On the determination of the friction-caused energy losses and its potential for monitoring industrial tribomechanical systems

The paper refers to a new method to quantify the energy losses due to frictional effects and imperfections in contacts in the case of real industrial tribomechanical systems. Whereby energy losses represent an integral indicator of quality of the real industrial tribomechanical system, in terms of the characteristics of the contact element materials, geometric accuracy, and manufacturing and assembly errors. This paper presents a complex theoretical model based on the differential equation of motion of a real tribomechanical system down a steep plane. The outputs of the theoretical model are exact mathematical expressions that define the current values of the coefficient of friction and the friction-caused energy losses. The measuring system enables the quantification of current values of the distance traveled per unit of time. Based on a series of experimentally determined values of distance traveled per unit of time, the values of energy losses of the real industrial tribomechanical system are determined using the developed theoretical model and the appropriate software support. The obtained results indicate a high reliability, a large potential, and a wide range of possible applications of the proposed method.


Introduction
As is known, friction represents the resistance to relative motion, be it sliding, rolling, or combined sliding-rolling motion, of a pair of surfaces of two bodies.Friction opposes the relative motion of the body.Friction, as a complex phenomenon, has been the subject of investigation by many researchers for more than 400 years.The friction coefficient represents the main characteristic of every tribological system [1].
Having knowledge of friction coefficient values is of utmost importance during the development, maintenance, and optimization of contact pairs, which are to be implemented in industrial systems.Based on the analysis of literature sources and existing tribometer solutions, in accordance with ASTM and ISO standards, it can be concluded that measurement methods and devices significantly affect the reliability of measurement results.
By investigating the possibilities of designing reliable and economical solutions for tribodiagnostic equipment that provide high measurement accuracy in testing finished industrial products, such as linear guides, many researchers have investigated and published their solutions.
Xu et al. [22] propose an analytical calculation method for predicting the time-varying friction of the roller linear motion guide.The model is formulated considering the combined effect of the time-varying contact load, contact area, and lubrication viscosity variations between the roller and raceway.Soleimanian and Ahmadian [23] consider friction effects in pre-sliding and sliding regimes of lubricated linear roller guideway systems to provide a dynamic model of the machine tool element.They applied single harmonic excitation forces at low frequencies to the rail and measured the force and response signals.
Oh et al. [24] developed a testbed to measure the friction force of the linear motion ball guide with preload, compare measured frictional forces with predicted values, and analyze the friction force components.Krampert et al. [25] integrated sensors into a linear guide and implemented a method for capturing the load on a rolling element linear guide by measuring the stresses resulting from the rolling element contact at the side of the runner block.They also compared obtained measurements with analytical and finite element method simulation.Considering that linear guideways have predominantly replaced box ways in industrial machinery to facilitate linear directional motion, Whitican et al. [26] address the problem of guides' nonlinearity.They developed a fixture allowing a physical connection with a servo-hydraulic shaker and tested a model for studying the form of the nonlinearity using the acceleration surface method.The structure was instrumented with triaxial accelerometers, and various excitation regimes were attempted on the structure.
Based on the analysis of previous investigations, this study proposes a novel theoretical model for determination of kinetic friction coefficient and presents experimental results.The proposed method is based on measuring two fundamental physical quantities, i.e., the distance traveled between fixed sensors and the time detected by sensors when a body passes through its reaction zones while sliding or rolling down an inclined plane-based test rig.The advantage of the proposed method is that the measurement process does not introduce any excitation into the measuring system, which results in high measurement accuracy.Furthermore, the location of the entire measuring system outside the friction zone opens a wide range of possibilities in terms of reliable and economical design solutions for tribodiagnostic equipment, especially when it comes to tribological tests in a controlled environment, such as tests at high temperatures or tests in a vacuum.

