Continuous formulation of the layout of a hydromechanical transmission

Mechanism and Machine Theory 133 (2019) 545–558

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Continuous formulation of the layout of a hydromechanical transmission Antonio Rossetti a, Alarico Macor b,∗ a b

National Research Council – Construction Technologies Institute, Padova, Corso Stati Uniti, 4, 35127 Padova, Italy Department of Engineering and Management, University of Padova, Stradella S. Nicola, 3, 36100 Vicenza, Italy

a r t i c l e

i n f o

Article history: Received 6 September 2018 Revised 30 November 2018 Accepted 4 December 2018

Keywords: Hydromechanical transmission Layout optimization Planetary gear

a b s t r a c t In this paper, an analytical formulation of the layouts of three-shaft hydromechanical transmissions is presented. The formulation was obtained starting from the Willis equation for the planetary gears. Its graphical representation results in a polar plot dependent only on the angular coordinate: by varying the latter between 0 and 2π , all the transmission layouts can be continuously described. In this way, the layout, together with the transmission ratios of the gears and the displacements of the hydraulic machines, can enter as a degree of freedom in the problem of mathematical optimization of the hydromechanical transmissions. The potential of this analytical formulation will be shown by studying the case of a transmission for a forklift of 75 kW. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Hydromechanical transmissions (HMTs) are increasingly used in heavy-duty vehicles and agricultural machinery due to their better driving comfort compared to traditional transmissions and to the possibility of managing the engine at its best efficiency [1]. HMTs are structured according to three-shaft configurations, known also as input coupled (IC) and output coupled (OC), or to four-shaft configurations, known as dual stage and compound [2–4]. In the first two configurations, the connections between the epicyclical gear and the transmission shafts produce six different layouts, in the second two they even produce 36 layouts. Each of these layouts behaves differently, with different efficiency. This great variety of layouts requires suitable methodologies to support the design choices. This problem has been addressed in different ways in the literature. Many authors studied and designed simple and complex transmissions with a priori structure [5–9]. Sung et al. [10] defined 12 layouts of IC and OC transmissions and by means of the network analysis identified the best ones as those limiting the recirculation power. Linares et al. [11] systematized the theory of three-shaft configuration and inserted it in a procedure in which the layout must be chosen in advance. Rotella and Cammalleri [12,13] proposed a kinematic analysis focused to prioritize functionalities over the actual gear set layout. The strength of the analysis is the formal language to discuss the kinematic behavior of simple and complex transmissions; its weakness is the lack of flexibility in the management of mechanical gear losses and in the complete description of Continually Variable Transmission (CVT) losses as a function of the working point.

∗

Corresponding author. E-mail address: [email protected] (A. Macor).

https://doi.org/10.1016/j.mechmachtheory.2018.12.006 0094-114X/© 2018 Elsevier Ltd. All rights reserved.

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Nomenclature D F T p P Q Rwheel V v = V/Vmax x, y z

displacement of the hydraulic units [m3 ] traction force [N] torque [Nm] pressure [Pa] point P, power [W] flow rate wheel radius [m] vehicle speed [m/s] nondimensional speed of the vehicle [–] cartesian coordinates of the functional representation plane teeth number of a gear

Greek letters η efficiency [–] ηg elementar gear pair efficiency [–] η0 planetary gear efficiency [–] ηpgr planetary gear reducer efficiency [–] ηdiff differential gear reducer efficiency [–] λ viscous losses coefficient [–] μ single objective criteria [–] p pressure difference between outlet and inlet of a pump [bar] θ angular coordinate of the functional representation plane τ gear ratio [–] τ0 τ 0 = –zr /zs standing gear ratio of the planetary gear [–] ω rotational speed [rad/s] Subscripts s, c, r referred fmp referred out referred axle referred Hy referred I referred II referred

to to to to to to to

the solar, carrier, ring of the planetary gear the full mechanical point the output gearbox, including the differential gear the wheel axle the hydraulic units the hydraulic machine connected to the Internal Combustion Engine by means of an ordinary gear the hydraulic connected to the driveline through the planetary gear

Acronyms HMT Hydro Mechanical Transmission IC Input Coupled OC Output Coupled CVT Continuously Variable Transmission ICE Internal Combustion Engine

