Considerations for Determining the Coefficient of Inertia Masses for A

sensors

Article Considerations for Determining the Coeﬃcient of Inertia Masses for a Tracked Vehicle

1 1 1 2, Octavian Alexa , Iulian Coropet, chi , Alexandru Vasile , Ionica Oncioiu * 1 and Lucian S, tefănit, ă Grigore 1 Military Technical Academy “FERDINAND I”, 39-49 George Cos, buc Av., 050141 Bucharest, Romania; [email protected] (O.A.); [email protected] (I.C.); [email protected] (A.V.);

[email protected] (L.S, .G.) 2 Faculty of Finance-Banking, Accountancy and Business Administration, Titu Maiorescu University, 040051 Bucharest, Romania * Correspondence: [email protected]; Tel.: +40-372-710-962

Received: 8 August 2020; Accepted: 28 September 2020; Published: 29 September 2020

Abstract: The purpose of the article is to present a point of view on determining the mass moment of inertia coefficient of a tracked vehicle. This coefficient is very useful to be able to estimate the performance of a tracked vehicle, including slips in the converter. Determining vehicle acceleration plays an important role in assessing vehicle mobility. Additionally, during the transition from the Hydroconverter to the hydro-clutch regime, these estimations become quite difficult due to the complexity of the propulsion aggregate (engine and hydrodynamic transmission) and rolling equipment. The algorithm for determining performance is focused on estimating acceleration performance. To validate the proposed model, tests were performed to determine the equivalent reduced moments of inertia at the drive wheel (gravitational method) and the main components (three-wire pendulum method). The dynamic performances determined during the starting process are necessary for the validation of the general model for simulating the longitudinal dynamics of the vehicle. Finally, the differential and algebraic equations of the virtual model approximate more accurately the actual process of the operation of the vehicle. The virtual model, through the data obtained from the simulation process, allows for the determination, indirectly, of the variation of the mass moment of inertia coefficient and its expression of approximation.

Keywords: tracked vehicle; mobility; stability; acceleration; torque converter; engine; coeﬃcient of inertia masses; hydro-clutches

1. Introduction The acceleration resistance generated by the inertial forces during the starting process of the vehicle depends not only on its weight but also on the rotating masses, from the traction engine elements to the vehicle tracks [1]. According to the relation of the moments of inertia δ [-] [2], the resistance generated by the inertia of the rotating elements depends not only on the mass but also on their radius. In the literature [3–5], the effect generated by the moving masses of translation and rotation is materialized in the form of the mass moment of inertia coefficient δ [6]. The mobility and stability of a vehicle on wheels or tracks are influenced by the moment of inertia [7] but also by the place (on the ground) where gravity acts [8,9]. In [10] is presented a database of the National Highway Traffic Safety Administration (NHTSA), which presents a methodology for calculating not only moments of inertia and center of gravity but also the physical characteristics of the vehicle.

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tracks [10–12]. The Matlab programming environment—SIMULINK and SIMSCAPE modules—was usedThe to developmentdevelop the ofgeneral the virtual simulation simulation model model. The aimed simulation at modeling model, the main which parts refers of a trackedto the vehicle,determinationSensors such 2020,as 20of, the xthe FOR 8power PEER V engine, REVIEW flow, theis built hydromechanical modularly, and transmission, the data on the the characteristics final transmission, of the2 of physical and 35 the trackscomponents, [10–12 ].used The for Matlab the construction programming of the environment—SIMULINK hydromechanical transmission, and SIMSCAPE are found modules—was in [13,14]. The usedmodular-typetracks to develop [10–12]. simulation the The general Matlab model simulation programming was generated model. envi Theronment—SIMULINK due simulation to the fact model, that itand whichwill SIMSCAPE be refers able to modules—was be the modified determination later, ofdepending theused power to on flow,develop the ischaracteristics builtthe general modularly, ofsimulation the and respecti the model datave on.components, The the simulation characteristics which model, come of thewhich from physical refers othercomponents, toequipment the determination of the power flow, is built modularly, and the data on the characteristics of the physical usedmanufacturers. for the construction of the hydromechanical transmission, are found in [13,14]. The modular-type components, used for the construction of the hydromechanical transmission, are found in [13,14]. The simulation model was generated due to the fact that it will be able to be modified later, depending on modular-typeThe virtual simulationsimulation model model was ha generateds as input due elements to the fact the that experi it will bemental able to data be modified provided later, by the themanufacturer, characteristicsdepending theon the ofmain thecharacteristics mass respective and dimensional of components, the respecti characteristicsve which components, come fromof which the other vehicle, come equipment from gearshift other manufacturers. times,equipment obtained experimentallymanufacturers.The virtual after simulation experimental model determination has as input of elementsthe pressure the variation experimental in the data hydraulic provided control by thesystem manufacturer, atThe changing virtual the simulationgears main [3], mass andmodel the and ha positions dimensional as input of elementsthe characteristicsaccelerator the experi pedal.mental of the The data vehicle, output provided gearshiftelements by the times,of the obtainedvirtualmanufacturer, model, experimentally generated the main after aftermass experimental andrunning dimensional the determinationprogra characteristicsm (simulation of of thethe pressurevehicle,process), gearshift variation show times, the in variation theobtained hydraulic over controltime experimentallyof systemthe main at changingdynamic after experimental characteristics gears [3], determination and theof the position tracked of the of pressure thevehicle accelerator variationand of the pedal. in themass hydraulic The moment output control of elements inertia ofcoefficient, thesystem virtual andat model,changing are shown generated gears in [3], Figure and after the1. running position the of programthe accelerator (simulation pedal. The process), output elements show the of variation the overvirtual time of model, the main generated dynamic after characteristics running the progra of them tracked (simulation vehicle process), and of show the mass the variation moment over of inertia time of the main dynamic characteristics of the tracked vehicle and of the mass moment of inertia coefficient, and are shown in Figure1. coefficient, and are shown in Figure 1.

Figure 1. General virtual simulating model of the tracked vehicle [1]. FigureFigure 1. 1.General General virtual virtual simulatingsimulating model model of of the the tracked tracked vehicle vehicle [1]. [1]. The evaluation of the parameters necessary to determine the moments of inertia by the The evaluation of the parameters necessary to determine the moments of inertia by the gravitationalThe evaluation method of Figure the parameters 2 is based necessary on detaching to determine the tracks the momentsfrom the ofdrive inertia wheel by theand gravitational attaching a methodgravitational Figure2 methodis based Figure on detaching 2 is based theon detaching tracks from the tracks the drive from wheel the drive and wheel attaching and attaching a cylindrical a cylindricalcylindrical device. device. Its Itsrole role is isto to wind wind a a cable cable around it,it, which which has has a weighta weight hanging hanging from from the other the other device. Its role is to wind a cable around it, which has a weight hanging from the other end. end. end.

Figure 2. Schematic for the installation for measuring the moments of inertia of the transmission of a tracked vehicle, using the gravitational method. Figure 2. Schematic for the installation for measuring the moments of inertia of the transmission of a tracked vehicle, usingusing thethe gravitationalgravitational method.method.

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The weight is hung by means of a pulley of a crane-type support. The working principle is as follows: the driver connects a stage of the hydromechanical transmission with an external pressure source, releases the brakes and the weight begins to fall. Knowing the weight value, the radius of the cylindrical device, the distance traveled by the weight and the fall time, analytical calculations can be performed to determine the moment of inertia [15]. Performing the same operations for both drive wheels will make it possible to determine the overall moment of inertia. The experimental determination of the mass moment of inertia coefficient [16] aims to highlight its implications for the dynamic performance of the vehicle, in particular for the variation of speed, acceleration and space traveled during the starting process. Another parameter necessary to be verified is the dependence of the mass moment of inertia coefficient on the transmission ratio or the slips in the Hydroconverter. The measured values of the moments of inertia of the parts that make up the main assemblies of the vehicle, which are in rotational motion, become input data in the general model for simulating the longitudinal dynamics. Based on these, the differential and algebraic equations of the virtual model approximate more precisely the actual operation process of the vehicle. There are a multitude of models for predicting longitudinal dynamics [17]. These models are found between the following extremes: the model regarding the mobility of vehicles on tracks [18] and the one that studies the interaction of the wheels (tracks) with the ground [19–24]. The final model (1) resulted from the processing of models for wheeled vehicles, as there is no unitary model for tracked military vehicles. Using the SIMSCAPE programming language, an own library was developed, which made the dynamic connection between armored housing, crawler propeller and ground. This model calculates, among other things, the forward resistances, which were not found in the Matlab toolbox. Based on these models, fast and robust algorithms were developed for evaluating acceleration performance and speeds under certain conditions. Tanks are complex structures, which create difficulties in establishing models. Among the algorithms developed and implemented in software, we mention: MOSES [21], NTVPM [22–24] and the least square method (LSM) [25]. The main new ideas of this paper can be summarized as follows: the building of a virtual model for the operation simulation of the accelerated motion regime of a hydromechanical transmission, monobloc, which equips a vehicle in operation; obtaining the analytical–numerical relation of the variation of the coefficient of inertia masses (δ) as a function of the pump speed; introduction of the correlation coefficient for an experimentally obtained quantity; development of two experimental methods for determining the coefficient of inertia masses; the concept developed and validated experimentally can be applied even in real operating conditions specific to military applications. The paper is structured as follows: Section2 addresses the hydrodynamic transmission model. Section3 presents the algorithm for the analytical determination of the coe fficient of inertia masses. Section4 provides the model for simulating the variation of the coe fficient of inertia masses. The experimental methods used to determine the same coefficient are presented in Section5. Section6 presents future developments. Finally, the conclusions of this paper are given in Section7.

2. Background To estimate the longitudinal dynamics of vehicles on tracks equipped with hydromechanical transmissions, a virtual model of operation for the accelerated motion regime was developed. The transmission is of hydromechanical type, monobloc. The gears are engaged under load by actuating friction elements in the gearbox. Power is transmitted on one ﬂow while driving in a straight line and on two power ﬂows while turning. Rectilinear driving and turning are performed according to the graph representing the evolution for turn commands in Table1. Sensors 2020, 20, 5587 4 of 31 Sensors 2020, 20, x FOR PEER REVIEW 4 of 35

Table 1.1. Graph representing thethe evolutionevolution forfor turnturn commands.commands.

