最大加工直徑320mm的臥式車床的主運(yùn)動(dòng)系統(tǒng)設(shè)計(jì)【Dmax=320mm Nmin=85rmin φ=1.26 Z=10】

最大加工直徑320mm的臥式車床的主運(yùn)動(dòng)系統(tǒng)設(shè)計(jì)【Dmax=320mm Nmin=85rmin φ=1.26 Z=10】,Dmax=320mm Nmin=85rmin φ=1.26 Z=10,最大加工直徑320mm的臥式車床的主運(yùn)動(dòng)系統(tǒng)設(shè)計(jì)【Dmax=320mm,Nmin=85rmin,φ=1.26,Z=10】,最大,加工 課程設(shè)計(jì)報(bào)告 320mm的臥式車床的主運(yùn)動(dòng)系統(tǒng)設(shè)計(jì) 5 設(shè)計(jì)任務(wù)書 車床的主參數(shù)（規(guī)格尺寸）和基本參數(shù)如下： 完成最大加工直徑 320 mm 的臥式車床的主運(yùn)動(dòng)系統(tǒng)設(shè)計(jì)，主要參數(shù)如下： 轉(zhuǎn)速級(jí)數(shù) Z=10 公比 1.26 最低轉(zhuǎn)速 nmin=85轉(zhuǎn)/分 工件最大回轉(zhuǎn)直徑 D(mm) 正轉(zhuǎn)最低轉(zhuǎn)速 nmin( ) 轉(zhuǎn)速級(jí)數(shù) 公比 320 85 10 1.26 目 錄 設(shè)計(jì)任務(wù)書 2 目 錄 4 第1章 設(shè)計(jì)要求及目的 6 第2章 機(jī)床主參數(shù)的確定 8 2.1 確定轉(zhuǎn)速范圍 8 2.2 確定電動(dòng)機(jī)型號(hào) 8 2.3擬定機(jī)床傳動(dòng)方案 9 2.3.1 傳動(dòng)系統(tǒng)擴(kuò)大順序的安排 10 2.3.2 繪制結(jié)構(gòu)網(wǎng) 10 2.3.3 傳動(dòng)組的變速范圍的極限值 11 2.3.4最大擴(kuò)大組的選擇 11 2.4 繪制轉(zhuǎn)速圖和傳動(dòng)系統(tǒng)圖 12 2.5 確定各變速組此論傳動(dòng)副齒數(shù) 13 第3章 傳動(dòng)件的計(jì)算 14 3.1 帶傳動(dòng)設(shè)計(jì) 14 3.2選擇帶型 15 3.3確定帶輪的基準(zhǔn)直徑并驗(yàn)證帶速 15 3.4確定中心距離、帶的基準(zhǔn)長度并驗(yàn)算小輪包角 16 3.5確定帶的根數(shù)z 17 3.6確定帶輪的結(jié)構(gòu)和尺寸 17 3.7確定帶的張緊裝置 17 3.8 驗(yàn)算主軸轉(zhuǎn)速誤差 19 3.9 計(jì)算轉(zhuǎn)速的計(jì)算 20 3.10 齒輪模數(shù)計(jì)算及驗(yàn)算 21 3.11 傳動(dòng)軸最小軸徑的初定 26 第4章 主要零件的設(shè)計(jì)與驗(yàn)算 30 4.1齒輪強(qiáng)度的校核驗(yàn)算 30 4.2 軸的校核 31 4.3 軸承壽命校核 33 參考文獻(xiàn) 35 第1章 設(shè)計(jì)要求及目的 一、設(shè) 計(jì) 任 務(wù)： 完成最大加工直徑 320 mm 的臥式車床的主運(yùn)動(dòng)系統(tǒng)設(shè)計(jì)，主要參數(shù)如下： 轉(zhuǎn)速級(jí)數(shù) Z=10 公比 1.26 最低轉(zhuǎn)速 nmin=85轉(zhuǎn)/分 二、設(shè) 計(jì) 要 求： 1.完成傳動(dòng)系統(tǒng)設(shè)計(jì)：方案合理，運(yùn)動(dòng)設(shè)計(jì)和動(dòng)力計(jì)算正確； 2.完成主軸箱草圖和展開圖設(shè)計(jì)（計(jì)算機(jī)打印A0）：布局合理、結(jié)構(gòu)緊湊，內(nèi)容表達(dá)完整、正確，圖紙規(guī)范整潔。 3.完成1張主軸箱的剖面圖（A1）：能綜合反映各軸的空間位置，操縱機(jī)構(gòu)安排合理、表達(dá)清楚，主軸中心高適當(dāng)； 4.完成主軸零件圖設(shè)計(jì)（A1）：結(jié)構(gòu)合理，形位公差和表面粗糙度等技術(shù)要求標(biāo)注合理，尺寸標(biāo)注完整正確。 5.完成計(jì)算說明書一份（25頁）：包括目錄、設(shè)計(jì)任務(wù)書、總論或前言（概述）、參數(shù)、運(yùn)動(dòng)設(shè)計(jì)的分析和擬定、動(dòng)力計(jì)算、結(jié)構(gòu)的選擇和分析及必要的說明、設(shè)計(jì)心得體會(huì)、參考文獻(xiàn)（書目前排列序號(hào)，以便正文引用）。要求條理清楚，計(jì)算、分析準(zhǔn)確。 機(jī)床技術(shù)參數(shù)有主參數(shù)和基本參數(shù)，他們是運(yùn)動(dòng)傳動(dòng)和結(jié)構(gòu)設(shè)計(jì)的依據(jù)，影響到機(jī)床的加工能力、決定和影響其他基本參數(shù)的依據(jù)，如車床的最大加工直徑，一般在設(shè)計(jì)題目中給定，基本參數(shù)是一些加工件尺寸、機(jī)床結(jié)構(gòu)、運(yùn)動(dòng)和動(dòng)力特性有關(guān)的參數(shù)，可歸納為尺寸參數(shù)、運(yùn)動(dòng)參數(shù)和動(dòng)力參數(shù)。 通用車床工藝范圍廣，所加工的工件形狀、尺寸和材料各不相同，有粗加工又有精加工；用硬質(zhì)合金刀具又用高速鋼刀具。因此，必須對(duì)所設(shè)計(jì)的機(jī)床工藝范圍和使用情況做全面的調(diào)研和統(tǒng)計(jì)，依據(jù)某些典型工藝和加工對(duì)象，兼顧其他的可能工藝加工的要求，擬定機(jī)床技術(shù)參數(shù)，擬定參數(shù)時(shí)，要考慮機(jī)床發(fā)展趨勢(shì)和同國內(nèi)外同類機(jī)床的對(duì)比，使擬定的參數(shù)最大限度地適應(yīng)各種不同的工藝要求和達(dá)到機(jī)床加工能力下經(jīng)濟(jì)合理。 機(jī)床主傳動(dòng)系因機(jī)床的類型、性能、規(guī)格和尺寸等因素的不同，應(yīng)滿足的要求也不一樣。設(shè)計(jì)機(jī)床主傳動(dòng)系時(shí)最基本的原則就是以最經(jīng)濟(jì)、合理的方式滿足既定的要求。在設(shè)計(jì)時(shí)應(yīng)結(jié)合具體機(jī)床進(jìn)行具體分析，一般應(yīng)滿足的基本要求有：滿足機(jī)床使用性能要求。首先應(yīng)滿足機(jī)床的運(yùn)動(dòng)特性，如機(jī)床主軸油足夠的轉(zhuǎn)速范圍和轉(zhuǎn)速級(jí)數(shù)；滿足機(jī)床傳遞動(dòng)力的要求。主電動(dòng)機(jī)和傳動(dòng)機(jī)構(gòu)能提供足夠的功率和轉(zhuǎn)矩，具有較高的傳動(dòng)效率；滿足機(jī)床工作性能要求。主傳動(dòng)中所有零部件有足夠的剛度、精度和抗震性，熱變形特性穩(wěn)定；滿足產(chǎn)品的經(jīng)濟(jì)性要求。傳動(dòng)鏈盡可能簡(jiǎn)短，零件數(shù)目要少，以便節(jié)約材料，降低成本。 36 第2章 機(jī)床主參數(shù)的確定 2.1 確定轉(zhuǎn)速范圍 根據(jù)=1.26 因?yàn)橐阎? ， 查標(biāo)準(zhǔn)數(shù)列表取最大轉(zhuǎn)速 因?yàn)?1.26=1.064，根據(jù)【1】表3-6標(biāo)準(zhǔn)數(shù)列。首先找到最小極限轉(zhuǎn)速85，再每跳過3個(gè)數(shù)取一個(gè)轉(zhuǎn)速，即可得到公比為1.26的數(shù)列： 85，106，132，170，212，265，335，425，530，670 2.2 確定電動(dòng)機(jī)型號(hào) 合理地確定電機(jī)功率N，使用的功率實(shí)際情況既能充分的發(fā)揮其使用性能，滿足生產(chǎn)需要，又不致使電機(jī)經(jīng)常輕載而降低功率因素。 目前，確定機(jī)床電機(jī)功率的常用方法很多，而本次設(shè)計(jì)中采用的是：估算法，它是一種按典型加工條件（工藝種類、加工材料、刀具、切削用量）進(jìn)行估算。根據(jù)此方法，中型車床典型切削條件下的用量： 1）主（垂直）切削力： 2）切削功率： N切 = 3）估算主電動(dòng)機(jī)功率： 根據(jù)以上條件，選定主電機(jī)： a.電機(jī)功率N： 根據(jù)機(jī)床切削能力的要求確定電機(jī)功率。但電機(jī)產(chǎn)品的功率已經(jīng)標(biāo)準(zhǔn)化，因此，按要求應(yīng)選取相近的標(biāo)準(zhǔn)值。 N=5.5kw b.電機(jī)轉(zhuǎn)速n 電機(jī)的轉(zhuǎn)速選擇的是： n=1440r/min 這個(gè)選擇是根據(jù)電機(jī)的轉(zhuǎn)速與主軸最高轉(zhuǎn)速nmax和Ⅰ軸的轉(zhuǎn)速相近或相宜，以免采用過大的升速或過小的降速傳動(dòng)。 c.電機(jī)的安裝和外形 根據(jù)電機(jī)不同的安裝和使用的需要，有四種不同的外形結(jié)構(gòu)，用的最多的有底座式和法蘭式兩種。本次設(shè)計(jì)的機(jī)床所需選用的是外行安裝尺寸之一。具體的安裝圖可由手冊(cè)查到。 d.常用電機(jī)的資料 根據(jù)常用電機(jī)所提供的資料，選用： Y132S-4 2.3擬定機(jī)床傳動(dòng)方案 級(jí)數(shù)為Z的傳動(dòng)系統(tǒng)由若干個(gè)順序的傳遞組組成，各傳動(dòng)組分別有Z1、Z2、Z3、…個(gè)傳動(dòng)副.即 Z=Z1Z2Z3… 傳動(dòng)副數(shù)為使結(jié)構(gòu)盡量簡(jiǎn)單以2或3為適合，即變速級(jí)數(shù)Z應(yīng)為2和3的因子： 即 Z=2a3b 實(shí)現(xiàn)12級(jí)主軸轉(zhuǎn)速變化的傳動(dòng)系統(tǒng)可以寫成多種傳動(dòng)副的組合: 1) 12=3×4 2) 12=4×3 3) 12=3×2×2 4) 12=2×3×2 5) 12=2×2×3 方案1）和方案2)可省掉一根軸。