so we will continue with what we were doing
in the last class we were looking at a broad
picture or perspective of vehicle dynamics
so we were looking at how we are going to
approach the subject of vehicle dynamics we
said that for us though there is a vehicle
it has its components and so on
when we are studying vehicle dynamics we said
the center of this whole thing is the mathematical
model so mathematical model comes from our
good old euler-newton equations and this has
an input and an output remember that when
we looked at the dynamics okay which is defined
by using these mathematical equations they
are classified into what we called as longitudinal
dynamics 
lateral dynamics and vertical dynamics okay
we said that we classify the dynamics what
we are going to study using this mathematical
model into a longitudinal lateral and vertical
dynamics we also said that for these of understanding
its effect we may most of the times delineate
or decouple them and study them in isolation
is that correct? does it not have an effect
or one has an effect on the other? yes it
is possible there is an effect
but in order to understand the subject most
of the time we will be decoupling the effects
of all these 3 right there are very important
things that will happen like low transfer
and all that we will understand it as we go
along we also said that the input for this
model is the driver’s input through the
steering or the acceleration and braking okay
in other words how the driver interacts with
the vehicle?
right that is what we are going to use as
an input we said that we cannot look at every
scenario by the driver and we will have some
test conditions okay in order to understand
the behavior of the vehicle of course we said
that the vehicle itself which goes into this
mathematical model will be defined by means
of certain parameters okay what we call as
kinematic and compliance parameters mass moment
of inertias compliances stiffnesses and so
on
the output from this model as we said yesterday
or in terms of displacements accelerations
okay velocities and so on we said that what
comes out as an output has an effect on the
occupants okay and we are going to study that
not to a great extent but at least as an introduction
we are going to study how this is going to
have an effect on the occupants okay so this
is the broad i would say basis under which
we are going to study the subject
we also said that we can again look at it
from a different perspective and call this
as driving dynamics 
okay safety 
and ride comfort 
when we look at this from a different perspective
same problem going to look at it from a different
perspective okay so it is not organized like
this it is not that longitudinal dynamics
is driving dynamics lateral dynamics is safety
vertical dynamics is ride comfort
it is to a certain extent it can be looked
at it like that but they are not necessarily
a clear demarcation like what we have here
so we would look at the safety during driving
okay both in the longitudinal as well as in
the lateral dynamics right okay we will continue
now with this short introduction and we are
now going to look at this mathematical model
okay and already we know or we had a very
very very simple model in the last class
we are now going to extend this simple model
into not a very complex model but just the
same model simple model making it more elaborate
so we are not going to make it very complex
we are going to use that same equations f=ma
okay so in other words we directly plunge
into what is called as the longitudinal dynamics
and look at a very simple mathematical model
that we will be using in order to understand
the longitudinal dynamics
whenever we talk about dynamics 2 things that
comes to our mind the very first thing are
the forces that are going to act on this vehicle
and the other is of course the acceleration
or deceleration and so on and our good friend
f=ma from newton is going to be of great help
okay in writing down this mathematical model
okay so the first thing is first so what are
the forces that are acting on the vehicle
okay?
