干摩擦的非線性動(dòng)力學(xué)外文翻譯@中英文翻譯@外文文獻(xiàn)翻譯

第 1 頁(yè) 共 10 頁(yè) 干摩擦的非線性動(dòng)力學(xué) Franz-Josef Elmer 瑞士巴塞爾，巴塞爾大學(xué)，物理學(xué)院 CH-4056 1996 年 11 月 5日收到來(lái)稿， 1997 年 5 月 21 日決定發(fā)表 摘要 研究一個(gè)有牽引力的彈簧 木塊系統(tǒng)以一個(gè)恒定的速度在一個(gè)表面上運(yùn)動(dòng)所受到的干摩擦動(dòng)力學(xué)。一個(gè)很普遍且符合摩擦學(xué)現(xiàn)象規(guī)律的動(dòng)力學(xué)推理正在研究中（此規(guī)律為：靜止時(shí)受靜摩擦力影響，運(yùn)動(dòng)速度受動(dòng)摩擦力影響）。共有三種可能發(fā)生的運(yùn)動(dòng) :粘 滑運(yùn)動(dòng)、連續(xù)滑動(dòng)、以及無(wú)粘性的振動(dòng)?，F(xiàn)在將要闡述地方以及全球的一些令人矚目的人所提出的分歧觀點(diǎn)以及他們的觀點(diǎn)極 其相似的不穩(wěn)定觀點(diǎn)。 庫(kù)侖關(guān)于干摩擦的 l定律被應(yīng)用了 200 多年。他規(guī)定摩擦力等于由物體本身材料所決定的摩擦系數(shù)與正壓力的乘積。靜摩擦系數(shù)（靜摩擦力是使物體由靜止開(kāi)始滑動(dòng)的力）通常等于或大于動(dòng)摩擦系數(shù)（動(dòng)摩擦力是使物體以一個(gè)恒速度運(yùn)動(dòng)的力）。 一個(gè)機(jī)械系統(tǒng)受到的干摩擦力是非線性的，因?yàn)閹?kù)侖定律把動(dòng)摩擦力和靜摩擦力區(qū)分開(kāi)了。如果動(dòng)摩擦系數(shù)小于靜摩擦系數(shù)，在一個(gè)滑動(dòng)并且粘住和滑動(dòng)轉(zhuǎn)化十分規(guī)則就如 l所說(shuō)的表面上，粘 滑運(yùn)動(dòng)就會(huì)發(fā)生。這種急變的運(yùn)動(dòng)發(fā)生在日常生活中的每一天，例如開(kāi)關(guān)門(mén)和拉小提琴。 即使庫(kù)侖定律 是很簡(jiǎn)單并且很容易確定的（工程上一些計(jì)算都依據(jù)這些公式），它關(guān)于靜摩擦產(chǎn)生的原因要求并不嚴(yán)格，因?yàn)殪o摩擦只是一個(gè)過(guò)程，它的作用不會(huì)涉及到平衡。因此不必為庫(kù)侖定律經(jīng)常應(yīng)用于實(shí)驗(yàn)這個(gè)事實(shí)背離而讓我們感到驚訝。典型的背離例子是下面所列出的：（ i）靜摩擦力是變化的而且是隨著靜止時(shí)間的延續(xù)而逐漸增加的 2， 3，即兩個(gè)相接觸的滑行表面而沒(méi)有發(fā)生相對(duì)運(yùn)動(dòng)。（ ii）動(dòng)摩擦力決定于滑動(dòng)速度；對(duì)于一個(gè)很大的速度，它可以近似地看作是線性地增加，這個(gè)速度就像是在 粘性摩擦力場(chǎng)那樣。在達(dá)到很大速度之前，摩擦力首先減小，直到減小到最小值，然后再繼續(xù)增大 3， 4。在有潤(rùn)滑油的臨界條件下（即是在滑動(dòng)的表面有極少的起潤(rùn)滑作用的單分子層存在）摩擦力將以一個(gè)很小的速度再次減?。ㄒ?jiàn)圖 1） 5， 6。動(dòng)摩擦系數(shù)作為影響滑動(dòng)速度的因素，至少有一個(gè)極值。動(dòng)摩擦力可能大于靜摩擦力，但是在物體將要 第 2 頁(yè) 共 10 頁(yè) 滑動(dòng)的瞬間，動(dòng)摩擦力始終是小于或等于靜摩擦力的。 本文的目的是針對(duì)非線性動(dòng)力學(xué)中在某一個(gè)程度上狹隘的諸如上面提到的干摩擦定律提出一個(gè)自由的論點(diǎn)。本文將拋開(kāi)在著作 1,3,4,6-9中已提出的明確 的定律。關(guān)于摩擦力的現(xiàn)象學(xué)的定律只是在肉眼所見(jiàn)的程度內(nèi)，這就意味著用顯微鏡可見(jiàn)的精微的程度將遠(yuǎn)遠(yuǎn)超過(guò)肉眼可見(jiàn)的程度。在本文的結(jié)束語(yǔ)中將要給出一個(gè)為什么這種假設(shè)總不是有效的簡(jiǎn)單論點(diǎn)。要去揭示這種宏觀現(xiàn)象的無(wú)根據(jù)性，因此去了解這種時(shí)空分離的假設(shè)下的完整的動(dòng)力學(xué)知識(shí)是很重要的。 圖 1 (a)圖為典型的速度影響的動(dòng)摩擦 定律的示意性草圖 (b)圖為有潤(rùn)滑油的臨界條件下的系統(tǒng) 圖 2 干摩擦諧波振蕩器 第 3 頁(yè) 共 10 頁(yè) 對(duì)于不同的干摩擦定律有兩個(gè)很重要的眾所周知地前提。（ i） 摩擦系數(shù)只能在儀器內(nèi)部測(cè)量（比如表面力測(cè)量?jī)x 10或摩擦力顯微鏡 11）。下面我們將會(huì)發(fā)現(xiàn)系統(tǒng)的運(yùn)動(dòng)狀態(tài)主要受摩擦力和儀器影響。例如粘 滑運(yùn)動(dòng)狀態(tài)就比變化速度作用下的狀態(tài)難于直接得出動(dòng)摩擦系數(shù)。因此，測(cè)量?jī)x器的影響是不能忽略掉的。（ ii）在粒狀材料中干摩擦也是一個(gè)重要的物理量 12。一些相互作用的雙尺是否合力作用的展開(kāi)話題將久遠(yuǎn)地影響庫(kù)侖動(dòng)力學(xué)定律的修改。 在兩個(gè)滑動(dòng)表面的機(jī)械環(huán)境下（比如儀器）肉眼可見(jiàn)的自由程度很大。最重要的一個(gè)是側(cè)面的。這里僅討論在單一程度 范圍內(nèi)描述的具體系統(tǒng)。圖 2 明確地說(shuō)明了儀器的構(gòu)成。諧波振蕩器是這樣組成的：一個(gè)木塊（質(zhì)量為 M）由一個(gè)彈簧（倔強(qiáng)系數(shù)為 k）與一個(gè)固定施力系統(tǒng)連接（見(jiàn)圖 2）。木塊與一個(gè)以恒定速度 v0 滑動(dòng)的表面接觸。木塊和滑動(dòng)表面之間的交互作用是靜止時(shí)的靜摩擦力 )(stickS tF和運(yùn)動(dòng)時(shí)的動(dòng)摩擦力 FK(v)的合力。在列寫(xiě)運(yùn)動(dòng)狀態(tài)方程時(shí)，我們必須區(qū)分木塊是處于粘住狀態(tài)還是滑動(dòng)狀態(tài)。如果它處于粘住狀態(tài)，它的位移 x將隨時(shí)間線性地增加，直到彈簧的彈力 )(kx 大于靜摩擦力SF。 因此 kttFxifvxrS /)(0 (1a) 當(dāng) ttr 時(shí)木塊在前一次滑動(dòng)后再一次回到原狀態(tài)。 如果木塊滑動(dòng)，運(yùn)動(dòng)方程為 )()( 00 xvFxvs i g nkxxM K （ 1b） 如果 kFxorvxS /)0(0 ， sign(x)表示 x的坐標(biāo)。 我們的研究將以庫(kù)侖定律關(guān)于持續(xù)靜和動(dòng)摩 擦的研究為基礎(chǔ)。當(dāng)0vx時(shí)，系統(tǒng)的狀態(tài)就像一個(gè)干燥的諧波振蕩器在平衡位置附近以 kFK / 這么大的位移振蕩。因而，有很多解決振動(dòng)的方法。在下面我們將要看到一些在速度影響下的動(dòng)摩擦情況下依然存在的方法。木塊的平衡位置在 k/)v(0KFx ,它被稱(chēng)為連續(xù)變化狀態(tài)。 第 4 頁(yè) 共 10 頁(yè) 每一個(gè)具有能夠使振動(dòng)的最大速度大于 v0 的初始狀態(tài)都將導(dǎo)致在一個(gè)有限時(shí)間內(nèi)實(shí)現(xiàn)粘 滑狀態(tài)的轉(zhuǎn)變。而滑動(dòng)狀態(tài)是不受初始狀態(tài)影響的，將以0,/ vxkFx S 開(kāi)始運(yùn)動(dòng)。然而，粘 滑系統(tǒng)定義了一個(gè)具有說(shuō)服力的相位周期變化的規(guī)律。