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IFAC PapersOnLine 52-14 (2019) 183–188

Translationalized Tire-Road Friction Translationalized Tire-Road Friction Translationalized Tire-Road Friction Formulation for Linear DC Machine Translationalized Tire-Road Friction Formulation for Linear DC Machine Formulation for Linear DC Machine Powered Mine Locomotives Formulation for Linear DC Machine Powered Powered Mine Mine Locomotives Locomotives Powered Mine Locomotives ∗∗ Mohlalakoma T. Ngwako ∗∗ Otis T. Nyandoro ∗∗

Mohlalakoma T. T. ∗ Otis∗∗∗ ∗∗ ∗ Mohlalakoma John T. Ngwako Ngwako Otis∗∗∗ T. Nyandoro Nyandoro ∗∗ van Coller Mohlalakoma T. Ngwako Otis T. Nyandoro John van Coller ∗ ∗∗ ∗∗∗ ∗∗∗ Mohlalakoma John T. Ngwako Otis T. Nyandoro van Coller John van Coller ∗∗∗ John van Coller ∗ ∗ (e-mail: [email protected]). (e-mail: [email protected]). ∗ ∗∗∗ (e-mail: [email protected]). (e-mail: [email protected] )) (e-mail: [email protected]). ∗∗ (e-mail: [email protected] ∗ ∗∗ ∗∗∗ ∗∗ (e-mail: [email protected]). (e-mail: [email protected] (e-mail: [email protected]) (e-mail: [email protected] )) ∗∗∗ (e-mail: [email protected]) ∗∗ ∗∗∗(e-mail: [email protected] ) ∗∗∗ (e-mail: [email protected]) [email protected]) (e-mail: ∗∗∗ School of Electrical and [email protected]) Information Engineering, University of (e-mail: School Electrical Engineering, University School of of Witwatersrand, Electrical and and Information Information Engineering, University of of Johannesburg, South Africa School of Electrical and Information Engineering, University of Witwatersrand, Johannesburg, South Africa School of Witwatersrand, Electrical and Information Engineering, University of Witwatersrand, Johannesburg, South Africa Africa Johannesburg, South Witwatersrand, Johannesburg, South Africa Abstract: A A comprehensive comprehensive friction friction model model for for aa linear linear DC DC machine machine powered powered mine mine locomotive locomotive Abstract: Abstract: A comprehensive friction model for aa linear DC machine powered mine locomotive is presented. The friction model presented is an extension of an existing friction model based on on Abstract: A comprehensive friction model for linear DC machine powered mine locomotive is presented. friction model presented is an extension of an existing friction model based Abstract: AaThe comprehensive friction model for a linear DC machine powered mine locomotive is presented. The friction model presented is an extension of an existing friction model based on slip. Hence, piece-wise dry friction model is developed. The friction and linear DC machine is presented. The friction model presented is an extension of an existing friction model based on slip. Hence, a piece-wise dry friction model is friction and linear DC machine is presented. friction model presented isThe an extension ofThe anshows existing friction model based on slip. Hence, aThe piece-wise dry friction model is developed. developed. The friction and linear DC machine dynamics are also simulated and presented. friction curve that there are two friction slip. Hence, a piece-wise dry friction model is developed. The friction and linear DC machine dynamics are also simulated and presented. The friction curve shows that there are two friction slip. Hence, a piece-wise dry friction model is developed. The friction and linear DC dynamics are also alsostatic simulated and presented. The friction friction curve shows shows that there there are two twomachine friction regimes namely; friction and kinetic friction. The results show the locomotive tends dynamics are simulated and presented. The curve that are friction The regimes namely; static friction and kinetic friction. results show that there the locomotive tends dynamics are also simulated and presented. The friction curve are two friction regimes static and kinetic friction. The results show the tends to move move namely; slow when thefriction friction force is incorporated incorporated and thatshows there that are more more losses when the regimes namely; static friction and kinetic friction. The results show that the locomotive locomotive tends to slow when the friction force is and that there are losses when the regimes namely; static friction and kinetic friction. The results show that the locomotive tends to move slow when the friction force is incorporated and that there are more losses when the voltage is low. For a voltage of 1V, a velocity of 3.3 m/s is found for a friction-less locomotive to move slow when the friction force is incorporated and that there are more losses when the voltage is low. For voltage of a velocity of 3.3 found for aa friction-less to move when friction force incorporated andis there more losseslocomotive when the voltage isslow low. For a athe voltage of 1V, 1V, a is velocity ofa locomotive 3.3 m/s m/s isthat found for are friction-less locomotive while for a steady-state velocity of 2.1 m/s for with friction losses. voltage is low. For a voltage of 1V, a velocity of 3.3 m/s is found for a friction-less locomotive while for a steady-state velocity of 2.1 m/s for a locomotive with friction losses. voltage is low. For a voltage of 1V, a velocity of 3.3 m/s is found for a friction-less locomotive while for for a a steady-state steady-state velocity velocity of of 2.1 2.1 m/s m/s for for aa locomotive locomotive with with friction friction losses. losses. while © 2019, (International Federation of Automatic Hosting by friction Elsevier losses. Ltd. All rights reserved. while forIFAC a steady-state velocity of 2.1 m/s for aControl) locomotive with Keywords: Linear Linear DC DC machine, machine, mine mine locomotive, comprehensive friction model. locomotive, comprehensive friction Keywords: model. Keywords: Linear Linear DC DC machine, machine, mine mine locomotive, model. Keywords: locomotive, comprehensive comprehensive friction friction model. Keywords: Linear DC machine, mine locomotive, comprehensive friction model. 