Kinematic and dynamic analysis of a planetary gear box in steady-state working regime

The paper presents a method for analysis of planetary gear boxes in order to calculate the speed and torque of the epicyclic gear trains elements. Based on these findings, the power flow within the gear box is determined. Additionally, the mechanical efficiency is estimated in order to assess the overall efficiency of the transmission.


Introduction
Planetary gear boxes consist of epicyclic gear trains, clutches and brakes connected so a structure with 2DOF or 3DOF results.Usually, 2DOF epicyclic gear trains are used, but also epicyclic gears with more than 2DOF may be used too (i.e.Ravigneaux mechanisms).
The assessment of the planetary gear box imposes the determination of kinematics (speed, overall ratio), power flows and overall efficiency [1], for all the stages the gear box could realize.The above mentioned data calculated for steady state working regime could be used for evaluation of the solution analysed, and as reference base for analysis of transitory regimes (such as shifting).
Kinematic issues of planetary gear box are, at least partially, solved using graphical methods (such as speed vector planes [2], [3]) or by using variants of bond graph theory (an exhaustive presentation of various variants as well as of their capabilities is presented in [4]).The above mentioned methods for analysing the planetary gear boxes involves the usage of specialized knowledge and present real difficulties in the case of structures with more than 2 epicyclic gear trains.Analytical methods for studying planetary gear train structures are detailed in [5] without direct application to planetary gearboxes.A useful approach is applied in [6], consisting in combining analytical method with block diagrams; this approach allowed to analyse the working possibilities of power-split infinitely variable transmissions.
Frequently, the gear trains design parameters have to be optimized; for ordinary gear trains, there are optimization methods which allow the calculation of gear teeth numbers, as it is presented in [7].For the planetary gear boxes, no reference was identified for the calculation of planetary gear design parameters which provide imposed gear box ratios.
An optimal design math model of epicyclic gear train based on the planet gear design features is detailed in [8]; the model is solved using the differential evolution algorithm programmed in Visual Basic 6.0, proving the interest in using the computer for analysing the planetary gears.For Ravigneaux mechanisms, a variant of the optimisation algorithm is detailed in [9].
The inverse kinematic planetary gear unit is presented in [10] without taking into consideration the aspects regarding the planetary gear boxes.The method presented below is simple, uses basic knowledge regarding the epicyclic power trains, may be used for complex structures of gear boxes and permits an exhaustive characterization of kinematics and power flows of planetary gear boxes, including the calculation of overall mechanical efficiency.This method is based on analytical approaches detailed in [11], [12] and [13], and provide a comprehensive analysis as well as the solving of kinematic inverse problem.
In order the present the proposed methods, a hydro-mechanical transmission type ZF-ECOMAT 7 HP 602 was selected.
The transmission of the series ZF-ECOMAT 7 HP 600 is largely used in trucks, buses as well as in military vehicles.
The transmission consists of: x torque converter (TC) of which stator could became freed do to one-way clutch FW; the turbine of torque converter may be connected directly to impeller by coupling the clutch ATC; x planetary gearbox which includes four epicyclic gear trains, four brakes and tree clutches.The kinematic diagram of the transmission type ZF-ECOMAT 7 HP 602 is shown in Figure 1.

