8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
1/28
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.
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
2/28
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
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
3/28
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.
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
4/28
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
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
5/28
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
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
6/28
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
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
7/28
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 + =
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
8/28
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=
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
9/28
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
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
10/28
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 =
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
11/28
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
12/28
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
+ =
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
13/28
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 + + + + =
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
14/28
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
+
=+ + +
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
15/28
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
8/14/2019 Mathematical Modeling Sequential Gear Shifting Mechanism
16/28
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