Theoretical basis
The first published theoretical research related to the determination of kinetic friction coefficient via the dynamic equation of motion for a body moving down an inclined plane was published by Leonard Euler in 1748 [27].
Unfortunately, it is safe to say that this method has not experienced a broader expansion in the scientific field, especially in the design field of modern tribodiagnostic equipment.Papers based on (or related to) Euler's research are mainly published in journals focused on physics education.One of a few papers based on Euler's idea to determine the kinetic friction coefficient using the differential equation of motion was published in a thematic journal in the field of tribology [28].
From theoretical, experimental, and technological aspects, a significant advantage of the proposed method is that it relies on measuring the three fundamental physical quantities (i.e., the mass, time, and distance traveled).Following Euler's idea, a group of authors have conducted research [29] that represents one of a few published experimental verifications of Euler's method for determining the kinetic friction coefficient.The authors have upgraded Euler's theoretical model by considering the effects of air resistance force on measurement errors, which are not to be neglected at higher velocities and low friction coefficient values on an inclined plane.
The upgraded Euler's theoretical model, which is planned to be verified experimentally, is based on the measurement of time required for the body of mass m to travel the distance s down an inclined plane, as illustrated in Fig. 1. Figure 1 shows the directions of active and resistive forces acting on a body moving down an inclined plane.The difference compared to the original Euler method is that the air drag force is also taken into consideration.
In the case from Fig. 1, the differential equation of motion down an inclined plane is as follows: The expression F w = kv 2 represents the air drag force.Using the following relations between the derivations we get the equation: ( where C is the drag coefficient, ρ w is the air density, and A is the front surface of the body.Integral of Eq. 3 is as follows: The solution ( 5) can be written in the following form: where: E k is the kinetic energy of the body on an inclined plane, A m� ⃗ g is work done by the Earth's gravitational force in the direction of motion, A �⃗ F  is work done by the fric- tion force, and A �⃗ F w is work done by the air drag force.
Work done by the Earth's gravitational force equals: The diagram of the change in friction force as a function of the distance traveled by the body moving down an inclined plane is given in Fig. 2. The maximum value of the friction force corresponds to the beginning of the motion, i.e., to the static friction.During the motion, the velocity increases while the friction force decreases, as represented by the general diagram of the change in the friction force.
Work done by the friction force A �⃗ F w , that is the energy E spent on overcoming the friction force (Fig. 2), can be written in the following form: where is the mean value of the friction coefficient on the distance s.
As stated above, the air drag force �⃗ F w depends upon the squared velocity.The general diagram of the change in ) A m� ⃗ g = mg sin s. ( A �⃗ velocity of the body on an inclined plane is given in Fig. 3.  where v is the average velocity of the body, which satisfies Eq. 9.
By integrating Eq. 6, considering that v = ds dt , we get an equation: Further integration is as follows: Here, R 1 (denoting the integration constant) is equal to zero, considering the initial conditions (for v = 0 and s = 0 ).Equation 11represents the distance function in the presence of resistance forces (friction and drag).If we consider an idealized case, where no resistance forces are acting on the body, Eqs. 10 and 11 are defined by the following: By dividing Eq. 10 by Eq. 12 and Eq.11 by Eq. 13, we get Eqs.14 and 15: Equation 16 is obtained from the condition that a body moving down an inclined plane in the presence of resistance travels the same distance as a body moving down an inclined plane without resistance.Equation 16 defines the relations between the times required for a body to travel the same distance in two considered cases Let us consider the time ratio t i t , in the case of the same distance traveled ( s = s i ).If we know (measure) the time t for which the body travels the path s in the presence of resistance, then we also know the time ratio t i t because the time t i can be calculated from Eq. 13:

If the time ratio
t is denoted by o and included in Eq. 16; then, Eq. 18 is obtained under the condition s = s i If we start from Eq. 1 and divide it by the quantity mgsin , we obtain the following: Equation 19 is valid for any value of and v so that we can write 2 = 2 0 .Now we can obtain the values of acceleration, velocity, and distance traveled for the case of body motion down an inclined plane in the presence of friction and air resistance, using the system of Eq. 20: By substituting the first two equations of the system of Eq. 20 into Eq.19, we obtain the formula for calculating the coefficient of friction as a function of time, that is, as a function of sliding velocity, as follows: After substituting a part of the Eq.21 by the third equation from the system of Eq. 20, we get: Based on expression (8) for the coefficient of friction μ, it is possible to determine the energy spent on friction as a function of the sliding distance: In Eq. 24, we can substitute the friction coefficient by Eq. 23: a = a i ( taking into account that the inclination angle is constant during the experiment, we can write Eq. 25 in the following form: By integrating Eq. 26, we get the formula for calculating the energy spent on overcoming the friction force: The developed theoretical model enables the determination of the physical-theoretical dependence, based on two measured values (sliding distance s and sliding time t) and experimental conditions -quantities.So, we can express coefficient of friction μ as function = m, C w , w , A, , v, s, t .Significant scattering of the friction coefficient μ that occurs during the measurement of friction force and normal load on any type of measuring instrumentation (from basic tribometer configurations to sophisticated tribometers) is caused by the process dynamics, measuring device errors, and surface microgeometry.

