Dynamic behavior and modelling of the pre-sliding static friction.pdf

Wear 242 Ž2000. 1–17 www.elsevier.comrlocaterwear

Dynamic behavior and modelling of the pre-sliding static friction Chen Hsieh ) , Y.-C. Pan High Precision Flexible Structure Control Lab, Institute of Aeronautics and Astronautics, National Cheng Kung UniÕersity, Tainan, Taiwan 701 Received 7 October 1998; received in revised form 7 November 1999; accepted 8 November 1999

Abstract One of the main difficulties in dealing with positioning problems with extremely high precision is the static friction and the induced nonlinear behavior existing in the mechanism. In this research, the phenomenological study of the static friction behavior before normal slip starts is conducted for direct drive torque motor systems. It is found that the qualitative behavior of static friction is consistent. Furthermore, based on the experimental results published in the literature, the pre-sliding behavior is similar for various mechanical mechanisms including friction between two simple materials and friction of complicated systems that have many friction sources. It shows that the pre-sliding motion is mainly the combination of two kinds of motion: plastic deformation that demonstrates creep and work hardening, and nonlinear spring deformation that demonstrates hysteresis loop with special memory and wipe out effects. These behaviors are very close to those of most plastic materials. A new static friction model that consists of a plastic module and a nonlinear spring module is introduced. This model is of a simple structure and is proved to match all those qualitative behaviors. Parameter estimation and simulation algorithms are also described. Simulation results show that this model does coincide with those experimental results conducted by the authors as well as by other investigators. q 2000 Published by Elsevier Science S.A. Keywords: Static friction; Pre-sliding behavior; Plastic deformation; Nonlinear spring deformation

1. Introduction Extremely high precision positioning control has become more and more important in the fields of measurement, assembly, and manufacturing. One of the main difficulties in dealing with such kind of problems is the static friction and the induced nonlinear behavior existing in the mechanism. A widely accepted concept of static friction is the resistant force between two contacting surfaces that is equal but opposite to the applied external force. Furthermore, there is no displacement before maximum static friction is achieved. w1–7x This Amontons– Coulomb type static friction model is usually treated as a dead zone type nonlinear system in control field. This assumption is appropriate for the stick–slip motion analysis or for the positioning control problem when the preci-

) Corresponding author. Tel.: q886-6-2757575 ext. 63670; fax: q8866-2389940. E-mail address: [email protected] ŽC. Hsieh..

0043-1648r00r$ - see front matter q 2000 Published by Elsevier Science S.A. PII: S 0 0 4 3 - 1 6 4 8 Ž 0 0 . 0 0 3 9 9 - 9

sion requirement is not high. However, it is not good enough when we talk about micro-, nano-, or pico-positioning, where any subtle movement is important. By static friction, stiction phase or pre-sliding phase discussed in this paper we mean the friction force and the period of motion before the normal slip starts. In this phase, the shear stress between two contacting surfaces is not big enough to break the contacting linkage. As will be revealed in the following sections, the behavior during stiction phase is, in fact, the result of a nonlinear dynamic system. It is the main purpose of this research to conduct a phenomenological study of the system dynamic behavior during stiction phase based on the experimental observations conducted in our lab as well as those results reported in the literature. A new model is also introduced which matches all the qualitative behavior of the static friction we found in these experiments. In the Section 2 we first describe the experimental results conducted in this research. Static friction behavior is then analyzed. Based on these results, we will give a review of existent static friction models. A new static friction model that satisfies all qualitative stiction proper-

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ties is then introduced in Section 3. Parameter estimation method is given in Section 4. In Section 5, simulation algorithm and simulation results are discussed. Section 6 is the conclusion.

