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With their advantages, continuously variable transmissions have gained more popularity in the last decade by their use in mechanical transmission systems. The present paper aims to analysis the efficiency of the transmission based on the mechanical efficiency of the planetary gear train integrated in such transmission. In this analysis, we consider the mechanical efficiency of the transmission has been determined considering how the efficiency of the CVT members changes as a function of the operating conditions. The efficiency of the planetary gear train as a function of the configuration, speeds in his three input/output shafts, and also with respect to the power flow type. Results are compared with those obtained from other methods performance evaluation of the transmission, available in the literature.

The function of a vehicle transmission is to adjust the traction available at the output shaft of the drive engine to suit the vehicle, the surface, the driver and the environment. One of most important decisive effect of the transmission is the fuel consumption. The need of reducing the fuel consumption has become nowadays a very important factor. Historically, to meet this objective, vehicles are usually equipped with gearboxes with more ratios while adjusting at best the situation of the ground to cross with the drive engine regime. The major drawbacks concerning gearboxes can be linked to the holes when moving from one report to another, and to the large number of ratios that have to be installed, especially for heavy vehicles. A way that could overcome these disadvantages, at least partially, is a continuously variable transmission using a speed variator that permitting to vary the output speed continuously, eliminating the discontinuity in the speed variation. However, these transmissions are limited in use due to their low capacity transmission. A best solution, which seems to be the most effective, consists in the use of continuously variable power split transmissions (CVPST), or infinitely variable transmission (IVT) ensures a continuously ratio transmission and an infinite ratio range. These kinds of transmissions are mainly composed of one or more planetary gear trains (PGT), a unit of speed variator (Continuously Variable Unit, CVU) and one (or more) ordinary gear trains with fixed ratio (FR). It should be noted that a classical planetary gear train operates with two degrees of freedom and three running shafts. The unit speed variator (CVU) carries out the control by imposing an angular velocity on one of the three running shafts of the PGT. These continuously variable transmissions are classified into two main categories: The case of transmission “Input-coupled (IC)”, (

It should be noted that the planetary gear train plays a fundamental role in the setting up of these power split transmissions. Its efficiency, which could be very low for some operating conditions, is a major factor in the performance assessment of the transmission in which the PGT is integrated. Some authors have considered this efficiency as constant for the whole modeling in the total transmission efficiency. Yan and Hsiech [

of the PGT constant as the same as for the CVU and FR. Mantriota et al. [3-5] have used an approach presented by Pennestri et al. [_{PGT}, τ_{FR} and τ_{CVT} be the stationary gear ratio of the PGT, the ratio of the ordinary gear and speed ratio of the variator, respectively.

As for organization of the paper, the next sections present all the basic elements we have adopted in this work, namely the efficiency of a PGT which is the central and the efficiency of the variator component of the transmission. The fifth section describes the proposed algorithm, to estimate the overall efficiency of the transmission for different configurations, basing on a kinematic and energetic analysis. Some applications with results compared with the literature are reported in Section 6. Finally, some concluding remarks in the last section.

In this work, planetary gear trains (PGT), characterized by its stationary gear ratio τ_{PGT}, operates with two degrees of freedom and comprise two suns gears (i) and (j), a carrier (k) and planets (S). It should be noted here that one or the two sun gears can be ring(s). Let ω_{i}, ω_{j} and ω_{k} be, respectively, the angular velocities of the two suns gears (i, j) and the carrier (k), respectively. Similarly, we denote by, T_{i}, T_{j} and T_{k} torques on the links (i), (j) and on the carrier shaft (k), respectively. Let P_{i}, P_{j} and P_{k} be the powers on the sun gear (i), the sun gear (j) and on the carrier (k), respectively. We gather all equations and relationships useful in the determination systematically of the mechanical efficiency of a PGT:

• Willis’ equation:

• Equilibrium equation:

• Powers on the three running shafts (i, j, k):

• The greatest power (P) of the three powers that crosses the three running shafts of the PGT. Each index i, j and k can take the values 3, 5 and 6 (

• The relative power, defined in the relative movement, such as:

or (5)

• The exponent q can have the values 1 or −1, according to the sign of the power ratio following [

• The mechanical efficiency of the PGT id defined by the ratio of the power on receiver shaft(s) by the

power on drive one(s):

• A power which comes to the PGT block is counted algebraically positive (driving(s) shaft(s)) and a power that leaves is counted negative (driven(s) shaft(s)).

