## 1 Introduction

In the text of González-Fernández [1] can be found a detailed description of the fundamentals of the method and the first applications in different fields of science and engineering: electrochemical processes, transport through membranes, heat transfer, etc. After this date, the Network Simulation Method (NSM), has been applied to new problems developing models not covered by a specific new text, so that the person concerned should refer to the specific scientific publications or doctoral theses in the research group ‘Simulation Networks’, from University of Cartagena (UPCT), or research groups working with this method at the Universities of Granada, Jaen and Murcia: [2–7]. In this regard it should be mentioned applications in the fields of fluid flow to transport (of mass or heat), inverse problem in heat transmission, magneto-hydrodynamic flow, mechanical vibrations, tribology, dry friction, membrane transport, elasticity, development of specific numerical calculation programs, etc. NSM had already been successfully applied in various fields of engineering, such as heat transfer [7, 11], electrochemical reactions [12, 13], transport across membranes [14], inverse problems [15–18], ion transport, magneto-hydrodynamics [19], problems coupled flow and transport [17,20–22]; Alhama, Soto and al 2012), and others [23–26]; all these works describe nonlinear transport processes. In addition, recently, it has been applied to mechanics of deformable solids [27], and dry friction at the atomic level.

On the other hand, there have been various programs that use the NSM as a tool for numerical calculation: PRODASIM for designing simple fins [28], PROCCA_09 for design and optimization of thermal problems [29], FATSIM_A for simulating flow fluids with solute transport problems [30], FAHET for simulation of flow fluids with heat transfer problems [31], EPSNET_10 for simulation problems elasticity [32] and OXIPSIS_12 for simulating corrosion problems [33].

The formal equivalence between the network model and the physical process is that both are governed by the same differential equations in finite difference in space, covering both the elementary cell volume or as to the boundary conditions. However, time remains continuous variable in the design model.

The formal approach, which is the basis for the development of the problems, is the ‘Network Theory’ of Peusner [34], in which his ‘thermodynamics of networks’ is supported; this theory, in turn, is based on the theory of circuits from a generalization of their conjugate variables, electric current and potential difference. Network models are for Peusner an accurate representation of the mathematic characteristic of the processes described. Thus, the variable velocity and displacement, characteristics of the problems, must satisfy Kirchhoff’s laws and their relationships determine the corresponding circuit elements. However, in each individual process and once conjugate variables chosen, the information on which circuit elements involved in the network model and how they connect with each and other, is obtained from the mathematical model and not on considerations of physical type on the role playing these variables.

In summary, in network theory, the feasibility of a network model involves:

- The existence of an independent network of time
- The existence of a conserved quantity called flow associated with each branch, connecting nodes and obeys the laws of Kirchhoff currents (LCK)
- The existence of a magnitude that satisfies criteria of uniqueness associated with each node, and obeys the Kirchhoff voltage law (KVL)

It is going to be showed different models of application of the method to different fields of engineering to be presented in the second section. In the third, the governing equations, and its transposition into network models in the fourth. Applications are going to be shown in the fifth section, concluding with the conclusions in the sixth.

## 2 Multi-Scale Problems

To show the most practical aspects of the method, applications at different scales of the same physical problem, dry friction, will be described. This allows us to probe the flexibility of the method to exploit those portions of code developed for a given application to another which has quite different elements. This capability allows efficiently address problems that occur at very different scales. From this perspective, it is interesting to work with elements, circuit components, which can represent very different scales phenomena.

From this perspective there are several problems that must be addressed from very different scales, so even coupled, such as stress analysis in ceramic coatings of metal parts. In this type of problems the most important difficulty is to exchange information between macro and micro scales.

However, in this work, in order not to overcomplicate the exhibition uncoupled problems have been chosen. Specifically, there has been the analysis of two devices: atomic force microscopes, dry friction at the atomic scale, and Girling brakes, dry friction in a device of industrial application. The first problem is formulated considering physical associated with crystalline structures under study of the mentioned microscope, while the second is the object of mechanics more applied.

