Thermal and efficiency characterization of a low-backlash planetary gearbox

8/17/2019 Thermal and efficiency characterization of a low-backlash planetary gearbox

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Original Article

Thermal and efficiency characterizationof a low-backlash planetary gearbox:An integrated numerical-analyticalprediction model andits experimental validation

Franco Concli

Abstract

In the automation filed, low-backlash gearboxes are required to guarantee precise positioning. For such kind of appli-cations, planetary speed reducers represent one of the most attractive solutions. This type of gearing ensures at thesame time high power density and reduction ratios. On the other side, the compactness of the solutions leads to highoperating temperatures. For this reason, it is important to be able to quantify the power dissipation and the operatingtemperatures already in the design stage, therefore to be able to find the best compromise between the load carryingcapacity and the maximum transmittable power due to thermal limitations. For this reason, an innovative calculationmethod capable to quantify the efficiency under different operating conditions and the related operating temperatureswas developed. Experimental tests were performed under different operating conditions to validate the predictions. Thecomparison shows good agreement.

Keywords

Gears, efficiency, temperature, computational fluid dynamics, experiments

Date received: 16 June 2015; accepted: 18 November 2015

Introduction

The increasing demand of power transmission cap-

ability in more and more reduced spaces represent a

big challenge for the gearbox manufacturers. This fact

is even more severe in the field of automation where

the miniaturization of the robotic systems is unrest-

rainable. In the field of packaging, one of the most

appreciated solution for the torque and speed conver-

sion is the planetary gearing. In the most widely usedconfiguration, such kind of kinematic consists in two

gears (one external gear called sun and an internal one

called ring-gear) mounted concentric. Additional

gears called planets engage with both the sun and

the ring-gear. The planets have, unlike the sun gear

that has a pure rotation, a rototranslating motion

because the ring-gear does not rotate. The planets

are mounted on a rigid structure called planet-carrier

that is able to transform their rototranslation into a

pure rotation of the output shaft. Depending on the

number of teeth of the gear, the total reduction ratio

can vary a lot making this solution profitable for vari-

ous applications. In planetary gearboxes, unlike in

fixed axis gear systems, additional power losses are

induced by the roto-translation of the planets.

Furthermore, due to the high power density, the

heat-exchange area is reduced and high temperatures

can arise limiting the possible operating field.

Being able to quantify the power dissipation and to

predict the operating temperature allows to find out

the best design compromise between the load carrying

capacity and the maximum transmittable power due

to thermal limitations.

For this purpose, Bonfiglioli Mechatronic Research(BMR) has developed a new hybrid calculation method

based on both analytical relations and numerical

results. The model was first validated with the data

obtained with dedicated measurements on a real gear-

box and successively systematically applied for the

complete characterization of the operating field of the

gearbox.

Reseach & Development Dept, Bonfiglioli Mechatronic Research, Italy

Corresponding author:Franco Concli, Bonfiglioli Mechatronic Research, via Fortunato Zeni, 8,

36068 Rovereto (Tn), Italy.

Email: [email protected]

Proc IMechE Part J:

J Engineering Tribology

0(0) 1–10

! IMechE 2015

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Power loss sharing

The mostly wide accepted power loss classification1

subdivides the losses (PL) according to the mechanical

components such as gears, bearings, seals, and

other components like clutches or synchronizers, if

present (subscripts G, B, S, and X, respectively).

Furthermore, the losses are subdivided into load-dependent and load-independent (subscript 0).

It should be pointed out that also the so-called load-

independent losses that are basically related to the

interaction with the lubricant and the sliding of

the seals are indirectly related to the transmitted

torque that induces a change in the operating tempera-

ture that, in turn, produces a change in the lubricant

properties. Despite this little discrepancy, in the follow-

ing, this nomenclature will be accepted and used.

PL ¼ PLG þ PLG0 þ PLB þ PLB0 þ PLS 0 þ PLX ð1Þ

Load dependent power loss of gears

(gear meshing losses) P LG

The gear meshing losses are generated in the contact

between the gear flanks due to relative sliding and

rolling. Except when the contact point P coincide

with the pitch point C (Figure 1(a) and (b)), in fact,

the gear flanks velocities have different tangential

components.