Mathematical optimization could play a key role in design support, as shown by the authors in [14,15] and by Petterson and Kroos for four-shaft configurations [16]. However, although the procedure is sophisticated, nothing guarantees that the a priori chosen transmission layout is the most efficient one. A step forward in this direction is the paper [17], in which graph theory was used in order to include the transmission layout into an optimization problem. While graph theory allows a systematic discussion of every possible layout, optimization has to be carried out iteratively for every case such in [18,19]. Furthermore, the absence of an inherent order prevents any general discussion of the layout rationalization. The present paper will focus on the definition of an innovative procedure to rationalize the effect of the planetary gear layout in HMT, which allows manifesting the underlying order between different configurations. The approach is based on a graphical representation of the planetary gear on a two-axis Cartesian system. Different layouts and different standing gear ratios will define a closed curve around the point [1,1] with three asymptotes, allowing the use of a continuous variable to cycle between the planetary gear layouts. The paper will first present the formalization of this approach and will explain the practical application of this representation to IC and OC transmissions. Finally, the IC configuration design will be discussed for a heavy vehicle, adopting the proposed representation to organically discuss the results.

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Table 1 Points in defining the planetary gear plane equation in {ωs , ωc , ωr } R3 .

P0 P1 P2

ωs

ωc

ωr

0 1 1

0 1 / ( 1 − τ0 ) 0

0 0 1/τ 0

Table 2 Coordinates of six points in R2 representing the planetary gear different layouts with τ0 = −2.

Prcs Pcrs Psrc Prsc Pcsr Pscr

[ [ [ [ [ [

x

y

τ0 /(τ0 − 1 ); (τ0 − 1 )/τ0 ; 1/τ 0 ; τ 0; 1 − τ0 ; 1 / ( 1 − τ0 ) ;

τ0

1 − τ0 1 / ( 1 − τ0 ) τ0 /(τ0 − 1 ) (τ0 − 1 )/τ0 1/τ 0

] ] ] ] ] ]

[ [ [ [ [ [

x

y

0.67; 1.50; −0.50; −2.00; 3.00; 0.33;

−2.00 3.00 0.33 0.67 1.50 −0.50

] ] ] ] ] ]

2. Functional design plane of three-shaft planetary gear 2.1. Punctual representation of the planetary gear The velocity equation of a simple planetary gear states the velocity of one of the axes given the other two and the standing gear ratio τ 0 of the gear:



( 1 − τ0 ) ;

−1;

τ0



  ωs ωc = 0 ωr

(1)

As this equation defines a plane in a three-dimensional space, the same relation can be stated using three non-aligned points belonging to the plane. As Eq. (1) is a homogenous equation, the origin points belong to the plane, and under this assumption it can be identified with only two characteristic points. For the present study, these points were defined by choosing a reference shaft, which is assumed to have constant speed equal to 1 and by computing the speed of each remaining shaft when the other is still. Assuming for example the sun as reference, Eq. (1) can be represented by the three points in Table 1. The same information can be compressed focusing on the dashed cells, ignoring all the null and unitary values in the table. In this way, the three points P0 , P1 , P2 can be reduced to only one point in a two-dimensional space, keeping track of the order used for the notation:



Pscr = 1/(1 − τ0 );

1 / τ0



(2)

where the first letter in the subscript refers to the shaft rotating at unitary velocity, while the second and the third are reported in the same order of the corresponding columns of the table. As a general notation, the point Pabc can be obtained from Eq. (1) as:



Pabc = x;

y



=



ωb /ωa q |ωc =0 ;

ωc /ωaq |ωb =0



(3)

Permuting the values used for x, y, and z, six different points in two-dimensional space can be obtained. The six points are shown in Table 2 along with the points for τ0 = −2. It is important to note that all these six points are a different representation of the same Eq. (1). As will be shown later, they are useful for visualizing the different layouts that the gear can assume in a mechanism. 2.2. Development of a Cartesian representation of the planetary gear The connection between the planetary gear shafts and the transmission shafts affects the overall transmission performance. So, all the shafts must be clearly identified on both sides. On the planetary gear side, shafts are inherently identified by their nature as: sun, ring and carrier shafts. On the system side, the shafts can be identified using their function or the main element they are driving/driven by. When three-shaft HMTs are concerned, these three shafts are (ignoring for the moment the presence of intermediate spur gears): the shaft derived from the Internal Combustion Engine (ICE); the shaft connected to CVT; and the shaft connecting to the driving axle of the vehicle. In order to maintain a general notation, the shafts connected to the planetary gear will be marked using the capital letters A, B, C. One of these three axes has to be used as reference. The reference shaft will be chosen as the one providing