all of us have experienced this the first
okay we are looking at the external forces
external force that acts on the vehicle okay
is what is called to say aerodynamic force
which we would call as ra okay the other force
next force which is important to us which
is acting on this body is the gravitational
pull or gravitational force okay and that
write it as mg or w and that can of course
be resolved into 2 directions like this and
of course this is equal to theta s
so if i call this as w then we have what is
this? w cos theta and w sin theta you may
have a trailer you know towing with this vehicle
which is called as a drawbar so if you have
a vehicle or if you have another trailer with
you then there will be a drawbar pull which
we would call as fd clear apart from this
the vehicle itself is going to give us some
force okay some nice guys some not so nice
guys
so if i am going to accelerate we need what
is called as the traction force okay so the
traction force is now going to act in that
direction so let us say let us put the traction
force in the front and the rear okay let us
look at this tractive force and call this
as ff and fr of course in every vehicle ff
and fr will not act together or it may be
a front wheel drive or it may be a rear wheel
drive and so on
so depending upon the front and the rear wheel
drive okay you will have either of the one
or if the 4 wheel drive then you will have
all these things apart from this what are
the very important forces okay which consumes
our fuel okay is what is called as rolling
resistance of the tire rolling resistance
of the tire okay acts opposite to we will
understand this rolling resistance in a minute
it acts opposite okay to this tractive force
okay
and in fact it is something like a braking
force that acts on the vehicle right now let
us first understand see up to this it is not
very difficult to understand all the forces
that are going to act okay what i am going
to do is very simple i am going to find out
the reactions of the front and the rear okay
let us say that we are accelerating a force
we are playing a tractive force so we can
say that i have a d'alembert’s force
actually it is not a force it is a pseudo
force okay which can be written as w/g*a okay
it is not a good practice actually to put
this d'alembert’s force it is nice to write
f=ma but then when i take some moments then
it becomes easier for me to have a force there
and that is the reason why i have a force
and call as the d'alembert’s force okay
now all these forces are familiar to you
in order to take the moments of course you
need some dimensions right so let us call
the dimensions something like this let us
say that length=l1 and that length=l2 l1 is
the distance from the front wheel to the cg
location l2 is the distance from the cg location
to the rear axle rear wheel and let the total
length of the vehicle l1+l2 let it be called
as l the other thing that is important to
us is the heights
so let us call this height as ha and let us
call the cg location height=h and let me call
that height to be hd you know how to determine
the 2 w’s or the reactions of the wheels
wf and wr if i want to find out wf i take
a moment about wr okay with proper signs i
can determine wf in other words wf*l=whatever
the moments that are due to the other things
okay
but before we go further there are 2 comments
that are important to us 1 is the system that
we are going to use in this course the x y
and the z direction that we are going to use
in this course okay
this comes out of an iso standard and we call
the direction which is along the direction
of travel okay as x perpendicular like that
as y and the other direction normal to the
ground as z okay so longitudinal lateral lateral
and vertical directions of course you know
that there are motion okay the angular motion
along these directions for example the angular
motion in the direction of x okay
in other words that angular motion okay along
the direction of x okay let me take that as
to be a correct one let us say positive okay
that is positive is called as what is that
angular motion called as? roll so this is
the roll and the angular motion okay here
which is in the y direction that is that angular
motion is called as the pitch and that is
what we call as yaw right so let us say colloquially
pitching okay that is moving in that direction
right okay
so that is the first thing the second is let
us go into the details of what is called as
rolling resistance rolling resistance is today
very very important for fuel consumption especially
in trucks can you imagine that the rolling
resistance whose origin are the tires consumes
nearly 30% of the fuel of the vehicle we are
going to do quite a bit of tire dynamics in
this course but let us understand what is
rolling resistance? and how do we get?
quickly we will go into details later just
to understand because i am putting a force
there so you have to understand what this
rolling resistance is okay now there is a
misnomer many students assume that the rolling
resistance is just the frictional resistance
of the tire absolutely not it is not the frictional
resistance rolling resistance comes from the
property of the elastomer or rubber 
which is the material of the tire
elastomer or rubber as it is called that is
what goes into the manufacture of the tire
elastomers have a property called viscoelasticity
usually depicted by a dashpot in order to
understand the effects clear now what is this
viscoelasticity and how does that going to
have an effect? will see that in a minute
as i said we will elaborate it later any material
can be looked at it is a small here
any material can be looked at from 3 simple
models one a spring other a dashpot and third
one is what i would call as a friction suppose
i say that a material is purely elastic okay
then you can say that the material can be
represented by means of a spring okay this
is not a very correct representation we are
not going into too much of details we can
say that okay spring a linear spring especially
okay is good enough to model say a linear
plastic material okay
so it is something like an understanding of
the material behavior a linear spring where
the force is proportional to the displacement
with the stiffness k can be looked as if it
is a material and k is something like e okay
so when i leave the force the spring comes
back to its original position and that is
what we usually call as elastic okay now to
this we can add other material behaviors
for example if you look at elastomers elastomers
are of course elastic and then viscous behavior
okay so in other words i can model elastomer
or i can understand elastomer as if it is
made up of a spring and a dashpot okay this
dashpot can either be attached in parallel
to look at it okay or we can understand the
behavior by attaching it like this and so
on there are names to these models okay
kelvin and maxwell models but we are not going
into details of this models okay we are putting
this in order to understand okay the behavior
for example if you have a metal which you
are taking into the plastic region then i
can model this metal using that spring and
the friction element okay so you can you know
join together in parallel or series and so
on you know these elements you can join them
and then write a mathematical equation which
can form the basis of the constitutive equation
or stress-strain behavior okay of the material
clear now we are not going into this as i
told you into the characteristics and i am
not going to write down equations here we
will understand only the elastomer part in
this case maybe pass a comment afterwards
about this friction and why friction is used
to model what we call as plasticity?