This produce sliding between the teeth. In a

loaded contact and in presence of friction, this

causes power dissipation.

PLG ¼ F R wi ¼ F R g ð2Þ

The actual force F R tangent to the teeth flank can be

expressed as the product of a function f L and the forcetangential to the pitch circle F bt. f L takes into account

the force repartition due to the teeth stiffness variation

during the contact and the actual number of engaged

gears. For 14"42, f L can be approximated as

reported in Figure 1(b). Under this hypothesis, the

mean gear meshing power losses can be expressed as,

PLG ¼ F bt

pb

Z Di Ai

i f L,i g,i dx þ

Z E i 1Di 1

i 1 f L,i 1 g,i 1dx

¼ F t

pb

mt

cosðwÞZ

E i

Ai

f L,i g,i

t

dx ¼ P m H v

ð3Þ

It seems that the power loss due to the sliding between

the teeth is directly proportional to the transmitted

power. This is true as long as the frictional coefficient

is assumed as constant. If the frictional coefficient is a

function of the temperature, the relation between the

load dependent power losses of gears and the trans-

mitted power is no more linear. In the present study,

for the estimation of the frictional coefficient, the rela-

tion proposed by ISO standard2 was used (4) even if

many other formulations are available like those

Figure 1. (a) Speed in different positions along the line of action; (b) velocities, friction coefficient, and force repartition trends along

the line of contacts; and (c) instantaneous gear mesh power loss.

2 Proc IMechE Part J: J Engineering Tribology 0(0)




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proposed by ISO6336,3 Michaelis et al.,4 and Ho ¨ hn

and Michaelis.5

¼ 0:048 F t= cosðtÞð Þ=b

2 t sinðwtÞ c

0:20:05oil Ra

0:25 ð4Þ

in which F t is the force at pitch circle, t the transversepressure angle, wt the working pressure angle, oil the

dynamic viscosity of the oil at operating temperature,

the equivalent radius of curvature at the pitch point

of contact, and Ra the arithmetic averaged roughness.

The above-described model can be applied also to

planetary gears providing the relative velocities. The

mechanical power losses due to rolling are related to

the shear stress in the EHL (elastohydrodynamic lubri-

cation) film. Even if literature provides detailed models

for the prediction of such losses like those proposed

by Talbot and Kahraman6 and Li and Kahraman,7

according to the findings of Thirumurugan andMuthuveerappan8 and Andersson9 the rolling power

losses are of an order of magnitude lower than the

sliding power losses. For this reason, this contribution

was neglected in this analysis.

Load independent power losses

of gears P LG0

Small planetary gearboxes used in the automatic

industry are almost always dip lubricated. The inter-

action of the gears with the oil involves in any case

losses, but in planetary reducers, the phenomenon is

more severe due the presence of the planet-carrier andthe rototranslation of the planets. Literature provides

many different empirical models for the prediction of

this kind of losses for simple rotating gears such as

those proposed by Ohlendorf,10 Dawson,11 and

Mauz.12 Different authors studied also the lubrication

of a gear pair13 observing that the trend of the losses is

significantly affected by many different operating par-

ameters. A huge amount of different equations were

provided also by other authors,14–17 but since these

models are derived from experiments, they are applic-

able or give reliable results only as far as the actual

geometry and operating conditions are similar to thatused in the experiments. To overcome this problem,

Seetharaman and Kahraman18 proposed a physics-

based fluid mechanics model able to predict spin

power losses of gear pairs due to oil churning and

windage.

More recently, different authors have applied single

phase computational fluid dynamics (CFD) simula-

tions to overcome this limitation.19–28 The author

has already experience with both multiphase gear

simulations29 and with the modeling of planetary sys-

tems.30–32 CFD simulations seems to provide extre-

mely accurate results not only for ordinary gears but

also for the complex motion of planetary gears.On this basis, the author has adapted an open

source CFD code released under the GNU license

for the simulation of planetary gears.33 The code

relies on the numerical solution of some equilibrium

equations representing conservation laws of physics

such as mass (5) and momentum (6) conservation.

In addition, the presence of two phases implies to

include an additional transport equation of a scalar

quantity that represents the volume fraction of oneof the two phases.