ExecutionExecution Elements Elements

ASS ASD ARM ARm ASS ASD ARM ARm Rectilinear Rectilinear Turn right bigTurn radius right big radius Progression ProgressionTurn right smallTurn radius right small radius Turn left big radiusTurn left big radius Turn left smallTurn radius left small radius Rectilinear Rectilinear Turn right bigTurn radius right big radius Reversing ReversingTurn right smallTurn radius right small radius Turn left big radiusTurn left big radius Turn left smallTurn radius left small radius

According toto the the general general kinematic kinematic diagram diagram (Figure (Figure3), the 3), hydromechanical the hydromechanical transmission transmission consists of:consists input of: mechanism input mechanism Mc_In, distribution Mc_In, distribution mechanism mechanism I MD I, distribution I MD I, distri mechanismbution mechanism II MD II, II drive MD mechanismII, drive mechanism of the hydraulic of the coupling hydraulic of thecoupling MACH of fan, the intermediate MACH fan, mechanism intermediate Mc_I, mechanism Hydroconverter Mc_I, HC,Hydroconverter inverter mechanism HC, inverter MI, CVP mechanism planetary MI, gearbox, CVP planetary MV turning gearbox, mechanism MV turning and right mechanism and left MID and andright MIS and planetary left MID and summation MIS planetary mechanisms. summation mechanisms.

Figure 3. Kinematic diagram of the hydromechanical transmission [[4].4].

3. Theoretical Considerations Regarding Mass Moment of Inertia CoeCoefficientﬃcient δ δ The algorithm for determining the analytical expressionexpression that approximates the mass moment of inertiainertia coecoefficientﬃcient is basedbased onon mathematicalmathematical models that describedescribe the dynamicdynamic operation of the main subsystems of a vehicle.vehicle. In order to carry out the calculations, it is necessary to elaborate the general nodal scheme of the vehicle,vehicle, composedcomposed ofof individualindividual nodalnodal schemesschemes (of(of thethe substructures)substructures) [[25].25]. There areare twotwo variants variants of of obtaining obtaining the the analytical analytic expressional expression of the of dynamic the dynamic moment moment at the wheel:at the onewheel: in aone hydrodynamic in a hydrodynamic regime regime (Figure 4(Figure) and one 4) and in a one mechanical in a mechanical regime (Figure regime5 (Figure). 5). The solution of operation in a hydrodynamic regime involves the application of the fundamental laws of the network model on the motor circuit–transmission–drive wheel.

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FigureFigure 4 4. Generalized nodal scheme of a vehicle on tracks in the case of hydromechanical operation.

Figure 4

Figure 5. Generalized nodal scheme of a vehicle on tracks in the case of mechanical operation.

The solution of operation in a hydrodynamic regime involves the application of the fundamental laws of the network model on the motor circuit–transmission–drive wheel. The relation for determining the torque at the wheel in the case of a hydrodynamic regime is given by (1) [4]: " !# dωp dω ( ) 2 ( ) ( ) r [ ] Merm = Mrm i icd ηcd i ITR i + IEP Kh Nm (1) i − i· · · · dωt · dt where: (M ) = (i ) i i (η ) η η M [Nm], (i ) = (i ) i i [ ], rm i cv i mi dl cv i mi dl Ht " cd i # cv i mi dl · · · h · i· · h i· · − 2 2 Ib5 2 (ηcd)i = (ηcv)i ηmi ηdl [ ], IEP = Ib4 i ηR kg m , (ITR)i = Ii + Imi + 2 kg m . R (i ) (ηcv) · · − · · · cv i· i · The second solution for operation in a mechanical regime involves following a diﬀerent route composed of ground–tracks–drive wheel. The relation for determining the momentum at the drive wheel in the case of a mechanical regime is given by (2): 2 dωr Merm = (Mrm) i (ηt) (It) [Nm] (2) IVm IVm − t IV· IV· IVm· dt

dωr Merm = eFp rrm + Ipr + Rpr rrm [Nm] (3) · · dt · where: (it) = iR (icv) i i [ ]; (ηt) = ηR (ηcv) η η [ ]. IV · IV· mi· dl − IV · IV· mi· dl − For the operation of the vehicle in a mechanical regime, the expression for determining the torque in dynamic mode is given by (3). From Equations (1)–(3), the following relations can be deduced for the dynamic traction force:

Ipr dv eF = eF + + R = R + R + R + F [N] (4) t p 2 pr s α a jA rrm · dt where F [N] is the load factor of the inertial ﬂow of the vehicle, jA 1 dv F = (M + ms) [N] (5) jA · dt and for the two modes of operation we will have:1

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dv FMmN=+⋅()[] (5) jA ş dt and for the two modes of operation we will have: Sensors 2020, 20, 5587 6 of 31 • hydromechanical operation: 2 ⋅η() ()icd cd dω hydromechanical operation: i i p dv • ()FF=−() ⋅() I +⋅⋅⋅ IK[] N (6) tti ii2  TREPhω rddtrm  t i2 (η ) " # cd i cd i dωp dv eFt = (Ft) · (I ) + I K [N] (6) • i i 2 TR i EP h mechanical operation—HC blocked:− rrm · · · dωt · dt

mechanical operation—HC blocked: ()i2 ⋅η() • cd cd IV dv ()FF =−() IV ⋅⋅() I[] N (7) ttIVm IVm 2 t IVm i2rdtrm (η ) cd IV cd IV dv eFt = (Ft) · (It) [N] (7) IVm IVm 2 IVm Since the power flows from the engine and− the onerrm from the· track· dt must be equal, this results in (4) = (6) = (7), which is the equation of dependence between the traction force specific to the steady stateSince of operation the power and flows the dynamic from the propulsion engine and forc thee. one By from replacing the track the mustdynamic be equal, propulsion this results force inin (4)the= dependency(6) = (7), which equation, is the equationthe analytical of dependence expression betweenof the traction the traction force is force specific specific to the to thestationary steady statemode of of operation operation and and the the dynamic results. propulsion On the other force. hand By replacing, the traction the dynamic force specific propulsion to the force stationary in the dependencyoperating regime equation, is also the defined analytical by th expressione equation of of the the traction traction force balance: is specific to the stationary mode of operation and the results. On the other hand, the traction force specificdv to the stationary operating regime is also defined by the()FRRRR equation=++++δ⋅⋅ of the tractionα balance: () M[] N (8) tprsii a adt dv Identifying the terms of( Fthet) two= Rdefiningpr + Rs + relationsRα + Ra of+ the(δ) tractionMa force[N] in the stationary operating(8) i i· · dt regime (Figure 6), the analytical expression of (δ) for the mechanical and hydromechanical operation is determined:Identifying the terms of the two defining relations of the traction force in the stationary operating regime (Figure6), the analytical expression of ( δ) for the mechanical and hydromechanical operation ()i2 ⋅η() is determined: mIş pr tt IV ()δ=++ +2 IV ⋅() [] − 1 i (ηt) It (9) IVm ms ⋅⋅Ipr22t IV IV IVm (δ) =1 +MMraarmarm+ + Mr· (I )  [ ] (9) IVm  2 2 t IVm  Ma Ma r Ma r ·  − · rm · rm 2 mI()i 2⋅η()" dω# ş prI cdii ( cdηcdi) d p ()δ=+1 m +s pr +cd i i ⋅()IIK + ⋅ ⋅ωp [] − (10) (δii) = 1 + + 22+ · (ITR) + I EPK h [ ] (10) i MMr⋅⋅2 Mr2 TR i EP h d ω aarmarmMa Ma rrm Ma rrm ·· · dωt t− · ·

FigureFigure 6.6 Dependence between pump speed and turbine speed.

FollowingFollowing thethe replacementreplacement ofof thethe expressionsexpressions forfor thethe angularangular velocitiesvelocities of thethe pumppump (Figure(Figure7 7)) andand thethe turbineturbine (Figure(Figure8 )8) in in (9) (9) and and (10), (10), the the ﬁnal final relation relation of ofδ δfor for hydromechanical hydromechanical operation operation is: is: 2 ( ) " # m Ipr icd ηcd i dnp (δ) = 1 + s + + i· (I ) + I K [ ] (11) i 2 2 TR i EP h Ma Ma r Ma r · · · dnt − · rm · rm Sensors 2020, 20, x FOR PEER REVIEW 7 of 35

()i2 ⋅η() mIş prcd cd i  dn p ()δ=+1 + +i ⋅()IIK + ⋅ ⋅[] − (11) ii⋅⋅22TR EP h MMraarmarm Mr dn t  ()= ( ) nps n tinterp cspline n t , n c , n t , n c  2 ()i ⋅η() ()  mIprcd cd i  dnn ps t (12) δ=++()nIIKn1 ş +i ⋅() +⋅⋅ () [] − tTREPhsti ⋅⋅22i  MMraarmarm Mr dn t

The variation (δ) is not important for speeds between 0 and 700 rpm, regardless of the floor on which the vehicle operates (Figure 7). During this interval, there are large slips in the hydroconverter, which corresponds to the specific situation of starting the vehicle. For speeds in the range of 700–2160 rpm, (δ) increases. This range corresponds to the area in which the aggregate operates as a hydroconverter. An area in which the value (δ) is decreasing appears during the transition from Sensorshydroconverter2020, 20, 5587 to clutch mode. This is due to the decrease in the slip rate of the pump in relation7 of to 31 the turbine (expressed by the derivative of the pump speed in relation to the turbine speed), as a result of the transition of the hydro unit from the hydroconverter to clutch mode. Since we have an  nps(nt) = interp[cspline(nt, nc), nt, nc]  MBS, we consider, for the study of dynamismI (i2 perf) (ηormances,) that the algorithmsdn (n ) for identifying(12) the  ms pr cd i· cd i ps t  [δ(nt)] = 1 + + 2 + 2 (ITR) + IEP Khs(nt) [ ] parameters, namely thei LSM, areMa easyMa rto implement.Ma r This methodi allows usdn tot update the current · rm · rm · · · − estimates of the tracked parameters [25].

Figure 7. The dependence between the transformation ratio and turbine speed. Sensors 2020, 20, x FORFigure PEER 7. REVIEWThe dependence between the transformation ratio and turbine speed. 8 of 35

The “atypical” variation of (δ) is also due to the mode of transition from one regime to another of the transformation ratio function. The turbine speed is directly proportional to the speed of the vehicle, therefore, the dependence (δ) can also be expressed as a function of the speed of the vehicle, → by changing the variable nvt , so that the speed of the vehicle can be determined according to turbine speed: π⋅nr ⋅ π⋅ nr ⋅ ()vv=→=trm() trmm ii30⋅⋅⋅()iii 30 ⋅() is (13) cvii m頳 lcd

FigureFigure 8. 8.VariationVariation of ofthe the mass mass moment moment of of inertia inertia coefficient coefficient (δ ()δ depending) depending on on the the turbine turbine speed. speed. The variation (δ) is not important for speeds between 0 and 700 rpm, regardless of the floor on which Therefore, the expression of (δ) can be defined as a function of v (Figure 9) for the four stages of the vehicle operates (Figure7). During this interval, there are large slips in the hydroconverter, which the CVP: corresponds to the specific situation of starting the vehicle. For speeds in the range of 700–2160 rpm, (δ) increases. This range corresponds to the area in which the aggregate operates as a hydroconverter. An area in which the value ( )() is2 decreasing⋅η() appears during the transition from hydroconverter to mIδ icd cd 1 dnv() δ=++() ş pr +i i ⋅() +⋅ () ⋅ ⋅ps [] − clutchvIIKv mode.1 This is due to the decrease in the slipTR rate of EP the pump hs in relation to the turbine (expressed (14) i MMr⋅⋅22 Mri 30⋅()i dv by the derivativeaarmarm of the pump speed in relation to the turbine speed), ascd ai result of the transition of ⋅π⋅ the hydro unit from the hydroconverter to clutch mode. Since we have3, 6 anrrm MBS, we consider, for the study of dynamism performances, that the algorithms for identifying the parameters, namely the The higher values of (δ) in the lower stages are due to the transmission ratio of the mechanical LSM, are easy to implement. This method allows us to update the current estimates of the tracked transmission and the equivalent reduced moment of inertia to the wheel axle ()I . parameters [25]. TR The ?atypical? variation of (δ) is also due to the mode of transition from one regime to another of the transformation ratio function. The turbine speed is directly proportional to the speed of the vehicle, therefore, the dependence (δ) can also be expressed as a function of the speed of the vehicle,

Figure 9. Variation of the mass moment of inertia coefficient depending on the speed of the vehicle.

