### Collegio di Ingegneria Informatica, del Cinema e Meccatronica

### Department of Control and Computer Engineering

### Master of Science in Mechatronic Engineering

### Master of Science Thesis

### Tribometer Set-up and Friction Coefficient in Elastomers of

### Sealing Systems

### Advisors:

### prof. Luigi

### Mazza

### prof. Guido

### Belforte

### Candidate:

### Jairo Alejandro

### Cort´

### es Guti´

### errez

Abstract iv

Introduction vi

1 Theoretical Apsects 1

1.1 Friction Models . . . 2

1.1.1 Leonardo da Vinci . . . 2

1.1.2 Guillaume Amontons . . . 2

1.1.3 Leonhard Euler . . . 2

1.1.4 Coulomb Model . . . 3

1.1.5 Stiction Force . . . 4

1.1.6 Viscous Friction . . . 4

1.1.7 Stribeck Effect . . . 4

1.1.8 Karnopp Model . . . 7

1.1.9 Dahl Model . . . 7

1.1.10 Lund-Grenoble (LuGre) Model . . . 8

1.2 Contact Surface . . . 9

1.2.1 Elastic Contact . . . 10

1.2.2 Plastic Contact . . . 11

1.3 Tribology . . . 12

1.3.1 Effects of Tribosystem Variables on Friction . . . 13

1.3.2 Effects of Roughness, Load and Contact Pressure . . . 14

1.3.3 Effects of Temperature . . . 15

1.3.4 Effect of the Slinding Velocity . . . 17

2 The Tribosystem 25 2.1 Initial Conditions . . . 25

2.1.1 Modifications . . . 26

2.2 Elements of the Tribosystem . . . 27

2.2.1 Base . . . 27

2.2.2 Pneumo-hydraulic Cylinder . . . 28

2.2.3 Spherical Joints . . . 28

2.2.4 Specimen Holder . . . 29

2.2.5 Linear Guide . . . 29

2.2.6 Vertical Pneumatic Cylinder (pressure) . . . 30

2.2.7 Control Unit with Pneumo-hydraulic Cylinder . . . 31

2.2.8 Friction Plate . . . 32

2.2.9 Sensors . . . 33

2.2.10 Electric Cylinder . . . 34

2.2.11 Control Unit with the Electric Cylinder . . . 35

2.3 Sensors Calibration . . . 37

2.4 Set-Up for Testing . . . 39

2.4.1 Precautions . . . 41

2.4.2 Clean the tribosystem . . . 42

2.5 Friction Force Computation . . . 42

2.6 Hardware and Software for Data Acquisition . . . 43

2.6.1 Matlab Script . . . 46

2.7 Tribometers in the Market . . . 49

2.7.1 CSM Instruments . . . 49

2.7.2 Phoenix Tribology Ltd . . . 53

2.8 Standard Guides and Test Method . . . 62

2.8.1 Digital Data Acquisition in Wear and Friction Measurements (G 163 - 99) . . . 62

2.8.2 Static and Kinetic Coefficients of Friction of Plastic Film and Sheeting (D 1894 - 01) 63 2.8.3 Measuring and Reporting Friction Coefficients (G 115 - 98) . . . 65

3 Results 69 3.1 Pneumo-hydraulic cylinder results . . . 69

3.1.1 Polyurethane TPU 8001 . . . 70

3.1.2 NBR . . . 72

3.1.3 HNBR 8001 . . . 73

3.2 Electric Cylinder . . . 74

3.2.1 Polyurethane TPU 8001 . . . 74

3.2.2 NBR . . . 76

3.2.3 HNBR 8001 . . . 77

3.3 Test with Weights . . . 78

3.4 Lubricated Tests . . . 80

3.4.1 Polyurethane . . . 82

3.4.2 NBR . . . 84

3.4.3 HNBR 8001 . . . 85

3.4.4 Totally Lubricated . . . 86

3.4.5 Continuous Tests . . . 86

3.4.6 Comparison Graphs . . . 93

4 Conclusions 97 A Load Cell Calibration 101 A.1 UMM-K 50 . . . 101

A.2 FN3030A2-200 . . . 102

B Matlab Scripts 103 B.1 FrictionComputation.m . . . 104

B.2 Calibrate.m . . . 107

B.3 MeanNormal.m . . . 109

B.4 Separate.m . . . 111

C Electronic Schematics 112 D Datasheet 117 D.1 Vertical Pneumatic Cylinder (pressure) . . . 117

D.2 Normal Load Cell . . . 125

D.3 Velocity and Position Transducer . . . 126

D.4 Liner Guide . . . 130

D.5 DSP 100-24 . . . 134

D.6 FN3030-200 and FN3030A2-200 . . . 137

D.7 DAQ National Instruments . . . 141

Tribology comes from the Greek word,“tribos”, and meaning “rubbing”or “to rub”. And from the suffix, “ology”that means “the study of”. Therefore, tribology is the study of rubbing. Tribology is present in the mechatronic systems of every area of engineering and industry. Mechatronics systems integrate many elements (or parts), in order to create the whole system. In general, every system has surfaces in contact and in relative motion (e.g. sliding, rolling, impacting), leading to frictional forces between the bodies in contact. Those frictional forces have been studied since over three centuries ago, by many physicists, mathematicians. They discovered that the friction force has a nonlinear behavior, and varies depending on the sliding velocity, normal force, wear, lubrication, temperature, roughness and so on. Each scientist performed tests and developed his own model, but not taking in to account all the factors mentioned before, in order to generate easy and reliable models.

Nowadays, the friction is one of the major problems in the biggest companies, because the friction force generates wear, errors in position, waste of energy, and therefore losses in the income. Therefore, a reduction of the friction force and wear leads to increases in service life, less downtime and lower operating costs, which can add up to tremendous savings.

The objective of this thesis is to, perform the set-up of a tribometer available at the Department of Mechanical and Aerospace Engineering (DIMEAS) laboratory of the Polytechnic University of Turin. Turning in to the detail, the tribometer was found with a pneumo-hydraulic actuator, that is called first configuration. With this configuration, in order to perform the data acquisition, the compatible sensors for each task were searched, then they were repaired and each one was calibrated. The tribometer is capable to acquire the sliding velocity (mm/s), position (mm), tangential and normal forces (N). To com-plete the set-up of this configuration, a program in Labview capable to acquire and save the respective signals in the adequate way was developed. In the same way, a Matlab script was developed to calculate the friction coefficient. The model used to calculate friction coefficient was the simplest but reliable one, developed by Coulomb, in which only takes in to account the normal force and the tangential force. Finally, were done some tests in order to verify the behavior of the tribometer and to compare the results with a previous configuration.

Then, the actuator was replaced for a electric one. This is the second configuration, and the set-up took longer. Talking from the mechanical point of view, it was necessary to develop some mechanical elements and modify others, in order to couple the tribometer with the new actuator. The velocity and position transducer was moved and placed in a new place, while the other sensors remained equal. From the electrical point of view, there were two boxes that control the actuator. And after a detailed analysis of the electric schematic, it was possible to connect and put in to work the actuator. Also was solved an-other problem, with removing two resistors in order to reach high velocities during the automatic cycles. The Labview program was modified to permit acquire the data in automatic way when a continuous test is desired. It permits to choose the number of cycles to acquire and the interval of time between each acquisition. The acquired cycles are calculated with the velocity and the distance of one stroke, because the distance of one stroke is always fixed. In this way, with those values is easy to calculate the time, in which Labview should acquire the data.