@hui i

@hxi i ¼ 0 ð5Þ

@ hui ið Þ

@t þ

@ hui ihui ið Þ

@x j ¼

@ h pið Þ

@t

þ @

@x j

@hui i

@x j

@hu j i

@xi

@ ij

@x j

ð6Þ

The above equations have been solved numerically

adopting Gauss scheme. The domain for the simula-tion is represented by the internal volume of the planet-

ary gearbox. From the author experience, the main

contribution to the PLG0 losses is given by the planet-

carrier rotation and the related motion of the planets.

For this reason, to speed up the calculation and to

reduce the computational effort, the sun-gear and the

ring-gear have been modeled without teeth. The effect-

iveness of such assumption was already validated in

previous works.25–27 Even if in such assumption

implies that the oil squeezing losses (oil pocketing)

are neglected, it is known that this phenomena assumes

importance only in case of injection lubrication while

becomes negligible on case of splash lubrication.With this simplification, it is possible to reproduce

the motion of the planet-carrier without topological

changes in the mesh and, therefore, run transient

simulations without the need of remeshing. All the

boundaries have been modeled as no-slip boundaries

(r p ¼ 0, v ¼ vwall ). Figure 2 shows the analyzed

planetary speed reducer and example of the mesh

adopted for the calculations.

Parametric analysis have been performed to find

out the best compromise between the results accuracy

and the computational effort. The final meshes

adopted have approximately 1 M cells for the smallestgearbox size and 3 M cells for the biggest size. In both

cases, the domain was meshed using the blockMesh

and the snappyHexMesh utilities of OpenFOAM.33

To achieve temporal accuracy and numerical stability

when running the simulations, a Courant number of 1

was adopted. The Courant number is defined as,

Co ¼ t U j j

x ð7Þ

where t is the time step, jU j is the magnitude of the

velocity through the cell, and x is the cell size in

the direction of the velocity. The time step is thereforedetermined considering the smaller cell size in the

mesh and the maximum value in the velocity field.

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The Reynolds unresolved terms have been neglected

and the model considered laminar.

To post-process the results, a dedicated utility

(written in Cþþ) was developed. This utility, starting

from the velocity and pressure fields and from the

mixture properties (calculated for each cell i with the

CFD code), calculates the resistant torque (viscous ð Þ

and inertial ( p) contribution) on the gear/planet car-

rier due to the presence of the lubricant mixture. The

basic relation on which the utility is based are

T LG0 ¼X

i

mixi mixi U i xi

Ai ri ð8Þ

T LG0 p ¼X

i

pi Ai ri ð9Þ

where the subscript i refers to the i th cell, mixi and

mixi are the viscosity and the density of the mixture in

the cell calculated as an averaged mean value of the

properties of the different phases, Ai is the area of the

cell corresponding to the boundary, and ri is the radial

distance of the cell from the axis.

Simulations were performed for all the sizes of gearbox and for different rotational speeds, static

lubricant levels and temperatures so to have a com-

plete characterization of the gearbox behavior in any

operating conditions. Figure 3(a) represents an exam-

ple of load independent power loss map calculated

with the CFD approach. Figure 3(b) represents an

example of lubricant distribution and velocities

inside the gearbox. This capability of the code ensures

the possibility not only to calculate the power dissi-

pation but also to get additional information about

the lubrication inside the gearbox.

The simulations were performed on a 108GFLOP

workstation. The characterization of the wholedomain has taken approximately 6 weeks that is a

significant amount of time but less than the standard

time required to manufacture and test physical proto-

types. As described in ‘‘General efficiency model’’ sec-

tion, once the operating field was characterized, the

results have been interpolated to have analytical equa-

tions directly applicable in the design practice to simu-

late the behavior of the gearbox under specific

working conditions.

Load power loss of bearings P LB and P LB0

As for the gears, also the bearing power losses can

be subdivided into load-dependent and load-

independent. The load dependent power loss dependson the sliding between the bearing elements such as

rings and rolling elements or between the journal and

the bearing in journal bearings. The load independent

power losses, in turn, depends from the interaction

with the lubricant (the oil that lubricates the gears

or grease applied directly on the bearing) and from

the sliding between the rings and the bearing seals if

present. Bearing manufacturers have many decades of

experience in this filed and have already presented

reliable prediction models.34,35 For this reason, no

further research and validation test were made on

bearings and the SKF model was applied for the cal-culation of the bearing power losses.

