation and distribution, and the hoists emergency braking. An analytic solution of temperature field was method. Furthermore, simulation experiments of temperature variation regularities of temperature field and internal temperature gradi- time, by simulating hoists emergency braking condition, the experiments were also conducted. It is found, by comparing simulation results at ScienceDirect Engineering is the distance between a point of brake shoe and the friction sur- face. As for the geometric structure and parameters shown in Fig. 2, its seen that a6r6 b,06u6u 0 ,06z6l. It is clear that the brake shoes temperature T is the function of the cylindrical coor- dinates (r,u,z) and the time (t). According to the theory of heat conduction, the differential equation of 3-D transient heat conduc- tion is gained as follows: o 2 T or 2 1 r oT or 1 r 2 o 2 T ou 2 o 2 T oz 2 1 a oT ot ; 1 wherea is the thermal diffusivity,a = k /(qC1 c); k is the thermal con- ductivity; q is the density; c is the specific heat capacity. 2.2. Boundary condition 2.2.1. Heat-flow and its distribution coefficient It is difficult for friction heat generated during emergency brak- ing to emanate in a short time, so it is almost totally absorbed by brake pair. As the brake shoe is fixed, the temperature of the fric- tion surface rises much sharply, and this eventually affects its tri- bological behavior more seriously. In order to master the real temperature field of the brake shoe during emergency braking, the heat-flow and its distribution coefficient of friction surface must be determined with accuracy. According to the operating condition of emergency braking, suppose that the velocity of brake disc decreased linearly with time, the heat-flow is obtained with the form q s r;tk C1lC1 pC1 v 0 C11C0 t=t 0 k C1lC1 p C1 w 0 C1 r:1C0 t=t 0 ; 2 where q is the heat-flow of friction surface; p is the specific pressure betweenbrakepair;v 0 andw 0 istheinitiallinearandangularvelocity ofthebrakedisc;listhefrictioncoefficientbetweenbrakepair;t 0 is the whole braking time, k is the distribution coefficient of heat-flow. Suppose the frictional heat is totally transferred to the brake shoe and brake disk, and the distribution coefficient of heat-flow is obtained according to the analysis of one-dimensional heat con- duction. Fig. 3 shows the contact schematic of two half-planes. Under the condition of one-dimensional transient heat conduc- tion, the temperature rise of friction surface (z = 0) is obtained with the form DT q k p p 4at p q pqck p 4t p ; 3 where q is the heat-flow absorbed by half-plane. And the heat-flow is gained from Eq. (3) p p respectively. According to Eq. (5), the distribution coefficient of Z.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937 933 Fig. 1. Schematic of hoists braking friction pair. Fig. 2. 3-D geometrical model of brake shoe. heat-flow entering brake shoe is obtained with the form k q s q a q s q s q d 1C0 q d q s q d 1 C0 1 q s q d 1 1 C0 1 1 q s csks q d c d k d C16C171 2 : 6 2.2.2. Coefficient of convective heat transfer on the boundary With regard to the lateral surface and the top surface of the brake shoe, their coefficients of convective heat transfer are ob- tained, respectively, according to the natural heat convection boundary condition of upright plate and horizontal plate h l 1:42DT l =L l 1 4 ; 7a h u 0:59DT u =L u 1 4 ; 7b q pqckDT= 4t: 4 Suppose the two half-planes has the same temperature rise on the friction surface, and then the ratio of heat-flow entering the two half-planes is given as q s q d pq s c s k s p DT= 4t p pq d c d k d p DT= 4t p q s c s k s p q d c d k d p ; 5 where the subscript s and d mean the brake shoe and brake disc, Fig. 3. Contact schematic of two half-planes. Engineering where the subscript l and u represent the lateral surface and the top surface, respectively; h is the coefficient of convective heat transfer on the boundary, DT is the temperature difference between the boundary and the ambient, L is the shorter dimension of the boundary. 