1. INTRODUCTION INTRODUCTION Other design design considerations considerations which which could could be be useful useful in in 1. Other 1. INTRODUCTION INTRODUCTION Other design considerations which be could be useful useful in minimizing the friction force would deciding whether 1. Other design considerations which could be in minimizing the friction force would be deciding whether 1. INTRODUCTION Other design which could be useful in minimizing theconsiderations friction force would be deciding whether the mine locomotive should have sliding or rolling friction. minimizing the friction force would be deciding whether the mine locomotive should have sliding or rolling friction. minimizing the friction force would be deciding whether the mine locomotive should have sliding or powered rolling friction. friction. In this journal paper, a linear DC machine trolley the mine locomotive should have sliding or rolling In this journal paper, a linear DC machine powered trolley the mine locomotive rolling friction. In this journal paper,should linearhave DCsliding machine powered trolley locomotive Szklarski [1969] model and or a powered comprehensive In this journal paper, aa[1969] linear DC machine trolley Knowing the force required to move (dispatch) an unlocomotive Szklarski model and a comprehensive In this journal paper, a linear DC machine powered trolley Knowing the force required to move (dispatch) an unlocomotive Szklarski [1969] model and a comprehensive friction model model of the the[1969] locomotive isand presented. A linear linear Szklarski model is a comprehensive Knowing the force required required to move move (dispatch) (dispatch) an undergroundthe locomotive is of of paramount paramount importance. The Knowing force to an The un- locomotive friction of locomotive presented. A locomotive Szklarski modelitis aa high comprehensive derground locomotive is importance. friction model of the[1969] locomotive isand presented. A linear linear DC machine is preferred because has efficiency. friction model of the locomotive presented. A Knowing the force required to move (dispatch) an The underground locomotive is of of paramount paramount importance. The dispatch force should exceed the friction force where the derground locomotive is importance. DC machine is preferred because it has a high efficiency. modelis of the locomotive presented. A linear dispatch force should exceed the friction force where the DC machine preferred because itis has has high efficiency. efficiency. Using electric locomotives also reduces reduces the operating costs DC machine islocomotives preferred because it aa operating high derground locomotive is motion of paramount importance. The dispatch force should the exceed the offriction friction force where wheremine the friction friction force force opposes the underground underground dispatch force should exceed the force the Using electric also the costs DC machine is preferred because it has a high efficiency. friction opposes the motion of the mine Using electric locomotives also reduces the operating costs significantly due to less money being used for ventilation Using electric locomotives also reduces the operating costs dispatch force should exceed the friction force where the friction force opposes the motion of the underground mine locomotive. This allows for the movement movement of the the locomolocomofriction forceThis opposes thefor motion of the underground mine Using significantly due to less money being used for ventilation electric locomotives also reduces the Dammers operating costs locomotive. allows the of significantly due to being for and the the related ventilation infrastructure et al. significantly due to less less money money being used used for ventilation ventilation friction force opposes the motion ofsystems the underground mine locomotive. This allows for the movement movement ofdynamic the locomolocomotive. The friction force in practical is and locomotive. This allows for the of the and related ventilation infrastructure Dammers et al. tive. The friction force in practical systems is dynamic and significantly due to less money being used for ventilation and the related ventilation infrastructure Dammers et al. [2016]. According to Kurnia et al. [2014], 60% of and the related ventilation infrastructure Dammers et al. locomotive. This allows for the movement of the locomotive. The friction force in practical practical systems is dynamic dynamic and [2016]. According to Kurnia et al. [2014], 60% of mine it has a static and kinetic region Chou [2004]. The static tive. The friction force in systems is and mine theAccording related infrastructure Dammers et al. it has a static and kinetic region Chou [2004]. The static [2016]. According to attributed Kurnia et toal. al.ventilation [2014], 60% of With mine operational costsventilation are costs. [2016]. to Kurnia et [2014], 60% of mine tive. friction force in practical systems is dynamic and and it hasThe static and kinetic region Chou [2004]. The static friction force region would be the region region where the dispatch it has aaforce static and kinetic region Chou [2004]. The static operational costs are attributed to ventilation costs. With [2016]. According to Kurnia et al. [2014], 60% of mine friction region would be the where the dispatch operational costs are attributed to ventilation costs. With linear DC DCcosts