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Fig. 1. System with planetary gear and related Cartesian representation.

the power to the other or the one which is maintained at constant velocity. The remaining two shafts will be used on a logarithmic Cartesian plane while the punctual representation of the speed equation shown in Eq. (3) will be plotted. This representation allows to convey a high level of information in one graph. The layout in Fig. 1 will be used as an example to clarify the construction of the plot and its reading, assuming τ0 = −2. First of all, the reference shaft on the system should be identified: A will be used in this case, assuming it as a driving shaft. The remaining shafts will be used in order on the Cartesian plane. The punctual notation in Eq. (3) can be applied to the example, taking care of reporting the planetary shafts in the same order used for the corresponding system shafts, starting from the reference shaft A and proceeding with the first and the second axes of the Cartesian plane (B and C respectively). As in the presented layout, the carrier is connected to A, the sun to B and the ring to C, the planetary gear will be represented as:



(τ0 − 1 )/τ0

Pcsr = 1 − τ0 ;



(4)

leading, for the chosen standing gear ratio, to the point Pcsr = [3; 1.5]. As the order of the planetary gear shafts used in the punctual representation is related to the configuration, then to the system shafts, Eq. (3) can be rewritten in terms of the system shafts:



Pabc = x;

y



=



 ωB /ωAq ω =0 ; C

  ωC /ωAq ω =0 B

(5)

where the small case letters a, b, c are the planetary gear shafts connected to A, B, C respectively. The point coordinates assume then a physical meaning: • •

the value along the shaft B axis is the speed ratio between A and B when ωC = 0; the value along the shaft C axis is the speed ratio between A and C when ωB = 0.

The position of the point Pcsr in Fig. 1 allows to immediately understand that, for the considered layout, the planetary gear, driven by the shaft A, will work as a speed multiplier toward both shafts B and C when the other shaft is still. 2.3. Continuous representation of the planetary gear layouts One of the most important properties of the punctual representation of the planetary gear is that it depends both on the value of the standing gear ratio and on the chosen layout. This allows to compare different layouts on the same Cartesian plane. Fig. 2a reports the functional representation of the six layouts as a function of the standing gear ratio τ 0 . Dashed lines are used to draw unfeasible solutions, i.e. τ 0 > − 2 and τ 0 < − 12, while solid lines are used instead in the range between − 2 and − 12, which can be realized using common planetary gears. When plotted using the continuous representation, the six layouts show strong relationships between each other, defining a three-branch hyperbola on the plane. In particular: - Each layout is continuous with another one, for τ 0 → − 1; - Each layout share an asymptote with another one, for τ 0 → − ∞; - Accounting for the points at infinity and the whole range of τ 0 = − 1 … − ∞, the six layouts draw a continuous and closed curve, named C, around the point [1, 1], where:

C ∈ R2 , C = f (Layout,

τ0 ), Layout ∈ [csr, crs, . . . , rsc], τ0 ∈ [−∞, − 1]

(6)

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Fig. 2. Continuous representation of planetary gear layouts: (a) functional representation for an IC configuration; (b) parametrization of curves in Fig. 2a.

Where the parametric expression of the point P of the C curve can be obtained using the expression given for x and y in Table 2, as follow:

⎧ ⎨[τ0 / (τ0 − 1 ), τ0 ] if Layout = rcs P ∈ C, P = [(τ0 − 1 ) / τ0 , 1 − τ0 ] if Layout = crs ⎩

(7)

...

As a result of this last observation, the resulting curve C of Fig. 2a can be parametrized using a proper function between

R and R2 , i.e. using just one parameter instead of the two used in Eq. (6).