2.2.3. Initial and boundary condition Contact surface between brake shoe and brake disc is subjected to continuous heat-flow q s during emergency braking process. Brake shoes boundaries are of natural convection with the air. The boundary and initial condition can be represented by C0k oT or h 1 T h 1 T 0 f 1 t; r a; t P0; 0 6u6u 0 ; 0 6 z 6 l; 8a k oT or h 2 T h 2 T 0 f 2 t; r b; t P0; 0 6u6u 0 ; 0 6 z 6 l; 8b C0k oT oz h 3 T q s h 3 T 0 f 3 t; z 0; t P0; 0 6u6u 0 ; a 6 r 6 b; 8c k oT oz h 4 T h 4 T 0 f 4 t; z l; t P0; 0 6u6u 0 ; a 6 r 6 b; 8d C0k 1 r oT ou h 5 T h 5 T 0 f 5 t; u 0; t P0; 0 6 z 6 l; a 6 r 6 b; 8e k 1 r oT ou h 6 T h 6 T 0 f 6 t; u u 0 ; t P0; 0 6 z 6 l; a 6 r 6 b; 8f Tr;u;z;tT 0 ; t 0; a 6 r 6 b; 0 6u6u 0 ; 0 6 z 6 l; 8g where T 0 is the initial temperature of the brake shoe at t =0. 2.3. Integral-transform solving method Integral-transform method has two steps for solving the prob- lem. Firstly, only by making suitable integral-transform for space variable, the original equation of heat conduction could be simpli- fied as the ordinary differential equation with regard to the time variable t. Then, by taking inverse transform with regard to the solution of the ordinary differential equation, the analytic solution of the temperature field with regard to the space and time vari- ables could be obtained. Integral-transform method is applied to solve Eq. (1) with boundary condition Eq. (8). By integral-transform with regard to the space variables (z,u,r) in turn, their partial differential could be eliminated”. Writing formulas to represent the operation of taking the inverse transform and the integral-transform with re- gard to z, these are defined by Tr;u;z;t X 1 m1 Zb m ;z Nb m Tr;u;b m ;t; 9 Tr;u;b m ;t Z l 0 Zb m ;z 0 C1Tr;u;z 0 ;tdz 0 ; 10 934 Z.-c. Zhu et al./Applied Thermal where Tr;u;b m ;t is the integral-transform of T(r,u,z,t) with regard to z; Z(b m ,z) is the characteristic function, Z(b m ,z)= cosb m (l C0 z); b m is the characteristic value, b m tanb m l = H 3 , and H 3 h 3 k ; N(b m ) is the norm, 1 Nb m 2 b 2 m H 2 3 lb 2 m H 2 3 H 3 . Submit Eq. (10) into Eqs. (1) and (8), the following equations is obtained: o 2 T or 2 1 r oT or 1 r 2 o 2 T ou 2 f 3 k cosl C1 b m C0b 2 m C1 Tr;u;b m ;t 1 a oTr;u;b m ;t ot ; 11a C0k oT or h 1 T C22 f 1 t; r a; t P0; 0 6u6u 0 ; 11b k oT or h 2 T C22 f 2 t; r b; t P0; 0 6u6u 0 ; 11c C0k 1 r oT ou h 5 T C22 f 5 t; u 0; t P0; a 6 r 6 b; 11d k 1 r oT ou h 6 T C22 f 6 t; u u 0 ; t P0; a 6 r 6 b; 11e Tr;u;b m ;t Z l 0 Zb m ;z 0 C1T 0 dz 0 ; t 0; a 6 r 6 b; 0 6u6u 0 : 11f In the same way, the inverse transform and the integral-transform with regard to u and r are defined by Tr;u;b m ;t X 1 n1 Uv n ;u Nv n e Tr;v n ;b m ;t; 12 e Tr;v n ;b m ;t Z u 0 0 u 0 C1Uv n ;u 0 C1Tr;u 0 ;b m ;tdu 0 ; 13 where e Tr;v n ;b m ;t is the integral-transform of Tr;u;b m ;t with re- gard to u; U(v n ,u) is the characteristic function, U(v n ,u)=v n C1 cosv n u + H 5 C1 sinv n u; v n is the characteristic value, tanv n u 0 vnH 5 H 6 v 2 n C0H 5 H 6 H 5 h 5 k ;H 6 h 6 k ; N(v n ) is the norm, 1 Nvn 2 v 2 n H 2 5 C1 u 0 H 6 v 2 n H 2 6 C16C17 H 5 hi C01 . e Tr;v n ;b m ;t X 1 i1 R v c i ;r Nc i e T v c i ;v n ;b m ;t; 14 e T v c i ;v n ;b m ;t Z b a R v c i ;r 0 C1 e Tr 0 ;v n ;b m ;tdr 0 ; 15 where e T v c i ;v n ;b m ;t is the integral-transform of e Tr;v n ;b m ;t with regard to r; R v (c i ,r) is the characteristic function, R v (c i ,r)=S v C1 J v (c i C1 r) C0 V v C1 Y v (c i C1 r), J v (c i C1 r) and Y v (c i C1 r) are the Bessel functions of the first and second kind with order v, where S v c i C1Y 0 v c i C1bH 2 C1Y v c i C1b; U v c i C1J 0 v c i