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by changingFigure the variable 8. Variationn tof thev mass, so moment that the of inertia speed coefficient of the ( vehicleδ) depending can on be the determined turbine speed. according to → turbine speed: Therefore, the expression of (δ) πcann ber defined as a functionπ nof vr (Figurem 9) for the four stages of ( ) = t rm ( ) = t rm the CVP: v i · · v i · · (13) 30 (icv) i i → 30 (i ) s · i· mi· dl · cd i Therefore, the expression of (δ) can be deﬁned as a function of v (Figure9) for the four stages of mI()i2 ⋅η() dnv() the CVP: ş pr cdi cd i 1 ps δ=++()vIIKv1 + ⋅() +⋅ () ⋅ ⋅[] − (14) i MMr⋅⋅22 MrTRi EP hs 30⋅()i dv aarmarm  cd i  2  m Ipr i (ηcd)i  ⋅π⋅ dnps(v)  s cd i·  3, 6 rrm 1  [δ(v)] = 1 + + + (ITR) + IEP K (v)  [ ] (14) i 2 2  i hs 30 (i )  Ma Mδ a rrm Ma rrm · · · · cd i · dv  − The higher values of ( ) in· the lower stages· are due to the transmission3,6 ratioπ rrm of the mechanical ·()· transmission and the equivalent reduced moment of inertia to the wheel axle ITR .

FigureFigure 9. Variation 9. Variation of the of massthe mass moment moment of of inertia inertia coeffi coeﬃcientcient depending depending on the on speed the speed of the ofvehicle. the vehicle.

The higher values of (δ) in the lower stages are due to the transmission ratio of the mechanical transmission and the equivalent reduced moment of inertia to the wheel axle (ITR).

4. Concepts for Simulating the Variation of the Mass Moment of Inertia Coefficient δ In order to indirectly determine the mode of variation of (δ), some of the data obtained from simulation process of the longitudinal dynamics of a vehicle are needed, namely the signals specific to the terms that are found in the differential equation of motion:

dv g F(t)t Rprsaα(t) m F(t)t Rprsaα(t) a(t) = = − δ(t) = − [ ] (15) 2 dt δ(t) · Ga s ⇒ a(t) Ma − · The traction force signal modeling is performed by utilizing, with the SIMULINK programming language, the ratio between the torque at the drive wheel, determined as a result of the product between the signals speciﬁc to the torque at the engine M_util, the transformation ratio of the hydroconverter Kh, transmission ratios and element yields arranged between the engine and the drive wheel and the signal input for modeling the radius of the drive wheel rrm. The forward resistance force signal is obtained by summing the thrust resistance force R_propulsor, the rolling resistance force R_sol, the air resistance force R_aer and the climbing resistance force R_alpha (Figure 10). Longitudinal acceleration signal acceleration, that of the mass of the vehicle Ma, as well as the signals introduced by the blocks that model the traction force and the one of forward resistance, are used to model the expression of the mass moment of inertia coeﬃcient (Figure 11). Sensors 2020, 20, x FOR PEER REVIEW 9 of 35

4. Concepts for Simulating the Variation of the Mass Moment of Inertia Coefficient δ SensorsIn 2020 order, 20, xto FOR indirectly PEER REVIEW determine the mode of variation of (δ), some of the data obtained 9from of 35 simulation process of the longitudinal dynamics of a vehicle are needed, namely the signals specific 4. Concepts for Simulating the Variation of the Mass Moment of Inertia Coefficient δ to the terms that are found in the differential equation of motion: In order to indirectly determine the mode of variation of (δ), some of the data obtained from Ft()−− R () t Ft() R () t simulation process dvof the longitudinal g tt dynamicsprsaαα of a vehicle are needed, namely prsa the signals specific at()== ⋅ m  δ=() t [] − δ⋅() s2 () (15) to the terms that aredt found in t the differential Gaa equation of motion: a t M Ft()−− R () t Ft() R () t The traction forcedv signal g modelingtt is performedprsaαα by utilizing, with the SIMULINK prsa programming at()== ⋅ m  δ=() t [] − (15) language, the ratio betweenδ⋅() the torque at the drive swheel,2 determined ()as a result of the product dt t Gaa a t M between the signals specific to the torque at the engine M _ util , the transformation ratio of the The traction force signal modeling is performed by utilizing, with the SIMULINK programming hydroconverter Kh , transmission ratios and element yields arranged between the engine and the language, the ratio between the torque at the drive wheel, determined as a result of the product r drivebetween wheel the andsignals the specific signal inputto the fortorque modeling at the theengine radius M of_ util the , drivethe transformation wheel rm . The ratio forward of the resistance force signal is obtained by summing the thrust resistance force R_ propulsor , the rolling hydroconverter Kh , transmission ratios and element yields arranged between the engine and the resistance force Rsol_ , the air resistance force Raer_ and the climbing resistance force drive wheel and the signal input for modeling the radius of the drive wheel rrm . The forward R _ alpha (Figure 10). resistance force signal is obtained by summing the thrust resistance force R_ propulsor , the rolling

Sensorsresistance2020, 20force, 5587 Rsol_ , the air resistance force Raer_ and the climbing resistance 9force of 31 R _ alpha (Figure 10).

Figure 10. Modeling of traction force and forward resistance forces.

Longitudinal acceleration signal acceleration, that of the mass of the vehicle Ma, as well as the signals introduced byFigureFigure the blocks 10.10. Modeling that model of traction the trac forcetion and force forward and resistanceresistance the one forces.offorces. forward resistance, are used to model the expression of the mass moment of inertia coefficient (Figure 11). Longitudinal acceleration signal acceleration, that of the mass of the vehicle Ma, as well as the signals introduced by the blocks that model the traction force and the one of forward resistance, are used to model the expression of the mass moment of inertia coefficient (Figure 11).

Sensors 2020, 20, x FOR PEERFigure REVIEW 11. Modeling the mass moment of inertia coeﬃcient. 10 of 35 5. Experimental MethodsFigure Used 11. toModeling Determine the mass the Massmoment Moment of inertia of coefficient. Inertia Coe ﬃ cient δ 5. Experimental Methods Used to Determine the Mass Moment of Inertia Coefficient δ Experimental methods highlight the mode of variation of (δ) by determining the values of Experimental methods highlight the mode of variation of (δ) by determining the values of equivalentequivalent moments moments of of inertiaFigure inertia 11. entering enteri Modelingng Expression Expression the mass moment (16)(16) (Figure(Figure of inertia 12).12). coefficient.

FigureFigure 12. 12.The The variation variation graph graph ofof ((δδ)) duringduring the simulation simulation of of the the starting starting process. process.

For the experimental determination of the equivalent reduced moments of inertia values Ipr, IEP, ITR and It, two methods were used: the gravitational one and the three-wire suspension.  2 ⋅η() mI()icd cd  dω δ= +ş +pr +i i ⋅() + ⋅ ⋅ p [] − iTREPh1 22IIK  MMr⋅⋅ Mri d ω  aarmarm t (16) ()i2 ⋅η()  mIş pr tTIV δ=++1 +IV ⋅()I [] −  IVm⋅⋅22 t IVM  MMraarmarm Mr

5.1. The Gravitational Method The gravitational method is a means of determining the equivalent reduced moment of inertia at the drive wheel of all rotating parts, arranged between the Hydroconverter turbine and the drive wheel. The equipment and materials used consist of an RS 38 incremental encoder, an encoder mounting device, a Monarch-type ATC frequency meter, a DMC 9012A measuring bridge, a mobile stand to check the hydraulic installation, a Power Book 1400cs portable electronic computer, connection cables, extension cords and power cables from the 220 V mains, a drum, a stopwatch and an experimental assembly (Figure 13). The experimental assembly consists of a metal frame, a pulley, and a metal box in which marked weights are inserted. The cable is attached at one end to the metal box and at the other end by a drum fixed to the gear of the drive wheel. The drum is designed to avoid uneven winding of the cable and the friction of the cable by the structural elements of the vehicle.

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For the experimental determination of the equivalent reduced moments of inertia values Ipr, IEP, ITR and It, two methods were used: the gravitational one and the three-wire suspension.  I (i2 ) (η ) dω  = + ms + pr + cd i· cd i ( ) + p [ ]  δi 1 M 2 2 ITR i IEP Kh dω  a Ma rrm Ma rrm · · · t − · 2· (16)  I (i ) (ηT)  ms pr t IV· IV  δIVm = 1 + + 2 + 2 (It) [ ] Ma Ma r Ma r IVM · rm · rm · − 5.1. The Gravitational Method The gravitational method is a means of determining the equivalent reduced moment of inertia at the drive wheel of all rotating parts, arranged between the Hydroconverter turbine and the drive wheel. The equipment and materials used consist of an RS 38 incremental encoder, an encoder mounting device, a Monarch-type ATC frequency meter, a DMC 9012A measuring bridge, a mobile stand to check the hydraulic installation, a Power Book 1400cs portable electronic computer, connection cables, extension cords and power cables from the 220 V mains, a drum, a stopwatch and an experimental assembly (Figure 13). Sensors 2020, 20, x FOR PEER REVIEW 11 of 35

Figure 13. The experimental device used for determination of (δ). Figure 13. The experimental device used for determination of (δ). To measure the speed of the drive wheel, an encoder is mounted on an adjustable support on The experimental assembly consists of a metal frame, a pulley, and a metal box in which marked the drive wheel on the other side of the vehicle (Figure 14). weights are inserted. The cable is attached at one end to the metal box and at the other end by a drum ﬁxed to the gear of the drive wheel. The drum is designed to avoid uneven winding of the cable and the friction of the cable by the structural elements of the vehicle. To measure the speed of the drive wheel, an encoder is mounted on an adjustable support on the drive wheel on the other side of the vehicle (Figure 14).