Due to the electric actuator, all the acquired signals were affected by line noise. Therefore, in the Matlab script to manage the data, a low pass butter-worth filter was developed in order to smooth the signals. After the complete set-up of the tribometer, tribological tests on materials used for common pneumatic sealing systems, e.g. HNBR 8001, Polyurethane TPU 8001 and NBR were performed. These tests were carried out among the materials cited before and stainless steel or aluminum, common

rials for piston-rod and cylinder barrel of pneumatic cylinders. This in order to simulate the real contact between a seal and either a cylinder barrel or a rod in a pneumatic cylinder.

Trying to be close with the real life, continuous and lubricated tests were performed. Those to un-derstand the behavior of the friction coefficient versus for example temperature or sliding distance. All the tests, done during this thesis, will help to know if the materials and geometry used for a specific task will work as expected or maybe is better to use different materials to ensure if possible, a better realization of the desired task. Since the friction force can not be linearized, if it is studied thoroughly, more information will be collected and greater will be the possibility to create digital control algorithms, which guarantee better performance. Therefore all the tests develop during this thesis will help for fur-ther investigations or even performing a friction force model.

Tribology comes from the Greek word, “tribos”, meaning “rubbing”or “to rub”. And from the suffix, “ology”that means “the study of”, therefore, tribology is the study of rubbing. Tribology is virtually present in every area of engineering and industry, briefly for example: aerospace, agriculture, automotive, dental implants, energy, fabric clothing, food processing, military, pharmaceutical, sports, and so on. All those applications have one common thing,the mechatronics systems.

Mechatronics systems integrate many elements (or parts), in order to create the whole system. In general, every system has surfaces in contact and relative motion (e.g. sliding, rolling, impacting), espe-cially in the actuators. However, there is a large variety of actuators, in order to satisfy all the industrial and engineering applications.

As an example, pneumatic, electric, hydraulic and pneumo-hydraulic cylinders are actuators, that convert their supply energy into a certain work. The work done by the actuators always implies relative motion. When two surfaces are in relative motion always is present the friction force. In cylinders, to reduce the friction between the seals and the rod or the piston are used lubricants, that in some appli-cations due to the product contamination, is necessary to reduce or even delete the lubricants, this in theory increases the friction force and the wear, but analyzing the chemical and physical processes during the movement between the surfaces in contact, is possible to produce materials that without lubricants, the friction force and the wear will not increase.

Over three centuries ago, Da Vinci that is considered as the father of the tribology, Amonton and Coulomb, began to study the friction force as a linear behavior and they found that friction is pro-portional to the normal force, independent of the apparent contact area and independent of the sliding velocity, i.e the friction was studied at the macroscopic level. Nowadays the friction is being studied at microscopic level, and appeared the dynamic models, where the roughness, temperature, sliding veloci-ties, wear, lubrication and so on, play an important role when the friction is being calculated. Friction creates heat, promotes wear, and wastes power, so the reduction of friction, by any means, is vital. Reduction of friction and wear leads to increases in service life, less downtime and lower operating costs, which can add up to tremendous savings.

Because of all those aspects mentioned above, make that the friction force has a non linear behavior. This behavior has a negative impact over the performance of controlled mechanical systems. Therefore when an accurate position is demanded, the tribology plays an important role generating sophisticated tools and many compensation models. Friction modeling and identification can enable proper com-pensation, which improves the precision of the position control. The non-linear nature of the friction causes that, a classic PID , placed in a control loop, is not sufficient to achieve high performance to track the trajectory, in all mechanical systems consisting of masses connected by means of transmis-sions or somewhat elastic. Since the friction force can not be linearized, if it is studied thoroughly, more information will be collected and greater will be the possibility to create digital control algorithms, which guarantee better performance. Characterize and understand properly the tribosystem will increase the odds to success, because it is possible to make the right choice for materials, contact geometry and chemistry, and make the appropriate measurements to give the desired answer of the mechatronic design.

The objective of this thesis is firstly tune a tribosystem available at the Department of Mechanical and Aerospace Engineering (DIMEAS) laboratory of the Polytechnic University of Turin, and perform experimental tests with two different actuators, pneumo-hydraulic and electric cylinder. Then, test dif-ferent materials as HNBR (with and without lipocer), Polyurethane (with and without lipocer) and NBR

(with and without lipocer), against two sliding plates made by anodized aluminum and stainless steel. These tests will be done due to these materials are found between the seals and the rods or the pistons of the linear actuators (e.g. cylinders), and compare the results, with the tests done by another student with the same materials, but with a diverse configuration of the tribosystem.

The first section of the Chapter 1 presents the most important theoretical aspects taking into account during the development of this thesis. It starts with the most significant friction models, beginning with the classic and static models and ends with the dynamic models. The second section talks about the contact mechanics, referring to the real area in contact when one surface is against to other. All the theory about contact mechanics was developed by physicists, but the main model was done by Hertz applied to static, quasi static or elastic contact. Also there were studies about plastic contact. The third section presents an information about the importance of the tribology in the present life, and the effects of tribosystem variables in the friction force, as roughness, normal load, contact pressure, temperature, sliding velocity. In relation with the last variable, the sliding velocity, there is an important phenomenon present in quasi all the mechatronic systems called Stick Slip, and this is the major problem when is talking about precision and exact position. Therefore doing a control or a robust mechanism to avoid this phenomenon, there will not be a wrong or undesired behavior.

The first section of the Chapter 2 presents a review of the initial conditions of the tribometer, as was found, and the restoration process, done in order to put working the tribometer. Basically all this chapter talks about the elements of the tribometer and their main characteristics, the calibration of the sensors, the set up for testing in safe and correct way. Since it is impossible to measure directly the friction force from the sensors of the tribometer, therefore was used an easy model to compute the friction force, and to perform this, was developed a program using Matlab to process the acquired data and calculate the friction force. Finally, this chapter presents a briefly information about the tribometers that can be find in the market, this to understand how the big companies develop the tribometers and how they work.

Chapter 3 presents the results of the set of tests, that were done during all this thesis, with the electric and pneumo-hydraulic cylinder. Are shown also another tests performed with weights, these in order to compare with the tests done with the vertical pneumatic cylinder (pressure) to corroborate that the force exerted by the vertical pneumatic cylinder (pressure) is the desired one, and therefore be sure that all the tests were carried out in the right way. Finalizing with lubricated and dry continuous tests.

### Theoretical Apsects

Mechanical systems are actually very complexes to analyze due to their variety and all the factors that can influence them. They play a very important role in the field of industry therefore must be analyzed carefully. An important phenomenon to analyze is the friction, which is present, e.g. between the mov-ing parts of the actuators, electric motors, transmissions, bearmov-ings, gears and seals. This phenomenon creates heat, promotes wear, and wastes power, so the reduction of friction is vital. To reduce it, first is necessary to know where it originates, how is its behavior and how to measure it. That is why through the years a variety of friction models have been developed.

Different models of friction can be presented and discussed because the friction force have different behavior depending on the conditions in which it is working , for example in function of the lubrication regime, of the sliding velocity, of the roughness, of the applied force or even of the temperature. The nonlinearity of the friction force, joined to the extreme importance of this phenomenon in many appli-cations, for decades has been justified conducting various studies to define mathematical models.

The mathematical modeling of this phenomenon, particularly important in the dynamic simulation of mechanical systems (with special regard to the servo systems), should be able to describe the mechan-ical behavior of the element subject to a friction force, discriminating between four possible kinematic situations listed below:

❼ Mechanic element initially at rest that must remain so.

❼ Mechanic element initially at rest that must move.

❼ Mechanic element initially in mechanical movement that must remain so.