## 3 Atomic Scale Problem

Atomic scale dry friction phenomenon is analyzed through Network Simulation Method implemented models, specifically ‘dry friction between microscope tip and smooth surface at atomic-scale’.

This study will analyze the phenomenon of dry friction at the atomic scale using models implemented by the network simulation method, namely the microscope frictional force (FFM), evolution of the microscope scanning force (SFM), increases research capacity of phenomena at the atomic scale, key issue for understanding the origin and nature of the fundamental laws of friction. The analysis of the behavior of these point contacts is done by a two-dimensional model of friction at the atomic scale. The response of the tip is the ‘stick-slip’ type [35].

After a detailed review of the literature a number of techniques and materials for analysis have been selected. The cases are:

- NaF (001): it is a surface with translational symmetry. In this case the FFM is used [36]
- Highly Oriented Pyrolytic Graphite (HOPG) is well-defined structures sheets, often chemically stable. In this case an SFM and LFM are used [37]
- Graphite: the AFM is used [38]

### 3.1 The Governing Equations

The AFM, SFM, FFM tip physical scheme on a sample surface model, Fig. 1, assumes one rigid sliding body connected by three springs (one for each spatial direction) to the mobile microscope support. Each spring reflects the elastic interaction between the two surfaces during the contact [36, 38]. The sliding body involves inertial effect and the interaction with the sample surface.

The hypotheses currently assumed in FFM, AFM and SFM models are: (i) the microscope support is rigid and moves at a constant velocity, *v _{M}*, (ii) the cantilever and the tip show elastic behaviour, and (iii) interactional force between the tip and the sample surface is a function of the relative distances between the tip and the atoms forming the sample surface.

Several forces act on the tip in x-direction: inertial force represented by *m* · (*d*^{2}*x _{t}*/

*dt*

^{2}), where

*x*is the absolute displacement of the tip of mass m; elastic force from the spring, represented by the term

_{t}*k*· (

_{x}*x*−

_{M}*x*); and damping force represented by the term

_{t}*c*· (

_{x}*dx*/

_{t}*d*). The coefficients of these forces are constants. Similar forces act in the other directions. In addition, the interaction force between the tip and the sample surface is implemented from a surface potential [36, 38]. Thus, the surface potential for NaF with an FFM tip, as used by Hölscher et al. [36] is:

_{t}where the potential amplitude, *V*_{0}, is 1*eV* , while *a _{x}* and

*a*, the double of the lattice constant, are 4.62 for both parameters. Hölscher et al. [37] use a slight modification of Eq. 1 to study HOPG with a SFM tip:

_{y}where *V*_{0} is 0.5*eV* and the lattice constant, a, is 2.46Å. Sasaki et al. [38] used the Lennard-Jones potential to study the graphite with an AFM tip:

where the strength of the potential, *ε*, is 0.87381 · 10^{−2}*eV* , and the value of the distance between the tip and the surface atom at which the potential is zero, *σ*, is 2.4945Å. The distance between the tip and the *i* − th atom of the surface, *r*_{0i}, is calculated from the *i* − th atom position inside the lattice, which is defined by its constant, 2.46 Å, and the nearest distance between 2 carbon atoms, 1.424Å. Hölscher et al. used the same potential with the same sample surface and microscope [39], but *σ* is equal to 3.4Å.

As a result of the balance of the forces on the tip, Fig. 1, the following equations can be written:

where *x _{M}*,

*y*and

_{M}*z*are the microscope support displacements over the three coordinates. The movement of the microscope support is a set of trips in the x-direction. For each of these trips, the initial condition of tip displacement in the x-direction is null, and in the z-direction is null except for the AFM tip on graphite sample. In the y-direction, the initial condition for this displacement is the same as for the microscope support. The initial condition of tip velocity is null for each direction.

_{M}## 4 Macroscopic Scale Problem

The physical phenomenon of dry friction between sliding bodies is an important subject in mechanical engineering and many studies have been devoted to it [40–44]. An example of dry friction is found in the Antilock Braking System (ABS), which prevents the wheels from locking up and stops a vehicle at a much faster rate and with better control than a driver could manage. The simplest antilock braking device is represented by a nonlinear bi-dimensional dynamic system.

