Design and characterization of an instrumented slider aimed atmeasuring local micro-impact forces between dry rough solids
Camille Grégoire, Bernard Laulagnet, Joël Perret Liaudet, Thibaut Durand, Manuel Collet, Julien Scheibert
IInstrumented slider to measure local micro-impact forces between dry rough solids
Camille Gr´egoire, Bernard Laulagnet, Jo¨el Perret-Liaudet, Thibaut Durand, Manuel Collet, and Julien Scheibert ∗ Univ Lyon, Ecole Centrale de Lyon, ENISE, ENTPE, CNRS,Laboratoire de Tribologie et Dynamique des Syst`emes LTDS UMR5513, F-69134, Ecully, France Laboratoire Vibrations Acoustique, INSA Lyon, 25 bis avenue Jean Capelle 69621 Villeurbanne Cedex, France (Dated: September 16, 2020)Sliding motion between two rough solids under light normal loading involves myriad micro-impactsbetween antagonist micro-asperities. Those micro-impacts are at the origin of many emerging macro-scopic phenomena, including the friction force, the slider’s vibrations and the noise radiated in thesurroundings. However, the individual properties of the micro-impacts ( e.g. maximum force, po-sition along the interface, duration) are essentially elusive to measurement. Here, we introduce aninstrumented slider aimed at measuring the position and the normal component of the micro-impactforces during sliding against a rough track. It is based on an array of piezoelectric sensors, eachplaced under a single model asperity. Its dynamical characteristics are established experimentallyand compared to a finite elements model. We then demonstrate its relevance to tribology by us-ing it against a track bearing simple, well-defined topographical features. The measurements areinterpreted thanks to a simple multi-asperity contact model.
Keywords: Sliding rough contact ; Array of piezoelectric sensors ; Experimental tribology ; Roughness noise; Local force measurements
I. INTRODUCTION
Contacts between solids are submitted to complexforce fields developing along the interface, the charac-teristics of which depend, among others, on the exter-nal loading, the macroscopic geometry and the surfacetopographies (see e. g. [1] for a recent review). Be-cause those force fields directly control several key tri-bological phenomena including energy dissipation, wearor sliding-induced vibrations, the measurement of localcontact forces is highly desirable. However, in general,such measurements remain challenging due to the diffi-culty to place non-invasive local force sensors in the closevicinity of the interface. Still, with the goal of better un-derstanding the elementary mechanisms occurring alongfrictional interfaces, a number of local force measurementmethods have been proposed in the literature.One strategy is to embed one or several force sensorsinside the bulk of one of the two solids in contact. Eachsensor, with a lateral size l , probes the stress field at acertain depth h below the surface. Those non-vanishinglength scales are responsible for the limited spatial resolu-tion of the measurement: first, the local stress at a depth h is an average over the surface stresses within a regionof typical extension h ; second, the stresses at depth arethemselves integrated over the sensor area. Overall, oneexpects the sensor output to be sensitive to the inter-facial forces acting in a region of typical lateral exten-sion l + h . Various technologies have been tested for theembedded sensors, including strain gauges [2], piezoelec-tric [3] or piezoresisting elements [4], pressure-sensitiveelectric conductive rubber (PSECR) [5] and MEMS [6], ∗ [email protected] with applications not only to tribology, but also to hapticsensors (see [7] for a review). For instance, millimeter-sized MEMS forces sensors embedded at the base of arough elastomer slab of millimetric thickness have beenused not only to investigate the pressure and shear stressfields at the contact between the rough slab and rigidsmooth sliders [8, 9], but also to unravel the effect of fin-gerprints on the tactile perception of fine textures [10].Another strategy is to probe the rough contact inter-face in a minimally-invasive way, for instance using animaging technique through a transparent material. Theshear stress field can be accessed by inversion of the in-plane displacement field [11, 12], which can be obtainedby following the motion of appropriate tracers, either in-corporated on purpose [11, 13] or naturally present inthe image due to, e.g., the surface roughness [12]. Thespatial resolution of such measurements is typically lim-ited by both the depth of the tracers and their inter-distance (although the latter limitation can be signifi-cantly reduced in steady state conditions [11]), or by thesize of the correlation box when a digital image correla-tion (DIC)-like method is used to measure the displace-ments [12]. If the surface topography is specifically en-gineered as a population of well-defined micro-spheres,both the shear and normal forces on each micro-contactcan be estimated based on its in-plane displacement andtrue contact area [14].In this work, our objective is to propose a measurementtool useful to better understand the phenomena occur-ring at the sliding interface between two rough metal-lic surfaces under light normal loading. In such lightlyloaded interfaces, the contact pressure is much smallerthan the materials’ elastic moduli, so that each micro-contact induces local displacements which remain negligi-ble compared to the size of the largest asperities in the to-pographies. In those conditions, the interactions between a r X i v : . [ phy s i c s . c l a ss - ph ] S e p two rough surfaces is expected to be essentially geomet-rical and to be localised at the (almost undeformed) tipof a few individual asperities at each instant. The slidingmotion of a rough slider on a rough track thus induces amotion normal to the interface, as the slider passes overa succession of random asperities of the track [15].When the slider and/or track are not perfectly rigid,and when the sliding speed is large enough, the interac-tion between the two rough surfaces is a succession ofmicro-impacts between antagonist asperities, which trig-ger a vibration of the solids, through not only their rigidbody modes, but also their other eigenmodes. Overall,those vibrations are the source of the so-called roughnessnoise [16], the empirical laws of which have been studiedextensively [17–20]. However, such empirical laws remainunexplained, and both numerical modelling [21] and sta-tistical analyses [22] point towards the challenging needto better describe the forces involved in the individual,sub-millisecond-lived micro-impacts.To address this challenge, none of the two measure-ment strategies mentioned above (embedded sensors andoptical monitoring) are suitable. First, metals beingopaque to light, imaging methods cannot be applied. Sec-ond, the non-vanishing size of the region probed by asensor buried inside the solid material generally encom-passes a significant number of surface asperities, thus im-peding identification of individual asperity/asperity im-pacts. Here, to overcome these difficulties, our strategyhas been to design a slider equipped with an array of well-defined surface asperities, each of them being monitoredby a dedicated force sensor (see Fig. 1). The expectedshort duration of micro-impacts, typically shorter than amillisecond [21], requires a large measurement bandwidthto be time-resolved, which led us to choose piezoelectricelements as sensors. FIG. 1. Sketch of an instrumented slider (top) on a roughtrack (bottom). The slider’s surface is equipped with severalpotential model asperities (blue spherical caps), each of thembeing monitored by a dedicated local force sensor.
In this Methods paper, our scope is not to solve theproblem of the roughness noise and sliding-induced vibra-tions under light normal loading, but to present a new in situ measurement tool which will be useful for thatpurpose in future works. The paper is organised as fol-lows. First, we describe the main elements of conceptionof the slider, its practical realisation, and its calibration(section II). Then, we demonstrate its tribological rele-vance by using it on a track with simple topographicalfeatures (section III), and by interpreting the signals with a simple multi-asperity model (Appendix).