Considering the case of α = π/2
If we multiply the Eq.22 by mgcos , we obtain the equation: If we consider the case = ∕2 , the friction force is F = 0 , sinα = 1, and, in the general case, it is valid that mg ≠ 0 .Then, Eq. 28 becomes: From the first equation of the system (20) and for considering case = ∕2 , Eq. 29 is reduced to: When we include expression (30) for 0 2 in (29), we get the following: (29) Considering that a = dv∕dt , Eq. 31 becomes: Equation 32 can be written in the following form: In the general case, the following expression has a constant value so that we can denote it by k 1 : Now, expression (33) can be written as: The solution of the differential Eq. 35 is of the form: The limit value of the velocity equals: According to Eq. 34 k 1 > 0 and k 2 = 2g k 1 , g > 0 , we can solve Eq. 37 for limit value of the velocity: The obtained expression (38) for calculating the limiting velocity v g is well-known from theoretical mechanics.It refers to the free fall of a body in the air, where, after reaching the limiting velocity, the body continues to move at a uniform (limiting) velocity.That indicates that an inclined plane of an arbitrary angle of inclination represents a general physical model of body movement. (31)

Considering the case of the limiting velocity
To support the statement, that an inclined plane of an arbitrary angle of inclination represents a general physical model of body movement, let us consider the case of the limiting velocity of a body moving down an inclined plane in the presence of friction.We start from Eq. 28 and the conditions for the velocity to reach its limiting value ( 0 <  < ∕2 ; F = 0 ).From Eq. 29, it follows: From the previous condition follows the expression for the calculation of the limiting velocity at which the friction force has zero value: The first solution of Eq. 40 refers to the state of rest at = 0 and v = 0 .From Eq. 40 follows that = 0 and 0 2 = 1.The developed theoretical model enables the determination of the physical-theoretical dependence, based on two measured values (sliding distance s and sliding time t) and experimental conditions -quantities.So, we can express coefficient of friction μ as function = m, C w , w , A, , v, s, t .
In general, depending on tribomechanical characteristics of the body, the quantity 0 2 can have an infinite number of values.At very small values of the friction coefficient, the 0 2 approaches value 1. Theoretically, it is a case of sliding of ideally smooth surfaces which are inert in terms of mutual adhesive action.In real technical systems, there is an infinite number of contact pairs in which 0 <  0 2 < 1. Changes in the friction coefficient and the velocity for three different values of the inclination angle are shown in Fig. 4. The parameter 0 2 was determined experimentally.At the moment of reaching the critical speed ( v = v g ), the body continues its motion with constant acceleration: This value of acceleration corresponds to the limiting velocity ( v = v g ) at which the contact between the body and the ramp separates.