但有一個(gè)傳動(dòng)組有四個(gè)傳動(dòng)副。若用一個(gè)四聯(lián)滑移齒輪，則將大大增加其軸向尺寸；若用兩個(gè)雙聯(lián)滑移齒輪，則操縱機(jī)構(gòu)必須互鎖以防止兩個(gè)滑移齒輪同時(shí)嚙合。將使得結(jié)構(gòu)比較復(fù)雜。故在此不予采用。 按照傳動(dòng)副“前）多后少”的原則選擇Z=3×2×2這一方案，但主軸換向采用雙向片式摩擦離合器結(jié)構(gòu)，致使Ⅰ軸的軸向尺寸過大，所以此方案不宜采用，加之主軸對(duì)加工精度、表面粗超度的影響最大。因此在主軸的傳動(dòng)副不宜太多，故方案5）亦不采用。而應(yīng)先擇12=2×3×2。 綜上所述： 方案4） 12=2×3×2 是比較合理的 2.3.1 傳動(dòng)系統(tǒng)擴(kuò)大順序的安排 12=2×3×2的傳動(dòng)副組合，其傳動(dòng)組的擴(kuò)大順序又可以有6種形式： 1) 12=21×32×26 2) 12=21×34×22 3) 12=23×31×26 4) 12=26×31×23 5) 12=22×34×21 6) 12=26×32×21 以上各種結(jié)構(gòu)式方案中，由于傳動(dòng)副的極限傳動(dòng)比和傳動(dòng)組的極限變速范圍的限制，一般升速時(shí)。極限變速范圍。 檢查傳動(dòng)組的變速范圍時(shí)，只需檢查最后一個(gè)擴(kuò)大組。因其他傳動(dòng)組的變速范圍都比他小。由式 對(duì)于方案2）和 方案5）有：,則對(duì)于方案2）和 方案5）不予考慮。 對(duì)于其余方案有：。然而在可行的結(jié)構(gòu)式方案1）、3）、4）、6）中，為了使中間軸變速范圍最小，在各方案同號(hào)傳動(dòng)軸的最高轉(zhuǎn)速相同時(shí)，變速范圍越小，最低轉(zhuǎn)速越高，轉(zhuǎn)矩越小，傳動(dòng)件尺寸也就越小。比較方案1）、3）、4）、6），方案1）的中間傳動(dòng)軸變速范圍最小，方案1）最佳。但由于Ⅰ軸裝有摩擦離合器，在結(jié)構(gòu)上要求有一齒輪的齒根圓大于離合器的直徑因而采用方案3）12=23×31×26 最佳 2.3.2 繪制結(jié)構(gòu)網(wǎng) 由上選擇的結(jié)構(gòu)式12=23×31×26 畫其結(jié)構(gòu)圖如下： 圖2.1結(jié)構(gòu)網(wǎng) 2.3.3 傳動(dòng)組的變速范圍的極限值 齒輪傳動(dòng)最小傳動(dòng)比Umin1/4,最大傳動(dòng)比Umax,決定了一個(gè)傳動(dòng)組的最大變速范圍rmax=umax/umin。 因此，要按照下表，淘汰傳動(dòng)組變速范圍超過極限值的所有傳動(dòng)方案。 極限傳動(dòng)比及指數(shù)X,X,值為： 表2.1 公比 極限傳動(dòng)比指數(shù) 1.41 X值：Umin==1/4 4 X'值：Umax=x, =2 2 (X+ X')值：rmin=x+x｀=8 6 2.3.4最大擴(kuò)大組的選擇 正常連續(xù)的順序擴(kuò)大組的傳動(dòng)的傳動(dòng)結(jié)構(gòu)式為： Z=Z1[1]Z2[Z1]Z3[Z1Z2] 最后擴(kuò)大組的變速范圍 按照r原則，導(dǎo)出系統(tǒng)的最大級(jí)數(shù)Z和變速范圍Rn為： 表2.2 Z3 2 3 1.41 Z=12 Rn=44 Z=9 Rn=15.6 最后擴(kuò)大組的傳動(dòng)副數(shù)目Z3=2時(shí)的轉(zhuǎn)速范圍遠(yuǎn)比Z3=3時(shí)大 因此，在機(jī)床設(shè)計(jì)中，因要求的R較大，最后擴(kuò)大組應(yīng)取2更為合適。 同時(shí)，最后傳動(dòng)組與最后擴(kuò)大組往往是一致的。安裝在主軸與主軸前一傳動(dòng)軸的具有極限或接近傳動(dòng)比的齒輪副承受最大扭距，在結(jié)構(gòu)上可獲得較為滿意的處理，這也就是最后傳動(dòng)組的傳動(dòng)副經(jīng)常為2的另一原因。 2.4 繪制轉(zhuǎn)速圖和傳動(dòng)系統(tǒng)圖 （1）選擇電動(dòng)機(jī)：采用Y系列封閉自扇冷式鼠籠型三相異步電動(dòng)機(jī)。 （2）繪制轉(zhuǎn)速圖： （3）畫主傳動(dòng)系統(tǒng)圖。根據(jù)系統(tǒng)轉(zhuǎn)速圖及已知的技術(shù)參數(shù)，畫主傳動(dòng)系統(tǒng)圖如圖2-3： 1-2軸最小中心距：A1_2min>1/2(Zmaxm+2m+D) 軸最小齒數(shù)和:Szmin>(Zmax+2+D/m) 2.5 確定各變速組此論傳動(dòng)副齒數(shù) (1)Sz100-124,中型機(jī)床Sz=70-100 (2)直齒圓柱齒輪Zmin18-24,m4 圖2-3 主傳動(dòng)系統(tǒng)圖 （7）齒輪齒數(shù)的確定。變速組內(nèi)取模數(shù)相等，據(jù)設(shè)計(jì)要求Zmin≥18~24,齒數(shù)和Sz≤100～124，由表4.1，根據(jù)各變速組公比，可得各傳動(dòng)比和齒輪齒數(shù)，各齒輪齒數(shù)如表2-2。 表2-2 齒輪齒數(shù) 傳動(dòng)比 基本組 第一擴(kuò)大組 第二擴(kuò)大組 1:1.26 1:2 1:3.16 1:1.26 1:1.58 1.58:1 1：1.58 代號(hào) Z Z Z Z Z Z Z Z’ Z5 Z5’ Z Z Z7 Z7’ 齒數(shù) 33 42 25 50 18 57 30 38 26 42 61 38 38 61 第3章 傳動(dòng)件的計(jì)算 3.1 帶傳動(dòng)設(shè)計(jì) 輸出功率P=5.5kW,轉(zhuǎn)速n1=1440r/min,n2=670r/min 計(jì)算設(shè)計(jì)功率Pd 表4 工作情況系數(shù) 工作機(jī) 原動(dòng)機(jī) ⅰ類 ⅱ類 一天工作時(shí)間/h 10~16 10~16 載荷 平穩(wěn) 液體攪拌機(jī)；離心式水泵；通風(fēng)機(jī)和鼓風(fēng)機(jī)（）；離心式壓縮機(jī)；輕型運(yùn)輸機(jī) 1.0 1.1 1.2 1.1 1.2 1.3 載荷 變動(dòng)小 帶式運(yùn)輸機(jī)（運(yùn)送砂石、谷物），通風(fēng)機(jī)（）；發(fā)電機(jī)；旋轉(zhuǎn)式水泵；金屬切削機(jī)床；剪床；壓力機(jī)；印刷機(jī)；振動(dòng)篩 1.1 1.2 1.3 1.2 1.3 1.4 載荷 變動(dòng)較大 螺旋式運(yùn)輸機(jī)；斗式上料機(jī)；往復(fù)式水泵和壓縮機(jī)；鍛錘；磨粉機(jī)；鋸木機(jī)和木工機(jī)械；紡織機(jī)械 1.2 1.3 1.4 1.4 1.5 1.6 載荷 變動(dòng)很大 破碎機(jī)（旋轉(zhuǎn)式、顎式等）；球磨機(jī)；棒磨機(jī)；起重機(jī)；挖掘機(jī)；橡膠輥壓機(jī) 1.3 1.4 1.5 1.5 1.6 1.8 根據(jù)V帶的載荷平穩(wěn)，兩班工作制（16小時(shí)），查《機(jī)械設(shè)計(jì)》P296表4， 取KA＝1.1。即 3.2選擇帶型 普通V帶的帶型根據(jù)傳動(dòng)的設(shè)計(jì)功率Pd和小帶輪的轉(zhuǎn)速n1按《機(jī)械設(shè)計(jì)》P297圖13－11選取。 根據(jù)算出的Pd＝6.05kW及小帶輪轉(zhuǎn)速n1＝1440r/min ，查圖得：dd=80～100可知應(yīng)選取A型V帶。 3.3確定帶輪的基準(zhǔn)直徑并驗(yàn)證帶速 由《機(jī)械設(shè)計(jì)》P298表13－7查得，小帶輪基準(zhǔn)直徑為80～100mm 則取dd1=100mm> ddmin.=75 mm（dd1根據(jù)P295表13-4查得） 表3 V帶帶輪最小基準(zhǔn)直徑 槽型 Y Z A B C D E 20 50 75 125 200 355 500 由《機(jī)械設(shè)計(jì)》P295表13-4查“V帶輪的基準(zhǔn)直徑”，得=224mm ① 誤差驗(yàn)算傳動(dòng)比： （為彈性滑動(dòng)率） 誤差 符合要求 ② 帶速 滿足5m/s300mm，所以宜選用E型輪輻式帶輪。 總之，小帶輪選H型孔板式結(jié)構(gòu)，大帶輪選擇E型輪輻式結(jié)構(gòu)。 帶輪的材料：選用灰鑄鐵，HT200。 3.7確定帶的張緊裝置 選用結(jié)構(gòu)簡(jiǎn)單，調(diào)整方便的定期調(diào)整中心距的張緊裝置。 對(duì)帶輪的主要要求是質(zhì)量小且分布均勻、工藝性好、與帶接觸的工作表面加工精度要高，以減少帶的磨損。轉(zhuǎn)速高時(shí)要進(jìn)行動(dòng)平衡，對(duì)于鑄造和焊接帶輪的內(nèi)應(yīng)力要小, 帶輪由輪緣、腹板（輪輻）和輪轂三部分組成。帶輪的外圈環(huán)形部分稱為輪緣，輪緣是帶輪的工作部分，用以安裝傳動(dòng)帶，制有梯形輪槽。由于普通V帶兩側(cè)面間的夾角是40°，為了適應(yīng)V帶在帶輪上彎曲時(shí)截面變形而使楔角減小，故規(guī)定普通V帶輪槽角 為32°、34°、36°、38°（按帶的型號(hào)及帶輪直徑確定），輪槽尺寸見表7-3。裝在軸上的筒形部分稱為輪轂，是帶輪與軸的聯(lián)接部分。中間部分稱為輪幅（腹板），用來聯(lián)接輪緣與輪轂成一整體。 表 普通V帶輪的輪槽尺寸（摘自GB/T13575.1-92） 項(xiàng)目 ? 