A source of the nonlinearity of this system is the relationship between the sliding process and the dynamic friction coefficient. Most classical friction models map the friction coefficient against the relative velocity in a static way. The shape of such characteristics is the same for qualitatively different dynamic responses (periodic, quasi-periodic, chaotic or stochastic), which means that the friction force is independent of the character of the motion, i.e., insensitive to the system dynamics (insensitive friction models). The term frictional memory [45] has been coined to refer to a pure delay between relative velocity and friction force [45] without reference to any specific model. Thus, hysteretic effects of frictional memory, such as frictional lag [46], pre-sliding displacement [47, 48], non-reversibility [49], break-away force [50, 51] and Stribeck effect [52], could explain it. The inclusion of these effects in friction models causes the friction characteristics to become motion-sensitive (sensitive friction models). It should be pointed out that the strong nonlinearity of the relative velocity near zero is the main problem when modelling the numerical switching algorithms [53]. Two examples of insensitive friction models are the classical Coulomb model [54], with a constant value of the kinetic friction coefficient, and the well-known Popp-Stelter approach with a nonlinear kinetic friction and Stribeck effect [14]. A model with symmetrical non-reversible characteristic and an exponential function of the relative acceleration is an example of the sensitive friction model [55].

Awrejcewicz used a physical model of a braking system for each wheel, which involves nonlinear coefficients [56–61]. To be more precise, a Girling ‘duo-servo’ brake [62], Fig. 2. To simplify the study of the system, it is usual to consider a sliding block, which represents the brake pad, and a rotational crank, Fig. 3, [56,63–65]. The sliding block is jointed to the chassis by a linear spring, with a stiffness *k*_{1}, and to the crank by two springs, with stiffness *k*_{2} and *k*_{3} for the horizontal and vertical directions, respectively. The elastic interaction between the brake pad and the drum brake is neglected. The belt velocity is represented by *v _{dr}*.

The solution of the dynamic problems is very sensitive to the range of the parameters involved in the problem: (*i*) parameters in dynamic friction coefficient, (*ii*) translational stiffness between the sliding surfaces and (*iii*) constant velocity.

The scheme used to analyse this mechanical system is the same employed in other friction problems studied by the authors [66, 67], and is at the core of a software application under development. For this reason, an exhaustive review of the state of the art is necessary, systematizing the different formulation of friction processes: Atomic Friction Microscope models [66] and macroscopic models [67].

The aim of this work is to use the rules of network simulation [1] to design a reliable and efficient network model for the stick-slip phenomenon, that can be simulated in commercial software of circuit analysis such as PSpice [68, 69]. The model, which provides both transient and steady state solutions, is based on the formal equivalence between the finite-difference equations of the mathematical and network models, an equivalence that goes beyond the scope of classical electrical analogy of many books [70, 71], which mainly deal with linear problems. The network method is a numerical tool whose efficiency has been demonstrated in many science and engineering problems, including heat transfer, electrochemistry, fluid flow and solute transport, inverse problems, etc. [72–75]. Once the equivalence between electric and mechanical variables has been chosen, linear terms of the PDE are easily implemented by linear electrical devices, such as resistors, capacitors and coils, while non-linear and coupled terms are implemented using auxiliary circuits or controlled current and voltage sources. The last are a special kind of source whose output can be defined by software as a function of the dependent or independent variables determined in any node or any electrical component of the model. In addition, boundary and initial conditions - whether linear or not - are also immediately implemented by suitable electric components. Once the network model has been designed, it is run with no need for other mathematical manipulations since the simulation code does this. The well-tried and powerful software for circuit simulation, PSpice, requires relatively short computing times thanks to the continuous adjustment of the internal time step required for the convergence. In addition, the solution simultaneously provides all the variables of interest: displacements and velocities.