2. Properties of static friction From material point of view, strain always exists when stress exists. This is also true for the pre-sliding phase of two contacting surfaces. This unravels the deficiency of the traditional concept adopted by the Amontons–Coulomb type friction models, where there is no displacement during the static friction phase. This small amount of displacement is indeed negligible in macroscopic point of view. However, it is vital to precision positioning systems. It is noted that, as static friction is concerned, the displacement discussed here is not due to the deformation of the bulk material of two contacting bodies but the deformation of the junctions between two contacting surfaces. w8–11x These junctions, with a total cross-sectional area much smaller than that of the apparent contacting area, take all the load, which make them deform much easier than the bulk material does and, hence, play the main role in the pre-sliding phase. In 1899, Stevens w12x, with the help of an interferometer, found that elastic displacement does exist before the normal slip occurs. Many other investigators w10,13–19x have also found similar results for metalto-metal contacts and other mechanisms. These behaviors can clearly be demonstrated by the following experimental results. 2.1. Description of experiments The block diagram of the system used in this research to conduct experiments is shown in Fig. 1Ža.. A schematic diagram of the mechanical structure is shown in Fig. 1Žb.. The plant is a direct drive DC torque motor 1 mounted on ordinary ball bearings and is driven by a linear amplifier. Two optical encoders with different resolutions are mounted to measure the rotor position independently. The resolution of the coarse encoder is 86 400 countrrev or, equivalently, 15 arc-secondsrcount, while that of the fine encoder is 1 620 000 countsrrev or, equivalently, 0.8 arcsecondrcount. The outputs are compared to guarantee that the position data acquired are correct. The input torque is indirectly obtained by measuring the armature current through the motor. The torque will then be the product of this current and the motor torque constant. Since the range

1 In this research, three motors are investigated. They are QT6202 and QT6205 made by Inland Motor, and RTM-6026 made by Racing Electronic. In this paper, data for RTM-6026 are presented.

Fig. 1. Ža. Block diagram of system hardware. Žb. Schematic diagram of the test platform.

of angular displacements of the rotor in the experiments is less than 1% of one cycle of the motor ripple which is the result of the discrete commutation bars, the variation of the torque constant due to the ripple is negligible. The bias effect due to ripple and the reluctance force of the motor will not influence the results described in this paper. The data acquisition and control is accomplished by a PC486r66 together with a DrA converter and an ArD converter. Both DrA and ArD converters have a resolution of 12 bits each. The motor parameters such as moment of inertia of the rotor and torque constant are identified using the method developed by Hsieh et al. w17x. It is noted that, under the load applied in this research, the maximum possible twist of the shaft, from end to end, would be less than two arc-seconds. This amount is much smaller than what is detected. Hence, the displacement measured is mainly due to the friction effect at the contacting region. It is also noted that, in this system, many parts contribute the friction force. They include the friction of ball bearings, the friction between brushes and commutator bars and the friction in the encoders. These frictions can be divided into two categories, rolling friction and sliding friction. However, the former is of an order much smaller than that of the latter and can be neglected 2 w8x. In this research we did not try to distinguish the individual contribution of each friction source but treated them as a combined result as what happens in most machines. It is interesting to find and will be shown later that there is no difference between such kind of friction and those frictions in other cases as reported in the literature as long as sliding friction is concerned. The experiments can be divided into two categories: dynamic tests and static tests.

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Even in a ball bearing, the main friction comes from the sliding friction between spacer and balls.

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Fig. 2. Results of experiment 1 Ždesired angle — 24 arc-seconds..

2.1.1. Dynamic tests Experiment 1 Žramp input.: We pre-assign a desired rotor position, then gradually increase the current with a constant rate. The current is cut once the desired position is reached. Same procedure is repeated for 5 s. Typical response with a desired angle of 30 counts Ž24 arc-seconds. is shown in Fig. 2. Experiment 2 Žstep-like input.: Same as experiment 1, except that the final current is held once the desired angle is reached. Typical response with a desired angle of 15 counts Ž12 arc-seconds. is shown in Fig. 3. Experiment 3 Žpulse input.: A 10-Hz periodic pulse signal with varying magnitude as shown in Fig. 4Ža. is applied. The duty cycle is 20%. The result is shown in Fig. 4Žb.. Short vibration is excited after each pulse input. The steady state response after each pulse is extracted from the raw data and shown below it.

Fig. 4. Ža. Input pulse sequence of experiment 3. Žb. Results of experiment 3.