II. DESCRIPTION AND CALIBRATION OFTHE SLIDER
Before describing the details of the developed instru-mented slider (section II A), let us note that generalmicro-impact forces are expected to have random direc-tions with respect to the interface. Indeed, the tangentplane of each micro-contact can very well be differentfrom the average plane of the macroscopic contact inter-face. In addition, each micro-contact is submitted notonly to repulsion forces normal to the local contact tan-gent plane to avoid interpenetration of the solids, but alsoto friction forces acting along the local contact tangentplane. Thus, one would ideally want to access the threecomponents of the micro-contact forces, which would re-quire three co-localised sensors associated to each modelasperity at the slider’s interface. In this first work, toavoid an excessive complexity in the design and realisa-tion of the slider, we decided to introduce only one sensorunit per asperity, to measure only the projection of themicro-impact forces on the normal to the slider’s averageplane.One could imagine that the slider may simply be pre-pared with an array of model asperities, each of whichbeing directly attached to a dedicated piezoelectric ele-ment (like in Fig. 1). However, during sliding on a roughfrictional interface, the piezoelectric elements could besubmitted to non-negligible tangential forces, which canhave two different undesired consequences. First, a givenpiezoelectric element is generally sensitive to all direc-tions of an external stimulus (although with different sen-sitivities), which makes it impossible to separate the dif-ferent contributions from a single charge output [23]. Sec-ond, piezoelectric ceramics can easily be damaged by aneven modest shear stimulus [24]. Thus, a large part of thechosen design will be justified by the necessity to exposethe piezoelectric elements to normal forces only. In prac-tice, we have tested various design options, each of whichhas been evaluated using a finite element model (FEM)of the full instrumented slider (section II B). The finaldesign has thus been selected after a FEM-assisted trial-and-error procedure. The predicted capabilities of theslider, in particular in terms of locality and bandwidthof the measurements, are furthermore affected by the restof the measurement chain (section II C), and will finallybe compared to actual measurements and calibrations insection II D. The signal analysis allowing efficient local-isation and force estimates of individual micro-impactsare discussed in section II E.
A. Slider’s specifications and mechanical assembly
Figure 2 (bottom right) shows an exploded view ofall the mechanical parts constituting the instrumented
FIG. 2. Main: Final design and dimensions of the slider (using the general tolerances ISO 2768mK). Bottom right: exploded3D view of the various mechanical parts constituting the instrumented slider. 1: Aluminum base. 2: Piezoelectric elements.3: Sphere-ended aluminum fitting parts. 4: Hollow cylinder-shaped spacer. 5: M3 screw (4 mm long). 6: Hollow carbon steelspring plate. 7: M3 screw (10 mm long). 8: Carbon steel top plate, featuring 8 threaded hollow cylinders on its bottom, and 9sphere holders on its top. slider. The sensing units (label 2 in Fig. 2) are ninecircular plate-shaped elements (12.7 mm in diameter,1 mm thick) of a piezoelectric ceramic (PZ27, Ferrop-erm F1270508) with the electrodes on the flat surfaces.The piezoelectric elements are first glued with an epoxyglue at the surface of a thick anodised aluminum (2017A)base (label 1). On top of each of them is then glued afitting part (label 3) having a cylindrical base with thesame diameter as the piezoelectric element, topped bya cylindrical body with a smaller diameter, itself endedby a spherical cap (radius of curvature 7.5 mm). Notethat a precise in-plane positioning of the piezoelectric el-ements and their associated fitting parts during gluingwas achieved using a dedicated 3D-printed positioningpart. To minimise the thickness of the glue films, glu-ing was performed under a dead weight of 2.6 kg. Theresulting altitudes of the summits of the nine sphericalcaps was finally measured with an interferometric pro- filometer, and found to deviate from their mean plane byless than 46 µ m each (32 µ m standard deviation).The top surface of the slider (on which the model as-perities will be attached), is a carbon steel plate of thick-ness 0.5 mm (label 8). It is pressed onto the nine spheri-cal caps (label 3) using a thinner (0.25 mm thick) carbonsteel spring plate (label 6) featuring nine holes throughwhich the spherical caps can pass and come into contactwith the top plate. The spring plate is first mounted onthe base (label 1) through eight top-headed screws (label7), each passing through a cylindrical spacer (label 4).The top plate is then attached to the spring plate witheight down-headed screws (label 5) screwed into eightthreaded cylinders spot-welded on the bottom surfaceof the top plate. Those screws can be actuated fromthe bottom surfaces of the base, through eight dedicatedthrough-holes.The spring plate serves two crucial roles. First, its rela-tively small bending stiffness is used to pull the top platein contact with all nine spherical caps, although each ofthem has a slightly different altitude. In practice, thedesign of Fig. 2 includes an initial 300 µ m gap betweenthe spring plate and the threaded cylinders attached tothe top plate. So, by fine tuning the angles of all eightindividual down-headed screws (label 5), a pre-load canbe applied and adjusted on each spherical cap. The pre-load is chosen large enough so that the contact betweenspherical caps and top plate is never lost, even in thecase of transient negative loads, for instance during po-tential vibrations of the top and/or spring plates. Notethat the sphere/plane geometry of the contact imposesa well-defined location of the normal force on the fittingpart, which, for a pure normal load applied by the topplate, eliminates any torque on the surface of the piezo-electric element. Second, the large in-plane stiffness ofthe spring plate essentially prevents any lateral motionof the top plate relative to the spherical caps, even incase a significant shear load is applied to the top plate.The spring plate is thus the main design element ensur-ing that the piezoelectric sensors are exposed to almostpurely normal stimuli.The top surface of the top plate further features ninespot-welded cylinders with an upper conical hole, intowhich model asperities can be fixed. The base also fea-tures two lateral arms, which will prove useful to push theslider during tribological experiments (see section III).The grooves on the top surface of the base serve as chan-nels to fit the cables connecting the nine piezoelectric ele-ments, as further described in section II C. The fact thatthe piezoelectric elements are glued above the groovesand not between two perfectly flat surfaces may cause anon-linearity of the sensors, but we have checked that, inour case, this effect remains negligible (see section II B).All dimensions of the slider can be found in the drawingsof Fig. 2. FIG. 3. View of the slider indicating the labels of all asperi-ties, used all along the article. TABLE I. Individuals coordinates of the model asperities la-beled 1 to 9 in Fig. 3. x - and y -positions (in mm) are relativeto the center of mass of the slider. The altitudes (in µ m) arerelative to the asperities’ mean plane.Asperity Position x Position y Altitude1 17.42 15.95 1072 17.53 -1.12 -303 17.45 -17.87 -964 0.59 16.09 145 0.44 -1.04 -306 0.43 -17.99 557 -16.44 16.00 -688 -16.49 0.92 -469 -16.42 -18.11 94
Finally, a stainless steel sphere with a radius of curva-ture of 0,75 mm is glued in each conical hole of the topplate, so that the slider features nine spherical asperitiesat its surface. A picture of the complete slider is shown inFig. 3. In the conditions used in the rest of this study, allcontact interactions between the slider and a track willonly occur through one or several of those nine potentialspherical asperities. Their individual altitudes have beenmeasured with an interferometric profilometer, and areprovided in Table II A with respect to their mean plane.Their standard deviation is 72 µ m while their maximumdifference is 203 µ m. B. Finite element analysis
To design the instrumented slider, we built several fi-nite element (FE) models of the slider, with different ar-chitectures and dimensions. Here, we present the staticand modal characteristics predicted by the FE model ofour final design solution, described in section II A.The entire instrumented slider was discretized andanalysed using the numerical software package Ansys. Asshown in Fig. 4, all parts of the slider were discretizedinto tetrahedral 3-D elements. They possess 10 nodes,3 degrees-of-freedom per node corresponding to transla-tions in the 3 directions, and such that displacementshave quadratic approximations. Elements are used withhomogeneous and isotropic elastic solid behaviour, ex-cept for the piezoelectric sensors, which are anisotropic.The material properties of all materials are reported inTab. II. The total final number of elements is up to340000, leading up to 610000 nodes and about 1.8 milliondegrees-of-freedom. The most critical part of the modelis the discretization of the spring plate (label 6 in Fig. 2),because a minimum number of elements in the thicknessis required. To determine this minimum number, we haveperformed two preliminary convergence tests. The firsttest is based on the modal analysis of a simply supported,0.25 mm thick, 60 ×
60 mm steel plate, for which naturalfrequencies and modal shapes are analytically known. Agood agreement was obtained with a 0.1 mm element size,leading to relative errors less than 0.2% for the first tennatural frequencies and well-reproduced modal shapes.The second test consisted in the convergence of natu-ral frequencies of a 0.25 mm thick, 60 ×
60 mm free-freeplate with the same holes as in our final design. Again, a0.1 mm element size was found adequate. Thus, we used0.1 mm-sized elements to discretize both the top plate(label 8, 147000 elements) and the spring plate (label 6,66000 elements), while 2 mm elements were used for thebase (label 1, 70000 elements) and for each sensor (la-bel 2, 350 elements), long screw (label 7, 1800 elements),small screw (label 5, 1300 elements) and spacer (label4, 200 elements). For the spherical-capped fitting parts(label 3, 3300 elements), the element size was 2 mm ex-cept for the cap, which had an element size of 0.5 mm.Junctions between parts are modelled as fully bondedcontacts, except for the connection between the top plateand spherical caps. For the latter, we used unilateral nor-mal contact and Coulomb friction with a friction coeffi-cient of 0.5, a conservative value larger than the expectedrange for aluminum/steel contacts [25]. a.b. FIG. 4. Mesh of the slider. a: full mesh. b: zoom on onesensing unit (label 3 in Fig. 3).