這和系統(tǒng)的狀態(tài)像干燥的諧波振蕩器的系統(tǒng)狀態(tài)并不是矛盾的。原因是：如果它在相位空間采樣，相位的變化范圍已被約定在了一條直線上如 (la)式所示。粘 滑狀態(tài)要求動(dòng)摩擦力KF必須小于靜摩擦力。通常粘住狀態(tài)時(shí)間 )/()(20kvFFt KSs tic k 遠(yuǎn)遠(yuǎn)大于滑動(dòng)狀態(tài)時(shí)間 kMkMvFFt KSs l i p /)()a r c ta n (2 2/110 。粘 滑狀態(tài)的最大振幅（即 )(max txt ）函數(shù)是一個(gè)關(guān)于 v0的不規(guī)律變化的函數(shù)，其中 v0是由 kFS /決定的，而且初始狀態(tài) 00 v。這個(gè)表達(dá)式對(duì)于一個(gè)速度影響的動(dòng)摩擦同樣適用。 未經(jīng)修改的庫(kù)侖定律致使以任何一個(gè)滑動(dòng)速度 v0 的持續(xù)變化狀態(tài)與粘 滑狀態(tài)是同時(shí)存在的。在一個(gè)速度影響的動(dòng)摩擦系統(tǒng)的一般范圍內(nèi)，這種雙穩(wěn)態(tài)都將存在，但是速度 v0要有一個(gè)嚴(yán)格的限制范圍。特別是當(dāng)在粘 -滑運(yùn)動(dòng)狀態(tài)出現(xiàn)臨界速度cv時(shí)。比如日常的一個(gè)現(xiàn)象： 快速地開(kāi)關(guān)門(mén)就可以消除它的吱吱的聲音。 我們?yōu)榱烁康厝ソ鉀Q線性運(yùn)動(dòng)的 FK關(guān)于 v 的狀態(tài)方程 0,)(0 vFvF KK。 方程（ lb）將變形為能夠容易解決的不干燥諧波振蕩器的方程。用一個(gè)具有權(quán)威性的持續(xù)變化狀態(tài)代替擺動(dòng)變化的解決辦法。如果軌跡0)0(,/)0( vxkFx S ,t0 時(shí)不再粘住，粘 滑狀態(tài)將會(huì)消失。臨界速度cvv 0由兩個(gè)方程00 )(,/( vtxkFtx s lipKs lip 得出。這樣將推出兩個(gè)關(guān)于slipt和cv非線性數(shù)學(xué)方程。當(dāng) Mk 時(shí)，可以近似認(rèn)為 )(2 0 kMFFv KSc （ 2） 臨界速度cv在粘 滑狀態(tài)的性質(zhì)探討中是很重要的一個(gè)物理量，因?yàn)閺乃臏y(cè)量方法中我們可以間接地知道有關(guān)機(jī)械裝置的干摩擦的知識(shí)（見(jiàn)論點(diǎn) 6）。 接下來(lái)討論變化狀態(tài)的)(vKF，正如圖 1 的例子 一樣。假設(shè)靜摩擦力SF始終是存在 第 5 頁(yè) 共 10 頁(yè) 的。0v為任何值時(shí)連續(xù)滑動(dòng)狀態(tài)都是存在的，但它僅僅在 0/)(00 dvvdFF KK時(shí)是穩(wěn)定的。在 )(vFK達(dá)到一個(gè)極值時(shí)這種穩(wěn)定狀態(tài)改變并且 Hopf 分歧出現(xiàn)。在接近極值并與連續(xù)滑動(dòng)狀態(tài)有很小的背離時(shí)， .)()()( /0 ccetAk vFtx tMkiK （ 3） 由振幅決定（標(biāo)準(zhǔn)形式） 13 AAMkMFiMFkAMFdtdA KKK 222 )(4(2 （ 4） 如果關(guān)于動(dòng)摩擦的極值的第三種說(shuō)法是絕對(duì)的，而 分歧是超臨界的，另外如上面所提到的眾所周知的觀點(diǎn)，另一種觀點(diǎn)產(chǎn)生了。這里稱(chēng)為擺動(dòng)滑動(dòng)狀態(tài)。它是一個(gè)最大的速度總是小于 v0 的這個(gè)有限的循環(huán)周期。這樣木塊決不會(huì)粘住。它的頻率由左手邊 (1b)的諧波振蕩器粗略地給出。動(dòng)摩擦的第二種說(shuō)法對(duì)于非線性頻率去諧是有效的。值得一提的是粘 滑振蕩器的頻率通常是遠(yuǎn)遠(yuǎn)小于擺動(dòng)變化狀態(tài)的。這種擺動(dòng)狀態(tài)與雷利的周期方程 130)( 3 uuuu 十分相似，事實(shí)上，雷利方程是（ 1b）方程的一個(gè)特例。因?yàn)閯?dòng)摩擦，幾種穩(wěn)定和不穩(wěn)定周期循環(huán)可能存在。通過(guò)改變 v0，產(chǎn)生和消除相互作用來(lái)承受分歧點(diǎn)。 應(yīng)該強(qiáng)調(diào)一點(diǎn)由（ 4）描述的 Hopf的 分歧觀點(diǎn)和 Heslot et al 3評(píng)述的 Hopf的 分歧觀點(diǎn)是沒(méi)有關(guān)系的 。后者的評(píng)述在一次政體（稱(chēng)為爬行的政體）上提出，（ 1a）是不適用的（同下面關(guān)于干摩擦定律有效性一致）。 一個(gè)擺動(dòng)變化狀態(tài)只有在它的最大速度小于由粘性狀態(tài)（ 1a）決定的滑動(dòng)速度 v0時(shí)才存在。擺動(dòng)變化狀態(tài)和粘住狀態(tài)是怎么樣相互作用影響 粘 滑運(yùn)動(dòng)？為了回答這個(gè)問(wèn)題，我們相反地計(jì)算點(diǎn)的軌跡 ),/)0(lim00 vkF K與 (lb)一致。三種具有說(shuō)服力的不同的軌跡是可能存在的。 （ 1） 利用相反軌道法采樣粘住狀態(tài)。他們一起定義了一個(gè)有界限的導(dǎo)致非線性軌道的初始狀態(tài)的裝置。這套裝置的界限專(zhuān)門(mén)稱(chēng)為粘 滑狀態(tài)邊界；它是一個(gè) 第 6 頁(yè) 共 10 頁(yè) 不可能存在的軌道，但是它把粘 滑振動(dòng)和非粘 滑狀態(tài)這兩個(gè)難以分開(kāi)的狀態(tài)區(qū)分開(kāi)了。 （ 2） 相反軌道法向內(nèi)部盤(pán)旋接近于一個(gè)非穩(wěn)定狀態(tài)或非連續(xù)滑動(dòng)狀態(tài)。此外所有的初始狀態(tài)除這些抵制狀態(tài)外都有一個(gè)固定的粘 滑周期。 （ 3） 這種相反軌道法向外盤(pán)旋 趨向于無(wú)窮，粘 滑狀態(tài)是不會(huì)發(fā)生的。 兩種局部分歧是由可能存在的：如果相反軌跡改變了從第一種情況到第三種情況的固定粘 滑循環(huán)，使粘 滑界限消失。從第一種情況到底二種情況的這種粘 滑界限也被非穩(wěn)定連續(xù)狀態(tài)或擺動(dòng)滑動(dòng)狀態(tài)或是它以消失變?yōu)榉€(wěn)定連續(xù)狀態(tài)或擺動(dòng)滑動(dòng)狀態(tài)。從第二種情況到第三種情況的轉(zhuǎn)變是不可能發(fā)生的。見(jiàn)圖 3， 對(duì)于 )(vFK 的一個(gè)特殊值，將有兩種分歧觀點(diǎn)產(chǎn)生。第一個(gè)是在 v0 0.059,0.082,0.966。第二個(gè)是在 v0 0.162,0.785。這個(gè)例子說(shuō)明了不 斷增加 v0,粘 滑運(yùn)動(dòng)可能消失也可能再產(chǎn)生。 此外著名的雙穩(wěn)態(tài)的粘 滑運(yùn)動(dòng)和連續(xù)滑動(dòng) 3，連續(xù)滑動(dòng)狀態(tài)，幾種擺動(dòng)滑動(dòng)狀態(tài)，和粘 滑振蕩器的穩(wěn)定性都是有可能的（見(jiàn)圖 3）。最后所有的觀點(diǎn)都認(rèn)為除了連續(xù)滑動(dòng)狀態(tài)，非常大的滑動(dòng)速度將會(huì)消失因?yàn)閯?dòng)摩擦力將要充分地影響這個(gè)很大的速度。 超強(qiáng)的過(guò)阻尼極限（ i.e. kMdvvdFk /)(適用于任何 v除了在極值里微小的距離）導(dǎo)致了時(shí)間的分離。對(duì)于一個(gè)相位圖上的任一點(diǎn)（ ),( xx ，并且0vx的狀態(tài)將快速地向點(diǎn) ),( vx 變化， v 由 0)(),(00 vvFvvFkx kk。