第 1 頁 共 10 頁 干摩擦的非線性動力學 Franz-Josef Elmer 瑞士巴塞爾，巴塞爾大學，物理學院 CH-4056 1996 年 11 月 5日收到來稿， 1997 年 5 月 21 日決定發表 摘要 研究一個有牽引力的彈簧 木塊系統以一個恒定的速度在一個表面上運動所受到的干摩擦動力學。一個很普遍且符合摩擦學現象規律的動力學推理正在研究中（此規律為：靜止時受靜摩擦力影響，運動速度受動摩擦力影響）。共有三種可能發生的運動 :粘 滑運動、連續滑動、以及無粘性的振動。現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以及他們的觀點極 其相似的不穩定觀點。 庫侖關于干摩擦的 l定律被應用了 200 多年。他規定摩擦力等于由物體本身材料所決定的摩擦系數與正壓力的乘積。靜摩擦系數（靜摩擦力是使物體由靜止開始滑動的力）通常等于或大于動摩擦系數（動摩擦力是使物體以一個恒速度運動的力）。 一個機械系統受到的干摩擦力是非線性的，因為庫侖定律把動摩擦力和靜摩擦力區分開了。如果動摩擦系數小于靜摩擦系數，在一個滑動并且粘住和滑動轉化十分規則就如 l所說的表面上，粘 滑運動就會發生。這種急變的運動發生在日常生活中的每一天，例如開關門和拉小提琴。 即使庫侖定律 是很簡單并且很容易確定的（工程上一些計算都依據這些公式），它關于靜摩擦產生的原因要求并不嚴格，因為靜摩擦只是一個過程，它的作用不會涉及到平衡。因此不必為庫侖定律經常應用于實驗這個事實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間的延續而逐漸增加的 2， 3，即兩個相接觸的滑行表面而沒有發生相對運動。（ ii）動摩擦力決定于滑動速度；對于一個很大的速度，它可以近似地看作是線性地增加，這個速度就像是在 粘性摩擦力場那樣。在達到很大速度之前，摩擦力首先減小，直到減小到最小值，然后再繼續增大 3， 4。在有潤滑油的臨界條件下（即是在滑動的表面有極少的起潤滑作用的單分子層存在）摩擦力將以一個很小的速度再次減小（見圖 1） 5， 6。動摩擦系數作為影響滑動速度的因素，至少有一個極值。動摩擦力可能大于靜摩擦力，但是在物體將要 第 2 頁 共 10 頁 滑動的瞬間，動摩擦力始終是小于或等于靜摩擦力的。 本文的目的是針對非線性動力學中在某一個程度上狹隘的諸如上面提到的干摩擦定律提出一個自由的論點。本文將拋開在著作 1,3,4,6-9中已提出的明確 的定律。關于摩擦力的現象學的定律只是在肉眼所見的程度內，這就意味著用顯微鏡可見的精微的程度將遠遠超過肉眼可見的程度。在本文的結束語中將要給出一個為什么這種假設總不是有效的簡單論點。要去揭示這種宏觀現象的無根據性，因此去了解這種時空分離的假設下的完整的動力學知識是很重要的。 圖 1 (a)圖為典型的速度影響的動摩擦 定律的示意性草圖 (b)圖為有潤滑油的臨界條件下的系統 圖 2 干摩擦諧波振蕩器 第 3 頁 共 10 頁 對于不同的干摩擦定律有兩個很重要的眾所周知地前提。（ i） 摩擦系數只能在儀器內部測量（比如表面力測量儀 10或摩擦力顯微鏡 11）。下面我們將會發現系統的運動狀態主要受摩擦力和儀器影響。例如粘 滑運動狀態就比變化速度作用下的狀態難于直接得出動摩擦系數。因此，測量儀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話題將久遠地影響庫侖動力學定律的修改。 在兩個滑動表面的機械環境下（比如儀器）肉眼可見的自由程度很大。最重要的一個是側面的。這里僅討論在單一程度 范圍內描述的具體系統。圖 2 明確地說明了儀器的構成。諧波振蕩器是這樣組成的：一個木塊（質量為 M）由一個彈簧（倔強系數為 k）與一個固定施力系統連接（見圖 2）。木塊與一個以恒定速度 v0 滑動的表面接觸。木塊和滑動表面之間的交互作用是靜止時的靜摩擦力 )(stickS tF和運動時的動摩擦力 FK(v)的合力。在列寫運動狀態方程時，我們必須區分木塊是處于粘住狀態還是滑動狀態。如果它處于粘住狀態，它的位移 x將隨時間線性地增加，直到彈簧的彈力 )(kx 大于靜摩擦力SF。 因此 kttFxifvxrS /)(0 (1a) 當 ttr 時木塊在前一次滑動后再一次回到原狀態。 