中型普通車床主軸箱設(shè)計(jì)【Dmax=350mm Nmin=132r-minN=4KW φ=1.41 Z=8】
中型普通車床主軸箱設(shè)計(jì)【Dmax=350mm Nmin=132r-minN=4KW φ=1.41 Z=8】,Dmax=350mm Nmin=132r-min N=4KW φ=1.41 Z=8,中型普通車床主軸箱設(shè)計(jì)【Dmax=350mm,Nmin=132r-min,N=4KW,φ=1.41,Z=8】,中型
寧XX大學(xué)
課程設(shè)計(jì)(論文)
中型普通車床主軸箱設(shè)計(jì)（題目35）
所在學(xué)院
專 業(yè)
班 級(jí)
姓 名
學(xué) 號(hào)
指導(dǎo)老師
年 月 日
5
摘 要
設(shè)計(jì)機(jī)床主軸箱時(shí)首先利用傳動(dòng)系統(tǒng)設(shè)計(jì)方法求出理想解和多個(gè)合理解。根據(jù)數(shù)控機(jī)床主傳動(dòng)系統(tǒng)及主軸功率與轉(zhuǎn)矩特性要求，分析了機(jī)電關(guān)聯(lián)分級(jí)調(diào)速主傳動(dòng)系統(tǒng)的設(shè)計(jì)原理和方法。從主傳動(dòng)系統(tǒng)結(jié)構(gòu)網(wǎng)入手，確定最佳機(jī)床主軸功率與轉(zhuǎn)矩特性匹配方案，計(jì)算和校核相關(guān)運(yùn)動(dòng)參數(shù)和動(dòng)力參數(shù)。本說明書著重研究機(jī)床主傳動(dòng)系統(tǒng)的設(shè)計(jì)步驟和設(shè)計(jì)方法，根據(jù)已確定的運(yùn)動(dòng)參數(shù)以變速箱展開圖的總中心距最小為目標(biāo)，擬定變速系統(tǒng)的變速方案，以獲得最優(yōu)方案以及較高的設(shè)計(jì)效率。在機(jī)床主傳動(dòng)系統(tǒng)中，為減少齒輪數(shù)目，簡(jiǎn)化結(jié)構(gòu)，縮短軸向尺寸，用齒輪齒數(shù)的設(shè)計(jì)方法是試算，湊算法，計(jì)算麻煩且不易找出合理的設(shè)計(jì)方案。本文通過對(duì)主傳動(dòng)系統(tǒng)中三聯(lián)滑移齒輪傳動(dòng)特點(diǎn)的分析與研究，繪制零件工作圖與主軸箱展開圖及剖視圖。
關(guān)鍵詞 分級(jí)變速；傳動(dòng)系統(tǒng)設(shè)計(jì),傳動(dòng)副，結(jié)構(gòu)網(wǎng)，結(jié)構(gòu)式，齒輪模數(shù)，傳動(dòng)比
目 錄
摘 要 2
目 錄 4
第1章 緒論 6
1.1 課程設(shè)計(jì)的目的 6
1.2課程設(shè)計(jì)的內(nèi)容 6
1.2.1 理論分析與設(shè)計(jì)計(jì)算 6
1.2.2 圖樣技術(shù)設(shè)計(jì) 6
1.2.3編制技術(shù)文件 6
1.3 課程設(shè)計(jì)題目、主要技術(shù)參數(shù)和技術(shù)要求 6
第2章 車床參數(shù)的擬定 8
2.1車床主參數(shù)和基本參數(shù) 8
2.2擬定參數(shù)的步驟和方法 8
2.2.1 極限切削速度Vmax、Vmin 8
2.2.2 主軸的極限轉(zhuǎn)速 8
2.2.3 主電機(jī)功率——?jiǎng)恿?shù)的確定 9
2.2.4確定結(jié)構(gòu)式 9
2.2.5確定結(jié)構(gòu)網(wǎng) 9
2.2.6繪制轉(zhuǎn)速圖和傳動(dòng)系統(tǒng)圖 10
2.3 確定各變速組此論傳動(dòng)副齒數(shù) 10
2.4 核算主軸轉(zhuǎn)速誤差 11
第3章 傳動(dòng)件的計(jì)算 12
3.1 帶傳動(dòng)設(shè)計(jì) 12
3.2選擇帶型 13
3.3確定帶輪的基準(zhǔn)直徑并驗(yàn)證帶速 13
3.4確定中心距離、帶的基準(zhǔn)長(zhǎng)度并驗(yàn)算小輪包角 14
3.5確定帶的根數(shù)z 15
3.6確定帶輪的結(jié)構(gòu)和尺寸 15
3.7確定帶的張緊裝置 15
3.8計(jì)算壓軸力 15
3.9 計(jì)算轉(zhuǎn)速的計(jì)算 17
3.10 齒輪模數(shù)計(jì)算及驗(yàn)算 18
3.11 傳動(dòng)軸最小軸徑的初定 23
3.12 主軸合理跨距的計(jì)算 24
第4章 主要零部件的選擇 25
4.1 軸承的選擇 25
4.2 鍵的規(guī)格 25
4.3變速操縱機(jī)構(gòu)的選擇 25
第5章 校核 26
5.1主軸合理跨距的計(jì)算 26
5.2 軸承壽命校核 27
第6章 多片式摩擦離合器的計(jì)算 28
6.1 摩擦離合器的選擇與驗(yàn)算 28
6.1.1按扭矩選擇 28
6.1.2外摩擦片的內(nèi)徑d 28
6.1.3選擇摩擦片尺寸 28
6.1.4計(jì)算摩擦面的對(duì)數(shù)Z 28
6.1.5摩擦片軸向壓力 29
第7章 摩擦離合器(多片式)的計(jì)算 29
結(jié) 論 32
參考文獻(xiàn) 33
第1章 緒論
1.1 課程設(shè)計(jì)的目的