❼ Mechanic element initially in mechanical movement that must stop.

The adoption of a model capable to describe correctly these dynamics, greatly increases the accuracy and confidence of the simulation results, providing a valuable tool to investigate critical and unwanted operating conditions, because these are harmful for the accuracy of the dynamic response and the proper functioning of the system.

Currently, the numerical simulation techniques are used to explain the friction phenomenon by in-troducing simplifying assumptions to overcome computational problems, but most of them used in the literature, led back to mathematical models, which although be the most powerful purely linear, still suffering certain deficiencies.

This section describes some of the main numerical models of dynamic and static friction in the time domain. Showing the structure of its calculation algorithm, the simplifications introduced and the corresponding strengths and weaknesses.

### 1.1

### Friction Models

### 1.1.1

### Leonardo da Vinci

Leonardo da Vinci (1452-1519) was the first who made studies on the problem of friction, he studied the friction experiments similar to those still used at present (inclined plane, block weights pulled by a rope and a pulley, see Figure 1.1), and discusses various systems to decrease friction (ball bearings, lubrication). He stated that:

❼ Friction force is independent of the contact area. ❼ Friction force is directly proportional to the load.

These works of Da Vinci were never published, so he did not have much influence until Amontons discovered his papers.

Figure 1.1: Da Vinci experiments

### 1.1.2

### Guillaume Amontons

Guillaume Amontons (1699) two centuries after, rediscovered the papers where Leonardo da Vinci wrote the basic laws of friction and presents a paper at the Royal Academy of Paris, which sets out the first two classical laws on friction, should be noted that these are identical to those mentioned by Leonardo da Vinci.

❼ Amontons’ 1st Law: Friction force is independent of the contact area. ❼ Amontons’ 2nd Law: Friction force is directly proportional to the load.

He also added that “the resistance caused by friction is almost the same for iron, copper, lead, wood, in any combination, if the surfaces are lubricated with pig fat; and this resistance is approximately equal to one third of the load ” [2]. He stated that the cause of friction in the roughness of the surface, considered as asperities, is due to the force required to overcome the asperities, as on an inclined plane, and thus the force depends precisely on the total weight and not the area of contact.

### 1.1.3

### Leonhard Euler

did not provide particularly significant results, but developed the first mathematical treatment of an explanatory model of friction, with a theoretical distinction between static and kinetic friction. He also introduced the well known symbol for the friction coefficientµ. [16]

µ= F

N (1.1)

### 1.1.4

### Coulomb Model

Coulomb (1736-1806) took the results of Amontons and studied the sliding friction, rolling and ropes and applications to machines. Coulomb distinguished between static friction and kinetic friction, and studied them separately. He investigated the dependence of static friction from four causes: the nature of the materials in contact, the area of the surfaces, the applied load, the contact time; and the dependence of kinetic friction of the first three previous cases and of the speed. He found always an accurate relation of proportionality between frictional force, either in the presence or absence of sliding velocities. Therefore he defined two regimes depending of the sliding velociy.

❼ Static Friction: It is a reaction force that prevents the movement, it does not have a default value, it can assume all the values up to a breakaway force, this breakaway force is called FS, it is the

limit where the sliding begins. Thus the static fricition coefficient is:

µS =

FS

N (1.2)

❼ Kinetic Friction: It is the force that opposes to the movement when two objects are moving relative to each other. The force is constant while the sliding condition remains and it has the same sign of the velocity. The kinetic friction coefficient is denoted asµK , and is usually less than the static

friction coefficient for the same materials. The kinetic friction force or Coulomb friction can be described as:

FC=µK·N·sgn(v) (1.3)

Coulomb’s model can be applied when working at high sliding velocities, but it presents drawbacks at low velocities where the transition from static to kinetic regime occurs, on the other hand when the system is at rest the friction force can be null or can take value between−FC to FC. Regardless of the

problems that has the model, it is used due to its simplicity. He established the third law of the friction:

❼ Coulomb’s Law: Kinetic friction is independent of the sliding velocity (Figure 1.2).

### 1.1.5

### Stiction Force

Stiction force or also called static friction, describes the friction force at rest (i.e. no relative motion). Morin (1833) stated that the static friction force at rest is higher than the Coulomb friction (i.e. Kinetic Friction). The static friction counteracts external forces below a certain level and thus keeps an object from sliding [17]. However, if the external force is too large that overcomes the static friction the sliding starts and occurs an intermittent motion called stick-slip (Section 1.3.4.2), therefore the static friction can be described as a function of the external force, not of the velocity.

F =

(

Fe ifv= 0 and |Fe|< FC

FS·sgn(Fe) ifv= 0 and |Fe| ≥FC

(1.4)

WhereFe is the external force,FC Coulomb friction andFS the stiction force.

### 1.1.6

### Viscous Friction

In real systems, dry friction is always accompanied with viscous friction, due to the majority of machines are lubricated with oil or grease. These lubricants are widely used because they provide a fluid barrier between rubbing metal parts that exchanges dry friction for viscous friction and vastly reduces wear. [4]

In the 19th century Reynolds (1866) developed the theory of hydrodynamics, leading to expressions for the friction force caused by the viscosity of lubricants between surfaces in relative motion [17]. The viscous friction is directly proportional to the sliding velocity (Figure 1.3.a) and usually is described as:

Fv=B·v·N (1.5)

WhereB is the viscous friction coefficient and v the sliding velocity. Viscous friction is often combined with Coulomb and Stiction friction as shown in Figure 1.3. However, these models do not capture all the dynamic behavior of friction.

a) Viscous c) Coulomb + Viscous + Stiction

*B N*

b) Coulomb + Viscous

Friction force Friction force Friction force

Sliding velocity Sliding velocity Sliding velocity

Figure 1.3: Friction Models

### 1.1.7

### Stribeck Effect

In most of the real systems is applied lubricant to increase the performance of the system (i.e. to reduce the wear and the friction). However, Stribeck (1902) observed for lubricated systems at low velocities that the friction force decreases continuously with increasing velocities [17], and he found that this behavior could not be modeled by Coulomb+Viscous+Stiction model. He discovered four regimes before reach the Coulomb friction (i.e. at very low velocities) that were calledStribeck effect or Stribeck curve(Figure 1.4), it explains all the dynamic behavior of the friction at very low velocities.

Figure 1.4: Generalized Stribeck curve

as shown in Figure 1.8, but when the force is removed they recover. Therefore the tangential force is:

Ft(x) =−ktx (1.6)

Figure 1.5: Idealized contact between asperities

Figure 1.6: Asperity deformation under applied force

Where Ft is the tangential force, xis the displacement andkt is the stiffness of the contact that

Figure 1.7: True sliding condition

❼ Boundary Lubrication: In this second regime, due to the low velocity, it is not enough to build a fluid film between the surfaces, and therefore solid-to-solid contact will remain. The friction force decreases with increasing velocity but generally is assumed that friction in boundary lubrication is higher than for fluid lubrication (regimes III and IV). Fein (1991) defined “boundary lubrication as a condition of lubrication in which friction and wear between two surfaces in relative motion are determined by the properties of the surfaces, and by the properties of the lubricant other than viscosity” [13]

❼ Partial Fluid Lubrication: The behavior in this regime is governed by a combination of boundary and fluid film effects. If the load pressure is too high and the relative velocity between surfaces is too low, the lubricant is brought into the load bearing region. However, due to the load pressure some lubricant is expelled, but a small fluid film with a thickness proportional to the viscosity and velocity will remain. Solid-to solid contact prevails if the film thickness is smaller than the height of the asperities. As the sliding velocity increases, the film is sufficiently thick and the separation is complete, therefore the load is fully supported by the fluid.