符號(hào) 槽型 Y Z A B C D E 基準(zhǔn)寬度 b p 5.3 8.5 11.0 14.0 19.0 27.0 32.0 基準(zhǔn)線上槽深 h amin 1.6 2.0 2.75 3.5 4.8 8.1 9.6 基準(zhǔn)線下槽深 h fmin 4.7 7.0 8.7 10.8 14.3 19.9 23.4 槽間距 e 8 ± 0.3 12±0.3 15±0.3 19±0.4 25.5 ± 0.5 37± 0.6 44.5 ± 0.7 第一槽對(duì)稱面至端面的距離 f min 6 7 9 11.5 16 23 28 最小輪緣厚 5 5.5 6 7.5 10 12 15 帶輪寬 B B =( z -1) e + 2 f ? z —輪槽數(shù) 外徑 d a 輪 槽 角 32° 對(duì)應(yīng)的基準(zhǔn)直徑 d d ≤ 60 - - - - - - 34° - ≤ 80 ≤ 118 ≤ 190 ≤ 315 - - 36° 60 - - - - ≤ 475 ≤ 600 38° - ＞ 80 ＞ 118 ＞ 190 ＞ 315 ＞ 475 ＞ 600 極限偏差 ± 1 ± 0.5 V帶輪按腹板（輪輻）結(jié)構(gòu)的不同分為以下幾種型式： （1） 實(shí)心帶輪：用于尺寸較小的帶輪(dd≤(2.5～3)d時(shí))，如圖7 -6a。  （2） 腹板帶輪：用于中小尺寸的帶輪(dd≤ 300mm 時(shí))，如圖7-6b。 （3） 孔板帶輪：用于尺寸較大的帶輪((dd－d)＞ 100 mm 時(shí))，如圖7 -6c 。 （4） 橢圓輪輻帶輪：用于尺寸大的帶輪(dd＞ 500mm 時(shí))，如圖7-6d。 （a） （b） （c） （d） 圖7-6 帶輪結(jié)構(gòu)類型 根據(jù)設(shè)計(jì)結(jié)果，可以得出結(jié)論：小帶輪選擇實(shí)心帶輪，如圖（a）,大帶輪選擇腹板帶輪如圖（b） 3.8 驗(yàn)算主軸轉(zhuǎn)速誤差 由于確定的齒輪齒數(shù)所得的實(shí)際轉(zhuǎn)速與傳動(dòng)設(shè)計(jì)的理論轉(zhuǎn)速難以完全相符，需要驗(yàn)算主軸各級(jí)轉(zhuǎn)速，最大誤差不得超過±10(ψ-1)%。 主軸各級(jí)實(shí)際轉(zhuǎn)速值用下式計(jì)算 n實(shí)=nd×(1-ε)×u1×u2×u3×u4 其中： ε—滑移系數(shù)ε=0.2 u1、 u2 、u3 、u4分別為各級(jí)的傳動(dòng)比 轉(zhuǎn)速誤差用主軸實(shí)際轉(zhuǎn)速與標(biāo)準(zhǔn)轉(zhuǎn)速相對(duì)誤差的絕對(duì)值表示 ⊿n=｜∣≤±10(ψ-1)% 實(shí)際轉(zhuǎn)速及轉(zhuǎn)速誤差如下: 表2.5各級(jí)傳動(dòng)組的轉(zhuǎn)速誤差 主軸轉(zhuǎn)速 n1 n2 n3 n4 n5 n6 n7 n8 n9 n10 理論轉(zhuǎn)速 85 106 132 170 212 265 335 425 530 670 實(shí)際轉(zhuǎn)速 85.8 107.2 134.5 173.2 214.6 266.4 336.4 418.5 535.6 674.3 轉(zhuǎn)速誤差 （%） 0.7 0.4 0.2 0.32 0.17 0.32 0.32 0.46 0.27 0.3 故轉(zhuǎn)速誤差滿足要求。 3.9 計(jì)算轉(zhuǎn)速的計(jì)算 （1）主軸的計(jì)算轉(zhuǎn)速nj，由公式n=n得，主軸的計(jì)算轉(zhuǎn)速nj=198.305r/min， 取212r/min。 (2). 傳動(dòng)軸的計(jì)算轉(zhuǎn)速 軸3=132 r/min 軸2=212 r/min，軸1=670r/min。 （2）確定各傳動(dòng)軸的計(jì)算轉(zhuǎn)速。各計(jì)算轉(zhuǎn)速入表3-1。 表3-1 各軸計(jì)算轉(zhuǎn)速 軸 號(hào) Ⅰ 軸 Ⅱ 軸 Ⅲ 軸 計(jì)算轉(zhuǎn)速 r/min 670 212 132 （3） 確定齒輪副的計(jì)算轉(zhuǎn)速。齒輪Z裝在主軸上其中只有212r/min傳遞全功率，故Zj=212r/min。 依次可以得出其余齒輪的計(jì)算轉(zhuǎn)速，如表3-2。 表3-2 齒輪副計(jì)算轉(zhuǎn)速 序號(hào) Z Z Z Z Z n 670 212 212 132 212 3.10 齒輪模數(shù)計(jì)算及驗(yàn)算 （1）模數(shù)計(jì)算。一般同一變速組內(nèi)的齒輪取同一模數(shù)，選取負(fù)荷最重的小齒輪，按簡(jiǎn)化的接觸疲勞強(qiáng)度公式進(jìn)行計(jì)算，即mj=16338可得各組的模數(shù)，如表3-3所示。 根據(jù)和計(jì)算齒輪模數(shù)，根據(jù)其中較大值取相近的標(biāo)準(zhǔn)模數(shù)： =16338=16338mm ——齒輪的最低轉(zhuǎn)速r/min； ——頂定的齒輪工作期限，中型機(jī)床推存：=15～24 ——轉(zhuǎn)速變化系數(shù)； ——功率利用系數(shù)； ——材料強(qiáng)化系數(shù)。 ——（壽命系數(shù)）的極值 齒輪等轉(zhuǎn)動(dòng)件在接取和彎曲交邊載荷下的疲勞曲線指數(shù)m和基準(zhǔn)順環(huán)次數(shù)C0 ——工作情況系數(shù)。中等中級(jí)的主運(yùn)動(dòng)： ——?jiǎng)虞d荷系數(shù)； ——齒向載荷分布系數(shù)； ——齒形系數(shù)； 根據(jù)彎曲疲勞計(jì)算齒輪模數(shù)公式為： 式中：N——計(jì)算齒輪轉(zhuǎn)動(dòng)遞的額定功率N=? ——計(jì)算齒輪（小齒輪）的計(jì)算轉(zhuǎn)速r/min ——齒寬系數(shù), Z1——計(jì)算齒輪的齒數(shù)，一般取轉(zhuǎn)動(dòng)中最小齒輪的齒數(shù)： ——大齒輪與小齒輪的齒數(shù)比，=；（+）用于外嚙合，（-）號(hào)用 于內(nèi)嚙合： 命系數(shù)； ：工作期限 ， =； ==3.49 ==1.8 =0.84 =0.58 =0.90 =0.55 =0.72 =3.49 0.84 0.58 0.55=0.94 =1.80.84 0.90 0.72=0.99 時(shí)，取=，當(dāng)<時(shí)，取=； ==0.85 =1.5； =1.2 =1 =0.378 許用彎曲應(yīng)力，接觸應(yīng)力，() =354 =1750 按接觸疲勞計(jì)算齒輪模數(shù)m 1軸由公式mj=16338可得mj=2.34mm，取m=3mm 2軸由公式mj=16338可得mj=2.31mm，取m=3mm 3軸由公式mj=16338可得mj=2.83mm，取m=3mm 根據(jù)有關(guān)文獻(xiàn)，也為了便于統(tǒng)一，在這里傳動(dòng)齒輪統(tǒng)一取m=3 表3-3 模數(shù) 組號(hào) 基本組 第一擴(kuò)大組 第二擴(kuò)大組 模數(shù) mm 3 3 3 （2）基本組齒輪計(jì)算。 基本組齒輪幾何尺寸見下表 齒輪 Z1 Z1` Z2 Z2` Z3 Z3` 齒數(shù) 33 42 25 50 18 57 分度圓直徑 99 126 75 150 54 171 齒頂圓直徑 105 132 91 156 60 177 齒根圓直徑 91.5 118.5 67.5 142.5 46.5 163.5 齒寬 24 24 24 24 24 24 按基本組最小齒輪計(jì)算。小齒輪用40Cr，調(diào)質(zhì)處理，硬度241HB～246HB，平均取260HB，大齒輪用45鋼，調(diào)質(zhì)處理，硬度229HB～246HB，平均取240HB。計(jì)算如下： ① 齒面接觸疲勞強(qiáng)度計(jì)算： 接觸應(yīng)力驗(yàn)算公式為 彎曲應(yīng)力驗(yàn)算公式為： 式中 N----傳遞的額定功率（kW），這里取N為電動(dòng)機(jī)功率，N=5kW; -----計(jì)算轉(zhuǎn)速（r/min）. =212（r/min）; m-----初算的齒輪模數(shù)（mm）, m=3（mm）; B----齒寬（mm）;B=24（mm）; z----小齒輪齒數(shù)；z=18; u----小齒輪齒數(shù)與大齒輪齒數(shù)之比,u=3.16; -----壽命系數(shù)； = ----工作期限系數(shù)； T------齒輪工作期限，這里取T=15000h.; -----齒輪的最低轉(zhuǎn)速（r/min）, =500（r/min） ----基準(zhǔn)循環(huán)次數(shù)，接觸載荷取=，彎曲載荷取= m----疲勞曲線指數(shù)，接觸載荷取m=3；彎曲載荷取m=6; ----轉(zhuǎn)速變化系數(shù)，查【5】2上，取=0.60 ----功率利用系數(shù)，查【5】2上，取=0.78 -----材料強(qiáng)化系數(shù)，查【5】2上， =0.60 -----工作狀況系數(shù)，取=1.1 -----動(dòng)載荷系數(shù)，查【5】2上，取=1 ------齒向載荷分布系數(shù)，查【5】2上，=1 Y------齒形系數(shù)，查【5】2上，Y=0.386； ----許用接觸應(yīng)力（MPa）,查【4】，表4-7，取=650 Mpa； ---許用彎曲應(yīng)力（MPa），查【4】，表4-7，取=275 Mpa； 根據(jù)上述公式，可求得及查取值可求得： =635 Mpa =78 Mpa （3）第一擴(kuò)大組齒輪計(jì)算。 擴(kuò)大組齒輪幾何尺寸見下表 齒輪 Z4 Z4` Z5 Z5` 齒數(shù) 30 38 26 42 分度圓直徑 90 114 78 126 齒頂圓直徑 96 120 84 132 齒根圓直徑 82.5 106.5 70.5 118.5 齒寬 24 24 24 24 （4）第二擴(kuò)大組齒輪計(jì)算。 擴(kuò)大組齒輪幾何尺寸見下表 齒輪 Z5 Z5` Z6 Z6` 齒數(shù) 61 38 38 61 分度圓直徑 183 114 114 183 齒頂圓直徑 189 120 120 189 齒根圓直徑 175.5 106.5 106.5 175.5 齒寬 24 24 24 24 按擴(kuò)大組最小齒輪計(jì)算。小齒輪用40Cr，調(diào)質(zhì)處理，硬度241HB～246HB，平均取260HB，大齒輪用45鋼，調(diào)質(zhì)處理，硬度229HB～246HB，平均取240HB。 