Load independent power loss of seals P LS0

The seals losses arise due to the relative sliding

between the shafts and the seals themselves. The mag-

nitude is dependent from the rotational speed, the

friction coefficient, and the contact pressures. One of

the most accepted model to predict such contribution

is that proposed by Niemann and Winter.1 The power

dissipation can be calculated according to (10).

PLS 0 ¼ lL d 2

n

1000 ð10Þ

Figure 2. (a) Section view of the analyzed planetary gearbox; (b) boundary corresponding to the housing, the bearing, and the sun

gear; and (c) boundaries corresponding to the planet carrier and to the planets.





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where lL is a coefficient that describes the operating con-ditions and varies with temperature and wear of the seal,

d is the seal diameter, and n the rotational speed. This

equation gives the results shown in Figure 3(c).

General efficiency model

The above-described models allow the calculation of

the power losses only if the lubricant temperature is a

priori known. It is not possible to calculate this tem-

perature directly, but it is possible to rate its iteration.

The total power loss PL is in fact dissipated in the

surrounding environment by heat exchange. This phe-nomenon is governed by the general heat transfer

equation (14).

Q ¼ A T ¼ PL ð11Þ

in which Q is the total transferred heat, A the exchange

surface, the heat transfer coefficient ½ KgKs3

, andT the

temperature difference between the surface and the sur-

rounding environment. The heat transfer coefficient

was established experimentally with different tests in

which both the power losses and the operating tem-

peratures were measured and results in 24.7 KgKs3. This

value is consistent with the suggested values provided

by literature. It appears that by increasing the operat-ing temperature, the heat flux increases linearly while

the power losses decrease monotonically. This con-

sideration ensures the possibility to find the equilib-

rium between the dissipated power PL and the

removed heat Q with an iterative procedure.

By integrating the prediction models together and

by implementing an iterative algorithm for the calcula-

tion of the operating temperature (that implies a new

calculation of the power loss, etc.), it is possible to com-

pletely characterize the gearbox in terms of mean oper-

ating temperature and efficiency starting only from

the geometrical data and the operating conditions.For this purpose Scilab,36 an open source software

for numerical computation, was used. All the above

presented equations were implemented in the code.

Regarding the CFD results, all the simulated condi-

tions were collected in a database and an interpolating

equations for each gearbox geometry was derived and

included in the code. In this way, simulating the oper-

ating temperature and efficiency takes less than 20 s.

For new geometries for which CFD simulations

should be performed, the computational time required

increases but remains in any case less than 1 week for

the characterization of the whole operating field (1 h

for each simulation), ensuring results in a shorter waywith respect to experimental tests in which the test rig

Figure 3. (a) Example of load independent gear power loss map obtained with 12 CFD simulations; (b) lubricant-air interface inside

the gearbox and velocity contours on different planes; and (c) seal power losses.

Concli 5




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setup and the tests take approximately 1 month (1 day

for each measurement).

Model validation

Once the model was implemented, dedicated experimen-

tal tests were performed to validate the method. Themeasurements were performed on an energy-closed-loop

test rig, and its configuration is schematically described

in Figure 4. A permanent magnet servomotor (9.) con-

trolled by a servoinverter is connected to the tested

gearbox (6.). The housing of the gearbox is mounted

on a flange. The output shaft of the gearbox is con-

nected to a second gearbox used as multiplier (5.).

Another permanent magnet servo motor (10.) con-trolled by a second servoinverter is connected to the

Figure 4. (a) Schematic layout of the test rig and (b) test rig.

Table 1. Test summary.

Test # !in (rpm) T 1 (Nm) L (ml) Test # !in (rpm) T 1 (Nm) L (ml)

1 3000 16 130 7 1000 16 130

2 3000 0 130 8 1000 0 130

3 3000 16 93 9 1000 16 93

4 3000 0 93 10 1000 0 935 3000 16 60 11 1000 16 60

6 3000 0 60 12 1000 0 60





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slave gearbox and operate as generator. Torque meas-

urement shafts (4. and 7.) at the input and output of

the tested gearbox measure the transmitted torque

(0.05% F.S.) and rotational speed to determine the

transmitted power. The output torque is feedback con-

trolled with the measurement of the torque meter (7.),

the input speed is feedback controlled with the motor

encoder. The shafts are connected with metal bellows

couplings.