We evaluated the relationship between a static loadapplied to each surface asperity and the resulting forcein all nine sensors. For such static finite element calcula-tion, we assumed that the base is perfectly rigid. So, weremoved it from the model and instead imposed clampedboundary conditions on the corresponding faces of theother parts usually in contact with the base. The con-tact between the top plate and each of the nine spherical caps was pre-loaded in a similar way as in the assembly ofthe experimental slider (see section II A): the 300 µ m airgap between the spring plate and the threaded cylindersof the top plate was first replaced by the same volumeof steel ; then all top-headed screws and spacers (labels7 and 4 in Fig. 2) were displaced downward by 300 µ m.The resulting deformations in the spring plate have anamplitude of 300 µ m, while the (stiffer) top plate deformsby no more than about 30 µ m. The generated pre-loadswere about 16 N on sensors 1, 3, 7 and 9, 12 N on sen-sors 2, 4, 6, 8 and 7 N on sensor 5. For each surfaceasperity i , a simulation with an additional vertical force F i =10 N applied on it was carried out, and the trans-mitted force in all sensors was calculated. Fig. 5 sum-marises the results as a static response matrix H . Eachelement H ij represents the resulting vertical force on the j th sensor induced by the force applied on the i th spheri-cal cap. H is a strictly diagonally dominant matrix, with (cid:80) j (cid:54) = i | H ij | < . | H ii | and H ii > . F i . In conclu-sion, a static force applied on a given surface asperity istransmitted to the corresponding sensor with less than5% rejection on the other sensors, which was consideredclose enough to the desired behaviour for our final design. FIG. 5. Forces (in N) on the piezoelectric sensors when astatic vertical force of 10 N is applied to a single surface as-perity, in our FE model. Row: label of the loaded asperity.Column: label of the sensor.
Similarly, we applied tangential loads between 1 and100 N on each of the surface asperities, and looked at theinduced tangential forces in all piezoelectric sensors. Thelatter forces have been found to be always smaller than9 % of the applied load. The equivalent shear stress wasalways below 0.1 MPa, well below the yield strength ofthe piezoelectric sensors (about 40 MPa). This result in-dicates that our design is expected to be effective in pro-tecting the piezoelectric elements from potentially dam-aging shear forces.To test the potential influence of the grooves abovewhich the piezoelectric sensors are glued (see e.g.Fig. 4(b)), we performed the following additional staticFE simulations, using the extension ACT Piezo & MEMSof Ansys. We considered a single sensing unit made of onepiezoelectric element (label 2 in Fig. 2, piezoelectric mod-uli e =-3.09 C.m , e =16.0 C.m , e =11.64 C.m , andrelative permittivities at constant strain (cid:15) S ,r =1129.69and (cid:15) S ,r =913.73, equipotential condition on both faces)and its spherically-capped fitting part (label 3), bonded TABLE II. Materials properties used in the Finite Element modelMaterials Part label in Fig. 2 Young Modulus E ( GP a ) Poisson’s ratio ν Density ρ ( kg/m )Aluminum 1, 3 71 0,33 2770Carbon steel 4, 5, 6, 7, 8, 9 200 0.3 7850PZ 27 2 E xx = E yy = 66 . E zz = 84 . either on a flat solid (reference case) or on a grooved solid.In both cases, a nominal normal force was applied overa circle of radius 1 mm around the apex of the sphericalcap, together with an additional tangential load (along x or y ) in the range [0–0.1] N and/or an additional torque(around x or y ) in the range [0–0.1] N/m. In all thosecases, the difference between the reference and groovedcases was found less than 3.6% for the electric potentialdifference (voltage) generated between the two faces ofthe piezoelectric element, suggesting a negligible influ-ence of the groove on the force measurement.To qualify the frequency domain on which the instru-mented slider conveniently responds, and its dynamicperformance, we then performed a (purely mechanical,i.e. not including piezoelectricity) modal analysis for thefull model (i.e. now including the base). The bottom faceof the base has fully constrained displacements, and, toensure linearity of the calculations, the contacts betweensphere-ended fitting parts and top plate are now fullybonded over a circle of radius 0.47 mm. Fig. 6 shows theeigenshape of the two first eigenmodes (eigenfrequenciesin the range 1.5–1.9 kHz). They consist of corner modesin the top and spring plates, i.e. their maximum deflec-tions are observed on two opposite corners of the plateswith antisymmetric displacements. The differences be-tween first and second modes are due to the presenceor not of screws at the most deformed corners. Modaldisplacements at the locations of the nine contacts arevery small for these eigenmodes, due to the high verti-cal stiffness of the piezoelectric ceramic and fitting partcompared to that of the top and spring plates.Figure 7 presents all the calculated eigenfrequencies inthe range [0-10] kHz. As can be seen, the eigenfrequen-cies are not evenly distributed over the frequency range,but form several mode packs. For plates in flexural vibra-tion, the modal density is expected to be constant versusfrequency [26]. However, mode packing arises when thesystem has a sort of spatial periodicity [27], as is the casewith the regular arrangement of connections between thetop and spring plates, and between the spring plate andbase. Such a design is thus presumably responsible forthe observed heterogeneous modal density.The first mode pack is found at around [1.5–2.2] kHz,so that we can assume a static behaviour of the instru-mented slider in the range [0–1.5] kHz. From this pointof view, the adopted design is expected to enable moni-toring of impacts of duration about and below the mil-lisecond, one of the desired features mentioned in theintroduction. Based on the static results of Fig. 5, we a. b. FIG. 6. Eigenshapes of the first (a, 1556,2 Hz) and second (b,1845,6 Hz) eigenmodes of the slider, as calculated with ourFE model. frequency (Hz)
FIG. 7. Eigenfrequencies of the slider up to 10 kHz, as calcu-lated with the FE model with the bottom face fixed. then expect that, below 1.5 kHz, the rejection on sensor j (cid:54) = i of an impact occurring on asperity i will remain lessthan about 5%.To better assess the expected dynamical behaviour ofthe slider, we finally calculated the force frequency re-sponse of all sensors to a harmonic vertical stimulus ona single asperity, using the mode superposition method.The results are shown on Fig. 8 in the case of a stimuluson asperity 1. As expected, the force frequency responseof sensor 1 is very close to 1 for all frequencies, becausethe force on asperity 1 is directly transmitted to sensor 1with little loss through the other sensors. Indeed, below1.5 kHz, the force frequency responses of all other sensorsare smaller than -55 dB. Interestingly, the two sensorswith the largest response are sensors 2 and 4, which as thenearest neighbours of asperity 1. The responses are actu-ally frequency-dependent, and in particular peaks can beseen in the vicinity of the eigenfrequencies, particularlyaround 2, 3 and 8 kHz. Nevertheless these peaks remainsmall (maximum value of -30 dB), meaning that, even inthe highly dynamic frequency domain, the excited sensoris still predicted to capture most of the imposed dynam-ical force. frequency (Hz) -100-80-60-40-20020 d B sens.1 sens.2 sens.3 sens.4 sens.5sens.6 sens.7 sens.8 sens.9 Eigenfrequencies FIG. 8. Force frequency response of all sensors when asperity1 is excited, expressed in dB (20 log (cid:2) Force responseExcitation (cid:3) ). Dottedvertical lines: eigenfrequencies predicted by the FE modalanalysis.