❼ Full Fluid Lubrication: In this regime there is no solid-to-solid contact, separation is complete and the load is fully supported by the fluids, due to the thickness of the lubricant film is large enough. The viscous friction term dominates the friction force and can be considered proportional to the sliding velocity.

Figure 1.8: I,II) Static Friction, Boundary Lubrication III)Partial Fluid Lubrication IV) Full Fluid Lubrication

The Stribeck effect can be modeled by:

F =

F(v) ifv6= 0

Fe ifv= 0 and|Fe|< FC

FS·sgn(Fe) ifv= 0 and|Fe| ≥FC

WhereFe is the external force andF(v) is an arbitrary function that generally is represented as:

F(v) =FC+ (FS−FC)e−(

v vs)

δs

+µvv (1.8)

WhereFCis the Coulomb friction,FS is the static friction,δsis an exponent that depends on the contact

geometry andvsis the Stribeck velocity.

### 1.1.8

### Karnopp Model

Karnopp (1985) improved Coulomb model in order to take into account stiction and Stribeck effect. He developed a model including the friction force at zero velocity (i.e. no relative motion) and he avoided the discontinuity problem between slip, stick and transition phases. The model defines a zero velocity inter-val,−∆V <|v|<∆V, called stick-slip zone. During this interval the friction force; is modeled as being equal to applied force until a saturation limit and is treated as linear with the velocity [17](Figure 1.9). Once the velocity is safely away from zero, the coulomb+viscous friction is used.

F =

Fslip=

(

FC+Bv ifv≥∆V

−FC+Bv ifv≤ −∆V

Fstick=

(

min(Fe, FS) if −∆V < v <∆V,Fe≥0

max(Fe, FS) if −∆V < v <∆V,F e≤0

(1.9)

The model was used since it allow efficient computer simulations but the drawback is that the external force is an input to the model that is not always given.

Figure 1.9: Karnopp Model

### 1.1.9

### Dahl Model

Dahl(1968) proposed a new friction model, which describes the friction during the stiction, he experi-mented with ball bearings trying to explain that behavior. At low amplitude and low frequencies of the applied force he noticed that in ball bearings the behavior is elastic and spring-like, because he noted that when the applied force is not larger than the breakaway force once it is released the bonds will re-turn to the original state. On the other hand, if the applied force is larger than the breakaway force the bonds will deform analogously to plastic deformation (i.e. permanent displacement), if the applied force is removed, the bonds will not return to exactly the previous state, this is an hysteresis behavior. That is why he compares the transition from static to kinetic friction to that of elastic to plastic deformation.

to the strain or relative displacement between the bodies. When the bodies are stressed until the yield point, the material will deform permanently, as referred before, it can not recover its initial state. If the applied force is held, a maximum friction force is reached, and the two bodies start to break free and a plastic deformation takes place. Therefore was a good analogy because, it has the same hysteresis behavior as was referred before with the friction force-displacement analysis. [10] [11]

Taking in to account these both analysis, Dahl assumed that friction force is not only a function of the velocity but the velocity as well. He expressed the friction force as:

dF(x)

dx =σ

1−_{F}F

C

sgn( ˙x)

n sgn

1−_{F}F

C

sgn( ˙x)

(1.10)

Whereσ the stiffness parameter (i.e. the slope) at equilibrium pointF = 0, FC Coulumb friction (i.e.

yield point), nis a material dependent parameter which is 0≤n≤1 for brittle materials andn≥1 is for more ductile like materials. Whenn= 1 the equation 1.10 can be written as:

dF dt =

F dx·

dx

dt =σx˙− F

FCσ|x˙| (1.11)

With this expression Dahl was able to model presliding and hysteresis, as shown in Figure 1.10.

Figure 1.10: Dahl Model

### 1.1.10

### Lund-Grenoble (

### LuGre

_{) Model}

LuGre(1995) model was important because, it incorporated many of the observed features of frictional behavior, found years ago (e.g. Stribeck effect, stick-slip and viscous friction). Also his work, as the Dahl model, focused on small displacements.

He used a simple method to explain the friction force behavior on two rough surfaces. At macroscopic level, the contacts of two rough surfaces can be represented with elastic bristles (Figure 1.11), the bristles on one surface are bonded with opposing bristles from the other surface. When there is a relative motion between the surfaces, the bristles deflect spring-like, in this way the friction force contributed by each bristle is assumed to be proportional to the strain on the bristle [1]. When the applied tangential force is large enough, the strain on any particular bristle exceeds a certain level then the bond is broken, therefore this leads to begin the sliding regime, should be clarified that each bristle will slip in a random manner due to the roughness of the surface.

The LuGre model stated that friction force varies with not only the average bristle deflection z, but also brush and bristle deflection velocitiesv and ˙x.

Average bristle deflectionz is given by

dz dt =v−

|v|

Figure 1.11: LuGre model analogy of contact between two rough surfaces.

Wherev is the relative velocity between the two surfaces and S(v) is a parametrized function that de-scribes the Stribeck effect.

g(v) = 1

σ0

FC+ (FS−FC)e−(

v vs)

2

(1.13)

WhereFC is the Coulomb friction force,FS is the static friction force (Stiction force),σ0 is the bristle

stiffness, andvsis the Stribeck velocity.

Finally the LuGre friction froce is given by

FL=σ0z+σ1z˙+σ2v (1.14)

Where σ1 is a damping coefficient and σ2 viscous coefficient. The LuGre friction force behavior can

be obtain from Equation 1.14, where the stiction force will be greater than the force for any other velocity. While at steady state, the LuGre model behaves as the Coulomb+Viscous model [10]. The LuGre model relates the friction force with the relative velocity, the position as the Dahl model (i.e. Hysteresis behavior) and with the viscosity. With the same model both the presliding and the sliding regimes are described. However, the model has limitations to work, e.g. it does not separate between dry or lubricated friction, but it is useful for control applications and allows efficient computer simulations.

### 1.2

### Contact Surface

then is fully plastic contact.

In summary, the theory of this section has an important role on tribology. On a microscopic scale it provides the essential theory for any relevant discussion of the nature of asperity contacts. On a macro-scopic scale, the theory is important for machines with non conforming contact. The importance of calculate the real area of contact, the shear stress, is crucial to calculate the friction coefficient, outlined later in Section 1.3.1.

### 1.2.1

### Elastic Contact

The contact between two bodies may be, from the geometrical point of view, conforming or non-conforming. A conforming contact occurs when the surfaces of two bodies fit together exactly, such as in bearings (between the bearing and the pivot). If the shapes of the bodies are different, instead, the contact is non-conforming and, theoretically, the contact occurs only on a point or along a line; a point contact occurs, for example, in rolling bearings (between ball and inner ring) and a line contact occurs in the gears (between tooth and tooth).

In the case of conforming contact, the nominal contact area (An) is finite and is easy to calculate. In

fact, even in the case of non-conforming contact, the nominal contact area is finite, due to the deformation of the two bodies in contact.