### 4.1 The Governing Equations

Several forces act on the block: inertial force, represented by *x* is the absolute horizontal displacement of the block of mass *m*, and the elastic forces from the two springs, represented by the terms −*k*_{1} · *x* and −*k*_{2} · (*x* − *y*). Similar forces act on the crank: inertial force, represented by *y* is the absolute vertical displacement of the horizontal arm tip of the crank with inertial moment of *J*, and the elastic forces from the two springs, represented by the terms −*k*_{3} · *y* · *L* and −*k*_{2} · (*y* − *x*) · *L*. The coefficients of these forces are constant. In addition, the interaction force between the block and the belt is implemented by [76, 77].

In Eq. 5, the first equation represents the stick phase, when the block has the same velocity that the belt. In this case, it is not necessary to solve the dynamic equation to get information about the block velocity. The second equation represents the slip phase, with the block slipping, and the dynamic equation is used in order to obtain the acceleration. The friction force is governed by Coulomb’s Law, which involves step function.

From Eq. 5, it is possible to distinguish two physical phases: stick and slip. Stick is controlled by the following single equation

whereas slip phase is controlled by a more complex equation. Thus, as a result of the balance of the forces on the block and on the crank, the following equations can be written

where *F _{k}* is a more complex version of

*F*, which involves the relative velocity effect.

_{f}As other authors, we also consider *k*_{2} = *k*_{1} [56, 63–65]

The parametrized Eq. 7 is given by

In the Eq. 8, *ε* · *v _{rel}* is a nondimensional magnitude, and so

*ε*has the dimension s/m.

The last term of the first equation of Eq. 8 represents *F _{k}*/

*k*

_{1}. The friction force magnitude is calculated from the normal reaction, an addition of weights and a spring force. The sign function is assessed by arctangent.

To simplify Eq. 8, it is necessary to define a nondimensional time,

where *J*/*m* · *L*^{2} is the parameter ξ, and *k*_{1} + *k*_{3}/*k*_{1} is the nondimensional parameter κ. Eq. 9 can be written as

The initial horizontal displacement of the block is null, as is the vertical displacement of the horizontal arm tip of the crank. The initial velocity of the block is the belt velocity and the vertical velocity of the horizontal arm tip of the crank is null.

## 5 Network Models

The solution of the problems mentioned above involves solving the equations developed further considering the boundary and initial conditions for each case. In the following paragraphs the network model corresponding to the governing equation for each of the problems mentioned is deduced.

### 5.1 Atomic Force Microscope

The initial conditions related to displacements, *x _{M}* −

*x*=

_{t}*y*−

_{M}*y*=

_{t}*z*−

_{M}*z*= 0, and velocities,

_{t}*dx*/

_{t}*dt*=

*dy*/

_{t}*dt*=

*dz*/

_{t}*dt*= 0, are introduced in the specifications of the initial conditions of the capacitors (initial voltage value) and of the coils (initial current value), respectively. The whole network model is now run in a Duo Core, using the code PSpice [68,69] and very few rules are required since the model contains very few electrical devices.

The design of a reliable network model requires a formal equivalence between the equations of the model and those of the process, including the initial conditions. Since Eq. 4, the basic network model is related to this equation, to which initial conditions must be added. Basic rules for designing the model can be found in González-Fernández [1]. The first step is to choose the equivalence between mechanical and electric variables (different choices give rise to different networks). For the problem under consideration, the following equivalence is established: *x _{t}* (tip displacement in the x-direction) ≡

*q*(electric charge in the network), or

*dx*/

_{t}*dt*(tip velocity in the x-direction) ≡

*dq*/

*dt*=

*i*(electric current at the network). In the other directions, similar equivalences are applied.

Now, each term of the Eq. 4 is considered as a voltage in an electric component whose constitutive equation is defined by the mathematical expression of such a term. Hence, the equation itself can be considered as a voltage balance (second Kirchhoff’s law) of a network that contains as many electrical components connected in series as there are terms in the equation. The linear term of Eq. 4, *d*^{2}*x _{t}*/

*dt*

^{2}, is associated to a coil (inductance) since the constitutive equation of this component is

*V*=

_{L}*L*(

*di*/

*dt*) =

*L*(

*d*

^{2}

*q*/

*dt*

^{2}), with

*L*= 1

*H*. The same electrical element is used for the other two directions. The non-linear terms cannot be implemented directly, and require the use of controlled current (or voltage) sources or auxiliary circuits. The first are devices whose output is defined by software as an arbitrary, continuous mathematical function of the dependent variables: voltages at any node of the network or the current in any component. This capability, which is beyond the scope of the classical electrical analogy that appears in most text books, makes the network method an interesting and efficient tool in the field of numerical computation.