2.1.2. Static tests In this category of experiments, the input consists of a piecewise constant current; each input value will be held for 2 s before it is changed to the next value. The rotor positions at the end of each 2-s interval are recorded. The 2-s interval was chosen empirically so that the response almost settled down. Typical torque-versus-displacement

curves are shown in Figs. 5–7. The process of each test is marked in alphabetic order. Similar experimental results are also observed in many different types of mechanism, such as the static friction of metal-to-metal contact ŽFig. 5 of Ref. w10x, Figs. 7 and 9 of Ref. w15x., ball bearing friction of a CMG ŽFigs. 1–6 of Ref. w18x., precision ball screw friction in an x–y table ŽFig. 15a of Ref. w19x. and rolling ball guide friction in a linear motion motor system ŽFig. 6 of Ref. w16x.. Some of these results are friction between simple materials while

Fig. 3. Results of experiment 2 Ždesired angle — 12 arc-seconds..

Fig. 5. Static response, case 1.

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the others, similar to the direct drive motor system used in this research, are combined effects of many friction sources. Consequently, these phenomena are common pre-sliding behavior for most mechanical contacts. However, there are still exceptions. For example, the friction force between the atomically flat surface of muscovite mica and Si 3 N4 needle shows a periodic force w20x, which is quite different from what we discussed here. Of course, the microstructure in this case is also quite different from that of an ordinary mechanical contact. 2.2. Static friction properties Fig. 7. Static response, case 3.

Based on these experimental results, we try to draw the following conclusions. 2.2.1. QuantitatiÕely P1. Static friction is time and position Õarying. Many investigators w13,15x have found that it is very hard to get quantitatively repeatable results. This is true even for well-prepared metal-to-metal friction experiments w15x, and not to mention the mechanisms in commercial machines. This variation can be attributed to many reasons, such as temperature, humidity, varying load, aging, inevitable misalignment of axis, periodically varying motor reluctance force, uneven distributed lubricant, cold welding after long dwell time and the destroy and reconstruction of the microstructure between two contacting surfaces. The variation, say stiction breakaway force for example, can be as large as several times from one point to another. P2. The static friction dynamics is temporarily time-inÕariant and locally position-inÕariant. By temporarily time-invariant we mean that the system properties remain constant during short interval of interest. This short interval is usually long enough for control purposes. By locally position-invariant we mean that the system properties remain unchanged if it stays in the pre-sliding range of any individual point. This property is the reason that we can obtain repeatable qualitative responses. Property P2 is also important to control system design. It allows the control engineer to

Fig. 6. Static response, case 2.

assume that the system parameter uncertainty is time-invariant. 2.2.2. QualitatiÕely The qualitative behavior of static friction is very consistent and repeatable. Few exceptions were found. These exceptions are temporary and minor and can be reasonably attributed to inevitable local structure variation. The behavior can clearly be divided into two parts, the plastic deformation and the nonlinear spring deformation as described below. 2.2.3. Plastic deformation Plastic deformation creates permanent displacement and is characterized by creep motion and work hardening. Creep motion is defined to be the continuous deformation under constant load. This motion can clearly be seen in Fig. 3. Due to work hardening, the deformation rate is decreasing and finally stops.3 It is noted that creep also happens when the force is intermittent as shown in Fig. 4Žb.. As work hardening is concerned, it is found that any applied force corresponds to a maximum possible work hardening ŽMPW.. Fig. 8 in Ref. w15x shows the relation between the MPW and the applied force for the case of metal-to-metal friction. It has been pointed out in Ref. w15x that the slope of the applied force with respect to MPW gets to infinity as the force approaches zero. The work hardening is close to isotropic, i.e., the amount of work hardening accumulated in one direction also applies to the other direction. This phenomenon can more easily be seen in Fig. 15Žh., which is a numerical duplicate of the experimental result shown by Fig. 9 of Ref. w15x using the model introduced in the next section. As a conclusion of the above description, we can treat creep as the transition response of the plastic deformation, and the accumulated amount of creep from both directions

3 If the applied force is close to the breakaway force, the creep may slow down first, then increase gradually and finally normal slip occurs, which is similar to the creep motion of plasticity at high temperature.

