International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
ROLLOVER STABILITY ANALYSIS OF
COMMERCIAL VEHICLE
Venkata Ramesh Chaganti1 , Dr.A.Siva Kumar2, G.Hima Bindu3 and Dr.K.Vijaya Kumar Reddy4
1
M.Tech Student, Department of Mechanical Engineering, MLR Institute of Technology,
Dundigal, Hyderabad-43, Andhra Pradesh, India.
2
Professor & HOD, Department of Mechanical Engineering. MLR Institute of Technology,
Dundigal, Hyderabad-43, Andhra Pradesh, India.
3
Asst. Professor, Department of Mechanical Engineering. MLR Institute of Technology,
Dundigal, Hyderabad-43, Andhra Pradesh, India.
4
Professor, Department of Mechanical Engineering. JNTUHCEH, Kukatpally,
Hyderabad-500085 Andhra Pradesh, India.
ABSTRACT
Rollover stability of commercial vehicle is concerned with the study of various vehicle parameters in relation to lateral
acceleration, yaw rate , roll dynamics and their influence on rollover stability. These parameters are examined to evaluate the
severity of the rollover of commercial vehicles. In this work the mathematical models are used to simulate these models using
SIMULINK software. The vehicle is firstly consider as simplified and then simulated as bicycle model with 2 DOF (lateral slip
and yaw). Further a 3 DOF model (lateral slip, yaw and roll) is constructed and also simulated using Simulink. The results of
these two models are compared with existing published work and showed good accuracy. Additionally a PID (Potential,
Integration, Derivation) control system has incorporated in for the 2 and 3 DOF truck models to control yaw, slip and roll.
Keywords: PID(Potential, Integration, Derivation),DOF(degrees of freedom),SIMULINK
1. INTRODUCTION:
When The vehicles which are used for transportation of freights via roads across the world are called as commercial
vehicles. Commercial vehicle undergoing modification in design, power train, suspension, etc. according to the human
requirements. But one of the astonishing fact is there exists a proportionality between the number of vehicles
manufactured and the number of vehicles involved with the fatal accidents. This is in the inadequate safety features
equipped in the vehicle and the careless driving of the driver.
According to the information by US National Highway Traffic Safety administration there are so many accidents caused
due to the inferior roll stability of heavy vehicles compared with light vehicles made them more likely to roll over[1].
These technologies are playing significant role in controlling the vehicle under braking but no significant contribution is
made against the vehicle rollover[2].
2. OBJECTIVE:
In this work an attempt is made to study the influence of yaw and lateral acceleration, on the rollover stability of heavy
vehicle and create a mathematical model using MAT LAB/SIMLINK in order to predict and optimize the commercial
vehicle stability. To incorporate a yaw control system to the models developed employing SIMULINK as described below
 Review the theory of vehicle dynamics and its application on commercial vehicles.
 Develop mathematical models for 2 and 3 degree of freedom system with PID and without PID which simulates lateral
acceleration, yaw, and rollover of typical commercial vehicle.
 Use MATLAB/SIMULINK to simulate the mathematical equation and predict various responses
3. M ATHEMATICAL M ODELING:
3.1. Mathematical modeling of roll dynamics
3.1.1. Static vehicle model
Roll model with tire and suspension compliance as shown in Figure 1 :
The relation among loads on wheels , lateral force and roll angle could be achieved as
K∅×∅∅+Ws(h c-h u)∅u+Ws×ay(h c-hu)+Fz×ay×h u=(Fz/2+Kt×∅u×T)(T-h u×∅u)-(Fz/2-Kt×∅u ×T) ;
The relation among the lateral acceleration and angle of roll is given by Ws×ay×h r=K∅×∅s=Ws×h r×∅ ;
At the point where tire lift-off the normal forces at that tire leads to zero Fz/2-Kt×∅UL×T=0
Volume 2, Issue 9, September 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Roll model of a Single unit vehicle given in Figure 2 :
The balance of roll moment about the roll centre of mass which is not supported by springs is specified as ay=([K∅-ms g
h r(1+fj)]∅+Wi Ti (1+fj ))/(ms h r+(ms qi h ci+mui hui+muj huj ))
in which ay is lateral acceleration; Fi are vertical tire loads; h is the height of cg; T is the track width; W is the weight of
the vehicle; ∆y is the lateral motion of the cg relative to the track ; ∅ is the roll angle of the vehicle; kt = tire stiffness in ;
T = half track width in; ∅L is the roll angle at the point where wheel lifts
Figure 1 Vehicle while taking a steady turn
Figure 2 Single unit vehicle coordinate system employed
for analysis of stability of roll
3.1.2. Dynamic vehicle modeling
The dynamic model structure as displayed in Figure3 has a number of inputs and outputs. A commercial vehicle with
Three Degrees of Freedom (lateral velocity ‘v’, the yaw rate and roll angle ‘∅' ) moving forward on the flat road with a
unvarying speed ‘U’ is taken in to account.