Kinematics
The proposed method consists of several successive steps.
The first step is devoted to transforming the kinematic diagram into nodal diagram applying the following rules: x each brake and clutch are represented by kinetic nodes, with the following notation: F1, F2, F3, F4 (brakes) A1, A2 and A3 (clutches); x each epicyclic gear train is represented by a generalized node (a circle) with the following notations: constant of the mechanism indexed j, noted Kj : , where indexes 2, 1, 0 refer respectively to the ring gear, sun gear and planet carrier, while z represents the tooth number; sun gear: 1 (inside the circle); 1 3 1 3 2 j j (outside the circle); ring gear: 2 (inside the circle); 2 3 1 3 1 j j (outside the circle); planet carrier: 0 (inside the circle); 0 3 1 3 3 j j (outside the circle).x the input shaft is noted a, and the output shaft is noted b.The results of this step are presented in Figure 2. The second step of the method deals with the calculation of the speed of all central elements of the transmission, indexed 1 to 12.The kinematic equations may be grouped as following: x Willis equations written for the four epicyclic mechanisms [12] (four equations): x equations for permanently connected central elements, six equations may be written: ; ; The equation for the input element is: In equation ( 3) it was adopted the value 1 for the input speed in order to facilitate comparative analysis of the speeds of the output elements in all stages of the gearbox.
The equation types ( 1), ( 2) and (3) totalize 11 equations; according to Figure 2, there are 13 central elements, indexed from 0 to 12.The two necessary equations for assembling a determined system of equations results from the conditions imposed for obtaining the desired stage of the gearbox: the coupling of two friction elements (clutch, brake); the coupling sequences are detailed in Table 1.
Adding to the equations ( 1), ( 2) and ( 3) one of the equations ( 4), ( 6), ( 8), ( 10), ( 12), ( 13), ( 15) or (17), a system of 13 linear equations resulted; the system of equations was solved symbolically using the matrix method detailed in [11].Using symbolic calculus in PTC Mathcad Prime it was possible to obtain the relations which allow the calculation of the ratio of all gearbox stages.
The 3 rd step of the proposed method consists of the estimation of numerical values of epicyclic gear trains constants (K1…K4) based on the existing gear ratios (values provided by the manufacturer).It may be observed that there are seven such relations involving only four unknown parameters; consequently, an overdetermined non-linear system resulted, usually without unique accurate solution.Nevertheless, it is noticed that relations (5), ( 7), ( 9) and ( 11) contain all the unknown parameters and, consequently, may be used for the estimation of the epicyclic gear constants by imposing the actual values of the gearbox ratios (inverse kinematic problem).
The calculated constants are the following: Due to the presentation of data regarding the gearbox ratios with only two digits, it is presumed that the calculated values for epicyclic gear train constants will generate error in the case of calculation of estimated gearbox ratios.The results of the calculations are presented in Table 2.The magnitude of the errors is low enough to allow further usage of the estimated values for Kj for calculation of central element speeds; the results of such calculations are presented in Table 3. From the Table 3 results that the maximum speed occurs for sun gear of K1, K2 and K3 epicyclic gear trains; this aspect doesn't imply negative behaviour of gearbox due to reduced rotation inertia of sun gears.The maximum relative speed of planet gear type elements occurs in 7 th gear for K1 and K2 sun gears.
Another kinematics issue refers to the speed variation of the central elements during the shifting; this issue is important in order to assess the inertia moments of central elements, a special attention being paid to ring gear and to carrier due to their bigger inertia.It is assumed that during the shifting process the output speed of the gearbox doesn't change (the vehicle speed is constant); consequently, at an upshift, the input speed of gearbox decreases with the ratio of the gearbox speed ratios existent before and after the shifting process.The relative reduction of the input shaft speed at up-shifting is presented in Table 4.
From the above data, it may observe that a maximum variation of speed occurs for the sun gear of K4 epicyclic train.For the ring gears, the maximum speed variation occurs also for K4.Considering the low input speed used in the first stage (usually, the drivers uses the first stage only for starting the acceleration process and prefers to shift up without reaching the maximum speed of the engine), the magnitude of the speed variation decreases accordingly with the low input speed.

Power flow
The next step of gearbox analysis deals with the calculation of torques acting on outer elements.
It is noticed that the ramified nodes included in the nodal diagram (Figure 2) obey to two main laws: x energy conservation -the sum of power flows acting on the ramified node is null; x torque equilibrium -the sum of torques acting on the ramified node is null.
Due to the fact that the ramified nodes imply torque distribution, it is necessary to index all the ramified nodes, as well as braches connected to it; the results of these operations are presented in Figure 3.
For each of the four epicyclic gear trains, the following two equations regarding torque distribution may be written [11]: in total, eight equations.The torque equilibrium law for ramified nodes indexed p, q…w, z conducts to the following equations respectively:

T T T T T T T T T T T T T T T T T T T T T T T T T T T T , (20)
in total, nine equations.The input element is indexed 12; we assume that the input torque is: In order to obtain a stage of the gearbox, two friction elements have to be coupled and only these friction elements could transfer torque; all other friction elements cannot transfer torque, and, consequently the torques of their input/output shafts (modelled as branches) are null.These considerations lead to the specific equations written (for clutches and brakes respectively) for each stage: in total eight equations for each stage.
Assembling for each stage the equation types ( 19), ( 20), ( 21) and one of the equations groups (22), a system of 26 equation results.It became possible to calculate the power flow of element j using the relation: The ratio relations presented in Table 1 allow identifying the epicyclic gear trains which take part in the power transmission.Consequently, for the first two stages the power flows through a single epicyclic gear train used as ordinary gear.For the 5 th stage, the planetary gearbox works "frozen", all its elements having the same speed.
The results obtained for power flow in remaining stages are presented in Figure 4 to Figure 8.It may observe that in Reverse stage, despite the three epicyclic gear trains participate in power transmission, actually there is a serial configuration consisting of three gears which transmit the same amount of power.
Analysing the power flow in the 4 th stage, it could observe that element 6 transmits a power flow greater than the input power (with 17.2%); that means the mechanisms K3 and K4 faces a closed loop power flow.

Overall mechanical efficiency of the gearbox
The efficiency of the epicyclic gear train is analysed by various papers in order to emphasise the influence of the mechanism constants, torque, speed, function (differential/summing) on the efficiency [14], [15], [16].
A common denominator of these papers is the need to previously know the power flow through each epicyclic gear train, as well as the difficulties in "assembling" the individual power losses into overall efficiency of the gearbox.
Instead, a generalized method, presented in [13], was adopted in order to reduce the volume of calculations as well as to avoid the possible error propagation from previous calculations.
In general, for a planetary gearbox, the ratio of the stage, denoted is, is a rational function of epicyclic gear trains constants Kj: ( ) ) For the planetary gearbox analysed, the above relation may be written as: ( ), 1,...,8; 1,...,4 ) where the Reverse stage was considered as the 8 th stage.
For the epicyclic gear train with the carrier blocked, the mechanical efficiency, noted K0, may be calculated using the same formulas as for ordinary gears (having fixed rotational axis); for the purpose of the present paper, it was adopted the value: 0 0.965 K [11].The overall efficiency in the stage s of the gearbox may be calculated using the following relation: , ( ) , 1,...,8; 1,...,4 ( ) ) where the functions fs(Kj) are presented in Table 1 (the relations ( 5), ( 7), ( 9), ( 11), ( 14), ( 16) and ( 18 ) The derivatives from the relations (28) were calculated symbolically using PTC Mathcad Prime, forwarded by numerical calculation of efficiency considering the calculated values of ratios presented in Table 1.
Mechanical efficiency in the 5 th stage is assumed to be 1 because the power is transmitted without any transformation of torque and speed through the gearbox.The obtained results are presented in Table 5. x the low efficiency occurs for stages (1, 2, 3 and R) in which the power transmits using a unique circuit, with lowest value for Reverse stage in which three epicyclic gear trains are connected in series (see Figure 8); x the highest efficiency is recorded in 6 th gear due to the circulation of power flows through three epicyclic gear trains (see Figure 6) without closed loop circulation as in the case of 4 th stage (see Figure 5).It is mentioned that this method is only a rough estimate which allows identifying situations in which the overall efficiency of the gear box is very low due to closed loop power flow circulation.

Conclusion
The proposed methodology for analysing the planetary gearboxes has the following main advantages: x allows the fast calculation of the gear ratios, and power flow for multi-path power flow (power split) transmissions, with direct applicability for analysing the automatic transmission; x allows the extensive usage of computers and appropriate commercial software for performing a comprehensive analysis of planetary gear boxes, resulting time saving and high accuracy; x allows the fast evaluation by computation of the overall efficiency and power loss of the gearbox; x may serve as first estimation for more accurate calculations of efficiency taking into consideration the influence of torque and speed on power losses; x may serve as basis for assessing the dynamic behaviour of the gearbox during stage shifting process.

Figure 3
Figure 3 Nodal diagram for torque calculation

Table 1
Stages of the planetary gearbox: coupled friction elements, ratios

Table 2
Error resulted by using calculated values of Kj 1 st gear 2 nd gear 3 rd gear 4 th gear 5 th gear 6 th gear 7 th gear

Table 3
Calculation results of central elements speedUnitElement 1 st gear 2 nd gear 3 rd gear 4 th gear 5 th gear 6 th gear 7 th gear R

Table 4
Speed variation due to up-shifting

Table 5
Overall efficiency of the planetary gearbox denotes the epicyclic gear trains that are not involved in power flow It may observe the following aspects: a