mathematical modeling sequential gear shifting mechanism

8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism

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MATHEMATICAL MODELING OF A SEQUENTIAL TYPE

GEAR SHIFTING MECHANISM OF AN AUTOMOBILE

TRANSMISSIONBy Priyanshu AgarwalDesign Engineer, R&D, Transmission Division,

Bajaj Auto Ltd.

ABSTRACT: There are numerous variables involved in the design of a sequential type gearshifting mechanism. In order to judge the effectiveness of such a mechanism it is essential todevelop a mathematical model that can quantify it. In this paper a mathematical model of a

sequential type gear shifting mechanism is developed. The mathematical model deals with theforces acting within the system while it is under a static equilibrium. Further, the efficacies of two

such mechanisms are then evaluated for comparison and the superiority of one over the other is

proved. In addition to this, a parameter study is then carried out using MATLAB to individually

assess the effect of each parameter on the output so as to provide a mathematical rationale at the

design stage itself.

KEYWORDS: mathematical model, design variables, drum, fork, fork roller, fork rail, drumtorque, axial force, efficacy of mechanism.

1. INTRODUCTION

Gear Shifting is a sophisticated phenomenon present in all automobiles equipped with manual

transmissions. A mechanism that can effectively handle the issue of gear shifting is always in

demand. Apart from shifting the gear, the effort it brings for the driver on the gear shifting lever isvital for a gear shifting mechanism. In order to reduce the effort needed to effect a gear change it is

imperative to mathematically evaluate all the design variables linked with the shifting elements.

There are two types of shifting elements involved in gear shifting: Internal shifting elements and

external shifting elements [1]. Internal shifting elements are the elements that are inside the transmission, such as selector

forks, fork rails, drum etc.

External shifting elements are the elements that are outside the transmission such as gearshifting lever, cable controls etc.

The mechanism described here consists of internal shifting elements and the mathematical model

presented deals with the design variables related to these elements.


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2. MECHANISM DESCRIPTION

Figure 2.1 An illustration of the Sequential Gear Shifting Mechanism

2.1 Construction

The mechanism shown above consists of the following major components:

Drum (Surface Cam)

Forks

Fork Rail

Fork Roller (sits on fork lug)

Star Wheel

Shift Lever-1 (Guitar shaped with a welded splined shaft)

Shifter Lever-2 (Cross Lever)

Shift Lever-1 Torsion Spring

Tension Spring

Fork RailFork

Fork Roller

Drum

Star Wheel

Shift Lever-2

Shift Lever-1Tension Spring

Torsion Spring


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The Fork legs rests on the gear collar and the fork is constrained to move on the fork rail. The

interface between the fork and the rail can be provided with plain bearings which can significantly

reduce the coefficient of friction between the rail and the fork. The material used for such plainbearings is Daidyne for which the coefficient of friction lies in the range of 0.01 to 0.1 [2] as

compared to 0.16 for a lubricated steel to steel contact. The increase in friction increases the drum

torque significantly which can reduce the mechanism efficiency of torque to force conversion.

Further the fork is provided with a lug that protrudes from its head. The head is also provided witha roller that can roll on the profile carved on the drum, which is a surface cam with a groove. The

roller is provided for converting sliding friction to rolling friction between the fork lug and the

drum groove. In addition to this, the drum is provided with a star wheel having projected lugs on its

front face on which the shift lever-2 rests. Shift lever-2 is pivoted on shift lever-1 and is connectedto it with a tension spring. Shift lever-1 consists of a guitar shaped sheet body with a splined shaft

welded to it. A torsion spring is provided on shift lever-1 so as to reposition the lever after shifting

the gear.

2.2 Working

The shift lever-1 is rotated with the help of a push-pull type wire. As shift lever-1 rotates, it rotates

the shift lever-2 which in turn rotates the star wheel with a fixed angle (60 degrees). Since, the star

wheel is bolted with the drum, it is also indexed with 60 degrees. Due to this fixed rotation of the

drum the forks engaged in the grooves on the drum surface, with the help of a roller, are moved

axially on the shifter rail. This axial movement of the fork leads to the axial movement of theshifting sleeve. The desired direction of the fork movement can be governed by engraving a

corresponding groove on the drum surface. Now the magnitude of drum toque required depends on

the various parameters involved e.g. groove ramp angle, drum radius, shifter rail diameter, forkdimensions etc. The shifting feel is hampered in case the required drum torque exceeds a certain

value. In addition to this, in order to convert this system into an automated shifting mechanism the

drum torque required should be minimized so as to actuate the mechanism with the smallest

possible actuator.

Considering the importance of drum torque required, it is imperative to develop a mathematical

model so as to study the influence of the various design variables involved and thus, minimize the

drum toque.

3. THE DEVELOPED MATHEMATICAL MODEL

The developed mathematical model is a simplified form of the actual mechanism as described in

section 2. Only the following components are selected for developing the model:

1. Drum2. Fork Roller3. Fork4. Fork RailFurther since at a time only one fork will come into action, hence only one fork is considered in

the mathematical model.


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Figure 3.1 Simplified representation of the mechanism with external forces & moments acting on

it.