Figure 14. Mounting the RS-38 optical encoder on the right-hand drive wheel.

The driving of the RS38 optical encoder is performed by means of a connecting shaft, provided with an elastic coupling, compensating for coaxial deviations, connected to the shaft of the final transmission. In Figure 15 we represent the measuring chain.

Sensors 2020, 20, x FOR PEER REVIEW 11 of 35

Figure 13. The experimental device used for determination of (δ). Sensors 2020, 20, 5587 11 of 31 To measure the speed of the drive wheel, an encoder is mounted on an adjustable support on the drive wheel on the other side of the vehicle (Figure 14).

Figure 14. Mounting the RS-38 optical encoder on the right-hand drive wheel. Figure 14. Mounting the RS-38 optical encoder on the right-hand drive wheel. The driving of the RS38 optical encoder is performed by means of a connecting shaft, provided Thewith driving an elastic of thecoupling, RS38 compensating optical encoder for coaxia is performedl deviations, by connected means of to a connectingthe shaft of the shaft, final provided with antransmission. elastic coupling, In Figure compensating15 we represent the for measuring coaxial deviations,chain. connected to the shaft of the ﬁnal Sensors 2020, 20, x FOR PEER REVIEW 12 of 35 transmission. In Figure 15 we represent the measuring chain.

δ FigureFigure 15. 15.Measuring Measuring chain chain ((δ):): DMC 9012A 9012A bridge, bridge, frequency frequency meter, meter, laptop. laptop.

5.1.1.5.1.1. The The Classical Classical Gravitational Gravitational Method Method The testsThe tests are are performed performed under under the the following following condit conditions:ions: the the engine engine is stopped is stopped and anddisconnected disconnected fromfrom the transmission.the transmission. As As stopping stopping the the engineengine results results in in a alack lack ofof pressure pressure in the in thehydraulic hydraulic control control system, an external pressure source is used to connect the transmission gears (Figure 16). system, an external pressure source is used to connect the transmission gears (Figure 16).

Sensors 2020, 20, 5587 12 of 31 Sensors 2020, 20, x FOR PEER REVIEW 13 of 35

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FigureFigure 16.16. ExternalExternal gearboxgearbox couplingcoupling system.system. Figure 16. External gearbox coupling system. TheThe testtest consistsconsists inin thethe descentdescent ofof the metal box, in a uniformly accelerated regime, withwith aa known The test consists in the descent of the metal box, in a uniformly accelerated regime, with a known accelerationacceleration aa,, untiluntil itit crossescrosses thethe heightheight HH andand reachesreaches thethe ground.ground. The rotational motionmotion ofof thethe drivedrive acceleration a, until it crosses the height H and reaches the ground. The rotational motion of the drive wheel and the rest of the transmission components has the character of a uniformly accelerated motion, wheelwheel and andthe therest rest of ofthe the transmission transmission componencomponentsts hashas the the character character of ofa uniformly a uniformly accelerated accelerated characterizedmotion,motion, characterized characterized by the angular by by the the accelerationangular angular acceleration accelerationε (Figure εε 17 (Figure(Figure). 17). 17).

Figure 17. The classical gravitational method, the representation of forces and moments. Figure 17. The classical gravitational method, the representation of forces and moments.

Figure 17. The classical gravitational method, the representation of forces and moments.

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Thus, we can deﬁne the equations that describe the two uniformly accelerated movements, of rotation and translation, for two diﬀerent loads, m1 and m2:   m1 g T1 = m1 a1 m2 g T2 = m2 a2  · − · · − ·  Iechiv_rm ε1 = Mt1 M f r Iechiv_rm ε2 = Mt2 M f r [Nm] (17)  · − · −  M = T r Mt = T r t1 1· 2 2· The following assumptions are made: the losses generated by friction are identical regardless of the value of the weight of the metal box and the movement of the box, the drum is uniformly accelerated, and the initial speed is zero. The hypotheses show:  h i h i  a = 2H m a = 2H m  1 2 s2 2 2 s2  t t2  1 h i h i (18)  ε = 1 2H 1 ε = 1 2H 1  1 r t2 s2 2 r t2 s2 · 1 · 2

From Relations (17) and (18), the expression of the equivalent moment of inertia is found, reduced to the drive wheel: 2 2 2 2 2 2 r t t (G G ) r G2 t G1 t h i 1 2 2 1 · · 1 − · 2 2 Iechiv_classical method = · · · − kg m (19) 2 H t2 t2 − g t2 t2 · · · 1 − 2 · 1 − 2 The experimental data (Table2) are entered in Relation (19) and the values corresponding to the equivalent moments of reduced inertia at the drive wheel are obtained (Table3).

Table 2. Experimental data obtained by applying the classical gravitational method.

CVP Gear Sample Number Mass (m1) (kg) Mass (m2) (kg) Time (t1) (s) Time (t2) (s) 1a_gr1 500 — 160.0 — 2a_gr1 500 — 160.4 — 3a_gr1 500 — 161.1 — I gear 1b_gr1 — 600 — 34.90 2b_gr1 — 600 — 34.90 3b_gr1 — 600 — 34.65 1a_gr2 500 — 144.0 — 2a_gr2 500 — 143.6 — 3a_gr2 500 — 146.2 — II gear 1b_gr2 — 600 — 31.80 2b_gr2 — 600 — 28.50 3b_gr2 — 600 — 27.85 1a_gr3 500 — 124.0 — 2a_gr3 500 — 114.4 — 3a_gr3 500 — 118.7 — III gear 1b_gr3 — 600 — 24.05 2b_gr3 — 600 — 24.35 3b_gr3 — 600 — 22.15 1a_gr4 500 — 7.460 — 2a_gr4 500 — 7.798 — 3a_gr4 500 — 7.929 — IV gear 1b_gr4 — 600 — 4.1 2b_gr4 — 600 — 3.6 3b_gr4 — 600 — 3.9 1a_neu 500 — 6.12 — 2a_neu 500 — 6.78 — 3a_neu 500 — 6.43 — Neutral 1b_neu — 600 — 3.12 2b_neu — 600 — 3.00 3b_neu — 600 — 3.20 Sensors 2020, 20, 5587 14 of 31 Sensors 2020, 20, x FOR PEER REVIEW 15 of 35

Table 3. Equivalent reduced moment of inertia at the drive wheel determined using the classical Table 3. Equivalent reduced moment of inertia at the drive wheel determined using the classical gravitational method. gravitational method. Value of the Equivalent Moment of Inertia Reduced at the Drive Wheel Value of the Equivalent Moment of Inertia Reduced at the Drive Wheel (kg m2) (kg·m· 2) EquivalentEquivalent moment moment of of inertia inertia I gear I gear II gear II gear III IIIgear gear IV IV gear gear Neutral Neutral Iechiv_rm_metoda_clasica 8735 6508 4122 1046 678 Iechiv_rm_metoda_clasica 8735 6508 4122 1046 678

5.1.2.5.1.2. Computer-Assisted Computer-Assisted Gravitational Gravitational Method Method TheThe experimental experimental determination determination of the the equivalent equivalent reduced moment of inertia at the drive drive wheel wheel IITRTR usingusing the the computer-assisted computer-assisted gravitational gravitational method method is is like like the the classical classical method. method. The The stages stages of of the procedure for determining the moments of inertia an andd the resulting equations are different. diﬀerent. In this situation,situation, only only one one weight weight of of mass mass m11 isis used, used, which which falls falls freely freely for for the the distance distance H,, generating generating a a uniformlyuniformly accelerated motion, characterized by the acceleration acceleration aa.. Additionally, Additionally, due due to to the the stretching stretching phenomenon, tension tension appears appears in in the the metal metal cabl cablee that acts on the drum with the moment Mt and imprintsimprints the the angular angular acceleration ε1 onon the rotating parts. The The angular acceleration ε1 andand the the linear linear accelerationacceleration a areare quantitiesquantities valid valid only only during during the the acceleration acceleration phenomenon, phenomenon, from from the the moment moment the boxthe boxis released is released until until the groundthe ground is touched is touched by the by lowerthe lower part part of the of boxthe (Figurebox (Figure 18). 18).

FigureFigure 18. 18. Computer-assistedComputer-assisted gravitational gravitational method, method, re representationpresentation of forces and moments.

FromFrom the moment moment the the box box touches touches the the ground, ground, the the deceleration deceleration phenomenon phenomenon begins, begins, the the tension tension inin the the metal metal cable cable has has zero zero value and the drum an andd the rotating elements are characterized by by the deceleration ε . deceleration ε22.   m g T = m a [Nm]  1 1 1 mgT⋅−· − = maNm ⋅·[]  111Iechiv_rm ε1 = Mt M f r [Nm]  ⋅ε· = −− (20) IMMNmMt = T r [Nm] []  echiv_1 rmh1· i t fr  =⋅[]m (20) MTrNmat = ε1 r 2  · s The rotational movement of thear=ε⋅ drum,m including the elements arranged between the drum and  s2 the turbine of the Hydroconverter, due to the constant friction moment Mfr, is gradually slowed down, ﬁnallyThe reaching rotational rest. movement In this situation, of the drum, the equation including describing the elements the decelerationarranged between phenomenon the drum of and the thedrum turbine and the of rotatingthe Hydroconverter, parts is: due to the constant friction moment Mfr, is gradually slowed  down, finally reaching rest. In this situation, Iechiv_rm theε2 =equationM f r [describingNm] the deceleration phenomenon  · h i − (21) of the drum and the rotating parts is: = dω 1  ε dt s2

Sensors 2020, Sensors20, 5587 2020, 20, x FOR PEER REVIEW 16 of 35 15 of 31

IMNm⋅ε =− [ ]  echiv_2 rm fr Equations (20) and (21) result in: ω ε=d 1 (21)  dt s2 m1 r (g ε1 r) h 2i Equations (20) andIechiv (21)_ resultrm_as_ in:computer = · · − · kg m (22) ε1 ε2 · mr⋅⋅() g −ε⋅− r I =⋅11kg m2 The tests are performed successivelyechiv___ rm as computer for operationε−ε in the neutral position and in the(22) four gears of 12 the planetary gearbox. The data acquisition system measures the values of the angular velocity of the drive wheel untilThe tests it stops. are performed The measurements successively for are operation repeated in the three neutral times position for and each in stagethe four of gears the planetary of the planetary gearbox. The data acquisition system measures the values of the angular velocity of gearbox (Figurethe drive 19 wheel). This until method it stops. doesThe measurements not require are measurements repeated three times with for two each di ﬀstageerent of masses.the It is enough thatplanetary the measurements gearbox (Figure are19). performed This method with does onlynot require one mass, measurementsm2. To conﬁrmwith two thedifferent quality of the data obtainedmasses. during It is enough the previous that the measurements tests, in the are case performed of I and with II gears, only one the mass, mass m2m. To1 isconfirm used. the quality of the data obtained during the previous tests, in the case of I and II gears, the mass m1 is used.