To implement the term *x _{t}*, the value of this variable must be determined by integrating the current in the network model,

*dx*/

_{t}*dt*. To this end, an auxiliary circuit is implemented, Fig. 4(b), in the models of FFM on the NaF sample; Fig. 5(c) in the model of AFM on a graphite sample. The terms

*y*and

_{t}*z*are implemented in similar network models.

_{t}*F*

_{1x}is the controlled current source whose output is the current,

*dx*/

_{t}*dt*, that crosses the voltage generator (used as an ammeter),

*V*. The voltage of this generator is zero so as not to disturb the network model of 4. The current of

_{x}*F*

_{1x}is integrated using a capacitor with

*C*

_{x1}=

*C*

_{y1}=

*C*

_{x}= 1

*F*; in this way, the voltage through this capacitor,

*V*

_{Cx1}=

*V*

_{Cy1}=

*V*

_{Cx}=

*C*

_{x−1}∫ (

*dx*/

_{t}*dt*)

*dt*, is simply the variable

*x*(a resistor of very high value,

_{t}*R*, is needed for electrical continuity).

_{INF}Once the variable *x _{t}* has been determined, a controlled voltage source is used to implement the term

*k*· (

_{x}*x*−

_{M}*x*) in the model

_{t}*E*

_{2}in the models of FFM on NaF sample. For the models of AFM on graphite, the controlled voltage source is

*E*

_{Tx1}. The variable

*x*is obtained as the product of the microscope support velocity and time, and is provided by the auxiliary circuit shown in Fig.4(c). The term related to

_{M}*E*

_{1}and

*E*

_{1a}, for FFM on the NaF sample. For the models of AFM on graphite sample, the controlled voltage source is

*E*

_{Tx2}. When the potential used is a Lennard-Jones potential, an auxiliary circuit is implemented, Fig. 5(b), for the model of AFM on graphite sample. The terms related to

*c*in the models of FFM on NaF sample. The last term of the other equations in Eq. 4 is implemented in similar form. The initial conditions related to displacements,

_{x}*x*−

_{M}*x*=

_{t}*y*−

_{M}*y*=

_{t}*z*−

_{M}*z*= 0, and velocities,

_{t}*dx*/

_{t}*dt*=

*dy*/

_{t}*dt*=

*dz*/

_{t}*dt*= 0, are introduced in the specifications of the initial conditions of the capacitors (initial voltage value) and of the coils (initial current value), respectively. The whole network model is now run in a Duo Core, using the code PSpice [68, 69] and very few rules are required since the model contains very few electrical devices.

### 5.2 Girling Brake

The design of a reliable network model requires the formal equivalence between the equations of the model and those of the process, including the initial conditions. The basic network model is related to Eq. 10 to which initial conditions must be added. Basic rules for designing the model can be found in González-Fernández [1]. The first step is to choose the equivalence between mechanical and electrical variables (different choices give rise to different networks). For the problem under consideration, the following equivalences are established: *x *(horizontal displacement of the block) ≡ *q*_{1} (electric charge in the network), *y* (vertical displacement of the horizontal arm tip of the crank) ≡ *q*_{2} (electric charge in the network), or *dx*/*dτ* (velocity of the block) ≡ *dq*_{1}/*dt* = *i*_{1} (electric current at the network), *dy*/*dτ* (vertical velocity of the horizontal arm tip of the crank) ≡ *dq*_{2}/*dt* = *i*_{2} (electric current at the network).