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Fig. 8. Simulation of the static test in Fig. 5 using Dahl’s model with gs 2063, i s 2, Fc s 0.5.

is equal to the extent of work hardening. No more plastic motion can be created if the MPW associated with the applied force is smaller than the current amount of work hardening. Otherwise, creep always exists even if the force applied is smaller than the maximum force ever applied. 2.2.4. Nonlinear spring deformation Nonlinear spring deformation is the part of the pre-sliding motion where plastic deformation is excluded. It can easily be seen when no more creep and work hardening happens and usually appears in the form of hysteresis loops ŽFigs. 5, 6, 15Žd,e,g,i,j,k ... This motion is characterized by two properties as follows. Ži. Suppose the trend of the applied force is reÕersed, i.e., from increasing force to decreasing force or conÕerse. Then, the deriÕatiÕe of force with respect to displacement after reÕerse is a positiÕe and non-increasing function of the distance measured from the reÕerse point only and is independent of the position and force of the reÕerse point. This can be seen in Figs. 5–7. Examine Fig. 5, if we translate curve hi so that point h coincides with points e and b, separately, we find that curve hi matches curves ef and bc. Furthermore, if we turn curve hi clockwise 1808 so that hi is upside down, then coincide point h with, respectively, points f, c and the original point of i, we find that hi coincides with curves fg, cd and ij. Same phenomenon can be found in Figs. 6 and 7. This property induces the interesting memory phenomenon of static friction; each branch of the force–displacement curve tries to recover its earlier one reverse point. For example, points b, e and h in Fig. 5 are recovered as the forces are recovered. Žii. If a hysteresis loop is completed. In this case, the curÕe recoÕers an earlier reÕerse point, which is then the intersection of seÕeral branches going in the same direction. The slope of the curÕe after that point will follow that of those intersecting branches which has the smallest slope at that point. For example, in Fig. 6, point f recovers point d. Curve fg, instead of following ef, follows cd. Similar phenomenon can be found in Figs. 5 and 15Ži, j.. This property together with property Ži. causes two conse-

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quences. Ž1. Every reverse point will be recovered once the force resumes the corresponding value no matter how many reverses may happen before this moment. Ž2. When a particular hysteresis loop is completed, the memory of the two reverse points that bound the hysteresis loop will be wiped out. For example, when the hysteresis loops b–c–d, e–f–g and h–i–j in Fig. 5, and loops d–e–f and b–c–g in Fig. 6 are completed, the influence of these loops disappear just as they have never happened. It is noted that nonlinear spring deformation is not reversible, it is neither linear elastic nor nonlinear elastic, which are usually reserved for reversible processes. However, when the motion is extremely small, the nonlinear spring can be linearized as an elastic spring and the system works as a mass–spring–damper system as shown in Fig. 2. Though the friction study in this research is phenomenological in nature, we are still interested in the following two questions. Why do different mechanisms exhibit the same pre-sliding dynamic behavior? Why are these behaviors? These are still open questions. The following observation and studies, however, may throw some light on the answer. But more rigorous proofs are needed. Examine the pre-sliding motion, it is found that it is same as the structure behavior of most plastic materials. w21–23x Typical behaviors of plasticity such as creep, work hardening, Bauchinger effect, hysteresis loop also consists of the pre-sliding motion. It is noted that the theory of plasticity is limited to a continuous structure while friction is involved in the interaction of two contacting surfaces. Though they are quite different, the coincidence of their behavior implies that they are closely related. The traditional and superficial concept of contacting topology as turning a series of mountains and valleys upside down and sitting it on another series of mountains and valleys but remaining as two different parts can hardly explain these behaviors. Fortunately, Bawden and Tabor w8x together with other researchers w9–11,24x have proved that the

Fig. 9. Ža. Model of friction. Žb. Static friction model.

