Proble m 4.1
Explain why the term Nf+Nr of Equations 4.19 and 4.20 is not necessarily equal to Wcos  and
discuss the conditions of equality.
1.4
a/l = 0.40
1.2
1
kF+kR
0.8
0.5
1.4
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.7
0.75
0.8
0.85
0.9
0.65
0.7
0.75
Coefficient of adhesion
0.8
0.85
0.9
a/l = 0.45
1.2
1
0.5
2
0.55
0.6
0.65
a/l = 0.50
1.5
1
0.5
0.55
0.6
Figure S4.1a The variation of k F  kR with p at h/l =0.3
1.03
1.02
h/l = 0.30
1.01
1
kF+kR
0.99
0.5
1.1
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.7
0.75
0.8
0.85
0.9
0.65
0.7
0.75
Coefficient of adhesion
0.8
0.85
0.9
h/l = 0.40
1
0.9
0.5
1.15
0.55
0.6
0.65
1.1
h/l = 0.50
1.05
1
0.5
0.55
0.6
Figure S4.1b The variation of k F  kR with p at a/l =0.4
a/l = 0.40
a/l = 0.45
a/l = 0.50
Coefficient of adhesion
1
0.8
0.6
0.4
0.2
0
0.2
0.25
0.3
h/l
0.35
Figure S4.1c The variation of p of Equation (** )
Proble m 4.2
0.4
Repeat Example 4.3.2 and show that for lower adhesion coefficients of below 0.7, FWD vehicle
can have better gradeability results than RWD vehicle.
25
20
a/l = 0.40
FWD
RWD
Gradable slope (deg)
15
10
0.5
30
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.7
0.75
0.8
0.85
0.9
0.65
0.7
0.75
Coefficient of adhesion
0.8
0.85
0.9
a/l = 0.45
20
10
0.5
40
0.55
0.6
0.65
30
a/l = 0.50
20
10
0.5
0.55
0.6
Figure S4.2
Proble m 4.3
For a vehicle with information given in table below,
a) Find an expression for the overall high gear ratio for the case in which the maximum
vehicle speed occurs at the engine speed corresponding to the maximum engine torque.
b) Repeat (a) for the case the maximum vehicle speed occurs at the engine speed
corresponding to the maximum engine power.
Table P4.3 Veh icle info rmation of Problem 4.3
1
2
3
4
5
Aero Drag Coefficient
Rolling Resistance Coefficient
Tire Ro lling Radius
High Gear Ratio
Maximu m engine torque
c
f
R
n
6
Engine speed at max torque
T

7
Torque at max engine power
TP
8
Engine speed at max power
P
9
Vehicle mass
m
Result: (a) T n 3  fmgRn 2  cR ( R* ) 2  0
Proble m 4.4
For the vehicle of Example 4.3.3 design the highest gear ratio in the manner described below and
compare the result with those of Examples 4.3.3-4.3.4.
First design gear 4 at the engine speed 10% above the speed at the maximum power and then
design gear 5 with a 25% overdrive.
Proble m 4.5
Prove Equation 4.37 for the overdrive gear ratio by writing the kinematic equation of vehicle
motion. State the assumptions involved in this process.
Proble m 4.6
In a 4WD vehicle the wheel torques at front and rear axles are distributed such that the ratio of
front to rear axle torques is given by r.
a) Assume the rear wheels are at the point of slip and derive an expression for the maximum
negotiable slope of the vehicle (assume equal gear ratios for front and rear and ignore the
rolling resistance).
b) Repeat (a) assuming the front wheels are at the point of slip.
c) For both cases of (a) and (b) derive expressions for the limits of torque ratio r.
d) Use numerical values 2.5, 0.5, 1.2, 0.8 and 0.02 for l, h, a,  and f R to evaluate the limits
of r.
