孫付春，李玉龍，文昌明，鐘 飛

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齒輪泵側(cè)隙卸荷的界定標(biāo)準(zhǔn)與驗(yàn)證

孫付春，李玉龍，文昌明，鐘 飛

（成都大學(xué)機(jī)械工程學(xué)院，成都 610106）

為解決當(dāng)前泵用齒輪副側(cè)隙大、小界定含糊的問題，基于側(cè)隙傳動(dòng)與困油的性能要求，從雙齒嚙合區(qū)內(nèi)的2困油容積連通和單齒嚙合區(qū)卸荷的性能完善方面，通過困油循環(huán)及困油過程的分析，建立出2類區(qū)域內(nèi)的困油流量及峰值，推導(dǎo)出卸荷用側(cè)隙、連通用側(cè)隙及其均值和峰值；并進(jìn)行實(shí)例運(yùn)算和驗(yàn)證分析。結(jié)果表明：卸荷區(qū)與連通區(qū)的困油流量峰值比為3，前者的卸荷負(fù)擔(dān)最大；連通區(qū)的真正連通，所需側(cè)隙高達(dá)2.41 mm，實(shí)際上并不存在；卸荷側(cè)隙大于連通側(cè)隙，以連通側(cè)隙作為側(cè)隙大與小的分界點(diǎn)，卸荷側(cè)隙作為上限值的界定可行；計(jì)算與試驗(yàn)的側(cè)隙誤差為7.5%，比較吻合，且上限值有20%的安全裕度，比較可靠等。泵用側(cè)隙的界定為大、小側(cè)隙的正確區(qū)分提供了參考，也可為后續(xù)的相關(guān)研究提供參考。

泵；振動(dòng)；齒輪泵；連通側(cè)隙；卸荷側(cè)隙；大小界定；困油流量

0 引 言

外嚙合齒輪泵（簡稱齒輪泵）是一種用于泵送工作油液的動(dòng)力泵，有著極其廣泛的應(yīng)用[1]。結(jié)構(gòu)上，一對(duì)外嚙合齒輪副為其核心部件。該齒輪副在常規(guī)的動(dòng)力傳遞中，為利于嚙合齒廓之間形成潤滑油膜、避免因輪齒摩檫發(fā)熱膨脹而卡死，非工作齒廓之間常留有一定的齒側(cè)間隙，簡稱側(cè)隙；但側(cè)隙同時(shí)也會(huì)產(chǎn)生齒間沖擊，影響齒輪傳動(dòng)的平穩(wěn)性。故在滿足傳動(dòng)性能要求的基礎(chǔ)上，一般采用0.0050.01（為模數(shù)，mm）的小側(cè)隙[2]，但設(shè)計(jì)上仍按無側(cè)隙對(duì)待。不過作為常規(guī)齒輪傳動(dòng)在泵中的特殊應(yīng)用，雖然大側(cè)隙能有效改善困油性能，卻同時(shí)也造成了容積效率下降、振動(dòng)加劇、噪聲增加[3-5]；反之，容積效率提高和振動(dòng)減緩、噪聲下降[6-8]，同時(shí)也降低了困油性能[9-12]，所以泵用側(cè)隙同樣會(huì)受到一定的限制，一般采用0.010.08的有側(cè)隙[13]，或參照推薦值選擇[14]。

目前，在關(guān)于容積效率和困油性能等的文獻(xiàn)研究中，一般都需要計(jì)算通過側(cè)隙處的介質(zhì)交換流量，但針對(duì)某一具體的側(cè)隙值，文獻(xiàn)[13]雖然給出了有側(cè)隙或者無側(cè)隙的界定標(biāo)準(zhǔn)，但側(cè)隙究竟屬于小側(cè)隙還是大側(cè)隙，一直很模糊。更多的是以小側(cè)隙的名義默認(rèn)雙齒嚙合區(qū)內(nèi)2困油容積的非連通問題[15-17]；或以大側(cè)隙的名義默認(rèn)雙齒嚙合區(qū)內(nèi)2困油容積的連通、單齒嚙合區(qū)域的無困油問題等[5,13,18]。而偏偏大、小側(cè)隙的界定又直接涉及到后續(xù)不同的計(jì)算方法[13,19-20]，例如，卸荷槽形位計(jì)算的不同方法[13,21-24]等，這些恰恰被大多數(shù)的研究所忽略。即針對(duì)泵用的這對(duì)齒輪副，何為側(cè)隙的大、小，截至目前為止，學(xué)術(shù)界并沒有給出具體的劃分界定標(biāo)準(zhǔn)。為此，論文擬從雙齒嚙合內(nèi)2個(gè)困油容積的連通和單齒嚙合區(qū)內(nèi)困油性能的完善角度，對(duì)此進(jìn)行深入的研究，并給出相應(yīng)的界定標(biāo)準(zhǔn)。