The equations of motion used
for modeling
withofthe
above stated
is dynamic
given model
Figure
3 schematic
input-output
of condition
the rollover
m((v+Ur) . )+(amf-bmr ) (r+ms ) h r ∅..
= Fyf+Fyr
(amf-bmr )(v. +Ur)+Izz r. +Ixz ∅..
= aFyf-bFyr +Mzf+Mzr
Ixx ∅.. +D∅ ∅. +(K∅-ms gh r )∅+ms h r (v. +Ur)+Ixz r. = df Fyf+dr Fyr
In which, a and b are the distance from the vehicle CG to the front and rear axles. m, ms, mf and mr are the complete
mass, vehicle sprung mass, unsprung mass at the front and rear
K∅, and D∅ are the tortional stiffness co-efficient and
tortional damping co-efficient.Ixx, Izz and Ixz are the vehicle roll inertia, vehicle yaw inertia and product of vehicle rollyaw inertia. ∅,∅. and ∅.. are the vehicle sprung mass roll angle about the roll axis and its derivatives. hf is the sprung
mass CG height above the roll axis. df=h cf-ha and dr=hcr-h a are the scrub derivatives of front and rear scrub. h a is the
reference roll centre height and h c is the roll centre height at the axle.
3.2. Rollover simulations using Simulink software
The vehicle modeling is generally done mathematically as it plays a significant role in analyzing the dynamics of the
vehicle, such as visualizing the data, analyzing the data and computing numerically. Software of Various types are
employed for modeling. In this paper I had chosen SIMULINK for modeling and analysis. It is a podium for designing
the dynamic systems by model based, which is incorporated in MATLAB. Simulink is employed for mathematical
modeling of the 2 and 3 DOF models a step steering input and a forward velocity which is constant thought out the
modeling is given as inputs. The outputs of lateral velocity, yaw rate, roll angle verses time are plotted in graphs. Though
the TWO DOF model is built and the results are conferred. The 3 DOF model is built in order to analyze the roll. Inputs
taken are given Table1
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Table 1 Inputs for Simulink
Parameter
m (kg)
ms(kg)
Izz (kg m2 )
Ixx (kg m2 )
Ixz (kg m2 )
a(m)
b(m)
c(m)
Cαf (N/Radians)
Cαr (N/Radians)
Cαm (N/Radians)
h s(m)
hcf(m)
hcr(m)
h c(m)
h r(m)
k∅f (N-m/degrees)
k∅r (N-m/degrees)
k∅m (N-m/degrees)
k∅ (N-m/degrees)
D∅f (N-ms/degrees)
D∅r (N-ms/degrees)
D∅m (N-ms/degrees)
D∅ (N-ms/degrees)
g(m/s2 )
Tf(m)
Tr(m)
Tm(m)
Iwf(kg-m2)
Iwm(kg-m2)
Iwr(kg-m2)
Ksf(N/m)
Ksr(N/m)
Kt(N/m)
rd(m)
Description
Total mass of the vehicle
Sprung mass of the vehicle
Inertia yaw moment of sprung mass
Inertia roll moment of sprung mass
Yaw roll product of inertia of sprung mass
Distance from front axle to CG
Distance from rear axle to CG
Distance from middle axle to CG
Cornering stiffness of the front axle.
Cornering stifnees of the rear axle
Cornering stiffness of the middle axle
Height of CG above the ground
Front roll centre height
Rear roll centre height
Roll centre height of the vehicle
Roll axis height
Cornering stiffness of the front axle
Cornering stiffness of the rear axle
Cornering stiffness of the middle axle
Total vehicle roll stiffness
Damping coefficient of Front axle
Damping coefficient of Rear axle
Damping coefficient of middle axle
Damping coefficient of entire vehicle
Acceleration due to gravity
Track width of front axle
Track width of rear axle
Track width of middle axle
Front wheel moment of inertia
Middle wheel moment of inertia
Rear wheel moment of inertia
Spring stiffness of the front axle
Spring stiffness of the rear axle
Stiffness of the tyre
Unloaded tyre radius
Values
14193
12487
34917
24201
4200
1830
2.520
1.220
10157.81
13665.9
11321.31
0.83
0.53
0.53
0.578
1.15
1000
1500
1500
4000
260
200
200
660
9.81
2.040
1.850
1.850
2.040
1.850
1.850
3,78,666
10,83,004
10,00,000
0.524
3.2.1. Two DOF vehicle model in Simulink
The initial step in the modeling process is the building of 2 DOF model (slip and yaw). A step steering input and a
constant forward velocity are given as inputs to attain lateral acceleration and yaw rate as outputs as displayed in figure4 .