Result: (a) tan    (1  r )
a  hf R
(d) r ≤ 0.57 and r ≥ 0.57
l   (1  r )h
70
65
55
Not acceptable
50
45
40
Max possible grade
(deg)
Slope angle (deg)
60
( Atan P)
35
Front w heel skid
30
Rear w heel skid
25
20
0
Limit for 'r'
0.2
0.4
0.6
Torque ratio (r)
0.8
1
Figure S4.6 The variation of slope with the torque ratio of Problem 4.6
Proble m 4.7
A method shown in Figure P4.7 is proposed for the evaluation of intermediate gearbox ratios. It
includes two low speed levels for the engine with the definition of  L    L (  1 ),
2
1
a) Find expressions for C g1 and C g2
b) Find expressions for n2 , n3 and n4 in terms of C gi .
c) Examine the difference between the average of
C g1 and
C g2 with C gp of geometric
progression.
d) Show that for   1 this method is identical to conventional geometric progression.
H
 L2
 L1
V1
V2
V3
V4
V5
Figure P4.7 Engine- vehicle speed diagram of Problem 4.7
Results: (a) Cg1   Cgp and C g 2 
1

C gp
Problem 4.8
A method is proposed for the evaluation of intermediate transmissions ratios presented in Figure
P4.8. With the assumption of  L    L (  1 ),  1  t , n1 / n5  N ,
2
1
a) Find an expression for calculation of C g   H in terms of N and t (or α).
 L1
b) Find expressions for n2 , n3 and n4 in terms of known parameters.
c) Show that for   1 the above results are identical to conventional geometric progression.
H
 L2
 L1
V1
V2
V3
V4
V5
Figure P4.8 Engine-vehicle speed diagram of Problem 4.8
Result: (a) Cg4  tC g3  tC g2  N 2  0
H
 L1
0
1
2
V1
3
5
6
4
V2
V3
7
8
V4
 L2
V5
Figure S4.8 Speed diagram of Problem 4.8
Proble m 4.9
Repeat Problem 4.8 for the following engine-vehicle speed diagram.
H
 L2
 L1
V1
V2
V3
V4
V5
Figure P4.9 Engine-vehicle speed diagram of Problem 4.9
Result: (a) Cg4  t (Cg3  Cg2  Cg  1)  N  0
Proble m 4.10
In a vehicle clutch, the inner and outer disk radii are r and R respectively, while the maximum
spring force is F  .
a) Write an expression for ΔT, the difference between delivered torque at the two cases of
constant pressure and uniform wear in terms of the uniform wear torque.
b) Calculate the ratio
r
for the three cases of ΔT= 1%, 5% and 10% of the torque of constant
R
wear.
c) Draw the variation of ΔT/Tuw versus
1
4 Rr 
T
Result: (b)  
2  uw
 3 3( R  r ) 
r
R
0.35
0.3
0.25
T
Tuw
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
kr
Figure S4.10 The variation of
T
versus k r of Problem 4.10
Tuw
Proble m 4.11
For the clutch of Problem 4.10,
a) Is there an answer for r for having equal torques at both cases of uniform pressure and
uniform wear? Explain why?
b) With  being the coefficient of friction, find an expression for the maximum difference
between the clutch torques in the two cases.
Result: (b)
1
F * R
3
Problem 4.12
If the sum of groove angles on the clutch plate lining is  (radians) show that the actual
maximum pressure on the material is
1
1

2
times its theoretical value with no grooves, for both
uniform pressure and uniform wear.
Proble m 4.13
During the clutch release in gear 1 the clutch force is increased linearly from zero to the maximum
of 5000 N. The variations of the engine and clutch rotational speeds are of the form shown in
Figure P4.13.

1500
e
c
t*
t
Figure P4.13 Engine-clutch speed diagram of Problem 4.13
For information given below determine the clutch efficiency.
Table P4.13 Information for Prob lem 4.13
1
2
3
4
Parameter
Engine maximu m torque
*
Lockup time t
Average clutch radius
Dynamic coefficient of friction
Value
100
1
20
0.4
Unit
Nm
s
cm
-
5
Engine rotating inertia
0.25
kg m2
Proble m 4.14
The driver of a vehicle decides to gearshift from gear 1 to gear 2 when the travelling speed reaches
36 km/h. At the time driver releases the clutch pedal the engine is idling.
a) Determine the rotational speeds of engine, clutch plate and driving wheels at the time
clutch starts to be released (use Table P4.14).
b) Specify the torque flow direction if the driver:
1) Just releases the clutch pedal
2) Increases the engine speed before releasing the clutch pedal
c) Plot a rough variation of the engine and clutch plate speeds versus time during the gearshift
for two cases at which the driver attempts to:
1) Accelerate right after gearshift
2) Maintain a uniform speed
d) Is it possible to gearshift without clutching? Explain how.