C. Related instrumentation
In order to acquire the time-evolution of the nine pres-sure signals, a shielded cable (AC-0005-K, Br¨uel & Kjær)was first welded on each piezoelectric element, before thegluing step described in section II A. Both welding pointsare located on the same face of the piezoelectric plate-shaped elements, thanks to one of the electrodes beingwrapped around the thickness of the element. Thus, forthe gluing step, the extra thickness due to the two weld-ing points and the emerging cable can be fitted into thegrooves present on the top surface of the base.The cable conveying the charges generated by eachof the nine piezoelectric elements is connected to oneinput channel of a conditioning amplifier (type 2694A, Br¨uel & Kjær, bandwidth [1 Hz–50 kHz]), through acharge-to-DeltaTron converter (type 2647, Br¨uel & Kjær,1 mV/pC, bandwidth [0.17 Hz–50 kHz]). The outputs ofthe conditioning amplifier are finally acquired using a32-channels recorder and analyser (OR38, OROS). Theacquisition rate is set to 25 kHz, ensuring that frequen-cies up to 12.5 kHz are adequately acquired. Note thatthe overall bandwidth of the measurement chain is [1–12500] Hz, which implies that the average value of thepressure signal is not available. In the dynamical condi-tions of a rough/rough contact sliding at sufficiently highspeeds and involving impacts in the millisecond range,as targeted with this instrumented slider, this is not alimitation. However, for combined low sliding speedsand large wavelengths of the topography, like in someof the test conditions explored in section III B, the lower-frequency, information-bearing contents of the pressuresignals may be filtered out from the outputs.The multi-channels acquisition device is used not onlyfor the pressure outputs from the piezoelectric sensors,but also for the other measured signals (impact-hammerforce, see section II D and friction force, see section III),so that all of them are acquired with the very same timeframe.
D. Dynamical characterisation
To calibrate and characterise the dynamical capabili-ties of our instrumented slider, we conducted a series ofexperiments in which the slider was first clamped to thetable. Then each of its nine individual surface asperitieswas submitted to short stimuli by an impact hammer(PCB piezotronics, model 086C03) equipped with a steelimpact cap. Ten impacts have been performed on eachasperity, and for each, the response of all nine sensorshas been monitored.Figure 9 shows the results of a typical experiment. Theimpact duration is of order 0.5 ms, with a maximum forceof order 100 N. The response of the piezoelectric sensorlocated just below the impacted surface asperity is essen-tially identical to that of the hammer. As a consequence,for each impact, we could determine a single scalar co-efficient relating the force amplitude of the hammer’sresponse to the voltage amplitude from the stimulatedpiezoelectric sensor. For each sensor, the sensitivity valuewas then taken as the average of those coefficients overthe ten impacts on it. Note that those coefficients havebeen applied to all signals in Fig. 9.When an impact is made above one of the nine piezo-electric sensors, the eight other sensors have a non-vanishing oscillating response, as seen in Fig. 9. Thisis an undesired feature of our instrumented slider, dueto residual coupling between sensors. Those mechani-cal couplings are presumably due to the finite stiffnessof the top plate of the slider, through which normal vi-brations can propagate and affect all sensors[28]. Thisinterpretation is consistent with the results of the modal -1 0 1 2 3 time (s) -3 -20020406080100120 f o r c e ( N ) sensor 1sensor 2sensor 3sensor 4sensor 5sensor 6sensor 7sensor 8sensor 9impact hammer FIG. 9. Typical response of all nine piezoelectric sensors whenasperity 1 is stimulated with an impact hammer. Grey: forcevs time in the hammer. Black dashed (other colors, solid):concurrent force in piezoelectric sensor 1 (in the other eightsensors). analysis presented in section II B, where it is seen thatthe first eigenmodes of the slider are related to deforma-tions of the top plate involving its whole surface, and notlocalised around a single sensor. In addition, the firstpredicted eigenmodes have frequencies expected between1.5 and 2 kHz, which can easily be excited by the con-sidered impacts of duration of the order of, but shorterthan, 1ms. And indeed, in the various coda of Fig. 9,one can see oscillations with a typical frequency of ordera few kHz (see, e.g., the coda of sensor 1 at about 3 kHz).The frequency contents of the signals illustrated inFig. 9 can be seen in Fig. 10, which shows the averagePower Spectral Density (PSD) over the ten shocks per-formed on asperity 1. It appears that the spectra of thehammer and of the directly excited sensor are equal in thewhole frequency range up to 10 kHz, meaning that me-chanical energy is actually injected and accurately mea-sured by the piezoelectric sensor up to 10 kHz. Note thatthe shape of the PSD, with its local minima at about 4700and 7300 Hz, is fully consistent with that of a half-sinewave of duration 0.33 ms, which is a good approximationof the force signal in the impact hammer. As expected,for most of the frequencies in the range [1–1000] Hz, theother spectra are much weaker, by more than one orderof magnitude. However, the spectra of the sensors thatare not directly excited by the hammer exhibit a series ofpeaks, whose amplitudes almost reach that of the excitedsensor. Those peaks are interpreted as the eigenmodes ofthe slider, which is substantiated by the good matchingbetween the frequency bands in which the peaks in thePSD are found, and those in which the eigenmodes havebeen predicted using the modal analysis of section II B frequency (Hz) P o w e r s pe c t r a l den s i t y ( N ² / H z ) sensor 1sensor 2sensor 3sensor 4sensor 5sensor 6 sensor 7sensor 8sensor 9impact hammerEigenfrequencies FIG. 10. Average PSDs of the nine output signals from thepiezoelectric sensors and from the impact hammer, over tenimpacts on asperity 1, one of which is shown in Fig. 9. Dottedvertical lines: eigenfrequencies predicted by the FE modalanalysis of section II B. (see Fig. 8 and dotted vertical lines in Fig. 10). Note thatFig. 10 is actually in nice qualitative agreement with theFE predictions of Fig. 8: sensors 2 and 4 are also amongthe ones having the strongest response when asperity 1is excited, and the signals of the non-excited sensors isenhanced in the same frequency bands (especially vis-ible around 2 and 3 kHz). Overall, we emphasise thatthe large difference in amplitude between the excited as-perity and the others, for all frequencies below 2 kHz, isa significant success, because it opens to possibility toidentify the asperity on which the impact is made fromthe unsollicited ones.
