孫章棟,朱才朝,劉懷舉,劉明勇,顧宗琳
(重慶大學(xué)機(jī)械傳動(dòng)國家重點(diǎn)實(shí)驗(yàn)室,重慶 400030)
擺線針輪傳動(dòng)線接觸彈流潤滑分析
孫章棟,朱才朝,劉懷舉,劉明勇,顧宗琳
(重慶大學(xué)機(jī)械傳動(dòng)國家重點(diǎn)實(shí)驗(yàn)室,重慶 400030)
對(duì)擺線針輪線接觸彈流潤滑進(jìn)行數(shù)值分析,得出嚙合過程中最小膜厚,與經(jīng)驗(yàn)公式對(duì)比,驗(yàn)證模型的正確性。以擺線輪θ=arccosK1處潤滑狀態(tài)作為判斷依據(jù),研究載荷、轉(zhuǎn)速和流變指數(shù)對(duì)擺線針輪傳動(dòng)潤滑狀態(tài)的影響規(guī)律。結(jié)果表明:隨著轉(zhuǎn)速提高,二次壓力峰高度降低并向入口區(qū)移動(dòng),膜厚相應(yīng)增加;而隨著載荷增加,接觸區(qū)變寬,二次壓力峰增加且向出口區(qū)移動(dòng),膜厚略微減小;流變指數(shù)n增加,二次壓力峰先增加后減小,最后趨近于Hertz壓力,并向出口區(qū)移動(dòng),膜厚相應(yīng)減小。討論了短幅系數(shù)k1對(duì)潤滑的影響,表明在滿足設(shè)計(jì)要求情況下,短幅系數(shù)k1減小,有利于提高潤滑性能。
擺線針輪;脂潤滑;數(shù)值解;Ostwald
擺線針輪行星傳動(dòng)以其結(jié)構(gòu)緊湊、傳動(dòng)比大、傳動(dòng)精度高和承載能力大等優(yōu)點(diǎn),而被廣泛應(yīng)用。通常使用脂來實(shí)現(xiàn)潤滑。漸開線齒輪潤滑問題已進(jìn)行廣泛的研究,但有關(guān)擺線針輪潤滑研究少有報(bào)道。擺線針輪潤滑不同于漸開線:首先擺線針輪傳動(dòng)嚙合原理以及幾何特征有別于漸開線傳動(dòng);其次齒擺線輪與針輪同時(shí)嚙合齒數(shù)多于漸開線,同時(shí)嚙合齒數(shù)理論上可以達(dá)到一半擺線輪齒數(shù);最后漸開線使用油進(jìn)行潤滑,脂的流變特性有別于油。彈流潤滑性能的好壞對(duì)擺線針輪傳動(dòng)的性能和使用壽命都有顯著影響。開展擺線針輪傳動(dòng)彈流脂潤滑數(shù)值分析具有重要的意義。
國內(nèi)外學(xué)者對(duì)齒輪彈流潤滑做了大量研究。Liu[2-3]對(duì)線接觸直齒輪的混合潤滑模型,討論了轉(zhuǎn)速及不同粗糙度對(duì)直齒輪潤滑的影響;Zhu等[4]討論了不同轉(zhuǎn)速工況下由混合潤滑過渡到干摩擦接觸的潤滑情況;Chu等[5]研究了接觸面的液面滑移和流體特性對(duì)潤滑的影響。Yang等[6]對(duì)斜齒輪進(jìn)行彈流潤滑分析,討論了螺旋角對(duì)潤滑的影響;Liu等[7]首次分析了有限長線接觸的熱解,并與無限長熱解作對(duì)比;Zhu等[8]采用有限長線接觸模型分析斜齒輪穩(wěn)態(tài)熱彈流潤滑問題,討論載荷與轉(zhuǎn)速工況變化對(duì)斜齒輪潤滑溫度及齒面剪切力的影響。劉曉玲等[9]展開對(duì)指數(shù)率非牛頓流體在線接觸條件下的彈流潤滑數(shù)值研究,應(yīng)用多重網(wǎng)格法和多重網(wǎng)格積分法數(shù)值求解了Newton流體和Ree-Eyring流體線接觸等溫和熱彈流潤滑問題。于玫等[10]探討了不同工況下熱效應(yīng)對(duì)脂潤滑彈流數(shù)值解的影響。李媛等[11]探討脂潤滑條件下激光加工微凹坑表面的摩擦特性,表明適當(dāng)?shù)陌伎由疃瓤梢愿纳票砻娴哪Σ撂匦院蜏囟忍匦?。白新瑞等?2]基于有限長線接觸彈流潤滑理論,求得乏油條件下圓柱滾子軸承彈流潤滑的完全數(shù)值解。楊沛然等[13]做了大量脂潤滑方面的研究工作。
Mihailidis等[14]推導(dǎo)出適合非牛頓流體的乏油廣義雷諾方程。由于擺線針輪傳動(dòng)通常是用潤滑脂實(shí)現(xiàn)潤滑,論文以O(shè)stwald模型對(duì)擺線針輪傳動(dòng)線接觸彈流脂潤滑問題進(jìn)行數(shù)值分析研究,研究載荷、轉(zhuǎn)速和流變指數(shù)對(duì)擺線針輪傳動(dòng)潤滑狀態(tài)的影響規(guī)律,討論設(shè)計(jì)參數(shù)短幅系數(shù)k1對(duì)潤滑的影響。
1.1 嚙合分析
擺線針輪傳動(dòng)同時(shí)嚙合的齒數(shù)多,如圖1所示。在嚙合過程中,擺線輪曲率半徑ρ變化,針齒半徑rz為定值,從而可得嚙合過程中的當(dāng)量曲率半徑為:
式中“+”號(hào)為用于外凸齒廓嚙合;“-”號(hào)為用于內(nèi)凹齒廓嚙合;ρ為有關(guān)嚙合相位角的函數(shù)。
圖1 擺線輪與針齒嚙合示意圖Fig.1 Mathematical model of the cycloid drive
由運(yùn)動(dòng)學(xué)分析可知,擺線輪與針齒在嚙合點(diǎn)處的線速度是相等的,設(shè)u1為擺線輪在嚙合點(diǎn)處的線速度,u2為針齒在嚙合點(diǎn)處的線速度,由圖1所示u1=u2=um=,對(duì)潤滑油的卷吸速度:
1.2 潤滑模型
基于Ostwald模型的潤滑脂本構(gòu)方程為:
式中:m為粘性函數(shù);n為流變指數(shù)
采用廣義Reynolds方程[15],引入等效粘度η*,對(duì)于Ostwald模型,其等效粘度為:
引入等效粘度可將潤滑脂的非牛頓流體轉(zhuǎn)換成牛頓流體,方便采用牛頓流體解法求解。利用Hertz接觸參數(shù)對(duì)潤滑模型進(jìn)無量綱化,無量綱量參考量:
式中x為流動(dòng)方向,z為膜厚方向,b為Hertz接觸半寬,h為油膜厚度,p為油膜壓力,PR為最大Hertz壓力,R為等效曲率半徑,η0為環(huán)境粘度,E為等效彈性模量。
無量綱Reynolds方程
式中ρ0為潤滑油環(huán)境密度。
無量綱Reynolds方程的邊界條件為:
式中H0為常數(shù)。
無量綱密度方程
無量綱載荷方程
將式(7)~式(9)離散化,建立擺線針輪傳動(dòng)彈流脂潤滑數(shù)值仿真模型,通過數(shù)值直接迭代法對(duì)彈流潤滑中的膜厚和壓力進(jìn)行數(shù)值求解。Gauss-Seidel迭代法用于求解低壓區(qū)壓力,Jacobi雙極子迭代法用于求解高壓區(qū)壓力[16]。
壓力求解無量綱計(jì)算域范圍X=[XinXout]=[-4,1.5],油膜厚度無量綱計(jì)算范圍Z=[0,1]。X方向的節(jié)點(diǎn)數(shù)1 025,油膜厚度Z向節(jié)點(diǎn)數(shù)11。數(shù)值迭代精度為:壓力相對(duì)誤差小于10-6。
選取擺線針輪行星傳動(dòng)嚙合副為研究對(duì)象,其嚙合副與潤滑相關(guān)參數(shù)如表1。
表1 擺線針輪傳動(dòng)副與潤滑相關(guān)參數(shù)Tab.1 The cycloid drive parameters and properties of the lubricant
圖2 單位嚙合力隨嚙合相位變化Fig.2 The variation of contact force with meshing process
圖3 等效曲率半徑和卷吸速度變化曲線Fig.3 The variation of reduced curve radius and rolling speed
擺線輪與針齒滾動(dòng)線接觸嚙合,在嚙合過程中,單位嚙合力F、等效曲率半徑和卷吸速度隨著嚙合相位變化。圖2給出單位嚙合力隨嚙合相位變化曲線,圖3給出卷吸速度和等效曲率半徑隨嚙合相位變化曲線。
設(shè)定輸入轉(zhuǎn)速500 r/min,輸出力矩420 N/m。選取某一嚙合相位處進(jìn)行分析,其潤滑脂膜壓力、膜厚如圖4所示。在出口區(qū)處,潤滑膜壓力出現(xiàn)明顯二次壓力峰,在二次壓力峰相對(duì)應(yīng)處,潤滑膜開始收縮,形成出口區(qū)的頸縮現(xiàn)象,頸縮處的膜厚為最小潤滑膜厚度。擺線輪與針齒滾動(dòng)嚙合剪切應(yīng)力如圖5所示,在潤滑膜出口區(qū)的頸縮位置出現(xiàn)剪應(yīng)力峰值,其它區(qū)域切應(yīng)力趨于零值,摩擦小,進(jìn)一步說明溫度對(duì)潤滑影響小。
擺線輪與針齒理論嚙合區(qū)間為0°~180°,而實(shí)際有效嚙合區(qū)間為30°~120°。計(jì)算最大壓力與最小膜厚隨著嚙合相位變化曲線如圖6。如圖7所示,與采用Dowson-Higginson潤滑油膜厚經(jīng)驗(yàn)公式計(jì)算結(jié)果變化趨勢相吻合,但比經(jīng)驗(yàn)公式偏大。由于Dowson-Higginson經(jīng)驗(yàn)公式用來計(jì)算潤滑油最小膜厚的,而修形齒廓最小膜厚數(shù)值解用來計(jì)算潤滑脂的,驗(yàn)證數(shù)值模型仿真結(jié)果正確性。從圖2、圖3和圖7可以得出在嚙合相位角θ=arccosK1=42.97°處為曲線拐點(diǎn),在嚙合過程中相位角θ=arccosK1=42.97°時(shí)油膜厚度最小,故以此相位處潤滑狀態(tài)作為整個(gè)嚙合過程中潤滑狀態(tài)的判斷依據(jù)。
