干摩擦的非线性动力学外文翻译@中英文翻译@外文文献翻译

第 1 页 共 10 页 干摩擦的非线性动力学 Franz-Josef Elmer 瑞士巴塞尔，巴塞尔大学，物理学院 CH-4056 1996 年 11 月 5日收到来稿， 1997 年 5 月 21 日决定发表 摘要 研究一个有牵引力的弹簧 木块系统以一个恒定的速度在一个表面上运动所受到的干摩擦动力学。一个很普遍且符合摩擦学现象规律的动力学推理正在研究中（此规律为：静止时受静摩擦力影响，运动速度受动摩擦力影响）。共有三种可能发生的运动 :粘 滑运动、连续滑动、以及无粘性的振动。现在将要阐述地方以及全球的一些令人瞩目的人所提出的分歧观点以及他们的观点极 其相似的不稳定观点。 库仑关于干摩擦的 l定律被应用了 200 多年。他规定摩擦力等于由物体本身材料所决定的摩擦系数与正压力的乘积。静摩擦系数（静摩擦力是使物体由静止开始滑动的力）通常等于或大于动摩擦系数（动摩擦力是使物体以一个恒速度运动的力）。 一个机械系统受到的干摩擦力是非线性的，因为库仑定律把动摩擦力和静摩擦力区分开了。如果动摩擦系数小于静摩擦系数，在一个滑动并且粘住和滑动转化十分规则就如 l所说的表面上，粘 滑运动就会发生。这种急变的运动发生在日常生活中的每一天，例如开关门和拉小提琴。 即使库仑定律 是很简单并且很容易确定的（工程上一些计算都依据这些公式），它关于静摩擦产生的原因要求并不严格，因为静摩擦只是一个过程，它的作用不会涉及到平衡。因此不必为库仑定律经常应用于实验这个事实背离而让我们感到惊讶。典型的背离例子是下面所列出的：（ i）静摩擦力是变化的而且是随着静止时间的延续而逐渐增加的 2， 3，即两个相接触的滑行表面而没有发生相对运动。（ ii）动摩擦力决定于滑动速度；对于一个很大的速度，它可以近似地看作是线性地增加，这个速度就像是在 粘性摩擦力场那样。在达到很大速度之前，摩擦力首先减小，直到减小到最小值，然后再继续增大 3， 4。在有润滑油的临界条件下（即是在滑动的表面有极少的起润滑作用的单分子层存在）摩擦力将以一个很小的速度再次减小（见图 1） 5， 6。动摩擦系数作为影响滑动速度的因素，至少有一个极值。动摩擦力可能大于静摩擦力，但是在物体将要 第 2 页 共 10 页 滑动的瞬间，动摩擦力始终是小于或等于静摩擦力的。 本文的目的是针对非线性动力学中在某一个程度上狭隘的诸如上面提到的干摩擦定律提出一个自由的论点。本文将抛开在著作 1,3,4,6-9中已提出的明确 的定律。关于摩擦力的现象学的定律只是在肉眼所见的程度内，这就意味着用显微镜可见的精微的程度将远远超过肉眼可见的程度。在本文的结束语中将要给出一个为什么这种假设总不是有效的简单论点。要去揭示这种宏观现象的无根据性，因此去了解这种时空分离的假设下的完整的动力学知识是很重要的。 图 1 (a)图为典型的速度影响的动摩擦 定律的示意性草图 (b)图为有润滑油的临界条件下的系统 图 2 干摩擦谐波振荡器 第 3 页 共 10 页 对于不同的干摩擦定律有两个很重要的众所周知地前提。（ i） 摩擦系数只能在仪器内部测量（比如表面力测量仪 10或摩擦力显微镜 11）。下面我们将会发现系统的运动状态主要受摩擦力和仪器影响。例如粘 滑运动状态就比变化速度作用下的状态难于直接得出动摩擦系数。因此，测量仪器的影响是不能忽略掉的。（ ii）在粒状材料中干摩擦也是一个重要的物理量 12。一些相互作用的双尺是否合力作用的展开话题将久远地影响库仑动力学定律的修改。 在两个滑动表面的机械环境下（比如仪器）肉眼可见的自由程度很大。最重要的一个是侧面的。这里仅讨论在单一程度 范围内描述的具体系统。图 2 明确地说明了仪器的构成。谐波振荡器是这样组成的：一个木块（质量为 M）由一个弹簧（倔强系数为 k）与一个固定施力系统连接（见图 2）。木块与一个以恒定速度 v0 滑动的表面接触。木块和滑动表面之间的交互作用是静止时的静摩擦力 )(stickS tF和运动时的动摩擦力 FK(v)的合力。在列写运动状态方程时，我们必须区分木块是处于粘住状态还是滑动状态。如果它处于粘住状态，它的位移 x将随时间线性地增加，直到弹簧的弹力 )(kx 大于静摩擦力SF。 因此 kttFxifvxrS /)(0 (1a) 当 ttr 时木块在前一次滑动后再一次回到原状态。 