Article citation information:

Drewniak, J., Garlicka, P., Kolber, A. Design for the bi-planetary gear train. Scientific Journal of Silesian University of Technology. Series Transport. 2016, 91, 5-17. ISSN: 0209-3324. DOI: 10.20858/sjsutst.2016.91.1.

 

 

Józef DREWNIAK[1], Paulina GARLICKA[2], Aleksandra KOLBER[3]

 

 

 

DESIGN FOR A BI-PLANETARY GEAR TRAIN

 

Summary. The article presents the design for a bi-planetary gear train. The project description is supplemented with calculations of kinematics, statics and meshing efficiency of the gear wheels included in the gear train. Excluded are calculations of strength and geometry of gears, shaft and rolling bearing, since they are similar to classical calculations for planetary gears. An assembly drawing in 2D and assembly drawings in 3D of the designed bi-planetary gear train are also shown. This gear train will form the main element of the research in hand.

Keywords: bi-planetary gear train, transmission ratio, meshing efficiency

 

 

1. INTRODUCTION

 

The structural diagram of a bi-planetary gear train, or double planetary gear train, is shown in Fig. 1a and Fig. 2 [1]. It consists of a main planetary gear set and an internal planetary gear set, which is known as an internal satellite system. The main planetary gear set comprises sun gear 1, satellites 2 and 6, external carrier arm h and ring gear 7, which is fastened to the frame, while the internal satellite system comprises sun gear 3, satellites 4 with internal carrier arm H, and ring gear 5, which is fastened to carrier arm h. In the bi-planetary gear train, the role of second toothed satellite 6 (Fig. 1b) fulfils the needs of the planetary gear train, known as the internal satellite system, with carrier arm H. H drives planetary gear 6, which already belongs to the main gear set. At the same time, rim gear 5 drives carrier arm h of the main planetary gear set.

 

 

Fig. 1. Kinematic diagrams of the bi-planetary and planetary gear trains [1]

 

A characteristic feature of this gear train is that the satellites within the satellite mechanism rotate around three axes – those of its own central planetary gear set and the main central bi-planetary gear train. This particular property of bi-planetary gear train is mainly used in face milling cutters for combined mining.

 

 

2. DESIGN CALCULATIONS

 

2.1. Kinematic diagram of the gear

 

The structure of the transmission is selected depending on the kinematic ratio. The values of output torque are equal to  and , respectively, when the value of input revolution is equal to .

For the required range of output speed , the mean transmission ratio and the acceptable range of ratios are:

,                                             (1)

.                     (2)

On basis that the demands correspond to the planetary gear, the kinematic diagram, as shown in Fig. 2, is according to [2, 3, 9].

This gear comprises the inner transmission (satellite planetary mechanism) of 3, 4, 5, H and of the outer transmission of 1, 2, 6, 7, h.

The degree of mobility of the transmission is:

                                          (3)

where ( number of moving parts); (number of pairs of 5-class (arm h on the right is oversized or redundant)); and(number of pairs 4-class (engagement of teeth, counted only once)).

 


Fig. 2. Kinematic diagram of the bi-planetary gear train

 

As the degree of mobility is , only one value of a speed output or input is needed to determine the speed of each of the transmission components.

 

2.2. Gear transmission ratio

 

According to the definition of the kinematic gear ratio,  (from pinion 1 to arm , when wheel 7 is fixed) is equal to [1, 3, 4, 5, 6, 7]:

,                                                        (4)

 

where  (the rotational speed of pinion 1 or input rotational speed of transmission) and  (the rotational speed of arm , that is, the transmission output rotational speed) (Fig. 1). In order to determine the kinematic ratio,  must first determine the base ratio of the inner transmission, assuming that the whole planetary gear rotational speed is equal to  (that is, to consider the kinematics of planetary transmission in relation to arm ). Then, the relative rotational speed of each wheel gear is equal to  () at the same time that  and  (because  and ).