C1aC0H 1 C1J v c i C1a; V v c i C1J 0 v c i C1bH 2 C1J v c i C1b; W v c i C1Y 0 v c i C1aC0H 1 C1Y v c i C1a; c i is the characteristic value which satisfies the equation U v C1 S v C0 W v C1 V v =0; N(c i ) is the norm, 1 Nc i p 2 2 c 2 i U 2 v B 2 C1U 2 v C0B 1 C1V 2 v , where B 1 H 2 1 c 2 i 1 C0v=c i a 2 C138 and B 2 H 2 2 c 2 i 1 C0v=c i b 2 C138. Finally, according to the above integral-transform, Eqs. (1) and (8) can be simplified as follows: d e T v dt ab 2 m c 2 i e T v Ac i ;v n ;b m ;t; t 0; 16a v v 29 (2009) 932937 e Tc i ;v n ;b m ;t e T 0 ; t 0; 16b where A(c i ,v n ,b m ,t)=g 1 + g 2 + g 3 , g 1 aC1 b C1 R v c i ;b k C1 e C22 f 2 a C1 R v c i ;a k C1 e C22 f 1 C18C19 ; g 2 Z b a v k C1 C22 f 5 C1 r 2 C1 R v c i ;rdr Z b a v C1cosv n u 0 H 5 C1sinv n u 0 k C1 C22 f 6 C1 r 2 C1 R v c i ;rdr; g 3 Z b a f 3 k C1 cosl C1b m C1 sinv n b m H 5 v 1C0 cosv n b m C20C21 C1 r C1 R v c i ;rdr: The solution e T v c i ;v n ;b m ;t can be gained by solving the Eq. (16).By taking the inverse transform with regard to e T v c i ;v n ;b m ;t according to Eqs. (9), (12) and (14), the analytic solution of brake shoes 3-D transient temperature field is obtained Tr;u;z;t X 1 m1 X 1 n1 X 1 i1 Zb m ;z Nb m Uv n ;u Nv n R v c i ;r Nc i e C0ab 2 m c 2 i t C1 e T v 0 Z t 0 e C0ab 2 m t 0 Ac i ;v n ;b m ;tdt 0 2 4 3 5 : 17 field is carried out with t 0 = 7.23 s. The change rules of temperature field and internal temperature gradient are analyzed. Whats shown in Figs. 59 are partial simulation results. What is shown in Fig. 5 is brake shoes 3-D temperature field when time is 7.23 s. It is seen from Fig. 5 that the highest temper- ature of the brake shoe is 396.534 K after braking, and its lowest temperature is 293 K. And the heat energy is mainly concentrated Fig. 5. 3-D temperature field of brake shoe (t = 7.23 s). Fig. 6. The change of temperature on friction surface with time t. Z.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937 935 Fig. 4. Half section view of brake shoes sample. Table 1 Basic parameters of brake pair and the emergency braking condition q (kg m C03 ) c (J kg C01 K C01 ) k (W m C01 K C01 ) T 0 (K) v 0 (m s C01 ) p (MPa) l 3. Simulation and experiment Fig. 4 shows the half section view of brake shoe sample. Line c and d are the center line and bottom line of the cross section, respectively. The sample dimension is: a = 137.5 mm, b = 162.5 mm, u 0 = 1/6 rad, l = 6 mm. The material of brake shoe and brake disc are asbestos-free and 16Mn, respectively. Their parameters and the condition of emergency braking are shown in Table 1. Suppose that the friction coefficient and the specific pressure are constant during emergency braking process. Based on the above analytic model, simulation of brake shoes 3-D temperature Brake shoe 2206 2530 0.295 293 10 1.38 0.4 Brake disc 7866 473 53.2 12.5 1.58 Fig. 7. The change of temperature on line d with time t. creases all the time when zP0.0006 m. Once the z is up to 0.002 m, the difference in temperature during brake is less than 3 K. It indicates that the heat energy focuses on the thermal effect layer, and its thickness is about 0.002 m. In order to prove the analytic model, experiments were carried out on the friction tester in Fig. 10. The experimental principle is as follows: when the brake begins, two brake shoes are pushed to brake the disc with certain pressure p and the temperature of point e on the friction surface is measured by thermocouple. Because the specimen thickness is too thin and the structure of the friction tes- ter is limited, it is difficult to fix the thermocouple in the brake shoe. Therefore, the thermocouple is fixed directly on the brake disc which is closed to point e shown in Fig. 10. Fig. 11 shows the temperatures change rules at point e under two situations of emergency braking. From Fig. 11, it is observed that