Numerical Design Method for CVT Supported in Standard Variable Speed Rubber V-Belts

applied sciences

Article Numerical Design Method for CVT Supported in Standard Variable Speed Rubber V-Belts

Ivan Arango 1,2 and Sebastian Muñoz Alzate 2,*

1 School of Engineering, Mechanical Engineering Department, EAFIT University, Medellín 050022, Colombia; iarango@eaﬁt.edu.co 2 Mechanical Engineering Department, Mechatronics and Machine Design Research Group, EAFIT University, Medellín 050022, Colombia * Correspondence: smunoza@eaﬁt.edu.co; Tel.: +57-(4)-261-95-00

Received: 6 August 2020; Accepted: 3 September 2020; Published: 8 September 2020

Abstract: The design of a V-belt continuously variable transmission (CVT) system is a complex problem due to the multiple interactions between components during its operation. Literature on CVT system design methods is scarce, and the vast majority of works include implicit equations that hinder applications at a basic design level. This research aims to introduce a numerical CVT design method for electric vehicles (EV) and internal combustion engine (ICE) vehicles considering each one of their components and using mechanical centrifugal actuators and a rubber V-belt. This design method is based on user needs, for which there are three main requirements: road speciﬁcations, vehicle characteristics, and expected performance. This method is focused on a transmission for a vehicle traveling on the same route constantly, such as public transport vehicles. From three-wheelers to medium cargo vehicles, there is a greatly diverse range of potential applications for using this method for each type of standard rubber V-belt.

Keywords: CVT; rubber V-belt; transmission; electric vehicle; powertrain

1. Introduction The implementation of continuously variable transmission (CVT) systems in transport vehicles has been growing during the last decades [1,2] due to the inherent advantages of using this type of transmission [3]. CVTs offer the option of continuously varying their speed ratio over a wide range of angular speeds, allowing for an infinite number of ratios to be obtained [4,5]. CVT transmissions in vehicles achieve lower values of polluting gases emissions [6–8], involve smaller parts, weigh less, and allow for better energy performance when compared to gearboxes [3,9]. These systems allow engine operation to be sustained at maximum efficiency, which improves energy consumption and enhances driving comfort either in electric vehicles (EV), internal combustion engine (ICE) vehicles, and hybrid vehicles [10,11]. Although CVTs have relatively low local efficiency compared to traditional vehicle transmissions [5], when operating in conjunction with an engine or motor, the system’s overall efficiency increases, due to CVTs operating more frequently in the regions of higher engine/motor efficiency [10,12,13]. The following articles present a component-based approach to address the design of this type of transmission. A taxonomic description of the most sensitive research in the field of CVT design is presented below. Certain parameters such as clamping force, slip between pulleys, torque load, and many others must be considered when designing a CVT, because the interactions between them affects the operation, efficiency, and conceptualization of the system [5,14–16]. The belt is the main component responsible for developing and interacting with the mentioned forces. Therefore, authors such as Kim et al. in [4] described a formula to easily determine the axial forces present in a rubber V-belt CVT with

Appl. Sci. 2020, 10, 6238; doi:10.3390/app10186238 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 6238 2 of 23 assumptions such as a negligible slip angle in the driven pulley active area and a constant belt tension on the driver pulley with an accuracy of 5%. Furthermore, they found that when it is desired to maintain a constant speed, the driver pulley axial force has a linear behavior against the increased system torque load. In [17], the authors compare three analytical rubber V-belt models; the Kim–Kim model, the Dittrich model, and the Cammalleri model, where the most sensitive results indicate that the Kim–Kim model describes the CVT transmission working accurately based on the experimental data. Other researchers such as Asayama et al. [18] developed an analytical model of metal belts behavior, in order to specify the variations on belt tension that take place when operating CVT. The study also allows the method to determine the clamping force under certain conditions, and thus determine the required pressure to prevent slippage on the belt. The efficiency of the CVT systems depends on the required clamping force values and the belt slip. Regarding the latter, in research by [19,20] the authors use metal V-belts and argue that the belt slip depends on the driven pulley capacity to transmit torque; considering a relation between torque, clamping force, the friction coefficient between the pulleys and the belt, and the speed ratio. In the operation of the metal belts CVT system, the metal belt passes through the idle sector of the pulley, a gap is generated between the members, generating a micro-slip and thus causing a loss in the torque transmission [14]. At low speed ratios this type of belt adds power transmission to the system, but at high ratios this capacity decreases [2]. Furthermore, the efficiency of a CVT decreases against high values of clamping force [21]. In chain CVT systems, the chain pins and the pulley plates are in contact. This interaction provides the power transmission from the driver pulley to the driven pulley. This leads to impacts when the chain pins and links pass in and out of the pulleys [2]. Although the belt is an important element in the design of the CVT, there are also other crucial elements that allow the change of the speed ratio and interaction with the belt. The CVTs speed ratio actuation and control are given by a variety of mechanisms present on the pulleys. The actuators design has been an important matter discussed by different authors. Cammalleri [22] presents a model to design the mechanical actuators for the driver and driven pulleys, being the actuators, respectively, a centrifugal roller and a helical torque cam with a compression torsion spring. The model presented allows calculation of the torque required by the system to be kept as close as possible to that provided by the engine/motor, thus increasing efficiency. The latter mechanisms are usually installed in low-powered vehicles. Complementing the above, different authors present models, designs and control strategies of these systems that allow pulleys to be operated with higher power systems, such as electric [23,24], hydraulic; electro-hydraulic [25,26], and electro-mechanical actuators [27]. These systems are approached from the control of the speed ratio change, and thus guarantee better performance and efficiency [28,29]. When analyzing the drive system, literature provides evidence that an electric motor delivers 100% of the torque when operated from rest while the efficiency curves present different topologies and values over 80% providing a better dynamic control capacity [30]. Zeraoulia et al. [31] define which characteristics, such as speed range and energy efficiency, are factors influenced by system dynamics and architecture, so the selection of the drive system requires specific emphasis on both variables. Additionally, Ruan et al. [5] discuss in their research, that the powertrain design specifications are more straightforward for an EV compared to an ICE. The complexity of a design method for CVT lies in factors such as component selection (belt, kind of actuators, pulleys, and engine/motor), interaction with other systems, and design parameters optimization. Different authors present several models and designs of CVT systems. Hofman et al. [32], present the powertrain design for hybrid vehicles, implementing and analyzing the best kind of CVT for each vehicle. Herein, fuel consumption is considered as the main design objective, where the proposed solution has an average error of less than 1.6%. However, they do mention that it is difficult to find an optimal overall design solution due to the interdependence of design options regarding topology of the drive train and its implications. Fahdzyana and Hofman [33], propose an integrated design method of an electromechanically actuated CVT and its control system, minimizing CVT mass up to 46.2%; tracking error, control effort up to 62%; and minimizing the distance between centers. However, they Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 22 Appl. Sci. 2020, 10, 6238 3 of 23 belt slippage. The authors also mention that a vehicle powertrain is a complex dynamic system due to the mathematical complexity of its designs. do not consider factors such as efficiency and control of belt slippage. The authors also mention that a Despite these proposals, some equations and approaches are complex and do not explicitly vehicle powertrain is a complex dynamic system due to the mathematical complexity of its designs. apply at a basic design level, and the methods do not address the design of the CVT from each of its Despite these proposals, some equations and approaches are complex and do not explicitly components in a practical manner. Therefore, this research aims to introduce a numerical CVT apply at a basic design level, and the methods do not address the design of the CVT from each of design method for ICEs and EVs considering each component, using mechanical centrifugal its components in a practical manner. Therefore, this research aims to introduce a numerical CVT actuators and standard rubber V-belts. The method is validated on simulations under real operation design method for ICEs and EVs considering each component, using mechanical centrifugal actuators conditions. The numerical design method is presented in Section 2. The methodology developed to and standard rubber V-belts. The method is validated on simulations under real operation conditions. evaluate the veracity of the method is explained in Section 3. The example analysis and discussion The numerical design method is presented in Section2. The methodology developed to evaluate the are shown in Section 4. Finally, Section 5 presents the conclusions. veracity of the method is explained in Section3. The example analysis and discussion are shown in 2.Section Numerical4. Finally, Continuously Section5 presents Variable the Transmission conclusions. (CVT) Design Method 2. NumericalThe design Continuously process is based Variable on Transmissionwhat the user (CVT) needs, Design mainly, Method on three requirements: road specifications, vehicle characteristics, and expected performance. This method focuses on a The design process is based on what the user needs, mainly, on three requirements: road transmission for a vehicle that is constantly traveling on the same route, such as public specifications, vehicle characteristics, and expected performance. This method focuses on a transmission transportation vehicles. for a vehicle that is constantly traveling on the same route, such as public transportation vehicles.

2.1. Initial Initial Requirements Requirements The firstfirst step step in thein designthe design process process implies implie gatherings gathering the essential the requirements.essential requirements. The topography The topographyof the route onof whichthe route you wanton which to operate you thewant vehicle to operate is required, the vehicle from this, is therequired, maximum from inclination this, the maximum inclination angle (θmax) is determined. Nevertheless, you can establish the maximum angle (θmax) is determined. Nevertheless, you can establish the maximum inclination degree the inclinationvehicle can degree climb. Twothe vehicle different can vehicle climb. top Two speed diff valueserent vehicle are also top required: speed casevalues 1—on are inclinedalso required: roads, case 1—on inclined roads, and case 2—on flat roads. Using vehicle conditions such as maximum and case 2—on flat roads. Using vehicle conditions such as maximum vehicle total mass (mv), vehicle vehicle total mass (mv), vehicle front area (Af), and wheel diameter (ϕwh), the system dynamics for front area (Af), and wheel diameter (φwh), the system dynamics for both cases are determined. both cases are determined. 2.2. Power Selection 2.2. Power Selection The selection of the electric motor or engine relies on the highest value the force’s distribution alongThe the selection longitudinal of the axis electric of the motor vehicle or (Figure engine1 reli). Consideres on the case highest 1 in value which the the force’s vehicle distribution climbs the alongsteepest the slope longitudinal of the route axis ( θofmax the) andvehicle case (Figure 2 where 1). the Consider vehicle case travels 1 in at which maximum the vehicle speed. climbs the steepest slope of the route (θmax) and case 2 where the vehicle travels at maximum speed.

Figure 1. ForcesForces distribution distribution along longitudinal axis of the vehicle for case 1. .. For case 1, the principle of forces is determined using Equation (1). Acceleration (x) is determined For case 1, the principle of forces is determined using Equation (1). Acceleration (ẍ) is by the need to go from zero to the chosen top speed. Poor acceleration makes traﬃc uncomfortable on determined by the need to go from zero to the chosen top speed. Poor acceleration makes traffic the road and a high acceleration demands a powerful engine/motor. uncomfortable on the road and a high acceleration demands a powerful engine/motor. .. Ft = mvx=+ Fclimb+ ++Fdrag++ Frolling, (1) 1 Ft1 mvx Fclimb Fdrag Frolling, (1) where the tractiontraction climbingclimbing forceforce isis FFclimbclimb,, the the aerodynamic aerodynamic force force is Fdrag andand the the force force due due to rolling resistance is Frolling. . For case 2, where the vehicle travels at the max speed,

=.. + + Ft2 mv x Fdrag2 Frolling2 , (2) Ft2 = mvx + Fdrag2 + Frolling2, (2)

Where Fdrag2 is the aerodynamic force and Frolling2 is the force due to rolling resistance for case 2. Where Fdrag2 is the aerodynamic force and Frolling2 is the force due to rolling resistance for case 2. Appl. Sci. 2020, 10, 6238 4 of 23

The power required by the engine/motor is determined using Equation (3), Ft is the greatest force . for both cases Ft1 and Ft2, and x is the linear speed of the vehicle corresponding to this force.

. Po = Ftx, (3)

The selected engine/motor power is determined by Po plus the sum of the additional power required by the vehicle’s accessories. From this selection, it is possible to obtain maximum torque Tomt at its respective speed ωmt, and torque Topow at maximum power at a speciﬁed speed.

