Optimum selection of mechanical gearbox ratiosFor minimization of the unobtainable power

SAID ALI HASSANDepartment of Operations Research and Decision Support,

Faculty of Computers and InformationCairo University, Cairo,

5 Tharwat Street, Giza, CairoEGYPT

Abstract: - Inspite the fact that conventional gearboxes of automobiles have several disadvantages, they can bemade more efficient if the number of gear ratios and their selection are properly chosen. Increasing the numberof speeds improves the vehicle's dynamic performance that is expected to be optimal with stepless continuouslyvariable speed drive. However, for a fixed number of speeds, selection of individual gear ratios plays animportant role in improving the vehicle's dynamic performance.

The following mathematical progressions are commonly used for determining the gear ratios of theautomobile transmission: arithmetic, harmonic, and geometric with constant or increasing roots. But thequestion is raised about the optimum selection of gear ratios, and if it is one of the used progressions or not.

The goal of this work is to use the optimization techniques to determine the best gearbox ratios. A non-linear programming model is introduced where the objective function represents the minimization of theunobtainable or wasted power. This wasted power is represented by the difference between areas under thecurves of discrete and continuous power transmission.

As a real application of the model, it is applied to a Jeep car with 4 gear ratios to show the effect ofchoosing different mathematical progressions on the vehicle's efficiency.

Key-words: - Mechanical gearbox ratios - Mathematical progressions - Wasted power in mechanicaltransmissions - Nonlinear models - Kuhn-Tucker optimality conditions.

1 IntroductionAutomobile transmission is required to provide thevehicle with tractive effort-speed characteristicssuitable for the largely changing load conditions.Between the many types of transmissions, themechanical with stepped gear ratios is still widelyused.

Mechanical gearbox has a highest, a lowest andintermediate gear ratios. The highest is determinedfrom the condition for maximum tractive effort, i.e.maximum load and grade-ability specified orlowest speed required. On the other hand, thelowest ratio is determined knowing the maximumrequired vehicle speed. The intermediate ratios areclassically chosen according to different types ofmathematical progressions.

The following mathematical progressions arecommonly used for determining the intermediategear ratios of the automobile transmission:

1. Arithmetic, 2. Harmonic,3. Geometric, and 4. Geometric with increasingroot.

Selection can also be done using a combinationof two different progressions. Choice of thesuitable progression depends upon vehicle type,specific power and operational demands. Forexample, for passenger cars having high specificpower, higher gears (i.e. lower ratios) are used. Forheavy-duty vehicles where load conditions aremore severe, low gears are used.

In arithmetic progression, high gears are widelyspaced and low gears are more close to each other.In harmonic progression, the high gears are closerwhile the low gears are widely spaced. Thegeometric progression stands as a compromisebetween them, while the geometric progressionwith increasing root is a compromise between thegeometric and the harmonic.

The relationships for calculating the individualratios according to the mentioned progressions areas follows:Arithmetic:

i1- i

2 = i

2 - i

3=..= in-1 - in = constant

Where, i1, i

2, i

3 are the ratios of 1st, 2nd and 3rd

speeds and in, in-1 are the ratios of the top andbefore the top speeds.Harmonic:

Geometric:

Geometric with increasing root:

,,.....,, 12

12

1

21

1-

-

-- === nn

n

n

n qii

qii

qii

and: constant..... 1

2

3

1

2 ==== -n

n

q

q

q

q

q

q

Where, q1, q

2,.........., q

n are the values of theincreasing root.

In [1], a comparison of the methods of gearratios spacing has been done. As a measure ofcomparison, the unobtainable power due totraction-speed characteristics of the mechanicaltransmission relative to the ideal one has beenconsidered. The comparison has been madeconsidering the data of a Jeep car and it has beenconcluded that the geometric progression wouldgive the least wasted power in comparison withother mathematical progression methods.

In [2], a comparison has been applied to 14small and medium class passenger cars to comparedifferent methods used for selection of gearboxratios. And in [3], the same comparison is made for7 trucks having different swept volumes andvarious Gross Vehicle Weights (GVW). It was alsoconcluded that selection of gearbox ratiosaccording to the geometric progression gavesmaller unobtainable power in comparison withother methods of selection.

The previous results show that the choice ofgearbox ratios according to the geometricprogression is better than other types ofmathematical progressions. However, nothing isproved about the global optimum choice of gearratios.

In this paper, a general mathematicalprogramming model is introduced to represent the

relation between the gear box ratios and theunobtainable power. The objective function is anon-linear function representing the value of theunobtainable (wasted) power that is needed to beminimized, while the constraints represent themathematical relations between different gearratios.

The main goal of this work is to conclude thevalues of the different gear box ratios as a result ofthe optimization technique that could be applied fordifferent types of vehicles with 4-speed gearboxwithout regard to the used mathematicalprogressions.

2 Computation of the wasted powerDuring gear changing in the mechanical gearbox, apart of power is wasted due to stepped powertransmission. Increasing the number of gear ratiosminimizes this power wasted and makes thetractive effort-speed diagram more close to theideal one. This ideal diagram represents acontinuous power transmission that isschematically shown in Figure 1 for a gearbox withfour gear ratios.