E. Impact localisation and force estimation
Based on the experimental results of section II D, wenow discuss further the expected capabilities of our in-strumented slider to locate in space and time, and tomeasure the amplitude of, a single impact among a ran-dom series of impacts. Given the large altitude of thenine spherical surface asperities with respect to the sur-face plate (about 3 mm), tracks with a regular topogra-phy (i.e. free from high-aspect-ratio asperities) will makecontact with the slider only at those surface asperities,and not via the plate. Spatially locating an impact thusreduces to identifying which surface asperity is involvedamong the nine possible ones.
Model of the sensors’ outputs
Assuming that the slider behaves linearly, the time out-put of the j th piezoelectric sensor, S j ( t ), can be writtenas: S j ( t ) = (cid:88) i =1 (cid:90) ∞ C ji ( τ ) F i ( t − τ )d τ , (1)where F i ( t ) is the force signal applied on the i th surfaceasperity, and C ji ( t ) is the response function of sensor j to an infinitely short impact on asperity i . Estimatingthe precise form of the 9 × C ji is a verychallenging task that we do not attempt to perform here.Instead, we will make some assumptions enabling simpli-fication of Eq. (1) in an analytical way, thus providing adirect understanding of its metrological implications.We first neglect any output from the piezoelectric sen-sors when no force is applied to the slider, i.e. we neglectthe codas in Fig. 9. During an impact, we already notedin section II D that the output of the impacted sensorfaithfully represents the time evolution of the externalforce. What can be seen from Fig. 9 is that, for all theother eight sensors, the output during the impact has abell shape with a small, negative amplitude. Althougheach of those signals has an individual extremum at a dif-ferent instant, we now further assume that they all havethe same shape as the external force (but different am-plitudes). The two above-mentioned assumptions allowus to reduce the response functions to: C ji ( τ ) = a ji δ ( τ ) , (2)where the coefficients a ji are assumed to be constants(unit V/N), and δ is the Dirac delta function. The co-efficients a jj correspond to the individual sensitivities ofthe piezoelectric sensors, whose (positive) value has beenestimated as described in section II D and used to plotsensor outputs in Newtons in Fig. 9. In a similar way, weevaluated the values of the a ji,i (cid:54) = j as the average value,over ten stimuli by the impact hammer on surface as-perity j , of the extremum value of the output of sensor i over the duration of the impact. If the instrumentedslider was perfect, we would have a ji = 0 for all i (cid:54) = j .In contrast, as seen in Fig. 9 from the signals of sensors2 to 8 during the impact, the a ji,i (cid:54) = j have non-vanishing(negative) values, which characterise cross-talk betweenthe outputs from different sensors.Inserting Eq. (2) into Eq. (1), and dividing by a jj , weobtain the following simplified relationship: S j ( t ) a jj = F j ( t ) + (cid:88) i =1 ,i (cid:54) = j a ji a jj F i ( t ) . (3)The quantity S j ( t ) a jj can be seen as the estimate of theimpact force, F j ( t ) , based on the sensor output, S j ( t ). Ifall a ji,i (cid:54) = j = 0, the estimate is accurate because S j ( t ) a jj = F j ( t ). In reality, the estimate is biased by an error E ( t ) = (cid:80) i =1 ,i (cid:54) = j p ji F i ( t ), which depends both on the amplitudesof the external forces, F i ( t ), and on the ratios of cross-talk over sensitivity coefficients, p ji = a ji a jj . The absolutevalues of all p ji are provided in Fig. 11. By definition,all diagonal elements are 100%. Except for sensor 4, allnon-diagonal terms remain smaller than 10%, indicatinga relatively small cross-talk between outputs.
8% 100% 10% 16% 10% 4% 3% 3% 4%
9% 5% 100% 20% 7% 5% 5% 8% 4%
6% 9% 4% 100% 10% 4% 8% 9% 4%
6% 8% 5% 12% 100% 4% 4% 7% 7%
4% 2% 4% 13% 5% 100% 3% 2% 4%
4% 4% 3% 6% 4% 4% 100% 10% 5%
4% 4% 4% 10% 5% 6% 6% 100% 7%
1% 2% 2% 1% 3% 3% 2% 2% 100%
FIG. 11. Matrix of the ratios | p ji | = (cid:12)(cid:12)(cid:12) a ji a jj (cid:12)(cid:12)(cid:12) . All values are in%. The grey level of a box is proportional to its value. Row:label of the impacted asperity. Column: label of the sensor. Localisation strategies
Let us first consider the case where the slider is submit-ted to a series of impacts happening sequentially, withoutany overlap in time: for each external force signal F j ( t )on asperity j , F i,i (cid:54) = j = 0 in the same time interval. Weare in the favourable case, already tested with the impacthammer, where signals consist of a succession of singleevents like the one shown in Fig. 9. For each, the asper-ity involved will be the one giving the largest signal (say j ); its amplitude will be accurately estimated by S j ( t ) a jj ,while the other signals (for i (cid:54) = j ) can be overlooked.The situation is more complex when several impactshappening on different asperities overlap in time. In sec-tion III A, we will argue theoretically and confirm ex-perimentally that the number of asperities in simultane-ous contact with a rigid track is not expected to exceedthree. For the rest of this section, we will thus considerthe unfavourable case of three simultaneous impacts, allwith a force evolution proportional to the same func-tion F ( t ). Without loss of generality, we will assumethat those three asperities are labelled 1 to 3, and that F ( t ) = F ( t ), F ( t ) = αF ( t ), F ( t ) = βF ( t ), with1 > α > β . Finally assume that all p ij are such that | p ji | ≤ p . In those conditions, Eq. (3) can be straightfor-0wardly recast into: S a = F ( t ) (1 ± p ( α + β )) , (4) S a = αF ( t ) (cid:18) ± p (1 + β ) α (cid:19) , (5) S a = βF ( t ) (cid:18) ± p (1 + α ) β (cid:19) , (6) S j a jj = ± F ( t ) p (1 + α + β ) , j > . (7)Those equations indicate a straightforward data analy-sis when three simultaneous impacts are suspected. Firstidentify the three sensor outputs giving the largest im-pact forces and record the three corresponding peak val-ues. The largest gives F , while the ratio of the other twoover F give α and β . With p = 0 . F is a conservative estimate ( p (1+ α + β ) < .