在曲線 )(0 xvFkx k 上的點(diǎn)，當(dāng)0kF 時(shí)時(shí)不穩(wěn)定的。他們分離了不同的 v 的求解辦法。駐留系統(tǒng)太久的快速運(yùn)動(dòng)之后將要沿著曲線 )(0 xvFkx k 變化。方向由 x 的符號(hào)決定。它也將到達(dá)穩(wěn)定連續(xù)滑動(dòng)狀態(tài)，或者是，接近一個(gè)極限值 KF ,它將突然地向曲線分歧轉(zhuǎn)變或者是向粘住狀態(tài)。如圖 l(b)所示動(dòng)摩擦定律在兩個(gè)極值之間的 v0,出現(xiàn)振動(dòng)滑動(dòng)狀態(tài)。這是一個(gè)不嚴(yán)密的振動(dòng)可能難以區(qū)別粘 滑擺動(dòng)狀態(tài)。如圖 l(a)中單一的最小速度為mvv的摩擦定律的情形下我們可以得到mvv 07的粘 滑狀態(tài)。在超強(qiáng)過(guò)阻尼界限任何雙穩(wěn)定狀態(tài)都將消 第 7 頁(yè) 共 10 頁(yè) 失，除了接近 )(vFK極值。 Yoshizawa 和 Israelachvili 14的實(shí)驗(yàn)是一致的，他們假設(shè)系 統(tǒng)處于超強(qiáng)過(guò)阻尼界限的摩擦定律如圖 1(a)7所示。 為了討論依賴(lài)靜止時(shí)間的靜摩擦力影響下的粘 滑狀態(tài)，我們建立了粘 滑坐標(biāo)系圖 3 典型的分歧觀點(diǎn)精確的動(dòng)摩擦力 )(vFK 見(jiàn)圖 l(b)。接下來(lái)的作用力由下式?jīng)Q定 : 2/)()()()( 221221 vvvvvvvF k ，其中05.0,2.0,1.0,3 21 vv 。由運(yùn)動(dòng)（ l） 的方程積分就可以得到結(jié)果。其他的參數(shù)如 1,40,1 kMFs。各段曲線表示了穩(wěn)定和不穩(wěn)定連續(xù)滑動(dòng)狀態(tài)（ CS），擺動(dòng)滑動(dòng)狀態(tài)（ OS），或者是粘 滑運(yùn)動(dòng)（ SS）。一系列曲線說(shuō)明 了粘 滑分界線。 第 8 頁(yè) 共 10 頁(yè) )(1 nn xTx ， nx 是開(kāi)始滑動(dòng)的位置。對(duì)于恒定不變的靜摩擦力在圖中規(guī)定為kFxT S /)( 。僅僅在滑動(dòng)狀態(tài)向靜止?fàn)顟B(tài)轉(zhuǎn)變的時(shí)候位移定義為 snx 。它是 nx 的函數(shù)，即 )(nsn xgx ,函數(shù) g通常是一個(gè)單調(diào)遞減的函數(shù)。靜止時(shí)間 sticknt是這個(gè)函數(shù)的最小實(shí)根 s tic knsns tic knS tvxktF 0)( （ 5） 這就定義了一個(gè)函數(shù) )( nsstickn xht 而且在 0SF時(shí)是單調(diào)遞減函數(shù)。這樣粘 滑圖可以這樣定義 kxghFxTS /)()( 。當(dāng)圖只有一個(gè)點(diǎn)時(shí)，粘滑運(yùn)動(dòng)就是存在的。 當(dāng) KF =常量 = )0(SF時(shí)，粘滑運(yùn)動(dòng)消失 )/()0()(2s u p00 ktFtFvv SStc 。對(duì)于非凸起函數(shù) FS(t), 在粘 滑圖中靜止時(shí)間導(dǎo)致鞍狀節(jié)點(diǎn)穩(wěn)定和不穩(wěn)定的固定點(diǎn)在非零值處出現(xiàn)了上確限。與凸起的 FS(t)8的情形相比在cvv 0時(shí)粘 滑運(yùn)動(dòng)有一個(gè)固定振幅。因?yàn)?T是一個(gè)單調(diào)遞增的函數(shù)，界限循環(huán)或平均混亂都是沒(méi)有可能的。 如果滑動(dòng) 靜止轉(zhuǎn)變不發(fā)生在 x 變?yōu)榕c v0（因?yàn)?kFxSns /)0(）相等的第一次。在這種情況下將得到一個(gè)由于非單調(diào)變化的 g 導(dǎo)致的非單調(diào)變化的 T。如果 )0(/)(SS FF 變得相當(dāng)大這樣過(guò)于發(fā)射是可能的。例如，對(duì)于一個(gè)持續(xù)變化的動(dòng)摩擦力如果)0(/)0(1)0(/)( FFFF KSS 過(guò)發(fā)射將要發(fā)生。對(duì)于實(shí)實(shí)在在的系統(tǒng)這種狀況是不可能發(fā)生的。注意混亂的可能性和對(duì)于不變的 FS的運(yùn)動(dòng) (l)的方程不可能表現(xiàn)出混亂狀態(tài)的事實(shí)是 不矛盾的。但是由于 FS的延遲使 (l)變?yōu)榱艘环N微分延遲方程。 利用干摩擦的現(xiàn)象學(xué)意味著我們把干摩擦看作是一個(gè)機(jī)械電路的元素并帶有非線性的速度量和力的性質(zhì)，比如，說(shuō)，在一個(gè)電路里的二極管。只要在肉眼可見(jiàn)的時(shí)間量程比任何互相作用的固體表面內(nèi)在的自由程度時(shí)間量程都大，這種看法就是可行的。但是有一種內(nèi)在的時(shí)間量程是分歧的，就是表面的相對(duì)速度變?yōu)?0：它等于表面特有的側(cè)部長(zhǎng)度值和相對(duì)速度的比值。這樣，任何動(dòng)摩擦定律 FK(v)都會(huì)變的殘缺不全如果 第 9 頁(yè) 共 10 頁(yè) t i m es ca l ecm i cr o s co p is ca l el en g t hcm i cr o s co p iv （ 6） 這個(gè)特有的長(zhǎng)度的范圍從幾微米到幾米。這可能是一個(gè)粗略的范圍，粗略的接觸范圍，表面粗糙度的相關(guān)長(zhǎng)度，或者是彈性相關(guān)長(zhǎng)度。任何干摩擦定律的局限性都不會(huì)涉及到擺動(dòng)滑動(dòng)狀態(tài)和持續(xù)滑動(dòng)狀態(tài)，只要它們的相對(duì)滑動(dòng)速度始終保持在比臨界速度 (6)大的范圍內(nèi)。但是在粘 滑運(yùn)動(dòng)中，粘住向滑動(dòng)轉(zhuǎn)變之后的瞬間和在滑動(dòng)向粘住轉(zhuǎn)變前的瞬間將大大影響粘住和滑動(dòng)的轉(zhuǎn)變 ，界面的動(dòng)力學(xué)行為細(xì)節(jié)會(huì)變得很重要。當(dāng)最大相對(duì)滑動(dòng)速度減少的時(shí)候這些細(xì)節(jié)資料的重要性將增加。例如， Heslot et al 3 用實(shí)驗(yàn)方法創(chuàng)立了十分完整的不同的行為在一個(gè)滑動(dòng)過(guò)程中當(dāng)最大相對(duì)速度比臨界速度值 (6)小。 本文在干摩擦能描述為速度依賴(lài)的動(dòng)摩擦和靜止時(shí)間依賴(lài)的靜摩擦的假設(shè)下討論了諧波振蕩器在一個(gè)固體平面上滑動(dòng)的非線性動(dòng)力學(xué)。除眾所周知的持續(xù)滑動(dòng)狀態(tài)和粘 滑振蕩器之外，建立了一種沒(méi)有粘住的擺動(dòng)滑動(dòng)狀態(tài)。所有這些分歧點(diǎn)都在圖 3中表示了出來(lái)。 致謝 我充滿(mǎn)感激地致謝托馬斯以及他的原稿讀物。本作品由瑞 士國(guó)家自然科學(xué)基金會(huì)支持。 