The wasted power is represented by thedifference of areas under the curves of tractiveeffort with ideal and stepped power transmission.To compute these areas, the equations of tractiveeffort-speed curves should be known.

Equation of the ideal curve is written as:

Fi=2700 . VBVP te //. 1max =h .......(1)Where:

Fi = available tractive effort at wheels forthe ideal curve, N

Pe max = maximum engine power,ht = total mechanical efficiency of the

running gear,V = vehicle speed, km/h,

B1 =2700. Pemax .ht = constant.Equation of the tractive effort for stepped

power transmission is written as:Ft = 2700 . Pe . ht / V = B2 . Pe / V ......(2)Where:

Ft = available tractive effort at wheels for thestepped curve, N Pe = engine power, HP B2 = 2700 . ht = constant.

Relationship between engine power and enginerevolutions can be expressed by the followingequation :

constant1

1

11111

2312

=--

==-=-ininiiii

constant1

3

2

2

1 =-===i n

i n

i

i

i

i

+

+

+

+=

432

54321 maxNNNN n

nA

n

nA

n

nA

n

nAAPP eeeeee

(3).

Where:A1, A2, A3, A4, and A5 = constantsdetermined by fitting the actual engine powercurve, ne = engine speed, r.p.m., n

N = engine speed at maximum power, r.p.m.Vehicle speed can also be expressed as a

function of engine speed and the engaged gear boxratio as follows:

go

de

ii

rnV

.

.377.0= , g = 1, 2,....., n ...........(4)

Where:V= vehicle speed, km/hrrd = wheel's dynamic radius, mio = final drive gear ratioig = engaged gear box ratio

From (4), we have:

Where: d

o

ri

B.377.03

= = constant.

Substituting ne from equation (5) into equation(3), we have :

Substituting Pe from equation (6) into equation(2), the equation for Ft is obtained as:

+

++= V

iBA

n

iBA

VA

PBFnN

g

N

gt e .

..... 2

2233321

2 max

]...

.. 3

4

44352

3

3334 V

n

iBAV

n

iBA

N

g

N

g +

)7..(.....1 34

523

42

321 ViCViCViCiCVCF ggggt ++++=

Where:C = B2. Pc max = constantC1 = A1.C = constantC2 = A2 . B3 . C/nN = constant

constant/.. 22333 == NnCBAC

constant/.. 33344 == NnCBAC

constant/.. 44355 == NnCBACThe wasted power ED is then calculated as

follows (Figure 1):

)(4

3

3

2

2

1

4

1

dVFdVFdVFdVFEV

V

V

V

V

V

V

Vi ttt ++-=D

(8)Where: V1=vehicle speed at which the tractive forces ofthe ideal and the first speed curves are equal =maximum vehicle speed at 1st gear, km/hr, V2 =maximum vehicle speed at 2nd gear, km/hr, V3=maximum vehicle speed at 3rd gear, km/hr,

V4 =maximum vehicle speed at 4th gear, km/hr.

3 Mathematical ModelFor a vehicle with 4-speed gearbox, theoptimization problem is formulated as follows:Find i2 and i3,

The objective function is formulated tominimize the unobtainable power:Minimize DE given by equation (8)subject to:

i2 < i1, i3 < i2 ,i4 < i3, i2, i3 > 0

Where i1 , i4 = known constantsPerforming the definite integrals in equation

(8), we obtain:( )

[ ( ) ( )1222121141

loglog

loglog

VViCVVC

VVBE

-+---=D

( ) ( )31323242122223 31

21

VViCVViC -+-+

( ) ] ( )[ ( )23322314142425 log log41

VViCVVCVViC -+---+

( ) ( )22233342223233 31

21

VViCVViC -+-+

( ) ]424343541

VViC -+

( ) ( )[ 3442341 loglog VViCVVC -+--( ) ( )23243442324243 3

121

VViCVViC -+-+

( ) ]434444541

VViC -+ (9)

)5( 377.0 3

== iVBi

iV gd

ge rn

[

] )6( 4

3

5

3

3

4

2

3

3

3

21max

+

+

+

+=

niBA

niB

AniB

A

niB

AAPP

N

g

N

g

N

g

N

g

e

V

VV

Ve

The expression for the maximum vehicle speedat different gearbox ratios is obtained from therelation:

ggo

dNg i

Dii

rnV ==

377.0,

g = 1, 2, 3, 4 .(10)

Where:

o

dN

irn

D377.0

=

After arranging equation (9) and substituting

4321 and ,,, VVVV from equation (10), we obtain thefollowing non-linear programming problem:Minimize:

( ) { ( )+---=D iiii CB E 41111 log log log log 4

-

-

-+

---

2

3

4

2

2

3

2

1

223

3

4

2

3

1

22 32

1ii

3ii

ii

ii

DCii

ii

DC

+

-

-

-+

3

3

4

3

2

3

3

1

234 33

1ii

ii

ii

DC

}

-

-

-+

4

3

4

4

2

3

4

1

245 34

1ii

ii

ii

DC .(11)

subject to:i2 i1 < 0 , i3 i2 < 0,i4 i3 < 0 , i2 , i3 > 0,i1 , i4 are known constants .. (12)