powered locomotivetonot not emittingcosts. any gases, gases, are attributed ventilation With it hasdoes aforce static and kinetic region Chouwhere [2004]. The static friction force would the the dispatch force notregion result in the thebe mine locomotive moving. This operational friction region would be the region region where the dispatch aa linear powered locomotive emitting any are attributed to ventilation costs. With force does not result in mine locomotive moving. This linear DCcosts powered locomotive not emittingare anyreduced gases, thelinear air pollutants in the mining environment aathe DC powered locomotive not emitting any gases, friction force region would be the region where the dispatch force does notthat result in the the mine locomotive moving. This operational would mean there is no net force on the locomotive. force does not result in mine locomotive moving. This air pollutants in the mining environment are reduced would mean that there is no net force on the locomotive. a linear DC powered locomotive not emitting any gases, the air pollutants in the mining environment are reduced and the miner workers are afforded a safe and healthy the air pollutants in the mining environment are reduced force does not result in the mine locomotive moving. This would mean friction that there there is would no net netbe force on the locomotive. locomotive. The kinetic kinetic forceis the on friction force from from the would mean that no force the and air thepollutants miner workers workers are afforded a safe safe and and healthy in theThis mining environment are reduced The friction force would be the friction force and the miner are afforded healthy working environment. addresses the occupational and the miner workers are afforded aa the safe occupational and healthy would mean thatlocomotive there no starts netbe force on theKnowing locomotive. The kinetic friction forceis would would be the friction force from from where the mine moving. the The kinetic friction force the friction force working environment. This addresses and theand miner workers are ISO afforded a the safe occupational and healthy where the mine locomotive starts moving. Knowing the working environment. This addresses the occupational health safety standard, 45001. working environment. This addresses The kinetic friction force would be the friction force from where the mine locomotive starts moving. Knowing the sourcesthe of mine the friction friction force starts is important important because it the al- health and where locomotive moving. because Knowingit safety standard, ISO 45001. environment. ThisISO addresses sources of the force is alhealth and and safety standard, standard, ISO 45001. the occupational health safety 45001. where the locomotive moving. Knowing sources ofa mine the friction friction force starts is important important because it the al- working lows for thorough friction analysis for the locomotive. sources of the force is because it alAdequate force is necessary to dispatch the the mine mine locolocolows for a thorough friction analysis for the locomotive. health and safety standard, ISO 45001. Adequate force is necessary to dispatch sources of the friction force is important because it allows for thorough friction analysis analysis for requirements the locomotive. locomotive. This for would make specifying specifying the power power of Adequate lows aa thorough friction for the Adequate force is necessary necessary to dispatch dispatch the mine locomotive in essence, the generated power would have to force is to the mine locoThis would make the requirements of motive in essence, the generated power would have to lows for a thorough friction analysis for the locomotive. This would make specifying the requirements of is necessary to dispatch the mine locothe locomotive locomotive possible. Subsequently, the efficiency and This would make specifying the power power requirements of Adequate motive in force essence, the adequate generated powerFor would have to be sufficient to achieve force. a linear DC motive in essence, the generated power would have to the possible. Subsequently, the efficiency and be sufficient to achieve adequate force. For a linear DC This would make the power requirements of motive the locomotive possible. efficiency and inwhere essence, the adequate generated power would have to performance of possible. the specifying mine Subsequently, locomotive arethe also determined. the locomotive Subsequently, the efficiency and be sufficient to the achieve adequate force. For linear DC machine locomotive trolley is positioned positioned on DC the be sufficient to achieve force. For aa linear performance of the mine locomotive are also determined. machine where the locomotive trolley is on the the locomotive efficiency and performance of possible. the mine mine Subsequently, locomotive arethe alsois determined. be sufficient to achieve adequate force. For a linear DC Suppose the underground mine locomotive powered by performance of the locomotive are also determined. machine where the locomotive locomotive trolley is positioned on the bar, the force might not be adequate. This necessitates machine where the trolley is positioned on the Suppose the underground mine locomotive is powered by bar, the force might not be adequate. This necessitates performance ofa the mine locomotive alsois Suppose the underground mine powered by the trolley is positioned onThis the a source source with with power rating of locomotive 3 kW kWare Polnik et al. [2014] [2014] Suppose the underground mine locomotive is determined. powered by machine bar, the where force might might not be beonadequate. adequate. This