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STATIC AND DYNAMIC ANALYSIS OF A

PLANETARY GEARBOX WORKING PROCESS

Marian Truţă1,a, Octavian Fieraru1,b , Radu Vilău1,c, Valentin Vînturiş1,d

and Marin Marinescu1,e

1Military Technical Academy, Blvd. George Coşbuc nb. 39-49, Bucharest, Romania

[email protected], [email protected], [email protected],

[email protected], [email protected]

Key words: planetary gearbox, power flow, efficiency, torque, power, nodal scheme.

Abstract:

Present paper focuses on the power circuits within planetary gearboxes, providing an originalmathematical model. This model is an excellent instrument to analyze and determine the torque,angular speed, power factors, global gear ratios and efficiency distribution. The analysis itself has

been developed for the second-gear, forward-motion of the gearbox. The analysis provides themathematical model of the gearbox’s way of working both in the static and dynamic modes.The method starts with issuing the equivalent graph (network structure) of the gearbox. Using thismodel, the gearbox is associated to a power driveline network that both transforms and guides the

power flow to the final transmission of the vehicle. Using the network structural model, the gearboxturns into a general nodal diagram (graph). Our study concerned both the static and dynamic modes.It consists in a mathematical determination of the cinematic (angular speed) and dynamic (torque)factors that charge the gearbox’s components. The mathematical model takes into account the

power losses and the inertia occurring within the entire network and their influences upon thegeneral power distribution.Using the model, we could get accurate results of the torque and power distribution. Moreover, themodel provides the map of the power distribution when the gearbox works in its second gear,emphasizing the difference between the analysis performed for the two above mentioned workingregimes (static and dynamic). The results can be further used as “entry points” for a much morecomplex mathematical model that describes the dynamic features of the vehicle.

Introduction

Planetary gearboxes are complex

mechanical structures. According to thetechnical references [4, 1], these gearboxes can be symbolically represented using typicalschematics, known as cinematic diagrams.Fig. 1 depicts the cinematic diagram of the

planetary gearbox following to be subject ofour study.

Analyzing the gearbox, one can notice itconsists of three simple planetary gear trains(K 1, K 2, K 3) and of four frictional elements:three multi-disc brakes (F1, F2, F3) and a multi-

disc clutch (Acv).The cinematic diagram provides full information about the shape and disposal of the gears but it

can’t offer information connected to the physical processes taking part when running. To emphasizethe physical processes and the power-flow modes, both for the stationary and the transient mode,

49 25

91

25

K 2K 3

9191

F2 F1F3

b

K 1

Acv

r

a

pq

Fig. 1- Cinematic diagram of the planetary

gearbox

Advanced Materials Research Vol. 837 (2014) pp 489-494Online available since 2013/Nov/08 at www.scientific.net © (2014) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.837.489

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a

p p b

a

q K 1

F2

12 0

a

K 2

12 0

Fig. 3 – Nodal graph for thesecond gear. Stationary mode.

we decided to use the nodal graphs. Technical reference [2] refers to them as a representationconcept that allows transforming the planetary gearbox into an energetic network. The nodal theoryassigns the planetary mechanisms to nodes and the shafts or other rigid connections to lines.

In the speed-up mode, unlike the stationary one, the angular velocities are varying. On the otherhand, the transient energy network reveals the need of consuming some energy amount and storing

it as inertial power-flow. This amount of energy is a result of the inertial masses of the spinning parts. As an outcome of this behavior, the general nodal graph that describes the transient mode

differs to the stationary one, as can be seen in fig .2.Starting from the general nodal graphs, issued both for the stationary and the transient modes and

keeping in mind the features of the nodes and lines within the energy network, the present paperwill deal with studying the stationary and transient modes of the TR-85M1 MBT’s gearbox . TheTR-85M1 is a main battle tank designed for the armed forces of Romania [7]. The gearbox is acomponent of the hidromecanical transmission THM-800, design by ICSITEM Bucureşti anddeveloped by Hidromecanica Braşov. Also, this paper provides a mathematical model that allowsan analytical determination of the torque and angular velocity along the elements running in the

second gear of the gearbox.Issuing the nodal graphs

The graphs of the transient and stationary modes derive from the general nodal graph with the(F2) brake applied. What we need is to emphasize the nodes and lines composing the drivelineunder these circumstances. That comes out of the cinematic analysis of the planetary gears.

a) The stationary mode. Cinematic analysis

Considering the (a, r, q) planetary gear, featured by (K 3) constant, one can notice that the sungear „a” is loaded by the motor, so it spins (it has a so-called motion law). The carrier is blocked bythe (F2) brake and it is connected to the (a, q, p) planetary gear. Hence, it has its own motion law.