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contacting mechanism is, in fact, composed of junctions which are the regions where real contacts of minute asperities occur. Cold welding and strong adhesion take place and the two materials, in effect, merge in one continuous solid as what is discussed in plasticity w25, p. 313x. They have also proved that the main contribution of friction force comes from the deformation of the junctions plus the force required plowing one surface on the other. As presliding motion is concerned, the deformation of the junctions is believed to dominate the phase. Since the total cross-sectional area of the junctions is much smaller than the apparent contacting area, they provide most of the displacement we observed during the pre-sliding range. This one-continuous-material concept can apply to simple metal-to-metal case as well as complicated cases as long as sliding friction is concerned and may provide an explanation to the coincidence of plasticity and pre-sliding motion. 2.3. ReÕiew of existent static friction models It is obvious that a model that can exhibit all static friction properties while maintaining simple structure for mathematical analysis is desired. Many static friction models have been proposed. Almost all models have demonstrated the mass–spring or mass–spring–damper property. Few of them tried to include the memory and wipe out effects of reverse points and no model, to our knowledge, has considered complete creep and work hardening phenomena. In the structure field, many nonclassical friction laws have been proposed in the study of contact problems. w24–29x Because of the feature of this type of problems, three issues, approaching, slip criterion and tangential motion under shear force, have been emphasized in these models. Ishigaki et al. w24x have derived the relation between approaching and applied normal load of the contacting of a ball and a plate. The theoretical results have a good consistence with the experimental data. Kikuchi and Oden w25x, Martins and Oden w26x and Oden and Pires w27x have discussed 3D cases. They provided both local and nonlocal rules for the slip criterion and assume that slip coincides with the direction of applied force. Fredriksson w28x and Curnier w29x, on the other hand, proposed associated and nonassociated slip potential function to determine the slip criterion and slip direction. The philosophy is similar to the potential function used in plasticity to determine yield point and direction for plastic flow. As tangential motion is concerned, these works adopt either rigid– perfect slip ŽCoulomb type friction. or elastic–perfect slip models. Fredriksson has also considered work hardening in his model. But hysteresis, memory effect and creep are not included in any of these models. In the control field, researchers pay most attention on the relation between tangential motion and applied shear force under constant normal load. It is restricted to uniaxial motion. This is also the main issue considered in this

paper. Among these existent models, Dahl’s model w30,31x, which was originally introduced in 1968, may be the most widely accepted one and has been analyzed and cited in many theoretical works w18,32x. Basically, this model gives the governing equation ŽEq. Ž2. of w30x. of a nonlinear spring as follows d Fs dx

sg 1y

Fs Fc

i

ž

sgn Ž x˙ . sgn 1 y

Fs Fc

/

sgn Ž x˙ . ,

Ž 1.

where g is the slope of the force–displacement curve at Fs s 0, and Fc is the Coulomb friction force. Creep and work hardening are not considered. In this model, the slope of the force–displacement curve is a function of the applied force Fs or, equivalently, it is a function of the distance measured from the reverse point and the value of force Fs at this switching instant w32x. Because of this, this model does not exhibit the special property of memorizing reverse points. More precisely, if the curve is reversed at a point with Fs s F1 , this point will not be recovered unless the curve is reversed again at Fs s yF1 and no other reverses are allowed before this reverse. A simulation of the experiment shown in Fig. 5 is demonstrated in Fig. 8. This deficiency has also been pointed out in a numerical simulation by Osborne and Rittenhouse w18x. They also pointed out that the slope of Dahl’s model gets into flat too quickly which makes it hard to match the experimental results. To compensate these deficiencies, they proposed a modified model which uses Dahl’s model to fit the basic function shape together with ‘‘large amount of logic and memory’’ to satisfy the special memory property. The logic and memory is not described in their report, and this model is only capable of simulation but can hardly be handled in any theoretical work. Haessig and Friedland w33x have presented two friction models in their work. The first one is referred to as the bristle model. The basic concept of this model is to represent the randomly distributed asperity contacts by randomly distributed bristle pairs. In this model, these bristle pairs work like linear springs. Each pair will break when a preassigned displacement is reached and a new pair is established then. The model is not designed for theoretical analysis and those friction properties listed in Section 2.2 can hardly be verified. The second one is the reset integrator model. As static friction model is concerned, this model is a standard mass–spring–damper system. Ro and Hubbel w34x also introduced a model when they worked on the positioning control problem of an x-table driven by a DC motor and a ball screw. Their friction model is as follows. Ff s Bu˙p q Bfr u˙p y u˙mdp q k frp Ž up y umdp . ,

ž

/

Ž 2.

where umdp is the micro dynamic reference position and is determined according to

u˙mdp s

½

0 Ž micro dynamics.

u˙p Ž macro dynamics.

P

Ž 3.

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Therefore, in stiction range, i.e., micro dynamics, the friction reduces to Ff s Ž B q Bfr . u˙p q K frp Ž up y umdp . ,

Ž 4.

a standard mass–spring–damper system. If umdp is redefined as follows

u˙mdp s

< u˙p < g u˙p

ž /

Ž up y umdp .

Ž 5.

and let z ' up y umdp , we have z˙ s u˙p y

< u˙p < g u˙p

z,

Ž 6.