Finally, analytic expressions of the mechanical efficiency of a given PGT, with respect to the power flow, the sign of the exponent (q) and the configurations are reported in

An important factor in power flow analysis within power split transmissions, called factor of power flow, is adopted to identify the type of power flows [_{out}) [10,11]. In the case of IC, where the power at the entrance noodle of the transmission is divided in two parts, the first across over the control branch through FR and CVU, as for the second directly into the PGT. The last behaves as a power collector (

According to the operating conditions, the three power flow Types I, II and III are defined by the following statements:

The variable element in this transmission is responsible for the greatest part of the mechanical losses because it is less efficient than conventional gears [

The efficiency of the CVU is computed according to the operating conditions. There are a number of papers in the literature, on experimental studies on the efficiency of V-belt. Using this Experimental results presented in [

and 6) related to the transmission ratio τ_{IVT}, refer to a specific value of the dimensionless output torque t. The parameter t points out, for every value of τ_{IVT}, the ratio between the IVT output torque and the transmissible maximum torque in condition of nonglobal sliding of the CVU [

Efficiency is a priority factor in the study of CVPSTs (IVTs). So, estimating the efficiency of a pre-designed variable transmission is a very important issue, as this will determine the prospective performance and the feasibility of the final design and the eventual prototype. Yan and Hsieh [_{CVT} and the planetary gear train η_{PGT} are constant. Mangialardi and Mantriota [3-5] applied the method for computing the efficiency of a PGT developed by Pennestri et al. [

compact. This work looks for configurations that the most suitable type of operation, knowing that the overall efficiency of the transmission function is a weighted performance of the planetary gear and that of the CVU. The weights are the weight fractions of power through the CVU and the PGT. In this study, we provide in the first step one algorithm of calculating the efficiency of the PGT (Section 2), then the overall efficiency equation of the transmission for different configurations of power flow Types I, II and III of the input-coupled architecture (IC) (

With a Type I power flow, in steady state, with η_{PGT} denoting the efficiency of the PGT train; based on the

power equation we have (

Namely: (11)

The equilibrium of the torques acting on the shafts of the PGT train is:

Thereby: (13)

Hence: (14)

Moreover (

with η_{FR} and η_{UVT} denoting the efficiency of the FR mechanism and of the CVU. Through these equations it is possible the determination of the parameters that characterize the performances of the CVPST (IVT).

A systematic algorithm to compute the mechanical efficiency of CVT (CVPST, IVT): An algorithm to calculate the mechanical efficiency of a given CVT is developed and proposed here. This algorithm consists in the following steps:

1) The PGT, CVU and FR are respectively identified by their characteristics values (τ_{PGT}, ρ), (τ_{UVT}, η_{UVT}) and (τ_{FR}, η_{FR}) respectively (to calculate ρ see Ref [

2) Identify the configuration (

3) For the kinematic situation, identify the types of power flows I, II or III by drawing the ratio according to τ_{IVT, CVPST} or τ_{UVT} (Equation (8));

4) The type(s) of power flow is now determined, identify then the sign of the exponent “q” thanks to formulas provided in

5) The overall mechanical efficiency of CVT can be calculated easily using the

In order to achieve the desired objective, by making more concrete the proposed algorithm on the mechanical efficiency of CVTs. once the type of transmission has been established. We are going to deal with two different applications.

In order to determine the performance of an overall architecture (IC) for different approaches, we propose two case studies of variable transmissions offered by the authors [3-5,9-11].

In this application, we consider a continuously variable transmission used in [9-11], it includes a simple PGT Type I [

• a belt mechanical variator with a speed ratio τ_{CVU} that varies between 0.5 to 2.5, and an efficiency η_{CVU} which is estimated at an average 0.8;

• a PGT with a gear ratio (τ_{PGT} = −3), its efficiency η_{PGT} will be calculated here;

• An ordinary gear with a constant ratio τ_{FR} = 2 and efficiency η_{FR} = 0.98.

_{IVT} = 0), thereby reversing the rotation of the output shaft. Furthermore, τ_{CVU} and τ_{IVT} may become directly or inversely proportional. In this locus, there will be change of power flow (see _{CVU} = 0.5), regardless of the configuration, value for CVPST (IVT) transmission ratio is equal the unit (Synchronization point).