參考文獻(xiàn) 1 Bowden F P and Tabor D 1954 Friction and Lubrication (Oxford: Oxford University Press) 2 Rabinowicz E 1965 Friction and Wear of Materials (New York: Wiley) 3 Heslot F, Baumberger T, Perrin B, Caroli B and Caroli C 1994 Phys. Rev. E 49 4973 4 Burridge R and Knopoff L 1967 Bull. Seismol. Soc. Am. 57 341 5 Bhushan B, Israelachvili J N and Landman U 1995 Nature 374 607 6 Berman A D, Ducker W A and Israelachvili J N 1996 The Physics of Sliding Friction ed B N J Persson and 第 10 頁(yè) 共 10 頁(yè) E Tosatti (Dordrecht: Kluwer Academic) 7 Persson B N J 1994 Phys. Rev. B 50 4771 8 Persson B N J 1995 Phys. Rev. B 51 13 568 9 Vetyukov M M, Dobroslavskii S V and Nagaev R F 1990 Izv. AN SSSR. Mekhanika Tverdogo Tela 25 23(Engl. transl. 1990 Mech. Solids 25 22) 10 Israelachvili J N 1985 Intermolecular and Surface Forces (London: Academic) 11 Mate C M, McClelland G M, Erlandsson R and Chiang S 1987 Phys. Rev. Lett. 59 1942 12 For an overview and more references on the physics of granular materials see Jaeger H M, Nagel S R and Behringer R P 1996 Physics Today 49 32 13 Kevorkian J and Cole J D 1981 Perturbation Methods in Applied Mathematics (New York: Springer) 14 Yoshizawa H and Israelachvili J 1993 J. Phys. Chem. 97 11 300 第 11 頁(yè) 共 10 頁(yè) Nonlinear dynamics of dry friction Franz-Josef Elmer Institut fur Physik, Universitat Basel, CH-4056 Basel, Switzerland Received 5 November 1996, in final form 21 May 1997 Abstract: The dynamical behaviour caused by dry friction is studied for a spring-block system pulled with constant velocity over a surface. The dynamical consequences of a general type of phenomenological friction law (stick-time-dependent static friction, velocity-dependent kinetic friction) are investigated. Three types of motion are possible: stickslip motion, continuous sliding, and oscillations without sticking events. A rather complete discussion of local and global bifurcation scenarios of these attractors and their unstable counterparts is present. Coulombs laws 1 of dry friction have been well known for over 200 years. They state that the friction force is given by a material parameter (friction coefficient) times the normal force. The coefficient of static friction (i.e. the force necessary to start sliding) is always equal to or larger than the coefficient of kinetic friction (i.e. the force necessary to keep sliding at a constant velocity). The dynamical behaviour of a mechanical system with dry friction is nonlinear because Coulombs laws distinguish between static friction and kinetic friction. If the kinetic friction coefficient is less than the static one, stickslip motion occurs where the sliding surfaces alternately switch between sticking and slipping in a more or less regular fashion 1. This jerky motion leads to the everyday experience of squeaking doors and singing violins. Even though Coulombs laws are simple and well established (many calculations in engineering rely on these laws), they cannot be derived in a rigorous way because dry friction is a process which operates mostly far from equilibrium. It is therefore no surprise that deviations from Coulombs laws have often been found in experiments. 第 12 頁(yè) 共 10 頁(yè) Typical deviations are as follows. (i) Static friction is not constant but increases with the sticking time 2, 3, i.e. the time since the two sliding surfaces have been in contact without any relative motion. (ii) Kinetic friction depends on the sliding velocity； for very large velocities, it increases roughly linearly with the sliding velocity like in viscous friction. Coming from large velocities, the friction first decreases, goes through a minimum, and then increases 3, 4. In the case of boundary lubrication (i.e. a few monolayers of some lubricant are between the sliding surfaces) it decreases again for very low velocities (see figure 1) 5, 6. The coefficient of kinetic