The equations employed for the modeling are given be
m(v. +Ur)+(amf-bmr) r. +ms h r ∅..
= Fyf+Fyr
(amf-bmr )(v. +Ur)+Izz r. +Ixz ∅..
=aFyf-bFyr+Mzf+Mzr
Figure 4 Two DOF model in Simulink
Volume 2, Issue 9, September 2013
Figure 5 Three DOF model in Simulink
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
3.2.2.Three DOF vehicle model in Simulink
The roll is taken in to account along with the lateral acceleration and yaw rate of the vehicle. Therefore the step steering
input and a constant forward are the inputs to the model . The lateral acceleration, the yaw rate and the roll rate are the
out puts from the model as displayed in Figure5. A step input is taken in this paper in order to see the effect of roll on the
vehicle. The equations employed as
m(v.+Ur)+(amf-bmr )r. ms h r ∅..
(amf-bmr )(v.+Ur)+Izz r.+Ixz ∅..
Ixx ∅..+D∅ ∅..+(K∅-ms ghr )∅+ms h r (v..+Ur)+Ixz r .
= Fyf+Fyr
=aFyf-bFyr+Mzf+Mzr
=df Fyf+dr Fyr
3.2.3.Designing of a PID control system
The PID (Proportional, Integration, controller is the simplest one, which can be implemented with ease. This control
system is extensively applied to many practical applications control system characteristics. A typical PID (or linear PID
controller) can be explained as:
U=kp (e+T1 ∫e+TD e. )
.
Where e, ∫e and e denotes the error, its error integration, and its error derivative, respectively. kp Denote the
proportional gain, T1 is the integral time constant, TD is the derivative time constant and U is the output of the controller
3.3.1.Trial and Error Tuning
The trial and error approach is extensively employed mainly in the process industries. A skilled engineer can normally
tune a loop of control swiftly depending on plant knowledge and a bit of trail and error.Tune the ‘P’ term initially (kp ) to
a sensible transient response keeping the values of ‘kt’ and ‘kd’ to zero.Vary the ‘I’ term (ki ) for suitable steady state
error keeping ‘(kd )’ =0.The value of ‘D’ term (kd ) can be varied if needed to elevate the transient response.
This trial and error method is adopted in this paper in order to reduce the lateral velocity and yaw rate for the 2 DOF
model.The values of kp,ki and kd used for the 2 dof model of slip and yaw are taken as kp=0.80 ; kt=0.001; kd =0.0013.2.
3.3.2.Two DOF vehicle model in Simulink with PID
PID control system (in green) in Figure 6 is incorporated to reduce the lateral velocity and yaw rate by measuring the
generated yaw rate and performing the Proportional, Integration, derivation and subtracts the generated outputs from the
initial generated yaw rate and lateral velocity to minimize them.. Thus in this paper an attempt is made to reduce the
lateral velocity of the heavy vehicle based on Active Yaw Control (AYC) system by taking the yaw output as input which
reduced the model complexity.
Figure 6 Two DOF model with PID
Figure 7 Three DOF model with PID
3.3.3. Three DOF vehicle model in Simulink with PID
Two PID control systems are incorporated in 3 DOF in Figure7 to reduce the lateral velocity, yaw rate and roll rate of the
vehicle. The first PID control system uses the yaw rate as an input to regulate or mitigate the lateral velocity and yaw rate.
The second PID control system uses the roll rate as an input to reduce the roll rate itself.The method of controlling used
in the PID control system is the Trail and error.
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Volume 2, Issue 9, September 2013
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4.RESULTS AND DISCUSSION:
4.1. Results and discussion of Two & Three DOF Model without PID:
Figure:8 describes the Step input with step time 5 sec, initial value 0 degrees, and final value 5 degrees, and sample time
of 1 sec is given. The analysis is done for the given forward speed of 13,5 m/s; The output of the 2 DOF model are
achieved by simulating the model for 10 seconds.