Eventually, the gear’s crown „r” is free since the (F3) is not applied; thus it spins freely. Accordingto the general rules, a free element leads to the conclusion that the planetary gear III is not a part ofthe driveline (no power flows through it). The sun gear of the (a, q, p) planetary gear has its motionlaw since it is loaded by the engine. Meanwhile, the (F2) brake stops the crown and its motion lawdelivers a null angular speed. Eventually, the powerreceived by the sun gear is sent to the carrier; hence, this

planetary gear is part of the driveline. Its constant is (K 2).The crown „p” of the (a, p, b) planetary gear is connectedto the carrier of the (K 2) planetary gear and it has its ownmotion law. Moreover, (K 2) planetary gear’s sun gear has adefined motion law. That leads to the conclusion that the

(K 1) planetary gear is also a component of the drivelinesince the power-flows arriving from the sun gear and thecrown are sent to the carrier. On the other hand, this carrieris connected to the output shaft of the gearbox (fig.3).

1

2 0

F3

Pa

ω b

K 3

a

p p b

a

qq

a

r K 2 K 1

ω a

F2 F1

Acv

1

2 0

1

2 0

Pb

F3

A4

Pa

ω ωω ω a

12 0

K 3

F2 F1

Acv

A3 A2 A0

A1

Pb

ω ωω ω b12 0

K 2

12 0

K 1

n4 n0n3

n1

n2

1

2 0

4

5 3

7

8 6

Fig. 2 – General nodal graphs. Stationary and transient modes

490 Modern Technologies in Industrial Engineering



A1

A4

ω ωω ω a

1

2 0K 3

A2F2 A0

ω ωω ω b1

2 0K 2

1

2 0K 1

n4 n0n3

n1

n2 ω ωω ω 2ω ωω ω 5ω ωω ω 8

8M

7M

6M5M

4M

3M2M

1M

0M bP

aP

Fig. 4 – Nodal graph for the second gear.Transient mode

Table 1 – Encoding the exterior elementsof the simple planetary gears

Exterior element Encoding rule K 1

code

K 2

code

K 3

code

Crown 3j-1,with j=1,2,3 2 5 8Sun gear 3j-2,with j=1,2,3 1 4 7

Planet wheels carrier 3j-3,with j=1,2,3 0 3 6

F3

∠∠∠∠b

K 3 ∠∠∠∠

∠∠∠∠7

∠∠∠∠8

K 2 ∠∠∠∠ K 1 ∠∠∠∠

∠a

F2 F1

Acv

12 0

n1

n3 n245 3

78 6

8 M

7 M

6 M 5 M

2 M

1 M

b

∠∠∠∠6 ∠∠∠∠5

∠∠∠∠412 M

10 M

4 M

3 M

11 M 9 M

a M

∠∠∠∠1

∠∠∠∠2∠∠∠∠3

∠∠∠∠11 ∠∠∠∠∠∠∠∠

∠∠∠∠12

∠∠∠∠10

Fig. 5 – Nodal graph of the gearbox

b) The transient mode. Cinematic analysis

The transient mode is featured by the inertial power-flows. Due to applying the (F2) brake, on thenodal graph will occur only the accumulating inertial flows (A0, A1, A2 and A4). Since the frictionelements (Acv, F1 and F3) don’t belong to the driveline, the graph doesn’t contain them.

The general nodal graph indicates that the power flows through the sun gears of the planetarygears (using simple node (n1)). According to the features of the simple nodes, the sun gears have thesame speed but are differently loaded.