1 有無側(cè)隙的現(xiàn)有界定

何謂有、無側(cè)隙，文獻(xiàn)[13]給出了如式（1）的界定式。設(shè)為公法線方向上的側(cè)隙值，mm；0為有、無側(cè)隙的分界值，mm。當(dāng)<0時(shí)，由于工作介質(zhì)通過側(cè)隙的阻力相當(dāng)大，介質(zhì)的通過流量被認(rèn)為是微乎其微接近于零，泵幾何尺寸上的計(jì)算，可按無側(cè)隙關(guān)系式來確定；否則，按有側(cè)隙關(guān)系式來確定。而有側(cè)隙又可分為大側(cè)隙和小側(cè)隙，但目前對(duì)此卻沒有明顯的界定，大小之分比較含糊與籠統(tǒng)。

式中為介質(zhì)動(dòng)力黏度，(N·s)/m2；為節(jié)圓圓周速度，m/s；p與p為泵的高、低壓腔的介質(zhì)壓力，Pa；為模數(shù)，mm。

2 困油循環(huán)及困油過程

圖1描述了齒輪泵的困油循環(huán)及困油過程。輪心分別記為1、2，偏向1、2側(cè)的2個(gè)困油容積、困油壓力，分別記為1、2，mm3和1、2，Pa。其中，圖1中a→b→c→d→e和d→e→f→a→b分別為1、2的困油循環(huán)；a→b→c→d→e→f→a為一個(gè)完整的困油過程。其中，a為1剛剛形成；b為2剛剛脫開；c為1在最小困油容積時(shí)的位置；d為2剛剛形成；e為1剛剛脫開；f為2在最小困油容積時(shí)的位置。a→b為單側(cè)隙雙齒嚙合區(qū)間，簡稱為側(cè)隙連通區(qū)；b→c→d和e→f→a為單側(cè)隙單齒嚙合區(qū)間，簡稱為側(cè)隙卸荷區(qū)。設(shè)剛形成1的嚙合點(diǎn)在1工作齒面上對(duì)應(yīng)的曲率半徑為，并以此作為困油過程的位置變量，記對(duì)應(yīng)于圖1中a~f的6個(gè)位置的分別用s~表示[1]，mm。其中

式中為理論嚙合線的長度，mm；p為基節(jié)，mm；ε為根切重合度[25]。由于泵用齒輪副允許存在一定的根切現(xiàn)象，2上的齒頂點(diǎn)能否參與嚙合，取決于根切程度[26]；它們均為齒形參數(shù)的函數(shù)。

注：1、2表示主、從齒輪的輪心；表示嚙合位置；表示側(cè)隙位置；p表示進(jìn)口壓力，Pa；p表示出口壓力，Pa；1、2表示偏向1、2側(cè)的困油容積，mm3；1、2表示1、2的困油壓力，Pa；為由主動(dòng)輪上嚙合點(diǎn)處的曲率半徑所定義的位置變量，mm；s表示1剛剛形成位置，mm；s表示2剛剛脫開位置，mm；s表示1的最小容積位置，mm；s表示2剛剛形成位置，mm；s表示1剛剛脫開位置，mm；s表示2的最小容積位置，mm；下同。

Note:1and2indicate two center points of driving gear and driven gear;indicates meshing point;indicates backlash point;pindicates inlet pressure, Pa;pindicates outlet pressure, Pa;1and2indicate the two different trapped-oil volumes on1side and2side, mm3;1and2indicate trapped-oil pressure of1and2, Pa;indicates the location variables defined as the curvature radius of meshing point of driving gear, mm;sindicates the location of1just formed, mm;sindicates the location of2just disappeared, mm;sindicates the location of1with minimum volume, mm;sindicates the location of2just formed, mm;sindicates the location of1just disappeared, mm;sindicates the location of2with minimum volume, mm; similarly hereinafter.