The gearbox surface and ambient temperatures are

measured by five sensors. To validate the calculations

for the whole operating field, tests were performed at

different speeds, with different static oil levels, with

and without load (Table 1). For the tests without

Figure 6. (a) Measured and (b) calculated power loss maps (T1 ¼ 16 Nm); (c) measured and (d) calculated temperatures maps

(T1 ¼ 16 Nm).

Figure 5. Comparison between the measured (Exp) and the calculated (Calc) power losses; the values above the bars represent the

temperatures.

Concli 7




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load, the coupling between the gearbox (9.) and the

torque-meter (7.) was disengaged, so that the powerthat enters the gearbox is totally dissipated: even if the

output shaft rotates, it does not transmit torque and,

therefore, power.

Results

Figure 5 shows a comparison between the calculated

values and the experimental results. The stripes repre-

sent the no-load working condition, whereas the solid

color represents the loaded conditions. The columns

with the error bars represent the experimental mea-

sured values. The bars corresponding to the calculatedvalues are subdivided to show the shear between the

losses. Concerning the calculated values, the colors

represent the losses generated by the bearings, the

seals, the meshing losses, and the load independent

losses of the gears starting from the bottom of the

column. Above the bars, the measured and predicted

temperatures are reported. It appears that the pre-

dicted values are compatible with the measurements.

The difference between the experimental and the com-

putational data is averaged 1% (max 17%) both in

terms of power losses and operating temperature

(max 10%).

Figures 6 and 7 show a comparison in terms of power loss (a and b) and temperature maps (c and

d), obtained by interpolating the measured (a and c)

and the calculated values (b and d) for the loaded

(Figure 6) and the unloaded (Figure 7) condition.The maps show that a reduction of the amount of

lubricant has a positive impact on the power loss espe-

cially for high rotational speeds, where the PLG0 con-

tribution is more significant.

While there is a direct relation between the calcu-

lated temperature and power loss maps, the measure-

ments show some fluctuation of the results that

should be attributed to the measurement uncertainty

and to the fact that even if the testing room was air-

conditioned, some minor variation in the environment

temperature was present and this has affected the

operating temperatures of the gearbox.Despite that, the maps (measured and calculated)

are very similar both for the loaded and unloaded

case, confirming the accuracy of the prediction

model that will be included in the company standard

design practice allowing to characterize the so-called

thermal limit already in the design stage (Figure 8).

Conclusions

The goal of this research was to find a model capable

to predict the efficiency and the operating temperature

of small planetary gearboxes. Some models available

in literature for the gear meshing, the bearing, andthe seal losses and some additional equations for the

load independent power losses of gears were used and

Figure 7. (a) Measured and (b) calculated load independent power loss maps (T2 ¼ 0 Nm); (c) measured and (d) calculated tem-

peratures maps (T2 ¼ 0 Nm).





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iteratively solved to estimate the operating tempera-

ture. The additional equations were obtained by inter-

polating the results of CFD numerical simulations

generated with a specifically developed tool. The

results of the global efficiency model were validated

with experimental measurements showing good

agreement.

This approach was successively used to character-

ize the complete operating field of the gearbox. It hasallowed to reduce significantly the effort needed to

generate the efficiency maps and the thermal charac-

terization of the whole gearbox gamma avoiding the

need of testing each size and each gear ratio in several

operating conditions. Because of the GNU license, in

fact, the calculation was significantly parallelized on

many CPUs allowing to perform the whole analysis

on a standard 108GFLOPS workstation in less than 1

week for each size. Moreover, the adoption of an

open-source tool gives the possibility to customize

the code for the specific industrial needs: a further

step will be the overcoming of the geometrical simpli-

fications introduced in the CFD model and the adop-tion of such tool to optimize the internal shape and

the lubricant circulation in the gearbox.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with

respect to the research, authorship, and/or publication of

this article.

Funding

The author(s) received no financial support for the research,

authorship, and/or publication of this article.

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