The second gear of the gearbox is engaged by applying the (F2) brake. At a closer examination ofthe (K 2) planetary gear one can notice that, blocking the crown, the power can be sent either fromthe carrier’s planet gears to the sun gear „4” or the other way around, from the sun gear to thecarrier „3”. So, this planetary gear is part of the driveline. As connected to the carrier of the (K 2)

planetary gear, the crown of the (K 1) planetary gear is also part of the driveline. Since its sun gear isloaded, one can say the power-flow is sent, via the planet gears, from the sun gear and the crown tothe carrier. As a result, (K 1) planetary gear is part of the driveline. When applying the torque

balance law to the splitting node (n4), it

comes out that the exterior element „8”is loaded by torque and it is alsospinning. The carrier is connected tothe (K 2) planetary gear’s crown, whichis blocked. Analyzing the (n1) node,we reach the conclusion that the (K 3)

planetary gear’s sun gear is loadedwith torque and also has a motion law.That also means the (K 3) planetarygear is part of the driveline (fig. 4).

Mathematical model for the gearbox’s operationModeling the gearbox’s operation assumes the analysis of the torque’s and angular speed’s ways

of transformation and the power-flows within the gearbox. The starting point is the input power,which will be conventionally considered positive. On the contrary, the output power is considerednegative. The analysis will be made using the matrix method for the stationary mode and classicmethod for the transient mode.

Stationary analysis of the gearbox

The matrix method of doesn’t need ahigh level of expertise to use it, is veryaccurate and time saving. Implementing

the algorithm requires the use of somerules to assign codes to external elementsof the planetary gearbox.

Following the implementation of these rules,the sun gear, the crown and the planet gear carrierwill be differentiated by means of the codes givenin the table 1.

The nodal graph from fig. 5 contains the newcodes of the exterior elements as well as thetorque and angular speeds featuring each element.

The angular velocities of the gearbox’s

components are achieved using the generalmotion equation of the exterior elements.

The equations generated by the law motionsare contained by a system of 13 equations with 13 unknown parameters:

Advanced Materials Research Vol. 837 491



1 1 1 2 1 1 4 2 3 5 6

2 3 4 2 5 1 10 6 12 5 11

3 6 7 3 8 2 9 11 1 7

1 0 1 0 0 01 0 0 0 01 0 0 0 0

b a( K ) K

( K ) K

( K ) K

ω ω ω ω ω ω ω ω ω ω ω

ω ω ω ω ω ω ω ω ω

ω ω ω ω ω ω ω ω

− + + + = = = − = − = − =

− + + + = − = − = − = − + + + = − = = − =

(1)

The system has been solved in MathCad and the solutions are given below:

2 3 9 8

1 4 7 10 5 6 11 120,384 0,215 0,5380

b a a a

a

ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω = = = = = −

= = = = = = = = (2)

The torque loading the gearbox’s elements can be determined by solving the general equationthat provide the torque distribution at the global nodes’ level, then the balance law of thehomogenous factors at the global and splitting nodes as well as the restrictions developed by thefriction elements. These equations are contained in a matrix system, given by C M D⋅ = , where „C ”is the coefficient matrix, “ M ” is the unknown parameters matrix and „ D” is the free terms matrix.

1

2

3

0 0 0 0 0 0 0 0 0 01 1 10 0 0 0 0 0 0 0 0 0 01

0 0 0 0 0 0 0 0 0 01 1 10 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 01 1 10 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 01 1 1 10 0 0 0 0 0 0 0 0 01 1 10 0 0 0 0 0 0 0 01 1 1 10 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 010 0 0 0 0 0 0 0 0 0 0 01

0 0 0 0 0 0 0 0 0 0 0 0 1

K

K

K

C

η

η

η

−

−

−=

1

2

3

4

5

6

7

8

9

10

11

12

00

0000100000

0

b M

M

M

M

M

M

D M M

M

M

M

M

M M

= =

(3)

The loads from the unknown parameter matrix can be computed in MathCAD with the followingfunction 1C D−

= ⋅ , where (C-1) is the inverse matrix of (C).The power distribution on the elements that compose the driveline will be provided by:

{ }0 1 12i i i

P M , where i ,ω = ⋅ ∈ … (4)

The results concerning the power expression on different elements of the driveline are:

( ) ( )( )

( )( ) ( )( )

11 1 2 2

1 1 2

1 1

3 3 4 41 2 1 2

5 6 7 8

10,562 0,426

1 1 1

0, 426 0,4381 1 1 1

0,976

cvII cvII acvII cvII a a a

cvII cvII cvII cvII

a a a a

b cvII a b a

K i P P i P P P P P P

K K K

K i K i

P P P P P P P P K K K K

P P P P P P P P P

η η η

η η

η η η η

η η η

η

⋅= → = = → =

+ + +

= − → = − = → =+ + + +

= − → = − = = = =

9 10 11 12 0 P P P = = = =

(5)

Consequently, when engaging the second gear of thegearbox, the driveline consists only of (K 1) and (K 2)

planetary gears. To reveal the power-flows, we have to putthem on the nodal graph (fig. 6).