friction as a function of the sliding velocity therefore has at least one extremum. The kinetic friction can exceed the static friction, but in the limit of zero sliding velocity it is still less than or equal to the static friction. The aim of this paper is to give a rather complete discussion of the nonlinear dynamics of a single degree of freedom for an arbitrary phenomenological dry friction law in the sense mentioned above. This goes beyond the discussion of specific laws found in the literature 1, 3, 4, 69. A phenomenological law for the friction force depends only on the macroscopic degrees of freedom. This implies that all microscopic degrees of freedom are much faster than Figure 1. Schematical sketches of typical velocity-dependent kinetic friction laws for (a) systems without and (b) systems with boundary lubrication. 第 13 頁(yè) 共 10 頁(yè) the macroscopic ones. At the end of this paper a simple argument will be given on why this assumption will not always be valid. To reveal this invalidity on the macroscopic level, it is therefore important to have a complete knowledge of the dynamical behaviour under the assumption that this timescale separation works. There are two other important reasons for knowing the consequences of the different dry friction laws. (i) Friction coefficients can only be measured within an apparatus (for example the surface force apparatus 10 or the friction force microscope 11). Below we will see that the dynamical behaviour of the whole system is strongly determined by the friction force and the properties of the apparatus. For example, stickslip motion makes it difficult to directly obtain the coefficient of kinetic friction as a function of the sliding velocity. Thus, the influence of the measuring apparatus cannot be eliminated. (ii) Dry friction also plays an important role in granular materials 12. An open question there is whether or not the cooperative behaviour of many interacting grains is significantly influenced by the dynamical behaviour due to modifications of Coulombs laws. The mechanical environment (e.g. the apparatus) of two sliding surfaces may have many macroscopic degrees of freedom. The most important one is the lateral one. Here only systems are discussed which can be well described by this single degree of freedom. Figure 2 schematically shows the apparatus. It is described by a harmonic Figure 2. A harmonic oscillator with dry friction. 第 14 頁(yè) 共 10 頁(yè) oscillator where a block (mass M) is connected via a spring (stiffness k) to a fixed support (see figure 2). The block is in contact with a surface which slides with constant velocity v0. The interaction between the block and the sliding surface is described by a sticking-time-dependent static friction force )(stickS tF and a velocity-dependent kinetic friction force )(vFK. For the equation of motion we have to distinguish whether the block sticks or slips. If it sticks, its position x grows linearly in time until the force in the spring (i.e. kx ) exceeds the static friction FS. Thus kttFxifvxrS /)(0 (1a) where ttr is the time at which the block has stuck again after a previous sliding state. If the block slips, the equation of motion reads )()( 00 xvFxvs i g nkxxM K (1b) if kFxorvxS /)0(0 where sign(x) denotes the sign of x. We start our investigation with Coulombs laws of constant static and kinetic friction.As long as 0vx the system behaves like an undamped harmonic oscillator with the equilibrium position shifted by the amount of kFK / . Thus, there are infinitely many oscillatory solutions. Below we will see that some of them may survive