Now, each term of Eq. 10 is considered as a voltage in an electric component whose constitutive equation is defined by the mathematical expression of such a term. Hence, the equation itself can be considered as a voltage balance (Kirchhoff’s second law) of a network that contains as many electrical components connected in series as there are terms in the equation. The linear terms of Eq. 10, *d*^{2}*x*/*dτ*^{2} and *d*^{2}*y*/*dτ*^{2}, are associated to two coils (inductances) since the constitutive equation of this component is *V _{L}* =

*L*(

*di*/

*dt*) =

*L*(

*d*

^{2}

*q*/

*dt*

^{2}), with

*L*= 1

*H*. Thus, the coils used in the network model have inductances of

*L*

_{1}= 1

*H*and

*L*

_{1A}= ξ =

*J*/

*mL*

^{2}. The non-linear terms cannot be implemented directly, and require the use of either controlled current (or voltage) sources or auxiliary circuits. The first are devices whose output is defined by software as an arbitrary, continuous mathematical function of the dependent variables: voltages at any node of the network or currents in any component. This capability, which is beyond the scope of the classical electrical analogy that appears in most text books, makes the network method an interesting and efficient tool in the field of numerical computation.

To implement the terms *x* and *y*, the values of these variables must be determined by integrating the currents, *dx*/*dτ* and *dy*/*dτ*, in the network model. To this aim, two auxiliary circuits are implemented, Fig. 6(b) and Fig. 6(h). *F*_{1} and *F*_{1A} are controlled current sources whose outputs are the currents, *dx*/*dτ* and *dy*/*dτ*, which cross the voltage generators (used as ammeters), *V*_{1} and *V*_{1A} respectively. The voltage of these generators is zero so as not to disturb the network model of Eq. 10. The current of *F*_{1} is integrated using a capacitor with *C*_{1} = 1*F*; in this way, the voltage through this capacitor, *R*_{INF1}, is needed as a requirement of electrical continuity). In the same way, the current of *F*_{1A} is integrated using a capacitor with *C*_{1A} = 1*F*, the voltage through which is the variable y (another resistor of very high value, *R*_{INF1A}, is needed as a requirement of electrical continuity).

Once these variables have been determined, a controlled voltage source is used to implement the term 2 · *x* − *y* for the model; this is *E*_{1}. In the same way, a controlled voltage source is used to implement the term −*x* + κ · *y* for the model; this is *E*_{1A}. The last term of the first equation in the system described in Eq. 10, *E*_{2}, and requires the variable *dx*/*dτ*. This is a problem because this variable is also the current that crosses this controlled voltage source. To solve this problem an auxiliary circuit represented in Fig. 6(f) provides the voltage used in *E*_{2}. To help this auxiliary circuit, another one is needed, Fig. 6(e), which calculates

The conditions that select the stick or the slip phase are implemented by two voltage controlled switches, *S*_{1} and *S*_{2}, following the flowchart represented in Fig. 10. To implement this, it is necessary to obtain the voltages that appear in it by using the auxiliary circuits represented in Figs. 6(c), 6(d) and 6(e). The voltage provided by controlled voltage source *E _{control}*, in Fig. 6(c), is a scalar associated to the slip phase when the voltage of

*E*, in Fig. 3(d), is bigger than the voltage of

_{Elas}*E*, Fig. 6(e). Otherwise, there are two possibilities: if the current that crosses

_{FRS}*L*

_{1}is not equal to the belt velocity, then the voltage of

*E*will provide the same value of this scalar of the last case; if not, another check will be necessary. When the voltage in the coil is not equal to zero, then this controlled voltage source provides the same value of this scalar, which is the slip phase. If this is not the case, stick phase arises. The voltage of controlled voltage source

_{control}*E*, in Fig. 6(d), provides the absolute value of the force in the horizontal springs, while controlled voltage source

_{Elas}*E*, in Fig. 6(e), provides the maximum friction force.

_{FRS}Thus, if *S*_{1} is closed (stick phase), then the current will be determined by the Eq. 6. On the other hand, if *S*_{1} is open *S*_{2} will be rapidly closed (slip phase) and the current will be determined by the Eq. 7.

The initial conditions of the problem are implemented by the initial conditions of the capacitors and coils. The full network model is run on a Core Duo processor, using PSpice code [69, 78, 79].