課程設(shè)計(jì)是在學(xué)完本課程后，進(jìn)行一次學(xué)習(xí)設(shè)計(jì)的綜合性練習(xí)。通過課程設(shè)計(jì)，使學(xué)生能夠運(yùn)用所學(xué)過的基礎(chǔ)課、技術(shù)基礎(chǔ)課和專業(yè)課的有關(guān)理論知識(shí)，及生產(chǎn)實(shí)習(xí)等實(shí)踐技能，達(dá)到鞏固、加深和拓展所學(xué)知識(shí)的目的。通過課程設(shè)計(jì)，分析比較機(jī)械系統(tǒng)中的某些典型機(jī)構(gòu)，進(jìn)行選擇和改進(jìn)；結(jié)合結(jié)構(gòu)設(shè)計(jì)，進(jìn)行設(shè)計(jì)計(jì)算并編寫技術(shù)文件；完成系統(tǒng)主傳動(dòng)設(shè)計(jì)，達(dá)到學(xué)習(xí)設(shè)計(jì)步驟和方法的目的。通過設(shè)計(jì)，掌握查閱相關(guān)工程設(shè)計(jì)手冊(cè)、設(shè)計(jì)標(biāo)準(zhǔn)和資料的方法，達(dá)到積累設(shè)計(jì)知識(shí)和設(shè)計(jì)技巧，提高學(xué)生設(shè)計(jì)能力的目的。通過設(shè)計(jì)，使學(xué)生獲得機(jī)械系統(tǒng)基本設(shè)計(jì)技能的訓(xùn)練，提高分析和解決工程技術(shù)問題的能力，并為進(jìn)行主軸箱設(shè)計(jì)創(chuàng)造一定的條件。
1.2課程設(shè)計(jì)的內(nèi)容
課程設(shè)計(jì)內(nèi)容由理論分析與設(shè)計(jì)計(jì)算、圖樣技術(shù)設(shè)計(jì)和技術(shù)文件編制三部分組成。
1.2.1 理論分析與設(shè)計(jì)計(jì)算
（1）機(jī)械系統(tǒng)的方案設(shè)計(jì)。設(shè)計(jì)方案的分析，最佳功能原理方案的確定。
（2）根據(jù)總體設(shè)計(jì)參數(shù)，進(jìn)行傳動(dòng)系統(tǒng)運(yùn)動(dòng)設(shè)計(jì)和計(jì)算。
（3）根據(jù)設(shè)計(jì)方案和零部件選擇情況，進(jìn)行有關(guān)動(dòng)力計(jì)算和校核。
1.2.2 圖樣技術(shù)設(shè)計(jì)
（1）選擇系統(tǒng)中的主要機(jī)件。
（2）工程技術(shù)圖樣的設(shè)計(jì)與繪制。
1.2.3編制技術(shù)文件
（1）對(duì)于課程設(shè)計(jì)內(nèi)容進(jìn)行自我經(jīng)濟(jì)技術(shù)評(píng)價(jià)。
（2）編制設(shè)計(jì)計(jì)算說明書。
1.3 課程設(shè)計(jì)題目、主要技術(shù)參數(shù)和技術(shù)要求
題目：中型普通車床主軸箱設(shè)計(jì)
題目35車床的主參數(shù)（規(guī)格尺寸）和基本參數(shù)如下：
工件最大回轉(zhuǎn)直徑
D(mm)
正轉(zhuǎn)最低轉(zhuǎn)速
nmin( )
電機(jī)功率
N（kw）
公比
轉(zhuǎn)速級(jí)數(shù)Z
350
132
4
1.41
8
33
第2章 車床參數(shù)的擬定
2.1車床主參數(shù)和基本參數(shù)
車床的主參數(shù)（規(guī)格尺寸）和基本參數(shù)如下：
工件最大回轉(zhuǎn)直徑
D(mm)
正轉(zhuǎn)最低轉(zhuǎn)速
nmin( )
電機(jī)功率
N（kw）
公比
轉(zhuǎn)速級(jí)數(shù)Z
350
132
4
1.41
8
2.2擬定參數(shù)的步驟和方法
2.2.1 極限切削速度Vmax、Vmin
根據(jù)典型的和可能的工藝選取極限切削速度要考慮：
允許的切速極限參考值如下：
表 1.1
加 工 條 件
Vmax(m/min)
Vmin(m/min)
硬質(zhì)合金刀具粗加工鑄鐵工件
30～50
硬質(zhì)合金刀具半精或精加工碳鋼工件
150～300
螺紋加工和鉸孔
3～8
2.2.2 主軸的極限轉(zhuǎn)速
計(jì)算車床主軸極限轉(zhuǎn)速時(shí)的加工直徑，則主軸極限轉(zhuǎn)速應(yīng)為
結(jié)合題目條件，取標(biāo)準(zhǔn)數(shù)列數(shù)值，
=132r/min
取
依據(jù)題目要求選級(jí)數(shù)Z=8， =1.41=1.065考慮到設(shè)計(jì)的結(jié)構(gòu)復(fù)雜程度要適中，故采用常規(guī)的擴(kuò)大傳動(dòng)。各級(jí)轉(zhuǎn)速數(shù)列可直接從標(biāo)準(zhǔn)的數(shù)列表中查出，按標(biāo)準(zhǔn)轉(zhuǎn)速數(shù)列為：
132，190，265，375，530，750，1060，1500
2.2.3 主電機(jī)功率——?jiǎng)恿?shù)的確定
合理地確定電機(jī)功率N，使機(jī)床既能充分發(fā)揮其性能，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。
根據(jù)題設(shè)條件電機(jī)功率為4KW
可選取電機(jī)為：Y112M-4額定功率為4KW，滿載轉(zhuǎn)速為1440r/min.