Figure 1.12: Elastic contact between two cylinders

An example of non conforming contact is, two cylinders in contact, as shown in Figure 1.12. Where

R1 andR2 are their respective radius, the length L of the contact and the normal force FN. Suppose

that the contact is elastic, withE1andE2the elastic modulus, and the surfaces are smooth, the nominal

area of contact (An) is:

An = 2aL (1.15)

Whereais [20]:

a=

r

8FNR′

πLE′ (1.16)

1

R′ =

1

R1

+ 1

R2

(1.17)

1

E′ =

1 2

1−ν12

E1 +

1−ν22

E2

(1.18)

Whereν1andν2 are the Poisson’s ratio. Therefore the nominal are of contact depends on the equivalent

radiusR′_{, i.e. it depends on the shape of the gap between the surfaces. Also}_{A}

n depends on the

equiv-alent elastic modulus E′_{, as one might expect, it is most strongly dependent on the elastic properties.}

With Hertz theory is possible to calculate also the maximum pressurepmax, the maximum shear stress

τmaxand the nominal pressure p0.

pmax= 2FN

πaL p0= 0,78pmax

τmax= 0,3pmax

These values are for the case of two cylinders, however in [20] there are more cases with whole the procedure.

### 1.2.2

### Plastic Contact

One of the most popular surface geometry models was developed by Greenwood and Williamson [8], who modeled contacts based on a distribution of asperity heights, this model predict that the area of real contact is nearly proportional to the load.

Consider now, a contact under very heavy loads and sliding condition. Fully plastic contact is then when the nominal pressurep0reaches a critical value, called yield pressure, and produces a yield stress.

If the applied load is removed first than reaches the yield point, i.e the contact is fully elastic, the ma-terial will return to its original shape (i.e. will return following the path ba), once the yield point is passed will be always plastic contact, and residual stresses will remain even after the external loads are removed(i.e. will return following the pathcb)(Figure 1.13) [21]. To predict the plastic flow deformations of the asperities depend on aplasticity index (ψ).

ψ=

E∗

H

r

σ∗

R (1.19)

WhereE∗_{=}_{E/}_{(1}_{−}_{ν}2_{),} _{R}_{the asperity radius,} _{σ}∗ _{the standard deviation of asperity heights, and also}

depends on the hardness (H), the elastic modulus(E) and Poisson’s ratio(ν), i.e. yielding process is material-dependent.

### 1.3

### Tribology

Tribology has been defined in 1965 as “the science and technology of interacting surfaces in relative motion and of the practices related thereto ”(Her Majestys Stationery Office, 1966) and modified later as “the science of behaviors of interaction surfaces in relative motion together with the active medium concerned (each of them is a tribo-element) in natural systems, their results and the technology related thereto”(Xie, 1996) [27]. Over the years the definition of tribology has changed, but doing a compilation and summary of all, can be defined as follows:

Tribology is defined as the science and engineering of phenomena such as, friction, wear and lubrica-tion, present between surfaces in relative motion.

Tribology took importance as a result of the recognition of these phenomena in engineering machines. The first friction theory was done by Leonardo da Vinci (Section 1.1.1) and he can be considered the father of the tribology. Over the years was developed important theories of friction, as mentioned in Section 1.1, which belong to the tribology. Most of them can be applied only to a specific branch of field for a specific target.

As consequence of friction and other phenomena, wear occurs on the contact surfaces. High friction force produces high shear stress that leads to wear, e.g. the friction produces heat which increases the temperature that is a contributor to the weakening(softening, melting, oxidation) of the surfaces [22]. Therefore due to the friction and the wear, the materials in contact can change their tribological prop-erties as, geometry, roughness, hardness. Hence, materials selection for tribological systems play vital role in the performance, operation and durability thereof.

Lubrication can be the solution to the problems caused by the friction and wear because the lubricants separate the two solid surfaces in relative motion against each other, this is the desired case but not the always present, because it is possible to have several types of lubrication during the relative motion, as mentioned in Section 1.1.7 (Stribeck effect), and this can influence both friction and wear and therefore the entire operation of the tribological system.

Figure 1.14: Tribosystem structure

tribosystem(Figure 1.14). [27]

In summary, tribology is present always where there is relative motion, and it is important in obtaining the best way (theory and application) to complete the desired function of tribosystems.

### 1.3.1

### Effects of Tribosystem Variables on Friction

Between a pair of materials, on average,µremains relatively constant satisfying certain contact condi-tions, therefore, it is important understand the effects of variables such as sliding speed, normal force (contact pressure), temperature, environment, and material properties on frictional behavior. Table 1.1 lists some of the factors that can affect the friction.

General Category Factor

Mechanical Contact geometry: macro, micro, nano

Load and contact pressure distribution at various scales Loading history

System dynamics: vibrations, stiffness, damping, hysteresis Type of motion and velocity profile

Materials Pairing of materials

Composition and purity of materials Adhesive characteristics

System dynamics: vibrations, stiffness, damping, hysteresis Type of motion and velocity profile

Elastic and plastic mechanical properties Property gradients in the near-surface

Thermophysical properties: thermal conductivity, thermal expansion, etc Method of creating the surface (Finishing, machining artifacts)

Residual stress state in the near-surface regions

Thermal effects Frictional heating

External heat sources Thermoelastic instability

Thermally induced phase transformations: softening, melting Thermal shock during cycling

Lubrication Quantity and means of supply

Regime of lubrication Properties of the lubricant Lubricant chemistry Lubricant ”aging” Filtration and cleanliness

Tribochemistry Relative humidity

Surface reactivity/catalysis Cleanliness

Composition of the surrounding environment Tribopolymerization

Friction polymer formation Oxides and tarnish films

Third bodies Transfer particle formation

Wear particle concentration and agglomeration Sizes, shapes, and morphology of particles External contaminants

Flow of third bodies in and out of contact

### 1.3.2

### Effects of Roughness, Load and Contact Pressure

Over three centuries ago, Da Vinci, Amonton and Coulomb found that friction is proportional to the normal force, independent of the apparent contact area and independent of the sliding velocity, i.e the friction was studied at the macroscopic level (waviness). Nowadays, the friction is being studied at microscopic level, and the roughness Ra appeared, which describes the surface texture and exists even

on very smooth surfaces.

Figure 1.15: Roughness and Waviness

It is possible that two surface textures can have the sameRa, but their frictional characteristics could

be different [15]. Due to the roughness the contact between the two bodies, does not take place over the entire geometric surface, but only on small areasAr, therefore the real contact areaAris much smaller

than the nominal contact areaAn as shown in Figure 1.16.

### N

Nominal Contact Area nom

True Contact Area A true

Figure 1.16: Contact Area

The real contact area Ar between two rough surfaces is proportional to the normal force

(Equa-tion 1.20), because the asperities are deformed in order to generate the real contact area capable of withstanding the normal force.

Ar=α·N (1.20)

Coulomb and Amonton friction laws affirmed that all surfaces are covered with a thin film of oxide [26], an adsorbed layer of water, or an organic film as shown in Figure 1.17. When the two rough surfaces slide against each other, the film will start to deform, and so allow the two asperities to slide past each other, the tangential friction force due to shearing the film,τ0, on the surface of all the contacting asperities is

therefore:

ReplacingAr of the equation 1.20:

Fr=τ0·α·N ⇒µ=τ0·α (1.22)

### Surface film,

### shear strenght

### t

0Figure 1.17: Roughness and organic film

When two surfaces are placed together under a normal force, the asperities will deform first elastically and then plastically as load increases [8]. On a plastically deforming the real areaA′

r, can be calculated

from the hardnessH.

Ar= N

H (1.23)

SolvingA′

r and replacing on equation 1.21:

Fr=

τ0

H ·N ⇒µ= τ0

H (1.24)

The hardnessH is the pressure required to resist penetration.