While the choice of the parametric form for the curve is not uniquely defined, for this paper the authors preferred the use of a polar representation, using the point [1, 1] as the pole and the angular coordinate as parameter (Fig. 2b)

C ∈ R2 , C := {θ ; r (θ ),

θ ∈ [0, 2π ]}

(8)

where r is a properly defined function of θ . Once the parametrization of Eq. (8) is used together with f −1 , the standing gear ratio and the layout can be expressed as results of the angle θ :

[Layout,

τ0 ] = f −1 (θ ; r (θ )), θ ∈ [0, 2π ]

(9)

This formulation handles the layout of a system in the form of a continuous function of the real parameter θ , allowing to discuss and compare easily the performances of different layout and designs. The set up and the results of an optimization problem taking advantage of this formulation will be shown later in this paper. 3. Application of the functional design plane to three-shaft hydromechanical continuous variable transmissions In order to show the potential of the functional design plane, the design results and the preliminary performances of a three-shaft input coupled transmission will be discussed. The three shafts to be connected to the planetary gear are: - the ICE shaft; - one end of the CVT unit; - the axle shaft. As the ICE is the prime mover of the system and it is supposed to be always running when the driveline is active, its shaft is a natural choice for the reference shaft (previously identified as A). The CVT and the axle shafts will be assigned to the first and second axes respectively, as shown in Fig. 3, where the powertrain layout is presented on the side of the corresponding curve on the functional design plane.

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Fig. 3. Functional design plane with powertrain layouts.

Now, the punctual representation acquires a more physical meaning, since it is related to two meaningful operating points of the transmission: the zero-speed condition and the full mechanical point. The latter is the operating condition in which no power flows through hydraulic CVT and all the power flows through the mechanical branch of the transmission. This is the most efficient operating condition. In particular rewriting Eq. (5) for the assumed axis convention:



Pabc = x;

y



=



 ωCV T /ωICEq ω

Axle =0

;

 ωAxle /ωICEq ω



CV T =0

(10)

- the x coordinate is the ratio between the hydraulic unit attached to the planetary gear and the engine speed when the vehicle is in the zero-speed condition; - the y coordinate is the ratio between the axle and the engine speed when the driveline in the full mechanical point. For every point of the continuous curve presented in Fig. 3, the preliminary design can be carried out as follows. The design procedure starts from the definition of two parameters: the values of θ and the non-dimensional values of the full mechanical point speed:

v f mp =

V f mp Vmax

(11)

First, the configuration and the standing gear ratio τ 0 were computed using the functions in Eq. (9) for the desired θ . Once the full mechanical point velocity is obtained from Eq. (11) for a given vfmp and maximum velocity Vmax , the axle gear ratio τ out can be obtained as follows:

τout =

V f mp /(2π Rwheel ) ωICE V f mp /(2π Rwheel ) 1 ωout = = ωaxle ωICE ωaxle ωICE y

(12)

taking advantage of the y coordinate defined in Eq. (10). The gear ratios τ I and τ II are designed to maintain the speed of the hydraulic units less than or equal to the design value ωHy Max . For the first hydraulic unit, given a constant speed of the engine, this condition leads to

τI =

ωHy Max ωICE

(13)

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Fig. 4. Main dimensions of the test vehicle (dimensions in mm). Table 3 Design data of the test vehicle. Vehicle model

Forklift

Vehicle data Vehicle mass Maximum speed Tire radius Maximum pulling force

10,560 kg 24 km/h 0.38 m 45.2 kN

Engine Maximum power Engine speed

75 kW 2400 rpm

which is independent from the design data. For the second hydraulic unit, the equation slightly changes, due to the different contribution of the planetary gear to the overall system in each configuration. τ II is computed for the highest value of ωCVT , whichever it occurs at zero or maximum velocity:

τII =

ωHy Max Vmax max(|ωCV T | )|VV = =0

(14)

Knowing the gear ratio τ II , the maximum torque TII V =0 at the hydraulic unit can be computed setting the maximum traction force at the wheels. The second unit displacement is then obtained limiting the maximum pressure difference to design value pmax

DII =

2π TII V =0  pmax

(15)

Finally, the first unit displacement is sized to guarantee the flow rate given by the second unit’s actual displacement and speed over the whole vehicle velocity range, obtaining:

DI = 2 π

QII

max

ωICE τI

(16)