3.1 Assumptions

1. The entire system is considered to be in a static equilibrium.

2. The distribution of axial force on fork legs is considered to be in the ratio of their arm lengths

from the rail center line.

3.2 Objective

To establish a relationship between the axial force to be developed on shifting sleeve and therequired drum torque.

TD

F a = Fa1 + F a2

F a1

F a2


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3.3 Inputs

The following inputs are considered for the model:

Table 3.3.1 Inputs for the mathematical model

The value offr for lubricated steel to steel contact is considered as 0.16 and for steel to aluminiumcontact is considered as 0.12.

The value ofrlfor lubricated steel to steel contact is considered as 0.16.

3.4 Design Variables

The developed mathematical model caters the following design variables:

Table 3.4.1 Design variables for the Mathematical Model

For clarification of the above design variables please refer to Fig.3.6.2, Fig.3.6.3, Fig. 3.7.3, Fig.

3.7.4 and Fig.3.7.5.

S NO. INPUTS DESIGNATION

1 Total Axial Force to be developed on Shifting Sleeve Collar Ffa

2 Coefficient of Friction between Fork & Rail fr

3 Coefficient of friction between Roller and Lug rl

S NO. DESIGN VARIABLES DESIGNATION

1 Length of Fork Leg Opposite to Lug Side A1

2 Length of Fork Leg on Lug Side A2

3 Fork Support Base Length from Fork Leg to opposite to Lug Side end B1

4 Fork Support Base Length from Fork Leg to Lug Side end B2

5 Distance of Lug from Rail Center Line C

6 Fork Lug Diameter dl

7 Fork Lug Roller Diameter dr

8 Fork Leg offset from Lug Center O3

9 Offset Distance of Fork Leg opposite to Lug Side from Shifting Sleeve Center E1

10Offset Distance of Fork Leg on Lug Side from Shifting Sleeve Center E

2

11 Fork Lug Angle from the ZF-axis as viewed in XY Plane.

12

Angle between 'Perpendicular to Radial Reaction from Shifting Sleeve' &

'Line joining Fork Leg Center to Rail Center'

13 Rail Diameter d

14 Groove Ramp

15 Drum Radius R


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3.5 Outputs

The various reactions, moments and friction forces acting within the system are considered to be

the output of the model with the Drum Torque, TD being the most important output.

The following are the outputs of the developed mathematical model:

Table 3.5.1 Outputs of the Mathematical Model

3.6 Drum Profile and Fork Roller Interaction

In order to study the interaction of the fork roller and the drum a developed view of the drumprofile is obtained by unwrapping the drum profile.

The forces acting on the drum are as shown in view V. These forces are responsible for developinga tangential force and an axial force on the drum. The tangential force is responsible for canceling

the moment, TD applied on the drum and hence,

S No. OUTPUTS Designation

1 Normal Reaction between Fork and Rail in Y direction at Fork Leg End Ny1

2 Normal Reaction between Fork and Rail in Y direction at opposite to Fork Leg End Ny2

3 Tangential Force on Fork Lug FT

4 Axial Force Fork on Fork Lug FA

5 Normal Reaction between Fork and Rail in X direction at Fork Leg End Nx1

6 Normal Reaction between Fork and Rail in X direction at opposite to Fork Leg End Nx2

7 Axial Force on Fork Leg at opposite to Lug Side End Ffa1

8 Axial Force on Fork Leg at Lug Side End Ffa2

9 Radial Reaction from Shifting Sleeve Collar at Fork Leg on Lug Side Fft

10 Normal Reaction between Roller & Lug R2

11 Normal Reaction between Drum & Roller R1

12 Friction Force at Drum & Roller interface fr

13 Friction Force at Roller & Lug interface fl

14 Friction Force between Fork & Rail X direction (away from leg) fx1

15 Friction Force between Fork & Rail X direction (towards leg) fx2

16 Friction Force between Fork & Rail Y direction at Fork Leg End fy1

17 Friction Force between Fork & Rail Y direction opposite to Fork Leg End fy2

18 Drum Torque Required TD


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Figure 3.6.1 Developed View of Drum Profile & Force Components acting on it

( )1 sin cos (3.6.1)D rT R f R = +

Further, the axial force, FD developed is to beresisted by the bearings holding the drum. Thus,

1 cos sin (3.6.2)D rF R f =

XRYRZR represents the roller coordinate system.

The sliding friction acting between the fork lugand roller under static equilibrium is related to

the normal reaction between the lug and the

roller as follows:

Figure 3.6.2 Free-body Diagram of Fork Roller

Considering static equilibrium along XR-axis

Considering static equilibrium along YR-axis

0xF =

0yF =

2 (3.6.3)l rl f u R=

1 2cos cos sin sin 0 (3.6.4)

l r R R f f + =


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1 2sin cos sin cos 0 (3.6.5)r l R f R f + + =

Considering static equilibrium about ZF-axis

0zM = (3.6.6)

2 2

lrr l

ddf f =

From equations (1), (2) & (4)

Figure 3.6.3 Forces & moments acting on Fork Lug

The forces acting on fork lug can finally be expressed as the tangential force (F T), axial force (FA)

and moment (Mfl) as mentioned in the following equations.