in the case of velocity-dependent kinetic friction. The equilibrium position of the block is k/)v(0KFx . It is called the continuously sliding state. Every initial state which would lead to an oscillation with a velocity amplitude exceeding v0 leads in a finite time to stickslip motion. Independent of the initial condition, the slips always start with0,/ vxkFx S . Thus, the stickslip motion defines an attractive limit cycle in phase space. This is not in contradiction with the 第 15 頁(yè) 共 10 頁(yè) fact that the system behaves otherwise like an undamped harmonic oscillator. The reason for that is that a finite bounded volume in phase space is contracted onto a line if it hits that part in phase space which is defined by (1a). Stickslip motion requires a kinetic friction FK which is strictly less than the static one. Usually the sticking time )/()(2 0kvFFt KSs tic k is much larger than the slipping time kMkMvFFt KSs l i p /)()a r c ta n (2 2/110 . The maximum amplitude of the stickslip oscillation (i.e. )(max txt) is a monotonically increasing function of v0 which starts at kFS / for v0= 0. This is also true for a velocity-dependent kinetic friction force. The unmodified Coulombs law leads to a coexistence of the continuously sliding state and stickslip motion for any value of the sliding velocity0v. In the more general case of a velocity-dependent kinetic friction this bistability still occurs but in a restricted range of v0. Especially, there will always be a critical velocity cvabove which stickslip motion disappears. This is an everyday experience: squeaking of doors can be avoided by moving them faster. In order to be more quantitative we solve the equation of motion for a linear dependence of KF on v, i.e. 0,)(0 w i t hvFvF KK.Equation (1b) becomes the equation of a damped harmonic oscillator which can be easily solved. Instead of a continuous family of oscillatory solutions we have an attractive continuously sliding state. Stickslip motion disappears if the trajectory with 0)0(,/)0( vxkFx S never sticks for t 0. The critical velocitycvv 0is defined by 00 )(,/( vtxkFtx s lipKs lip . It leads to two nonlinear algebraic equations forsliptandcv. For Mk the solution can be given approximately: )(2 0 kMFFv KSc 第 16 頁(yè) 共 10 頁(yè) (2) The critical velocity cv plays an important role in the discussion of the nature of stickslip motion, because its measurement tells us indirectly something about the mechanisms of dry friction (see the discussion in 6). Next we discuss a general non-monotonic)(vKFlike the examples shown in figure 1. The static friction SF is still assumed to be constant. The continuously sliding state exists for all values of 0vbut it is only stable if 0/)(00 dvvdFF KK. At an extremum of )(vFK the stability changes and a Hopf bifurcation occurs. Near the extremum and for small deviations from the continuously sliding state the dynamics of .)()()( /0 ccetAk vFtx tMkiK (3) is governed by the amplitude equation (normal form) 13 AAMkMFiMFkAMFdtdA KKK 222 )(4(2 (4) If the third derivative of the kinetic friction at an extremum is positive, the Hopf bifurcation is supercritical, and in addition to the well known attractors mentioned above, another type of attractor appears. Here it is called the oscillatory sliding state. It is a limit cycle where the maximum velocity always remains less than v0. Thus the block never sticks. Its frequency is roughly given by the harmonic oscillator of the left-hand side of (1b). The second derivative of the kinetic friction is responsible for nonlinear frequency detuning. Note that the frequency of the stickslip