同理根據(jù)基本組的計(jì)算， 查文獻(xiàn)【6】，可得 =0.62， =0.77，=0.60，=1.1， =1，=1，m=3.5，=355； 可求得： =619 Mpa =135Mpa 3.11 傳動(dòng)軸最小軸徑的初定 傳動(dòng)軸直徑按扭轉(zhuǎn)剛度用下列公式估算傳動(dòng)軸直徑： mm 其中：N—該傳動(dòng)軸的輸入功率 KW Nd—電機(jī)額定功率； —從電機(jī)到該傳動(dòng)軸之間傳動(dòng)件的傳動(dòng)效率的乘積 —該傳動(dòng)軸的計(jì)算轉(zhuǎn)速r/min —每米長度上允許的扭轉(zhuǎn)角(deg/m),可根據(jù)傳動(dòng)軸的要求選取如表3.2所示： 表3.2 剛度要求 允許的扭轉(zhuǎn)角 主 軸 一般的傳動(dòng)軸 較低的傳動(dòng)軸 0.5—1 1—1.5 1.5—2 對(duì)于一般的傳動(dòng)軸，取=1.5。取估算的傳動(dòng)軸長度為500mm。 對(duì)Ⅰ軸有： KW =670r/min 預(yù)取mm 對(duì)Ⅱ軸有：KW =1120 r/min mm 預(yù)取 對(duì)Ⅲ軸有： KW =140 mm 預(yù)取 采用花鍵軸結(jié)構(gòu)，即將估算的傳動(dòng)軸直徑d減小7%為花鍵軸的直徑，在選相近的標(biāo)準(zhǔn)花鍵。 =32×0.93=29.76 =38×0.93=35.34 =46×0.93=42.78 查表可以選取花鍵的型號(hào)其尺寸分別為 軸取 6-30×26×6 軸取 6-38×33×10 軸取 6-43×40×12 最脆弱軸的計(jì)算校核 對(duì)于傳動(dòng)軸，除重載軸外，一般無須進(jìn)行強(qiáng)度校核，只進(jìn)行剛度驗(yàn)算。 軸的抗彎斷面慣性矩（） 花鍵軸 = 式中 d—花鍵軸的小徑（mm）； i—花軸的大徑（mm）； b、N—花鍵軸鍵寬，鍵數(shù)； 傳動(dòng)軸上彎曲載荷的計(jì)算，一般由危險(xiǎn)斷面上的最大扭矩求得： = 式中 N—該軸傳遞的最大功率（kw）; —該軸的計(jì)算轉(zhuǎn)速（r/min）。 傳動(dòng)軸上的彎矩載荷有輸入扭矩齒輪和輸出扭矩齒輪的圓周力、徑向力，齒輪的圓 周力 式中 D—齒輪節(jié)圓直徑（mm）,D=mZ。 齒輪的徑向力： 式中 α—為齒輪的嚙合角，α＝20o； ρ—齒面摩擦角，； β—齒輪的螺旋角；β＝0 故N 花鍵軸鍵側(cè)擠壓應(yīng)力的驗(yàn)算 花鍵鍵側(cè)工作表面的擠壓應(yīng)力為： 式中 —花鍵傳遞的最大轉(zhuǎn)矩（）； D、d—花鍵軸的大徑和小徑（mm）； L—花鍵工作長度； N—花鍵鍵數(shù)； K—載荷分布不均勻系數(shù)，K=0.7～0.8； 故此花鍵軸校核合格 軸承疲勞強(qiáng)度校核 機(jī)床傳動(dòng)軸用滾動(dòng)軸承，主要是因疲勞破壞而失效，故應(yīng)進(jìn)行疲勞驗(yàn)算。其額定壽命的計(jì)算公式為： C—滾動(dòng)軸承的額定負(fù)載（N）,根據(jù)《軸承手冊(cè)》或《機(jī)床設(shè)計(jì)手冊(cè)》查取，單位用（kgf）應(yīng)換算成（N）； —速度系數(shù)， 為滾動(dòng)軸承的計(jì)算轉(zhuǎn)速（r/mm） —壽命系數(shù)， —壽命系數(shù)，對(duì)球軸承=3，對(duì)滾子軸承=； 工作情況系數(shù)，對(duì)輕度沖擊和振動(dòng)的機(jī)床（車床、銑床、鉆床、磨床等多數(shù)機(jī)床），； —功率利用系數(shù)，查表3—3； —速度轉(zhuǎn)化系數(shù)，查表3—2； —齒輪輪換工作系數(shù)，查《機(jī)床設(shè)計(jì)手冊(cè)》； P—當(dāng)量動(dòng)載荷，按《機(jī)床設(shè)計(jì)手冊(cè)》。 故軸承校核合格 第4章 主要零件的設(shè)計(jì)與驗(yàn)算 4.1齒輪強(qiáng)度的校核驗(yàn)算 ① 齒面接觸疲勞強(qiáng)度計(jì)算： 接觸應(yīng)力驗(yàn)算公式為 彎曲應(yīng)力驗(yàn)算公式為： 式中 N----傳遞的額定功率（kW），這里取N為電動(dòng)機(jī)功率，N=5kW; -----計(jì)算轉(zhuǎn)速（r/min）. =212（r/min）; m-----初算的齒輪模數(shù)（mm）, m=3（mm）; B----齒寬（mm）;B=24（mm）; z----小齒輪齒數(shù)；z=18; u----小齒輪齒數(shù)與大齒輪齒數(shù)之比,u=3.16; -----壽命系數(shù)； = ----工作期限系數(shù)； T------齒輪工作期限，這里取T=15000h.; -----齒輪的最低轉(zhuǎn)速（r/min）, =500（r/min） ----基準(zhǔn)循環(huán)次數(shù)，接觸載荷取=，彎曲載荷取= m----疲勞曲線指數(shù)，接觸載荷取m=3；彎曲載荷取m=6; ----轉(zhuǎn)速變化系數(shù)，查【5】2上，取=0.60 ----功率利用系數(shù)，查【5】2上，取=0.78 -----材料強(qiáng)化系數(shù)，查【5】2上， =0.60 -----工作狀況系數(shù)，取=1.1 -----動(dòng)載荷系數(shù)，查【5】2上，取=1 ------齒向載荷分布系數(shù)，查【5】2上，=1 Y------齒形系數(shù)，查【5】2上，Y=0.386； ----許用接觸應(yīng)力（MPa）,查【4】，表4-7，取=650 Mpa； ---許用彎曲應(yīng)力（MPa），查【4】，表4-7，取=275 Mpa； 根據(jù)上述公式，可求得及查取值可求得： =635 Mpa =78 Mpa 4.2 軸的校核 （a） 主軸的前端部撓度 （b） 主軸在前軸承處的傾角 （c） 在安裝齒輪處的傾角 E取為， ， 由于小齒輪的傳動(dòng)力大，這里以小齒輪來進(jìn)行計(jì)算 將其分解為垂直分力和水平分力 由公式 可得 主軸載荷圖如下所示： 由上圖可知如下數(shù)據(jù)：a=364mm,b=161mm,l=525mm,c=87mm 計(jì)算（在垂直平面） ,, ，， ，， 計(jì)算（在水平面） ,, ，， ，， 合成： 4.3 軸承壽命校核 由П軸最小軸徑可取軸承為7008C角接觸球軸承,ε=3；P=XFr+YFaX=1，Y=0。 對(duì)Ⅱ軸受力分析 得：前支承的徑向力Fr=2642.32N。 由軸承壽命的計(jì)算公式：預(yù)期的使用壽命 [L10h]=15000h L10h=×=×=h≥[L10h]=15000h 軸承壽命滿足要求。 參考文獻(xiàn) 1.段鐵群.《主軸箱設(shè)計(jì)》，科學(xué)出版社； 2.于惠力，向敬忠，張春宜.《機(jī)械設(shè)計(jì)》，科學(xué)出版社； 3.潘承怡，蘇相國. 《機(jī)械設(shè)計(jì)課程設(shè)計(jì)》，哈爾濱理工大學(xué)； 4.戴署.《金屬切削機(jī)床設(shè)計(jì)》，機(jī)械工業(yè)出版社； 5.陳易新，《金屬切削機(jī)床課程設(shè)計(jì)指導(dǎo)書》； 6、機(jī)床主軸、變速箱設(shè)計(jì)簡(jiǎn)明手冊(cè) 7、機(jī)械設(shè)計(jì)課程設(shè)計(jì) 8、金屬切削機(jī)床設(shè)計(jì) 9、機(jī)械制造裝備等 Bebek, Bearing load Bending stress beam is rate, parameter with the most important influence on design of the crankshaft. Results of bearing loads and web bending stresses are tabulated. must overall systems on parameters of the crankshaft system. Studies on crankshaft of internal combustion engines mainly fo- cus on vibration and stress analyses 19. Although stress analy- ses of crankshafts are available in literature, there are few studies on the effect of counterweight configuration on main bear- ing loads and crankshaft stresses. Sharpe et al. 10 studied balanc- ing of the crankshaft of a V-8 engine using a rigid crankshaft model tions are carried out at engine speed range of 10002000 rpm. Bending stresses at the centres of each web are also calculated. 2. Engine specifications The specifications of in-line six-cylinder diesel engine are given in Table 1. The 9.0 L engine crankshaft has eight counterweights at crank webs 1, 2, 5, 6, 7, 8, 11 and 12. 