2.2.4確定結(jié)構(gòu)式
已知Z=x3b
a、b為正整數(shù)，即Z應(yīng)可以分解為2和3的因子，以便用2、3聯(lián)滑移齒輪實(shí)現(xiàn)變速。
取Z=8級(jí) 則Z=22
對(duì)于Z=8可分解為：Z=21×22×24。
綜合上述可得：主傳動(dòng)部件的運(yùn)動(dòng)參數(shù)
=140 Z=8 =1.41
2.2.5確定結(jié)構(gòu)網(wǎng)
根據(jù)“前多后少” , “先降后升” , 前密后疏,結(jié)構(gòu)緊湊的原則,選取傳動(dòng)方案 Z=21×22×24,易知第二擴(kuò)大組的變速范圍r=φ(P3-1)x=1.414=3.95〈8 滿足要求，其結(jié)構(gòu)網(wǎng)如圖2-1。
圖2-1結(jié)構(gòu)網(wǎng) Z=21×22×24
2.2.6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)
2.3 確定各變速組此論傳動(dòng)副齒數(shù)
(1)Sz100-120,中型機(jī)床Sz=70-100
(2)直齒圓柱齒輪Zmin18-20,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≥18~20,齒數(shù)和Sz≤100～120，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
1:2
1:1
1:1.41
1.41：1
1：2.84
代號(hào)
Z
Z
Z
Z
Z
Z
Z
Z’
Z5
Z5’
Z
Z
齒數(shù)
48
48
32
64
42
42
35
49
55
39
25
69
2.4 核算主軸轉(zhuǎn)速誤差
實(shí)際傳動(dòng)比所造成的主軸轉(zhuǎn)速誤差，一般不應(yīng)超過±10(-1)％,即
〈10(-1)％=4.1％
各級(jí)轉(zhuǎn)速誤差
n
1500
1060
750
530
375
265
190
132
n`
1538.5
1098.9
769.3
548.6
384.5
274.3
198.8
136.5
誤差
2.5％
3.7％
2.5％
3.7％
2.5％
3.7％
2.5％
3.7％
轉(zhuǎn)速誤差小于4.1％，因此不需要修改齒數(shù)。
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=4kW,轉(zhuǎn)速n1=1440r/min,n2=1060r/min
3.1.1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＝4.4kW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=132mm
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。
3.8計(jì)算壓軸力
由《機(jī)械設(shè)計(jì)》P303表13－12查得，A型帶的初拉力F0＝130.59N，上面已得到=153.36o，z=6，則
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）
3.9 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=234r/min，
取265 r/min。
(2). 傳動(dòng)軸的計(jì)算轉(zhuǎn)速
軸3=750 r/min，軸2=1060 r/min，軸1=1060r/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
1060
1060
750
（3） 確定齒輪副的計(jì)算轉(zhuǎn)速。
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
1060
1060
750
750
265
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：=15～20
——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
6級(jí)材料的直齒輪材料選；20熱處理S-C59
=16338mm
=16338=2.6 mm
=275mm
=275 =2.2mm
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
2.5
2.5
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`
齒數(shù)
48
48
32
64
分度圓直徑
120
120
80
160
齒頂圓直徑
125
125
85
165
齒根圓直徑
113.75
113.75
73.75
153.75
齒寬
20
20
20
20
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～286HB，平均取260HB，大齒輪用45鋼，調(diào)質(zhì)處理，硬度229HB～286HB，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）. =265（r/min）;
m-----初算的齒輪模數(shù)（mm）, m=2.5（mm）;
B----齒寬（mm）;B=20（mm）;
z----小齒輪齒數(shù)；z=32;
u----小齒輪齒數(shù)與大齒輪齒數(shù)之比,u=2;
-----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
Z3
Z3`
Z4
Z4`
齒數(shù)
42
42
35
49
分度圓直徑
105
105
87.5
122.5
齒頂圓直徑
110
110
92.5
127.5
齒根圓直徑
98.75
98.75
81.25
116.25
齒寬
20
20
20
20
（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)
55
39
25
69
分度圓直徑
165
117
75
207
齒頂圓直徑
171
123
91
213
齒根圓直徑
157.5
109.5
67.5
199.5
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～286HB，平均取260HB，大齒輪用45鋼，調(diào)質(zhì)處理，硬度229HB～286HB，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
由【5】式6，傳動(dòng)軸直徑按扭轉(zhuǎn)剛度用下式計(jì)算：
d=1.64（mm）
或 d=91（mm）
式中 d---傳動(dòng)軸直徑（mm）
Tn---該軸傳遞的額定扭矩（N*mm） T=9550000；
N----該軸傳遞的功率（KW）
----該軸的計(jì)算轉(zhuǎn)速
---該軸每米長(zhǎng)度的允許扭轉(zhuǎn)角，==。
各軸最小軸徑如表3-3。
表3-3 最小軸徑
軸 號(hào)
Ⅰ 軸
Ⅱ 軸
最小軸徑mm
35
40
3.12 主軸合理跨距的計(jì)算
由于電動(dòng)機(jī)功率P=5.5kw，根據(jù)【1】表3.20，前軸徑應(yīng)為60~90mm。初步選取d1=80mm。后軸徑的d2=（0.7~0.9）d1，取d2=60mm。根據(jù)設(shè)計(jì)方案，前軸承為NN3016K型，后軸承為圓錐滾子軸承。定懸伸量a=120mm，主軸孔徑為30mm。
軸承剛度，主軸最大輸出轉(zhuǎn)矩T=9550=9550×=424.44N.m
設(shè)該機(jī)床為車床的最大加工直徑為350mm。床身上最常用的最大加工直徑，即經(jīng)濟(jì)加工直徑約為最大回轉(zhuǎn)直徑的50%，這里取55%，即180mm，故半徑為0.09m；
切削力（沿y軸） Fc==4716N
背向力（沿x軸） Fp=0.5 Fc=2358N
總作用力 F==5272.65N
此力作用于工件上，主軸端受力為F=5272.65N。
先假設(shè)l/a=2，l=3a=240mm。前后支承反力RA和RB分別為
RA=F×=5272.65×=7908.97N
RB=F×=5272.65×=2636.325N
根據(jù) 文獻(xiàn)【1】式3.7 得：Kr=3.39得前支承的剛度：KA= 1689.69 N/ ；KB= 785.57 N/；==2.15
主軸的當(dāng)量外徑de=(80+60)/2=70mm，故慣性矩為
I==113.8×10-8m4
η===0.14
查【1】圖3-38 得 =2.0，與原假設(shè)接近，所以最佳跨距=120×2.0=240mm
合理跨距為（0.75-1.5），取合理跨距l(xiāng)=360mm。
根據(jù)結(jié)構(gòu)的需要，主軸的實(shí)際跨距大于合理跨距，因此需要采取措施
增加主軸的剛度，增大軸徑：前軸徑D=100mm，后軸徑d=80mm。前軸承
采用雙列圓柱滾子軸承，后支承采用背對(duì)背安裝的角接觸球軸承。