now what is the difference between an elastic
material and viscoelastic material? sometimes
people call this as hyper elastic viscoelastic
material and so on
first of all let us understand that elastic
material is not necessarily linear it can
be nonlinear elastic as well so if i now have
a load deflection of the stress-strain curve
okay when i load a material which means that
i am applying forces i keep increasing the
force because of which the stress is increased
and there is increase in strain and so on
so when i load the material let us say that
the path taken by the stress-strain curve
is something like that it goes like this
when i unload an elastic material it would
actually all of you know that it would follow
the same path on the other hand a viscoelastic
material does not follow the path when it
is unloaded and would now follow a different
path okay and that amount of energy is lost
and usually called as hysteresis loss clear
so there is an amount of energy that is lost
is this same as plastic?
there is a subtle difference good difference
that though at the end of loading there is
a residual strain here the strain will come
back to 0 with time so time is an important
factor in viscoelasticity time and frequency
are important factors in viscoelasticity right
so in other words what i mean by time and
frequency are important is that the material
behavior is affected by the rate at which
you loaded or in other words the frequency
at which you loaded and so on okay
so time and frequency are important factors
so the first thing is that to conclude whatever
we have been saying that there is a loss of
energy when the material is loaded and unloaded
how is it going to affect us? why is it that
the tire should develop a rolling resistance
okay? and that is what we are coming now
now let us say that obviously all of you know
it but i am just reiterating what is well
known let us say that i have a tire we have
what are called as treads let us say that
tread okay and that is the ground so this
tread material as it approaches is going to
get let say that it gets compressed and then
again gets released okay why tread the material
inside the tire which we are going to see
what they are okay?
also gets compressed and released or in other
words there is a loading unloading cycle as
the tire rolls similar to what you see in
this stress-strain curve so in other words
if i go and sit here in this tread okay and
go through the cycle of rolling right i will
go through a compression and then whole compression
as i come near i get completely compressed
so go out you know the load on me gets released
so because of this cycle okay i lose energy
okay or there is hysteresis loss right now
who is going to compensate for this hysteresis
loss? because your vehicle has this tire and
tire is losing energy so who is going to compensate?
the vehicle has to compensate okay the vehicle
has to compensate so the first thing is that
because of the material of the tire there
is lot of advantages why then rubber you know
let us not talk about that
we have lot of advantages we will see that
okay so because of the material with which
this tire is made of we have hysteresis loss
and the loss has to be compensated by the
engine ultimately and so this opposes the
motion
now let us understand how did i get this force?
i said that there is a rolling resistance
force which let us call this as fr okay rolling
resistance force it can be the front and the
rear okay so how did i get this as a force?
okay so in order to understand this we have
to look at what is called as the contact patch
of the tire 
what is a contact patch?