Similarly, the relative rotational speed of inner arm H, in relation to external arm h, is . The basic kinematic ratio of the inner transmission, which consists of wheels 3, 4, 5 and arm H with known relative speed , ,  and , can be determined using the Willis formula (5) and expression (6) for the transmission ratio of wheels 3, 4 and 5, in relation to arm H as a function of the number of teeth (such as for the gears with fixed axes) [1]:

                                                       (5)

 

             (6)

Hence:

,                                                 (7)

because .

To calculate the requested transmission ratio of the bi-planetary gear train, it is firstly necessary to determine unknown relative rotational speeds  and  using formula (7) above, as a function of  (9) and  of the two conditions on the gear ratios  (8) (the ratio from arm H to wheel 7) and  (10) (the ratio from wheel 3 to wheel 1):

,                                              (8)

Hence

,                                         (9)

because .

Similarly,

,                                           (10)

Hence

,                            (11)

because  and .

Thus, after the subsequent transformations, the quotient of the number of teeth , according to formula (7), takes the following form (12):

.                                         (12)

Hence the formula for the ratio of the bi-planetary gear train (4) is as follows:

 

                                      (13)

The required value of the ratio obtained for the numbers of teeth of the bi-planetary gear train – namely,, , , , , , , is due to the following:

,                 (14)

That is, .

The actual value of the output speed is .

 

2.3. Engagement efficiency of the transmission

 

Determining the efficiency of the gear train only takes into consideration the losses incurred due to friction in the gear engagements, i.e., it does not include the friction that occurs between the bearings and the resistance of the splashing oil. Thus, according to formula (15), the efficiency of the designed bi-planetary gear train only depends on the base kinematic ratio  of the transmission and the base efficiency  [1, 6, 7]:

.                                                    (15)

The base efficiency is determined in relation to the serial sequence composed of the pair of wheels 1 and 2, planet gears 3, 4 and 5, and the last pair of wheels 6 and 7:

 

.                                                (16)

 

The efficiency of the pair of outer wheels 1 and 2, when considered in motion relative to arm h, is determined by the following known formula [3, 6]:

 

                     (17)

 

A similar formula determines the efficiency of the pair of inner wheels 6 and 7, when considered in motion relative to arm h [2, 3, 5, 6]:

 

.                       (18)

 

The base efficiency for the internal epicyclic gearing of 3, 4, 5 and H is determined by the following formula [2, 3, 5, 6]:

 

,                                                    (19)

where  (see formula (7)) and are the base transmission ratio and the base efficiency of the inner epicyclic transmission of 3, 4, 5 and H, respectively:

 

                                 (20)

where(base efficiency of wheels 3 and 4).

,                       (21)

 

where (base efficiency of wheels 4 and 5).

.                     (22)

Component values of the base efficiency for the assumed value of the coefficient of friction in engagement  are:

,            

,     ,

,       ,     .

Thus, the efficiency of the planetary gear is:

 

,                                  (23)

because:

 

And:

.

 

The value of efficiency of the bi-planetary gear unit is similar to the value of efficiency of the three-axes non-planetary gear train (), although it is greater than the efficiency value of the planetary gear, even when the transmission ratio is of less value.


2.4. Forces acting on the spur gear teeth

 

According to the principle of the operation of the bi-planetary gear train, the output torque  consists of three components:

a) Torque  generated by  satellite 2 [1, 2]:

 

,                            (24)

where  (the force acting on the axis of satellite 2) (Fig. 3); , (the pitch radiuses of sun gear 3 and the satellite 2)’; and (the radius of carrier arm h).

 


Fig. 3. The forces acting during the engagement of gears 1, 2, 3 and on left arm h

 

b) Torque  generated by rim gear wheel 5 (Fig. 3, Fig. 4):

 

,                      (25)

where ,  (the pitch radiuses of sun gear 3 and rim gear 5); and  (the circumferential force acting on gear 5).

 

c) Torque  generated by carrier arm H and satellite 6 (Fig. 5):

,                 (26)

 

where(the pitch radius of satellite 6); and  (the force acting on the axis of satellite 6).