2.3. Speed Ratio of CVT and Gearbox Once the engine/motor is selected, it is time to calculate the overall speed ratio of the system. A powertrain for EV usually consists of a gearbox, differential reducer, and a CVT [11,27] or a transaxle with a CVT. The first speed ratio is determined for case 2 when Rcvt_min has a minimum value of 1:1. For this case, the vehicle is at maximum linear speed and the engine/motor will be working at the maximum power zone. Towh is the wheel torque and η is the efficiency. It is possible to have a speed ratio of less than one, yet it is not efficient to increase angular speed to then reduce it again within the powertrain. TopowRcvt_minη Towh Ft2 = = , (4) (0.5φwh) (0.5φwh)

0.5φwhFt2 Rgear = , (5) TopowRcvt_minη Equation (9) is used to determine the maximum ﬁxed reduction system ratio (gearbox and/or diﬀerential reducer). However, its design is not discussed further in this article. For case 1, the vehicle uses the energy to overcome mainly its inertia on climbs. Using Rgear and Equations (6) and (7) the maximum CVT speed ratio Rcvt_max can be determined.

TomtRcvt_maxRgearη Towh Ft1 = = , (6) (0.5φwh) (0.5φwh)

0.5φwhFt1 Rcvt_max = , (7) TomtRgearη

2.4. Rubber V-Belt Selection CVT belts are commercially available, and feature both metric and imperial systems. The belt tension capacity (FTmax) is a function of the section dimensions. For a wider belt there is a tendency to have a greater number of reinforcement members; and at higher heights, it tends to have a thicker friction surface and a higher minimum diameter. The belt reinforcement’s quality indicates the speciﬁc tension value in (N/mm2) it can withstand. The maximum tension supported by the belt is the product of the sectional area multiplied by its maximum speciﬁc stress. Also, the belt must resist tensile forces such as working tension (FTwork), centrifugal force (FTce) tension which is a function of the radius, the linear density of the belt, the angular speed (rpm), and bending tension (FTdo) when passing through the pulleys. This latter tension is a function of the belt thickness and the radius pulleys (Figure2). Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 22

The power required by the engine/motor is determined using Equation (3), Ft is the greatest force for both cases Ft1 and Ft2, and ẋ is the linear speed of the vehicle corresponding to this force.

= Po Ft x , (3)

The selected engine/motor power is determined by Po plus the sum of the additional power required by the vehicle’s accessories. From this selection, it is possible to obtain maximum torque Tomt at its respective speed 𝜔mt, and torque Topow at maximum power at a specified speed.

2.3. Speed Ratio of CVT and Gearbox Once the engine/motor is selected, it is time to calculate the overall speed ratio of the system. A powertrain for EV usually consists of a gearbox, differential reducer, and a CVT [11,27] or a transaxle with a CVT. The first speed ratio is determined for case 2 when Rcvt_min has a minimum value of 1:1. For this case, the vehicle is at maximum linear speed and the engine/motor will be working at the maximum power zone. Towh is the wheel torque and η is the efficiency. It is possible to have a speed ratio of less than one, yet it is not efficient to increase angular speed to then reduce it again within the powertrain.

To R η To Ft = pow cvt_min = wh , 2 φ φ (4) (0.5 wh) (0.5 wh)

0.5φ Ft R = wh 2 , gear η (5) Topow Rcvt _ min

Equation (9) is used to determine the maximum fixed reduction system ratio (gearbox and/or differential reducer). However, its design is not discussed further in this article. For case 1, the vehicle uses the energy to overcome mainly its inertia on climbs. Using Rgear and Equations (6) and (7) the maximum CVT speed ratio Rcvt_max can be determined.

To R R η To Ft = mt cvt_max gear = wh , 1 φ φ (6) (0.5 wh) (0.5 wh)

0.5φ Ft R = wh 1 , cvt _ max η (7) Tomt Rgear

2.4. Rubber V-Belt Selection CVT belts are commercially available, and feature both metric and imperial systems. The belt tension capacity (FTmax) is a function of the section dimensions. For a wider belt there is a tendency to have a greater number of reinforcement members; and at higher heights, it tends to have a thicker friction surface and a higher minimum diameter. The belt reinforcement’s quality indicates the specific tension value in (N/mm2) it can withstand. The maximum tension supported by the belt is the product of the sectional area multiplied by its maximum specific stress. Also, the belt must resist tensile forces such as working tension (FTwork), centrifugal force (FTce) tension which is a function of the radius, the linear density of the belt, the angular speed (rpm), and bending tension (FTdo) when Appl. Sci. 2020, 10, 6238 5 of 23 passing through the pulleys. This latter tension is a function of the belt thickness and the radius pulleys (Figure 2).

Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 22

The sum of the working tension, the centrifugal force tension, and the bending tension must not be greater than the maximum tension of the belt (Equation (9)). Therefore, an iterative process based FigureFigure 2.2. WorkingWorking tensionstensions inin thethe rubberrubber V-belt.V-belt. on commercial references is presented to determine the minimum driver pulley diameter and the belt sectionThe sum selection of the working (Figure 3). tension, The minimum the centrifugal pulley force diameter tension, depends and the on bending CVT belts tension specifications, must not besuch greater as belt than thickness, the maximum topology, tension whether of the it belthas teeth (Equation on the (9)). inside Therefore, and outside an iterative face which process decrease based onthe commercial bending stress, references and the is side presented section to area determine of the belt. the minimumThe process driver begins pulley by selecting diameter an and initial the belt sectionsection selectionfrom where (Figure to obtain3). The transversal minimum pulleyarea value diameter (Abelt); depends section ondimensions CVT belts such specifications, as width ( suchWbel); asthickness belt thickness, (hbel); belt topology, angle (φ whetherbel); maximum it has tensile teeth on stress; theinside and belt and density outside (ρ facebel). Then, which these decrease tensions the bendingand the stress,minimum and operational the side section diameter area ofare the dete belt.rmined. The process The minimum begins by driver selecting pulley an initial diameter belt section(ϕ1_min) is from determined where to using obtain Equation transversal (8); area this value equation (Abelt was); section developed dimensions by iterating such as the width minimum (Wbel); thicknessdiameter (untilhbel); beltthe bending angle (ϕ belstress); maximum is significant. tensile The stress; diameter and belt can density also (beρbel checked). Then, thesein norms tensions and andstandards the minimum [34]. It operationalis verified diameterthat for the are enti determined.re range Theof motor minimum angular driver speeds pulley (rpm) diameter the (sumφ1_min of) istensions determined is not usinggreater Equation than the (8); tension this equation that the belt was withstand developed (Equation by iterating (9)). the Finally, minimum it is validated diameter untilthat the the recommended bending stress lateral is significant. pressure The is not diameter exceeded can either. also be checked in norms and standards [34]. It is verified that for the entire range of motor angular speeds (rpm) the sum of tensions is not greater φ =15.5+ (2.5h 1.3 ), (8) than the tension that the belt withstand (Equation1_min (9)). Finally,bel it is validated that the recommended lateral pressure is not exceeded either. > + + , FT max FTwork FTce FTdo (9) 1.3 The maximum stress that the beltφ1_ mincan= withstand15.5 + 2.5 ish bela function, of the material’s properties. (8)In Equation (10) this stress is calculated, where σbelt is the maximum tensile stress. FTmax > FTwork + FTce + FTdo, (9) = σ FT max Abelt belt , (10)

Figure 3. Belt selection ﬂowchart. Figure 3. Belt selection flowchart.

The working tension is given by Equation (11), where (ϕ1_min) is the minimum driver pulley diameter.

To , F = mt Twork φ (11) (0.5 1_ min )

Assuming that the non-tensioned side of the belt is not significant when compared to the tensioned side, due to the high friction coefficient μ between the pulley-belt and the belt angle (φbel), the approximation in Equation (12) is proposed. By calculating the working force it is possible to solve the minimum pulley diameter (ϕ1_min) using Equation (11). Appl. Sci. 2020, 10, 6238 6 of 23

The maximum stress that the belt can withstand is a function of the material’s properties. In Equation (10) this stress is calculated, where σbelt is the maximum tensile stress.

FTmax = Abeltσbelt, (10)

The working tension is given by Equation (11), where (φ1_min) is the minimum driver pulley diameter. Tomt FTwork = , (11) (0.5φ1_min) Assuming that the non-tensioned side of the belt is not signiﬁcant when compared to the tensioned side, due to the high friction coeﬃcient µ between the pulley-belt and the belt angle (φbel), the approximation in Equation (12) is proposed. By calculating the working force it is possible to solve the minimum pulley diameter (φ1_min) using Equation (11).

F F , (12) Twork ≈ 1 F eµτcc 1 = , (13) F2 sin(0.5ϕbel)

The tension produced by the centrifugal force is mainly given to the linear speed of the belt (VLbelt).

2 FTce = Abeltρbel(VLbelt) , (14)

The linear speed belt can be determined from the following expression, where ωmt is the angular speed at maximum engine/motor torque. π V = φ ω , (15) Lbelt 60 1_min mt

The belt density can be determined using Equation (16), where ρlbel is the belt’s lineal density from manufacturers. ρlbel ρbel = , (16) Abelt A bending stress is produced when the belt passes through the pulleys. This stress is a function of the relationship between the belt thickness and the belt diameter. εbb is the equivalent bending stress considering the location of the reinforcements in the transversal section and the belt teeth geometry.

hbel FTdo = εbb , (17) 0.5φ1_min

2.5. Pulley Center Distance and Belt Length The distance between centers of CVT pulleys (C) for vehicles should be kept to a minimum given the space in which they are located [33]. The limitation is given by the contact angle τcc between the belt and the smallest pulley, knowing that τcc 160 avoids the torque losses. An approach to ≥ ◦ determine C considering this constraint is to analyze the right triangle (Figure4) which is formed between the two pulleys at the max speed ratio using Equation (18).

[0.5(φ2_max φ1_min)] tan(10) = 0.176 = − , (18) C Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 22

≈ FTwork F1 , (12)

μ τ cc F1 = e , ϕ (13) F2 sin( 0.5 bel )

The tension produced by the centrifugal force is mainly given to the linear speed of the belt (VLbelt).

= ρ 2 FTce Abelt bel (VLbelt) , (14)

The linear speed belt can be determined from the following expression, where 𝜔mt is the angular speed at maximum engine/motor torque. π = φ ω , VLbelt 1_ min mt (15) 60

The belt density can be determined using Equation (16), where ρlbel is the belt’s lineal density from manufacturers. ρ ρ = lbel , bel (16) Abelt

A bending stress is produced when the belt passes through the pulleys. This stress is a function of the relationship between the belt thickness and the belt diameter. εbb is the equivalent bending stress considering the location of the reinforcements in the transversal section and the belt teeth geometry.

h , F = ε bel Tdo bb φ (17) 0.5 1_ min

2.5. Pulley Center Distance and Belt Length The distance between centers of CVT pulleys (C) for vehicles should be kept to a minimum given the space in which they are located [33]. The limitation is given by the contact angle τcc between the belt and the smallest pulley, knowing that τcc ≥ 160° avoids the torque losses. An Appl. Sci. 2020, 10, 6238 7 of 23 approach to determine C considering this constraint is to analyze the right triangle (Figure 4) which is formed between the two pulleys at the max speed ratio using Equation (18).