1 2 1 tan 1 (3.6.7)l

rl

r

d R R u

d

= +

2sin cos (3.6.8)

T lF R f = +

2cos sin (3.6.9)

A lF R f =

(3.6.10)2

l

f l

d

M f=


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3.7 Fork Free-Body Diagram

3.7.1 Coordinate System

The coordinate system XFYFZF defined for the fork is represented in figure 3.7.1. Axis ZF is alignedwith the fork rail axis and the axis YF is aligned with the line joining the shifting sleeve center to

the fork rail center. The remaining axis XF is perpendicular to the plane YFZF, thus formed.

Figure 3.7.1 Free-Body Diagram of Fork


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Figure 3.7.2 Cut Section View of Fork representing the reaction and friction forces acting between

the Fork and the Rail.

Since the fork will be sliding on the rail the friction forces acting on the fork can be established

using the following relationships:

1 1 (3.7.1) y fr y f u N =

2 2 (3.7.2) y fr y f u N =

1 1 (3.7.3) x fr x f u N =

2 2 (3.7.4) x fr x f u N =


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0yF = 1 2 sin sin 0 (3.7.6) Ny Ny FT Fft + + =

Considering static equilibrium about ZF-axis0

zM =

Figure 3.7.4 Free-Body Diagram of Fork representing forces in XFZF plane.

Considering static equilibrium along ZF-axis

0zF = 1 2 1 2 0 (3.7.8) A fa x x y yF F f f f f + + + + + =

Please note that the sign offx1, fx2, fy1, fy2 should be negative respectively in case the corresponding

reactionsNx1, Nx2, Ny1, Ny2 comes out to be negative.

Considering static equilibrium about YF-axis

0y

M =

( )1 1 2 2 3 2 2 1 1 1 2sin cos 0 (3.7.9)

2 A fa fa ft x x x x f

dF C F E F E F O N O N O f f M + + + + + + =

20 (3.7.7)

cos

ft

T

F AF C

+ =


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Figure 3.7.5 Free-Body Diagram of Fork representing forces in YFZF plane.

Considering static equilibrium about XF-axis

0xM =

The axial force acting on the fork legs can be summed up to give rise to the total axial force which

is to be developed.

1 2 (3.7.11) fa fa faF F F+ =

3 1 1 2 2 1 1 2 2 1 2cos sin sin 0 (3.7.10)2 2

A T fa fa y y y y f

d dF C F O F A F A N B N B f f M + + + + =


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Finally, as per assumption (2) i.e. the distribution of the axial force on fork legs will be in

proportion to their arm lengths from the rail center line.

3.8 Design Variable Optimization

Considering the number of design variables involved it is advisable to eliminate the redundantvariables so as to effectively deal with the problem. However, the above mathematical model was

developed without any such optimization so as to establish a more generic approach for solving the

problem in case any of the following mentioned relationships ceases to exist.

Figure 3.8.1 Design variables defining the fork geometry

A2

A1

XF

YF

ZF

C

2E1

DS

AC

2 2

1 1

12 2 2 2

1 1 2 2

(3.7.12)fa

fa

F A E

F A E A E

+

=+ + +


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It was observed that the design variables A1, A2, E1 & E2 are related and that by introducing two

more variables the total number of design variables can be optimized.

Table 3.8.1 Design variables added in the mathematical model

A1, A2, E1 & E2 can then be related using the following equations:

2 1sin (3.8.1)

2 2

C C

s s

A A A AD D

= =

The value ofevaluated from the above equation and can then be substituted in the followingequations to calculate A1, E1 & E2.

1 sin (3.8.2)

2

sC

DA A

= +

1 2 cos (3.8.3)2

sD

E E

= =

The following design variables are then eliminated:

SNO.

DESIGN VARIABLES ELIMINATED DESIGNATION

1 Arm Length of Fork Leg Force Opposite to Lug Side A1

2 Offset Distance of Fork Leg opposite to Lug Side from Shifting Sleeve Center E1

3 Offset Distance of Fork Leg on Lug Side from Shifting Sleeve Center E2

4

Angle between 'Perpendicular to Radial Reaction from Shifting Sleeve' & 'Line joining

Fork Leg Center to Rail Center'

Table 3.8.2 Design variables eliminated from the mathematical model

S NO. DESIGN VARIABLES ADDED DESIGNATION

1 Sleeve Mean Diameter (excluding chamfer & fillet) Ds

2 Distance of Fork Leg Contact Point Circle Center from Rail Center Ac


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The following are the constraints for the design variables:

The fork leg offset from fork lug center should always lie betweenB1 andB2.

1 3 2- (3.8.4) B O B

The lug diameter should always be less than the roller diameter. In case the roller diameter is equal

to the lug diameter that means no roller is used and the equations can simply be modified by

equating dl equal to dr.

(3.8.5)l rd d

The fork rail diameter should always be less than the dimension C as fork rail is to be

accommodated within this dimension.

(3.8.6)d C

mathematical modeling sequential gear shifting mechanism

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