圖1 困油循環(huán)及困油過程

Fig.1 Trapped oil circulation and trapped oil process

3 困油流量及峰值

3.1 側(cè)隙卸荷區(qū)

在側(cè)隙卸荷區(qū)內(nèi)，始終只存在由一個(gè)嚙合點(diǎn)和一個(gè)側(cè)隙點(diǎn)所圍成的單一困油容積1或2。每單一的困油容積及其變化率為

式中0為最小的困油容積，mm3；為齒數(shù)，稱1、2為1、2對(duì)時(shí)間的變化率，稱為困油流量，mm3/s，它們的峰值均為

式中Q為側(cè)隙卸荷區(qū)內(nèi)的困油流量峰值，mm3/s；為齒寬，mm；為角速度，rad/s。

3.2 側(cè)隙連通區(qū)

小側(cè)隙時(shí)，在圖1中[s,s]區(qū)間內(nèi)的1、2不能連成一體，并形成了2個(gè)單一的困油容積。此時(shí)，每個(gè)單一1或2均由一個(gè)嚙合點(diǎn)和一個(gè)側(cè)隙點(diǎn)所圍成。其間的困油流量與式（4）相同，峰值與式（5）相同。

大側(cè)隙時(shí)，1、2連成一體，該連體困油的體積、壓力設(shè)為和，且該連體與高、低壓油腔之間，分別被2個(gè)嚙合點(diǎn)所隔開。由于該2個(gè)嚙合點(diǎn)處的嚙合間隙極小，此時(shí)雖為大側(cè)隙，但困油現(xiàn)象仍始終存在。此時(shí)

式中為1、2的連體體積，mm3；為對(duì)應(yīng)于的困油流量，mm3/s；Q為側(cè)隙連通的困油流量峰值，mm3/s；Q為側(cè)隙卸荷的困油流量峰值，mm3/s；K為卸荷區(qū)和連通區(qū)困油流量的峰值比。

例取ε=1.1，則K=3，說明側(cè)隙卸荷區(qū)比連通區(qū)內(nèi)的困油壓力卸荷的負(fù)擔(dān)大很多。

4 側(cè)隙界定的理論計(jì)算

如果設(shè)困油流量以困油容積的膨脹為正，壓縮為負(fù)；困油壓力的變化率以壓力增大為正，降低為負(fù)；且僅考慮困油流量是產(chǎn)生困油壓力的唯一主因。那么，困油的體積彈性模量可定義為[1]

式中為體積彈性模量，Pa，in、Qout為外界流進(jìn)、流出困油容積的交換流量，mm3/ s；且記以流入為正。則

式中為對(duì)時(shí)間的變化率，Pa/s。

在側(cè)隙連通區(qū)內(nèi)，對(duì)于2個(gè)單一的1、2，各自與外界交換的總流量，包括通過側(cè)隙的流量（簡稱為側(cè)隙流量）、軸向（間隙）流量、嚙合（間隙）流量、卸荷槽的卸荷流量等，如將除側(cè)隙流量外的其他流量均以側(cè)隙流量的形式體現(xiàn)，則所折算出的連通側(cè)隙，如圖2所示，故該區(qū)間內(nèi)的困油為一封閉容積。由2>1，得

式中1、2為1、2對(duì)時(shí)間的變化率，Pa/s；Q、QΔpd為側(cè)隙處的剪切流量和壓差流量，mm3/s。

注：Q、Δpd表示側(cè)隙處的剪切流量和壓差流量，mm3·s-1；c表示連通側(cè)隙，mm；1、2和表示壓力變化率，Pa·s-1；1、2、表示容積變化率，mm3·s-1；下同。

Note:QandΔpdindicate shear flow and differential pressure flow through backlash gap, mm3·s-1; cindicates backlash for connection of two trapped-oil volumes, mm;1,2andindicate trapped-oil pressure rate of change over time, Pa·s-1;1and2,indicate trapped-oil volume rate of change over time, Pa·s-1; similarly here in after.