necessitates the introduction introduction of locomotive wheels the mine mine locomotive. bar, the force not This necessitates a a power rating of 3 Polnik et al. the of wheels on the locomotive. This Suppose the underground mine locomotive is powered by a source with a power rating of 3 kW Polnik et al. [2014] bar, the force might not be adequate. This necessitates and the friction losses and the overall losses of the machine aand source with a power rating of 3 kW Polnik et al. [2014] the introduction of wheels on the mine locomotive. This would mean that the locomotive trolley on a bar would the introduction wheels on the trolley mine locomotive. This the friction losses the overall losses of the machine would mean that thatofthe locomotive on aa bar bar would a source with powerand rating 3 kW Polnik et al. [2014] the and the friction losses and the overall of machine wheels onDC themachine mine locomotive. This sum up friction to an aassumed assumed power loss oflosses 200 W. This would and the losses and the of overall losses of the the machine would mean trolley on haveintroduction tomean be modified modified to alocomotive linear bara attached attached to would thatofthe the locomotive trolley on bar would would sum up to an power loss of 200 W. This would have to be to a linear DC machine bar to and the friction losseslocomotive and the overall of an the machine sum up to an an assumed power loss oflosses 200 W. This would would mean that the locomotive trolley on a bar would mean that the mine would have efficiency sum up to assumed power loss of 200 W. This would have to be modified to a linear DC machine bar attached to rolling wheels of a locomotive. This would introduce rolling have to be modified to a linear DC machine bar attached to mean that the mine locomotive would have an efficiency rolling wheels of locomotive. This would rolling sum to an assumed power loss of 200 This would mean that thefriction mine locomotive locomotive would have an efficiency towheels be modified to apower linearrequirements DC machine bardispatch attached to of 93 93 up %. The model formulated in W. thisan study tries have mean that the mine would have efficiency rolling wheels of aaa locomotive. locomotive. This would introduce introduce rolling friction. The force and to this rolling of This would introduce rolling of %. The friction model formulated in this study tries friction. The force and power requirements to dispatch this mean that the mine locomotive would have an efficiency of 93 %. The friction model formulated in this study tries rolling wheels of awould locomotive. This would introduce rolling to estimate the friction force of the mine locomotive in of 93 %. The friction model formulated in this study tries friction. The force and power requirements to dispatch this mine locomotive be lower. friction. The force and power requirements to dispatch this to estimate the friction force of the mine locomotive in mine locomotive locomotive would be lower. lower. of 93 %. by Theincorporating friction model formulated inand this study tries to estimate the friction force of mine locomotive in The forcewould and power requirements to dispatch this practice both thethe static kinetic fricto estimate the friction force of the mine locomotive in friction. mine would be mine locomotive be lower. practice by incorporating both the static and kinetic fricto estimate the friction force of the mine locomotive in mine The sliding friction of a linear DC machine machine is is often often repreppractice by incorporating incorporating both the static and kinetic friclocomotive would be lower. tion regimes. The linear DC machine is meant to generate practice by both the static and kinetic fricThe sliding friction of a linear DC tion regimes. The linear DC machine is meant to generate The sliding friction offriction linearChiasson DC machine machine is reppractice by incorporating both the mine static and kinetic fricresented as viscous [2005], Chapman The sliding friction of aa linear DC is often often reption regimes. The linear DC machine is meant meant to generate generate a motive force that allows for the locomotive to be tion regimes. The linear DC machine is to resented as viscous friction Chiasson [2005], Chapman a motive force that allows for the mine locomotive to be The sliding offriction a linear DC machine is often resented as friction viscous friction Chiasson [2005], Chapman tion regimes. The linear DCat machine istrack meant [2012]. However, viscous friction only represents the repdyas viscous Chiasson [2005], Chapman motive force that allows for themine mine locomotive to transported from one point the to to thegenerate desired aa motive force that allows for the mine locomotive to be be resented [2012]. However, viscous friction only represents the dytransported from one point at the mine track to the desired resented as viscous friction Chiasson [2005], Chapman [2012]. However, viscous friction only represents the dya motive force that allows for the mine locomotive to be namic friction of the machine. This misrepresents friction [2012]. However, viscous friction only represents the dytransported from one point at the mine track to the desired location. transported from one point at the mine track to the desired namic friction of the machine. This misrepresents friction location. However, viscous friction only represents the dynamic friction friction of the the machine. This misrepresents friction transported from one point at the mine track to the desired [2012]. namic of machine. This misrepresents friction location. location. namic friction of the machine. This misrepresents friction location.