Functionally, the gearbox works like a dual-powercircuit transmission, using a summing differentialmechanism at its output. With respect to the input power,the power loads on the planetary gears (K 1) and (K 2) are

positive (direct flow) and smaller than unit. Moreover,there is no overload of either of the two circuits.

Pt 1Pt 2

F2

1

2 0

1

2 0

Pa

w pw q w b

w a

a2 P

a1 P

b2 P

1 P

p2 P

F 2 P

Fig. 6 – Power-flows within thegearbox in its second gear




Transient (dynamic) mode analysis of the gearbox

We used now the classic analysis. It is used also by numerous references [1, 4]. The approach isdifferent since the inertial flows occur within the driveline due to the need of speeding-up thespinning parts, involving a supplement of power “losses”. The nodal graph given in fig. 4emphasizes the planetary gears composing the driveline for the second gear of the gearbox. Butthere is no information about the power-flows directions. To reveal this information we need todetermine the analytical expressions and the sign of the power loading each exterior element of thegearbox. The algorithm uses the analytical expressions of the angular velocities and total gear ratiosfor the stationary mode. The computation takes into account the power losses.

The torque is determined by using the torque balance and distribution laws on the global andsplitting nodes of the second gear driveline [1, 2], as follows:

( )

( )

( )

( )

( )

a6 F 2 5

j1 10 j0 b 3 2 4

8 j43 2 2 50 1 1 2

j 2 27 3 8

a j1 1 4 7 0 1 1 2

b 4 j0 03 j2 2 6 3 3 8 j4 4

d M M M 0 M I M M M 0 M 1 K M dt

M M 0 M K 1 K M M 1 K M d

M I M K M dt M M M M M 0 M K 1 K M

d d M I M M M 0 M K 1 K M M I

dt

ω η

η η η ω

η η η

ω ω η η

+ + = = ⋅+ + = = − +

+ == − += − +

= ⋅=

+ + + + == − +

= ⋅+ + = = − + = ⋅

dt

(6)

where ( I 0, I 1, I 2 and I 4) are theequivalent inertia moments of the

parts spinning with (ω b, ωa, ω2 and ω8) angular velocities.

The analytical expressions andthe flow-chart (fig. 7) of the

power-flows loading thegearbox’s components in its

second gear are given byreplacing the torque and angularspeed in the power definingequation [1]:

( )( )

( ) ( )

( ) ( )( )

0 4 21 5 6 22 2

3 2 2

4 20 1 2

3 2 2

4 21 1 2

1 3 2 2

12

0 0 01 1

1 1

1 1 1

b a cvII cvII a a F

cvII cvII

a cvII cvII a a

cvII cvII a a a

I I I P P I P P P

i K K K

I I P P I

K K K

i I I P P I

K K K K

K P

η η ε ω η η

η η ε ω η

η ε ω

η η

η

= − + + + = = =

+ +

= − − + + + + +

= − + +

+ + +

=

( ) ( ) ( )( )

( ) ( ) ( )

( )( ) ( )

4 21 2

2 1 3 2 2

1 4 23 1 7 42 2

2 1 3 1 2 3

1 4 24 1 2

1 2 3 1 2

1 1 1 1

1

1 1 1

1 1 1

cvII cvII a a a

cvII cvII

a a a a a

cvII cvII

a a a

i I I P I

K K K K K

K i I I P P I P I

K K K K K K

K i I I P P I

K K K K K

η ε ω

η η

η η ε ω ε ω

η η

η η ε ω

η η η

− + +

+ + + +

= − − + − = + + +

= − + −

+ + +

8 42

3

1a a P I

K ε ω

= −

(7)

The power is obtained as a rational expression that depends on (icvII ) - the total gear ratio in thesecond gear of the gearbox, (η cvII ) - the total efficiency of the gearbox in the same gear, the inertia

moment of the spinning parts, the partial efficiencies, the constants of the planetary gears and thespeed and acceleration of the gearbox’s input shaft.