The normal force on each real contact area, can produce local plastic deformation that favor cold welds between the asperities in contact and thus generate between these structural bridges. It must be said, however, that, at the same time, the plastic deformation produces an increase in the contact area A′

r, with A′r> Ar; therefore the normal force is reduced, thus stabilizing the cold welds. It is now

more understandable why a heavy body offers resistance to detachment. To put in motion the body, the force applied to move the body, must in fact be able to break the structural bridges generated for cold welding, whose total resistance is summarized by the maximum value of the static friction force. The microstructural process just described explains the dependence of sliding friction on the contact pressure, as shown in Figure 1.18.

Thus, the friction force is, proportional to the normal force, independent of contact area, sensitive to surface films, dependent of contact pressure and can be influenced by surface roughness [26], but also lay (the directionality of finnish marks or scratches) [7].

### 1.3.3

### Effects of Temperature

Thermodynamics laws require that energy produced must be dissipated to the surroundings:

Uin=Uout+Uaccumulated (1.25)

Figure 1.18: Friction due to adhesion by a rubber slider on a dry, rigid surface as a function of vertical contact pressure [3].

energy. In sliding tribological systemsUinis the product of F and the velocityv,Uout is thermal energy

produced by heat,Uaccumulatedis the energy consumed or stored in the material. The dissipated energy

as heat can affect the tribological properties (chemical, geometrical) of the materials. Increasing the temperature the hardness of the material is reduced [25], and also more phenomena:

❼ The shear strength of interfacial materials. ❼ The viscosity of liquid and solid lubricants.

❼ The tendency of the surfaces of materials to react with the surrounding environment to form films or tarnishes.

❼ The tendency of formulated liquid lubricants to change chemical composition(e.g., oxidize or change molecular weight).

❼ Wear processes, which affect surface roughness and traction.

❼ The tendency of materials to adhere and transfer to the rubbing partner. ❼ The ability of a surface to adsorb or desorb contaminants. [7]

Figure 1.19 shown how the flow of heat is greater in correspondence of the contact asperities, i.e. the highest temperature is reached in these areas, this temperature is calledflash temperature, and it is reached for few time due to the short contact duration between the asperities. The mean temperature on the superficial zones next to the asperities is calledmean surface temperature.

Figure 1.19: Flow Heat

The heating is also important in lubricated contacts. The evaluation of the temperature reached by the lubricant in use, is very important to establish the correct functionality of the lubricant itself, and therefore of the tribological system. The temperature influences markedly the viscosity of the lubricant and furthermore, the use of liquid lubricant is limited by the surface heating [24].

The thermal power accumulated by the lubricant is:

P =ρ·c·Q·∆T (1.26)

Where ∆T is the change in temperature,ρthe density, cthe specific heat capacity of the lubricant and

Qis the flow rate of lubricant expressed with the following relation in the case of flat surfaces:

Q=v·h0·L·K+ 1

K+ 2 (1.27)

Wherev is the sliding velocity. The lubricant are dragged into the contact region; as the input section, characterized by a heighth1, and the output section,h0, as shown in Figure 1.20, andK os equal to:

K= h1−h0

h0

(1.28)

The change in temperature of the lubricant is therefore:

∆T = Fr·v

ρ·c·Q (1.29)

To calculate the exact temperature must be iterated, because if the temperature increases the viscosity decreases.

In summary, the temperature noticeably affects the friction coefficient, because if very high tem-peratures are reached, may occur the above phenomena, and therefore change not only the tribological properties of the material, but also the properties of the whole tribological system and thereby affect the friction coefficient.

### 1.3.4

### Effect of the Slinding Velocity

X LUBRICANT

h0 h h h1

Z

V

Figure 1.20: Lubricant Film

increase when velocity increases, while with others decreases or even a more complex non-monotonous behavior [9]. It is possible to take in to account distinct effects to explain those behaviors, e.g. Stribeck effect (Section 1.1.7), Stick-Slip phenomenon and Prand-Tomlinson Model, which will be discussed later.

The increment of the sliding velocity has a direct influence in the temperature, either in lubricated (Equation 1.29) or unlubricated surface, and in the same way, temperature has an effect on the friction coefficient (Section 1.3.3), therefore with the change of the velocity a variation on the friction coefficient will be obtained as well, taking this into account, a continuous test for a couple of hours at high sliding velocities will not only cause an increment on the temperature, but also the wear will take an important role on the friction coefficient, due to tthe wear is associated to a formation of ferrous oxide on the surface that reduces the friction coefficient [14], the last phenomena occurs under unlubricated conditions.

### 1.3.4.1

### Prandtl-Tomlinson Model

Prandtl-Tomlinson model was done to try to explain the behavior of the friction on a nanometer scale.
It is the simplest usable model of a tribological system, it takes in to account two of the most important
fundamental properties of an arbitrary frictional system, without the conservative force, no static
fric-tion can exist, without damping, no macroscopic sliding fricfric-tional force may exist [19]. Later using the
model, was found some relations between sliding velocity and friction force. Gnecco showed that friction
force has logarithmic relation with the sliding velocity, at low velocities. Sang showed that friction is
proportional to (ln(v))2/3_{, where}_{v} _{is the relative velocity, but he worked at high velocities[12].}

The model considered the movement of a point mass on one dimension, in a periodic potential, exter-nal force and damping factor, this can be associated with movement of one asperity on a rough surface (Figure 1.21 and Figure 1.22).

Figure 1.21: Prandtl-Tomlinson model

Figure 1.22: Simplified Prandtl model in the case of a very soft spring (modeled here by a constant force)

mx¨=k(v0t−x)−ηx˙−N sin(ax) (1.30)

Wherexis the position of the mass, mits mass, k the stifness of the spring,η damping factor,N the periodic force and a the wave number. In the case of very soft spring, the motion over one or several periods does not change the spring force F , which can be considered to be constant.

mx¨=F−ηx˙−N sin(ax) (1.31)

When the mass is at rest and aF force is applied, the equilibrium position moves to thex, the equa-tion 1.32 describes that behavior.

F =N sin(ax) (1.32)

This equation has a solution only whenF≤N, therefore the force of static friction is:

Fs=N (1.33)

This is the minimum force with which the mass begins to move. If the mass is in motion and the force reduced, the mass will continue to move, even with a smaller force than the force of static friction, due to it already possesses the necessary energy generated by the inertia. Macroscopically, it means that the kinetic friction can be smaller than that of the static friction [19].

❼ Case of null damping

If damping is zero and the mass is already in motion, it will continue to move even without force, as mentioned above. Therefore the energy of the system at that moment:

E0= Z

C

F·dx=

Z

mx¨+N sin(ax)·dx

=m·v

2

2 −

N

a ·cos(ax)

(1.34)

Thus, the velocity depends on the position x.

v=

s

2

m

E0+

N

acos(ax)

(1.35)

In the presence of small damping, the motion is periodic if the work performed by the forceF over
a period of 2_{a}π is equal to the energy dissipated by the damping.

Z 2aπ

0

F·dx=

Z 2aπ

0

ηvdx

2πF k =η

Z 2aπ

0 s

2

m

E0+

N

acos(ax)

dx

Thus the value of the forceF is:

F= ηa 2π ·

r

2

m

Z 2aπ

0 r

E0+

N

acos(ax)dx (1.36)

To satisfy that the mass will always continue moving.

r

E0+

N

acos(ax)≥0 E0≥ −

N

acos(ax)

Therefore the minimum value of the energy Emin is:

Emin=

N a

Thus the minimum value of the force can be obtain substituting the minimum value of the energy on the equation 1.36.