圖5 擺線輪與針齒滾動(dòng)嚙合切應(yīng)力Fig.5 Cycloid gear and pin gear mesh rolling shear stress
圖6 中心壓力與最小膜厚隨嚙合相位變化Fig.6 The variation of center-pressure and min-film thickness
圖7 最小膜厚數(shù)值解與經(jīng)驗(yàn)公式比較Fig.7 Compare film thickness numerical solution with empirical formula
3.1 工況參數(shù)變化對(duì)潤滑影響
圖8~9給出了轉(zhuǎn)速變化對(duì)θ=42.97°處潤滑性能的變化規(guī)律。隨著轉(zhuǎn)速提高,入口區(qū)壓力增加,二次壓力峰高度減小,其位置向入口區(qū)移動(dòng),膜厚相應(yīng)增加,潤滑膜平行部分縮短,頸縮變得不明顯,表明擺線輪與針齒嚙合時(shí),卷吸速度增大,有利于形成良好的彈流潤滑膜。
圖10~11給出了載荷變化對(duì)θ=42.97°處潤滑性能的變化規(guī)律。隨著載荷增加,接觸區(qū)變寬,入口區(qū)壓力增加,二次壓力峰增加,其位置向出口區(qū)移動(dòng),膜厚略微減小,膜厚頸縮現(xiàn)象更加明顯。主要由于隨著載荷增加,油膜剛度變化較小,故膜厚變化不明顯。
3.2 流變指數(shù)n對(duì)潤滑影響
圖12、13是流變指數(shù)n對(duì)θ=42.97°處潤滑性能的影響規(guī)律。當(dāng)n<1時(shí),隨著流變指數(shù)n增加,入口區(qū)壓力降低,二次壓力峰增加,并向出口區(qū)移動(dòng),膜厚相應(yīng)降低;當(dāng)n>1時(shí),隨著流變指數(shù)n增加,入口區(qū)壓力略微降低,二次壓力峰減小并向出口區(qū)移動(dòng),最后二次壓力峰消失,接近Hertz壓力,膜厚相應(yīng)減小,頸縮也越來越不明顯,最后消失,潤滑狀態(tài)變差。
圖8 轉(zhuǎn)速對(duì)油膜壓力分布的影響Fig.8 Pressure distribution for different rotational Speeds
圖9 轉(zhuǎn)速對(duì)油膜厚度的影響Fig.9 Film shape for different rotational Speeds
圖10 載荷對(duì)油膜壓力的影響Fig.10 Pressure distribution for different loads
圖11 載荷對(duì)油膜厚度的影響Fig.11 Film shape for different loads
圖12 流變指數(shù)n對(duì)油膜壓力的影響Fig.12 Pressure distribution for different n
圖13 流變指數(shù)n對(duì)膜厚的影響Fig.13 Film shape for different n
3.3 短幅系數(shù)k1對(duì)潤滑的影響
圖14~15給出了短幅系數(shù)k1對(duì)擺線輪平均半徑處潤滑性能的變化規(guī)律。隨著短幅系數(shù)k1增加,在大部分嚙合區(qū)間范圍內(nèi)壓力增加,承載能力下降,膜厚減小。表明在滿足設(shè)計(jì)要求情況下,短幅系數(shù)k1減小,有利于提高潤滑性能。
圖14 中心壓力隨短幅系數(shù)k1變化Fig.14 The variation of center-pressure for different k1
圖15 最小膜厚隨著短幅系數(shù)k1變化Fig.15 The variation of min-film thickness for different k1