2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 187 Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2019 IFAC 187 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 2019 IFAC IFAC 187 Copyright © 187 10.1016/j.ifacol.2019.09.185 Copyright © 2019 IFAC 187

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2019 IFAC MMM 184 Mohlalakoma T. Ngwako et al. / IFAC PapersOnLine 52-14 (2019) 183–188 Stellenbosch, South Africa, August 28-30, 2019

in practical systems especially where the input is low. It is for this reason that a comprehensive friction model with both the static and dynamic friction is necessary. The tire-road (rolling) friction model by Pacejka and Sharp [1991] has the two friction regimes. This model could be extended to suit sliding friction for applications in linear DC machine powered locomotives. Therefore, the key contribution of this study is; translationalizing existing tireroad friction. The second contribution is; extending the linear DC machine model found in Chiasson [2005]. The rest of the journal paper is structured as follows; section 2 the literature review is presented, section 3, a linear DC machine powered model is presented , the friction formulations are found in section 4, in section 5 simulation results for the linear DC machine powered locomotive with friction losses is found. A conclusion is found in section 6.

2. LITERATURE REVIEW Several friction models such as Amonton, Dahl, Stribeck, LuGre, Coulomb and viscous friction models exist. The work done by Amonton is based on findings from an investigation performed on the motion of a sliding block on a surface Astrom [1995]. Amonton found that the friction force at a sliding surface is proportional to the normal load Korondi et al. [2014], Astrom [1995]. He also deduced that the friction force is independent of the contact surface area.

Dahl introduced a dynamic friction model which is able to model the pre-displacement and hysteresis van Geffen [2009]. In the Dahl friction model a spring-like behaviour during stiction is deduced. It is worthwhile to mention that the Dahl model is preferred for extensive hysteresis analysis Liu et al. [2015]. The Dahl model is an extension of Coulomb with a friction lag when the direction of motion changes Liu et al. [2015]. The model is a function of the velocity and displacement van Geffen [2009]. The limitation of the Dahl model is its inability to capture the Stribeck effect, static friction and stick-slip motion Liu et al. [2015], Korondi et al. [2014], Wojewoda et al. [2009]. The LuGre model is an extension of the Dahl friction model incorporating the Stribeck effect. However, the LuGre model failed to address the stick-slip motion van Geffen [2009]. The Pacejka friction model is another friction model which applies LuGre while capturing the slip dependence of the friction force Pacejka and Sharp [1991]. From the review on friction models it is clear that the dynamic nature of a system necessitates a dynamic friction model to capture the truest behaviour of a system. Without a dynamic friction model, a system would be exposed to variable environmental conditions whilst the friction model is stationary. This would be contradictory to the findings by Reynolds, Dahl and LuGre. 3. MODELING 3.1 System diagram

The Coulomb friction model is described by two regimes; static friction and kinetic friction Chou [2004]. The static friction is always greater than the kinetic friction Liu et al. [2015], Hashiguchi and Ozaki [2008]. The Coulomb friction model is based on the principle that the friction force is in the opposite direction to the velocity and that the friction force is also proportional to the normal force Liu et al. [2015], Korondi et al. [2014]. The friction force is independent of the magnitude of the velocity Liu et al. [2015], Korondi et al. [2014], Astrom and de Wit [2008]. Some of the of the limitations of the Coulomb model are; the dissipation nature of the friction is not demonstrated, and the model is based solely on the roughness theory Korondi et al. [2014]. The viscous model by Reynolds is often combined with the Coulomb friction model Liu et al. [2015]. In the viscous friction model the friction force is a function of the viscous coefficient and sliding speed Liu et al. [2015]. The major limitation of the viscous friction model is its poor representation of friction in the absence of a lubricant Liu et al. [2015]. The viscous friction model is described by a linear function which is proportional to the velocity Liu et al. [2015], Miller [1997]. In the work done by Stribeck, the conclusions of the friction model is based on experimental observations. Stribeck found that the friction force decreases with an increase in velocity van Geffen [2009]. This phenomenon is called the Stribeck effect Korondi et al. [2014]. Other friction models include the rate-and-state friction models where the friction force is a function of the velocity and the state of variables Astrom and de Wit [2008]. 188

Fig. 1. Linear DC machine powered mine locomotive on a rail track The linear DC machine powered mine locomotive comprises a trolley used to carry coal/ore where the trolley has a mass M, the trolley is attached to a conducting bar placed on a conducting rail. The bar has a mass m and linear DC machine circuit with a voltage source V is used to power the locomotive. Current i is flowing along the rail. 3.2 Rolling friction The rolling friction of a linear DC machine powered mine locomotive is due to the wheel-rail contact and wheel-bearing Nyandoro et al. [2011]. The rolling friction increases with an increase in load and decreases with a decrease in load. Another factor affecting the rolling