如果木块滑动，运动方程为 )()( 00 xvFxvs i g nkxxM K （ 1b） 如果 kFxorvxS /)0(0 ， sign(x)表示 x的坐标。 我们的研究将以库仑定律关于持续静和动摩 擦的研究为基础。当0vx时，系统的状态就像一个干燥的谐波振荡器在平衡位置附近以 kFK / 这么大的位移振荡。因而，有很多解决振动的方法。在下面我们将要看到一些在速度影响下的动摩擦情况下依然存在的方法。木块的平衡位置在 k/)v(0KFx ,它被称为连续变化状态。 第 4 页 共 10 页 每一个具有能够使振动的最大速度大于 v0 的初始状态都将导致在一个有限时间内实现粘 滑状态的转变。而滑动状态是不受初始状态影响的，将以0,/ vxkFx S 开始运动。然而，粘 滑系统定义了一个具有说服力的相位周期变化的规律。这和系统的状态像干燥的谐波振荡器的系统状态并不是矛盾的。原因是：如果它在相位空间采样，相位的变化范围已被约定在了一条直线上如 (la)式所示。粘 滑状态要求动摩擦力KF必须小于静摩擦力。通常粘住状态时间 )/()(20kvFFt KSs tic k 远远大于滑动状态时间 kMkMvFFt KSs l i p /)()a r c ta n (2 2/110 。粘 滑状态的最大振幅（即 )(max txt ）函数是一个关于 v0的不规律变化的函数，其中 v0是由 kFS /决定的，而且初始状态 00 v。这个表达式对于一个速度影响的动摩擦同样适用。 未经修改的库仑定律致使以任何一个滑动速度 v0 的持续变化状态与粘 滑状态是同时存在的。在一个速度影响的动摩擦系统的一般范围内，这种双稳态都将存在，但是速度 v0要有一个严格的限制范围。特别是当在粘 -滑运动状态出现临界速度cv时。比如日常的一个现象： 快速地开关门就可以消除它的吱吱的声音。 我们为了更定量地去解决线性运动的 FK关于 v 的状态方程 0,)(0 vFvF KK。 方程（ lb）将变形为能够容易解决的不干燥谐波振荡器的方程。用一个具有权威性的持续变化状态代替摆动变化的解决办法。如果轨迹0)0(,/)0( vxkFx S ,t0 时不再粘住，粘 滑状态将会消失。临界速度cvv 0由两个方程00 )(,/( vtxkFtx s lipKs lip 得出。这样将推出两个关于slipt和cv非线性数学方程。当 Mk 时，可以近似认为 )(2 0 kMFFv KSc （ 2） 临界速度cv在粘 滑状态的性质探讨中是很重要的一个物理量，因为从它的测量方法中我们可以间接地知道有关机械装置的干摩擦的知识（见论点 6）。 接下来讨论变化状态的)(vKF，正如图 1 的例子 一样。假设静摩擦力SF始终是存在 第 5 页 共 10 页 的。0v为任何值时连续滑动状态都是存在的，但它仅仅在 0/)(00 dvvdFF KK时是稳定的。在 )(vFK达到一个极值时这种稳定状态改变并且 Hopf 分歧出现。在接近极值并与连续滑动状态有很小的背离时， .)()()( /0 ccetAk vFtx tMkiK （ 3） 由振幅决定（标准形式） 13 AAMkMFiMFkAMFdtdA KKK 222 )(4(2 （ 4） 如果关于动摩擦的极值的第三种说法是绝对的，而 分歧是超临界的，另外如上面所提到的众所周知的观点，另一种观点产生了。这里称为摆动滑动状态。它是一个最大的速度总是小于 v0 的这个有限的循环周期。这样木块决不会粘住。它的频率由左手边 (1b)的谐波振荡器粗略地给出。动摩擦的第二种说法对于非线性频率去谐是有效的。值得一提的是粘 滑振荡器的频率通常是远远小于摆动变化状态的。这种摆动状态与雷利的周期方程 130)( 3 uuuu 十分相似，事实上，雷利方程是（ 1b）方程的一个特例。因为动摩擦，几种稳定和不稳定周期循环可能存在。通过改变 v0，产生和消除相互作用来承受分歧点。 应该强调一点由（ 4）描述的 Hopf的 分歧观点和 Heslot et al 3评述的 Hopf的 分歧观点是没有关系的 。后者的评述在一次政体（称为爬行的政体）上提出，（ 1a）是不适用的（同下面关于干摩擦定律有效性一致）。 一个摆动变化状态只有在它的最大速度小于由粘性状态（ 1a）决定的滑动速度 v0时才存在。摆动变化状态和粘住状态是怎么样相互作用影响 粘 滑运动？