 

Fig. 4. The forces acting during the engagement of gears 3, 4, 5 and on arm H

 


Fig. 5. The forces acting during the engagement of gears 6, 7 and on right arm h

 

As such, total output torque  is equal to:

 

.                                           (27)

 

Total output torque , with regard to engagement efficiency , is equal to:

 

.                             (28)

 

 

3. DESIGN BI-PLANETARY GEAR TRAIN  

 

3.1. Description of the gear train

 

As previously mentioned in this paper, the bi-planetary gear train consists of two planetary drive systems, i.e., the outer gear set, also known as the main gear set, and the inner gear set, which is known as a satellite planetary system. The main planetary gear set consists of sun gear 3, which meshes with the three satellites 4 mounted on three shafts, namely, pinions 10 of the satellite planetary system (Fig. 6). Pinions 10 represent the sun gear of the planetary satellite system. It drives the three satellites 8, which are still engaged with the gear wheel due to internal teeth 9 being fitted to one of the three holes of arm 11. Each satellite is mounted rotationally on axle 12, on which is supported one of the three internal arms H (according to Fig. 1). Each arm H is a part of second satellite 13, which already belongs to the main gear set.

 

 

Fig. 6. Assembly engineering drawing of the bi-planetary gear train


Satellite 13 is meshed with fixed internal gear wheel 14 and internally mounted on the right side of the hole in arm 11 of output shaft 15. Power is transmitted from the electric motor via shaft 1 and clutch 2 to sun gear 3, which is engaged with the three satellites 4 mounted on shaft 10 that transmits part of the torque along intermediate arm 11, then along output shaft 15, while the rest of the power is transmitted through gear wheel 9 to arm 11, then through internal arm 7 (satellite 13) to right arm 11.

 

3.2. The order of the assembly of the bi-planetary gear train elements

 

Engineering drawings of the assembly sequence, which allow for checking the correctness of the designed bi-planetary gear trains (Fig. 6) in terms of mounting options and possibly assessing its degree of difficulty, are included in Table 1.

 

Tab. 1.

Engineering drawings of the assembly sequence

 

a)

 

 

b)

 

 

c)

 

 

d)

 

 

 

 

 


 

e)

f)

 

g)

 

h)

i)

j)

k)

 

 

Note: not shown is the sequence of the assembly of simple elements, such as oil filler plugs, oil level and oil drain plug.


4. CONCLUSION

 

This paper presents the design of a bi-planetary gear train together with the calculation of the kinematics, statics and meshing efficiency of gear wheels. Omitted from the calculation are the geometry and strength of gears, shafts and rolling bearings, as these are recognized as being typical design calculations. 3D drawings of the order of assembly components have confirmed the correctness of the design of the bi-planetary gear train. The designed planetary gear is constructed as a prototype for transmission to the test bench in a laboratory for fundamental machine design. The main aim of this study is in comparing the dynamics of the transmission between fastened rim wheel 14 and the solution, when wheel 14 is suspended flexibly from the coupling gear (variant II of the solution).

 

 

References

 

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Received 20.12.2015; accepted in revised form 27.04.2016

 

 

Scientific Journal of Silesian University of Technology. Series Transport is licensed under a Creative Commons Attribution 4.0 International License

 



[1] Faculty of Mechanical Engineering and Computer Science, University of Bielsko-Biała, 2 Willowa Street,
43-309 Bielsko-Biała, Poland.
E-mail: jdrewniak@ad.ath.bielsko.pl.

[2] Faculty of Mechanical Engineering and Computer Science, University of Bielsko-Biała, 2 Willowa Street,
43-309 Bielsko-Biała, Poland. E-mail: pgarlicka@ath.bielsko.pl.

[3] Faculty of Mechanical Engineering and Computer Science, University of Bielsko-Biała, 2 Willowa Street,
43-309 Bielsko-Biała, Poland. E-mail: akolber@ath.bielsko.pl.