2019 IFAC MMM Mohlalakoma T. Ngwako et al. / IFAC PapersOnLine 52-14 (2019) 183–188 Stellenbosch, South Africa, August 28-30, 2019

friction is the uneven-ness of the rail track. Modelling the rolling dynamics of the linear DC machine powered mine locomotive would result in an accurate prediction of the system behaviour Z. Yang [2019]. 3.3 Physical model of a linear DC machine The linear DC machine power source physically comprises a DC source, a resistor, a conducting bar attached to the source and resistor, and a pair of conducting rails that allow for the movement of the bar. The distance between the pair of rails of the mining track, is marked as l in fig. 2 whereas the current, force applied to the bar, the source voltage and resistance are labelled i, F, V and R, respectively. The locomotive power supply circuit shown in fig. 2 is placed in a constant magnetic field, B Boldea and Nasar [1985]. This circuit represents the simplest model of an electromagnetic circuit. The electromagnetic circuit is analyzed using physics laws, and the equations underlying the behaviour of the electromagnetic circuit follow in section 3.4.

Fig. 2. Linear DC machines 3.4 Mathematical model The mathematical model as used in this paper is the same model as used by Chiasson [2005] and Chapman [2012]. The mathematical model of the linear DC machine is generally formulated using physics laws where the Lorentz law is used to find the force applied to the bar, the Faraday’s law is used to determine the induced voltage, and the motion of the bar is governed by Newton’s law. It is worthwhile to mention that the bar is positioned on a rail and the bar can move to the forward (right) and backward (left). The Lorentz force experienced by the bar is expressed by: F L = il(−ˆ y ) × B(−ˆ z) (1) = ilB x ˆ Where FL is the Lorentz force. The length of the bar is a vector thus, the length has a magnitude, l, and a direction, y ˆ. The magnetic field is also a vector in the zˆ direction, the positive direction of the magnetic is represented by the magnetic field going into the page ’x’ whilst, the magnetic field in the negative direction is represented by the magnetic field going out of the page ’o’. The force applied on the bar is in the x ˆ direction where right and left represent the positive and negative direction, respectively. The induced voltage is described by: e = lB x˙ (2) 189

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Where e is the induced voltage and x˙ 1 is the velocity of the bar. The current flowing along the electromagnetic circuit is found using Kirchoff’s laws: V −e (3) i= R The nature of the induced voltage is such that the induced voltage opposes the source voltage. As such, an increase in the induced voltage results in a decrease in current. To extend on the aforementioned, a decrease in the induced voltage results in an increase in current. The current is also dynamic in that it changes with a change in the velocity of the bar. For an ideal linear DC machine at steady-state, the induced voltage is constant. In reference to Eq.3, hence the current will also be constant. Remark: It should be observed that the dynamic nature of the current is due to the dynamic induced voltage. The induced voltage as seen in Eq.2 is a function of the velocity of the moving bar. The equation describing the motion of the bar follows the Newton’s laws and this is shown by: mt x ¨ − ilB = Fapp − FM L (Fapp , FL ) (4) Where the mt is total mass of the bar and mine trolley (mt = M + m), Fapp is the external force applied on the bar, FM L is the friction force which will be formulated by the author of this document extending the work done by Pacejka. The friction force as used by Chiasson [2005] is the viscous friction force. However, the friction force in this work will not assumed to be viscous friction but, the friction force will be represented as a function of the external applied force and the Lorentz force for a general case. The friction model formulation takes into consideration for the fact that some mine locomotives have external forces applied to them such as the gravitational force when the locomotive is on an inclined plane. A discussion on the friction model will be provided in the sections that follow. The physical model and mathematical model of the electromagnetic circuit analyzed in this document are presented. A description of the of these models is given. The mathematical model of the electromagnetic circuit is based on Lorentz law to determine the force exerted on the bar due to electromagnetic effects, Faraday’s law which states that the induced voltage is the rate of change of the magnetic flux is used to determine the the induced voltage, Kirchoff’s law is used to determine the current flowing in the electromagnetic circuit whilst, the Newton’s law of motion, in particular, the Newtons second law is used to model the motion of the moving bar. 4. LOCOMOTIVE FRICTION MODELING 4.1 Locomotive rolling friction When the wheels of the linear DC machine powered machine mine locomotive are in contact with the rail, rolling friction is introduced. The rolling friction model is formulated by Pacejka whose work is applied by Tsiotras [2000] and Nyandoro et al. [2011] . First, the total force applied on the locomotive is defined as: FA = Fapp + FL (5)