## 6 Simulation Results

### 6.1 Atomic scale friction simulation: atomic force microscope

The Network Simulation Method has been applied to sample surfaces and tips for which data are available: FFM tip on NaF [36], SFM tip on HOPG [37], and AFM tip on graphite [39]. The first sample has a cubic crystal net and the others have a hexagonal crystal net, which tests the potential of the network model. The characteristics of the tip, the test conditions and the sample size for the different tests are:

- the FFM tip on NaF sample is characterized by
*m*=_{x}*m*= 10_{y}^{−8}*kg*,*c*=_{x}*c*= 10_{y}^{−3}*N*·*s*/*m*and*k*=_{x}*k*= 10_{y}*N*/*m*. The microscope support moves with a constant velocity of 400 Å/*s*, and the results are depicted with a scan range of 20*x*20Å and scan angle of 0 [36]. The proposed model solves the Eq. 4 using the potential defined in Eq. 1. - the SFM tip on HOPG sample is characterized by
*m*=_{x}*m*= 10_{y}^{−8}*kg*,*c*=_{x}*c*= 10_{y}^{−3}*N*·*s*/*m*and*k*=_{x}*k*= 25_{y}*N*/*m*. The microscope support moves with constant velocity of 400 Å/*s*, and the results are depicted with a scan range of 20*x*20 Å and scan angle of 7. The surface of the sample is composed of 271 hexagons with a carbon atom in each vertex (600 altogether); and it is assumed that the sample surface has a very high stiffness [37]. - the AFM tip on graphite is characterized by
*c*=_{x}*c*= 0_{y}*N*·*s*/*m*,*k*=_{x}*k*= 0.25 − 2.5_{y}*N*/*m*,*k*= 0.25_{z}*N*/*m*, and a vertical displacement of −6Å for the tip. The results are depicted with a scan range of 9.8*x*8.5 Å and scan angle of 15 [38]. Besides,*m*=_{x}*m*= 10_{y}^{−6}*kg*and a constant velocity of 10Å/*s*for the microscope support, parameters not mentioned by Sasaki et al. [38], were assumed to obtain results similar to theirs.

Eq. 4 is solved in each test for the potential functions defined in the following equations:

- Eq. 1 in the FFM tip on NaF sample model
- Eq. 2 in the SFM tip on HOPG sample model
- Eq. 3 in the AFM tip on graphite sample model

In the last case, for graphite sample, only short computational times are necessary for the software to find the right values of the parameters not supplied by Sasaki et al. [38]. Fig. 7(b) shows the components of the tip elastic force, *F _{x}* and

*F*, in each position on the NaF sample surface, whose crystal net is shown in Fig. 7(a). The correspondence between both depicted images enables us to use this software as a tool to interpret those images from the microscope. Thus, this software is able to manage different theoretical crystal nets, even with crystallographic defects, providing the corresponding theoretical elastic forces, which can be compared with those from the microscope. The results of the simulation agree with those obtained by Hölscher et al. [36].

_{y}Fig. 8(b) shows the components of the tip elastic force, *F _{x}* and

*F*, in each position on the HOPG sample surface, whose crystal net is shown in Fig.8(a). The results of the simulation agree with those obtained by Hölscher et al. [37]. Besides, Fig. 9(b) shows the x-direction component of the tip elastic force,

_{y}*F*, in each position on the graphite sample surface, whose crystal net is shown in Fig. 9(a). In this case, a 3D graph is used to show the details better, to be more precise the stick - slip mechanism. The lines of the graph are traced at different y-coordinates, separated by a fixed step. When one of these lines crosses the sample surface at a critical position, the instabilities appear, Fig. 9(b). These critical positions have a relation with the minimal threshold of the distance between the tip and the atom on the surface. The results of the simulation agree with those obtained by Sasaki et al. [38].