圖2 連通側(cè)隙的計(jì)算

Fig.2 Gap calculation of backlash for connection

在側(cè)隙連通區(qū)內(nèi)的任何位置，由于困油流量與流進(jìn)、流出困油容積的各流量，始終處于平衡狀態(tài)，即Δp=p1?2≈1?p始終為一確定值。則2≡1≡0。

式中Δp為許可壓差，Pa，為節(jié)圓壓力角，rad，c為滿足Δp的側(cè)隙值，mm；為流量系數(shù)；為等效卷吸速度，mm/ s；為介質(zhì)密度，kg/m3。得

式中c,A和c,M為c均值和最大值，mm。其中，最大值的位置為=0.5(s+s)。

圖2中2個(gè)單一的1、2如能實(shí)現(xiàn)真正的連通，即需滿足Δp≡0，則真正連通下的側(cè)隙為

式中c為實(shí)現(xiàn)零壓差所需要的最小側(cè)隙，mm，為分度圓壓力角，rad。

例取=3 mm，=10，=20°，=30°，則c=2.41 mm，該值很大，小齒數(shù)的泵用齒輪副是無法實(shí)現(xiàn)的，即2個(gè)單一的1、2的真正連通是不存在的，只能是一定允許壓差下的近似連通。

在側(cè)隙卸荷區(qū)內(nèi)，對(duì)于單一的1或2，各自與外界交換的總流量，同樣也包括通過側(cè)隙的側(cè)隙卸荷流量、軸向流量、嚙合流量、卸荷槽的卸荷流量等，如將除側(cè)隙卸荷流量外的其他流量均以側(cè)隙卸荷流量的形式體現(xiàn)，則可折算出相應(yīng)的卸荷側(cè)隙。對(duì)于1，由[27]

得

式中Δp1為許可壓差，Pa，1為滿足Δp1時(shí)1所需的折算側(cè)隙的下限值，mm，當(dāng)1>p，取“＋”；否則，取“－”。

對(duì)于2，由

得

式中Δp2為許可壓差，Pa，2為滿足Δp2時(shí)2所需的折算側(cè)隙的下限值，mm，當(dāng)2>p，取“＋”；否則，取“－”。

取

式中c為c1和c2中的最大值，mm。且

式中c和c為c的均值和最大峰值，mm；K為c與c的比值，1max、2max為困油壓力1、2的最大峰值，Pa。

由于1max≈2max，所以K大于1，即c>c。定義

式中max與min為側(cè)隙的上、下限。

由此定義出側(cè)隙的大與小，max時(shí)為超大側(cè)隙，問題均得到了解決。

5 實(shí)例運(yùn)算及驗(yàn)證

文獻(xiàn)[17]提供了試驗(yàn)用側(cè)隙=30、50、200m的一組試驗(yàn)結(jié)果，如圖3a所示。試驗(yàn)用泵的幾何參數(shù)為模數(shù)4.75齒數(shù)10，齒頂圓直徑58 mm，中心距48.8 mm，=12.7 mm，由齒輪副三維模型測(cè)得0=20 mm3，對(duì)應(yīng)于試驗(yàn)用側(cè)隙分別為30、50、200m的壓力角為23°50′、23°40′、23°10′，出口壓力p為2、1、2 MPa。=870 kg/m3，=0.09 Pa·s，=0.62。由圖3a中，知轉(zhuǎn)速=1 000 r/min下困油壓力的峰值為2.7、3.7、7.3 MPa。另外，由矩形卸荷槽距中心線的距離B=6.4 mm，經(jīng)計(jì)算知采用的是大側(cè)隙卸荷槽，其卸荷效果一般[13]。

注：Δpd為連通許可壓差，Pa；Δps2為卸荷許可壓差，Pa；p2max為p2的最大值，Pa；Bd表示卸荷槽離中心線的距離，h表示試驗(yàn)用側(cè)隙，下同。