如果木塊滑動，運動方程為 )()( 00 xvFxvs i g nkxxM K （ 1b） 如果 kFxorvxS /)0(0 ， sign(x)表示 x的坐標。 我們的研究將以庫侖定律關于持續靜和動摩 擦的研究為基礎。當0vx時，系統的狀態就像一個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，有很多解決振動的方法。在下面我們將要看到一些在速度影響下的動摩擦情況下依然存在的方法。木塊的平衡位置在 k/)v(0KFx ,它被稱為連續變化狀態。 第 4 頁 共 10 頁 每一個具有能夠使振動的最大速度大于 v0 的初始狀態都將導致在一個有限時間內實現粘 滑狀態的轉變。而滑動狀態是不受初始狀態影響的，將以0,/ vxkFx S 開始運動。然而，粘 滑系統定義了一個具有說服力的相位周期變化的規律。這和系統的狀態像干燥的諧波振蕩器的系統狀態并不是矛盾的。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 滑狀態要求動摩擦力KF必須小于靜摩擦力。通常粘住狀態時間 )/()(20kvFFt KSs tic k 遠遠大于滑動狀態時間 kMkMvFFt KSs l i p /)()a r c ta n (2 2/110 。粘 滑狀態的最大振幅（即 )(max txt ）函數是一個關于 v0的不規律變化的函數，其中 v0是由 kFS /決定的，而且初始狀態 00 v。這個表達式對于一個速度影響的動摩擦同樣適用。 未經修改的庫侖定律致使以任何一個滑動速度 v0 的持續變化狀態與粘 滑狀態是同時存在的。在一個速度影響的動摩擦系統的一般范圍內，這種雙穩態都將存在，但是速度 v0要有一個嚴格的限制范圍。特別是當在粘 -滑運動狀態出現臨界速度cv時。比如日常的一個現象： 快速地開關門就可以消除它的吱吱的聲音。 我們為了更定量地去解決線性運動的 FK關于 v 的狀態方程 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。用一個具有權威性的持續變化狀態代替擺動變化的解決辦法。如果軌跡0)0(,/)0( vxkFx S ,t0 時不再粘住，粘 滑狀態將會消失。臨界速度cvv 0由兩個方程00 )(,/( vtxkFtx s lipKs lip 得出。這樣將推出兩個關于slipt和cv非線性數學方程。當 Mk 時，可以近似認為 )(2 0 kMFFv KSc （ 2） 臨界速度cv在粘 滑狀態的性質探討中是很重要的一個物理量，因為從它的測量方法中我們可以間接地知道有關機械裝置的干摩擦的知識（見論點 6）。 接下來討論變化狀態的)(vKF，正如圖 1 的例子 一樣。假設靜摩擦力SF始終是存在 第 5 頁 共 10 頁 的。0v為任何值時連續滑動狀態都是存在的，但它僅僅在 0/)(00 dvvdFF KK時是穩定的。在 )(vFK達到一個極值時這種穩定狀態改變并且 Hopf 分歧出現。在接近極值并與連續滑動狀態有很小的背離時， .)()()( /0 ccetAk vFtx tMkiK （ 3） 由振幅決定（標準形式） 13 AAMkMFiMFkAMFdtdA KKK 222 )(4(2 （ 4） 如果關于動摩擦的極值的第三種說法是絕對的，而 分歧是超臨界的，另外如上面所提到的眾所周知的觀點，另一種觀點產生了。這里稱為擺動滑動狀態。它是一個最大的速度總是小于 v0 的這個有限的循環周期。這樣木塊決不會粘住。它的頻率由左手邊 (1b)的諧波振蕩器粗略地給出。動摩擦的第二種說法對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遠小于擺動變化狀態的。這種擺動狀態與雷利的周期方程 130)( 3 uuuu 十分相似，事實上，雷利方程是（ 1b）方程的一個特例。因為動摩擦，幾種穩定和不穩定周期循環可能存在。通過改變 v0，產生和消除相互作用來承受分歧點。 應該強調一點由（ 4）描述的 Hopf的 分歧觀點和 Heslot et al 3評述的 Hopf的 分歧觀點是沒有關系的 。后者的評述在一次政體（稱為爬行的政體）上提出，（ 1a）是不適用的（同下面關于干摩擦定律有效性一致）。 一個擺動變化狀態只有在它的最大速度小于由粘性狀態（ 1a）決定的滑動速度 v0時才存在。