3) of theexperimental noise. In other words, all peaks larger thanthis value can be safely considered to be true impactsand not the results of some cross-talking between out-puts. All peaks below may be discarded. The amplitudeof the impact force on the successfully localised asperitiescan thus be conservatively estimated, using Eqs. (4), (5)and (6), to be F ± αF ± α % and βF ± β %,respectively. III. EXPERIMENTAL RESULTS
In order to illustrate the tribological relevance of theinstrumented slider described and characterised in sec-tion II, we now slide it on various model topographies. Astainless steel track has been textured with macroscopicobstacles, grooves and bumps of sufficient amplitude andlateral size to induce controlled changes in the distribu-tion of spherical asperities in contact during sliding. Fig-ure 12 shows a picture of the instrumented slider, laidon the textured track (upside down compared to, e.g.,Fig. 2). The latter is attached on a motorised (Kollmor-gen, AKM Series servo-motor) linear translation stage(Misumi LX26). When the track is translated, the dis-placement of the slider is prevented by a stopper, fixedin the laboratory frame, and equipped with a horizontalsteel cylinder (1 mm diameter) which comes into contactwith the vertical plane of the lateral arm of the slider’sbase which is close to asperity 4 (see Fig. 3). The tan-gential force exerted by the stopper is measured by apiezoelectric sensor (Kistler 9217A, with charge ampli-fier Kistler 5015A).The principle of the experiments is the following. Thenormal load on the interface, approximately P = 2 . V , in the range [0.5–5] mm/s. Two different il-lustration cases will be analysed in the following: when FIG. 12. Picture of the mechanical setup, showing the in-strumented slider laying under its own weight on a texturedmetallic track. The latter is translated horizontally at a con-stant velocity, V , while the slider is held essentially fixed inthe laboratory frame by a stopper. the slider explores a nominally flat part of the track (sec-tion III A) and when one of the spherical asperities passesabove a macroscopic hole (section III B). A. Sliding on a nominally flat surface
Figure 13 shows a typical recording of the nine piezo-electric sensors during steady sliding over a nominally flatpart of the textured track, at a velocity V = 0 . µ m. An estimate of the order of magnitudeof the normal force on the highest asperity necessary todeform it by such an amount (and thus, assuming thatthe slider cannot rotate, to allow both asperities to bein contact simultaneously) can be obtained using Hertz’stheory. With a radius of curvature R =0.75 mm, Young’s1
25 25.005 25.01 25.015 25.02 25.025 25.03 25.035 25.04 25.045 25.05
Time (s) -0.100.1 F o r c e ( N ) sens.1 sens.2 sens.3 sens.4 sens.5 sens.6 sens.7 sens.8 sens.9
21 21.5 22 22.5 23 23.5
Time (s) -0.100.1 F o r c e ( N )
20 25 30 35 40 45 50 55 60 65
Time (s) -0.200.2 F o r c e ( N ) a.b.c. FIG. 13. Typical time evolution of the asperity forces recorded during steady sliding on a nominally flat part of the track at V = 0 . modulus E =210 GPa and Poisson’s ratio ν =0.3 for theasperity, such a normal force is about 400 N, i.e. waylarger than the slider’s weight ( P =2.5 N). Thus, in thelight normal load conditions explored here, only threeasperities are expected to touch the track, to satisfy iso-static equilibrium, which explains our first observation.In order to predict which are the three asperities incontact among the nine potential asperities, a simplemulti-asperity contact model has been used, fully de-scribed in Appendix. The model is dynamic, but whenconstant stimuli are imposed, it provides results rele-vant to quasi-static configurations. In particular, whenthe model is used with a vanishingly small sliding veloc-ity, it predicts, based on the experimental in-plane po-sitions and altitudes of all nine spherical asperities (seeTab. II A), which are the three that are in contact witha given track topography, h ( x, y ). In the present case inwhich h can be assumed to be homogeneous, the modelpredicts that asperities 1, 7 and 9 are in contact to sat-isfy isostatic equilibrium. Those asperities are the same as the ones that are active in Fig. 13(a). The model inAppendix thus fully explains both the number and iden-tities of the asperities in contact.The second observation in Fig. 13(a) is the periodicityof the local force signals, at a frequency about 200 Hz.Inspection of the PSDs of all force signals for all ex-plored sliding velocities revealed a common frequencyband [150–250] Hz with a significantly larger amplitude,responsible for the oscillations in Fig. 13(a). The sliding-velocity-independence of this (high-)frequency band (seeFig. 14) suggests that the oscillation is due to eigenmodesof the experimental system. Since all eigenmodes ofthe clamped slider, identified in section II D, were above1 kHz, we presume that the 200 Hz mode is associatedwith the vibration of the bundles of cables (see Fig. 12),which is now possible with the slider freely standing onthe track.Another periodicity can be found in the force signalsof the three contacting asperities, at a larger time scale,as illustrated in Fig. 13(b). For V = 0 . FIG. 14. Evolution of the various frequencies identified inthe force signals, as a function of the sliding velocity. Pink:very low frequencies seen in Fig. 13(c). Red: low frequenciesseen in Fig. 13(b). Circles: fundamental. Squares (+): first(second) harmonic. The high-frequency band ([150–250] Hz)responsible for the oscillations seen in Fig. 13(a) is shown inblue and delineated with dashed lines. Solid and dotted lines:linear fits. served frequency is about 2 Hz. Such a low frequencysuggests that the nominally flat track surface is actuallydecorated with a periodic topography. This is indeedconfirmed in Fig. 15(a), which shows the topography ofa characteristic portion of the track surface. It is madeof characteristic trochoidal grooves originating from theface milling process used to prepare the surface. As seenin Fig. 15(b), those grooves have a period of 248 ± µ m(as measured from the corresponding peak in the Fouriertransform of the profile in Fig. 15(b)) and typical peak-to-peak amplitude 8 µ m. Assuming that the profile inFig. 15(b) would be a perfect sine wave, the radius ofcurvature of the bottom parts of those grooves would beabout 400 µ m, i.e. smaller than that of the spherical as-perities (750 µ m). Thus, the slider’s asperities cannotexplore the whole topography, but only the crests of thegrooves. More quantitatively, we calculated that a sphereof radius 750 µ m cannot go deeper than 6.5 µ m into theabove-mentioned sine wave without yielding an unphys-ical interpenetration of the two solids. We have thusapplied the model in Appendix to static topographies inwhich the track altitude at the location of asperities 1,7 and 9 were offset by either 0 or -6.5 µ m. We testedall eight combinations of those two offsets at the loca-tion of the three asperities, as estimators of the strongestpossible differences with respect to a perfectly flat track.Only two combinations led to a change in the predictedasperities in contact: asperity 4 takes over asperity 7when both asperities 1 and 7 are simultaneously above the deepest point of a groove. Given the small probabil-ity of such a situation, it is reasonable to consider it asirrelevant in practice. As a consequence, the amplitudeof the topography of the milling-induced grooves is notsufficient to induce detectable modifications of the asper-ities in contact with respect to a perfectly smooth track,which explains why the three asperities in contact (1, 7and 9) remain the same all along the sliding experimenton the nominally flat part of the track. FIG. 15. (a): 2D map of the topography of the nominally flatsliding surface, as measured from interferometric profilometry.(b): profile extracted along the segment drawn in the toppanel, which is locally orthogonal to the grooves.