經(jīng)計(jì)算得傳動(dòng)性能所要求的0.0050.01的側(cè)隙為23.75～47.5m；困油性能所要求的0.01～0.08的側(cè)隙為47.5～380m；0=3～7m。由此可見，泵用齒輪副均為有側(cè)隙齒輪副。

分別將Δp取2.7、3.7、7.3 MPa和Δp2取0.7、2.7、5.3 MPa，分別代入式（11）和式（18），計(jì)算結(jié)果，如圖4所示。其中，圖3a為試驗(yàn)結(jié)果，圖3b為擇算后的連通側(cè)隙，圖3c為擇算后的卸荷側(cè)隙，其最大值和均值，如表1所示。其中，d,和s,為由卸荷槽關(guān)閉點(diǎn)位置，即B=6.4 mm計(jì)算出的連通側(cè)隙和卸荷側(cè)隙。

困油壓力峰值多出現(xiàn)在卸荷槽關(guān)閉點(diǎn)附近[28-29]，動(dòng)態(tài)的流場(chǎng)分析也說明了這一點(diǎn)[30]，即此時(shí)的卸荷槽的卸荷流量近乎為零，嚙合流量本來就很小，也近乎忽略。因此，在卸荷槽關(guān)閉點(diǎn)附近，除側(cè)隙外的其他交換流量近乎為軸向流量。

表1 六類側(cè)隙值的計(jì)算結(jié)果

注：d,、d,表示連通側(cè)隙的均值和最大峰值；s,、s,表示卸荷側(cè)隙的均值和最大峰值；d,表示卸荷槽關(guān)閉點(diǎn)附近的連通側(cè)隙值；s,表示卸荷槽關(guān)閉點(diǎn)附近的卸荷側(cè)隙值；為大側(cè)隙時(shí)理論值與試驗(yàn)值的相對(duì)誤差，%。

Note:d,andd,indicate the mean value and the maximum peak value of backlash gap used for the connection of two trapped-oil volumes;s,ands,indicate the mean value and the maximum peak value of backlash gap used for trapped-oil relief;d,indicates the backlash gap used for the connection of two trapped-oil volumes when the trapped-oil relief was closed;s,indicates the backlash gap used for trapped-oil relief when the trapped-oil relief was closed;indicates the relative error between theoretical value and experimental value when large backlash,%.

表1中=200m下的擇算側(cè)隙值s,=215m，即其中的s,?15m為以軸向流量為主的其他流量的擇算量，其他流量忽略后的誤差為(s,?)/=7.5%。且max=239m≥=200m，說明有(max?)/≈20%的安全裕度，比較可靠。

圖3a中=30、50m本為小側(cè)隙，在所計(jì)算出的max中超過60%的部分，用于彌補(bǔ)其他流量，此時(shí)也就不存在所謂的連通問題。由于齒輪泵側(cè)隙常大于85m[14]，=30、50m在實(shí)際運(yùn)用中，并不常見。

在=30、50m的情況下，如采用min=62、76m作為實(shí)際的側(cè)隙值，則由式(15)的計(jì)算，原有的Δp為7.3、3.7 MPa，減小后為1.7、1.6 MPa，此時(shí)2個(gè)單一的1、2可視為近乎的連通。說明以min作為側(cè)隙大與小的界定參考，是可行的。

6 結(jié) 論

1）側(cè)隙卸荷區(qū)和側(cè)隙連通區(qū)的困油流量比為3，側(cè)隙卸荷區(qū)內(nèi)的卸荷負(fù)擔(dān)最大。側(cè)隙連通區(qū)的真正連通，所需側(cè)隙高達(dá)2.41 mm，實(shí)際上并不存在。

2）卸荷用側(cè)隙大于連通用側(cè)隙，以107m的連通側(cè)隙作為側(cè)隙大與小的分界點(diǎn)，以239m的卸荷側(cè)隙作為側(cè)隙上限的界定可行。計(jì)算側(cè)隙與試驗(yàn)側(cè)隙的誤差為7.5%，且約有20%的安全裕度，理論計(jì)算和試驗(yàn)結(jié)果比較吻合。

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Demarcated standard and verification of backlash relief in external gear pumps

Sun Fuchun, Li Yulong, Wen Changming, Zhong Fei

(,,610106,)