Fmin=

4

π ηN √

N am (1.37)

From the previous equation is possible to calculate the damping factor at which the force of kinetic friction would be equal to force of static friction is:

η= π 4

√

N am (1.38)

This is the damping factor obtained from the simplified model, solving the complete model in the same way the damping factor is [23]:

η= 2π

2_{N}

ka2 (1.39)

When η < 1 the movement is continuous and no dissipation occurs; when η > 1 the stick-slip behavior is found.

❼ Case of large damping

In case of large damping, the inertia can be neglected [6] from the equation 1.31, and becomes the equation of an over-damped motion.

0 =F−ηx˙−N sin(ax) (1.40) Where,

˙

x=dx

dt =

F−N sin(ax)

η (1.41)

dt= _{F} dx

−N sin(ax)

η

(1.42)

One spatial period is traversed on one period time.

Z T

0

dt=

Z 2aπ

0

dx

F−N sin(ax)

T = η

aN

2π

q

F N

2

−1

(1.43)

The average speed is:

¯

v= 2π/a

T = √

F2_{−}_{N}2

η (1.44)

Therefore, the forceF will depend on the average speed, as is shown in Figure 1.23

F=

q

N2_{+ (}_{η}_{v}_{¯}_{)}2 _{(1.45)}

Figure 1.23: Friction response for the Prandtl-Tomlinson model in the over-damped case [18].

### 1.3.4.2

### Stick-Slip

The stick-slip phenomenon occurs at low speeds and can not be justified by a simple friction model, e.g. Stiction or Coulomb model friction. The stick-slip is a typical oscillatory behavior of mechanical systems that contain frictional forces. The phenomenon appears as alternation of phases of arrest (stick) and sliding (slip).

Due to the adhesion and low sliding speed, the time contact between the asperities increases ,and favors the formation of tough junctions that induce high friction. Under the action of the tangential force applied, the junctions break and the sliding velocity increases suddenly. The junctions that are carry out immediately after the stick, are obviously less tough and the friction decreases. At this point the system decelerates to keep the sliding velocity constant. This slowdown can result in the formation of new tough junctions and the phenomenon is repeated. The stick-slip is then originated from the phenomena that occur in the contact areas, however, it is also influenced by the mechanical properties of the tribolog-ical system, which can attenuate the oscillations. The phenomenon of stick-slip can be minimized by reducing the adhesion, increasing the sliding velocity or increasing the stiffness of the tribological system.

Should be noted, finally, that in general, the friction coefficient is never a constant value during the sliding between two bodies, the friction coefficient can increase or decrease until reach a steady state; in any case, however, the coefficient of friction typically oscillates around a mean value. During the sliding, in addition, can be generated superficial heating which may cause transitions in the friction coefficient (Section 1.3.3).

A simple system that collects the essential aspects of the stick-slip, consists of a heavy block of mass

Figure 1.24: Mass-Spring system for Stick-Slip motion modelling

The end A of the spring is pulled with a constant velocity V, in order to create a shear stress on the contact surface of the block with the floor. The friction force between the two surfaces can be charac-terized by the values of static and dynamic friction coefficients.

Initially the block M is at rest (stick) and the spring elongates linearly with velocity v, the static friction force is equal and opposite to the tangential force kxof the spring and grows withxup to the valueM gµsin which the block is set in motion (slip). At this point the friction force falls to the dynamic

valueM gµdand remains there until the block stops again. The block accelerates due to the spring force

is no longer balanced, and exceeds the equilibrium position of the spring, after which decelerates and stops. That behavior can be explained by a mathematical procedure.

Applying Newton’s second law to the system at rest:

FS =FT (1.46)

WhereFS is the static friction force andFT is the tangential force (i.e. the spring force). At timet=t1,

the tangential force is given by

FT =k·x(t1) (1.47)

And the normal load is

N =M g (1.48)

Supposing that at timet=t1 the tangential force is the one required to set motion the block, the static

friction coefficient µs is defined as the ratio between the tangential force required to set in motion the

block and the normal load.

µs=

k·x(t1)

M g (1.49)

From timet1 the block starts to slide and is subject to the elastic force of the spring and the dynamic

friction force.

Fd=µd·N (1.50)

Whereµd is the dynamic friction coefficient. Applying Newton’s second law for the slip regime:

Mx¨=Fd−kx(t) (1.51)

The initial conditions

(

x(0) = µs·N k

˙

x(0) =v (1.52)

Becausex(t) is not the real position of the block, a new variabley(t) was created to work with its real position.

Therefore the Equation 1.51 becomes:

My¨=−k·y(t) (1.54)

And the initial conditions

(

y(0) = (µs−µd)·N k

˙

y(0) =v (1.55)

The Equation 1.54 is a second order differential equation with initial conditions, which its solution has the form of:

y(t) =Acos(ωt+φ) =Acos(ωt)cos(φ)−Asin(ωt)sin(φ) (1.56)

WhereAandφdepend on the initial conditions andw2_{=}_{k/M}_{. Therefore at time}_{t}_{= 0 the position:}

y(0) =A·cos(φ) =(µs−µd)·N

k ⇒A=

(µs−µd)·N

k·cos(φ) (1.57)

And the velocity

˙

y(0) =−A·sin(φ)w=v⇒sin(φ) =− v

Aw (1.58)

Substituting Equation 1.57 and 1.58 in Equation 1.56.

y(t) =(µs−µd)·N

k ·cos(wt) + v

w·sin(wt) (1.59)

WhereN =M g. Thereforex(t) is given by

x(t) = M·g

k (µs−µd)·cos(wt) + v

w·sin(wt) +

M ·g·µd

k (1.60)

The block stops again when its velocity returns to zero or when v is small compared to the average velocity of slipping(i.e. whenwt=π), in this moment the position is

x= M·g·(2µd−µs)

k (1.61)

Analysing the behavior at low velocities

v w =v

r

M k <<

(µs−µd)·N

k (1.62)

In this case the termsin(wt) can be neglected, and the position equation is given now by

x(t) =M ·g

k (µs−µd)·cos(wt) +

M·g·µd

k (1.63)

The time constants of stick and slip regimes are

TStick=

2·M ·g·(µs−µd)

k·v (1.64)

TSlip=π·

r

m

k (1.65)

### The Tribosystem

This chapter describes the conditions in which the tribosystem was found and the changes done to reach the objectives of this thesis. It continues with a description of the elements of the tribosystem, then the steps require to set-up the tribosystem for safe operation. Finally, it provides the methods and the tools used to compute the friction force.

### 2.1

### Initial Conditions

The first presentation of the bank was made by a student, which used a clamp where the specimens were held, one of the main features of the tribosystem is its flexibility and modular construction which allows to test different types of sealing materials, its drawbacks were that it could not do continuous and automatic measurements, short time contact and only out-stroke measurements. The tribosystem was done to study the friction coefficient and the stickslip phenomena of the materials which the seals of the pneumatic cylinders are made. The measurable quantities from the tribosystem were: speed, position, normal and tangential force, from the latter two, was possible to calculate the friction force. All the tests were done on flat specimens with prismatic shape and dimensions 10mm·10mm·2mm. The rubbers tested were:

1. Nitrile rubber NBR.

2. Hydrogenated nitrile rubber HNBR.

3. Polyurethane rubbers.

The contact surfaces, wherein the rubber were slipped, were metallic materials that are found in the pneumatic cylinders. The metal-rubber coupling were the following:

1. NBR - Anodized aluminum.

2. HNBR - Anodized aluminum.

3. Polyurethane rubbers - Stainless steel and anodized aluminum.

The tribosystem allowed to vary the sliding velocity, the length of the sliding and the pressure acting on the rubber. The purpose of the tribosystem was to test the rubbers above mentioned, in order to compare their friction coefficients from the untreated material and the lubricated one. The objective was to find a treatment that allows to obtain a friction coefficient without lubrication, comparable with the same material but in lubricated sliding conditions, this, in order to remove the lubricant, with all the advantages that this implies in pneumatic cylinders.