Figure 4. Distance between centers and τcc. Figure 4. Distance between centers and τcc. By accepting this approximation, it can be established that the length of the rubber V-belt is then: [0.5(φ −φ )] q tan(10) = 0.176 = 2_ max 1_ min , (18) 2 2 C L = 4C (φ max φ ) + 0.5[φ max(τcc_out + φ )τ ], (19) − 2_ − 1_min 2_ 1_min cc_in By accepting this approximation, it can be established that the length of the rubber V-belt is then:where φ2_max is the maximum driven diameter and it is determined with the expression (φ1_min Rcvt_max). The values for the driver and driven pulley’s contact angles are τcc_in and τcc_out respectively. The exact = 2 − ()φ −φ 2 + []φ τ + φ τ , angle of contact between theL belt4C and driver2 _ max 1 pulley_ min 0 is,.5 2 _ max ( cc _ out 1_ min ) cc _ in (19) Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 22   where ϕ2_max is the maximum driven diameter and φ it is determinedφ with the expression (ϕ1_min 1 2_max 1_min  Rcvt_max). The values for the driverτcc and_in = drπiven2 sinpulley’s−  contact− angles ,are τcc_in and τcc_out respectively. (20) −   φ −2φc   −1 2_ max 1_ min , The exact angle of contact between theτ belt= andπ − 2 driversin  pulley is,  cc_in  2c  (20) The exact angle of contact between the belt and driven pulley is, The exact angle of contact between the belt and driven pulley is,  φ φ 1 2_max 1_min  τcc_out = π + 2 sin−  φ −−φ  , (21) −1 2_ max 1_ min ,  τ = π + 2sin  2c  cc_ out  2c  (21)   2.6. Complementary Diameters

2.6. ComplementaryDimensions for Diameters the driver and driven pulleys can be determined by knowing the minimum and maximum speed ratios of the CVT system (Rcvt_min and Rcvt_max) (Table1). Dimensions for the driver and driven pulleys can be determined by knowing the minimum and maximum speed ratiosTable of 1. theContinuously CVT system variable (Rcvt_min transmission and Rcvt_max (CVT)) (Table pulley 1). diameters.

Table 1. ContinuouslySpeed Ratio variable transmission Diameters (CVT) pulley diameters.

RSpeedcvt_min Ratio Diametersφ1_max = φ2_min Rcvt_max φ2_max = φ1_min Rcvt_max Rcvt_min ϕ1_max = ϕ2_min Rcvt_max ϕ2_max = ϕ1_min Rcvt_max 2.7. Lateral Displacement, Tensile and Clamping Forces 2.7. Lateral Displacement, Tensile and Clamping Forces The belt angle ϕbel and the minimum and maximum radius of the pulleys determine the lateral displacementThe belt angleσLat (Figure φbel and5). the minimum and maximum radius of the pulleys determine the lateral displacement σLat (Figure 5).

Figure 5. LateralLateral displacement displacement scheme. scheme.

[0.5 (φ −φ )] tan(0.5ϕ ) = 1_ max 1_ min , bel σ (22) lat

Trigonometric relationships are constructed with the geometry of the pulley and the belt (Figure 6), which relate the parameters to the radial force (Frad), the normal force (FNor), the clamping force (Fcl) and the tangential tension to the belt centerline which is equivalent to the friction force (Ffric) between the belt and the pulley without considering slip [35]. The following equations describe these forces as a function of the tensions F1 and F2, F1 is calculated considering Equation (12).

 ΔF −  1  Δτ  ΔF = F + 1 2 sin cc , Nor  1  ()ϕ   (23)  2  sin 0.5 bel  2 

Δ Δτ Δ =  + F1−2   cc  , Frad 2F1  sin  (24)  2   2 

Δ Δτ Δ =  + F1−2   cc  , 2 ( Ffric ) F1 sin  (25)  2   2 

 ΔF −  cos()()0.5ϕ sin 0.5Δτ ΔF = F + 1 2 bel cc , cl  1  ()ϕ (26)  2  sin 0.5 bel Appl. Sci. 2020, 10, 6238 8 of 23

[0.5(φ1_max φ1_min)] tan(0.5ϕbel) = − , (22) σlat Trigonometric relationships are constructed with the geometry of the pulley and the belt (Figure6), which relate the parameters to the radial force (Frad), the normal force (FNor), the clamping force (Fcl) and the tangential tension to the belt centerline which is equivalent to the friction force (Ffric) between the belt and the pulley without considering slip [35]. The following equations describe these forces as a function of the tensions F1 and F2, F1 is calculated considering Equation (12). ∆F1 2 1 ∆τcc ∆FNor = F1 + − sin , (23) 2 sin(0.5ϕbel) 2 ∆F1 2 ∆τcc ∆F = 2 F + − sin , (24) rad 1 2 2 ∆F1 2 ∆τcc 2(∆F ) = F + − sin , (25) f ric 1 2 2 ∆F1 2 cos(0.5ϕbel) sin(0.5∆τcc) ∆Fcl = F1 + − , (26) 2 sin(0.5ϕbel) Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 22

FigureFigure 6. 6. RubberRubber V-belt V-belt system system forces. forces.

2.8.2.8. Misalignment Misalignment of of Pulleys Pulleys by by Diameter Diameter Change Change LateralLateral displacement isis not not the the same same when when the pulleys the pulleys change diameter.change diameter. This causes This a misalignment causes a misalignmentin the system, in which the issystem, calculated which using is calculated Equation (27) using for eachEquation pulley, (27) in orderfor each to verify pulley, if the in maximumorder to verifypermissible if the maximum value is exceeded. permissible Two value cases is are exceeded. analyzed, Two ﬁrst cases when are the analyzed, driver pulley first iswhen in its the minimum driver pulleydiameter is in and its theminimum driven pulleydiameter is maximum and the driven in diameter, pulley and is maximum the second in case diameter, when the and inverse the second occurs. caseThis when is done the to inverse consider occurs. the minimum This is done and to maximum consider the possible minimum ratios. and maximum possible ratios.

 σ +σ  ∠ = tan1−1 σlat11 +latσ2lat,2 ∠ =aligtan− |  | ,(27) (27) alig  c c 

ToTo prevent prevent belt belt crossing, crossing, diffe diﬀerentrent pulley anglesangles areare usedusedϕ φpull_1pull_1 andand φϕpull_2pull_2. One. One is is larger larger than than the the beltbelt angle angle ϕφbel and thethe otherother one one smaller, smaller, without without exceeding exceed theing degreesthe degrees that a ﬀthatect theaffect contact the betweencontact betweenthe belt the and belt the and pulley. the Thepulley. admitted The admitted angle variation angle variation can be betweencan be between two and two three and degrees. three degrees.

2.9.2.9. Force Force Balance Balance TheThe difference difference in in clamping clamping forces forces in in the the driver driver and and driven driven pulleys pulleys for for CVT CVT systems systems leads leads to to a a changechange in in the the ratio ratio RRcvtcvt. When. When the the lateral lateral force force of of the the driver driver pulley pulley exceeds exceeds the the driven driven one, one, a alateral lateral closingclosing displacement displacement in in the the driver pulleypulley occursoccurs andand the the belt belt occupies occupies a greatera greater radius. radius. Given Given that that the thelength length of theof the belt belt is fixed, is fixed, the the driven driven pulley pulley will will open open where where the the belt belt movers movers to a to lower a lower radius. radius. This Thischange change in radius in radius modifies modifies the ratiothe ratio between between them, them, increasing increasing the speedthe speed of the of driventhe driven pulley. pulley. InIn this this case, case, there isis aa setset of of centrifugal centrifugal elements elements to to develop develop these these forces forces in the in driverthe driver pulley. pulley. These Theseinvolve involve a ramp a whichramp which reorients reorients part of part the radial of the force radial (F cenforce) in ( anFcen axial) in an or lateralaxial or direction lateral direction to the pulley. to the pulley. For the driven pulley, this force is derived from a spring. The increase or decrease in this force is carried out by compressing the spring with a cam. To transmit more torque the cam compresses the spring which is actuated by the torque on the output shaft. These lateral forces have an additional function, they produce a lateral compression on the belt called clamping force (Fcl), which prevents the belt from slipping and losing efficiency. Figure 7 presents four possibilities for this balance of forces, which determines the dynamics of the mechanism. The numbers correspond to the positions of the torque and angular speed graph in Figure 8.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 22

Figure 6. Rubber V-belt system forces.

2.8. Misalignment of Pulleys by Diameter Change Lateral displacement is not the same when the pulleys change diameter. This causes a misalignment in the system, which is calculated using Equation (27) for each pulley, in order to verify if the maximum permissible value is exceeded. Two cases are analyzed, first when the driver pulley is in its minimum diameter and the driven pulley is maximum in diameter, and the second case when the inverse occurs. This is done to consider the minimum and maximum possible ratios.

 σ +σ  ∠ = −1 lat1 lat2 , alig tan   (27)  c 

To prevent belt crossing, different pulley angles are used φpull_1 and φpull_2. One is larger than the belt angle φbel and the other one smaller, without exceeding the degrees that affect the contact between the belt and the pulley. The admitted angle variation can be between two and three degrees.

2.9. Force Balance The difference in clamping forces in the driver and driven pulleys for CVT systems leads to a change in the ratio Rcvt. When the lateral force of the driver pulley exceeds the driven one, a lateral closing displacement in the driver pulley occurs and the belt occupies a greater radius. Given that the length of the belt is fixed, the driven pulley will open where the belt movers to a lower radius.

ThisAppl. change Sci. 2020, in10 ,radius 6238 modifies the ratio between them, increasing the speed of the driven pulley.9 of 23 In this case, there is a set of centrifugal elements to develop these forces in the driver pulley. These involve a ramp which reorients part of the radial force (Fcen) in an axial or lateral direction to theFor pulley. the driven For pulley,the driven this forcepulley, is derivedthis force from is derived a spring. from The a increase spring. orThe decrease increase in or this decrease force is carriedin this forceout by is compressingcarried out theby springcompressing with a cam.the spring To transmit with a more cam. torque To transmit the cam more compresses torque the the spring cam compresseswhich is actuated the spring by the which torque is actuated on the output by the shaft. torque on the output shaft. TheseThese lateral lateral forces forces have have an an additional additional function, function, they they produce produce a a lateral lateral compression compression on on the the belt belt calledAppl.called Sci. clamping clamping 2020, 10, x FORforce force PEER ( (FFcl clREVIEW),), which which prevents prevents the the belt belt from from slipping slipping and and losing losing efficiency. eﬃciency. Figure Figure9 of 2277 presentspresents fourfour possibilitiespossibilities for for this this balance balance of forces, of forces, which determineswhich determines the dynamics the dynamics of the mechanism. of the mechanism.The numbers The correspond numbers to correspond the positions to ofthe the positi torqueons and of the angular torque speed and graph angular in Figure speed8 .graph in Figure 8.

Appl. FigureSci. 2020 7., 10 (a, )x PossibleFOR PEER operational REVIEW situations: low speed, high torque (b); high speed, high torque (c);9 of 22 low speed, low torque (d); high speed, low torque (e).

Before calculating the parameters of the centrifugal actuator, certain data must be defined, which then results in a graph as presented in Figure 8. A blue curve is drawn on the motor torque graph, which proportionally corresponds to the lateral force required to transmit the torque without slippage. Additionally, the centrifugal force is shown in red; the desired component of the centrifugal force in the axial or lateral direction in orange; and the required lateral force on the driven pulley spring in green. Finally, in Figure 8 the ratio change zone is highlighted in a grey section. The designer chooses two engine speeds: the rpm 𝜔CVT_ini where the CVT should start changing ratio, usually close to the maximum torque, and a second speed 𝜔CVT_end where the speed ratioFigure change 8.8. should TheThe clamping clamping end, usually force force (blue), near(blue), the maximum centrifugalthe centrifugal engine force (red),fo efficiency.rce the (red), centrifugal the centrifugal force’s lateral force’s component lateral component(yellow) and (yellow) the spring and force the (green)spring areforce drawn (green) over are the drawn torque andover angular the torque speed and motor. angular speed motor. Before calculating the parameters of the centrifugal actuator, certain data must be deﬁned, which 2.10.then Driver results Pulley in a graph Actuators as presented in Figure8. A blue curve is drawn on the motor torque graph, which proportionally corresponds to the lateral force required to transmit the torque without slippage. The driver pulley for CVT systems has centrifugal actuators in conjunction with a spring that Additionally, the centrifugal force is shown in red; the desired component of the centrifugal force in generates the clamping force required to perform the transmission ratio change Rcvt [22]. As shown the axial or lateral direction in orange; and the required lateral force on the driven pulley spring in in Figure 9, there are two types of CVTs, each with advantages and disadvantages. The first type are green. Finally, in Figure8 the ratio change zone is highlighted in a grey section. The designer chooses weight and ramp systems and the second a flyweight cam system. two engine speeds: the rpm ωCVT_ini where the CVT should start changing ratio, usually close to the maximumFigure torque, 8. The andclamping a second force speed (blue),ωCVT_end the centrifugalwhere the fo speedrce (red), ratio the change centrifugal should force’s end, usually lateral near maximumcomponent engine (yellow) eﬃciency. and the spring force (green) are drawn over the torque and angular speed motor. 2.10. Driver Pulley Actuators 2.10. Driver Pulley Actuators The driver pulley for CVT systems has centrifugal actuators in conjunction with a spring that generatesThe driver the clamping pulley forcefor CVT required systems to perform has centrifugal the transmission actuators ratio in conjunction change Rcvt with[22]. a As spring shown that in generates the clamping force required to perform the transmission ratio change Rcvt [22]. As shown

in Figure 9, there are two types of CVTs, each with advantages and disadvantages. The first type are weight andFigure ramp 9. systems Centrifugal and actuation the second system: a flyweight (a) ramps cam and system.roller weights, (b) flyweight cam.