擺動變化狀態和粘住狀態是怎么樣相互作用影響 粘 滑運動？為了回答這個問題，我們相反地計算點的軌跡 ),/)0(lim00 vkF K與 (lb)一致。三種具有說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。他們一起定義了一個有界限的導致非線性軌道的初始狀態的裝置。這套裝置的界限專門稱為粘 滑狀態邊界；它是一個 第 6 頁 共 10 頁 不可能存在的軌道，但是它把粘 滑振動和非粘 滑狀態這兩個難以分開的狀態區分開了。 （ 2） 相反軌道法向內部盤旋接近于一個非穩定狀態或非連續滑動狀態。此外所有的初始狀態除這些抵制狀態外都有一個固定的粘 滑周期。 （ 3） 這種相反軌道法向外盤旋 趨向于無窮，粘 滑狀態是不會發生的。 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，使粘 滑界限消失。從第一種情況到底二種情況的這種粘 滑界限也被非穩定連續狀態或擺動滑動狀態或是它以消失變為穩定連續狀態或擺動滑動狀態。從第二種情況到第三種情況的轉變是不可能發生的。見圖 3， 對于 )(vFK 的一個特殊值，將有兩種分歧觀點產生。第一個是在 v0 0.059,0.082,0.966。第二個是在 v0 0.162,0.785。這個例子說明了不 斷增加 v0,粘 滑運動可能消失也可能再產生。 此外著名的雙穩態的粘 滑運動和連續滑動 3，連續滑動狀態，幾種擺動滑動狀態，和粘 滑振蕩器的穩定性都是有可能的（見圖 3）。最后所有的觀點都認為除了連續滑動狀態，非常大的滑動速度將會消失因為動摩擦力將要充分地影響這個很大的速度。 超強的過阻尼極限（ i.e. kMdvvdFk /)(適用于任何 v除了在極值里微小的距離）導致了時間的分離。對于一個相位圖上的任一點（ ),( xx ，并且0vx的狀態將快速地向點 ),( vx 變化， v 由 0)(),(00 vvFvvFkx kk。在曲線 )(0 xvFkx k 上的點，當0kF 時時不穩定的。他們分離了不同的 v 的求解辦法。駐留系統太久的快速運動之后將要沿著曲線 )(0 xvFkx k 變化。方向由 x 的符號決定。它也將到達穩定連續滑動狀態，或者是，接近一個極限值 KF ,它將突然地向曲線分歧轉變或者是向粘住狀態。如圖 l(b)所示動摩擦定律在兩個極值之間的 v0,出現振動滑動狀態。這是一個不嚴密的振動可能難以區別粘 滑擺動狀態。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。在超強過阻尼界限任何雙穩定狀態都將消 第 7 頁 共 10 頁 失，除了接近 )(vFK極值。 Yoshizawa 和 Israelachvili 14的實驗是一致的，他們假設系 統處于超強過阻尼界限的摩擦定律如圖 1(a)7所示。 為了討論依賴靜止時間的靜摩擦力影響下的粘 滑狀態，我們建立了粘 滑坐標系圖 3 典型的分歧觀點精確的動摩擦力 )(vFK 見圖 l(b)。接下來的作用力由下式決定 : 2/)()()()( 221221 vvvvvvvF k ，其中05.0,2.0,1.0,3 21 vv 。由運動（ l） 的方程積分就可以得到結果。其他的參數如 1,40,1 kMFs。各段曲線表示了穩定和不穩定連續滑動狀態（ CS），擺動滑動狀態（ OS），或者是粘 滑運動（ SS）。一系列曲線說明 了粘 滑分界線。 第 8 頁 共 10 頁 )(1 nn xTx ， nx 是開始滑動的位置。對于恒定不變的靜摩擦力在圖中規定為kFxT S /)( 。僅僅在滑動狀態向靜止狀態轉變的時候位移定義為 snx 。它是 nx 的函數，即 )(nsn xgx ,函數 g通常是一個單調遞減的函數。靜止時間 sticknt是這個函數的最小實根 s tic knsns tic knS tvxktF 0)( （ 5） 這就定義了一個函數 )( nsstickn xht 而且在 0SF時是單調遞減函數。這樣粘 滑圖可以這樣定義 kxghFxTS /)()( 。當圖只有一個點時，粘滑運動就是存在的。 當 KF =常量 = )0(SF時，粘滑運動消失 )/()0()(2s u p00 ktFtFvv SStc 。對于非凸起函數 FS(t), 在粘 滑圖中靜止時間導致鞍狀節點穩定和不穩定的固定點在非零值處出現了上確限。