To identify the highest value QII max , the two flow rates QII V =0 and QII V =Vmax at the extreme conditions of V = 0 and V = Vmax should be checked. For both these conditions, given the speed of the engine shaft and the outlet shaft, the following step are applied: - compute the actual torque on the wheels, given by the most stringent between the maximum pulling torque and the maximum power conditions; - using the planetary gear equilibrium equations compute the working condition of the CVT shaft ωCVT and TCVT ; - compute both the speed and the torque at the second unit using the value of τ II obtained in Eq. (14) (ωII = τII ωCV T and TII = TCV T /τII , respectively); - obtain the flow rate in the second hydraulic unit, assuming  p =  pmax :

QII =

TII |ωII |  pmax

(17)

3.1. Test vehicle A 75 kW forklift was selected as a test vehicle. Its main dimensions and design data are reported in Fig. 4 and Table 3, respectively. The design speed value and the maximum working pressure for the two hydraulic units are respectively ωHy Max = 30 0 0 rpm and Pmax = 400 bar.

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Fig. 5. Results of design procedure: (a) τ II gear; (b) τ out gear; (c) DI displacement; (d) DII displacement.

3.2. Preliminary design results discussion The results of the preliminary design can be easily rationalized for the considered test case in terms of the two design variables: θ and vfmp . In order to obtain a continuous map, the parameter θ was sampled in the whole range 0 . . . 2π , corresponding to the ideal values of the standing gear ratio between −1 …−∞, for all the six configurations. The nondimensional full mechanical point was varied between 0.1 and1. The gear ratio of the second hydraulic unit τ II , the output gear ratio τ out , and the hydraulic unit displacements DI DII are mapped in Fig. 5 as a function of the two design variables. Every map reports as secondary axis the layout name, using the three-letter labels reported in Fig. 3 and the absolute value of the corresponding standing gear ratio τ 0 . As expected, the continuous parameter θ allows to highlight the continuity underlying the different layouts, setting the proper order between different solutions. The different connections of the planetary gear lead to significant changes in both τ II and τ out . As visible in Fig. 5a, for a given θ , τ II increases as the non-dimensional full mechanical point speed vfmp increases from 0.1 to 0.5, while for vfmp > 0.5 remains constant. This is the visualization of the two possible cases in Eq. (14). In the first case, when vfmp < 0.5, the maximum speed of the CVT shaft is reached at the vehicle maximum velocity, i.e. ωCV T max = ωCV T |V =Vmax . The value of ωCV T |V =Vmax decreases as the vfmp increases, leading to higher values of τ II . Instead, for vfmp > 0.5, the maximum ωCVT is obtained when V = 0 and is independent by the full mechanical point value. The speed ωCVT can be

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both lower (τ II > 1) and higher (τ II < 1) than the maximum velocity of the hydraulic speed, with the higher values between the SCR and the SRC layouts (θ = 225◦ ) and a minimum for θ = 0◦ . The τ out is instead influenced by both the design parameters on the whole design space, and differently from the τ II , its resulting values are for the greatest part in the range 0.01 . . . 1, suggesting that the shaft exiting the planetary gear needs significant speed reduction to match the speed range of the axle shaft imposed by the application. The highest values are approximatively 1 and correspond to the maximum of the τ II ratio, between the SCR and the SRC layout. This trend is in accordance with Eq. (12), where τ out is expressed as a function of y−1 , noting that SCR and the SRC layouts are the ones with the lowest y. Maps in Figs. 5c and d demonstrate that the displacement of the hydraulic units is not sensitive to the planetary gear arrangement and varies only as a function of the full mechanical point velocity. This result is based on the preliminary design reported in Section 3, which does not account for the losses (Eq. (15)). The effects of the mechanical and hydraulic losses will be discussed in the following section. 3.3. Performance maps In order to relate the effect of the design on the transmission efficiency, the performance of the different solutions was obtained solving the mechanical equations for torque and speed, for different vehicle speeds assuming as boundaries for the ICE the maximum power and the nominal speed of 75 kW and 2400 rpm, respectively. Constant efficiencies were assumed for both mechanical and hydraulic components. While this assumption can prevent to obtain detailed results, it proved to be enough to fairly compare the preliminary design sketched according to Section 3.2. Elemental gear pair efficiency ηg was set to 0.98, and the number of pairs needed was computed assuming a maximum n n gear ratio of 1.5 for a single pair. The efficiencies of the gear ratios τ I and τ II resulted then in ητ I = ηg I and ητ II = ηg II , respectively, where nI and nII are the number of gear pairs involved. Planetary gear efficiencies were set to η0 = 0.975 and implemented in the torque system according to:



1 − τ0 η0−t

1

τ0 η0−t

0

0

⎧ ⎫

⎨Tc ⎬

1

⎩ ⎭

Tr

=0

(18)

Ts

where the exponent t depends on the power flow inside the gear and can be expressed as a function of the incoming power from the solar, measured using the carrier reference frame: t = sign(Ts (ωs − ωc ) ). The output gear efficiency ηout was defined as function of the overall gear ratio τ out , considering the block a proper combination of planetary gear reducers (η pgr = 0.985) and the differential gear (ηdi f f = 0.975). The reducer stages were obtained assuming the maximum reduction ratio of the single stage set to 6 and the differential gear ratio of 4. The viscous losses were implemented using as reference the shaft connecting the planetary gear to the axle gear reducer. The empirical value of λ = 0.1 [Nm rad−1 s] was shown to lead to consistent results for the considered test case, thus leading to the following formulation for ηout :

ηout

  λω n pgr = ηdi f f η pgt − T

(19)

where npgr is the number of planetary gear reducers involved and ω and T are, respectively, the rotational speed and the torque of the shaft entering the reducer. The efficiencies of the hydraulic components were set to 0.90 and 0.96 for the volumetric and hydromechanical efficiencies respectively. The solution of the mechanical system for the considered cases supplies the transmission efficiency along with some other indices useful to perform an optimal design. The average performances were obtained as a mean over the whole velocity range:

η¯ =



Vmax 0

ηdV/Vmax

(20)

and is reported in Fig. 6 as a function of the design parameters. The continuous representation allows to appreciate the effect of the layout on the average efficiency. The data clearly identify the best solutions and the feasible design areas. In particular, the SCR and the SRC layouts have the highest efficiencies for non-dimensional full mechanical points in the range between 0.3 and 0.45. Similar results were obtained by the authors for a tractor in [17]. In fact, the best three layouts in that paper are the same ones that provide the highest efficiency here, i.e. RCS, RSC, CRS. The other three layouts in [17] did not admit solution, i.e. they were not able to reach the prescribed maximum speeds. Conversely, the same three layouts are able here to satisfy the design speeds, even if with very low efficiency. This different behavior is to be found in the characteristics of the two vehicles and in the different way of describing the losses of hydraulic machines: in [17] an accurate loss model dependent on operating variables was used; in this work, instead, a constant efficiency was assumed.

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Fig. 6. Average efficiency: (a) whole map; (b) detail on the most performing solutions.

Fig. 7. Minimum displacement required to meet the design pulling force when losses are considered: (a) DI displacement; (b) DII displacement.

Excluding the SCR and the SRC layouts, the RSC is the only other one having average efficiency higher than 0.6, which is obtained in the range between 0.5 and 0.8 for vfmp . This area agrees with the design data presented in Fig. 5, as it coincides with the small number of the related output gear, then, with the higher ηout and a reasonable value for τ II . While some smooth gradients with sharp discontinuities are visible on the large scale view in Fig. 6a, the detail reported in Fig. 6b shows a significant amount of small intensity patterns when a higher resolution is applied. The cause of these sharp increases and decreases in the average efficiency is due to the discontinuous formulation used for the mechanical path efficiency, which is a discrete function of the number of mechanical gear pairs involved. The displacements of the two hydraulic machines (Fig. 7) are now different from those calculated under ideal conditions in Section 3.2. In order to meet the request for the maximum traction force, the recirculating power in the CVT must increase, increasing then the hydraulic units’ displacement, accordingly to Eq. (15). As can be seen from Fig. 7, the smallest displacements roughly correspond to the higher efficiency areas of SCR and SRC layouts. The average efficiency η¯ reported in Fig. 6 delivers an integral information of the average performance but does not contain the information to discuss the actual shape of the efficiency curve, in particular the efficiency trend at low speeds. The latter is an important feature for heavy-duty vehicles as it impacts directly on the traction force curve. Fig. 8 reports the typical trend of the traction force as a function of the velocity for a powertrain equipped with continuous transmission. The point marked as A defines the maximum velocity at which the maximum traction force is available

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555

Fig. 8. Traction force trend for low velocities.