_{x}### 6.2 Macroscopic scale friction: Girling brake

The model used in this numerical simulation follows the mechanism shown in Fig. 11, and is similar to that used by Awrejcewicz et al [56]. Its dimensions and properties are:

- crank arm length,
*L*= 0.105*m* - drum radius,
*R*= 0.11*m* - crank mass,
*m*= 0.215_{c}*kg* - block mass,
*m*= 0.174*kg* - spring stiffness,
*k*_{1}=*k*_{2}= 214.62*N*/*m*, and*k*_{3}= 107.31*N*/*m*

Let us assumed that the inertial moment of the crank, *J*, is equal to *m _{c}* ·

*L*

^{2}/3. The dimensionless parameters, from the data above, are: ξ = 1/3 ·

*m*/

_{c}*m*= 0.412 and κ = 1.5. The value of the modified gravity acceleration is:

*g*

^{′}=

*g*·

*m*/

*k*

_{1}= 0.01778

*m*.

The parameters that characterize the friction are:

- dimensionless accuracy parameter,
*ε*^{′}= 3.5121*e*7 - dimensionless parameter in dynamic friction coefficient,
*δ*^{′}= 3.5121

The static friction coefficient, *μ _{s}*, is calculated from a measure using a mass on the belt. A friction force of 1.6 N is obtained from a mass of 0.277

*kg*, which means a value of

*μ*equal to 0.59. The modified belt velocity used,

_{s}*v*

^{′}

_{dr}, is equal to -0.002788 m.

Fig. 12 shows the phase plane for the belt velocity used for a time span from 100 to 1,000. Stick phase can be recognized as a horizontal line with ordinate *V*_{1} = −0.002788*m*, Fig. 12. In addition, Fig. 13 shows the frequency spectrum of the signal *X*_{1} for a time span from 100 to 1,000.

A waveband arises centred on the value of 0.143433 Hz, a value that has no relation with belt velocity. Fig. 14 shows the Poincaré map using a period of 1/0.143433 for a time span from 0 to 3,200.

Fig. 14 shows the chaotic behaviour of the system. To quantify the chaos, an approximation to the maximum Lyapunov exponent is calculated from a time series of the solution by the software OpenTSTool [80], developed by Göttingen University. This approximate value is equal to 0.0514, which suggests chaotic behavior of the solution.

## 7 Conclusions

A numerical model based on the network method has been designed to simulate several types of microscope and sample surfaces with different crystal nets. The results were successfully compared with those of different authors obtained with different methodologies. The software is flexible enough to implement the different values of the main parameters with no important changes, and allows different types of graph to be used. Thus, this software is able to manage different theoretical crystal nets, even with crystallographic defects, providing the corresponding theoretical elastic forces that can be compared with those obtained by microscope. Furthermore, the short computational time necessary means that the software can be used to determine the values of the microscope parameters from experimental images of certain materials, whose data are not available. The calculated parameters allow other sample materials to be simulated. The final potential use of this software is to test the effect of microscope parameters on the quality of the image assisting to the use of the microscope.

Equations equivalent to those used by Awrejcewicz et al. have been developed, adapting them for the general forms used in the analysis of different friction problems using the network method. Unlike, in the scheme proposed by these authors, we do not include horizontal and vertical damping, thus increasing the instability of the system and the difficulty of solving the problem. To save computational time, several parameterizations have been considered. As a result, the parameters in the left term of the first equation of the system in Eq. 10 are cancelled, speeding up the solution. The use of nondimensional time, useful in the solution to other friction problems, should be emphasised. Finally, it is necessary to evaluate the performance of the different tools to identify the chaotic behaviour. For this reason, the authors proved the current techniques, selecting the Poincaré map, frequency spectrum and maximum Lyapunov exponent because they better demonstrate such behaviour. Since the system jumps between two phases and it is not possible to use a Jacobian-based method, which would be more reliable, a direct method is used to calculate the maximum Lyapunov exponent. As all the references mention, this algorithm depends strongly on the embedding dimension and the time delay. Thus, the maximum Lyapunov exponent is difficult to specify, but in this work we are more interested in knowing the relative chaotic level than the absolute value. Another point that introduces uncertainty in the calculation of this exponent is the sample frequency necessary to obtain the data used in the algorithm.

^{}

Communicated by Juan L.G. Guirao

We are grateful to Dr. J.A. Moreno, professor of the Mechanical Engineering Department in the Technical University of Cartagena, for some interesting and convenient tips about Network Simulation Method.

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