a patch that is formed obviously by contact
of the tire with the road in other words more
precisely it is the pressure distribution
at the contact right so we will see the 3-dimensional
pressure distribution later or rather 2-dimensional
pressure distribution later now let us understand
a section of this pressure distribution okay
so let us say that i come into contact at
that point and leave contact at that point
in other words that is where my contact is
it is not necessary that the contact pressure
exist only when the tire rolls when the vehicle
is stationery also you have contact pressure
let us for a moment stop the vehicle okay
and look at this contact patch so the contact
patch now is not that of a one tread okay
but there are number of treads so the contact
patch would look something like this
so in other words rubber is symmetrically
compressed about the center this is the vehicle
that is standing okay symmetrically compressed
about the center so whatever is the force
that is compressed okay by this and it has
to be it has to come out in the other side
okay whatever is compressed has to come out
the other side okay now let us understand
one or two more things about tires before
we go into the details
the first is that the tires that we use or
what is called as pneumatic tires okay 
in other words we inflate the tire to a particular
pressure right so many of you might have driven
a car even now when you go to a gas station
to fill or inflate your tire still you talk
in pounds per square inch units okay 32 psi
if you are driving a huge vehicle truck it
is 120 psi and so on right so let us go into
some details and look at the section okay
from this angle
so let us say that the tire okay that is how
it is deformed okay let us say that the tire
is deformed like this right so when you look
at it from this section or whatever be the
section that is how the tire is deformed okay
for a moment i am taking out the tread and
i am saying that the tire has a thickness
something like that right that is what the
inflation pressure okay which we have used
in order to inflate the tire okay
when we inflate the tire we get what is called
as inflation pressure so now we know very
well equilibrium equations we know very well
that whatever infinitesimal element you take
should be under equilibrium between the forces
that are acting on this infinitesimal elements
okay so obviously when i take an infinitesimal
element here okay i said contact pressure
is what is acting in that region right
so if i want this to be under equilibrium
or if i want it to be at equilibrium then
the pressure that is acting the contact pressure
that is acting should equilibrate the inflation
pressure okay when it is in full contact okay
so the contact pressure should be equal to
the inflation pressure okay contact pressure
should be equal to the inflation pressure
so strictly speaking the contact pressure
should have been uniform okay
but contact pressures are never uniform we
will see more about it a bit later because
of the local bending because of the bending
of the side walls these are called side walls
and that is why the contact pressure is never
uniform okay it has a particular shape we
will study this okay after 2 or 3 classes
right so now let me come back so in other
words there is a lot of theory as to how contact
pressure develops?
how contact pressure is distributed? whether
it is uniform? whether it is not uniform?
and all those things okay now here when i
talk about this i am only talking about the
pressure okay because of the tread as it travels
along or around the circumference okay so
here i am looking at the pressure on the tread
okay so the pressure on the tread compresses
okay goes to a maximum and then gets released
okay
so let us not right now confuse between this
and this we will come to that later so the
contact pressure what we are talking about
is because of the tread getting compressed
right when the tire is stationary okay then
we have a contact pressure something like
this because there are a number of treads
there is one tread that is getting compressed
another tread getting compressed a bit more
another tread much more another tread slightly
less and so on
so number of treads are involved at various
compressive positions okay and hence we have
a contact pressure like that of the treads
that are formed okay on the other hand let
us now roll the tire 
okay
now when i roll the tire let me follow a tread
so that is for a static so i am just removing
that let us now roll the tire when i roll
the tire one tread or one block that block
is what we are going to follow that block
gets compressed okay goes to a maximum compression
and then gets released so one block here okay
that block as i rotate as the tire revolves
goes into this position so same block goes
into this position maximum compression goes
out and gets completely released
so in other words a block gets loaded like
that okay and then gets unloaded now how is
that it is going to be unloaded? it is going
to be unloaded like this so unloaded like
this so as the blocks gets loaded and unloaded
these blocks loose energy or hysteresis develops
in these blocks clear it is not only the blocks
that gets compressed or loaded and unloaded
but the sides of the tire they also go through
the same thing
so in other words the sides of the tire gets
also loaded and unloaded and so on okay so
in other words this loading and unloading
cycle gives rise to this energy loss and that
has to be accounted for by the vehicle i am
just repeating that so that you understand
it and that is quite clear now how does this
loading unloading cycle affects that contact
patch okay in a very simple sitting as we
had seen
how does that gets affected? because the loading
cycle or loading path is different from the
unloading path? how does that get affected?
so it was symmetric when it was stationary
that is fine but when it gets loaded and unloaded
look at this carefully for the same strain
in the unloading path the stress is less okay
so now we are talking about the pressure okay
that is acting on the treads since for loading
and unloading they are different
this curve cannot be symmetric because both
of them are not the same so they cannot this
guy is due to loading and this is due to unloading
okay so they cannot be symmetric because i
am following the same tread okay which is
going through the cycle so it cannot be the
same so how it should be? this has to be a
different curve this has to be a different
curve so the 
curve actually shifts and becomes something
like this
because the loading curves are different from
unloading curve the curve of the symmetry
is lost becomes something like this if i now
say that the reaction force in order to support
this is not a very correct picture that is
why i introduced this inflation pressure keep
that in mind we will come back to this topic
again okay not a very correct picture we are
going to see very interesting things how inflation
pressure is going to act? and how actually
the tire carries a load?