Fig. 9. Transmission efficiency η (VA ) = VA /VA

id .

and corresponds to the first point at which the ICE is releasing the full power: for velocities lower than VA the ICE power is limited to not exceed the hydraulic unit’s maximum pressure. Velocity VA is a function of the transmission efficiency:

VA = VA

id

P

η (VA )

(21)

where VA id = ICE F is the ideal velocity for a transmission efficiency of 1. Thus, the parameter η (VA ) = VA /VA id can be assumed as an index representative of the transmission efficiency at low speeds and its capability to deliver the maximum pulling force over a wide (or narrow) velocity range. The map of this parameter is shown in Fig. 9. The full mechanical point is the design variable with the higher impact. As expected, the highest values are obtained when the full mechanical point, which is the highest efficiency condition, is approximatively set at velocity VA id . For the present case, in fact, VA id = 6 [km/h], corresponding to vA id = 0.25. The designs with vfmp greater than 0.5 are characterized by poor performances when evaluated using the η(VA ). This prevents any actual development of these designs. This is an important and complementary information as the average efficiency plot of Fig. 6 would suggests the existence of high performance solution for up to v f mp = 0.45. When the design variable θ is taken into account, the maximum η(VA ) area appears between θ = 210◦ − 240◦ for both SCR and SRC layouts, according to the interval identified for the maximum average efficiencies.

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A. Rossetti and A. Macor / Mechanism and Machine Theory 133 (2019) 545–558 Table 4 Scaling of the three objectives. variable

1 – best achievable result

0 – non-conformal solution

Average efficiency η¯ Low speed efficiency η(VA ) Total displacement

0.74 0.98 168 [cc]

0.70 0.60 200 [cc]

3.4. Use of the continuous representation as design tool for optimization problems and for optimal design The global optimization of an HMT should account for all the feasible solutions of the problem, thus implying that the all the possible layouts has to be considered. Assuming the gear ratios and the hydraulic units of the transmissions are sized to satisfy the design constraints of the transmissions such as the maximum speed, input power and pulling force as in Eqs. (13)–(17), the optimization statement can be written then as follow:

Find xopt =



layout,

 τ0 , v f mp which minimizes f (x )

(22)

where the function f(x) represents a specific objective of interest, such as the average HMT losses in the velocity range. The presence of the layout in the design vector prevents the direct application of common optimization algorithm to the problem in Eq. (22), as they are usually developed to handle numeric inputs. In previous work [17], this problem was solved rewriting the problem in Eq. (22) in an iterative form such as:

f or i = 1..6   Find xopti = layouti , τ0 , v f mp which minimizes f (x ) end f or   Find xopt ∈ xopt i which minimizes f (x )

(23)

where the optimization problem was solved independently for each one of the 6 layouts, and the global optimum was selected as the best of the local optima. The continuous description of the layout allows instead to solve directly the problem as a whole, as stated in Eq. (22), taking advantage of Eq. (9), obtaining:

Find xopt =



 θ , v f mp which minimizes f (x )

(24)

This procedure can be easily coupled with direct search algorithms as all the design variables are now numeric variables. Moreover, the presented approach can be used as a preliminary tool by designers, as the parameters and the performance maps shown in the previous chapter allow to simplify the comparison of different design choices. The full mechanical point and the layout can then be discussed on the basis of different objectives, such as the maximum average efficiency, the efficiency at low speeds, weight and encumbrance of the hydraulic CVT, represented by the hydraulic unit displacements. Each of these objectives can be represented in a scalar form by a parameter μ, which can be obtained by means of a non-dimensional scaling and then represented in a proper map. The values 1 and 0 are respectively assigned to the best and worst result, assuming a linear trend between the extreme limits. The values adopted in this paper for these limits are presented in Table 4. The three objectives can be combined by means of a geometrical average to obtain multiobjective design criteria, such as: 1

- maximum efficency and minumum weight criterion μ1 = (μη¯ · μDI +DII ) 2

1

- max. efficency and max. efficiency at low speed criterion μ2 = (μη¯ · μη (VA ) ) 2 1

- all three criteria μ3 = (μη¯ · μη (VA ) · μDI +DII ) 3 As an example, the resulting parameter μ3 is presented in Fig. 10, where the iso-lines of the average efficiency were also plotted to allow an easier comparison with Fig. 7b. The map clearly identifies the best compromise areas. The small acceptable range for the average efficiency limits the possible solutions to the configurations SCR and SRC. The request of a small total displacement and high performances at low speed is satified with v f mp = 0.25 ÷ 0.30.