與凸起的 FS(t)8的情形相比在cvv 0時粘 滑運動有一個固定振幅。因為 T是一個單調遞增的函數，界限循環或平均混亂都是沒有可能的。 如果滑動 靜止轉變不發生在 x 變為與 v0（因為 kFxSns /)0(）相等的第一次。在這種情況下將得到一個由于非單調變化的 g 導致的非單調變化的 T。如果 )0(/)(SS FF 變得相當大這樣過于發射是可能的。例如，對于一個持續變化的動摩擦力如果)0(/)0(1)0(/)( FFFF KSS 過發射將要發生。對于實實在在的系統這種狀況是不可能發生的。注意混亂的可能性和對于不變的 FS的運動 (l)的方程不可能表現出混亂狀態的事實是 不矛盾的。但是由于 FS的延遲使 (l)變為了一種微分延遲方程。 利用干摩擦的現象學意味著我們把干摩擦看作是一個機械電路的元素并帶有非線性的速度量和力的性質，比如，說，在一個電路里的二極管。只要在肉眼可見的時間量程比任何互相作用的固體表面內在的自由程度時間量程都大，這種看法就是可行的。但是有一種內在的時間量程是分歧的，就是表面的相對速度變為 0：它等于表面特有的側部長度值和相對速度的比值。這樣，任何動摩擦定律 FK(v)都會變的殘缺不全如果 第 9 頁 共 10 頁 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） 這個特有的長度的范圍從幾微米到幾米。這可能是一個粗略的范圍，粗略的接觸范圍，表面粗糙度的相關長度，或者是彈性相關長度。任何干摩擦定律的局限性都不會涉及到擺動滑動狀態和持續滑動狀態，只要它們的相對滑動速度始終保持在比臨界速度 (6)大的范圍內。但是在粘 滑運動中，粘住向滑動轉變之后的瞬間和在滑動向粘住轉變前的瞬間將大大影響粘住和滑動的轉變 ，界面的動力學行為細節會變得很重要。當最大相對滑動速度減少的時候這些細節資料的重要性將增加。例如， Heslot et al 3 用實驗方法創立了十分完整的不同的行為在一個滑動過程中當最大相對速度比臨界速度值 (6)小。 本文在干摩擦能描述為速度依賴的動摩擦和靜止時間依賴的靜摩擦的假設下討論了諧波振蕩器在一個固體平面上滑動的非線性動力學。除眾所周知的持續滑動狀態和粘 滑振蕩器之外，建立了一種沒有粘住的擺動滑動狀態。所有這些分歧點都在圖 3中表示了出來。 致謝 我充滿感激地致謝托馬斯以及他的原稿讀物。本作品由瑞 士國家自然科學基金會支持。 參考文獻 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 共 10 頁 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 共 10 頁 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 共 10 頁 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 共 10 頁 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 共 10 頁 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 共 10 頁 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 共 10 頁 (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 共 10 頁 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 共 10 頁 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 共 10 頁 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 共 10 頁 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 共 10 頁 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