2.10.1. Ramp and Weight System Design The method for achieving this centrifugal force, which is nevertheless changing and has a component in the axial direction, consists of placing three weights (mwe) on the driver pulley with the possibility of them moving radially. When the pulley is rotated, the weights will experience a

Figure 9. Centrifugal actuation system: (a) ramps and roller weights, (b) flyweight cam.

Figure 7. (a) Possible operational situations: low speed, high torque (b); high speed, high torque (c); low speed, low torque (d); high speed, low torque (e).

Figure 8. The clamping force (blue), the centrifugal force (red), the centrifugal force’s lateral component (yellow) and the spring force (green) are drawn over the torque and angular speed motor.

Appl.2.10. Sci.Driver2020 ,Pulley10, 6238 Actuators 10 of 23 The driver pulley for CVT systems has centrifugal actuators in conjunction with a spring that generates the clamping force required to perform the transmission ratio change Rcvt [22]. As shown Figure9, there are two types of CVTs, each with advantages and disadvantages. The ﬁrst type are in Figure 9, there are two types of CVTs, each with advantages and disadvantages. The first type are weight and ramp systems and the second a ﬂyweight cam system. weight and ramp systems and the second a flyweight cam system.

Figure 9. Centrifugal actuation system: (a) ramps and roller weights, (b) ﬂyweight cam. Figure 9. Centrifugal actuation system: (a) ramps and roller weights, (b) flyweight cam. 2.10.1. Ramp and Weight System Design 2.10.1. Ramp and Weight System Design The method for achieving this centrifugal force, which is nevertheless changing and has a The method for achieving this centrifugal force, which is nevertheless changing and has a component in the axial direction, consists of placing three weights (mwe) on the driver pulley with componenttheAppl. possibility Sci. 2020 in, 10 the, of x FOR themaxial PEER direction, moving REVIEW radially. consists Whenof placing the three pulley weights is rotated, (mwe) the on weightsthe driver will pulley experience with10 theof a22 possibilitycentrifugal of force, them then moving the weights radial willly. When be displaced the pulley in a rampis rotated, that has the a variableweights anglewill experience such that the a centrifugal force, then the weights will be displaced in a ramp that has a variable angle such that the axial component varies, as shown in Figure 10. axial component varies, as shown in Figure 10.

Figure 10. Diagram of axial force (F ) derived from the centrifugal force due to the ramp. Figure 10. Diagram of axial force (Fcl_wecl_we) derived from the centrifugal force due to the ramp. Knowing the maximum and minimum values in the change zone (Figure8) for angular speed, Knowing the maximum and minimum values in the change zone (Figure 8) for angular speed, displacements, forces, and speed ratio, ranges are established and divided into nwe range differentials. displacements, forces, and speed ratio, ranges are established and divided into nwe range In order to obtain the result, each weight’s radial displacement differential is associated with the differentials. In order to obtain the result, each weight’s radial displacement differential is associated required lateral force, a centrifugal force, an angular speed, a speed ratio, and the ramp angle is to be with the required lateral force, a centrifugal force, an angular speed, a speed ratio, and the ramp calculated. The location of the starting point for the first ramp differential is an input parameter in the angle is to be calculated. The location of the starting point for the first ramp differential is an input iterative method. For each iteration, the previous differential points are connected to the beginning of parameter in the iterative method. For each iteration, the previous differential points are connected the next one in order to generate continuity. The result is the ramp or profile of the actuator as shown to the beginning of the next one in order to generate continuity. The result is the ramp or profile of in the example in Section4. the actuator as shown in the example in Section 4. For the ramp’s design, an iterative calculation process is initiated with i = 1 to i = nwe using the For the ramp’s design, an iterative calculation process is initiated with i = 1 to i = nwe using the Equations (28)–(38) for each iteration, and considering the following: Equations (28)–(38) for each iteration, and considering the following:

The• speed The speed range range in which in which the CVT the CVT changes changes from from ratio ratioRcvt_max Rcvt_maxto Rto Rcvt_minand and (grey (grey zone zonein in • cvt_min FigureFigure8) to the 8) maximumto the maximum engine engine/motor/motor speed. speed. • The centrifugal force Fcen_i at each speed (Figure 8, red line). The centrifugal force Fcen_i at each speed (Figure8, red line). • • The clamping force weights value (Fcl_we_i) required at each speed (Figure 8, orange line). The clamping force weights value (Fcl_we_i) required at each speed (Figure8, orange line). Fcl_we_ini • Fcl_we_ini and Fcl_we_end are the initial and final force range. and Fcl_we_end are the initial and ﬁnal force range. • The radial displacement Rwe_i of the weights at each speed. Rwe_min and Rwe_max are the The radial displacement Rwe_i of the weights at each speed. Rwe_min and Rwe_max are the maximum • maximum and minimum radius displacement. and minimum radius displacement. The ramp’s orientation angle (λi) is calculated for each speed ωi of the driver pulley, where i is the iteration number. ω − ω Δω = m _ max CVT _ ini , i (28) nωe

ω = ω + Δω , i CVT _ini i i (29)

R − R Δ = ωe _ max ωe _ min , Rωe _ i (30) nωe

= + Δ , Rωe_i Rωe_ min Ri i (31)

F − F Δ = cl _ωe _ end cl _ωe _ ini , Fcl _ωe (32) nωe

= + Δ , Fcl _ωe_i Fcl _ωe_ini Fcl _ωe i (33)

mω 2 , F = e ()vl cen _ i ωe (34) Rωe _ i

−  Fcen_ i , λ = tan 1  i   (35)  Fcl _ωe _ i  Appl. Sci. 2020, 10, 6238 11 of 23

The ramp’s orientation angle (λi) is calculated for each speed ωi of the driver pulley, where i is the iteration number. ωm_max ωCVT_ini ∆ωi = − , (28) nωe

ωi = ωCVT_ini + ∆ωii, (29)

Rωe_max Rωe_min ∆Rωe_i = − , (30) nωe

Rωe_i = Rωe_min + ∆Rii, (31)

Fcl_ωe_end Fcl_ωe_ini ∆Fcl_ωe = − , (32) nωe

Fcl_ωe_i = Fcl_ωe_ini + ∆Fcl_ωei, (33)

mωe 2 Fcen_i = (vlωe) , (34) Rωe_i ! 1 Fcen_i λi = tan− , (35) Fcl_ωe_i

∆RiFcen_i ∆σlat_i = , (36) Fcl_ωwe_i Xi σlat_i = ∆δlat_i, (37) 1

Fcl_ωe_i MAωe = , (38) Fcen_i where vlωe is the tangential speed of weights and MAωe the weights mechanical advantage.

2.10.2. Centrifugal Flyweight Cam Design Flyweight cams which have a profile according to lateral force requirements are yet another mechanism found in the field of centrifugal actuators for driver pulleys. The operational principle is shown in Figure9. The cam pivots on its rotation axis at one end. The center of gravity is located next to the rotation axis and separated from the pulley rotation axis. Then the flyweight is exposed to a centrifugal force that tends to make it turn and displace. The cylindrical follower prevents the flyweight from turning freely. The flyweight and follower pivot shaft are attached to the pulley frame and the moving face, respectively (Figure9). When the flyweight tries to turn it produces a force that pushes the follower. This force experiments with a reaction that pushes the moving plate of the driver pulley until it moves and, thus, closes the space that the belt occupies; then the rubber V-belt occupies a greater radius, which produces the speed ratio change. Similar to the design of the ramps and weights, the cam profile is defined by the clamping force and the centrifugal force that occurs at the point of contact with the follower. Using the lever law, the equivalent centrifugal force Fcc and the lateral component of this force Fcl (Figure 11, blue arrow) are located at the point of contact between the cam and the follower. The method begins by drawing an auxiliary line through the pivot point of the flyweight and the center of gravity of the cam. An auxiliary coordinate system (Xˆ i) is created on this line (Figure 12). Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 22

ΔR F Δσ = i cen_i , lat _i (36) Fcl _ωwe_i

i σ = Δδ , lat _ i  lat _i (37) 1

F = cl _ωe _ i , MAωe (38) Fcen _ i where vl𝜔e is the tangential speed of weights and MA𝜔e the weights mechanical advantage.

2.10.2. Centrifugal Flyweight Cam Design Flyweight cams which have a profile according to lateral force requirements are yet another mechanism found in the field of centrifugal actuators for driver pulleys. The operational principle is shown in Figure 9. The cam pivots on its rotation axis at one end. The center of gravity is located next to the rotation axis and separated from the pulley rotation axis. Then the flyweight is exposed to a centrifugal force that tends to make it turn and displace. The cylindrical follower prevents the flyweight from turning freely. The flyweight and follower pivot shaft are attached to the pulley frame and the moving face, respectively (Figure 9). When the flyweight tries to turn it produces a force that pushes the follower. This force experiments with a reaction that pushes the moving plate of the driver pulley until it moves and, thus, closes the space that the belt occupies; then the rubber V-belt occupies a greater radius, which produces the speed ratio change. Similar to the design of the ramps and weights, the cam profile is defined by the clamping force and the centrifugal force that occurs at the point of contact with the follower. Using the lever law, the equivalent centrifugal force Fcc and the lateral component of this force Fcl (Figure 11, blue arrow) are located at the point of contact between the cam and the follower. The method begins by drawing an Appl. Sci. 2020, 10, 6238 12 of 23 auxiliary line through the pivot point of the flyweight and the center of gravity of the cam. An auxiliary coordinate system (X̂ i) is created on this line (Figure 12).