3D solid model of the crank- shaft is obtained using Pro/Engineer and is shown in Fig. 1. Sche- matic representation of the crankshaft is given in Fig. 2. Static * Corresponding author. Tel.: +90 212 359 7534; fax: +90 212 287 2456. Advances in Engineering Software 40 (2009) 95104 Contents lists available E-mail address: yasin.yilmazboun.edu.tr (Y. Yilmaz). being the main part responsible for power production. Crankshaft system mainly consists of piston, piston pin, con- necting rod, crankshaft, torsional vibration (TV) damper and fly- wheel. Counterweights are placed on the opposite side of each crank to balance rotating inertia forces. In general, counterweights are designed for balancing rates between 50% and 100%. For acceptable maximum and average main bearing loads, mass of counterweights and their positions are important. Maximum and average main bearing loads of an engine depend on cylinder pres- sure, counterweight mass, engine speed and other geometric study on effect of counterweight configuration on main bearing loads and crankshaft stresses is still needed. In this study, counterweight positions and masses of an in-line six-cylinder diesel engine crankshaft system are studied. Maxi- mum and average main bearing forces and crankshaft bending stresses are calculated for 12-counterweight configurations with a zero degree counterweight angle, and for eight-counterweight configurations with 30C176 counterweight angle for 0%, 50% and 100% counterweight balancing rates. Analyses are carried out using Multibody System Simulation Program, ADAMS/Engine. Simula- 1. Introduction New internal combustion engines power, good fuel economy, small engine harmless as possible to the environment. each component of the engine on its be investigated in detail. Crankshaft tion engines have important influence 0965-9978/$ - see front matter C211 2008 Elsevier Ltd. All doi:10.1016/j.advengsoft.2008.03.009 C211 2008 Elsevier Ltd. All rights reserved. have high engine size, and should be as Therefore, the effect of performance should of internal combus- engine performance and optimized counterweights to minimize main bearing loads. Stanley and Taraza 11 obtained maximum and average main bearing loads of four and six-cylinder symmetric in-line engines using a rigid crankshaft model and estimated ideal counterweight mass that resulted in acceptable maximum bearing load. Rigid crankshaft models that are used in counterweight analyses do not consider the effect of crankshaft flexibility on main bearing loads and can lead to considerable errors. Therefore, an extensive Crankshaft models Balancing rate Both configurations show the same trend. The load from gas pressure rather than inertia forces is the An investigation of the effect of counterweight load and crankshaft bending stress Yasin Yilmaz * , Gunay Anlas Department of Mechanical Engineering, Faculty of Engineering, Bogazici University, 34342 article info Article history: Received 11 February 2008 Received in revised form 17 March 2008 Accepted 24 March 2008 Available online 6 May 2008 Keywords: Counterweight configuration abstract In this study, effects of counterweight stress of an in-line six-cylinder ADAMS. In the analysis, rigid, rigid, beam and 3D solid models analyses. Twelve-counterweight terweight configurations with ing rates, are considered. It with increasing balancing Advances in Engineering journal homepage: rights reserved. configuration on main bearing Istanbul, Turkey mass and position on main bearing load and crankshaft bending diesel engine is investigated using Multibody System Simulation Program, and 3D solid crankshaft models are used. Main bearing load results of are compared and beam model is used in counterweight configuration configurations with a zero degree counterweight angle and eight-coun- 30C176 counterweight angle, each for 0%, 50% and 100% counterweight balanc- found that maximum main bearing load and web bending stress increase and average main bearing load decreases with increasing balancing rate. at ScienceDirect Software cate/advengsoft unbalance of each crank throw (with and w/o counterweights) is determined using Pro/Engineer and is given in Table 2. The balanc- ing system data for the crank train are given in Table 3. 3. Modeling of crankshaft system Using ADAMS/Engine, a crankshaft can be modeled in four dif- ferent ways: rigid crankshaft, torsionalflexible crankshaft, beam crankshaft and 3D solid crankshaft. Rigid crankshaft model is mainly used to obtain free forces and torques, and for balancing purposes. Torsionalflexible crankshaft model is used to investi- gate torsional vibrations where each throw is modeled as one rigid part, and springs are used between each throw to represent tor- sional stiffness. Beam crankshaft model is used to represent the torsional and bending stiffness of the crankshaft. Using beam mod- el bending stresses at the webs can be calculated 12. Table 1 Engine specifications Unit 9.0 L engine Bore diameter mm 115 Stroke mm 144 Axial cylinder distance mm 134 Peak firing pressure MPa 19 Rated power at speed kW/rpm 295/2200 Max. torque at speed Nm/rpm 1600/12001700 Main journal/pin diameter mm 95/81 Firing order 1-5-3-6-2-4 Flywheel mass kg 47.84 Flywheel moment of inertia kg mm 2 1.57E+9 Mass of TV damper ring kg 4.94 Mass of TV damper housing kg 6.86 Moment of inertia of the ring kg mm 2 1.27E+5 Moment of inertia of the housing kg mm 2 0.56E+5 Main Bearing #1 Main Bearing #2 Main Bearing #3 Main Bearing #4 Main Bearing #5 Main Bearing #6 Main Bearing #7 Counterweights Fig. 1. 