为了回答这个问题，我们相反地计算点的轨迹 ),/)0(lim00 vkF K与 (lb)一致。三种具有说服力的不同的轨迹是可能存在的。 （ 1） 利用相反轨道法采样粘住状态。他们一起定义了一个有界限的导致非线性轨道的初始状态的装置。这套装置的界限专门称为粘 滑状态边界；它是一个 第 6 页 共 10 页 不可能存在的轨道，但是它把粘 滑振动和非粘 滑状态这两个难以分开的状态区分开了。 （ 2） 相反轨道法向内部盘旋接近于一个非稳定状态或非连续滑动状态。此外所有的初始状态除这些抵制状态外都有一个固定的粘 滑周期。 （ 3） 这种相反轨道法向外盘旋 趋向于无穷，粘 滑状态是不会发生的。 两种局部分歧是由可能存在的：如果相反轨迹改变了从第一种情况到第三种情况的固定粘 滑循环，使粘 滑界限消失。从第一种情况到底二种情况的这种粘 滑界限也被非稳定连续状态或摆动滑动状态或是它以消失变为稳定连续状态或摆动滑动状态。从第二种情况到第三种情况的转变是不可能发生的。见图 3， 对于 )(vFK 的一个特殊值，将有两种分歧观点产生。第一个是在 v0 0.059,0.082,0.966。第二个是在 v0 0.162,0.785。这个例子说明了不 断增加 v0,粘 滑运动可能消失也可能再产生。 此外著名的双稳态的粘 滑运动和连续滑动 3，连续滑动状态，几种摆动滑动状态，和粘 滑振荡器的稳定性都是有可能的（见图 3）。最后所有的观点都认为除了连续滑动状态，非常大的滑动速度将会消失因为动摩擦力将要充分地影响这个很大的速度。 超强的过阻尼极限（ i.e. kMdvvdFk /)(适用于任何 v除了在极值里微小的距离）导致了时间的分离。对于一个相位图上的任一点（ ),( xx ，并且0vx的状态将快速地向点 ),( vx 变化， v 由 0)(),(00 vvFvvFkx kk。在曲线 )(0 xvFkx k 上的点，当0kF 时时不稳定的。他们分离了不同的 v 的求解办法。驻留系统太久的快速运动之后将要沿着曲线 )(0 xvFkx k 变化。方向由 x 的符号决定。它也将到达稳定连续滑动状态，或者是，接近一个极限值 KF ,它将突然地向曲线分歧转变或者是向粘住状态。如图 l(b)所示动摩擦定律在两个极值之间的 v0,出现振动滑动状态。这是一个不严密的振动可能难以区别粘 滑摆动状态。如图 l(a)中单一的最小速度为mvv的摩擦定律的情形下我们可以得到mvv 07的粘 滑状态。在超强过阻尼界限任何双稳定状态都将消 第 7 页 共 10 页 失，除了接近 )(vFK极值。 Yoshizawa 和 Israelachvili 14的实验是一致的，他们假设系 统处于超强过阻尼界限的摩擦定律如图 1(a)7所示。 为了讨论依赖静止时间的静摩擦力影响下的粘 滑状态，我们建立了粘 滑坐标系图 3 典型的分歧观点精确的动摩擦力 )(vFK 见图 l(b)。接下来的作用力由下式决定 : 2/)()()()( 221221 vvvvvvvF k ，其中05.0,2.0,1.0,3 21 vv 。由运动（ l） 的方程积分就可以得到结果。其他的参数如 1,40,1 kMFs。各段曲线表示了稳定和不稳定连续滑动状态（ CS），摆动滑动状态（ OS），或者是粘 滑运动（ SS）。一系列曲线说明 了粘 滑分界线。 第 8 页 共 10 页 )(1 nn xTx ， nx 是开始滑动的位置。对于恒定不变的静摩擦力在图中规定为kFxT S /)( 。仅仅在滑动状态向静止状态转变的时候位移定义为 snx 。它是 nx 的函数，即 )(nsn xgx ,函数 g通常是一个单调递减的函数。静止时间 sticknt是这个函数的最小实根 s tic knsns tic knS tvxktF 0)( （ 5） 这就定义了一个函数 )( nsstickn xht 而且在 0SF时是单调递减函数。这样粘 滑图可以这样定义 kxghFxTS /)()( 。当图只有一个点时，粘滑运动就是存在的。 当 KF =常量 = )0(SF时，粘滑运动消失 )/()0()(2s u p00 ktFtFvv SStc 。对于非凸起函数 FS(t), 在粘 滑图中静止时间导致鞍状节点稳定和不稳定的固定点在非零值处出现了上确限。