Figure 19. Testing using the computer-assisted gravitational method. Figure 19. Testing using the computer-assisted gravitational method. The processed data [26,27] allow for the graphical tracing of the variation of the angular velocity, The processedduring the dataacceleration [26,27 and] allow deceleration for the of graphical the transmission tracing parts, of the in the variation form of variation of the angular of the velocity, during theangular acceleration velocity of and the drive deceleration wheel (Figure of 20a–e). the transmission parts, in the form of variation of the angular velocity of the drive wheel (Figure 20a–e). A first analysis of the previous graphs shows an almost linear evolution of the angular velocity ω. The phenomenon is more evident in the acceleration phases of the first three gears and in all the deceleration phases of all gears (except their terminal part). The nonlinearity noticed in the final acceleration phase of the 4th gear and in the terminal phases of the decelerations of all gears (exponentially) is due to the frictions in the system, both the turbulent one in the Hydroconverter and the laminar one in the rest of the transmission. One of the hypotheses imposed is that friction has a linear character in relation to speed, so the frictions due to the turbulent flow phenomena in the Hydroconverter are linearized. This allowed for a linear interpretation of the angular velocity behavior.

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FigureFigure 20. Variation 20. Variation of the of the angular angular speed speed of of the the drive drive wheel. wheel. ((aa)) Operation in in I Igear; gear; (b ()b operation) operation in in II gear;II (c )gear; operation (c) operation in III gear;in III gear; (d) operation (d) operation in IVin IV gear; gear; (e )(e operation) operation in in neutral.neutral.

Sensors 2020, 20, x FOR PEER REVIEW 18 of 35

A first analysis of the previous graphs shows an almost linear evolution of the angular velocity ω. The phenomenon is more evident in the acceleration phases of the first three gears and in all the deceleration phases of all gears (except their terminal part). The nonlinearity noticed in the final acceleration phase of the 4th gear and in the terminal phases of the decelerations of all gears (exponentially) is due to the frictions in the system, both the turbulent one in the Hydroconverter and the laminar one in the rest of the transmission. One of the hypotheses imposed is that friction has a linear character in relation to speed, so the frictions due to the turbulent flow phenomena in the Hydroconverter are linearized. This allowed for a linear interpretation of the angular velocity behavior. SensorsThe2020 linearity, 20, 5587 of the behavior can be highlighted by several methods. One of these is the use17 of of 31 the correlation coefficient. In general, the correlation coefficient is used to determine how a signal measured at one point in a system tracks the evolution of another signal measured at either the same pointThe in the linearity system of or the at behavioranother. In can this be situation, highlighted theby closer several the methods.correlation One coefficient of these is is to the one, use the of betterthe correlation the signals coe arefficient. correlated. In general, The application the correlation of this coe methodfficient isin used this tocase determine aims to establish how a signal the linearitymeasured of at the one evolution point in aof system the angular tracks thevelocity evolution, by using of another the “vector” signal measured time, whose at either evolution the same is independent,point in thesystem “correlated” or at another.with the evolution In this situation, of the angular the closer velocity the correlationvector. A correlation coefficient coefficient is to one, asthe close better as thepossible signals to arethe correlated.unit value leads The application to the conclusion of this methodthat the indepen thisdent case variable aims to establish(angular velocity)the linearity also ofhas the a linear evolution evolution. of the To angular approximate velocity, the by angular using the velocity ?vector? for the time, two whose operating evolution modes is (acceleration,independent, deceleration), ?correlated? with the method the evolution of least of squares the angular is used velocity (Figure vector. 21a–e.). A correlation coefficient as close as possible to the unit value leads to the conclusion that the dependent variable (angular n velocity) also has a linear evolution. To approximate()−⋅ω−ω the() angular velocity for the two operating modes  ttii (acceleration, deceleration), the method of least= squares is used (Figure 21a–e). ()ω i 1 Ctcov , − ==Pn n 1 []−∈− [ ] Ccor ( ) 1,1 (23) ⋅ nn(ti t) ωi ω SSt ω i = 1 22− · − Ccov(t, ω) ()tt−ω−ωn 1 () Ccor = =s ii−s [ ] [ 1, 1] (23) ii==11n n St Sω P 2⋅ P 2 − ∈ − · (ti t) (ωi ω) nni =−−111− i = 1 − n 1 n 1 − · −

Figure 21. Cont. Sensors 2020, 20, x FOR PEER REVIEW 19 of 35 Sensors 2020, 20, 5587 18 of 31

Sensors 2020, 20, x FOR PEER REVIEW 20 of 35

FigureFigure 21. 21.Experimental Experimental data data processing. processing. Modeling Modeling accuracy accuracy parameters. parameters. ( a(a)) I I gear, gear, probe probe 3; 3; ( b()b II) II gear, probegear, 1; (probec) III gear,1; (c) III probe gear, 1; probe (d) IV 1; gear,(d) IV probe gear, 3;probe (e) neutral,3; (e) neutral, probe probe 1. 1.

The value of the coefficient of determination (24) is close to the unit value for all samples, which means that a high percentage of the experimental data are very close to the calculated values, as follows: • acceleration phase—95,72% in I gear, 96,76% in II gear, 97,41% in III gear, 96,73% in IV gear and 98,02% in neutral; • deceleration phase—93,99% in I gear, 97,95% in II gear, 98,36% in III gear, 98,50% in IV gear and 98,26% in neutral.

n ()ω−ωˆ  ii 2 =−i=1 [] − R 1 n (24) ()ω−ω  i i=1

Another relevant criterion for the quality of the approximation of the experimental data is the standard error ES (25) which falls within the allowed approximation limits: • I gear—2,96% acceleration phase and 3,65% deceleration phase; • II gear—2,71% acceleration phase and 3,55% deceleration phase; • III gear—2,49% acceleration phase and 3,17% deceleration phase; • IV gear—5,47% acceleration phase and 4,51% deceleration phase; • neutral – 4,72% acceleration phase and 4,86% deceleration phase.

n ()ω−ωˆ  ii 2 =−i=1 [] − R 1 n (25) ()ω−ω  i i=1

Based on the algorithm for processing and representing the angular velocities as well as the angular accelerations and decelerations [1,28–30], the corresponding values are calculated for each stage of the gearbox and for each test separately (Table 4). Sensors 2020, 20, 5587 19 of 31

The value of the coeﬃcient of determination (24) is close to the unit value for all samples, which means that a high percentage of the experimental data are very close to the calculated values, as follows:

acceleration phase—95.72% in I gear, 96.76% in II gear, 97.41% in III gear, 96.73% in IV gear and • 98.02% in neutral; deceleration phase—93.99% in I gear, 97.95% in II gear, 98.36% in III gear, 98.50% in IV gear and • 98.26% in neutral.

Pn (ωi ωˆ i) i = 1 − R2 = 1 [ ] (24) − Pn − (ωi ω) i = 1 − Another relevant criterion for the quality of the approximation of the experimental data is the standard error ES (25) which falls within the allowed approximation limits:

I gear—2.96% acceleration phase and 3.65% deceleration phase; • II gear—2.71% acceleration phase and 3.55% deceleration phase; • III gear—2.49% acceleration phase and 3.17% deceleration phase; • IV gear—5.47% acceleration phase and 4.51% deceleration phase; • neutral—4.72% acceleration phase and 4.86% deceleration phase. •

Pn (ωi ωˆ i) i = 1 − R2 = 1 [ ] (25) − Pn − (ωi ω) i = 1 − Based on the algorithm for processing and representing the angular velocities as well as the angular accelerations and decelerations [1,28–30], the corresponding values are calculated for each stage of the gearbox and for each test separately (Table4).

Table 4. The results obtained from the process of deriving the experimental data on angular velocity.

Angular Angular Weight Testing Gearbox CVP Testing Acceleration (ε1) Deceleration (ε2) (kg) (rad/s2) (rad/s2) m –600 Test 1a_et1 0.0100 0.2445 2 − m –600 Test 2a_et1 0.0163 0.1870 First gear 2 − m –600 Test 3a_et1 0.0146 0.1858 2 − m –600 Test 4a_et1 0.0140 0.1877 2 − m –600 Test 1a_et2 0.0190 0.2507 2 − Second gear m –600 Test 2a_et2 0.0240 0.2479 2 − m –600 Test 3a_et2 0.0251 0.2475 2 − m –600 Test 1a_et3 0.0336 0.3997 2 − Third gear m –600 Test 2a_et3 0.0343 0.4026 2 − m –600 Test 3a_et3 0.0395 0.3920 2 − m –600 Test 1a_et4 1.1604 0.6537 2 − Fourth gear m –600 Test 2a_et4 1.1614 0.6624 2 − m –600 Test 3a_et4 1.2037 0.6402 2 − m –600 Test 1a_etn 2.0304 0.9135 Neutral position 2 − m –600 Test 2a_etn 1.8918 1.1062 2 − Sensors 2020, 20, 5587 20 of 31

Due to the behavioral nonlinearities of the angular velocity, the last section of the evidence is not taken into account when obtaining the angular decelerations. The linear approximation of the entire deceleration area to the actual stop of the drive wheel, including non-linear areas, could introduce errors in the calculation of the angular deceleration. For this reason, the model is used only to describe the linear area of the data section, where the angular velocity decreases rapidly after an almost linear variation (Table5).

Table 5. Equivalent moment of inertia reduced at the drive wheel determined by the computer-assisted gravitational method.

Equivalent Reduced Moment of Inertia Value at the Drive Wheel (kg m2) · First Gear Second Gear Third Gear Fourth Gear Neutral Position Testing I_echiv Testing I_echiv Testing I_echiv Testing I_echiv Testing I_echiv 1a 7785.2 1a 5866.3 1a 3649.9 1a 846.33 1a 509.42 2a 7894.1 2a 5818.1 2a 3620.2 2a 841.77 2a 503.34 3a 7845.6 3a 5803 3a 3664.7 3a 831.70 3a —–

Iechiv_as_calc_et1 Iechiv_as_calc_et2 Iechiv_as_calc_et3 Iechiv_as_calc_et4 Iechiv_as_calc_neutru 7841 5829 3645 840 506

5.2. Three-Wire Suspension Method This method allows for the experimental determination of the moments of inertia for various rotating moving parts.