Appl. Sci. 2020, 10, x FOR PEER FigureREVIEWFigure 11. 11. LateralLateral force force principle principle by by flyweight ﬂyweight cam. cam. 12 of 22

The following step is to select a starting point (X̂ ini, Ŷini) in the flyweight profile. This point corresponds to the initial angle βini of the flyweight and the initial distance of the follower σfoll_ini. Figure 8 shows the ratio change sector in the grey area. Here, initial and final speeds between which the ratio change occurs is declared. Immediately associated with each speed is the change in lateral displacement for the moving face of the driver pulley. As in the design case of centrifugal weight and ramp actuator, ranges are also declared and divided into nwe range differentials. The lateral displacement of the follower center σlat is declared as an independent variable and the centrifugal force is calculated using the center of gravity location data, the flyweight angle βi-1, and the initial contact point of the flyweight and the follower (Xcpo, Ycpo). The differential’s inclination θs is calculated in the flyweightFigure 12. Parameterprofile ramp and variable using the naming centrifugal in flyweightflyweight force cam. (Fcen ) and the required clamping force (Fcl). If the center of gravity is located at a different site, the unusable weight should The following step is to select a starting point (Xˆ , Yˆ ) in the flyweight profile. This point be relocatedThe starting to different point placesof the whereramp profileit is ensure differentid thatali theis inilinked center to of the gravity endpoint is relocated of the previous to the correspondsiteration.Appl. Sci. 2020 In each, to10, thex iterationFOR initial PEER angletheREVIEW previousβini of thedifferential flyweight is andconnected the initial to the distance new one of in the such follower way thatσfoll_ini12 the of 22. initially chosen point (Xcg, Ycg). This implies that the mass (mlen) is constant. Figureend of8 the shows first theone ratio will changecoincide sector with in the the starting grey area. point Here, of the initial second and one. final As speeds shown between in Figure which 13, thewithin ratio the change procedure occurs there is declared. is a trigonometri Immediatelyc correlation associated between with each the speed flyweight is the cam change angle in lateralβ, the displacementcontact point forangle the φ movingcpo, and facethe oftangent the driver angle pulley. at the As contact in the point design θ. case With of these centrifugal three weightangles, andthe rampinitial actuator,or acquired ranges parameters, are also declared and the and previous divided iteration into nwe data,range the di ffsystemerentials. is solved. The method usedThe here lateral is implicit, displacement and several of the iterations follower are center madeσlat tois optimize declared the as anresult. independent variable and the centrifugal force is calculated using the center of gravity location data, the flyweight angle βi-1, and the initial contact point of the flyweight and the follower (Xcpo,Ycpo). The differential’s inclination θs is calculated in the flyweight profile ramp using the centrifugal force (Fcen) and the required clamping force (Fcl). If the center of gravity is located at a different site, the unusable weight should be relocated to different places where it is ensured that the center of gravity is relocated to the initially chosen point (Xcg,Ycg). This implies thatFigure the 12. mass Parameter (mlen) isand constant. variable naming in flyweight cam. The starting point of the ramp profile differential is linked to the endpoint of the previous iteration. In eachThe iteration starting the point previous of the diff ramperential profile is connected differenti toal the is newlinked one to in the such endpoint way that of the the end previous of the firstiteration. one will In coincideeach iteration with the previous starting point differential of the secondis connected one. Asto the shown new inone Figure in such 13 ,way within that the the procedureend of the there first isone a trigonometricwillFigure coincide 13. Trigonometric correlationwith the starting betweenrelationsh point theips of of flyweight thethe mechanism.second cam one. angle As βshown, the contact in Figure point 13, anglewithinϕcpo the, and procedure the tangent there angle is a at trigonometri the contact pointc correlationθ. With thesebetween three the angles, flyweight the initial cam orangle acquired β, the parameters,contactWith point the and clamping angle the previousφcpo force, and and iterationthe thetangent Equations data, angle thesystem (39)–(at the50), contact is solved.an iterative point The θ methodcalculation. With these used process three here is angles,is implicit, started the andwithinitial several i = or1 to acquired iterationsi = nwe. The parameters, are sub-index made to and ( optimizei) and the (iprevious− the1) represents result. iteration the actualdata, the and system last iteration, is solved. respectively. The method used here is implicit, and several iterations are made to optimize the result. = β Xcg_(i−1) Rcg sin ()i−1 , (39)

= β , Ycg_(i−1) Rcg cos ()i−1 (40)

= ω 2 ()− , Fcen_i mlen CVT _i X rax X cg()i−1 (41)

= ( ) , Fcc_i Fcen_i Ycg(i−1) Ycpo(i−1) (42)

θ = −1(), s_i tan Fcl_i Fcc_i (43) FigureFigure 13. 13.Trigonometric Trigonometric relationships relationships of of the the mechanism. mechanism. = ˆ 2 + ˆ 2 , Rcpo_i X i−1 Yi−1 (44) With the clamping force and the Equations (39)–(50), an iterative calculation process is started with i = 1 to i = nwe. The sub-index (i) and (i−ϕ1) represents= −1( ˆ ˆ the), actual and last iteration, respectively. cpo_ i tan X i Yi (45) = β Xcg_(i−1) Rcg sin ()i−1 , (39) Δ ˆ = ()Δ θ β ϕ , X i ftrig X i , s , i−1, cpo (46) = β , Ycg_(i−1) Rcg cos ()i−1 (40) Δ ˆ = ()Δ θ β ϕ , Yi ftrig Yi , s , i−1, cpo (47) = ω 2 ()− , Fcen_i mlen CVT _i X rax X cg()i−1 (41) ˆ = ˆ + Δ ˆ , Xi Xi−1 Xi (48) = ( ) , Fcc_i Fcen_i Ycg(i−1) Ycpo(i−1) (42)

θ = −1(), s_i tan Fcl_i Fcc_i (43)

= ˆ 2 + ˆ 2 , Rcpo_i X i−1 Yi−1 (44)

ϕ = −1( ˆ ˆ ), cpo_ i tan X i Yi (45)

Δ ˆ = ()Δ θ β ϕ , X i ftrig X i , s , i−1, cpo (46)

Δ ˆ = ()Δ θ β ϕ , Yi ftrig Yi , s , i−1, cpo (47)

ˆ = ˆ + Δ ˆ , Xi Xi−1 Xi (48) Appl. Sci. 2020, 10, 6238 13 of 23

With the clamping force and the Equations (39)–(50), an iterative calculation process is started with i = 1 to i = nwe. The sub-index (i) and (i 1) represents the actual and last iteration, respectively. −

Xcg_(i 1) = Rcg sin β(i 1), (39) − −

Ycg_(i 1) = Rcg cos β(i 1), (40) − − 2 Fcen_i = mlenωCVT_i Xrax Xcg(i 1) , (41) − − Fcc_i = Fcen_i Ycg(i 1)/Ycpo(i 1) , (42) − − 1 θs_i = tan− (Fcl_i/Fcc_i), (43) q 2 2 Rcpo_i = Xˆ i 1 + Yˆ i 1 , (44) − − 1 ϕcpo_i = tan− Xˆ i/Yˆ i , (45) ∆Xˆ i = ftrig ∆Xi, θs, βi 1, ϕcpo , (46) − ∆Yˆ i = ftrig ∆Yi, θs, βi 1, ϕcpo , (47) −

Xˆ i = Xˆ i 1 + ∆Xˆ i, (48) − Yˆ i = Yˆ i 1 + ∆Yˆ i, (49) − βi = βi 1 + ∆βi, (50) − where ωm_max is the maximum angular motor speed; Rcg is the radius from the rotation axis to the flyweight center of gravity; Xrax is the distance from rotation axis to flyweight fixed point; and Rcpo is the radius from rotation axis to the contact point between the flyweight and the follower.

2.11. Driven Pulley Spring Design The torque sensor cam is not a disk cam as shown for didactic purposes in Figure7, instead it is a cylindrical cam. The driven pulley spring must be pre-compressed to a given value Xp_sp and when the CVT is on its high ratio Rcvt_max it exerts a force equal to the amount required by the clamping force Fcl (Equation (52)). It also has to be able to be compressed to a length equal to or greater than the lateral displacement of the pulley σLat. The spring constant Ksp depends of the turn number (nsp); spring diameter (φsp); shear stress (G); wire spring diameter (dw); and the spring clamping force (Fcl_sp).

4 dw G Ksp = , (51) 8φspnsp

Fcl_sp Ksp = , (52) Xp_sp

2.12. Torque Sensing Cam Design The spring is required to provide enough lateral force so that the belt does not slip during the start of high torque. This requirement calls for a spring that does not decrease in force as the motor /engine speed increases, therefore, decreasing the amount of available torque, as shown in Figure 14. The eﬀect of having lateral force that is higher than required (Figure 14, blue line) results in greater friction losses between the belt and the pulleys and, therefore, in a drop in system eﬃciency. To solve this problem, engineers added a cam on the driven pulley located on the output shaft. This cam responds to shaft torque, compresses at high torque values, and decompresses at low torque values. The result is equivalent to having multiple springs, shown as a green dotted line in Figure 14. The result is that Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 22

ˆ = ˆ +Δ ˆ , Yi Yi−1 Yi (49)

β = β + Δβ , i i−1 i (50) where 𝜔m_max is the maximum angular motor speed; Rcg is the radius from the rotation axis to the flyweight center of gravity; Xrax is the distance from rotation axis to flyweight fixed point; and Rcpo is the radius from rotation axis to the contact point between the flyweight and the follower.

2.11. Driven Pulley Spring Design The torque sensor cam is not a disk cam as shown for didactic purposes in Figure 7, instead it is a cylindrical cam. The driven pulley spring must be pre-compressed to a given value Xp_sp and when the CVT is on its high ratio Rcvt_max it exerts a force equal to the amount required by the clamping force Fcl (Equation (52)). It also has to be able to be compressed to a length equal to or greater than the lateral displacement of the pulley σLat. The spring constant Ksp depends of the turn number (nsp); spring diameter (ϕsp); shear stress (G); wire spring diameter (dw); and the spring clamping force (Fcl_sp).

d 4G K = w , sp φ (51) 8 sp nsp

F = cl_ sp , Ksp (52) X p _ sp

2.12. Torque Sensing Cam Design The spring is required to provide enough lateral force so that the belt does not slip during the start of high torque. This requirement calls for a spring that does not decrease in force as the motor /engine speed increases, therefore, decreasing the amount of available torque, as shown in Figure 14. The effect of having lateral force that is higher than required (Figure 14, blue line) results in greater friction losses between the belt and the pulleys and, therefore, in a drop in system efficiency. To Appl. Sci. 2020, 10, 6238 14 of 23 solve this problem, engineers added a cam on the driven pulley located on the output shaft. This cam responds to shaft torque, compresses at high torque values, and decompresses at low torque whenvalues. the The torque result is is increased, equivalent both to having axes acquire multiple an springs, oﬀset angle shownεsp_i asand a green an axial dotted displacement line in Figureσaxi_i 14. thisThe isresult due tois thethat ramp when as the seen torque in Figure is incr 15eased,. both axes acquire an offset angle εsp_i and an axial displacement σaxi_i this is due to the ramp as seen in Figure 15.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 22 Figure 14. Force graph including the speed ratio change andand thethe sensingsensing torquetorque camcam eeffect.ﬀect.

The grey area of Figure 14 shows the ratio change zone. From there, the maximum and minimum values of torques; displacements; forces; reduction ratio; and angular speed are obtained and thus the ranges are established and divided into differentials nwe. Each lateral displacement differential for the cam follower is associated with its required lateral force, a torque on the driver and driven pulley, an angular speed, and a speed ratio, then the cam profile angle is calculated.

Figure 15. Torque sensing camcam diagram.diagram.

The greylocation area of Figurethe starting 14 shows point the for ratio the change first ramp zone. differential From there, is the an maximum input parameter and minimum in the valuesiterative of method. torques; displacements;For each iteration forces; to generate reduction cont ratio;inuity and the angular points speed of the are previous obtained differential and thus theare rangesconnected are establishedto the starting and points divided of into the di nextfferentials one. Thenwe .result Each lateralis the actuator displacement ramp di asfferential shown forin the camexample follower in Section is associated 4. The withdesign its of required the torque lateral sensor force, cam a torque (Equations on the (53)–(63)) driver and is driven divided pulley, into n anwe angularparts that speed, consider and the a speed following: ratio, then the cam profile angle is calculated. The• location The torque of the range starting to be point affected for the (Ti first). Tini ramp is the di initialfferential torque is an and input Tmin parameter the minimum in the torque. iterative method.• For The each spring iteration force range to generate (Fsp_i) continuityaccording to the the points torque. ofthe Fsp_ini previous is the initial differential spring are force. connected to the• starting The axial points displacement of the next of one. the Thecam result∆σaxi_i. is the actuator ramp as shown in the example in Section4. The design of the torque sensor cam (Equations (53)–(63)) is divided into nwe parts that − Δ = Tmin Tini , consider the following: Ti (53) nωe

The torque range to be aﬀected (Ti). Tini is the initial torque and Tmin the minimum torque. • T = T + ΔT i , The spring force range (F ) according toi theini torque.i F is the initial spring force. (54) • sp_i sp_ini The axial displacement of the cam ∆σaxi_i. T • F = i , ta _i (55) Rcam_ sp T T ∆T = min − ini , (53) i − Faxi _nmaxωe Faxi _ min , ΔF = sp _ i n (56) Ti = Tini + ∆Tωeii, , (54)

= T+i Δ , FFtasp__i =Fsp_ini Fsp,_i i (55)(57) Rcam_sp

FaxiF_max −FFaxi_min σ = = axi_ max axi_ min , ∆Fsp_iaxi_ max − , (56)(58) nωKesp

Fsp_i = Fsp_ini + ∆Fsp_ii, (57) σ Δ σ = axi _ max , axi _ i (59) Faxi_maxnωe Faxi_min σaxi_max = − , (58) Ksp i σ = Δσ , axi_i  lat _i (60) 1

−  Fsp _ i  , ψ = tan 1  ram _ i   (61)  Fta _ i 

F = ta_i , MAsen (62) Fsp_i

σ =σ , per_i axi_i MAsen (63) where Fta_i is the tangential force in the torque sensing cam; Rcam_sp is the torque sensing cam radius; Faxi_max is the maximum axial force of the torque sensing cam; Faxi_min is the minimum axial force of the torque sensing cam; and 𝜓ram_i is the angle of the torque sensing cam.