第4章 主要零部件的選擇
4.1 軸承的選擇
I軸：與帶輪靠近段安裝雙列角接觸球軸承代號(hào)7007C 另一安裝深溝球軸承6012
II軸：對(duì)稱布置深溝球軸承6009
III軸：后端安裝雙列角接觸球軸承代號(hào)7015C
另一安裝端角接觸球軸承代號(hào)7010C
中間布置角接觸球軸承代號(hào)7012C
4.2 鍵的規(guī)格
I軸安裝帶輪處選擇普通平鍵規(guī)格：
BXL=10X56
II軸選擇花鍵規(guī)格：
N× d×D×B =8X36X40X7
III軸選擇鍵規(guī)格：
BXL=14X90
4.3變速操縱機(jī)構(gòu)的選擇
選用左右擺動(dòng)的操縱桿使其通過桿的推力來控制II軸上的三聯(lián)滑移齒輪和二聯(lián)滑移齒輪。
第5章 校核
5.1主軸合理跨距的計(jì)算
設(shè)機(jī)床最大加工回轉(zhuǎn)直徑為?350mm,電動(dòng)機(jī)功率P=5.5kw，,主軸計(jì)算轉(zhuǎn)速為265r/min。
已選定的前后軸徑為:定懸伸量a=85mm。
軸承剛度，主軸最大輸出轉(zhuǎn)矩:
TIII ＝
設(shè)該車床的最大加工直徑為350mm。床身上最常用的最大加工直徑，即經(jīng)濟(jì)加工直徑約為最大回轉(zhuǎn)直徑的50%，這里取55%，即180mm，故半徑為0.09m；
切削力（沿y軸） Fc=250.346/0.09=2781N
背向力（沿x軸） Fp=0.5 Fc=1390N
總作用力 F==3109N
此力作用于工件上，主軸端受力為F=3109N。
先假設(shè)l/a=2，l=3a=255mm。前后支承反力RA和RB分別為
RA=F×=3109×N
RB=F×=3109×N
根據(jù)《主軸箱設(shè)計(jì)》得：=3.39得前支承的剛度：KA= 1376.69 N/ ；KB= 713.73 N/；==1.93
主軸的當(dāng)量外徑de=(85+65)/2=75mm，故慣性矩為
I==1.55×10-6m4
η===0.38
查《主軸箱設(shè)計(jì)》圖 得 =2.5，與原假設(shè)接近，所以最佳跨距=85×2.5=212.5mm
合理跨距為（0.75-1.5），取合理跨距l(xiāng)=250mm。
根據(jù)結(jié)構(gòu)的需要，主軸的實(shí)際跨距大于合理跨距，因此需要采取措施
增加主軸的剛度，增大軸徑：前軸徑D=85mm，后軸徑d=55mm。后支承采用背對(duì)背安裝的角接觸球軸承。
5.2 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+YFa
X=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。
第6章 多片式摩擦離合器的計(jì)算
6.1 摩擦離合器的選擇與驗(yàn)算
6.1.1按扭矩選擇
K=Kx9550 Nm
式中:
—離合器的額定靜力矩(Kgm) K—安全系數(shù)
—運(yùn)轉(zhuǎn)時(shí)的最大負(fù)載力矩
查《機(jī)械設(shè)計(jì)手冊(cè)》表，取K=2 =0.96
則K= =118.8 Nm
6.1.2外摩擦片的內(nèi)徑d
根據(jù)結(jié)構(gòu)需要采用軸裝式摩擦片，摩擦片的內(nèi)徑d應(yīng)比安裝在軸的軸徑大2～6mm,取d=35mm
6.1.3選擇摩擦片尺寸
尺寸如下表6.1所示
表6.1
片數(shù)
靜力矩
d
D
D1
B
b
9
60
35
90
98
30
10
6.1.4計(jì)算摩擦面的對(duì)數(shù)Z
式中：f-----摩擦片間的摩擦系數(shù)； [p]----許用壓強(qiáng)MPa；
D------摩擦片內(nèi)片外徑 mm； d-------摩擦片外片內(nèi)徑 mm；
----速度修正系數(shù)； -----接合面數(shù)修正系數(shù)；
-----接個(gè)次數(shù)修正系數(shù)； K------安全系數(shù)。
分別查表
～1.2 mm =35mm
1.0
=10
6.1.5摩擦片軸向壓力
計(jì)算軸向壓力Q
=3.14×1.0××
=5073N
第7章 摩擦離合器(多片式)的計(jì)算
設(shè)計(jì)多片式摩擦離合器時(shí)，首先根據(jù)機(jī)床結(jié)構(gòu)確定離合器的尺寸，如為軸裝式時(shí)，外摩擦片的內(nèi)徑d應(yīng)比花鍵軸大2～6mm，內(nèi)摩擦片的外徑D的確定，直接影響離合器的徑向和軸向尺寸，甚至影響主軸箱內(nèi)部結(jié)構(gòu)布局，故應(yīng)合理選擇。
摩擦片對(duì)數(shù)可按下式計(jì)算
Z≥2MnK/fb[p]
式中 Mn——摩擦離合器所傳遞的扭矩（N·mm）；
Mn＝955×η/＝955××11×0.98/800＝1.28×（N·mm）;
Nd——電動(dòng)機(jī)的額定功率（kW）；
——安裝離合器的傳動(dòng)軸的計(jì)算轉(zhuǎn)速（r/min）；
η——從電動(dòng)機(jī)到離合器軸的傳動(dòng)效率；
K——安全系數(shù)，一般取1.3～1.5；
f——摩擦片間的摩擦系數(shù)，由于磨擦片為淬火鋼，查《機(jī)床設(shè)計(jì)指導(dǎo)》表2-15，取f=0.08；
——摩擦片的平均直徑（mm）;
=（D+d）/2＝67mm;
b——內(nèi)外摩擦片的接觸寬度（mm）；
b=（D-d）/2=23mm；
——摩擦片的許用壓強(qiáng)（N/）；
＝＝1.1×1.00×1.00×0.76＝0.836
——基本許用壓強(qiáng)（MPa），查《機(jī)床設(shè)計(jì)指導(dǎo)》表2-15，取1.1；
——速度修正系數(shù)
＝n/6×=2.5（m/s）
根據(jù)平均圓周速度查《機(jī)床設(shè)計(jì)指導(dǎo)》表2-16，取1.00；
——接合次數(shù)修正系數(shù)，查《機(jī)床設(shè)計(jì)指導(dǎo)》表2-17，取1.00；
——摩擦結(jié)合面數(shù)修正系數(shù)，查《機(jī)床設(shè)計(jì)指導(dǎo)》表2-18，取0.76。
所以 Z≥2MnK/fb[p]＝2×1.28××1.4/（3.14×0.08××23×0.836＝11 臥式車床反向離合器所傳遞的扭矩可按空載功率損耗確定，一般取
＝0.4＝0.4×11＝4.4
最后確定摩擦離合器的軸向壓緊力Q，可按下式計(jì)算：
Q=b(N)＝1.1×3.14××23×1.00＝3.57×
式中各符號(hào)意義同前述。
摩擦片的厚度一般取1、1.5、1.75、2（mm），內(nèi)外層分離時(shí)的最大間隙為0.2～0.4（mm）,摩擦片的材料應(yīng)具有較高的耐磨性、摩擦系數(shù)大、耐高溫、抗膠合性好等特點(diǎn)，常用10或15鋼，表面滲碳0.3～0.5（mm）,淬火硬度達(dá)HRC52～62。
結(jié) 論
經(jīng)過課程設(shè)計(jì)，使我和同伴對(duì)主軸箱設(shè)計(jì)這門課當(dāng)中許多原理公式有了進(jìn)一步的了解，并且對(duì)設(shè)計(jì)工作有了更深入的認(rèn)識(shí)。懂得了理論和實(shí)踐同等重要的道理。理論能指導(dǎo)實(shí)踐，使你能事半功倍，實(shí)踐能上升成為理論，為以后的設(shè)計(jì)打下基礎(chǔ)。? 從校門走出后,一定要重視實(shí)踐經(jīng)驗(yàn)的積累,要多學(xué)多問。把學(xué)校學(xué)習(xí)的專業(yè)知識(shí)綜合的應(yīng)用起來，這非常重要。體會(huì)到把技術(shù)搞好就必須安心的學(xué)習(xí)，虛心向別人請(qǐng)教，耐心的對(duì)待每一個(gè)問題，不放過任何一個(gè)自己遇到的問題，要善于發(fā)現(xiàn)問題。
在設(shè)計(jì)過程中，我們得到了老師們的精心指導(dǎo)和幫助，在此表示衷心的感謝！由于我們的經(jīng)驗(yàn)尚淺，知識(shí)把握不熟練，設(shè)計(jì)中定有許多地方處理不夠妥當(dāng)，有些部分甚至可能存在錯(cuò)誤，希望老師多提寶貴意見。
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》；
湖南科技大學(xué)本科生畢業(yè)設(shè)計(jì)（論文）
Virtual Design and Optimization of Machine Tool Spindles
Y. Altintas(l), Y. Cao
Manufacturing Automation Laboratory, Department of Mechanical Engineering University of British Columbia, Vancouver, Canada
h t t p : / / w .mech. ubc.ca/-ma1
Abstract