Figure:9 describes the lateral velocity of the 2 DOF model start to vary at 5 sec in 10 seconds simulation for the given 5
degrees steering input and reaches a steady state value of 17 m/s at 5.5 sec and follows till 10 sec.
Figure 10 shows the yaw rate of the 2 DOF model start to vary in the same manner as the lateral velocity for the given
step steer and reaches a steady state value 6.5 deg/sec at 5.5 sec and follows the same till the 10 sec. Figure :11 shows the
roll angle of the 3 DOF model start to vary at 4.8 sec to the given step steer of 5 degrees and reaches a steady state value
of 0.5 deg at 5.5sec and follows till 10 sec or till the end of simulation.
Figure 8 The graph shows the step steering input
Figure 10 Variation of Yaw rate with respect to
time for step steering input
Figure 9 Variation of lateral vel with respect to
time for step steering input
Figure 11 Variation of roll angle with respect to time for
step steering input
4.2. Comparison of Two DOF model with PID and Two & three DOF model with PID
From the Figure12 it is learnt that variation of lateral velocity of the 2 DOF model with and without PID control system
for the given step steer . The lateral velocity without PID control system (blue line) rised from 0 at 5 sec to 17 m/s at 5.5
sec and remains unvaried till the end of simulation of 10 sec for the given steer. The lateral velocity of with PID control
system (green line) rises a maximum of 11.7 m/s at 5.7 sec and remains unvaried till the end of simulation. In
comparison with the conventional the lateral velocity of the 2 DOF model with PID is reduced to 35% for the same step
steer.
From the Figure13 it is observed that variation of yaw rate of the 2 DOF model with and without PID in the same
manner as the lateral velocity. The yaw rate of the model without PID varies from 0 at 5 sec and reaches a steady state of
5 deg/sec at 5.5 sec and remains unvaried till the end of simulation of 10 sec. The yaw rate of the model with PID control
system reaches a steady state of 1deg/ sec at 5.7 sec and remains unvaried till end of simulation.
The Figure14 shows the variation of roll of the 3 DOF model with PID and without PID control system for the given step
steer. The roll of the model without PID control system varied from ‘0’ at 5 sec and reached to ‘0.55’ deg/sec at 5.5 sec.
The roll of the model with PID control system varied from ‘0’ at ‘5’ sec and reached to ‘0.1’ deg/sec at 5.5 sec and
remained unvaried till the end of simulation.
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Therefore from the results shown in figures (12), (13), (14) it is evident that with the implementation of PID control
systems to the three DOF model the lateral velocity is reduced to 35% , yaw rate is reduced to 75% and roll rate is
reduced to 80%.
Figure 12 Lateral velocity with time to a step input
with control system feedback
Figure 13 Yaw rate with time to a step input
feedback of control system
with
Figure 14 Variation of the roll rate with time to a step input
with feedback of control system in three DOF
5. CONCLUSIONS:
 The two DOF and three DOF models built using truck data in Simulink made understand the variation of the lateral
velocity, yaw rate and roll of the truck for the given inputs of step steer and forward velocity.
 With the incorporation of the PID (potential, integration, derivation) the lateral velocity, yaw rate and roll of the truck
model are reduced significantly in order to improve the dynamics of the truck. in this the yaw rate and roll are reduced
to greater extent than the lateral velocity as the control system is build based on active yaw control technology .
REFERENCES
[1] Hussain. (2005). Modelling Commercial Vehicle Handling and Rolling Stability. Proceedings of the Institution of
Mechanical Engineers, Part K: Journal of Multi-body Dynamics (Volume 219, Number 4 / 2005), 1-13.
[2] Díaz,M.B. (2007). Active roll control using reinforcement learning for a single unit heavy vehicle. 12th IFToMM
World Congress, (pp. 1-6). Besançon (France).
[3] Hirokazu Okuyama, F. M. (2006). Stability Augumentation System For The Heavy Duty Commercial Vehicles Estimation Of The Gravity Position With Ar Method And Application For Antirollover.Japan: Toyota Central R&D
Labs, Inc.,http://www-nrd.nhtsa.dot.gov/pdf/nrd-01/Esv/esv16/98S4O06.PDF.
[4] (2013,
Sep
01).
Retrieved
from
PID
controller
tunning
a
short
tutorial:
http://saba.kntu.ac.ir/eecd/pcl/download/PIDtutorial.pdf .
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