oscillator is usually much smaller than the frequency of the oscillatory sliding state. This oscillatory state is similar to the limit cycle of Rayleighs equation 第 17 頁(yè) 共 10 頁(yè) 130)( 3 uuuu , in fact, Rayleighs equation is a special case of (1b). Depending on the kinetic friction, several stable and unstable limit cycles may exist. By varying v0 they are created or destroyed in pairs due to saddle-node bifurcations. It should be noted that the Hopf bifurcation described by (4) is not related to the Hopf bifurcation observed by Heslot et al 3 which occurs in a regime (called the creeping regime) where (1a) is not applicable (see also the discussion below about the validity of dry friction laws). An oscillatory sliding state exists only if its maximum velocity is smaller than the sliding velocity v0 because of the sticking condition (1a). How does the interplay of the oscillatory sliding states and the sticking condition lead to stickslip motion? In order to answer this question we calculate the backward trajectory of the point ),/)0(lim 00 vkF K in accordance with (1b). Three qualitatively different backward trajectories are possible. (1) The backward trajectory hits the sticking condition. Together they define a bounded set of initial conditions leading to non-sticking trajectories. The boundary of this set is called the special stickslip boundary； it is not a possible trajectory but it separates between the basins of attraction of the stickslip oscillator and the non-stickslip attractors. (2) The backward trajectory spirals inwards towards an unstable oscillatory or continuously sliding state. Again all initial states outside these repelling states are attracted by a stickslip limit cycle. (3) The backward trajectory spirals outward towards infinity, and stickslip motion is impossible. Two types of local bifurcations are possible: if the backward trajectory changes from case 1 to case 3 the stickslip limit cycle annihilates with the special stickslip boundary. For changes from case 1 to case 2 the special stickslip boundary is either replaced by an unstable continuous or oscillatory sliding state or it annihilates with a 第 18 頁(yè) 共 10 頁(yè) stable continuous or oscillatory sliding state. A change from case 2 to case 3 is not possible. Figure 3 shows, for a particular choice of )(vFK, both types of bifurcations. Here the first bifurcation type occurs at v0 0.059,0.082,and 0.966.The second type occurs at v0 0.162,and 0.785This example shows that for increasing v0 stickslip motion can disappear and reappear again. Besides the well known bistability between stickslip motion and continuous sliding 3, multistability between one continuously sliding state, several oscillatory sliding states, and one stickslip oscillator are possible (see figure 3). Eventually for large sliding velocities all attractors except that of the continuously sliding state will disappear because the kinetic friction has to be an increasing function for sufficiently large sliding velocities. The strongly overdamped limit (i.e. kMdvvdFk /)( for any v except in tiny Figure 3. Typical bifurcation scenarios for a particular kinetic friction force )(vFK of the 第 19 頁(yè) 共 10 頁(yè) intervals around the extrema) leads to a separation of timescales. From an arbitrary point ),( xx in phase space with0vxthe system moves very quickly into the point. ),( vx where v is a solution of 0)(),(00 vvFvvFkx kk. Points on the curve )(0 xvFkx k with 0kFare unstable. They separate basins of attraction of different solutions v. After the fast motion has decayed the system moves slowly on the