An integrated digital model of spindle, tool holder, tool and cutting process is presented. The spindle is modeled using an in-house developed Finite Element system. The preload on the bearings and the influence of gyroscopic and centrifugal forces from all rotating parts due to speed are considered. The bearing stiffness, mode shapes, Frequency Response Function at any point on the spindle can be predicted. The static and dynamic deflections along the spindle shaft as well as contact forces on the bearings can be predicted with simulated cutting forces before physically building and testing the spindles. The spacing of the bearings are optimized to achieve either maximum dynamics stiffness or maximum chatter free depth of cut at the desired speed region for a given cutter geometry and work-piece material. It is possible to add constraints to model mounting of the spindle on the machine tool, as well as defining local springs and damping elements at any nodal point on the spindle. The model is verified experimentally.
Keywords:
Spindle, Cutting, Vibration
1 INTRODUCTION
High-speed machining is widely used in industry due to increased manufacturing efficiency. However, high speed spindles have smaller shaft diameter and bearings which lead to chatter unless the spindle is designed to operate at the desired cutting conditions. Chatter leads to poor surface finish and overloads the bearings which shorten the spindle life [I] . The dimensions of the spindle shaft, and the stiffness, preload, and spacing of the bearings, tool geometry and holder, and work material affect the overall performance of the spindle during machining. The aim of the modeling study is to simulate the performance of the spindle and optimize its dimensions to achieve maximum dynamic stiffness and increased material removal rate.
Angular contact ball bearings are most commonly used in high-speed spindles due to their low-friction properties and ability to withstand external loads in both axial and radial directions [2]. The stiffness of the bearings is dependent on the contact angle, which in turn depends on the speed, contact loads between the balls and rings. Jones developed a general theory for the load-deflection analysis of bearings, including centrifugal and gyroscopic loading of the rolling elements under high-speed operation [3] which is used in this paper. The rotating shafts and stationary housing have been commonly modeled by Finite Element techniques [4, 5]. Most past research did not consider the nonlinear behaviour of the bearing stiffness. For example, Nelson [6] employed Timoshenko beam theory to establish the system matrices for analyzing the dynamics of rotor systems with the effects of rotary inertia, gyroscopic moments, shear deformation, and axial load, but the bearings are modeled as linear springs. As presented by Abele [7], the structural dynamics of spindles change at high speeds, which affect the location and shape of stability pockets [8].
This paper presents a general Finite Element model which can predict the stiffness of the bearings, contact forces on bearing balls, natural frequencies and mode shapes, frequency response functions and time history response under cutting loads. The model includes the bearing preload, rotating effects from both bearings and the spindle shaft. Henceforth, the paper is organized as follows.The nonlinear finite element model of the spindle shaft and bearings, which considers the bearing preload, gyroscopic and centrifugal speed affects, are presented. The Model of a spindle is experimentally verified in section 3. A Bearing spacing optimization method to obtain either maximum dynamics stiffness or maximum chatter free depth of cut for multiple flute cutters is presented in section 4.The paper is concluded with a summary of contributions.
2 FINITE ELEMENT MODEL OF SPINDLE SYSTEMS
Figure 1 shows the experimental spindle instrumented with non-contact displacement sensors along its shaft. The spindle has a standard CAT 40 tool holder interface with maximum 15000 rev/min speed, and driven by a 15kW motor connected to the shaft with a pulley-belt system.