X2 !2 X12 !2 X6 2 rrm rrm rrm h 2i Ipr = Iri + Ig + Irs kg m (26) · rri · rg · rrs · i = 1 i = 1 i = 1

 i2 η i2 η i2 η i2 η   I i2 η + R R I + R R I + R R I + R R I +   e R R i2η 5 i2η 6 i2η i2η 7 i2η i2η i2η 8   1 1 2 2 2 2 3 3 2 2 5 5 7 7   i2 η i2 η i2 η i2 η  h i I =  + R R I + R R I + R R I + R R I +  kg m2 (27) EP  i2η i2 η 15 i2η i2 η 16 i2η i2η 9 i2η i2η i2η 13   2 2 10 10 2 2 11 11 2 2 4 4 2 2 4 4 8 8  ·  i2 η i2 η i2 η   + R R I + R R I + R R I   i2 i2 i2 i2 14 i2 i2 i2 10 i2 i2 i2 i2 11  2η2 4η4 8η8 9η9 2η2 4η4 5η5 2η2 4η4 5η5 6η6 The method involves the use of a three-wire pendulum. The three-wire pendulum is a device consisting of a movable circular platform, a fixed disk and three thin cables, characterized by the same diameter and the same length. The fixed disk is rigidly attached to a metal beam. The cables are used to support the circular platform at three equidistant points. The cables have a threaded adjustment device at their ends, which allows for obtaining parallelism between the mobile circular platform, the fixed disc and the floor (Figure 22). The technical data of the pendulum used are platform radius R = 0.345 m, fixed disk radius r = 0.18 m, wire length L = 3.22 m, platform mass mp = 13.7 kg. Sensors 2020, 20, x FOR PEER REVIEW 22 of 35

ii22ηη i 2 η i 2 η Ii2η+RR I + RR I + RR I + RR I + eR R 2η56 2 η 22 ηη 7 222 ηηη 8 i11 i 22 ii 2233 iii 225577 iiii2222ηηηη IIIII=+RR + RR + RR + RR +kg⋅ m2 EP 22η η15 22 ηη 16 22 ηη 9 222 ηηη 13 (27) ii221010 ii 221111 ii 2244 iii 224488 ii22ηηi2η ++RRI RR II+ RR 2222ηηηη14 222ηηη10 2222 ηηηη 11 iiii22448899 i224455ii iiii 22445566

The method involves the use of a three-wire pendulum. The three-wire pendulum is a device consisting of a movable circular platform, a fixed disk and three thin cables, characterized by the same diameter and the same length. The fixed disk is rigidly attached to a metal beam. The cables are used to support the circular platform at three equidistant points. The cables have a threaded adjustment device at their ends, which allows for obtaining parallelism between the mobile circular platform, the fixed disc and the floor (Figure 22). The technical data of the pendulum used are platformSensors 2020 radius, 20, 5587 R = 0.345 m, fixed disk radius r = 0.18 m, wire length L = 3.22 m, platform mass21 mp of 31= 13.7 kg.

FigureFigure 22. Three-wireThree-wire pendulum. pendulum.

ToTo retrieve the necessary information to obtain the moments of inertia, in this case case the the oscillation oscillation period ofof thethe platform, platform, the the MicroStrain MicroStrain 3DM-GX1 3DM-GX1 inertial inertial sensor sensor is attached is attached to the to experimental the experimental device. device.The sensor The issensor centered is centered on the element on the subjectedelement subjec to theted experimental to the experimental process (Figure process 23 ).(Figure The 3DM-GX1 23). The 3DM-GX1Data Acquisition Data Acquisition and Display and Software Display sensorSoftware program sensor isprogram used to is process used to the process experimental the experimental data. data.Sensors 2020, 20, x FOR PEER REVIEW 23 of 35

Figure 23. The main elements of the three-wire pendulum measuring chain. Figure 23. The main elements of the three-wire pendulum measuring chain.

The experiment begins with the initial rotation of the platform at an angle of φ = 20...250. This causes the platform to rise to a height of h (Figure 24).

Figure 24. Schematic representation of the three-wire pendulum.

By releasing the platform, it begins to perform circular oscillating movements, described by the relation: 2π ϕ=⋅()tRsin ⋅ trad[]  (28) Tp

The platform moves in a rotational movement around the OO2 axis and, at the same time, performs a translational movement along the OO2 axis of amplitude h. In this movement, the mobile platform is subject to the law of the conservation of energy. For this reason, it is known that at points D and F (points of maximum height), the kinetic energy of the platform is zero and the potential energy is maximum. During the movement from point D to point A, which represents the equilibrium position, the potential energy Ep is transformed into kinetic energy Ec. When passing through the

Sensors 2020, 20, x FOR PEER REVIEW 23 of 35

Sensors 2020, 20, 5587 22 of 31 Figure 23. The main elements of the three-wire pendulum measuring chain.

TheThe experimentexperiment begins with the initial initial ro rotationtation of of the the platform platform at at an an angle angle of of φϕ = =20...2520 ...0. This25◦. Thiscauses causes the platform the platform to rise to to rise a toheight a height of h of(Figureh (Figure 24). 24).

FigureFigure 24.24. SchematicSchematic representationrepresentation ofof thethe three-wirethree-wire pendulum.pendulum.

ByBy releasingreleasing the platform, platform, it it begi beginsns to to perform perform circular circular oscillat oscillatinging movements, movements, described described by the by therelation: relation: ! 2π ϕ(t) = R sin t [rad] (28) · 2Tπp · ϕ=⋅()tRsin ⋅ trad[] (28) The platform moves in a rotational movement around the OO axis and, at the same time, performs Tp 2 a translational movement along the OO2 axis of amplitude h. In this movement, the mobile platform is subjectThe toplatform the law moves of the conservationin a rotational of movement energy. For around this reason, the OO it is2 knownaxis and, that at atthe points sameD time,and Fperforms(points ofa translational maximum height), movement the kinetic along the energy OO2 of axis the of platform amplitude is zero h. In and this themovement, potential the energy mobile is maximum.platform is Duringsubject theto the movement law of the from conservation point D to of point energy.A, which For this represents reason, it the is equilibriumknown that position,at points theD and potential F (points energy of maximumEp is transformed height), into the kinetickinetic energy energyE cof. Whenthe platform passing is through zero and the the equilibrium potential position,energy is the maximum. kinetic energy During is the maximum. movement On from the A point–F path, D to kinetic point energyA, which is transformedrepresents the into equilibrium potential energy.position, The the total potential energy energy of system Ep isE istransformed constant and into can kinetic be expressed energy as Ec the. When sum passing of the kinetic through energy the and the potential energy of the studied system:

  E = 1 I ω2 [Nm]  c 2  · Ep = m g h [Nm] E = Ec + Ep [Nm] (29)  · · h i ⇒  dϕ ( ) rad  ω = dt = ϕ t s

The height h, from a geometric point of view, is given by the length of the segment (OO1) and can be calculated as the diﬀerence between segment BC and segment BE.

2 ϕ 4 r R sin 2 h = · · · [m] (30) BC + BE

Given that angle ϕ varies in a very small range, it can be stated that, in terms of value, sin (ϕ) is approximately equal to the value of angle ϕ and that the length of segment BC is approximately equal Sensors 2020, 20, 5587 23 of 31 to that of segment BE. Taking into account these hypotheses, the analytical expression of the maximum height (h) becomes: r R ϕ2 h = · · [m] (31) 2 l · By replacing the analytical expression of the maximum height in the potential energy deﬁnition relation, a function is obtained that depends on the square of the angular displacement of the circular platform with respect to the equilibrium position ϕ(t). Applying the law of the conservation of energy and performing the required simpliﬁcation operations, the analytical expression of the moment of inertia is determined:

r R ϕ(t)2 = · · [ ] Ep m g 2 l Nm 1· · · 2 Ec = I ϕ(t) [Nm] h i 2 m g r R 2 2 dE dEp· · h i I = · · · T kg m (32) c + = dE Nm → 4π2 l · p · dt dt dt s · ϕ(t) = R sin 2π t [m] · Tp · It can be seen that, in terms of value, the moment of inertia depends on the dimensional and mass characteristics of the experimental device r, R, m and the period of oscillation T. Determining the moments of inertia of the parts requires knowing the moment of inertia of the device, ip. Its value is subtracted from the moment of inertia of the assembly, consisting of the measuring device and part Ip-i Sensors 2020, 20, x FOR PEER REVIEW 25 of 35 (Figure 25), resulting in the moment of inertia of the part (32).

Figure 25. Representation of the experimental procedure for determining moments of inertia. Figure 25. Representation of the experimental procedure for determining moments of inertia.

In both situations, the calculation algorithm requires determining periods Tp and Tp-i specific to the oscillation movements of the mobile platform and the platform element i assembly. Both oscillation periods Tp and Tp-i are determined by applying the relations:

ttpp−1 TsTs==[] − [] (33) ppnn1

In order to determine the oscillation times tp, tp-i and the number of oscillations n performed by the platform or the platform element i assembly, for the time periods tp or tp-i, the inertial sensor of the measuring chain is used—MicroStrain 3DM-GX1. For better accuracy of the results, the experimental process is repeated ten times, both for the platform and for the platform element i assembly. From the resulting data are extracted those necessary to determine the period of oscillation: oscillation time tp and oscillation time consumed by the element subjected to the measurement process tp-i for the execution of a fixed number n of oscillations—20 oscillations (Figure 26).

Sensors 2020, 20, 5587 24 of 31

In both situations, the calculation algorithm requires determining periods Tp and Tp-i speciﬁc to the oscillation movements of the mobile platform and the platform element i assembly. Both oscillation periods Tp and Tp-i are determined by applying the relations:

tp tp 1 Tp = [s] Tp 1 = − [s] (33) n − n

In order to determine the oscillation times tp, tp-i and the number of oscillations n performed by the platform or the platform element i assembly, for the time periods tp or tp-i, the inertial sensor of the measuring chain is used—MicroStrain 3DM-GX1. For better accuracy of the results, the experimental process is repeated ten times, both for the platform and for the platform element i assembly. From the resulting data are extracted those necessary to determine the period of oscillation: oscillation time tp and oscillation time consumed by the element subjected to the measurement process tp-i for the execution of a ﬁxed number n of oscillations—20 oscillations (Figure 26).

Table 6. Experimental data obtained with the three-wire pendulum method.

Period Moment of Inertia Ip Tested Element Sample Number Period Period or I (kg m2) p-i · j j Tp or Tp-i (s) Tp or Tp-i (s) 1 3.2530 2 3.2420 3 3.2395 4 3.2630 Three-wire pendulum 5 3.2410 3.2558 0.6959 platform 6 3.2752 7 3.2635 8 3.2575 9 3.2690 10 3.2540 1 3.0520 2 3.0465 3 3.0560 4 3.0480 3.0532 5.9727 Platform—tensioning 5 3.0490 wheel assembly 6 3.0475 7 3.0605 8 3.0635 9 3.0565 3.0532 5.9727 10 3.0520

The oscillation periods Tp and Tp-i are calculated using the arithmetic mean of the oscillation periods speciﬁc to the ten experimental samples.