To ascertain the topographical origin of the force oscil-lations seen in Fig. 13(b), we now compare the frequencyof those oscillations with that expected from sliding onthe grooves. We performed sliding experiments at differ-ent velocities, V , and for each we calculated the PSD. Inthe frequency range [1–100] Hz, they all contain a well-defined main peak (fundamental) and a series of har-monics. The latter are due to the non-harmonic shape ofthe periodic pattern. The frequencies of the fundamen-tal and of its two first harmonics are shown as a functionof V in Fig. 14. All three frequencies are found propor-tional to V , as demonstrated by the slope 1 in the log-logrepresentation of Fig. 14. This shows that the temporalperiodicity actually originates from a spatial periodicity.For the fundamental, the coefficient of proportionality,which corresponds to the spatial period, is found to be249 ± µ m. This value is in good agreement with the pe-riod of the grooves found by profilometry (248 ± µ m).A third periodicity can be seen in Fig. 13(c), as a verylow frequency modulation of the signal’s envelope. Asseen in Fig. 14, the frequency of this modulation is alsoproportional to the sliding velocity, thus indicating a to-pographical origin for it. The corresponding wavelength,estimated from the coefficient of proportionality of theassociated line in Fig. 14, is about 5.0 mm. This char-acteristic length scale can indeed be distinguished as awaviness in the topographical signal of Fig. 15. Such awaviness is classically attributed to an undesired coaxi-ality error between the centerlines of the spindle and themilling tool.3
70 80 90-505 10 -3
70 80 90-505 10 -3
70 80 90-505 10 -3
70 80 90-505 10 -3
70 80 90-505 10 -3
70 80 90-505 10 -3
70 80 90-505 10 -3
70 80 90-505 10 -3
70 80 90-505 10 -3 time (s) f o r c e ( N ) FIG. 16. Blue: Force in N on the nine asperities (asperity label indicated in top left corner of each panel) when asperity 7passes over a deep, 6 mm long groove at V = 0 . Overall, with the above analyses, we took advantageof undesired residual topographical features on the flatparts of the track to show the capabilities of our instru-mented slider. Not only is it able to detect those residualfeatures at different time/length scales, but, as desired,it also quantifies the elicited force fluctuations, and indi-cates which asperities are excited.
B. Sliding on a macroscopic topographic defect
In section III A, we have seen that, when the trackis nominally flat, asperities 1, 7 and 9 are in contactat all times. We will now consider a situation in whichthe asperities in contact must change during sliding, be-cause one of those three asperities passes over a hole deepenough to prevent any possible contact for a certain slid-ing distance. Such a situation is illustrated in Fig. 16,which shows the recorded force signals on all nine asperi-ties over a time window bracketing the passage of asperity7 over a 6 mm long hole.Before reaching the hole, the situation is the same asin section III A: asperities 1, 7 and 9 are in contact, asindicated by a significantly larger signal width than theother six, due to the high frequency oscillations seen inFig. 13(a). When asperity 7 passes over the hole, its force signal fluctuations suddenly die out, indicating that, asexpected, it has lost contact with the track. Simulta-neously, the force fluctuations on asperity 4 abruptlyincrease, showing that asperity 4 is now sliding on thetrack. Application of the model in Appendix to the newtopography seen by the slider consistently predicts thatasperities 1, 4 and 9 is the triplet satisfying the new iso-static equilibrium, thus explaining the switching of activ-ity between asperities 7 and 4. When asperity 7 reachesthe end of the hole, the slider and hence all force signalsrecover their initial configuration.We emphasise that, in Fig. 16, the fact that a givenasperity is in contact can only be seen from the ampli-tude of the force fluctuations discussed in section III A.Contact/non-contact states do not translate into a fi-nite/vanishing average level of the signals, because ofthe high-pass filtering effect of the conditioning ampli-fiers used on each sensor’s output. Those amplifiers areclaimed by the manufacturer to behave like a 1 Hz firstorder high-pass filter, meaning that, after about 1 s, anyconstant force applied on an asperity will be filtered outfrom the outputs, yielding force signals centered on zero.Nevertheless, the force signal on asperity 7 exhibits tran-sients at the edges of the hole, when asperity 7 gets outof (or back into) contact with the track, which are sig-natures of the corresponding changes in average normal4force on the asperity. Figure 17(a) overplots the tran-sients associated with the force drop at the entry of thehole, for all tested sliding velocities. As can clearly beseen, they all have the same amplitude and shape, withthe same characteristic timescale for the force relaxation.The very same phenomenology is observed in other ex-periments in which other asperities pass over the hole, asillustrated in Fig. 17(b) on the case of asperity 9.
Time (s) -0.2-0.100.10.20.3 F o r c e ( N ) a. Time (s) -1-0.500.5 F o r c e ( N ) b. Time (s) -1-0.500.511.5 F o r c e ( N ) c. FIG. 17. Force transients on asperity 7 (a) or 9 (b and c)when it suddenly looses (a and b) or gets back into contact(c) with the track for various sliding velocities. Dashed lines:instantaneous negative (a and b) or positive (c) force dropsfiltered by the second-order high-pass filter of Eq. (8).
The time evolution of all signals in Fig. 17(a) and (b)can be nicely captured as the response of the followingsecond-order high-pass filter F ( f ) = − f f c jfQf c − f f c (8)applied to a negative step-change (force drop) on the as-perity. Q =0.656 is the quality factor and f c =0.74 Hzis the cutoff frequency. Note that this cutoff frequency is close to the 1 Hz expected for the filtering effect ofthe conditioning amplifiers. The fact that the filter is ofsecond order rather than the expected first order is pre-sumably due to a non-negligible effect of the rest of themeasurement chain on the final outputs.Importantly, the peak in the filtered response is equalto the amplitude of the force step-change, meaning thatthe amplitude of the peak of the transient is actually ameasurement of the force drop undergone by the asper-ity when it start passing over the hole. In panel (a) ofFig. 17, one can thus estimate the force drop on asperity7 to be about -0.12 N, while it is about -0.96 N in panel(b) for asperity 9. Those values actually compare verywell with the force drops predicted by the model in Ap-pendix: -0.11 N for asperity 7, and -1.15 N for asperity9. Note that transients with the opposite sign are ob-served in the force signals when the underlying forcestep is opposite, for instance when asperity 9 suddenlygets back into contact on the other side of the hole(Fig. 17(c)). The transients are still well-captured by thefilter of Eq. (8) (shown only for V =5 mm/s on Fig. 17(c)).However, in contrast with when the asperities loose con-tact with the track, when they get back into contact,the amplitude of the transient exhibit a non-negligiblesliding-velocity dependence. It would correspond to apositive force-step of amplitude 0.8, 0.9, 1.1 and 1.25 Nfor V =0.5, 1, 2 and 5 mm/s, respectively. Such a velocitydependence is compatible with more violent lateral im-pacts of asperity 9 with the end corner of the hole as thesliding velocity is larger. This effect is not expected at theentry of the hole, which may explain why the same am-plitude was found for all sliding velocities in Figs. 17(a)and (b).Overall, the good agreement between the sensors’ out-puts and the filtered expected force jumps indicates thatour instrumented slider can efficiently be used to mea-sure the impact forces on its asperities. The results ofFig. 17 show that it is the case for abrupt step-changesin the force between two constant values. For more real-istic impacts characterised by a short contact duration,i.e. sufficiently shorter than the cutoff timescale of about1 s, the high-pass filter will deform only slightly the sig-nals, so that the sensors outputs are expected to provide,directly, a faithful image of the force evolution. To ver-ify this, we applied the filter of Eq. (8) onto half-sinemodel impacts with different durations between 10 s and0.1 ms. We found that the alteration of the force evolu-tion is negligible (less than 2% in amplitude) for impactsshorter than about 0.01 s. This result strongly suggeststhat, during sliding of two randomly rough metallic sur-faces, for which the typical impact duration is of the orderor less than the millisecond [21], our slider will provideaccurate measurements of the individual impact forces.5 IV. CONCLUSION
We have introduced a 6 cm-sized instrumented sliderbased on an array of piezoelectric sensors. It is able tomonitor the normal component of the individual contactforces on each of nine model asperities, enabling bothspatial localisation, and measurement of the amplitude,of the impacts on the slider’s surface. The slider behavesstatically below about 1.5 kHz, and rejects less than 5%of the force on a given asperity onto the other sensors.We showed how the slider can successfully be used tomeasure the force variations on each individual surfaceasperity elicited by either minute topographical featureson a nominally flat track, or by large defects such that anasperity suddenly loses (or gets back into) contact withthe track.Our instrumented slider thus appears as a promisingtool for a variety of future tribological studies involvingrealistic randomly rough surfaces. In particular, it shouldbe useful to better characterise the impacts at the originof the roughness noise, the source of friction-induced vi-brations, or the space-time patterns of the real contactfluctuations during sliding.