3. Methodology Appl. Sci. 2020, 10, 6238 15 of 23

σaxi_max ∆σaxi_i = , (59) nωe Xi σaxi_i = ∆σlat_i, (60) 1 F ! 1 sp_i ψram_i = tan− , (61) Fta_i

Fta_i MAsen = , (62) Fsp_i

σper_i = σaxi_iMAsen, (63) where Fta_i is the tangential force in the torque sensing cam; Rcam_sp is the torque sensing cam radius; Faxi_max is the maximum axial force of the torque sensing cam; Faxi_min is the minimum axial force of the torque sensing cam; and ψram_i is the angle of the torque sensing cam.

3. Methodology To compare and validate this proposed method, three types of test are performed: (a) a measurement and characterization of all the elements and components of a commercial CVT is elaborated. The values are used as parameters in the design using the equations presented. The capacity in torque, speed ratio, and others are compared with the manufacturers’ specifications. This testing is carried out on the test bench. (b) Using this method, a design for a powertrain composed of a rubber V-belt CVT and a differential reducer is presented. (c) The behavior of the designed CVT operating in an electric vehicle is simulated running a stringent route. To validate the proposed numerical design method, a computer simulation tool is developed which sequentially addresses the different steps presented. This tool solves the iterations and equations using Runge–Kutta’s method. For the selection of the commercial elements, a database is created taking the greatest amount of specifications that can be obtained, and then introduced into the computer simulation tool. With this application, a programming object that uses the results of the design is developed, and it performs as such in a complete vehicle model that includes other remaining objects. For the method development, some aspects were found in the literature and others had to be validated. The test bench is a device in which data from the road and the vehicle as weight of the vehicle along the road, number of random or fixed stops, and the desired cruise speed are introduced. The vehicle or powertrain is tested in the device to obtain information about torques, speeds, currents, and energy consumption. Figure 16 shows the elements that make up the vehicle and the dynamometer that allows for the acquisition of real-time data from the powertrain. Its characteristics and components are 1. Commercial CVT: Polaris, USA, sportsman 500 primary drive clutch 1996–2011; Team, USA, tied driven secondary clutch 421896, Gates, USA, 19G4006E G-force CVT belt; 2. Galoce, China, GTS100 torque sensor (150 Nm); 3. Electric motor and controller HPEVS, USA, AC20 + Curtis instruments, USA, 1238E-7621; 4. Galoce, China, Torque sensor GTS-100 torque sensor (1000 Nm); 5. Flowfit, UK, Gearbox 40-97001-3.8 gear ratio 3.8; 6. DANA SPICER, USA, 520290T-3X differential D-max gear ratio 4.3; 7. 2 Brakes load capacity 1400 Nm; 8. Wheels 255/70R17.5. These are part of a kit with various motors, gearboxes and some own made CVTs. Due to the maximum size of vehicles available, the variation of rubber V-belt CVTs is given only from a width of 32 mm until the industrial or agricultural belts with a width of 100 mm. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 22

To compare and validate this proposed method, three types of test are performed: a) a measurement and characterization of all the elements and components of a commercial CVT is elaborated. The values are used as parameters in the design using the equations presented. The capacity in torque, speed ratio, and others are compared with the manufacturers’ specifications. This testing is carried out on the test bench. b) Using this method, a design for a powertrain composed of a rubber V-belt CVT and a differential reducer is presented. c) The behavior of the designed CVT operating in an electric vehicle is simulated running a stringent route. To validate the proposed numerical design method, a computer simulation tool is developed which sequentially addresses the different steps presented. This tool solves the iterations and equations using Runge–Kutta’s method. For the selection of the commercial elements, a database is created taking the greatest amount of specifications that can be obtained, and then introduced into the computer simulation tool. With this application, a programming object that uses the results of the design is developed, and it performs as such in a complete vehicle model that includes other remaining objects. For the method development, some aspects were found in the literature and others had to be validated. The test bench is a device in which data from the road and the vehicle as weight of the vehicle along the road, number of random or fixed stops, and the desired cruise speed are introduced. The vehicle or powertrain is tested in the device to obtain information about torques, speeds, currents, and energy consumption. Figure 16 shows the elements that make up the vehicle and the dynamometer that allows for the acquisition of real-time data from the powertrain. Its characteristics and components are 1. Commercial CVT: Polaris, USA, sportsman 500 primary drive clutch 1996–2011; Team, USA, tied driven secondary clutch 421896, Gates, USA, 19G4006E G-force CVT belt; 2. Galoce, China, GTS100 torque sensor (150 Nm); 3. Electric motor and controller HPEVS, USA, AC20 + Curtis instruments, USA, 1238E-7621; 4. Galoce, China, Torque sensor GTS-100 torque sensor (1000 Nm); 5. Flowfit, UK, Gearbox 40-97001-3.8 gear ratio 3.8; 6. DANA SPICER, USA, 520290T-3X differential D-max gear ratio 4.3; 7. 2 Brakes load capacity 1400 Nm; 8. Wheels 255/70R17.5. These are part of a kit with various motors, gearboxes and some own made CVTs. Due Appl. Sci. 2020, 10, 6238 16 of 23 to the maximum size of vehicles available, the variation of rubber V-belt CVTs is given only from a width of 32 mm until the industrial or agricultural belts with a width of 100 mm.

Figure 16. Testing bench: active dynamometer. Using this test bench, a powertrain operated simulating diﬀerent torques and angular speed. Using this test bench, a powertrain operated simulating different torques and angular speed. Data obtained and the inferences from these test were used in order to make considerations, which Data obtained and the inferences from these test were used in order to make considerations, which were converted into algorithms or equations providing robustness to the design method and the were converted into algorithms or equations providing robustness to the design method and the computer tool. The design process of the gear systems was not evaluated in this analysis. The computer tool. The design process of the gear systems was not evaluated in this analysis. The following parameters can be measured on the test bench torque and speed at the input and output of following parameters can be measured on the test bench torque and speed at the input and output of the CVT shafts: angular and linear speed at the rollers where the wheels are located; brake pressure; the CVT shafts: angular and linear speed at the rollers where the wheels are located; brake pressure; brake temperature; motor temperature; and voltage and current of the energy system. This data are brake temperature; motor temperature; and voltage and current of the energy system. This data are collected by a Delta, Taiwan, PLC (DVP28SV211T) using digital and analog signals and then analyzed collected by a Delta, Taiwan, PLC (DVP28SV211T) using digital and analog signals and then in real-time in LabVIEW SP1 2019. The test bench and the powertrain can be conﬁgured with the components list in Table2.

Table 2. Test bench and powertrain components.

Components Technical Specifications AC20 Electric induction motor Peak power 65.3 HP @ 5000 rpm Electric induction motor Peak power 120 HP @ 4000 rpm Commercial CVT, flyweight cam driver pulley actuator and Input torque: 57 Nm, Rcvt_max = 3.38 torque cam sensing on driven pulley Own made CVT, Ramp and weight driver pulley actuator and Input torque: 500 Nm, Rcvt_max = 3.1 torque cam sensing on driven pulley Gearbox PTO 40-97001 Input torque 300 Nm, Rgear = 3.8 Own made Gearbox Input torque 900 Nm, Rgear = 3.1 Own made two stages Gearbox Input torque 1500 Nm, Rgear = 8.9 DANA SPICER 520290T-3X differential D-max Input torque 5200 Nm, Rgear = 4.3 Transaxle Input 1800 Nm, Rgear = 8.7 water-cooled brake Break power 140 KW at 70 km/h Wheels 255/70/R17.5

The route chosen to test the vehicle’s CVT (in the simulation software) has a distance of 15.704 km; a range of slopes from 4◦ to 16◦; a maximum speed of 80 km/h and an average speed of approximately 27.8 km/h, and conditions of active and passive load of 5500 N without variation over the entire route. Some of this data were collected from a previous study for a public transport vehicle that circulates every day on this same route. The concurrent vehicles on this route are powered by ICE, have a 5 shift-speed gearbox, and a power between 100–130 KW. Appl. Sci. 2020, 10, 6238 17 of 23

4. Example Analysis and Discussion A powertrain design composed of a CVT and a fixed gear transmission for a common city vehicle transmission is presented. Typically mountainous towns have small buses (22–28 passengers), delivery cars, and fire and police trucks, among others, with cargo capacities of approximately 3000 or 4000 kg. These vehicles do not exceed speeds of 80 km/h in their operation and their average operating speed is 30 km/h. Chosen routes have between 4 and 15 km of maximum travel, with inclination angles between 4◦ and 16◦. One of the reasons for these vehicles to be small is that they travel on city slopes and these roads are usually narrow and with low curvature radius. The characteristics used for the analysis of the vehicle powertrain are: gross vehicle mass of 5500 kg, aerodynamic coefficient of 0.45, front area 3.5 m2, air density 0.98 kg/m3, the rolling coefficient is 0.03, wheel diameter approximately 0.75 m (R17.5). The aim is for the loaded vehicle to achieve maximum speed on slopes within 5 s. This results in a climb acceleration of 1.7 m/s2 and a maximum speed of 30 km/h on slopes and 80 km/h on flat routes. Using equations (1)–(3) it is determined that the peak power for the vehicle is 130 KW on a 16◦ slope and 45 KW continuous power on flat routes. Recent recurring problems in vehicle design are the reduction of polluting emissions, increasing overall efficiency, and improving energy performance. A solution to this problem is the use of EV, therefore, this type of vehicle is the one chosen. New electric motors tend to operate at high rpm (>6000) to decrease the weight per power unit. However, for these angular speeds the belts would exceed the allowable tangential speed (approximately 30 m/s) and a fatigue of the V-belt could occur, as a result of the available tensile capacity being reduced by the effect of the centrifugal force. Therefore, a motor with a maximum rpm in the order of 4000 rpm should be selected with the requirements of having a continuous torque value of 500 Nm at 1500 rpm and a peak value of 840 Nm. After analyzing the possibilities in the market, the motor described in Figure 17 wasAppl. chosen.Sci. 2020, 10, x FOR PEER REVIEW 17 of 22

(a) (b)

FigureFigure 17.17. ElectricElectric motormotor curves:curves: (aa)) continuouscontinuous graph;graph; ((bb)) peakpeak graph.graph.

TheThe ratioratio ofof thethe fixedfixed transmissiontransmission system is obtained using Equations (4) (4) and and (5), (5), the the RRgeargear hashas a avalue value of of 6.2. 6.2. Configurations Configurations may may vary, vary, for for instance, instance, it it is is possible possible to to have a single gearboxgearbox andand aa didifferentialfferential or or a transaxle.a transaxle. According According to Equationsto Equati (6)ons and (6) (7)and the (7) maximum the maximum ratio of theratio CVT of istheR CVT_maxCVT is =RCVT_max1.94. Finally, = 1.94. the Finally, maximum the maximum speed ratio speed of the systemratio of is the 12. system In this case,is 12. the In system this case, is composed the system of an is electriccomposed motor, of an a CVT,electric and motor, a diff erentiala CVT, and as show a differential in Figure as 18 show. in Figure 18.