10 10 1 X j 1 X j Tp = Tp [s] Tp 1 = T [s] (34) 10· − 10· p 1 j = 1 j = 1 − Sensors 2020, 20, 5587 25 of 31 Sensors 2020, 20, x FOR PEER REVIEW 26 of 35

FigureFigure 26. 26. GraphicalGraphical representation representation of ofthe the experimental experimental data data and and the algorithm the algorithm for extracting for extracting the data the necessarydata necessary to determine to determine the period the period of oscillation, of oscillation, the thered red sample sample is istheir their average, average, and and each each color color representsrepresents the the samples samples in in Table Table6 6..

TheThe oscillation oscillation periods periods Tp are and replaced Tp-i are in calculated the relations using of thethe momentsarithmetic of mean inertia of the specific oscillation to the periodsplatform specificIp and to the ten platform experimental element samples.i assemblies Ip-i (Table6).  10h i 10 =⋅11mjjg r R 2 =⋅2 TTsTTs Ip = ·[]·2 · Tp −−kg m [] (34) pppp 4π l · 11· (35) 10 jj==11mp· i g r R 2 10h 2i  I = − · · · T kg m p i 4π2 l p i − · · − · The oscillation periods are replaced in the relations of the moments of inertia specific to the g r R h i platform Ip and to the platformI = elementI I i= assemblies· · m Ip-i T(Table2 m 6).T 2 kg m2 (36) i p i p 2 p i p i p − − 4π l· − · − − · ·  · Table 6. ExperimentalI data obtained(I with) the(i 2three-wire) (η ) pendulumdω method.  = + ms + pr_ exp + TR_ exp i + cd i· cd i p [ ]  δi 1 M 2 2 2 IEP_ exp Kh dω  a Ma rrm Ma rrm Ma rrm · · · t − · ·2 · Moment of Inertia(37)  Ipr (i ) (ηT)  ms _ exp t IV· IV Period  δIVm = 1 + + 2 + 2 (It) [ ] Ip Ma Ma r Ma r IVM Sample· rm · rm · − Tested Element Period Period or Ip-i (kg·m2) The final relation for determining theNumber variation of the coefficient of inertia masses contains data Tpj or Tp-ij Tp or Tp-i

obtained both from the gravitational method and from the(s) three-wire suspension(s) method (Figure 27). 1 3.2530 2 3.2420 Three-wire pendulum platform 3 3.2395 3.2558 0.6959 4 3.2630 5 3.2410

Sensors 2020, 20, x FOR PEER REVIEW 28 of 35 Sensors 2020, 20, 5587 26 of 31

FigureFigure 27. 27.The The variation variation of of the the coe coefficientﬃcient of of inertia inertia masses masses resulting resulting from from the the experimental experimental data data for for startingstarting the the tracked tracked vehicle. vehicle.

6.6. Discussion Discussion TheThe general general virtual virtual model model for for simulating simulating the the longitudinal longitudinal dynamics dynamics of of the the vehicle vehicle required required performingperforming some some tests. tests. TwoTwo directionsdirections werewere followed:followed: oneone aimedaimed atat determiningdeterminingthe the starting starting performances,performances, and and the the other other at obtaining at obtaining the equivalent the equivalent moments moments of inertia of that inertia entered that the entered expression the ofexpression approximation of approximation of the coefficient of th ofe inertiacoefficient masses. of inertia To obtain masses. the To variation obtain ofthe the variation coefficient of ofthe inertiacoefficient masses, of threeinertia distinct masses, methods three distinct were used: methods indirect, were based used: on theindirect, signals based generated on the from signals the interrogationgenerated from of the the virtualinterrogation model; of direct, the virtual which model; takes overdirect, the which data takes resulting over from the data the gravitationalresulting from andthethree-wire gravitational pendulum and three-wire tests; empirical, pendulum which istests; in fact empirical, an analytical–experimental which is in fact method.an analytical– experimentalAll these steps method. have led to the conclusion that this approach leads to smaller errors because there is no needAll these to rely steps on thehave results led to of the predictive conclusion models that th [31is –approach34]. leads to smaller errors because there is noTo need date, tothe rely proposed on the resu solutionslts of predictive in thisfield models have [31–34]. relied more on theoretical assumptions or computerTo date, simulations the proposed to demonstrate solutions the in eff thisectiveness field ha ofve virtual relied models more foron simulatingtheoretical the assumptions longitudinal or dynamicscomputer of simulations tracked heavy to vehicles demonstrate [32]. In the addition, effectiveness these vehicles of virtual can be models deployed for to simulating perform tasks the inlongitudinal unstructured dynamics environments, of tracked being heavy forced vehicles to move [32]. and In change addition, position these atvehicles speeds can characterized be deployed by to aperform strong start. tasks On in theunstructured other hand, environments, different obstacles being canforced appear, to move which and slows change the position dynamics at ofspeeds the movementcharacterized [33]. by a strong start. On the other hand, different obstacles can appear, which slows the dynamicsAs the of system the movement is considered [33]. non-deformable, vibrations were not taken into account during the measurements. For this reason, we appreciate that no additional errors were obtained, which would vitiate the results.

Sensors 2020, 20, 5587 27 of 31

Testing techniques will be further developed through the implementation of wireless technologies and artiﬁcial intelligence elements.

7. Conclusions The results obtained by the computer-assisted gravitational method are more accurate than those obtained by the classical gravitational method. This is because the computer-assisted gravitational method uses two sets of data to determine moments of inertia: those on the acceleration section and those on the deceleration section. Compared to this, the classical gravitational method uses only the data on the acceleration section. It was pointed out that in the first three gears of the gearbox, the time dependence of the angular velocity was linear, both on the acceleration zone and on the start zone of the deceleration process. Instead, in the 4th gear and in the final part of the measurement process, due to the intensification of the friction phenomenon, the allure of the angular velocity variation curve became exponential towards the end of the acceleration or deceleration process. To approximate the specific values of acceleration and angular deceleration, the experimental samples were linearized (first-order polynomials were used as approximants). Linearization resulted in constant accelerations and decelerations. The least squares method approximated in good and very good limits the experimental data obtained by the computer-assisted gravitational method (with errors below 6%). From the comparison of the values of moments of inertia with those modeled in 3D (for the parts tested with the three-wire pendulum), it was observed that the differences between them were small. For the parts that could not be mounted on the three-wire device, the 3D models were used, with the veracity of the data being assumed by the precision of the 3D models.

Author Contributions: Conceptualization, O.A., L.?.G., I.C. and A.V.; methodology, I.O.; software, O.A., I.C. and A.V.; validation, L.?.G., and I.O.; formal analysis, I.O.; investigation, O.A., L.?.G., I.C. and A.V. writing—original draft preparation, I.O. and L.?.G.; writing—review and editing, I.O. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

h m i a s2 longitudinal acceleration of the vehicle Ccor [ ] linear correlation coefficient − Ccov [ ] covariance coefficient − Ec [Nm] kinetic energy Ep [Nm] potential energy fprsαa [ ] the coefficient of the total advance resistance force − FjA [N] the inertial flow load factor of the vehicle Fjpr [N] the inertial forces of the propulsion system Fjs [N] the force of inertia of the track 0 Fp [N] stationary propulsion force eFp [N] dynamic propulsion force Ft [N] stationary traction force eFt [N] dynamic traction force h m i g s2 gravitational acceleration Ga [N] vehicle?s weight HC torque convertor (icv) [ ] the total kinematic transmission ratio in the gear (i = I ... IV) of the planetary i − gearbox i [ ] the absolute transmission ratio of the mechanical transmission cd − i [ ] the kinematic transmission ratio of the final drive dl − i [ ] the kinematic transmission ratio of the distribution mechanism R − i [ ] torque convertor kinematic transmission ratio h − Sensors 2020, 20, 5587 28 of 31 i [ ] the inverse of the kinematic transmission ratio of the torque convertor h0 − i [ ] the kinematic transmission ratio of the planetary summation mechanism mi − (it)IV the absolute transmission ratio for the operation of the transmission with the torque convertor blocked h i I kg m2 the sum of the moments of inertia of the rotating elements with angular velocity ω 0 · b h 2i I kg m the sum of the moments of inertia of the rotating elements with angular velocity ωa 1 · h 2i I2 kg m the sum of the moments of inertia of the rotating elements with angular velocity ω2 h · i I kg m2 the sum of the moments of inertia of the rotating elements with angular velocity ω 3 · 5 h 2i I kg m the sum of the moments of inertia of the rotating elements with angular velocity ω8 4 h · i I kg m2 the moment of inertia of the satellite plateaus of the planetary summation 21 · mechanisms h 2i I22 kg m the moment of inertia of the satellite plateaus of the final drive h · i I kg m2 equivalent moment of inertia reduced at the crankshaft b4 h · i I kg m2 equivalent moment of inertia reduced at the output shaft of the planetary gearbox b5 · h i I /I kg m2 equivalent moment of inertia reduced at the inlet shaft of the torque convertor EP EP_ exp · pump h 2i Ig kg m moments of inertia of the road wheel h · i I kg m2 the moments of inertia of the wheels contained in the tracked propulsion system ipr · h 2i Iji kg m equivalent reduced moments of inertia at the entry shaft of the planetary gearbox h · i I kg m2 equivalent reduced moments of inertia at the output shaft of the reversing mi · mechanism h 2i Ipr kg m moment of inertia of the propulsion system wheels reduced to the drive sprocket h · i I kg m2 moments of inertia of the tensioning wheel ri · h 2i Irs kg m moments of inertia of the support roller · h i I ... I kg m2 equivalent moment of inertia reduced to the drive sprocket axle TR TR_ exp · h 2i (It) kg m equivalent moment of inertia reduced to the axle of the drive sprockets in IVm · mechanical mode K [ ] torque convertor capacity factor − K [ ] constant of the planetary mechanism of the final drive dl − K [ ] torque convertor transformation ratio h − K [ ] the constant of the planetary summation mechanism mi − Ma tank mass mc [kg] armored hull mass M f r [Nm] shear moment MHp [Nm] the torque that loads the torque convertor pump shaft MHt [Nm] the torque that loads the torque convertor turbine shaft Mjpr [Nm] the torque reduced to the drive sprocket of the moments of inertial forces of the propulsion system wheels mpr [kg] propulsion system mass ms [kg] track mass 0 Mp [Nm] torque absorbed by torque convertor pump Mrm [Nm] stationary drive sprocket torque Merm [Nm] dynamic drive sprocket torque n [ ] number of oscillations − n [ ] number of experimental values −h rad i np s the torque convertor pump h rad i nt s angular speed of torque convertor turbine r [m] the radius of the drum on which the metal cable was wound R [m] amplitude of oscillating motion Ra [N] air resistance force rg [m] road wheel radius Sensors 2020, 20, 5587 29 of 31