The gear pumps are used for pumping the working fluid, and its key component is a pair of gear pairs. In the power transmission, the backlash of the gear pair is used to form the lubricating oil film to avoid sticking due to the friction and heat expansion of the gear teeth, but it also affects the stability of the oil film. The choice of backlash in the gear pump is also limited. The backlash of the gear has an influence on the trapped oil performance and volumetric efficiency of the gear pump, while the definition whether there is a backlash existed and the definition what is large backlash and what is small backlash is vague. Based on the common requirements of different backlash values to transmission performance and trapped oil performance, the special trapped-oil circulation and trapped-oil process were analyzed in this study, and the emphasis was on the double teeth meshing range and the single tooth meshing range. From the connection aspect of two different trapped-oil volumes in double teeth meshed range and the improvement aspect of trapped oil performance in single tooth meshed range, we used to separately calculate dynamic trapped-oil flow rate and its maximum value under the two different ranges of double teeth meshing and single tooth meshing, as well as the different formulas to separately calculate dynamic backlash values, its mean value, its lower limiting value used for the connection in double teeth meshed range, and the relief in single tooth meshed range. From such exercises, we derived the definition what was large backlash and what was small backlash. The backlash therefore was defined as, a small backlash was when the backlash value was less than the maximum peak of the backlash for connection, and when the backlash value was greater than the maximum peak of the backlash for connection and less than the maximum peak of the backlash for trapped oil relief, it was a large backlash, and when the backlash value was greater than the maximum peak of the backlash for trapped oil relief, it belonged to the large backlash. An instance of an external gear pump which backlash was 30, 50, 200m, was operated and its operation results were analyzed by the theory we developed. The results showed that when the trapped oil flow peak ratio of the unloading area and the connected area was 3, the unloading burden of the former was large. In fact, the really communicating to the communication area required up 2.41 mm backlash, which did not actually exist. So the gear pairs used in gear pumps was the gear pairs with backlash forever. The absolute connection of each other of two different trapped-oil volumes in double teeth meshed range and the absolute trapped-oil relief in single tooth meshed range were nonexistent, but only relatively existed under a certain permission pressure difference. As long as the trapped-oil relief requirement in single tooth meshed range was satisfied by an adopted backlash value, then the connection of two different trapped-oil volumes each other would naturally be met by the adopted backlash value. Backlash for trapped oil relief was larger than the backlash for connection, which can be used in definition what was large backlash and what was small backlash, and the backlash for trapped oil relief can be used as the upper limit. The error in the calculation and experiment was 7.5%, which was reasonable, and the upper limit of the safety margin was 20%, which was reliable. This research provided a reference for distinguishing large backlash between small backlash by Pump backlash defining, and which also provided a theoretical basis for the subsequent studies.

pumps; vibrations; gear pumps; backlash for connection; backlash for relief; demarcated size; trapped-oil flow rate

10.11975/j.issn.1002-6819.2017.20.008

TH325; TH137.3

A

1002-6819(2017)-20-0061-06

2017-04-18

2017-09-01

四川省自然科學(xué)重點(diǎn)資助項(xiàng)目（16ZA0382）；北京衛(wèi)星制造廠資助項(xiàng)目（20804）。

孫付春，博士，副教授，主要從事農(nóng)業(yè)機(jī)械方面的基礎(chǔ)研究與應(yīng)用開發(fā)。Email：fch.sun@163.com

孫付春，李玉龍，文昌明，鐘 飛. 齒輪泵側(cè)隙卸荷的界定標(biāo)準(zhǔn)與驗(yàn)證[J]. 農(nóng)業(yè)工程學(xué)報(bào)，2017，33(20)：61－66. doi：10.11975/j.issn.1002-6819.2017.20.008 http://www.tcsae.org

Sun Fuchun, Li Yulong, Wen Changming, Zhong Fei. Demarcated standard and verification of backlash relief in external gear pumps[J]. Transactions of the Chinese Society of Agricultural Engineering (Transactions of the CSAE), 2017, 33(20): 61－66. (in Chinese with English abstract) doi：10.11975/j.issn.1002-6819.2017.20.008 http://www.tcsae.org