3D solid model of the crankshaft. C3, C4, C5, C6 C1, C2, C7, C8 1, 6 3, 4 2, 5 C1 C2 C3 C4 C5 C6 1 2 Fig. 2. Eight-counterweight arrangement Table 2 Properties of the crank throws Throw 1 Throw 2 Mass (kg) 12.50 9.25 CG position from crank rotation axis (mm) 12.423 31.435 Static unbalance (kg mm) 155.265 290.767 96 Y. Yilmaz, G. Anlas/Advances in Engineering Software 40 (2009) 95104 C7 C8 3 4 5 6 of the 9.0 L engine crankshaft. Throw 3 Throw 4 Throw 5 Throw 6 12.50 12.50 9.28 12.55 11.967 11.966 31.027 11.702 149.734 149.734 287.871 146.856 Elastic 3D solid model of the crankshaft can be obtained using an additional finite element program. The procedure is lengthy and time consuming and usually one ends up with degrees of free- dom in order of millions. To simplify the finite element model, modal superposition technique is used. The elastic deformation of the structure is approximated by linear combination of suitable modes which can be shown as follows: u Uq 1 where q is the vector of modal coordinates andUis the shape func- tion matrix. Table 3 Crankshaft system data Crank radius (mm) 72 Connecting rod length (mm) 239 Mass of complete piston (kg) 3.42 Connecting rod reciprocating mass (kg) 0.92 Reciprocating mass (total per cylinder) (kg) 4.32 Connecting rod rotating mass (kg) 2.01 Y. Yilmaz, G. Anlas/Advances in Engineering An elastic body contains two types of nodes, interface nodes where forces and boundary conditions interact with the structure during multibody system simulation (MSS), and interior nodes. In MSS the position of the elastic body is computed by superposing its rigid body motion and elastic deformation. In ADAMS, this is performed using Component Mode Synthesis” technique based on CraigBampton method 13,14. The component modes contain static and dynamic behavior of the structure. These modes are con- straint modes which are static deformation shapes obtained by giving a unit displacement to each interface degree of freedom (DOF) while keeping all other interface DOFs fixed, and fixed boundary normal modes which are the solution of eigenvalue problem by fixing the entire interface DOFs. The modal transforma- tion between the physical DOF and the CraigBampton modes and their modal coordinates is described by 15 u u B u I C26C27 I0 U C U N C20C21 q C q N C26C27 2 where u B and u I are column vectors and represent boundary DOF and interior DOF, respectively. I, 0 are identity and zero matrices, respectively. U C is the matrix of physical displacements of the inte- rior DOF in the constraint modes. U N is the matrix of physical dis- Fig. 3. Model of the crankshaft system. placements of the interior DOF in the normal modes. q C is the column vector of modal coordinates of the constraint modes. q N is the column vector of modal coordinates of the fixed boundary nor- mal modes. To obtain decoupled set of modes, constrained modes and normal modes are orthogonalized. Elastic 3D solid crankshaft model of the 9.0 L engine is obtained in MSC.Nastran using modal superposition technique. First, 3D so- lid model of the crankshaft that is shown in Fig. 1 is exported to MSC.Nastran and finite element model of the crankshaft, which is characterized by approximately 300,000 ten-node tetrahedral ele- ments and 500,000 nodes is obtained. The modal model of the crankshaft is developed with 32 boundary DOFs associated with 16 interface nodes. Constrained modes obtained from static analy- sis correspond to these DOFs. Flexible crankshaft model is obtained through modal synthesis considering the first 40 fixed boundary normal modes. Therefore flexible crankshaft model is character- ized by a total of 72 DOFs. This model is exported to ADAMS/En- gine and crankshaft system model that is shown in Fig. 3 is obtained. 3D finite element model is run with ADAMS. 