論文基于Ostwald模型對(duì)擺線針輪線接觸脂潤滑彈流問題進(jìn)行研究,得出嚙合過程中最小膜厚值,并與經(jīng)驗(yàn)公式對(duì)比,驗(yàn)證數(shù)學(xué)模型的正確性。以擺線輪θ=arccos K1處潤滑狀態(tài)作為判斷依據(jù),研究載荷、轉(zhuǎn)速和流變指數(shù)對(duì)擺線針輪傳動(dòng)滑狀態(tài)的影響規(guī)律,表明隨著轉(zhuǎn)速提高,二次壓力峰高度降低并向入口區(qū)移動(dòng),膜厚相應(yīng)增加,潤滑膜平行部分縮短;而隨著載荷增加,接觸區(qū)變寬,二次壓力峰增加,其位置向出口區(qū)移動(dòng),膜厚略微減小,膜厚頸縮現(xiàn)象更加明顯;隨著流變指數(shù)n變化,當(dāng)n<1時(shí),隨著流變指數(shù)n增加,二次壓力峰增加,并向出口區(qū)移動(dòng),膜厚相應(yīng)降低;當(dāng)n>1時(shí),隨著流變指數(shù)n增加,二次壓力峰減小并向出口區(qū)移動(dòng),最后二次壓力峰消失,接近Hertz壓力,膜厚相應(yīng)減小。討論了短幅系數(shù)k1對(duì)潤滑的影響,隨著短幅系數(shù)k1增加,在大部分嚙合區(qū)間范圍內(nèi)壓力增加,承載能力下降,膜厚減小。在滿足設(shè)計(jì)要求情況下,短幅系數(shù)k1減小,有利于提高潤滑性能。
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Numerical analysis of elastohydrodynamic lubrication for cycloid drives
SUN Zhang-dong,ZHU Cai-chao,LIU Huai-ju,LIU Ming-yong,GU Zong-lin
(The State Key Laboratory of Mechanical Transmission,Chongqing University,Chongqing 400030,China)
Numerical analysis of grease elastohydrodynamic lubrication for cycloid drives was performed.The minimum film thickness of meshing process was obtained.Comparting it with that obtained from an empirical formula,the correctness of model was verified.The average radius of cycloid wheel's lubrication state was taken as the judgment basis,the effects of rotating speed,load and rheological index on grease EHL were analyzed.Results showed that with increase in rotating speed,the second pressure peak drops and moves toward the inlet,the film thickness increases;with increase in load,the contact field width increases,the second pressure peak rises and moves toward the out let,the film thickness decreases slightly;with increase in the rhoological index n,the second pressure peak firstly increases and then decreases,finally approaches Hertz pressure,and it moves toward the outlet,the film thickness decreases.The effect of the short ampliitude coefficient k1on lubrication was discussed,it was shown that under the conditions to meet design requirements;the decrease in k1can improve the lubrication performance.
Cycloid Drive;Grease Lubrication;Numerical Analysis;Ostwald
TH117
A
10.13465/j.cnki.jvs.2014.23.035
由國家自然科學(xué)基金(51175523);教育部博士點(diǎn)基金(20110191110039)資助
2013-07-08 修改稿收到日期:2013-12-12
孫章棟男,博士生
朱才朝男,博士,教授,博士生導(dǎo)師,1967年生