ACKNOWLEDGMENTS
We acknowledge support from Labex CeLyA of Uni-versit´e de Lyon, operated by the French National Re-search Agency (Grants No. ANR-10-LABX-0060 andNo. ANR-11-IDEX-0007). We thank the following col-leagues from LTDS for their help: St´ephane Lemahieu(welding and gluing of the piezoelectric sensors), Di-dier Roux (manufacturing), Matthieu Guibert (designand instrumentation), Youness Benaicha and Karl Lan-det (finite elements), Davy Dalmas (critical reading ofthe manuscript).
APPENDIX: MULTI-ASPERITY CONTACTMODEL
We model the instrumented slider as a rigid, homoge-neous parallelepiped submitted to the gravitational field (cid:126)g = − g (cid:126)e z (Fig. 18). We consider only three degrees-of-freedom: the vertical displacement of the centre of mass G of the slider, z along the axis (cid:126)e z , and its two rota-tions around the G (cid:126)e x and G (cid:126)e y axis, respectively φ and ψ . In contrast, the displacement of the centre of massalong the (cid:126)e x and (cid:126)e y axis, as well as the rotation aroundthe G (cid:126)e z axis are constrained. The slider is supported bya track moving along the (cid:126)e y axis at a constant velocity (cid:126)V = V (cid:126)e y . The track is textured and we assume that thetopography is described by the spatial function h ( x, y ).Interactions between the moving track and slider dependon the contact conditions at the nine hemispheric surfaceasperities of the slider, the summits of which are located at points P j ( j = 1 , .., FIG. 18. Sketch of the multi-asperity contact model.
Without loss of generality, the resulting macroscopicforce F z (cid:126)e z and torques L x (cid:126)e x , L y (cid:126)e y can be obtained fromthe local normal forces f j (cid:126)e z (with f j ≥
0) exerted by thetrack on the slider through its surface asperities, and fromthe distances between the corresponding contact points P j and the center of mass G . Letting (cid:126)GP j = x j (cid:126)e x + y j (cid:126)e y − z j (cid:126)e z j = 1 , .., z j > m ¨ z = − mg + F z ( f j ) = − mg + (cid:88) j =1 f j ,I x ¨ φ = L x ( f j , y j ) = + (cid:88) j =1 y j f j ,I y ¨ ψ = L x ( f j , x j ) = − (cid:88) j =1 x j f j , (10)where m , I x and I y are, respectively, the mass and mo-ments of inertia around (cid:126)e x and (cid:126)e y . FIG. 19. Sketch of the normal indentation at contact points.
To compute the local contact force on asperity j , weassume Hertzian behaviour [29] for its elastic part andnonlinear viscous damping for its dissipative part. Thus,we have: f j = (1 + α j ˙ δ j ) k j δ j / , (11)6where δ j represents the normal contact indentation (seeFig.19), which can be expressed, by choosing convenientorigins and under the assumption of small displacement,as follows: δ j = ( h j − z + z j + φy j − ψx j ) H ( h j − z + z j + φy j − ψx j ) , (12)where h j = h ( x j , y j − V t ) represents the local altitude ofthe textured track at the contact point P j , and H ( . ) isthe Heaviside function introduced to account for loss ofcontact. In practice k j is obtained from Hertzian theory,so k j = k = 43 √ RE ∗ , (13)with R the radius of the surface asperity, E ∗ the compos-ite modulus such that 1 /E ∗ = (1 − ν ) /E + (1 − ν ) /E ,with E , and ν , the Young moduli and Poisson coeffi-cients of the antagonist solids.Note that in Eq. (11), the dissipative force is assumedproportional to the elastic force, which we have obtainedfrom an equivalent viscous damping of the linearised dy-namic response around the static deformation. To thisend, we assume that (i) the slider’s motion is only gov-erned by the vertical displacement z ( φ = ψ = 0), (ii) thenumber of contacts n respects isostatic equilibrium un-der light load, so that n =3, and (iii) the initial distancesat the location of the 3 asperities ( h j + z j ) are all equalto D . Consequently, normal approaches are identical,leading to: δ j = ( D − z ) H ( D − z ) , (14)˙ δ j = − ˙ zH ( D − z ) . (15)By introducing the variable u = D − z , ˙ u = − ˙ z , ¨ u = − ¨ z and assuming permanent contact H ( u ) = 1, we obtainfor Eqs. (10), (11) and (12): − m ¨ u = − mg + (cid:88) i =1 (1 + α ˙ u ) ku / , (16) m ¨ u + 3 k (1 + α ˙ u ) u / = mg, (17) m ¨ u + K (1 + α ˙ u ) u / = N, (18)with N = mg and K = 3 k . The static equilibrium iseasily obtained as u s = ( N/K ) / . Introducing the newvariable q defined by u = u s (1 + q ), ˙ u = u s ˙ q and¨ u = u s ¨ q leads to:2 mu s q + Ku s / (1 + 2 αu s q )(1 + 23 q ) / = N. (19)Now we can linearise the motion equation around thestatic equilibrium q = 0, that is:(1 + 23 αu s ˙ q )(1 + 23 q ) / ≈ (1 + 2 αu s q )(1 + 23 q ) (20) ≈ q + 2 αu s q, (21) and 2 mu s q + Ku s / (1 + 2 αu s q + q ) = N. (22)With ω = Ku s / m , τ = ωt and () (cid:48) = ddτ (), we get: d qω dt + 2 αu s ω dqωdt + q = 0 , (23) q (cid:48)(cid:48) + 2 ζq (cid:48) + q = 0 . (24)So, the coefficient α can be related to an equivalent modalviscous damping ζ = αu s ω/
3, which gives α = 3 ζu s ω = √ ζK − / m / u − / s (25) α = √ ζK / m − / g − / = 3 / √ k / m − / g − / ζ (26) α = (1944 k m − g − ) / ζ (27)(28)Note that the dimension of α has been verified to be ∼ T L − and unit s.m − .In order to compute the dynamic response of the sliderunder the excitation of the moving track, a classical nu-merical time integration method is used, based on theexplicit velocity-Verlet scheme. In order to ensure thestability of this time-integration scheme, we must choosea time step ∆ t such as Ω∆ t <
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