Figure 1: Spindle system
Figure 2: Finite element model for spindle bearing system
The spindle is modeled by an in house developed Finite Element system dedicated for spindles as shown in Figure2. The Timoshenko beam theory is used to model the spindle shaft and housing. The black dots represent nodes, where each node has three translational and two rotational degrees of freedom. The pulley is modeled as a rigid disk. The spindle has two front bearings (BI and B2) in tandem and three rear bearings (B3, B4 and B5) in tandem. The preload is applied hydraulically on the outer ring defined as node A3, which can move along the spindle housing with nodes A4 and A5. The forces are transferred to inner rings B3 to B5 through bearing balls, then to the spindle shaft through inner ring B5. The forces are transmitted to front bearings by inner ring B I , which is also fixed to the spindle shaft, then to the housing by outer ring A2, which is fixed to the housing. The whole spindle is self-balanced in the axial direction under the preload. An initial preload is applied during the assembly and can be adjusted through the hydraulic unit. The tool is assumed to be rigidly connected to the tool holder which is fixed to the spindle shaft through springs with stiffness in both translation and rotation. Depending on the rigidity of the machine tool, the spindle housing can be rigidly fixed or elastically supported on the spindle head. The inner and outer rings are related by nonlinear equations from which bearing stiffness is obtained by solving equations of the system.
2.1 Equations of motion for the spindle shaft with rotating effects
The following discrete equations in matrix forms for the beam can be obtained using the finite element method:
where [M] is the mass matrix, [M]c is the mass matrix used for computing the centrifugal forces,[G]is the gyroscopic matrix which is skew-symmetric, [K] is the stiffness matrix, [K]P is the stiffness matrix due to the axial force P ,is the spindle speed, {q} is the displacement vector and {F} is the force vector that includes distributed and concentrated forces. The damping matrix is not included here and is estimated from experimentally identified modal damping ratios.
2.2 Nonlinear bearing model
The Hertzian contact theory is used to predict the bearing contact force and elastic ball displacements.
Figure 3: Bearing mode
The force acting on the bearing ring is:
where , and , are contact displacements between bearing balls and rings; θi, and θo are bearing contact angles;represent the displacement vectors for the nodes on the spindle shaft, inner ring, outer ring and spindle housing, respectively; are functions of ,respectively, depending on the configuration of bearing rings; Qi, and Q0, are contact forces; Fc, and Mg, are centrifugal force and gyroscopic moment depending on the spindle speed .The derivative of force with respect to the displacement is the bearing stiffness matrix:
where KI, and KO are 5 by 5 matrices. The bearing stiffness matrix depends on the displacements which are in turn affected by the stiffness of bearings, hence the system dynamics is nonlinear. By assembling all matrices of spindle shaft/housing, disk and bearings, the following general non-linear dynamic equations for the spindlebearing system can be obtained:
where [M] is the total mass matrix; [C] is the equivalent damping matrix including gyroscopic matrix; {F(t)} is the external force and {R(x)} is the internal force of the system which depends on the displacement {x}. The Newton-Raphson method is used to solve Eq.(4).
3 EXPERIMENTAL VERIFICATION
The nonlinear Finite Element model of the spindles is experimentally verified using an instrumented spindle. Arrays of non-contact displacement sensors are installed in the spindle housing in two radial directions along the shaft, and two axial displacement sensors are mounted close to the spindle nose. First, the spindle is hung using elastic strings as a free-free system as shown in Figure 4. The frequency response functions under different preloads are measured by performing the impact modal tests.
Figure 4: Experimental setup.
3.1 Frequency response function (FRF)
An impact force which is measured from a real impact blow test is applied at the spindle nose in the radial direction while the bearings are preloaded with a 500N force which changes the bearing contact angle as well as the bearing stiffness. Experimentally identified modal damping ratios 4% and 3% are used for the two dominate modes (506 Hz, 2685 Hz) respectively, and 3% is used for the rest of the modes. The FRF at the spindle nose is measured and also predicted by applying the same measured impact force to the nonlinear Finite Element model presented here. The FRF is calculated by using Fourier transforms of the simulated acceleration and input force. The measured and predicted FRF are shown in Figure 5, which is in good agreement. The two modes are most dominant at the spindle nose, which influence the machining stability most.The proposed model is able to predict the influence of preload accurately, which is quite important in designing and operating the spindle shafts at chatter vibration free spindle speeds.
Figure 5: FRF in the radial direction at the spindle nose
3.2 Effects of preload and speed on spindle dynamics
The bearing stiffness increases with the increasing preload, but decreases as the spindle speed increases. Figure 6 shows the relation among radial bearing stiffness, preload and spindle speeds for bearing number 1. The speed effects are more obvious at lower preloads. Since the bearing stiffness is difficult to measure experimentally, the validity of the mathematical model is measured from the accuracy of FRF prediction which agrees quite well here with measurements. In general, the natural frequencies of all modes increase with the preload due to increased bearing stiffness, but decrease with the spindle speed due to decreasing stiffness caused by centrifugal forces. The lower modes are most affected by the spindle speed. A sample relationship between the speed, preload and the first natural frequency is shown in Figure 7.
Figure 7: Natural frequency vs preload and spindle speed
3.3 Prediction of FRF with the tool
The tool-spindle connection is the main source of flexibility in practice, and it is also difficult to model due to unknown contact stiffness and damping at the tool holder joints [9]. As an example, two scenarios are tested: The elastic tool is rigidly connected, or connected via distributed springs at the spindle taper. The end mill has a diameter of 19.05 mm with a stick out of 55 mm, and is attached to the tool holder with a mechanical collet. The FRF at the tool tip for two interface connections is shown in Figure 8. The first mode is less affected, but the rigid tool connection leads to a higher second natural frequency than the spring connection. The first two modes match experimental results better with the spring connection since they are from the whole spindle system. However, the added spring to the tool interface brings a third mode which is not visible from the experiments. The results indicate that the tool-spindle interface mechanics and dynamics require more research since it has a strong influence on the dynamics of spindles during high speed machining.
Figure 8: FRF with the tool in radial direction at the tool tip.