与凸起的 FS(t)8的情形相比在cvv 0时粘 滑运动有一个固定振幅。因为 T是一个单调递增的函数，界限循环或平均混乱都是没有可能的。 如果滑动 静止转变不发生在 x 变为与 v0（因为 kFxSns /)0(）相等的第一次。在这种情况下将得到一个由于非单调变化的 g 导致的非单调变化的 T。如果 )0(/)(SS FF 变得相当大这样过于发射是可能的。例如，对于一个持续变化的动摩擦力如果)0(/)0(1)0(/)( FFFF KSS 过发射将要发生。对于实实在在的系统这种状况是不可能发生的。注意混乱的可能性和对于不变的 FS的运动 (l)的方程不可能表现出混乱状态的事实是 不矛盾的。但是由于 FS的延迟使 (l)变为了一种微分延迟方程。 利用干摩擦的现象学意味着我们把干摩擦看作是一个机械电路的元素并带有非线性的速度量和力的性质，比如，说，在一个电路里的二极管。只要在肉眼可见的时间量程比任何互相作用的固体表面内在的自由程度时间量程都大，这种看法就是可行的。但是有一种内在的时间量程是分歧的，就是表面的相对速度变为 0：它等于表面特有的侧部长度值和相对速度的比值。这样，任何动摩擦定律 FK(v)都会变的残缺不全如果 第 9 页 共 10 页 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） 这个特有的长度的范围从几微米到几米。这可能是一个粗略的范围，粗略的接触范围，表面粗糙度的相关长度，或者是弹性相关长度。任何干摩擦定律的局限性都不会涉及到摆动滑动状态和持续滑动状态，只要它们的相对滑动速度始终保持在比临界速度 (6)大的范围内。但是在粘 滑运动中，粘住向滑动转变之后的瞬间和在滑动向粘住转变前的瞬间将大大影响粘住和滑动的转变 ，界面的动力学行为细节会变得很重要。当最大相对滑动速度减少的时候这些细节资料的重要性将增加。例如， Heslot et al 3 用实验方法创立了十分完整的不同的行为在一个滑动过程中当最大相对速度比临界速度值 (6)小。 本文在干摩擦能描述为速度依赖的动摩擦和静止时间依赖的静摩擦的假设下讨论了谐波振荡器在一个固体平面上滑动的非线性动力学。除众所周知的持续滑动状态和粘 滑振荡器之外，建立了一种没有粘住的摆动滑动状态。所有这些分歧点都在图 3中表示了出来。 致谢 我充满感激地致谢托马斯以及他的原稿读物。本作品由瑞 士国家自然科学基金会支持。 参考文献 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 页 共 10 页 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 页 共 10 页 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 页 共 10 页 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 页 共 10 页 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 页 共 10 页 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 页 共 10 页 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 页 共 10 页 (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 页 共 10 页 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 页 共 10 页 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 页 共 10 页 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 页 共 10 页 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 页 共 10 页 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