Experimental setup
The experimental research plan foresees the development and implementation of a dedicated device a laboratory set, which would enable the determination of the coefficient of friction and energy losses occurring in real industrial tribomechanical systems.The designed device is planned to enable tests of linear roller guides of the "RAIL-TROLLEY" type in a wide range of industrial velocity and load values typical for such tribomechanical systems in industry.Figure 5 shows a schematic representation of the implemented device.The measurement setup of the device consists of a magnetic tape (4) located in the carrier (5) fixed to the platform (3), magnetic tape reader (sensor 6) based on the stationary part of the device structure and corresponding Arduino electronics (7) meant for data management and storage of experimentally obtained values of distance traveled "s" per unit of time, going from position A to position B (Fig. 5).
Loaded linear rolling guideway assembly initially starts moving from position "A" (Fig. 5).A relatively small value of inclined plane angle and small value of rolling friction force enable the guideway to be manually brought to initial position "A," without difficulties, even under substantial loading of the system.The forced stopping of the loaded guideway in position "B" is performed via mechanical and hydraulic shock absorbers (8).The foregoing presentation referred to the analysis of the conceptual solution for the device (Fig. 5).Globally, considerations are being made for a device solution that would simulate conditions identical to those in exploitation of real industrial tribomechanical systems (such as linear rolling guideways) and that would enable quantification of energy losses via a measurement setup and the presented theoretical model.Whereby the energy losses could be quantified at different levels and different types of loading of the rolling guideway assembly.
A combination of normal load and moment load along one and/or multiple axes represents the general case of loading on a guideway operating in industrial manufacturing systems.The possibility of simulating the exploitation conditions on a real industrial system and quantification of energy losses has significant qualitative advantages over model simulations, primarily in terms of the reliability of obtained results.A method based on the inclined plane principle clearly opens up a wide range of possible programs of experimental testing of real industrial guideways.
In the following presentation, the paper authors present basic characteristics of the preliminarily realized device intended primarily for experimental verification of the proposed method.
Figure 6 shows the CAD model of the developed device, while Fig. 7 presents a photograph of the device.
The first functional subassembly consists of a real industrial hardened and ground guideway assembly whose elements are a rail of type HGH30CA and length 2000 mm The second functional subassembly, consisting of two steel U-profiles, is formed by welding pipes with a crosssections of 80 × 60 × 3 mm and of certain lengths, and 500mm long steel plates with a cross-section of 100 × 10 mm (positions 8 and 9).On the side walls of the pipes, openings with a diameter of 20 mm were made at certain distances, which are used to connect the first and the second functional subassemblies and to form the required slope angle.The connection is made via threaded spindles with a diameter of 20 mm (position 10) passing through 20-mm openings and nuts (position 11), thereby forming closed-frame shaped stable structure (Fig. 6).The constructed profiles are mounted to the device chassis (position 13) via four 12-mm diameter openings on the U-profiles' plates and 12-mm diameter screws (position 11).
Final position is determined by a hydraulic shock absorber, attached to one of the U-profiles via the plate (position 14).Position 15 determines the location of the sensor (magnetic tape reader), while position 16 determines the location of Arduino control electronics meant for data management and storage of experimentally obtained values of distance traveled "s" per unit of time, by a guideway loaded with a desired load, in the entire range of movement speeds, from zero to the speed defined by the chosen inclination angle and the actual present forces of resistance.
A magnetic linear encoder contains a read head that slides on a magnetic tape.On the surface of the tape, there is a set of magnetically coded poles that enable the sensor system to determine the current position of a read head.The distance between those marks determines the resolution of the system.The denser they are, the higher the resolution.There is also a specially coded zero position reference mark.
A signal unit within the head (slider) emits a pulse signal that passes through the measuring tape.The signal is distorted while passing through the magnetic marks (poles).Each time the signal passes through a pole, the transducer reads the change in the magnetic field.After that, it compares the received data with the zero position and generates a corresponding digital signal that carries information about distance, displacement and its direction, and velocity.

Experimental results
Experimental research was performed primarily to verify the reliability of the presented theoretical model of the proposed method, that is, the reliability of the method when quantifying friction and energy dissipation parameters in real industrial tribomechanical systems.When it comes to tribological parameters, such as the coefficient of friction, the most effective indicator of the reliability of the method is a small dispersion of the obtained results with a large number of experiment repetitions.
The measuring system of the device, the Arduino electronics, and the software enable the quantification of the traveled distance at the corresponding computermonitored time, from the moment of motion initiation to the moment of forced stop, i.e., the end position.The obtained series of distance and time values correspond to the actual motion of a body down an inclined plane in the presence of friction force and air drag.For the identical time intervals for which the distance traveled in the presence of resistive forces was experimentally quantified, the s i was calculated based on Eq. 13 as the distance that the body would travel given the identical angle of inclination if there were no friction force and air drag.The results for three experiments and theoretical model are provided in Table 1.SI system units were used for variables:  Figure 8 shows experimentally obtained curves of distance as a function of time for 10 experiment repetitions in identical test conditions and identical contact loading.The same diagram also shows the theoretical curve of distance change with time according to Eq. 13.
The experimentally obtained curves of distance change with time resulted from ten repetitions of the experiment in the presence of the resistive friction force and air drag.As Fig. 8 shows, the curves deviate negligibly from one another.The theoretical curve of the distance change with time refers to the idealized case of the platform moving without the presence of the resistive friction force and air drag.As can be visually observed, the difference between the theoretical and experimentally obtained curves is significant.From the third equation within the system of Eq. 20, it follows that the value of the actual distance traveled in the presence of the resistive friction force and air drag can be determined by multiplying the distance defined by Eq. 13 with the square of the ratio of the ideal time (t i ) and the actual time (t), for any specific time-from the zero position to the moment of the forced stop of the motion.So, in Fig. 8, the curve of the ideal distance s i serves as a benchmark, indicating the time the body would travel the identical distance if there were no resistive friction force and air drag.The diagram in Fig. 9 indicates that the values of the friction coefficients obtained in ten independent experiments are concentrated in an interval narrower than the interval 0.0465-0.05,which confirms the repeatability and reliability of the measurement results.Furthermore, the diagram indicates a slight decrease in the coefficient of friction with an increase in the distance, i.e., the velocity, which confirms the well-known effect of the velocity change on the value of the friction coefficient.
Figure 10 indicates that in ten repetitions of the experiment, the maximum differences in energy dissipation due to friction do not exceed the value of 1 J.
The values of E μ in Fig. 10 are calculated, i.e., the diagram is drawn based on Eq. 27.The total work of the friction force A(F μ ) was also calculated for each experiment, based on Eq. 8, and the difference between E μ and A(F μ ) is negligible.To verify the model, the energy spent on  friction was also calculated as the difference between the potential energy E p at the beginning and the remaining kinetic energy E k at the end of the motion.Whichever of the three models is used, the calculated energy values are practically identical.The maximum differences in energy dissipation in all ten experiments are in the interval of 0.3-1 J, which makes 1.4-6%, depending on the experiment.The authors believe that these minor differences in energy dissipation primarily result from the change in the friction conditions in the guide itself, that is, in the contact between the ball and the rolling guide, caused by temperature and contact pressure changes when performing a series of ten experiments.