3.4 Bearing Force Prediction under Cutting Loads
The developed finite element system permits virtual cutting with the spindle. The milling forces whose peak value is about 1000N have been simulated for a 4 fluted end mill cutting AL7050 and are applied to the tool tip in three directions. A preload of 1500 N is first applied, followed by the cutting forces acting on the end mill after transient response due to the preload diminishes. The corresponding changes in the bearing stiffness as well as the contact forces experienced by the bearings are shown in Figures 9 and 10. The front bearings are most affected by the cutting forces which are periodic at tooth passing intervals. Since the axial force is opposite to the preload force when a right handed helix angle is used on the end mill, the front bearing stiffness decreases and the bearing contact forces increase under the cutting periodic load. If the axial force is larger than the bearing preload, the bearing stiffness can be lost momentarily. Due to periodicity of milling forces at tooth passing frequency, the bearing stiffness and contact forces change, which is a major nonlinearity in analyzing the dynamic behaviour of spindles during milling.
Figure 9: Bearing stiffness.
Figure 10: Contact forces on bearings.
4 SPINDLE DESIGN OPTIMIZATION
The spindles should be designed either to achieve maximum dynamic stiffness at all speeds for general operation, or remove maximum axial depth of cut at the specified speed with a designated cutter for a specific machining application. Although both criteria are implemented, the objective of cutting maximum material at the desired speed is presented here. The spindle modes are automatically tuned in such a way that chatter free pockets of stability is created at the desired spindle speed and depth of cut by optimizing the locations of bearings and the integral motor. The objective function is defined as follows:
where Wi and(aclim)i are the weight and critical depth of cut for the cutter respectively, which is evaluated by the stability theory developed by Altintas [10]; Nf is the total number of cutters with different flutes.
A motorized spindle with the tool is shown in Figure 11 where six design variables are defined. The required cutting conditions are listed in Table 1.
Figure 11: Initial design and design variables
Figure 12: Stability lobes before and after optimization
The chatter stability lobes for a four fluted cutter computed from the three initial design trials and the final optimized design are shown in Figure 12. The desired spindle speed is 9,000 rpm, and the minimum depth of cut is 3 mm. The cutting is not stable for all three initial designs, but it becomes stable after automatic optimization. The physics behind the optimization is to locate the natural frequency of the spindle at the desired tooth passing frequency and satisfy the dynamic stiffness requirement, which is done by automatic adjusting of the bearing spacing ( X l , X 2 , . . , X 6 ) .The algorithm allows the optimization of multiple cutters with different flutes at the desired speeds as well.
5 SUMMARY
A general finite element method, which can predict the static and dynamic behavior of spindle systems, is presented. The spindle and housing are modeled by Timoshenko beam elements. The gyroscopic and centrifugal effects from both spindle shaft and bearings are included in constructing the dynamic model of the spindle system. The nonlinear stiffness matrix for the angular contact bearings are established through the analysis of load deflection of bearings. Hertzian theory is used to determine the relationship between the contact force and displacement of bearing balls and rings which are considered as elastic elements. The stiffness matrix of the bearing, the contact angle, preload and deflection of spindle shaft and housing are all coupled in the Finite Element model of the spindle assembly. The simulated results are compared favorably well against experimental measurements conducted on an instrumented, industrial size spindle. The simulation shows that the rotational speed of the spindle shaft has a bigger influence on the lower natural frequencies. The proposed optimization method is used to achieve maximum depth of cut or dynamic stiffness by tuning of the spindle modes through optimizing the locations of bearings and the motor for motorized spindles. The overall design and analysis model allows virtual testing of spindles under simulated cutting forces. The dynamic behaviour of the spindle, contact loads experienced by the bearings, the displacements of the shaft at any point can be predicted under simulated cutting conditions. The proposed model can be used to improve the design of spindles for targeted machining applications.
6 REFERENCES
[1] Altintas, Y., Weck, M., 2004, "Chatter Stability of Metal Cutting and Grinding", Annals of CIRP, vol. 53/2, pp. 619-642.
[2] Weck, M., Koch, A,, 1993, "Spindle Bearing Systems for High Speed Applications in Machine Tools", Annals of CIRP, vol. 42/1, pp. 445-448.
[3] Jones, A. B., 1960, "A General Theory for Elastically Constrained Ball and Radial Roller Bearings Under Arbitrary Load and Speed Conditions," ASME J. Basic Eng., pp. 309-320.
[4] Jedrzejewski, J., Kowal, Z., Kwasny, W., Modrzycki, W., 2004, "Hybrid Model of High Speed machining Centre Headstock, Annals of CIRP, vo1.53/1. pp. 285-288.
[5] Zeljkovic, M., Gatalo, R., 1999, "Experimental and Computer Aided analysis of High-speed Spindle Assembly Behaviour", Annals of CIRP, vol. 48/1, pp. 325-329.
[6] Nelson, H. D., 1980, "A finite rotating shaft element using Timoshenko beam theory," ASME J. Mech. Des V0l.102, pp.793-803.
[7] Abele, E., Fiedler, U., 2004, "Creating Stability Lobe Diagrams during Milling", Annals of CIRP, vol. 53/1, pp. 309-312.
[8] Smith, S., Snyder, J., 2001, "A Cutting Performance Based Template for Spindle Dynamics", Annals of CIRP, VOI. 50/1, p.259-262.
[9] Rivin, E., 2000, "Tooling Structure - Interface Between Cutting Edge and Machine Tool", Annals of CIRP, VOI. VO1.49/2, pp. 591-643.