Table P4.14 Information for Prob lem 4.14
Engine idle speed
Tire Ro lling Radius
Overall ratios at 1st , 2nd and 3rd gears
1000
33
14, 9 and 6
Result: (a) e 1000 rpm , c  2578.3 rpm and w  286.5 rpm
rpm
cm


c
c
e
1000
1000
e
t
(1)
t
(2)
Figure S4.14 Typical speed variations for (1) accelerating and (2) maintaining speed right after
gearshift
Proble m 4.15
For a 4WD vehicle the CG is at equal distances from the front and rear axles and the CG height
to the ground is half of the same distance. For P=1, assume equal driveline efficiencies for
driving with front or rear wheels and determine the ratio of FWD to RWD low gear ratios
k
N L FWD
.
N L RWD
Result: k 0.6
Proble m 4.16
The gear ratio of a layshaft gearbox in gear 1 is 3.85:1. Two options with sub ratio combination
1.752.2 and 1.9252 are proposed for the determination of tooth numbers (at each case the left
figure is input gear mesh ratio and the second figure is the output gear mesh ratio). The distance
between the centrelines of the upper and lower shafts has to be larger than 100 mm but as small
as possible. Gear modules must be larger than 1mm with spacing of 0.25 mm (e.g. 1.25, 1.50,
etc). Find the tooth numbers for all 4 gears for both given options and select the best answer.
Table S4.16a Two alternatives ratios
1
2
N2
N1
N G1
N P1
n1
1.750
1.925
2.2
2.0
3.85
3.85
Table S4.16b Solutions for the alternative ratios of Table S4.16a
1
2
N2
N1
N G1
N P1
N1
N2
NP1
NG1
N
Ci
1.750
1.925
2.2
2.0
64
40
112
77
55
39
121
78
352
234
110
102.4
Proble m 4.17
For the vehicles with given properties in Table P4.17,
a. Determine the maximum grade each vehicle can climb.
b. At the grade found in (a) what percentage of maximum engine power is utilized at
speed of 30 km/h?
c. At the grade found in (a) what percentage of maximum engine torque is utilized at
speed of 30 km/h?
Table P4.17 Vehicle information of Problem 4.17
Parameter
1
2
3
Distribution of weight F/R
Rolling Resistance Coefficient
Tire Ro lling Radius
(%)
(m)
Vehicle 1
(RW D)
55/45
0.02
0.30
Vehicle 2
(FWD)
60/40
0.02
0.30
4
5
6
7
8
9
10
11
12
High Gear Ratio
Maximu m engine torque
Engine speed at max torque
Torque at max engine power
Engine speed at max power
Vehicle mass
CG height to ground
Friction coefficient
Wheel base
(Nm)
(rp m)
(Nm)
(rp m)
(kg)
(m)
(m)
13.0
150.0
3000
120.0
5000
1200
0.6
0.8
2.2
Proble m 4.18
The information for a passenger car clutch spring is given in Table P4.18.
a) Plot the FS -S and FB - B curves using the MG formulae.
b) Calculate the seesaw gain k s.
c) Plot the variation of clamp force inside the first plot of FS - S .
