Longitudinal Dynamics

 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