Rpr [N] propulsion system resistance force Rprsaα [N] total advance resistance force rrm [m] drive sprocket radius rri [m] tensioning wheel radius rrs [m] support roller radius Rs [N] runway resistance force Rα [N] uphill resistance force Rδ [N] acceleration resistance force sd [m] starting distance St [ ] standard deviations for the vector time t [s] − h rad i Sω [ ] standard deviations for the angular velocity ω − s t1 [s] times of lowering of mass weights m1 [kg] from height H [m] t2 [s] times of lowering of mass weights m2 [kg] from height H [m] td [s] starting time tp i ... tp [s] the times elapsed during the n oscillations of the circular platform and the platform − element i assembly Tp [s] the period of oscillation of the platform of the experimental device ts [s] gear changing time t [s] arithmetic means of experimental values - vectors t [s] h m i h km i v s ... V h vehicle speed δ [ ] coefficient of inertia masses − δ [ ] the coefficient of the inertial masses of rotation arranged upstream of the torque m_phc − convertor δpr [ ] the coefficient of inertial masses of rotation of the tracked propulsion system − δr [ ] the coefficient of inertial masses in rotational motion − δt [ ] the coefficient of inertial masses in translational motion − δ [ ] the coefficient of the inertial masses of rotation arranged downstream of the torque thc_rm − convertor. h rad i εp s2 angular acceleration of torque convertor pump h rad i εrm s2 angular acceleration of drive sprocket h rad i εt s2 angular acceleration of torque convertor turbine η [ ] mechanical transmission efficiency cd − (ηcv) [ ] the efficiency corresponding to the gear (i) of the planetary gearbox i − η [ ] the efficiency of the planetary mechanism of the final drive di − ηe [ ] efficiency of outer cylindrical − η [ ] torque convertor efficiency h − η [ ] efficiency of inner cylindrical i − η [ ] efficiency of bevel gears k − η [ ] the efficiency of the planetary summation mechanism and the lateral demultiplexer mi − ηmi t f [ ] efficiency of summation mechanisms and final transmissions − − η [ ] distribution mechanisms efficiency R − (ηt) [ ] transmission efficiency with the torque convertor blocked IV − ϕ(t)[m] elongation of the oscillating motion h rad i ω s the angular velocity h rad i ωa s the angular velocity of the input shafts of the gearbox h rad i ωb s the angular velocity of the output shafts of the gearbox h rad i ωe s angular speed at the crankshaft h rad i ωˆ i s the value estimated by the approximation model h rad i ωp s the torque convertor pump h rad i ωrm s angular speed of the drive sprocket h rad i ωt s angular speed of torque convertor turbine h rad i h rad i ω s arithmetic means of experimental values - vectors ω s Sensors 2020, 20, 5587 30 of 31

References

1. Alexa, O.; Tru?ă, M.; Marinescu, M.; Vilău, R.; V?nturi?, V. Simulating the longitudinal dynamics of a tracked vehicle. Adv. Mater. Res. 2014, 1036, 499–504. [CrossRef] 2. Ubysz, A. Problems of rotational mass in passenger vehicles. Proc. Probl. Transp. Probl. Int. Sci. J. 2010, 5, 33–40. 3. Grigore, L.?.; Priescu, I.; Grecu, D.L. Cap. 4 Terrestrial Mobile Robots, Applied Artificial Intelligence in Fixed and Mobile Robotic Systems; AGIR: Bucharest, Romania, 2020; p. 703. 4. Ciobotaru, T.; Alexa, O. Ingineria Autovehiculelor Militare cu ?enile. Vol. III: Transmisia. Frânele. Agregatul Energetic; Publishing House of the Military Technical Academy ?Ferdinand I?: Bucharest, Romania, 2019; p. 356. 5. Abbassi, Y.; Ait-Amirat, Y.; Outbib, R. Global modeling and simulation of vehicle to analyze the inertial parameters effects. Int. J. Model. Simul. Sci. Comput. 2016, 7, 1650032. [CrossRef] 6. U.S. Army Test and Evaluation Command Test Operations Procedure. Test Operations Procedure (TOP) 01-2-520 Moments of Inertia; Policy and Standardization Division (CSTE-TM); U.S. Army Test and Evaluation Command: Aberdeen Proving Ground, MD, USA, 2017; p. 41. 7. Zhu, T.; Zhang, F.; Li, J.; Li, F.; Zong, Z. Development Identification Method of Inertia Properties for Heavy Truck Engine Based on MIMS Test Rig. In Proceedings of the 4th International Conference on Mechatronics and Mechanical Engineering (ICMME 2017), MATEC Web of Conference, Kuala-Lumpur, Malaysia, 28–30 November 2018; Volume 153, pp. 373–380. 8. Dumberry, M.; Bloxham, J. Variations in the Earth?s gravity field caused by torsional oscillations in the core. Geophys. J. Int. 2004, 159, 417–434. 9. Matsuo, K.; Otsubo, T. Temporal variations in the Earth?s gravity field from multiple SLR satellites: Toward the investigation of polar ice sheet mass balance. In Proceedings of the 18th International Workshop on Laser Ranging, Fujiyoshida, Japan, 11–15 November 2013. 10. Koralewski, G. Modelling of the system driver—Automation—Autonomous vehicle—Road. Open Eng. 2020, 10, 175–182. [CrossRef] 11. Tru?ă, M.; Alexa (Fieraru), O.; Vilău, R.; V?nturi?, V.; Marinescu, M. Static and Dynamic Analysis of a Planetary Gearbox Working Process. Period. Eng. Adv. Mater. Res. 2013, 837, 489–494. 12. Technical Presentation Data of the Tank TR 85 M1, C.N.; ROMARM Bucharest Mechanical Plant Branch S.A.: Bucharest, Romania, 2020. 13. Technical Documentation of CHC 420CML Hydroconverter; Hidromecanica: Brasov, Romania, 2004. 14. The Measurement Sheets for the Approval of the Manufacturing Process of the 8VSA2T2M Engine Assembly by C.N.; ROMARM S.A.—Bucharest Mechanical Plant Branch: Bucharest, Romania, 2020. 15. Goelles, T.; Schlager, B.; Muckenhuber, S. Fault Detection, Isolation, Identification and Recovery (FDIIR) Methods for Automotive Perception Sensors Including a Detailed Literature Survey for Lidar. Sensors 2020, 20, 3662. [CrossRef] 16. Messina, M.; Njuguna, J.; Palas, C. Mechanical Structural Design of a MEMS-Based Piezoresistive Accelerometer for Head Injuries Monitoring: A Computational Analysis by Increments of the Sensor Mass Moment of Inertia. Sensors 2018, 18, 289. [CrossRef] 17. Ciobotaru, T.; Frunzeti, D.; Rus, I.; J?ntschi, L. The working regime analysis of a track-type tractor. Int. J. Heavy Veh. Syst. 2012, 19, 172–187. 18. Korlath, G. Mobility analysis of off-road vehicles: Benefits for development, procurement and operation. J. Terramech. 2007, 44, 383–393. [CrossRef] 19. Sandu, C.; Freeman, J.S. Military tracked vehicle model. Part II: Case study. Int. J. Veh. Syst. Model. Test. 2005, 1, 216–231. [CrossRef] 20. Galati, R.; Reina, G. Terrain Awareness Using a Tracked Skid-Steering Vehicle with Passive Independent Suspensions. Robot. AI Robot. Control Syst. 2019, 6, 1–11. [CrossRef] 21. ?zdemir, M.N.; Kili?, V.; ?nl?soy, Y.S. A new contact & slip model for tracked vehicle transient dynamics on hard ground. J. Terramech. 2017, 73, 3–23. 22. Wong, J.Y.; Wei, H. Evaluation of the effects of design features on tracked vehicle mobility using an advanced computer simulation model. Int. J. Heavy Veh. Syst. 2005, 12, 344–365. [CrossRef] Sensors 2020, 20, 5587 31 of 31

23. Rubinstein, D.; Hitron, R. A detailed multi-body model for dynamic simulation of oﬀ-road tracked vehicles, 14th International Conference of the ISTVS. J. Terramech. 2004, 41, 163–173. [CrossRef] 24. Kang, O.; Park, Y.; Park, Y. Look-ahead preview control application to the high-mobility tracked vehicle model with trailing arms. J. Mech. Sci. Technol. 2009, 23, 914–917. [CrossRef] 25. Ordaz, P.; Rodríguez-Guerrero, L.; Santos, O.; Cuvas, C.; Romero, H.; Ordaz-Oliver, M.; López-Pérez, P. Parameter estimation of a second order system via non linear identiﬁcation algorithm. IOP Publ. Conf. Ser. Mater. Sci. Eng. 2020, 844, 012038. [CrossRef] 26. Vilău, R.; Marinescu, M.; Alexa, O.; Oloeriu, F.; Tru?ă, M. Advantages of spectrally analyzed data. Stochastic models for automotive measured parameters. Adv. Mater. Res. 2014, 1036, 493–498. [CrossRef] 27. Vilău, R.; Marinescu, M.; Alexa, O.; Tru?ă, M.; V?nturi?, V. Diagnose method based on spectral analysis of measured parameters. Adv. Mater. Res. 2014, 1036, 535–540. [CrossRef] 28. Grewal, M.S.; Andrews, A.P. Kalman Filtering: Theory and Practice Using Matlab, 4th ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2015; p. 640. 29. Grigore, L.?.; Popa, D.; Ciobotaru, T.; V?nturi?, V.; Popoviciu, B. Considerations regarding the measuring the performance of a vehicle during braking on a slope extended. Adv. Mater. Res. 2013, 718–720, 490–495. [CrossRef] 30. Xu, X.; Dong, P.; Liu, Y.; Zhang, H. Progress in Automotive Transmission Technology. Automot. Innov. 2018, 1, 187–210. [CrossRef] 31. Huang, W.; Wong, J.Y.; Preston-Thomas, J.; Jayakumar, P. Predicting terrain parameters for physics-based vehicle mobility models from cone index data. J. Terramech. 2020, 88, 29–40. [CrossRef] 32. Lee, S.H.; Lee, J.H.; Goo, S.H.; Cho, Y.C.; Cho, H.Y. An Evaluation of Relative Damage to the Powertrain System in Tracked Vehicles. Sensors 2009, 9, 1845–1859. [CrossRef] 33. Ma, X.; Wong, P.K.; Zhao, J.; Xie, Z. Multi-Objective Sliding Mode Control on Vehicle Cornering Stability with Variable Gear Ratio Actuator-Based Active Front Steering Systems. Sensors 2017, 17, 49. [CrossRef] 34. Wi, H.; Park, H.; Hong, D. Model Predictive Longitudinal Control for Heavy-Duty Vehicle Platoon Using Lead Vehicle Pedal Information. Int. J. Automot. Technol. 2020, 21, 563–569. [CrossRef]

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