4. Forces acting on crankshaft system and balancing Forces in an internal combustion engine may be divided into inertia forces and pressure forces. Inertia forces are further divided into two main categories: rotating inertia forces and reciprocating inertia forces. The rotating inertia force for each cylinder can be written as shown below: F iR;j m R C1 r R C1 x 2 C1C0sinh j j cosh j k3 where m R is the rotating mass that consists of the mass of crank pin, crank webs and mass of rotating portion of the connecting rod; r R is the distance from the crankshaft centre of rotation to the centre of gravity of the rotating mass, x is angular velocity of the crankshaft, and h j is the angular position of each crank throw with respect to Top Dead Centre” (TDC). If there are two counterweights per crank throw, each counterweight force is given by 11 F CWi;j C0m CWi;j C1 r CWi;j C1 x 2 C1C0sinh j c i;j j cosh j c i;j k hi ; i 1;2 j 1;2;.;6 4 where c i,j is the offset angle of counterweight mass from 180C176 oppo- site of crank throw j”. There are two counterweights per throw. i” denotes the counterweight number. The counterweight size that is required to accomplish an assessed balancing rate is U CW K C1U Crank throw m cr-r C1 rC1cosc 2 5 where U CW is the static unbalance of each counterweight, U Crank_throw is the static unbalance of each crank throw, m cr-r is the mass of connecting rod rotating portion, r is the crank radius and K is the balancing rate of the internal couple due to rotating forces. From this formula follows the balancing rate for a given crankshaft and a given counterweight size: K 2 C1 U CW U Crank throw m cr-r C1 rC1cosc 6 For a standard in-line six-cylinder engine crankshaft with three pairs of crank throws disposed at angles of 120C176 that are arranged symmetrical to the crankshaft centre, rotating forces, and first and second order reciprocating forces are naturally balanced. This can be explained by the first and second order vector stars shown in Fig. 4. The six-cylinder crankshaft generates rotating and first Software 40 (2009) 95104 97 and second order reciprocating couples in each crankshaft half that balance each other but which result in internal bending moment. At high speeds, the two equally directed crank throws, 3 and 4 yield a high rotating load on centre main bearing. The rotating inertia force of each cylinder is usually offset at least partially by counterweights placed on the opposite side of each crank. In gen- eral, the counterweights are designed for balancing rates between 50% and 100% of the internal couple. Gas forces in cylinders are acting on piston head, cylinder head and on side walls of the cylinder. These forces are equal to F p;j C0 pD 2 4 C1P cyl;j hC0P cc;j hC138 k; j 1;2;.;6 7 1, 6 2, 5 3, 4 3, 4 1, 6 2, 5 Fig. 4. First and second order vector stars. 0 20 40 60 80 100 120 140 160 180 200 0 90 180 270 360 450 540 630 720 Crank Angle (degree) Pressure (bar) 1000rpm 1200rpm 1350rpm 1675rpm 2000rpm Fig. 5. Gas pressure values at different engine speeds for the 9.0 L engine. Bearing #1 0 25 50 75 100 125 150 0 120 240 360 480 600 720 Crank Angle deg Force kN Rigid Beam 3D solid Fig. 6. Forces acting on main bearing #1 for rigid, beam and 3D solid crankshaft models at 1000 rpm engine speed. Bearing #2 0 25 50 75 100 125 150 175 0 120 240 360 480 600 720 Crank Angle deg Force kN Rigid Beam 3D solid Fig. 7. Forces acting on main bearing #2 for rigid, beam and 3D solid crankshaft models at 1000 rpm engine speed. Bearing #3 0 25 50 75 100 125 150 0 120 240 360 480 600 720 Crank Angle deg Force kN Rigid Beam 3D solid Fig. 8. Forces acting on main bearing #3 for rigid, beam and 3D solid crankshaft models at 1000 rpm engine speed. Bearing #4 0 25 50 75 100 125 150 0 120 240 360 480 600 720 Crank Angle deg Force kN Rigid Beam 3D solid Fig. 9. Forces acting on main bearing #4 for rigid, beam and 3D solid crankshaft models at 1000 rpm engine speed. Bearing #5 125 150 Rigid Bam 3D solid 98 Y. Yilmaz, G. Anlas/Advances in Engineering Software 40 (2009) 95104 0 25 50 75 100 0 120 240 360 480 600 720 Crank Angle deg Force kN Fig. 10. Forces acting on main bearing #5 for rigid, beam and 3D solid crankshaft models at 1000 rpm engine speed. where D is cylinder diameter, P cyl is the gas pressure in the cylinder and P cc is the pressure in the crankcase. The gas forces are transmit- ted to the crankshaft through the piston and connecting rod. Cylin- der pressure curves for the 9.0 L engine studied under full load at different engine speeds are given in Fig. 5. Pressure curves are ob- tained using AVL/Boost engine cycle calculation program which simulates thermodynamic processes in the engine taking into ac- count one dimensional gas dynamics in the intake and exhaust sys- tems 16. 