d) Calculate the initial deflection  C*
Table P4.18 Clutch information of Problem 4.18
Parameter
Value Unit
1
Inner diameter Di
34
mm
2
Outer diameter Do
185
mm
3 Bellville inner diameter Db
151
mm
4
Spring thickness t
2.3
mm
5
Bellville height h
3.4
mm
*
6
3.4
mm
Set point deflection  S
7
Modulus of elasticity
206
MPa
8
Poison’s ratio
0.3
-
12.0
150.0
3000
120.0
5000
1150
0.6
0.8
2.2
600
Spring force (kgf)
500
400
Spring force
300
Bearing force
200
100
0
0
1
2
3
Displacement (mm)
4
5
Figure S4.18a Plots of the FS -S and FB - B variations
600
500
Maximum
Spring force (kgf)
400
300
Clamp force
200
100
0
Minimum
-100
0
1
2
3
Pressure plate displacement (mm)
4
5
Figure S4.18b The variation of clamp force
Proble m 4.19
For the vehicle with given specifications in Table P4.19.1, the engine is off. For the two cases of
uphill and downhill, determine the maximum grade vehicle can stop without slipping, if:
a. Only gear 1 is engaged.
b. Only the handbrake on front wheels is activated
c. Only the handbrake on rear wheels is activated
d. Only the footbrake is activated
e. Gear 1 is engaged together with the handbrake acting on front wheels
f.
Gear 1 is engaged together with the handbrake acting on rear wheels
Compare the results for both cases of FWD and RWD by filling in Table P4.19.2.
Table P4.19.1 Vehicle information of Problem 4.19
Parameter
unit value
1
2
3
4
5
6
7
8
9
Tire Ro lling Radius
High Gear Ratio
Engine braking torque @ 0 rp m
Vehicle mass
Maximu m road friction coef.
Weight distribution F/R (FWD)
Weight distribution F/R (RWD)
CG height to wheelbase ratio h / l
Coefficient of ro lling resistance fR
m
Nm
kg
0.30
14
30
1200
0.8
58/42
55/45
0.35
0.02
Table P4.19.2 Proposed table for filling in the results
Results for Maximum grade (deg)
Uphill
FWD
1
2
3
4
5
6
RWD
Downhill
FWD
RWD
Case a
Case b
Case c
Case d
Case e
Case f
Table S4.19 Maximum stopping grades
Results for Maximum grade (deg)
Uphill
1
2
3
Case a
Case b
Case c
FWD
5.7
19.13
25.3
RWD
5.7
19.13
25.3
Downhill
FWD
RWD
5.7
5.7
25.3
25.3
19.13
19.13
4
5
6
Case d
Case e
Case f
38
19.13
27.1
38
21.1
19.13
38
25.3
21.1
38
27.1
25.3
Proble m 4.20
The intention is to investigate the existence of a certain grade and friction coefficient for which
both FWD and RWD vehicles with same properties generate equal traction forces.
a) In the expression for the tractive forces of FWD and RWD vehicles ignore the hf R term
and find a condition for friction coefficient which guarantees equal traction forces for the
both cases.
b) Using the result of (a) prove that for both cases: tan  0.5P .
a
b
c) Use  0.45 and  2.0 and calculate the values of μP and θ.
l
h
d) Retain the hf R term and repeat case (a) and show that the result in (b) is still valid and that
4
the friction coefficient must satisfy the condition:  P  2 f R 
if values in (c) are used.
11
Proble m 4.21
In the expressions for the tractive forces of FWD and RWD vehicles, ignore the hf R term and
derive equation for slope in the form of tan  
c
d / P  e
a) Find values c, d and e for both cases given
and then,
a
b
 0.45 and  2.0 .
l
h
b) From a mathematical point of view investigate the possibility of a maximum
grade each type of vehicle can negotiate by differentiating θ with respect to μP
and find θmax for each case.
c) Draw the variation of θ versus μP for both cases (evaluate θ values for μP up to 4).
d) What would be the values of θmax for each case in the real practice? (Suggest a
practical μmax ).
100
80
60
Slope angle (deg)
40
20
0
-20
-40
FWD
RWD
-60
-80
-100
0
0.5
1
1.5
2
2.5
Coefficient of adhesion
3
3.5
4
Figure S4.21 Theoretical variation of slope with μP
Proble m 4.22
In the derivation of Equations 4.212 and 213 the limitation on the clutch torque was not included.
Clutch torque Tc
Derive the equations for the clutch efficiency with considering this limitation.
Tmax
T*
T(t)
tT
tr
tL
Ti me t
Figure S4.22 Clutch torque variations during release