ž /

and the friction force Ff s Bu˙p q Bfr z˙ q k frp z.

Ž 7.

That is exactly the model introduced by Canudas de Wit et al. w35x. In their model g Ž u˙p . is defined as ˙

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k frp g u˙p ' Fc q Ž ts y Fc . eyŽ u p r Õ s . ,

ž /

Ž 8.

where ts and Õs are the maximum static friction and Stribeck velocity, respectively. As static friction is concerned, they have proved that their model can be reduced to a special case of Dahl’s model if g Ž u˙p . ' Fcrk frp and B s Bfr s 0. Futami et al. w16x found that the static friction could be divided into two regions when they studied the nanometer dynamics of a linear motion motor. When the displacement is restricted in 100 nm, the friction model is a standard linear spring. When the displacement is between 100 nm and 100 mm, the model is a nonlinear spring. However, the mathematical model of this nonlinear spring is not given in their report. It will be clear in the following sections that the linear spring in the first region is in fact the linear part of the nonlinear spring around a reverse point. There is no damper in their model. The experiment results conducted by Futami et al. w16x and Osborne and Rittenhouse w18x are also numerically duplicated using the model introduced in the next section and are shown in Fig. 15Ži. and Žj.. As mentioned earlier, the static friction behavior is very close to that of most plastic solids. It may be helpful, then, to know how the behavior of plastic solid is modeled in the literature. It turns out that the standard way w21–23x to do this is to first define a simplified plastic element, which usually consists of an elastic region followed by either a linear or nonlinear work hardening region. Then, depending on the accuracy required, many basic elements with different parameters are combined to approximate the real response. When a cyclic force is applied, then, isotropic, kinematic or independent hardening rules are adopted. In these three rules, only kinematic hardening rule considers the Bauchinger effect to its full extent while the isotropic rule completely neglect this effect. Time-dependent creep

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response is usually discussed separately and is not included in these models. An important difference between the creep in plasticity and that in pre-sliding motion is that in the former case the creep, except at high temperature, usually takes a long time to progress, but it progresses much faster in the latter case.

3. Modeling of static friction As pointed out by Prager and Hodge w22x ‘‘a mathematical theory attempting to take account of all mechanical phenomena observed in the plastic range would not be practical’’. This, of course, also applies to the case of pre-sliding motion. This is mainly because of the nonlinearity of the stress–strain relation and the discontinuous change of system states due to work hardening and reverse of deformation. However, as discussed in Section 2.2, the motion can clearly be divided into two categories, i.e., plastic deformation and nonlinear spring deformation. This implies that though a simple mathematical model is impossible, it is possible to be expressed by a mechanical model. This model is shown in Fig. 9Ža.. This model consists of four elements — a plastic module, a nonlinear spring module, a viscous damper and a hook. These elements are massless and, hence, do not really exist. They are phenomenological elements. Each element exhibits a special mechanical property and is described by a simple mathematical expression. The combined result then matches the complete pre-sliding behavior. For convenience, this model is described as a system in linear motion and the following analysis that is based on this model gives the relations between force, mass, linear displacement and velocity. It should be clear that the analysis and concepts given here are also applicable to a rotational motion. In this case, torque, moment of inertia, angular displacement and angular velocity should replace force, mass, linear displacement and velocity, respectively. The hook at the left end sticks at the position where it is if the internal force is less than the breakaway force, i.e., the maximum static friction. Richardson and Nolle w36x and Johannes et al. w37x have shown that the breakaway force decreases exponentially under high rate of applied force. Once the hook starts to move the tangential contacting force between the hook and ground can be described as a function of velocity. This kind of functions describes the dynamic friction and have been proposed by many investigators as mentioned in the beginning of this paper and in the reference of the review paper w38x. In this research, we restrict our study to the stiction range only. In this case, the model is reduced to that shown in Fig. 9Žb.. The viscous damper Cs in the model is a standard damper, which has a linear relation between force and velocity. The justification of the existence of Cs will be clear in a later analysis. The plastic module and nonlinear spring module are described below.

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Fig. 10. Plastic module.

3.1. Plastic module Consider the module shown in Fig. 10, in which s is the applied force and x p is the elongation of the module. The governing equation is defined as follows

°x˙ s ~ ¢x˙ s h

p

a Ž f Ž < s