5. Main bearing loads: comparison of crankshaft models Main bearing loads are calculated using ADAMSs rigid, beam and 3D solid crankshaft models and compared. In the rigid model, no vibration effects are considered which can lead to considerable errors if vibration effects have a major role on the system (like in multithrow crankshafts). To consider vibration effects beam crank- shaft model is used and main bearing loads and bending stresses at webs are calculated. Rigid model assumes crankshaft to be stati- cally determinate and reaction force of any given bearing depends on the load exerted on the throws adjacent to that bearing. Beam model assumes the crankshaft to be statically indeterminate and the load exerted on a throw affects all bearings. Analyses are car- ried out at an engine speed range of 10002000 rpm. A more sophisticated 3D solid hybrid model that combines FE with ADAMS is used to check the results obtained by beam model. Maximum main bearing load occurs at bearing number two at Bearing #6 0 25 50 75 100 125 150 0 120 240 360 480 600 720 Crank Angle deg Force kN Rigid Beam 3D solid Fig. 11. Forces acting on main bearing #6 for rigid, beam and 3D solid crankshaft models at 1000 rpm engine speed. Bearing #7 0 25 50 75 100 125 150 0 120 240 360 480 600 720 Crank Angle deg Force kN Rigid Beam 3D solid Fig. 12. Forces acting on main bearing #7 for rigid, beam and 3D solid crankshaft models at 1000 rpm engine speed. Bearing #1 40 50 60 70 80 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) Maximum Bearing K=0% K=50% K=100% Force (kN) Fig. 13. (a) Maximum and (b) average bearing forces at Bearing #2 120 130 140 150 160 K=0% K=50% K=100% Maximum Bearing Force (kN) 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) Fig. 14. (a) Maximum and (b) average bearing forces at Y. Yilmaz, G. Anlas/Advances in Engineering Software 40 (2009) 95104 99 an engine speed of 1000 rpm, therefore results are plotted in Figs. 612 for 1000 rpm only. Rigid crankshaft model overestimates the maximum main bearing load at bearings 1 and 7 with respect to beam and flexible crankshaft models. However it underestimates the maximum main bearing load at other bearings. For example at bearing 2, beam model gives a maximum main bearing load that is 50% more than that of rigid models because the beam model as- sumes the crankshaft to be statically indeterminate and considers Bearing #1 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) 0 5 10 15 20 Average Bearing K=0% K=50% K=100% Force (kN) bearing #1 for 12-counterweight configurations. Bearing #2 20 25 30 35 40 K=0% K=50% K=100% 1000 1200 1400 1600 1800 2000 Average Bearing Force (kN) Crank Angular Velocity (rpm) bearing #2 for 12-counterweight configurations. bending vibrations. Maximum main bearing load difference of beam and 3D solid models is approximately 5%. Main bearing loads for beam and 3D solid crankshaft models are generally in good agreement. In bearings 3, 5 and 6, 3D solid model gives larger bear- ing loads at firing positions of the cylinders that are not adjacent to bearing. Because obtaining elastic 3D solid models for different counterweight configurations is difficult and time consuming, and beam model gives equally valid results, beam model is used Bearing #3 100 110 120 130 140 K=0% K=50% K=100% Bearing #3 20 25 30 35 40 K=0% K=50% K=100% Maximum Bearing Force (kN) 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) Average Bearing Force (kN) Fig. 15. (a) Maximum and (b) average bearing forces at bearing #3 for 12-counterweight configurations. Bearing #4 60 70 80 90 100 110 120 K=0% K=50% K=100% Bearing #4 10 15 20 25 30 35 40 K=0% K=50% K=100% Maximum Bearing Force (kN) 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) Average Bearing Force (kN) Fig. 16. (a) Maximum and (b) average bearing forces at bearing #4 for 12-counterweight configurations. Bearing #6 120 130 140 K=0% K=50% K=100% Bearing #6 35 40 45 50 K=0% K=50% K=100% Bearing #5 100 110 120 130 140 K=0% K=50% K=100% Bearing #5 20 25 30 35 40 K=0% K=50% K=100% Maximum Bearing Force (kN) 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) Average Bearing Force (kN) Fig. 17. (a) Maximum and (b) average bearing forces at bearing #5 for 12-counterweight configurations. 100 Y. Yilmaz, G. Anlas/Advances in Engineering Software 40 (2009) 95104 100 110 Maximum Bearing Force (kN) 1000 1200 1400 1600 1800 2000 Crank Angular Velocity (rpm) Fig. 18. (a) Maximum and (b) average bearing forces at 20 25 30 1000 1200 1400 1600 1800 2000 Average Bearing Force (kN) Crank Angular Velocity (rpm) bearing #6 for 12-counterweight con