2019 IFAC MMM 186 Mohlalakoma T. Ngwako et al. / IFAC PapersOnLine 52-14 (2019) 183–188 Stellenbosch, South Africa, August 28-30, 2019

Where FA represents the total force applied on the locomotive. The rolling friction in this study is an extension of the work done by Pacejka and the rolling friction is described by: F A FA µ(FA ) = 2FA0 2 0 2 (6) FA0 + FA Where µ is friction coefficient. (7) Fr = µ(FA )FN Where Fr is rolling friction and FN is the normal force. 4.2 Bar sliding friction Up until now, the friction force mathematical model is represented as Ff r with the details of the model not being provided. The Dry friction force Kluge et al. [2015] is the force opposing the motion of the bar. The effects of the friction force are present when the Lorentz force or an external applied force (such as gravity) acts on the bar. Hence, the model formulated for the friction force is such that it is a function of the Lorentz and an external applied force. The friction force formulation is obtained using curve fitting where the static and kinetic friction are represented using a piece-wise function. Reference Hashiguchi and Ozaki [2008] tells us that the friction coefficient of a body is high when a body at rest starts sliding, this is known as the static friction. As the body continues moving, the friction coefficient starts decreasing up until a stationery value is reached, at this stage, the kinetic friction is reached Hashiguchi and Ozaki [2008]. The idea of the friction force dynamic model is an extension of the approach taken from Nyandoro et al. [2011], Tsiotras [2000]. The dynamic friction model formulated in this study is an extension of Pacejka’s friction model and the LuGre model Tsiotras [2000], Korondi et al. [2014], Astrom and de Wit [2008]. The Pacejka’s friction model used to represent rolling friction is such that the friction is a function of the slip coefficient and the model is formulated assuming steady state conditions Tsiotras [2000]. However, it should be noted that the model in this study does not have slip. The LuGre model is a dynamic model, and it represents the Stribeck, Coulomb and static friction Korondi et al. [2014], Astrom and de Wit [2008]. The resulting dynamic friction model representing the sliding friction is described by: 0 ≤ F ≤ FN Fo F A k 1 Ff r (FA ) = FN ( F A ) 2 + k FN FN ≤ F ≤ ∞ FN

3

Where Ff r (FA ) is the sliding friction force, FN is the threshold static friction force (also the normal force) and kn are constants.

In fig.3 a dynamic friction curve for friction force based on the Pacejka and LuGre is shown. The solid lines on the curve represent sliding friction while the dotted lines on the curve represent rolling friction. Tsiotras states that the friction force in real life systems is a function of the speed and normal force Tsiotras [2000]. The aforementioned is in accordance to Amonton whose work shows that the friction force is proportional to the normal force and also 190

Reynolds, Dahl and LuGre’s work where friction force is a function of velocity van Geffen [2009], Astrom [1995], Dahl [1968], Liu et al. [2015]. The friction model formulated in this study incorporates the speed of the bar (the current is a function of the speed) and the normal force, thereby mimicking practical systems. From the curve provided, it is deduced that the friction force opposing the motion of the bar starts by increasing with an increase in total applied force (FA ) until the threshold friction force is reached. The threshold friction force is the peak friction force observed on the dynamic curves. The friction regime from when there is no applied up until the peak friction is reached represents the static friction. As the applied force continues increasing, the friction force starts decreasing. As the friction force leaves the static friction regime, the friction force enters the kinetic friction regime where in this case, the kinetic friction is represented by a decrease in friction force starting from the peak friction. The results obtained from the friction formulation are in agreement with Hashiguchi and Ozaki [2008], Nyandoro et al. [2011], Tsiotras [2000]. It is worthwhile to mention that the results of the dynamic friction curve comply with the findings by Amonton, Stribeck, Coulomb, Dahl and most importantly the LuGre control group. Remark: To re-emphasize the correctness of the friction model, the results in fig.3 show that the friction force decreases with an increase in velocity. This complies with the results from work done in other friction studies Hashiguchi and Ozaki [2008], Nyandoro et al. [2011], Tsiotras [2000]. Another important observation in fig.3 is that the sliding friction force (denoted by solid lines) is greater that the rolling friction force (denoted by dotted lines) for the same load. The linear DC machine has a bar that slides in principle, which means a high friction force. To minimize the friction force, the mine locomotive should have wheels. This would mean that the linear DC machine would be used to pull the wheel of the mine locomotive. This shown in the dynamic friction force curve where the maximum friction force for a full load of a mine locomotive based solely on sliding is 1.1 pu. For a mine locomotive with wheels, the sliding friction of the linear DC machine should not be ignored hence, the full load friction force would be the sum of rolling friction full load and no load sliding friction. It worthwhile to mention that the no load friction force is represented as 0.05 of load for illustration purposes, this is because the mine locomotive still experiences a friction force opposing its motion even when it is not loaded. Therefore the maximum friction force would be 0.79 pu. Therefore, a mine locomotive with wheels is preferred. The next step would be to approximate the desired friction model based on the results shown in fig.3. The desired friction model is a combination of sliding friction and rolling friction. The friction force of a bar sliding on a rail has thus been formulated using extension methods of the analysis provided by Pacejka and LuGre friction models. An analysis and curve of the friction model formulated is in agreement of the findings by LuGre.