Figure 18. Electric powertrain.

The belt section is determined using the iterative process in Figure 3, for which the commercial belt section that most closely approximates the parameters is the W100 reference. It has a width Wbel of 100 mm, a thickness hbel of 32 mm, a belt angle φbel of 30°, and a cross-sectional area of 3050 mm2. The maximum tension present in the belt is 11,858 N determined by the sum of the three tensions. Assuming that the belt tensile strength (σbelt) is 4 N/mm2, the minimum cross-sectional area of the section is obtained using the Equation (10) of 2965 mm2, afterwards the aforementioned belt is selected as it is the closest to this value. The tensions that interact on the belt are the working tension, the tension due to the centrifugal force, and the bending tension. Using Equation (11) it is determined that the value of the working tension is 8840 N with a starting torque of 840 Nm and a diameter ϕ1_min of 193 mm. A more precise value of this tension is obtained using Equation (12) with a result of 8750 Nm, considering a friction coefficient μ between the belt and the pulley of 0.5. The tension due to the centrifugal force adds a value of 995 N (Equation (14)) with a belt density of 1.32 kg/dm3 (Equation (16)). Belts with deep and short teeth have low bending stress [36], considering a bending stress of 15 N/mm2 the bending tension is 2013 N (Equation (17)). The minimum diameter ϕ1_min for a V-belt with a thickness of 32 mm is 193 mm (Equation (8)). With the value of the maximum CVT speed ratio Rcvt_max, it is determined that the maximum driven pulley diameter is 384 mm. The pulleys diameter to develop a minimum ratio of 1:1 is 278 mm. The minimum pulley center distance C is 300 mm (Equation (18)) using the τcc_in ≥ 160° approximation. However, the pulleys are almost rubbing against each other, and to correct this the contact angle is increased to 166° obtaining a value of C of 400 mm which allows for an increase in the tensile capacity. Finally, the belt’s length is 1674 mm. Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 22

(a) (b)

Figure 17. Electric motor curves: (a) continuous graph; (b) peak graph.

The ratio of the fixed transmission system is obtained using Equations (4) and (5), the Rgear has a value of 6.2. Configurations may vary, for instance, it is possible to have a single gearbox and a differential or a transaxle. According to Equations (6) and (7) the maximum ratio of the CVT is Appl. Sci. 2020, 10, 6238 18 of 23 RCVT_max = 1.94. Finally, the maximum speed ratio of the system is 12. In this case, the system is composed of an electric motor, a CVT, and a differential as show in Figure 18.

Figure 18. Electric powertrain. The belt section is determined using the iterative process in Figure3, for which the commercial The belt section is determined using the iterative process in Figure 3, for which the commercial belt section that most closely approximates the parameters is the W100 reference. It has a width Wbel belt section that most closely approximates the parameters is the W100 reference. It has a width Wbel2 of 100 mm, a thickness hbel of 32 mm, a belt angle ϕbel of 30◦, and a cross-sectional area of 3050 mm . of 100 mm, a thickness hbel of 32 mm, a belt angle φbel of 30°, and a cross-sectional area of 3050 mm2. The maximum tension present in the belt is 11,858 N determined by the sum of the three tensions. The maximum tension present in the belt is 11,858 N determined2 by the sum of the three tensions. Assuming that the belt tensile strength (σbelt) is 4 N/mm , the minimum cross-sectional area of the Assuming that the belt tensile strength (σbelt) is 4 N/mm2, the minimum cross-sectional area of the section is obtained using the Equation (10) of 2965 mm2, afterwards the aforementioned belt is selected section is obtained using the Equation (10) of 2965 mm2, afterwards the aforementioned belt is as it is the closest to this value. selected as it is the closest to this value. The tensions that interact on the belt are the working tension, the tension due to the centrifugal The tensions that interact on the belt are the working tension, the tension due to the centrifugal force, and the bending tension. Using Equation (11) it is determined that the value of the working force, and the bending tension. Using Equation (11) it is determined that the value of the working tension is 8840 N with a starting torque of 840 Nm and a diameter φ1_min of 193 mm. A more precise tension is 8840 N with a starting torque of 840 Nm and a diameter ϕ1_min of 193 mm. A more precise value of this tension is obtained using Equation (12) with a result of 8750 Nm, considering a friction value of this tension is obtained using Equation (12) with a result of 8750 Nm, considering a friction coeﬃcient µ between the belt and the pulley of 0.5. The tension due to the centrifugal force adds a coefficient μ between the belt and the pulley of 0.5. The tension due to the centrifugal force adds a value of 995 N (Equation (14)) with a belt density of 1.32 kg/dm3 (Equation (16)). Belts with deep and value of 995 N (Equation (14)) with a belt density of 1.32 kg/dm3 (Equation (16)). Belts with deep and short teeth have low bending stress [36], considering a bending stress of 15 N/mm2 the bending tension short teeth have low bending stress [36], considering a bending stress of 15 N/mm2 the bending is 2013 N (Equation (17)). tension is 2013 N (Equation (17)). The minimum diameter φ1_min for a V-belt with a thickness of 32 mm is 193 mm (Equation (8)). The minimum diameter ϕ1_min for a V-belt with a thickness of 32 mm is 193 mm (Equation (8)). With the value of the maximum CVT speed ratio Rcvt_max, it is determined that the maximum driven With the value of the maximum CVT speed ratio Rcvt_max, it is determined that the maximum driven pulley diameter is 384 mm. The pulleys diameter to develop a minimum ratio of 1:1 is 278 mm. The pulley diameter is 384 mm. The pulleys diameter to develop a minimum ratio of 1:1 is 278 mm. The minimum pulley center distance C is 300 mm (Equation (18)) using the τcc_in 160◦ approximation. minimum pulley center distance C is 300 mm (Equation (18)) using the τcc_in ≥ 160° approximation. However, the pulleys are almost rubbing against each other, and to correct this the contact angle is However, the pulleys are almost rubbing against each other, and to correct this the contact angle is increased to 166 obtaining a value of C of 400 mm which allows for an increase in the tensile capacity. increased to 166°◦ obtaining a value of C of 400 mm which allows for an increase in the tensile Finally, the belt’s length is 1674 mm. capacity. Finally, the belt’s length is 1674 mm. The lateral belt displacement in the driver pulley is 23.57 mm and 29.95 mm in the driven pulley (Equation (22)), and this causes the belt to be crossed by 0.69◦ (Equation (27)). To enhance alignment, the driven pulley angle is reduced to 28◦, ergo its lateral displacement is reduced to 26.42 mm and the misalignment to 0.4◦, which reduces the crossing belt by 57.97%. This modiﬁcation generates a small distortion between the lateral forces on both pulleys when the belt passes from one to the other. The maximum normal lateral pressure reaches 9 kg/cm2. The lateral force on the belt is 7600 N on average (Equations (13) and (23)) or 8600 N as a result of the integral (Equation (26)) and a 160◦ contact angle. The lateral force to prevent the belt slip is in the range from 8540–1415 Nm (Figure 19b). Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 22

The lateral belt displacement in the driver pulley is 23.57 mm and 29.95 mm in the driven pulley (Equation (22)), and this causes the belt to be crossed by 0.69° (Equation (27)). To enhance alignment, the driven pulley angle is reduced to 28°, ergo its lateral displacement is reduced to 26.42 mm and the misalignment to 0.4°, which reduces the crossing belt by 57.97%. This modification generates a small distortion between the lateral forces on both pulleys when the belt passes from one to the other. The maximum normal lateral pressure reaches 9 kg/cm2. The lateral force on the belt is 7600 N on average (Equations 13 and 23) or 8600 N as a result of the integral (Equation 26) and a 160° contact angle. The lateral force to prevent the belt slip is in the range from 8540–1415 Nm (Figure 19b). Figure 17 shows the CVT ratio change zone, starting at the point where the torque starts decreasing (1500 rpm) and ending at the sector where the motor’s efficiency is the highest (2500 rpm). The torque range is 840 to 200 Nm. In this case, for the design of the centrifugal actuator, we have a ramp and weight system. Using the Equations (28)–(38) we can find the form of the ramp, the values and relationships between the centrifugal and clamping force, and the angle and mechanical advantage as a rotational speed function. The data obtained are initial ramp radius 40 mm and a final ramp radius of 120 mm, lateral displacement of 23.57 mm, and the mass of the weights is found Appl. Sci. 2020, 10, 6238 19 of 23 by iterations. If the weight grows the lateral displacement grows, and vice versa. For the required displacement, the total weight is 288 g for the three weights.

(a) (b) (c)

FigureFigure 19. 19.( a()a The) The proﬁle profile of theof rampthe ramp on which on which the weight the weight moves radially;moves radially; (b) the available(b) the available centrifugal forcecentrifugal and the force required and the clamping required forceclampi asng a force function as a offunction the rpm; of the (c) rpm; mechanical (c) mechanical advantage advantage (MA) or relationship(MA) or relationship between thebetween forces the and forces the rampand the angle ramp to angle develop to develop the clamping the clamping force. force.

FigureFor low 17 rpm shows the the centrifugal CVT ratio force change is low zone, (285 starting N) but atthe the required point where clamping the torque force is starts high decreasing(8540 N) (1500so the rpm) mechanical and ending advantage at the must sector be high where (30:1). the motor’sThis is achieved efficiency by isusing the a highest near radial (2500 angle rpm). (1.9°) The torqueon the range ramp. is The 840 opposite to 200 Nm. situation In this occurs case, for at thehigh design speed, of high the centrifugalcentrifugal actuator,force (2096 we N) have and a low ramp andclamping weight force system. (1415 Using N). In the this Equations case, the (28)–(38)angle will we turn can to find the theright form with of a thehigher ramp, value the regarding values and relationshipsthe radial axis between (46.2°). the centrifugal and clamping force, and the angle and mechanical advantage as a rotationalFor the speed driven function. pulley, Thea spring data obtainedand a cylindrical are initial cam ramp are radius designed 40 mm to regulate and a final the ramp clamping radius offorce 120 mm,according lateral to displacement the torque and of the 23.57 change mm, andin di theameter mass for of the the pulley. weights The is found driven by pulley iterations. spring If is the weightcompressed grows at the maximum lateral displacement torque applying grows, a maximum and vice versa.lateral Forclamping the required force. As displacement, the torque changes the total weightthe spring is 288 expands g for the and three at the weights. same time, it responds to displacement due to the speed ratio change thatFor compresses low rpm it the by centrifugala maximum force of 24 is mm. low With (285 N)a minimum but the required spring diameter clamping of force 70 mm, is high and (8540 using N) Equations (51) and (52), the spring constant is determined with a value of 140 N/mm. The so the mechanical advantage must be high (30:1). This is achieved by using a near radial angle (1.9◦) displacement of the spring is 60 mm and the free length is 175 mm, the diameter of the spring wire is on the ramp. The opposite situation occurs at high speed, high centrifugal force (2096 N) and low 14 mm made of piano wire rope that provides a torsional stress of 727 N/mm2 at maximum clamping force (1415 N). In this case, the angle will turn to the right with a higher value regarding the compression, and a 10 mm pre-compression for minimum clamping force. The cam that compresses radial axis (46.2 ). the spring is designed◦ with Equations (52)–(62). The torque range when the CVT changes from For the driven pulley, a spring and a cylindrical cam are designed to regulate the clamping maximum to minimum speed ratio is 1620–400 Nm. The diameter of the cam is designated a value of force according to the torque and the change in diameter for the pulley. The driven pulley spring is 150 mm. compressed at maximum torque applying a maximum lateral clamping force. As the torque changes the spring expands and at the same time, it responds to displacement due to the speed ratio change that compresses it by a maximum of 24 mm. With a minimum spring diameter of 70 mm, and using Equations (51) and (52), the spring constant is determined with a value of 140 N/mm. The displacement of the spring is 60 mm and the free length is 175 mm, the diameter of the spring wire is 14 mm made of piano wire rope that provides a torsional stress of 727 N/mm2 at maximum compression, and a 10 mm pre-compression for minimum clamping force. The cam that compresses the spring is designed with Equations (52)–(62). The torque range when the CVT changes from maximum to minimum speed ratio is 1620–400 Nm. The diameter of the cam is designated a value of 150 mm. In this case, the graphs should be read backwards from 50–0 mm axial displacement (Figure 20), which is equivalent to the initial condition in the driver pulley. An angle of 69.35◦ is used to balance the maximum torque (1620 Nm) which is equivalent to 21,600 N of tangential force with the clamping force (8540 N) and the mechanical advantage (2.65). Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 22 Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 22