4 Finding the Optimal SolutionThe Kuhn-Tucker conditions (necessary conditionsfor optimality) are [4]:

-+

-+

-

42

33

31

223

432

23

21

2232

2

3

12

1ii

ii

DCii

ii

DCii

iDC

02152

43

41

324

5 =-+

-+ mm

ii

ii

DC .(13)

-+

-+

-

43

34

32

233

433

24

22

3232

3

4

22

1ii

ii

DCii

ii

DCii

iDC

03253

44

42

334

5 =-+

-+ mm

ii

ii

DC .(14)

0

,0

,0

34

23

12

---

ii

ii

ii

.. (15)

( )( )( ) 0

0

0

343

232

121

=-=-=-

ii

ii

ii

mmm

. (16)

0,,

0,

321

12

mmmii

(17)

The unique solution which satisfies the Kuhn-Tucker conditions for all vehicle types (i.e.independent of the values of the constants C

2, C

3,

C4, C

5, and D) is that m1 = m2 = m3 = 0, and all the

values inside brackets in equations (13) and (14)=0, this will lead to:

3122 iii = , and 42

23 iii = .

This can be written as:

1)constant(4

3

3

2

2

1 >=== Ki

i

i

i

i

i

Which satisfies the geometric progression.This is the optimal solution for the non-linear

programming problem since it is a unique solutionthat satisfies the Kuhn-Tucker necessary conditionsfor optimality.

5 An Example of ApplicationA Jeep car is considered as an actual example ofapplication. The car has the following main data:

Max power = 112 HP at 3600 rpmMax torque = 281 N.m at 1600 rpmMain gear box:

1st speed ratio, i1 = 3,1

4th speed ratio, i4 = 1

Axle ratio io = 3.73

Wheel dynamic radius rd = 0.35 m

To show the effect of choosing different valuesfor the gearbox ratios, we will calculate theindividual gear ratios of a four-speed transmissionselected according to the usual mathematicalprogressions. The ratios of the top and first speedsare kept as those for the actual gear box, theywould have the values indicated in Figures 2 a, b, c,and d.

The traction-speed curves corresponding toeach of these progressions and the expected wastedpower for the Jeep car are shown also in the samefigures.

A computer FORTRAN program is written tocalculate the predicted power losses. The results areshown in table 1.

6 ConclusionA nonlinear programming model is introduced todetermine the gear ratios of a 4-speed automobiletransmission. The objective function represents theunobtainable (wasted) power due to steppedtraction-speed characteristics given by themechanical transmission relative to the ideal onewith continuous power transmission. The problemconstraints represent the mathematical relationsbetween different gearbox ratios.

The optimal solution of the problem proves thatchoosing the gear ratios according to the geometricprogression gives the global minimum wastedpower.

The presented example considering the data ofa Jeep car with 4-speed gear box showedthat the arithmetic and harmonic progressions giveapproximately 35% higher power waste withrespect to the geometric progression, while thegeometric progression with increasing root givesonly 5% higher power waste.

Table 1: Predicted power losses

GREAT AREA SMALLAREA

DIFFERENGE

2708.1749659.2979689.1299776.738

Geometric losses =

2685.5539390.8989416.5089499.230841.148

22.621268.398272.621277.508

2708.17413665.7898874.8206584.551

Harmonic losses =

2685.55312823.8168675.0826512.1721136.711

22.621841.973199.73872.379

2708.1746534.8368798.12113792.203

Arithmetic losses =

2685.5536463.2628597.07012929.7971157.652

22.62171.574201.050862.406

2708.17411064.1029629.2898431.770

Mod. Geom. losses =

2685.55310641.8919361.6608265.395878.836

22.621422.211267.629166.375

References:[1] S.Shaaban, and S.A. Hassan, " Comparison ofmethods of selecting gear ratios of automobiletransmission", 1st Conference on Applied MechanicalEngineering, Military Technical College, Cairo, Egypt,1984.[2] S.Shaaban, and S.A.Hassan, " Application ofthe methods of selection of gear box ratios topassenger car", 2nd Conference on AppliedMechanical Engineering, Military TechnicalCollege, Cairo, Egypt, 1986.[3] S.Shaaban, and S.A.Hassan, "Analysis of themethods of spacing of mechanical transmissiongear ratios applied to trucks", The 3rd Conferenceon Theoretical and Applied Mechanics, Academy ofScientific Research and Technology, Cairo, Egypt,1988.[4] H.A. Taha, Operations Research, AnIntroduction, 5th Edition, Macmillan PublishingCompany, New Yourk, 1992.[5] John David Associates, " Getting rid of the gearbox", Engineering Application, England, April1983.[6] B.S. Gottfried, Theory and problems ofprogramming with FORTRAN, Mc Graw-Hillcompany, 1992.[7] S.Lipschutz, and A. Poe, Programming withFORTRAN including structured Fortran, Shaum'soutline series in Computers, McGraw-Hill BookCompany, 1991.

Fig. 1: Ideal and stepped traction-speed curves.