After, other student modified the tribosytem and he removed the clamp and put a specimen holder held up by tensioned cables, in order to do possible, continuous and automatic measures, longer time contact, in-stroke and out-stroke measures, but without modify the essence of the tribosystem. The idea

was good but the tribosystem was found without load cells. The tribosystem was put in working, but were never taken measurements, therefore the drawbacks for the friction measure were never seen.

Figure 2.1: Tribosystem with Pneumo-hydraulic Cylinder

The pneumatic circuit of the tribosystem is shown in Figure 2.2, but were separated the pneumatic connections for the cylinder that are shown in Figure 2.3, in order to show a detailed information about the connection. The reference signals come from the control unit (Figure 2.12).

Figure 2.2: Tribosystem pnuematic circuit

### 2.1.1

### Modifications

The tribosystem had been out of use for a long period, but as mentioned before was known that worked. Therefore the modifications were done to put the tribosystem working, first it was cleaned rigorously, then took a long period of time to search and select the proper elements.

AVV

RIEN_VEL

AVV

FUO_VEL

FUO

### A

### A

Figure 2.3: Cylinder pnuematic circuit

acquires the data. Also was added a supply voltage (±15V) to supply the load cell that measures de normal force. To connect it, was done a box where there are all the necessary connections for its correct operation. The pressure regulator of the vertical pneumatic cylinder (pressure) (i.e. the one that does the normal force), had to be changed, because it was not working.

The main modification of the tribosystem was change the actuator, which led to small changes in the structure (Figure 2.4). It was done to ensure constant velocities, to work at high velocities, to have an automatic operation, and also to work at very low velocities and thereby check the stick-slip phenomenon. Also was designed the outside enclosure, it works as a protection to the user and also for the tribosystem. If the user opens one fence while the tribometer is working, the tribosystem will stop immediately.

### 2.2

### Elements of the Tribosystem

This section describes the main characteristics of the elements that compose the tribosystem. The main elements are: the base, pneumo-hydraulic cylinder or electric cylinder, spherical joints, specimen holder, linear guide, vertical pneumatic cylinder, the control unit and all the sensors. As mentioned before the tribosystem suffered modifications in this way, first are describe the elements that belongs to the first configuration and then are described the new elements (e.g. electric cylinder). Bellow are listed the main elements of the first configuration (Figure 2.1):

### 2.2.1

### Base

Figure 2.4: Tribosystem with electric cylinder

Figure 2.5: Tribosystem base

### 2.2.2

### Pneumo-hydraulic Cylinder

The pnuemo-hydraulic cylinder US1250300D61C (Figure 2.6) was chosen in first instance to guarantee operation at low and constant velocities.

The cylinder is braked with oil using four valves, two for the output stroke, and two for the input stroke, working in parallel, it can work in fast or slow mode. To control the position of the piston were mounted two limit switches in the aluminum body of the cylinder.

### 2.2.3

### Spherical Joints

Figure 2.6: Pneumo-hydraulic Cylinder

Figure 2.7: Spherical Joints

### 2.2.4

### Specimen Holder

The specimen holder (Figure 2.8) has a rectangular hole, on where are placed the specimens. A tensioned
cables that are connected to the loads cells hold up the specimen holder. It was found without any
doc-umentation, was not known its weight, neither its volume or the material with which it was made. The
only way to know the material with which it was made, was calculating the density. Therefore the design
was done in Solidworks to know its volume (V = 1.29527·10−4_{·}_{m}3_{), then was weighed (}_{m}_{= 1}_{.}_{0333}_{·}_{Kg}_{).}

Knowing those two values, the density is defined by

ρ=m

V ⇒ρ= 7977.4873· m3

Kg (2.1)

The density obtained is related to iron. To avoid that the vertical pneumatic cylinder (pressure) transmits a horizontal resistance force, an air bearing was interposed between the load cell and the specimen holder. With this design of the holder is easy to change at anytime the specimen.

### 2.2.5

### Linear Guide

The linear guide used was HIWIN model HGW-CA (Figure 2.9). It was designed with high load capacity and rigidity, it was chosen due to its features and advantages, here some of them:

Figure 2.8: Holder

• Long life with high motion accuracy: Rolling contact has little wear; therefore, machines can achieve a long life with higly accuracy motion.

• High speed motion is possible with a low driving force: Because linear guideways have little friction resistance, only a small dirving force is needed to move a load.

• Equal loading capacity in all directions: This special linear guideways can take loads in either the vertical or horizontal directions.

• Easy installation and lubrication: Installing the linear guide is fairly easy, following the recom-mended installation procedure. For the lubrication, grease can be easily supplied through the grease nipple on the linear guideway block.

• Self-aligning capability: By design, the circular-arc groove has contact points at 45 degrees, it can absorb installation errors due to surface irregularities.

• High rigidity in all four directions: Because of the four-row design at 45 degrees each one, the linear guideway has equal load ratings in the radial, reverse radial and lateral directions.

Figure 2.9: LinearGuide

### 2.2.6

### Vertical Pneumatic Cylinder (pressure)

To generate the normal force (FN) was used a double acting pneumatic cylinder Camozzi, model

• Pressure: 1−10 bar

• Velocity: 10−1000mm/s(without load).
• Temperature: 0−80◦_{C.}

Figure 2.10: Vertical Pneumatic Cylinder (pressure)

### 2.2.7

### Control Unit with Pneumo-hydraulic Cylinder

The purpose of the Control Unit (Figure 2.11) is to set in motion the tribosystem, to activate the vertical pneumatic cylinder (pressure) or to choose between in-stroke, out-stroke, fast or slow motion.

Figure 2.11: Control Unit

Each push button has its own function:

• Emergency: As its name indicates is used for any emergency, when it is activated, automatically there will not be supply for the relays and therefore the main control will turn off completely and the valves closed.

• ON-OFF selector: This selector turn on or turn off the main control. When it is turned on, all the relays and some sensors are supplied. No matter if the switch is turned on and emergency is enabled, the main control will turn on never.

• Rientro-Fuoriuscita: These push buttons are to set input (Rientro) or output (Fuoriuscita) stroke.

• Rientro lento-veloce: These push buttons are to set the input stroke velocity slow(lento) or fast (veloce).

• Fuoriuscita lento-veloce: These push buttons are to set the output stroke velocity slow(lento) or fast (veloce).

• Pressore OFF-ON: These push buttons are to activate or deactivate the vertical pneumatic cylin-der (pressure), the one that does the normal force (Fn).

• Avvio-Stop: These are push buttons to set in motion (Avvio) or stop the tribosystem.

Inside the control unit there is a pneumatic circuit (Figure 2.12). It has relays, push buttons and leds. Outside, to complete all the pneumatic circuit of the tribosystem, there are the pneumatic valves, the actuators, the pressure regulators and the limit switches. The outputs of the control unit are the reference signals to switch the state the pneumatic valves, and the inputs are the two signals coming from the limit switches.

Figure 2.12: Control unit pnuematic circuit

Also inside the control unit there is a DC voltage supply, TDK Lambda DSP 100-24. It converts the voltage from 240V AC to 24V DC in order to supply all the relays. Its main characteristics are: high reliability, low power consumption, over-current and over-voltage protection is incorporated and a LED indicator gives instant visual confirmation of output status and easy installation. Whole datasheet can be found in the Appendix D.5.