curve )(0 xvFkx k . The direction is determined by the sign of x . It either reaches a stable continuously sliding state, or, near an extremum ofkF, it jumps suddenly to another branch of the curve or to the sticking condition. For kinetic friction laws of the form shown in figure 1(b) with v0 between the two extrema, we get an oscillatory sliding state. It is a relaxation oscillation which may be difficult to distinguish from a stickslip oscillation. In the case of a friction law with a single minimum at mvvas shown in figure 1(a) we get stickslip motion for mvv 07. In the strongly overdamped limit any multistability disappears except near the extrema of )(vFK . The experiments of Yoshizawa and Israelachvili 14 are consistent with the assumption that the system is in a strongly overdamped limit with a friction law as shown in figure 1(a) 7. In order to discuss the influence of a stick-time-dependent static friction on the 第 20 頁(yè) 共 10 頁(yè) stickslip behavior we define a stickslip map )(1 nn xTx , where nxis the position of the block just before slipping. For constant static friction the map reads kFxTS /)( . The position just at the time of the slip-to-stick transition is defined by snx. It is a function ofnx, i.e. )(nsn xgx , where g is usually a monotonically decreasing function. The sticking time stickntis the smallest positive solution of s tic knsns tic knS tvxktF 0)( (5) This defines a function )( nsstickn xht which is a monotonically decreasing function due to 0SF .Thus the stickslip map is given by kxghFxT S /)()( . If the map has one fixed point, then stickslip motion exists. ForSK Ftco n sF tan, stickslip motion disappears if )/()0()(2s u p 00 ktFtFvv SStc . For a non-convex )(tFS , the supremum occurs at a non-zero value of the sticking time leading to a saddle-node bifurcation of a stable and an unstable fixed point of the stickslip map. Atcvv 0the stickslip motion has a finite amplitude, in contrast to the case of a convex )(tFS8.Because T is a monotonically increasing function, limit cycles or even chaos are not possible. If the slip-to-stick transition does not happen at the first time when x becomes equal to v0 (because of kFxSns /)0() chaotic motion may occur 9. In this case we get a non-monotonic T due to a non-monotonic g. Such over-shooting is only possible if )0(/)(SS FF becomes relatively large. For example, for a constant kinetic friction over-shooting occurs if )0(/)0(1)0(/)( FFFFKSS . For most realistic systems 第 21 頁(yè) 共 10 頁(yè) this condition is not satisfied. Note that the possibility of chaos is not in contradiction to the fact that the equation of motion (1) with constant FS cannot show chaotic motion. But the retardation of FS turns (1) into a kind of differential-delay equation. Using phenomenological dry friction means that we treat dry friction as an element in a mechanical circuit with some nonlinear velocity-force characteristic such as, say, a diode in an electrical circuit. This treatment is justified as long as the macroscopic timescales are much larger than any timescale of the internal degrees of freedom of the interacting solid surfaces. But there is one internal timescale which diverges if the relative velocity between the surfaces goes to zero: it is given by the ratio of a characteristic lateral length scale of the surface and the relative sliding velocity. Thus, any kinetic friction law )(vFK becomes invalid if t i m es ca l ecm i cr o s co p is ca l el en g t hcm i cr o s co p iv (6) The characteristic length scale ranges from several micrometres to several metres. It may be the size of the asper