Pττττ2 Pττττ1A1

A4

ω ωω ω a

A2F2 A0

ω ωω ω bn4 n0n3

n1

n2 ω ωω ω 2ω ωω ω 5ω ωω ω 8

8P

7P

6P 5P

4P

3P 2P

1P

0P bP

aP

K 1

1

2 01

2 0K 2

1

2 0K 3

Fig. 7 – Power flow-chart in the transient mode of thesecond gear of the gearbox

Advanced Materials Research Vol. 837 493



Analyzing the power flow-chart we can notice that some part of the input power is consumed bythe accumulator inertial flux (A1) and by the sun gears of the planetary gears. The inertial power-flow is due to the energy consumption to speed-up the input shaft, the sun gears and the componentsof the clutch (Acv).

The planetary gear (K 3) acts as a reducing gear. The input power-flow is completely consumed

to speed-up the crown and the (F3) brake’s discs. The power losses within the energy network arerepresented by the accumulator flux (A4). The planetary gear (K 2) acts as a reducing gear. Its crownis braked and the power runs from the sun gear “4” to the carrier, via its planet gears. Compared tothe input power, the output one is diminished due to the friction in gear meshing. The power lossesare depicted as Pτ2 and turns into heat.

The output power-flow of the (K 2) planetary gear isn’t completely used to feed the crown of the(K 1) planetary gear. Part of it is used to speed-up the crown of the (K 1) planetary gear, the discs ofthe (F1) brake as well as the carrier of the (K 2) planetary gear. The energy consumed to speed-up theabove mentioned elements is given by the inertial accumulator flux (A2).

From the nodal graph, one can notice that (K 1) planetary gear is fed with power through its sungear as well; hence, it works as a summing mechanism. Pτ1 gives its total power losses.

The power emerges from the gearbox, so it is an output power. It is directed to the planetarysumming gear sets. We can also notice that the power loading the carrier of (K 1) planetary gear isn’tcompletely sent to the summing mechanisms’ crowns. Part of it is used to speed-up differentspinning parts; hence it is accumulated by the inertial power flux (A0).

Conclusions

At a glance, we can conclude that the nodal graph of the transient mode differs from the one ofthe stationary mode. When working in the transient mode, the gearbox also involves the planetarygear (K 3) besides (K 1) and (K 2). Moreover, inertial flows (A0, A1, A2 and A4) are also involved. The

planetary gear (K 3) takes its share on the nodal graph due to the inertial power-flow that loads thecrown with the torque (M j4).

On the other hand, the torque and angular velocities distribution differs between the two modes.Moreover, for the stationary mode, we could see that the torque or the power distribution dependsonly on the planetary gear’s constants and their partial efficiencies. Meanwhile, in the transientmode, the inertia moments of the spinning parts claim their share as well, since they depend on theacceleration of the input element’s acceleration. Eventually, they lead to the inertial power-flowswithin the energy network.

The paper defines a theoretical model that presents the distribution of the power flow and itcalculated the efficiency of the gearbox in the second gear. A part of the data obtained whith thismodel is partially verified using the experimental data obtained from the experimental studies. Thetheoretical model also will be validated in a future paper using experimental data.

References[1] T. Ciobotaru, L. Grigore, V. Vînturiș and L. Loghin, Transmisii planetare pentru autovehiculemilitare, Editura Academiei Tehnice Militare, Bucharest, 2005; [2] M. Gorianu, Mecanica autovehiculelor cu roți și șenile, Editura Academiei de Înalte StudiiMilitare, Bucharest, 1992;[3] T. Ciobotaru, V. Vînturiș and M. Oană, Acceleration process of the tracked vehicle withhydrodynamic transmission / Article, CONAT20105002, 2010;[4] M. Gorianu, I. Bun, G. Arniceru and I. Filip, Mecanica autovehiculelor rapide pe șenile, EdituraAcademiei de Înalte Studii Militare, Bucharest, 1977;[5] I. Bun, Transmisia autovehiculelor pe șenile, Editura Academiei Tehnice Militare, Bucharest,

1983;[6] M. Gorianu, I. Vesa, Construcţia și calculul automobilului - Transmisia, Editura Academiei deÎnalte Studii Militare, Bucharest, 1972;[7] http://umbucuresti.ro/Produse/Tanc-TR85M1/4630.




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Static and Dynamic Analysis of a Planetary Gearbox Working Process

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static and dynamic analysis of a planetary gearbox workimg process_amr_vol 837_pag 489_anul...

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