Statistical analysis of experimental results
The presented theoretical-experimental method is based on the law of distance traveled change with time for a real tribomechanical system.The distance change law was defined by Eq. 20 within the chapter concerning theoretical considerations, whereby distance traveled in the presence of resistive force is defined as a product of theoretical distance (s i ) and the ratio of theoretical and real time (t i /t).Experimental data processing was performed in the software package "STATISTICA" using the Gauss-Newton method of nonlinear regression.Equation 20, derived from theoretical considerations chapter, was taken as a base function for experimental data processing.For the purpose of statistical analysis of experimental data, two constants (C1 and C2) were added to Eq. 20, determined by the processing program itself.Hence, data processing was performed using base function given as: Thereby, the obtained values of constants point to the degree of agreement between the theoretical model and experimentally determined values of distance traveled per unit of time.By processing experimental data regarding the ten performed experiments, obtained values of constants were found to be similar to theoretical ones (C 1 = 1; C 2 = 2-Eq.20), and coefficients of correlation were close to 1.The following example of output results regarding one of the ten performed experiments shows exactly that (Fig. 11).

Discussion
Friction phenomenon and energy losses due to friction have represented a lively research area for centuries [30], and still do nowadays [31].Friction, as a complex phenomenon, has been the subject of investigation by many researchers for more than 400 years.The friction coefficient represents the main characteristic of every tribological system, as is given in more detail in the review of literature sources.
Authors of this paper have for years followed in the footsteps of nineteenth-century Euler's research results regarding kinetic friction [27] and published their own research, as can be seen in references [28,29].Whereby, the Euler's method was generalized in terms of possible application and upgraded in terms of experimental verification of the method [28,32,33].This paper primarily presents the generalized theoretical model of an inclined plane, which includes the actually present drag, besides just friction.The presented theoretical model, through Eqs.23 and 27, enables the calculation of current values of friction coefficient and energy spent on overcoming friction resistance as a function of all relevant process parameters, whereby the ξ 0 factor is experimentally determined.
The ξ 0 factor represents a ratio between real time spent traveling down an inclined plane (in the presence of resistive friction forces and drag) and travel time without resistive Fig. 10 Experimental curves of energy spent (dissipated) on distance traveled Fig. 11 One example of output results displays from data processing in "STATISTICA" software package forces present (the idealized case).Meaning that by only measuring two basic physical quantities (distance and time) during the process of a body traveling, current values of friction coefficient and values of energy spent could be quantified.Whereby, partial levels of energy spent on overcoming frictional resistance and energy spent on overcoming drag would be exactly quantified as functions of velocity and other process parameters.
The realized device and applied measurement setup enable a large amount of experimental data or strings of distance traveled and time values to be obtained in a short time period, order of a second, thereby achieving a high level of reliability of the measurement method.It should be emphasized that the applied measurement setup does not introduce excitation or error into the measurement system, unlike the sensors used for quantification of resistive forces.The performed experimental studies, statistical data processing, and obtained results (diagrams in Figs. 8, 9, and 10) all point to the high reliability of the method.
The presented method opens up a wide space for testing industrial guides.For a start, the authors plan to focus on determining the influence of the normal symmetrical load of the guide on the coefficient of friction and energy losses.By attaching additional masses m to the platform (position 4 in Fig. 6), the presented method, the developed device, and the measuring system allow tests to be performed in a wide range of normal symmetrical loads of the linear guide.
Changing the normal load to a certain level leads to a change in the level of energy losses caused by friction and elastoplastic deformations in the contact area between the ball and the track.These tests can give a much more comprehensive picture of the influence of the normal load level of the guide on the value of the coefficient of friction.They can also be beneficial for grading the quality of different guides regarding their geometric accuracy, material quality, and heat treatment.
The authors also plan to direct future experimental research toward determining the effects of the normal symmetrical load combined with the moment load along the axes.By applying appropriate technological solutions on the developed device and by adding eccentric masses and connecting them to the platform, it is possible to simulate the combined loads along the axes.Schematic representations of the planned experiments are given in Fig. 12.
In real industrial systems, there are rare examples of linear rolling guides that can be said to function in the absence of a dynamic component of the normal load or dynamic load by the moment of force along one of the guide's axes.It should also be highlighted that regarding the geometry of the linear rolling guides, in addition to the static load capacity from the aspect of moment load on all three axes, renowned manufacturers declare the permitted levels of the dynamic load capacity.However, the manufacturers of guides of this type do not provide The authors of the paper believe that the presented method and the developed device could be for quantifying friction parameters during dynamic moment load simulation along one or more axes.Inertial forces can efficiently achieve the dynamic moment load along the chosen axes.The idea is that the drive units with accumulator power supply serve as inertial mass carriers that can be attached to the platform in.By choosing the mass m, the radius R, the corresponding values of the coordinates, and the angular velocity ω, it is possible to simulate the dynamic loads in a broader range during the industrial guide's movement down an inclined plane.Schematic representations of the planned experiments are given in Fig. 13.