[10] Altintas Y., Budak E., 1995, "Analytical Prediction of Stability Lobes in Milling", Annals of CIRP, 44/1, pp.357-366
- 23 -
虛擬機(jī)床主軸的設(shè)計(jì)和優(yōu)化
英屬哥倫比亞大學(xué)機(jī)械工程系與加拿大溫哥華大學(xué)制造自動(dòng)化實(shí)驗(yàn)室
Y. Altintas(1), Y. 曹
h t t p : / / w .mech. ubc.ca/-mal
摘要
本文呈現(xiàn)了一個(gè)主軸, 工具架、工具和切削過程一體化的數(shù)字模型。主軸是在一個(gè)內(nèi)部開發(fā)的有限元建模系統(tǒng)建模的。本文涵蓋了軸承預(yù)負(fù)荷，以及旋轉(zhuǎn)部件的轉(zhuǎn)速給陀螺和離心力所造成的影響。主軸上的每一點(diǎn)的軸承剛度、模態(tài)、頻率響應(yīng)函數(shù)可以預(yù)測(cè)出來。沿主軸軸的靜態(tài)和動(dòng)態(tài)變形量以及接觸力軸承可以用模擬預(yù)測(cè)切削力在身體上構(gòu)建和測(cè)試紡錘波。對(duì)軸承的間距進(jìn)行優(yōu)化可以使動(dòng)態(tài)剛度最大或最大震顫免費(fèi)深度削減速度所需的地區(qū)對(duì)于一個(gè)給定的刀具幾何形狀和工件材料。添加約束模型在機(jī)床主軸的安裝,以及定義本地彈簧和阻尼元素在主軸上的任何節(jié)點(diǎn)是可行的。模型驗(yàn)證實(shí)驗(yàn)。
關(guān)鍵詞：主軸、切割、振動(dòng)
1介紹
由于生產(chǎn)效率的增加，高速切削廣泛應(yīng)用于工業(yè)。然而, 高速運(yùn)轉(zhuǎn)的主軸要配置引起震顫的直徑較小的軸和軸承,除非該主軸是專為在理想的切削條件下運(yùn)行而設(shè)計(jì)的。震顫導(dǎo)致不良表面光潔度，使軸承超載，從而會(huì)縮短主軸的壽命[1]。在機(jī)械加工中，影響主軸的總體性能的因素有主軸的尺寸、剛度、預(yù)加載,各軸承的間距,刀具幾何形狀，工具架以及加工材料。建模研究的目的是模擬主軸的性能和優(yōu)化其尺寸,以達(dá)到最大程度的動(dòng)態(tài)剛度和增加材料去除率。
角接觸球軸承最常用在高速主軸中，由于其低摩擦性能和承受工作載荷的能力為軸向和徑向[2]。軸承的剛度取決于接觸角,反過來依賴于速度、接觸球和環(huán)之間的負(fù)載。瓊斯發(fā)展了撓度曲線分析的一般理論。用來分析軸承,包括高速運(yùn)轉(zhuǎn)的軸承滾子的離心和旋轉(zhuǎn)運(yùn)動(dòng)[3]，這些在本文中有使用。轉(zhuǎn)動(dòng)軸和固定箱體通常通過有限元建模技術(shù)來建模[4, 5]。過去的絕大多數(shù)研究沒有考慮軸承剛度的非線性行為。例如，納爾遜[6]采用的一種Timoshenko梁理論建立了系統(tǒng)矩陣分析轉(zhuǎn)子系統(tǒng)的動(dòng)力學(xué)，其受到轉(zhuǎn)動(dòng)慣量，陀螺力矩,剪切變形、軸承和軸向負(fù)荷的影響,但被建模為線性彈簧。正如Abele[7]提出的主軸結(jié)構(gòu)動(dòng)力在速度較高的情況下發(fā)生改變，這將影響穩(wěn)定容器的位置和形狀[8]。
本文提出一種通用有限元模型，它可以預(yù)測(cè)軸承的剛度,接觸力軸球承、固有頻率和振型、頻率響應(yīng)函數(shù)和在切削負(fù)載下為響應(yīng)次數(shù)計(jì)數(shù)。模型包括軸承預(yù)負(fù)荷、軸承和軸的旋轉(zhuǎn)帶來的影響。此后,本文組織如下文：本文展示了軸的和軸承的非線性有限元模型,考慮了軸承預(yù)負(fù)荷,陀螺和離心速度的影響。第三節(jié)實(shí)驗(yàn)驗(yàn)證了主軸模型。第四節(jié)提出了軸承間距優(yōu)化方法為獲得最大動(dòng)力剛度或最大振顫槽刀具。本文還包括對(duì)貢獻(xiàn)的總結(jié)。
2主軸系統(tǒng)的有限元模型
圖1是裝備非接觸式位移傳感器的實(shí)驗(yàn)主軸。主軸有標(biāo)準(zhǔn)的CAT40刀架接口，最大轉(zhuǎn)速為15000轉(zhuǎn)速/分鐘,由15千瓦電機(jī)驅(qū)動(dòng)，它與軸和傳動(dòng)皮帶系統(tǒng)息息相關(guān)。主軸根據(jù)內(nèi)部開發(fā)的轉(zhuǎn)為主軸設(shè)計(jì)的有限元建模系統(tǒng)開模,如圖2所示。Timoshenko 梁理論是用來模擬心軸（錠桿）和外殼。黑色圓點(diǎn)代表節(jié)點(diǎn),每個(gè)節(jié)點(diǎn)有三個(gè)平移和兩個(gè)旋轉(zhuǎn)自由度。皮帶輪被建模為一個(gè)剛性圓盤。軸有兩個(gè)前軸承(BI和B2)串聯(lián)和三個(gè)后輪軸的(B3,B4和B5)。外環(huán)上的預(yù)加載應(yīng)用液壓A3定義為節(jié)點(diǎn),可以沿著軸住房節(jié)點(diǎn)A4和A5。部隊(duì)轉(zhuǎn)移到內(nèi)部環(huán)B3通過軸承球B5,然后通過內(nèi)圈B5主軸軸。部隊(duì)傳送到前線軸承內(nèi)圈的我,這也是固定在主軸,然后外環(huán)A2的住房,這是固定的住房。整個(gè)主軸平衡軸向方向的預(yù)加載。初始預(yù)加載在組裝和應(yīng)用可以通過液壓調(diào)節(jié)單元。工具被認(rèn)為是剛性連接的的工具架固定在主軸軸平移和旋轉(zhuǎn)通過彈簧剛度。根據(jù)機(jī)床的剛度,可以嚴(yán)格固定軸住房或彈性支承主軸頭。相關(guān)的內(nèi)環(huán)和外環(huán)的非線性方程組,得到軸承剛度通過求解方程的系統(tǒng)。
Figure 1: Spindle system
圖一：主軸系統(tǒng)
Spindle nose：主軸端部 tool：刀具 tool-holder:刀架
housing：外罩 shaft：主軸 Hydraulic fluid：液壓流體 bearing：軸承 pulley：齒輪
收藏