you know we are going to get to details there
okay so we will come to that a bit later but
let us now understand this from a different
angle okay and give an explanation only to
the rolling resistance we will refine it as
we go along so if this is the load that is
acting on the tire then the load is now equilibrated
from the ground or in other words that is
the load that is going to act okay which opposes
the weight
now since this symmetric distribution is affected
what will be my resultant force due to this
contact with the ground? the resultant force
which is developed due to this compression
which opposes the load that is on the tire
would now get displaced and hence actually
instead of acting right at the center the
load now acts okay away from the center right
and that is how the load acts
when it acts away from the center then if
i now look at that load with respect to this
center not only i am going to equilibrate
this load with this force but i am also creating
an additional effect correct so what is that
additional effect? that will be a torque that
will acting or a movement that will be acting
like that right watch carefully that the moment
is now going to oppose the motion of the tire
okay
so there is an opposing force or opposing
moment that is acting okay now i do not want
to put that moment here i know that the moment
opposes the motion so i just want to replace
this moment by means of a force that is acting
here okay because that will oppose the motion
of the vehicle so i replace this moment which
in reality exist because of viscoelasticity
by a force here okay
and call this as rolling resistance force
and say that this force rolling resistance
force creates the same moment which opposes
in other words fr*r=this into this we will
give names to that in a minute so first let
us understand the philosophy of development
of a rolling resistance force
so the philosophy of this rolling resistance
force to summarize is the viscoelastic behavior
of the elastomer which means that there is
a loss of energy which means that the symmetric
contact pressure distribution when its stationary
gets affected or in other words it becomes
skewed okay and this skewed distribution produces
a normal force okay which not only opposes
or not only supports the vehicle or the tire
but also creates a moment which opposes the
motion and the opposing moment or motion or
torque is now also depicted as a force which
opposes the motion of the vehicle or the tire
and we call that as the rolling resistance
force clear okay so that is why we have a
rolling resistance force the rolling resistance
force of course you can see this very clearly
rolling resistance force since it comes out
of a moment which supports the weight w okay
so this force has to be proportional to w
right so we usually write the rolling resistance
force to be a rolling resistance coefficient
multiplied by w obviously the rolling resistance
opposes the vehicle motion and hence is not
“a good force” okay it is not hitting
us to travel actually it is opposing you since
it is opposing you or opposing the motion
of the vehicle we consume energy we have to
overcome that like you have the aerodynamic
forces we have rolling resistance forces which
opposes the motion
interestingly note that when the vehicle brakes
this rolling resistance force would act in
the same direction as that of the braking
force which is now going to flip and act from
the other direction so rolling resistance
force aids in braking and opposes traction
clear okay so the first thing you would tell
that why not i completely reduce rolling resistance
go to 0? is it possible? how low you can go?
there are lot of issues we will come to that
later when we talk about tire mechanics so
first things first so that is the rolling
resistance force which is written in terms
of rolling resistance coefficient and w okay
our next step is to find out wf and wr okay
and wf determined by taking a moment about
the point a and wr determined by taking a
moment about the point b okay so on one hand
we have wf*l=on the other hand you are going
to write down the moment due to the forces
okay so you know this very well so wf*l is
in the clockwise direction so accordingly
put the forces and the moment okay
rather the moment due to the forces put the
signs properly and we will see how we end
up with this equation in the next class we
are going to make some assumptions okay with
respect to these heights we would see that
usually in a passenger car these heights are
almost the same and when you make an assumption
that ha=h=hd okay that makes our life simple
one of the things which is obvious which all
of us experience which you would immediately
notice is that wf and wr is going to get affected
when a vehicle is accelerating okay
or in other words that is what is called as
a load transfer you would have noticed this
when you go in a vehicle in a car obviously
all of us know that very simple mechanics
that when you accelerate you tend to fall
back and when you brake you tend to fall forward
or in other words there is a load transfer
to the axles as well another very interesting
effect so we will write down this equation
we will find out wf and wr then we will look
at traction and braking and so on okay we
will stop here and we will continue in the
next class