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Fig. 3. Dynamic friction curve 4.3 Desired locomotive friction model The resulting friction model of the mine locomotive is the sum of the no load linear DC sliding friction force of the bar and the rolling friction force. (9) FM L = Ff rN L (FA ) + Fr Where FM L is the resulting friction force of the mine locomotive and Ff rN L (FA ) is the no load sliding friction force.

Coulomb, the friction force is in the direction opposing the motion of the bar. Hence, the velocity where friction force is present is slightly lower than for a case where there is no friction force. As a result of the lower velocity in a system with friction force, the steady state current in fig.6 is slightly higher compared to a case where there is no friction force. This is because the current is a function of the velocity as shown by Eq.3. The equation shows that the current decreases with an increase velocity whilst the current also increases with a decrease in velocity. Also, the displacement of the linear DC machine shown in fig.5, in the presence of friction force is lower due to the bar moving at a lower velocity. Another interesting thing deduced from fig.7 is that the load has an effect on the friction force. In fig.7, the velocity dynamics of the mine locomotive are shown for load ranges of (no − load, 0.25 × (f ullload), 0.5 × (f ullload), 0.75 × (f ullload), (f ullload). It is deduced from these velocity dynamics that a loaded mine locomotive moves slower compared to a a no-load mine locomotive which has the highest velocity compared to all the other mine locomotives. The load is also found to affect the velocity of the locomotive and the conclusion is that the higher the load, the slower the motion of the mine locomotive.

4.4 Linear DC machine model Now that the dynamic friction model of the mine locomotive has been formulated, substituting the dynamic friction model into Eq.4 yields: m¨ x − ilB = Fapp − FM L

(10) Fig. 4. Velocity dynamics of the mine locomotive

5. SIMULATION RESULTS The simulation results are found in this section. It should be noted that a small-scale mine locomotive is simulated for purposes of proof of concept. The following parameters are used for simulation purposes: Table 1. mine locomotives parameters

Locomotive mass (kg) Bar mass (kg) Load (N) Magnetic field (B) Rail separation length (l) Resistor (R) Input voltage (V) Applied force (N)

1 kg 0.25 kg 12.25 N 0.5 T 0.6 m 100mΩ 1V 0N

Fig. 5. Displacement of the mine locomotive

The simulation results are a comparison of the linear DC behaviour for a case where there is friction force and when there is no friction force. The friction force formulated in sec.4 is in accordance with the work done by Amonton where the friction force is proportional to the normal force. From the results shown in fig.7 it is deduced that the linear DC machine has a lower steady state velocity when the friction force is incorporated in the system model. By 191

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Fig. 6. Current dynamics of the linear DC machine

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Fig. 7. Velocity dynamics of the mine locomotive for variable loads Variable voltage results of the mine locomotive are also presented. The voltage is varied from 1 V - 3 V. The results show that the system with a voltage of 1 V, 2V, 3 V respectively, moves at 37.5%, 15.6% and 12%, respectively slower than the no-load mine locomotives. This justifies the need for a comprehensive friction model.

Fig. 8. Velocity dynamics of the mine locomotive for variable voltage input 6. CONCLUSION A friction model for a linear DC machine powered mine locomotive is formulated and presented. The friction model accounts for the sliding friction of the a linear DC machine and the rolling friction of the wheels of the locomotive. The linear DC machine model by Chiasson [2005] is extended with a detailed friction model. Simulations result show that the locomotive tends to move slower when the friction force effects are taken into consideration and that a friction model is needed especially when modelling smallscale systems. Further research work will include stick-slip effects, thermal modeling and practical implementation of the locomotive. REFERENCES Astrom, K. (1995). Control of systems with friction. 1–8. Swedish Research Council for Eng. Science. Astrom, K. and de Wit, C.C. (2008). Revisiting the lugre model stick-slip motion and rate dependence. BME MOGI. Boldea and Nasar, S. (1985). Linear motion electromagnetic systems. John Wiley Sons, Inc. Chapman, S. (2012). Electric machinery fundamentals. 191–264. Mcgrawhill. Chiasson, J. (2005). Modeling and high performance control of electric machines. 1–400. IEEE Power engineering. 192

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