the temperature at point e in- at first, then decreases; the highest temperature by simula- is lower than and also lags behind the experimental data. In 11a, the simulation temperature reaches the maximum K at 3.6 s while the experimental data comes up to the 435.65 K at 3.8 s. In Fig. 11b, the simulation result the maximum 469.55 K at 4.5 s while the experimental comes up to 479.68 K at 5 s. It is seen from Fig. 11, the temper- measured by experiment is lower than simulation results at Engineering 29 (2009) 932937 Fig. 8. The change of temperature gradient on line c with time t. 936 Z.-c. Zhu et al./Applied Thermal on the layer of friction surface (named thermal effect layer), which indicates the thermal diffusibility of the brake shoe is poor. In or- der to mater the temperature change rules of friction surface dur- ing emergency braking process, the variation of friction surfaces temperature with time t is simulated. What is shown in Fig. 6 re- veals that the temperature of friction surface increases firstly, then decreases. This is because that the speed of brake disc is high in the beginning and this results in large heat-flow while the coefficient of convective heat transfer is low on the boundary at the moment, so the temperature increases; at the late stage of brake the heat- flow decreases with the speed while the coefficient of convective heat transfer is high due to large difference in temperature on the boundary, which leads to decreasing in temperature. Figs. 6 and 7 reflect the temperature change rules in the radial dimension: the temperature at the outside of brake shoe is higher than that in- side, and the outside temperature changes more greatly. Fig. 8 demonstrates the change rules of the temperature gradi- ent along the direction z. The highest temperature gradient of the friction layer is up to 3.739 C2 10 5 K/m and decreases sharply along the direction z. The lowest value is only 4.597 C2 10 C011 K/m. In the beginning the temperature gradient of thermal effect layer is the highest while the temperature is close to the surrounding temper- ature. As the brake goes on, the temperature gradient decreases gradually until the end. Fig. 9 shows the change of temperature at different depth on the line c with time t. The temperature de- creases sharply with the increasing z, and the boundary condition has litter influence on the inner temperature. The temperature in- then it inverses. This is because the thermocouple itself ab- heat energy from the brake shoe in the beginning, then re- to the brake shoe when the temperature decreases. on between the experimental data and the simulation re- indicates that the simulation shows good agreement with the nt, and the errors of their highest temperature are 1.99% Fig. 9. The change of temperature at different depth on the line c with time t. Fig. 10. Schematic of friction tester. creases tion Fig. 427.14 maximum reaches data ature first, sorbs leases Comparis sults experime Fig. 11a. Temperatures change rules at point e with time t (p = 1.38 MPa, v 0 =1- 0 m/s). beginning the temperature gradient of thermal effect layer change rules of brake shoes 3-D transient temperature field during emergency braking. Acknowledgements This project is supported by the Key Project of Chinese Ministry of Education (Grant No. 107054) and Program for New Century Excellent Talents in University (Grant No. NCET-04-0488). Z.-c. Zhu et al./Applied Thermal Engineering 29 (2009) 932937 937 was the highest, the temperature increased swiftly; as the braking process going on, the temperature gradient decreased while the temperature increased; the boundary and 2.16%, respectively. It indicates that the analytic solution of 3- D transient temperature field is correct. 4. Conclusion (1) The theoretical model of 3-D transient temperature field was established acco