Conclusions
In this paper, a method of experimental determination of kinetic friction coefficient of real industrial rolling guideways was presented.The method is based upon the principle of following the law of change in distance traveled with time, in the case of a loaded real industrial guideway traveling down an inclined plane (rail of the guide).This method enables the testing of real industrial tribomechanical systems with a simulation of different types of loading.The obtained results point to the high reliability of the proposed method and open up a whole specter of possible applications, in terms of tribological parameter quantification for real tribomechanical pairs while simulating real load levels and types, which offers significant advantages in qualitative terms.Quantification of friction of coefficient and energy spent values can present a unique indicator of quality of the industrial tribomechanical pair itself.Therefore, this method can be used for quality grading of standard industrial linear guideways.The authors' future research directions will be focused on examining real industrial guideways under simulated combined load and moment loads along axes.Research will also be focused on the implementation of the method for testing real industrial guideways under simulated dynamic loads.

Figure 3
provides the general diagram of the change in velocity as the function of the distance traveled by the body on an inclined plane.Work done by the air drag force A �⃗ F w can be written as:

Fig. 1 Fig. 2 Fig. 3
Fig. 1 Motion of a body along an inclined plane-analysis of forces

Figure 5
gives schematic representation of a real industrial guideway (1) based along the ξ axis, forming a certain angle α with the horizontal plane.The α angle can be selected from a certain range of values, thereby allowing testing with a wide range of speeds.The cart (2) rolls along the guideway (1).A platform (3) is attached to the cart (2) through which, using the guideway assembly, different levels of normal load and moments along ξ, η, and ζ axis can be simulated.Desired normal load levels are achieved through the cart and platform mass, as well as additional masses, which act at the center of gravity (CG) of the cart-and-platform rolling assembly.Desired moment load along the axes ξ and ζ is achieved via the additional masses "m" and coordinate values of ξ and ζ, referring to CG of additional masses (Fig. 5 detail S1-S1 and detail S2-S2).With appropriate technical solutions, it is possible to simulate the load along the ζ axis and the moment load along the η axis (Fig. 5 detail S2-S2).

Fig. 4
Fig. 4 Changes in the friction coefficient depending on the velocity, for three different values of the inclination angle

Fig. 5 Fig. 6
Fig. 5 Schematic representation of the implemented device

Fig. 7
Fig. 7 Photograph of the realized device

Figures 9 and 10
show graphs of friction coefficient change as a function of velocity and graphs of energy spent on distance traveled for all 10 performed experiments.Curves were plotted according to Eqs. 22 and 27.

Fig. 8 Fig. 9
Fig. 8 The theoretical curve of distance change with time and experimental curves of distance change with time in presence of the resistive friction force and drag

Fig. 12
Fig. 12 Schematic representations of the planned experiments to investigate the effects of combined load and moment loads

Fig. 13
Fig. 13 Schematic representations of the planned experiments to investigate the effects of dynamic loads