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)”, (Figure 1) in which one shaft of CVU is connected to the input one of the transmissions while the “Output-Coupled, OC”, (Figure 2), corresponds to the case of that one shaft of the CVU is related to the output one of the transmission [1]. Taking into account of the direction of power flow that can occur between the three components, these transmissions are classically divided into three types for the case “IC” (Figure 1) and three others for the case “OC”: Type I, Type II and Type III, (Figure 2) [1,2].

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 [1] and Schembri et al. [2] performed a preliminary analysis of two classes, Input-coupled and Output-Coupled, used in an infinitely variable transmission considering the efficiency

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. [6]. In Ref [6] the authors considered the planetary gear train as composed of two ordinary gear mechanisms, and, the efficiency of the PGT is computed basing on this assumption. In this work, our objective is to establish the overall efficiency of the transmission, based on a simple algorithm to assess the mechanical efficiency of planetary gear trains which operate as differential mechanism [7]. This paper aims also to compare ours results with those available in the literature, otherwise obtained by other approaches. In order to illustrate the strengths of the present approach, two comprehensives applications are presented. The first deals with a continuously variable transmission consisting of a speed variator, a mechanism gears with fixed ratio and a planetary gear trains of Type I [8]. The second is the same as the first, the difference lies in the type planetary gear train which is of Type IV [8]. These transmissions can perform as continuously variable power split transmissions (CVPST), or infinitely variable ones (IVT), according to the type of powers flow required by the design variables. Let τ_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 (Figure 3 and 4), according to the configuration of the transmission (see Table 1):

• 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 [7]:

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

Table 1. Transmission ratio in the six possible connection arrangements in a planetary gear trains.

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 Table 2. The configurations of the PGT, which determine the input, output and floating shafts (Figure 1) are illustrated in Table 1 below.

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 [9] (Figures 1 and 2). The type of power flow is conventionally defined by the ratio of the power through the control circuit by the output power (P_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 (Figure 1, Type III). In this case of operating, the transmission behaves as a power split transmission. In the two other cases, the power flow leading to a power recirculation (Figure 1, Type I, II). According to Figures 1 and 2, transmissions with a Type III power flow are more desirable because there is no power recirculation in the transmission circuit. In Type I and Type II power flows, certain transmission components are exposed to a power load greater than that of the input load. Distinction between the three power types could be made thanks to the so-called circulating power factor, defined by [9-11]:

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 [9]. The type of variable element chosen for this particular application was the metal pushing belt CVU because it offers acceptable efficiency values compared to other variable elements [3, 5].

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 [3], derived from a series relations to compute the efficiency of CVU, which interpolates the experimental data according to the input torque of the CVU, the angular velocity of the driving pulley CVU, and the overall ratio of the transmission. In this work, the curves (Figures 5

Table 2. Mechanical efficiency of the PGT for the three power flow types and configurations.

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 [5].

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 [1] and subsequent integrations by Mangialardi and Mantriota [12], Zhang and Leduc [13] and S. Schembri et al. [2] have examined the efficiency of an IVT assuming that the efficiency of the CVU η_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. [6], which is not systematic and its application to more complex models of transmissions would be rather cumbersome. The CVPST (IVT) system efficiency is a function of the individual component efficiencies and the ratio of power flowing through the individual branches of the system. Often the performance of a CVU is improved by coupling it to a planetary gear train. Knowing that CVUs have generally a lower efficiency than PGTs, the best performance is obtained with a low ratio of power through the variable branch. Therefore, the flow Type III promotes high transmission efficiency and low dissipation due to the power recirculation. In addition, dimensions of CVUs are strictly related to the maximum power and torque required during the operating conditions. So that this type of flow leads to more technological solutions are more

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) (Table 3).

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

Table 3. Total efficiency of the transmission according to power flow type.

power equation we have (Figure 1):

Namely:   (11)

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

Thereby:  (13)

Hence:    (14)

Moreover (Figure 1, Type of power flow I)

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 [14]);

2) Identify the configuration (Table 1);

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 Table 2. The mechanical efficiency of the PGT can be calculated easily, using the Table 2, where all possible cases are illustrated;

5) The overall mechanical efficiency of CVT can be calculated easily using the Table 3.

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].