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Fig. 10. Optimization parameter μ3 .

4. Conclusions The paper presented a mathematical formalization to organize in a continuous form the different layouts that can be obtained combining the three shafts of a planetary gear with a mechanical system. The graphic representation of the formalization results in a polar plot that collects in a single real parameter the effects of both the configuration and the transmission ratio of the planetary gear. This plot was then applied to compare all the possible designs and related performances of an input coupled hydromechanical transmission of a 75 kW forklift. For this test case, the best overall performance is obtained when the sun is connected to the internal combustion engine, the ring to the continuously variable transmission and the carrier to the output shaft; furthermore, the full mechanical point must be chosen in the range between 0.3 and 0.45 of the maximum vehicle speed. If instead the design criterion is the reduction of the overall size of the hydraulic units and the performances at low speed, the full mechanical point must be chosen in the interval 0.25–0.30. While the current paper is limited to simple three shaft planetary gear train, ad focused in the specific case of the Input Coupled configuration, the graphic representation can be applied to other mechanical systems which include three shaft planetary gear. In HMT, for example, the Output Coupled transmission. Further development will include the application to compound configuration, where two three-shaft planetary gear are present as well as one brake and clutch couple, and to the dual stage configuration, that includes a Ravigneaux gear. The procedure is particularly well suited to be coupled with continuous optimization algorithms and more stringent optimizations than those outlined in the paper, provided that adequate loss models are adopted for the components. Acknowledgements The authors gratefully acknowledge the financial support from University of Padova (DOR 2016). References [1] K.T. Renius, R. Resch, Continuously Variable Tractor Transmissions, ASAE Disting. Lect. 29 (2005). [2] J.H. Kress, Hydrostatic Power-Splitting Transmissions for Wheeled Vehicles – Classification and Theory of Operation, SAE Tech. Pap. 680549 (1968). doi:10.4271/680549. [3] B. Carl, M. Ivantysynova, K. Williams, Comparison of Operational Characteristics in Power Split Continuously Variable Transmissions, SAE Tech. Pap. 20 06-01-3468 (20 06). doi:10.4271/20 06- 01- 3468. [4] F. Jarchow, Leistungsverzweigte Getriebe (Power Split Transmissions), VDI-Z 106 (1964) 196–205. [5] P. Casoli, A. Vacca, G.Berta, S. Meleti, M. Vescovini. (2007) "A Numerical Model for the Simulation of Diesel/CVT Power Split Transmission" SAE paper no. 2007-24-137, SAE-NA ICE2007, 17-20 Settembre, 2007 Capri-Napoli. ISBN: 978-88-900399-3-0. DOI:10.4271/2007- 24- 0137. [6] C. Blake, M. Ivantysynova, K. Williams, Comparison of operational characteristics in power split continuously variable transmissions, Commercial Vehicle Engineering Congress & Exhibition (SAE 2006), SAE, October 2006 Technical Paper Series 2006-01-3468. [7] L. Mangialardi, G. Mantriota, Power flows and efficiency in infinitely variable transmissions, Mech. Mach. Theory 34 (1999) 973–994, doi:10.1016/ S0094-114X(98)00089-5. [8] G. Mantriota, Performances of a series infinitely variable transmission with type I power flow, Mech. Mach. Theory 37 (2002) 579–597, doi:10.1016/ S0 094-114X(02)0 0 017-4. [9] G. Mantriota, Performances of a parallel infinitely variable transmissions with a type II power flow, Mech. Mach. Theory 37 (2002) 555–578, doi:10. 1016/S0 094-114X(02)0 0 018-6. [10] D. Sung, S. Hwang, H. Kim, Design of hydromechanical transmission using network analysis, Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 219 (2005) 53–63, doi:10.1243/095440705X6406.

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