Appl. Sci. 2020, 10, 6238 20 of 23 Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 22

(a) (b) (c) (a) (b) (c) Figure 20. (a) Cylindrical cam profile, (b) tangential and axial force as a function of axial Figure 20. (a) Cylindrical cam profile, (b) tangential and axial force as a function of axial displacement, (c) relationship between forces and profile angle as a function of axial displacement. displacement,(a) (c) relationship between forces( band) profile angle as a function of axial(c) displacement. In this case, the graphs should be read backwards from 50–0 mm axial displacement (Figure 20), FigureFigureIn this 20. 20. (case,a )(a Cylindrical) Cylindricalthe graphs cam camshould profile, profile, (beb) read tangential (b) backwardstangential and axial and from force axial 50–0 as aforce functionmm asaxial a of functiondisplacement axial displacement, of axial (Figure 20), which is equivalent to the initial condition in the driver pulley. An angle of 69.35° is used to balance which(cdisplacement,) relationship is equivalent ( betweenc) relationship to the forces initial between and condition profile forces angle andin asthe profile a functiondriver angle pulley. of as axial a function An displacement. angle of axial of 69.35°displacement. is used to balance the maximum torque (1620 Nm) which is equivalent to 21,600 N of tangential force with the the maximum torque (1620 Nm) which is equivalent to 21,600 N of tangential force with the clampingIn this force case, (8540 the N) graphs and theshould mechanical be read backwardsadvantage from(2.65). 50–0 mm axial displacement (Figure 20), clampingFinally, thisforce powertrain (8540 N) and is simulated the mechanical in a vehicle advantage with the(2.65). above characteristics. The route has a whichFinally, is equivalent this powertrain to the initial is simulated condition in in a vehiclethe driver with pulley. the above An angle characteristics. of 69.35° is usedThe routeto balance has a total distanceFinally, of 15.704this powertrain km, inclinations is simulated vary between in a vehicle 4◦ and with 16 ◦the, a realabove travel characteristics. time of 33.8 min,The route and the has a totalthe distancemaximum of 15.704torque km, (1620 inclinations Nm) which vary is betweenequivalent 4° andto 21,600 16°, a Nreal of travel tangential time offorce 33.8 withmin, theand averagetotal distance speed of of the 15.704 route km, is 27.80 inclinations km/h. It vary is necessary between to4° establishand 16°, a that real these travel data time were of 33.8 collected min, and theclamping average forcespeed (8540 of the N) route and theis 27.80 mechanical km/h. It advantage is necessary (2.65). to establish that these data were collected in athe previous average exploration speed of the of route several is 27.80 routes km/h. in a It city is ne withcessary mountainous to establish topographic that these data conditions were collected and in a previousFinally, explorationthis powertrain of several is simulated routes in in a avehicle city with with mountainous the above characteristics. topographic The conditions route has and a busin weights a previous corresponding exploration to of mass several public routes transport in a city vehicles. with mountainous As a result topographic of this study conditions a dynamic and bustotal weights distance corresponding of 15.704 km, to inclinations mass public vary transpor betweent vehicles. 4° and 16°, As a a real result travel of thistime studyof 33.8 a min, dynamic and simulationbus weights software corresponding for vehicles to was mass developed, public transpor and usedt vehicles. for elaborating As a result the simulations of this study for a this. dynamic simulationthe average software speed of for the vehicles route is was 27.80 developed, km/h. It is and necessary used for to establishelaborating that the these simulations data were for collected this. simulationThe linear software vehicle speedsfor vehicles distribution was developed, along the and route used can for be elaborating observed in the Figure simulations 21. In this for case, this. in aThe previous linear vehicleexploration speeds of severaldistribution routes along in a citythe routwithe mountainouscan be observed topographic in Figure conditions21. In this case,and the powertrainThe linear design vehicle is based speeds on distribution a vehicle that along does the not rout operatee can be high observed speeds, in and Figure which 21. operatesIn this case, thebus powertrain weights corresponding design is based to onmass a vehicle public thattranspor doest notvehicles. operate As higha result speeds, of this and study which a dynamic operates betweenthe powertrain 20 and 40 kmdesign/h for is half based ofits on travel a vehicle time. that This does type not of transportoperate high is presented speeds, inand cities which that operates have betweensimulation 20 and software 40 km/h for vehiclesfor half wasof its developed, travel time. and Th usedis type for ofelaborating transport the is presentedsimulations in for cities this. that servicesbetween from 20 feeder and 40 buses km/h to for other half vehicles of its travel of greater time. a fflThuence.is type of transport is presented in cities that have servicesThe linear from vehicle feeder speeds buses distribution to other vehicles along ofthe greater route canaffluence. be observed in Figure 21. In this case, thehave powertrain services design from feeder is based buses on ato vehicle other vehiclesthat does of not greater operate affluence. high speeds, and which operates between 20 and 40 km/h for half of its travel time. This type of transport is presented in cities that have services from feeder buses to other vehicles of greater affluence.

FigureFigure 21. 21. LinealLineal vehicle vehicle speed. speed. Figure 21. Lineal vehicle speed. The powertrain is composed of a CVT with the speed ratio characteristics present in Figure 22, the The powertrain is composed of a CVT with the speed ratio characteristics present in Figure 22, CVT operatesThe powertrain on an average is composed ratio ofFigure 1.597. of a CVT21. Lineal with vehicle the sp speed.eed ratio characteristics present in Figure 22, the CVT operates on an average ratio of 1.597. the CVT operates on an average ratio of 1.597. The powertrain is composed of a CVT with the speed ratio characteristics present in Figure 22, the CVT operates on an average ratio of 1.597.

Figure 22. CVT speed ratio during the route. FigureFigure 22. CVT22. CVT speed speed ratio ratio during during the the route. route. The CVT system conﬁgurationFigure allows22. CVT forspeed the ratio currents during achieved the route. by the electric motor to remain below the values of its continuous curve, and the high current peaks occur during short distances and short time (Figure 23). Appl. Sci. 2020, 10, x FOR PEER REVIEW 20 of 22

Appl. Sci.The2020 CVT, 10, system 6238 configuration allows for the currents achieved by the electric motor to remain21 of 23 below the values of its continuous curve, and the high current peaks occur during short

(a) (b)

FigureFigure 23.23. (a) Motor current current on on the the travel travel road road and and road road topography, topography, (b (b) )Motor Motor current current on on time. time.

TheThe totaltotal routeroute hashas anan energy consumption of of 22. 22.39193919 KWh. KWh. In In the the simulation, simulation, the the regenerative regenerative brakebrake waswas not not included included in thein the system. system. We assumeWe assume that ifthat it is if considered, it is considered, performance performance will improved. will improved. 5. Conclusions 5. Conclusions A method of sequential equations is presented to enable the implementation of a computer applicationA method and toof performsequential the equations analysis and is presented assignment to of enable all dimensional the implementation and operational of a computer parameters forapplication a rubber V-beltand to CVT perform design. the The analysis increasing and useassignment of vehicles of thatall dimensional only travel specificand operational routes and haveparameters low dynamic for a rubber demands V-belt is addressed CVT design. by theThe methodincreasing by providinguse of vehicles an advantage that only fortravel the specific design of vehiclesroutes and that have operate low at dynamic their highest demands possible is addressed efficiency. by The the reason method is thatby providing the sequential an advantage design process for basedthe design on user of needs, vehicles in termsthat operate of road, at vehicle their characteristics,highest possible and efficiency. desired performance The reason integratesis that the the systemsequential in a design way that process each particularbased on user need needs, can be in analyzed terms of road, from allvehicle aspects characteristics, that influence and the desired results. performanceA CVT systemintegrates design the system becomes in aa complexway that problemeach particular due to need the number can be analyzed of interactions from all and relationshipsaspects that influence between the forces results. in the pulleys. Due to the strong interrelationship between parameters and variableA CVT values,system implicitdesign becomes equations a arecomplex present problem in the designdue to thatthe excludenumber analyticalof interactions methods and as practicalrelationships tools. between Using numericalforces in the methods pulleys. and Due equations, to the strong the interrelationship proposed method between provides parameters simplified equationsand variable and values, data for implicit developing equa ations design are andpresent solution in the for design the complex that exclude interactions analytical in the methods CVT design. as practicalThe desiredtools. Using vehicle numerical behavior methods generates and variations equations, in actuators’ the proposed geometry. method However, provides the simplified numerical equations and data for developing a design and solution for the complex interactions in the CVT method allows for a fast convergence in the approximation of the result in the design of either type of design. pulley actuator. Compared to an ICE, which at low rpm has low torque, an electric motor provides The desired vehicle behavior generates variations in actuators’ geometry. However, the maximum torque and generates a minimum centrifugal force in the CVT driver pulley. This difference numerical method allows for a fast convergence in the approximation of the result in the design of requires the driver ramp to have a different profile. either type of pulley actuator. Compared to an ICE, which at low rpm has low torque, an electric Commercial rubber V-belts have a wide range of applications in the transport industry, especially motor provides maximum torque and generates a minimum centrifugal force in the CVT driver in EV. Using the method for each type of standard rubber V-belts, a diverse range of applications can pulley. This difference requires the driver ramp to have a different profile. be addressed in vehicles from three-wheelers to medium cargo vehicles as presented in the results. The Commercial rubber V-belts have a wide range of applications in the transport industry, advantageespecially ofin usingEV. Using a CVT the in method a vehicle for is theeach reduction type of standard in the size rubber of the V-belts, motor and a diverse its characteristics. range of Comparedapplications to can a fixed be transmission,addressed in thevehicles vehicle from would three-wheelers have lower speedsto medium on plane cargo routes vehicles or would as needpresented a larger in motorthe results. to be The able advantage to start on of inclined using a routes.CVT in Furthermore, a vehicle is the the reduction system would in the size also of operate the outsidemotor and the maximumits characteristics. efficiency Compared range. to a fixed transmission, the vehicle would have lower speeds on plane routes or would need a larger motor to be able to start on inclined routes. Author Contributions: Conceptualization, I.A.; methodology, S.M.A.; software, I.A. and S.M.A.; validation, I.A. andFurthermore, S.M.A.; formal the analysis,system would I.A. and also S.M.A.; operat investigation,e outside the I.A. maximum and S.M.A.; efficiency writing—original range. draft preparation, S.M.A.; writing—review and editing, I.A. and S.M.A.; supervision, I.A.; project administration, I.A. All authors Author Contributions: Conceptualization, I.A.; methodology, S.M.; software, I.A. and S.M.; validation, I.A. and have read and agreed to the published version of the manuscript. S.M.; formal analysis, I.A. and S.M.; investigation, I.A. and S.M.; writing—original draft preparation, S.M.; Funding:writing—reviewThis research and editing, was I.A. funded and byS.M.; Departamento supervision, I.A.; Administrativo project administration, de Ciencia, I.A. Tecnolog All authorsía e have Innovaci readón (COLCIENCIAS)and agreed to the (Grant published No 033-2018);version of andthe manuscript. EAFIT University. Conflicts of Interest: The authors declare no conflict of interest. Funding: This research was funded by Departamento Administrativo de Ciencia, Tecnología e Innovación (COLCIENCIAS) (Grant No 033-2018); and EAFIT University.

Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2020, 10, 6238 22 of 23

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