dear friends welcome back to one more ah lecture

on engineering system design in the last few classes we have been discussing about the

modeling techniques employed for ah engineering system design what kind of engineering methods

can be used at various stages of system design we discussed about the data modeling process

modeling and behavior modeling and ah today we will ah talk more about modeling of a physical

systems especially the dynamics of physical systems

whenever we make some product or whenever we design a product we need to look at the

performance of the product even before we really make it so in order to make sure that

whatever the product we are designing should work as per the requirement we need to create

some models and then test it and ensure that it actually performs the way we want in some

cases it may be a a simple just to check the the shape and its ah wait and other features

but in some other cases it may be the performance parameters like the acceleration time or the

desolation time or the time to reach a particular state or what will be the maximum attainable

ah parameter we can achieve so these are the things which we need to analyze

using models so the modeling and simulation basically the ah system modeling and simulation

looks at the possibility of developing models for the physical systems and testing them

through simulations to study their behavior we can actually use ah various kinds of ah

modeling techniques you can use the physical modeling or iconic model where you actually

we make some small prototypes and then text them or we can actually do some experiments

simple experiments and then study the behavior or we can actually create some mathematical

models and ah apply some non mathematical techniques to simulate it and study the behavior

so there are various ways of ah doing this ah modeling and simulation we will look at

some of these methods and then see how this can be applied for system engineering of course

ah the focus of this ah course is not to go into the details of this modeling techniques

but basically to tell you that ah these are the modeling techniques existing and ah what

are the basic principles applied for such modeling methods

so we will be briefly going through these methods but not into the details or in depth

study will not be carried out if you want to know more about ah those methods you can

always refer to some other courses which are readily available or methods ah available

on the web or there are some video courses also available so you can use any of these

ah resources and learn more about this methods so let us ah look at the methods existing

or you need to see why we actually we need the models and what kind of ah models can

be used so here the uses of models in engineering

are basically to seek answers to some of the questions will the system as designed will

work so we are designing some system and we need to see whether it will really work or

not and which of the two system designs is the better suppose you have many designs or

alternatives are existing so which one will really work and do i adequately understand

the system and as well as what kind of trade off i can have so these are the basic purposes

of making the model so we need to check whether the system will work or whether the understood

the system completely or whether we can have a ah choice between two models or whether

we can have a a trade off in some of the aspects so this can be studied using engineering models

and coming to the methods of modeling we have ah various methods of ah modeling the system

or the physical systems so you can go for a heuristic modeling basically depending on

our ah on heuristics and our own ah intuition and imagination we can actually create the

models and then test them to see the behavior and the other one is known as the mathematical

modeling in mathematical modeling we can actually write down the mathematical equations corresponding

to the physical model and then study the behavior another method is known as physical system

modeling or bond graph modeling so we can see the physical system modeling method is

ah bond graph is one of the methods of physical system modeling and then this bond graph method

we normally what do is look at the physical system and develop the corresponding model

which actually have a one to one correspondence with the model and that is easy to understand

and this method is used for multi domain systems when you have a system with ah various disciplines

like a mechanical engineering electronics communication software

so for such systems we can use the physical system based modeling what bondgraph is one

of the methods then there are other methods like ah dimensional analysis the numerical

modeling you might a heard about finite difference and finite element methods and these are the

modeling techniques we can employee for modeling of the system and then ah once you have a

model whether it is a mathematical model or a physical system based model we can actually

use it for simulation and we can carry out the time domain analysis as well as frequency

domain analysis to find out the behavior so the model is basically converted to a simulation

model the physical system model or the mathematical model will be converted to a simulation model

and using the simulation model we can predict the performance in time domain or in frequency

domain and then see what kind of parameters we need to modify in order to get the desired

performance so that is the use of modeling and simulation in ah system engineering so

this is normally used when we go for the actual design of the system so initial design stages

like a functional decomposition and ah the design of ah architecture we dont go for this

kind of ah analysis but when we really make this design the components and the subsystems

we need to check the performance of these components and subsystems and we go for this

modeling and simulation and then analyze the performance

let us go through one or two ah modeling methods ah to show how this can be used for ah our

application heuristic modeling as i mentioned it is a common sense or minimum cost physical

modeling so here ah we dont go for any detailed modeling of the system we apply some common

sense or make some ah simple models using minimum cost as a criterion and then develop

the model so that is heuristic modeling common sense or minimum cost modeling

just ah tell you a case study where this kind of ah minimum cost physical modeling can be

used this was a a real ah situation or a real project where actually some common sense modeling

was applied to get the work done without going for a detailed analysis we mentioned about

the atlas missile project in one of the lectures so this was a project managed by us air force

and corps of engineers for developing a series of family of missiles for the us air force

so this a missiles ah need to have some installation support so the install missile and supporting

equipment in a vertical underground concrete silo

so the supporting equipment need to be inserted or to be placed on a underground concrete

silo missile to be lowered to the silo through open doors at the ground so once you have

a concrete silo we need to lower the missile to the concrete silo through a open door at

the ground one of the propellant lines that is prefabricated piping sections could not

be maneuvered into the propellant systems shaft so there was a propellant system shaft

which actually should carry the propellant lines but they were not able to maneuver this

propellant system shaft through the silo or to the ground silo they created

so they tried many ways to do this but they could not bring this propellant system shaft

in to the silo because the propellant lines were actually obstructing the moment of the

ah system shaft but they want to do the complete section then it would i mean cost around three

hundred thousand us dollars for seventy sites so that was the problem if they could not

do this to they could not maneuver this particular ah shaft into the silo then there were to

redo the complete section and for all the sites it may cost around the three hundred

thousand dollars so they could not do it for many days and

they were trying what to do with this particular problem and then this ah heuristic modeling

method ah came to their help they created a very simple model of the pipeline as well

as the shaft and then tried ah on the ground many ways of inserting this into the ah silo

so they maneuvered this ah shaft ah under various orientations and various positions

and they would find that there is one position where actually they can bring this into the

shaft or the shaft can be brought inside the silo

so this was ah tested using very simple ah prototype very low cost model and they found

out the method to insert it and then they simply applied the same principle and then

they could maneuver this ah shaft in to the silo and could solve the problem so this kind

of modeling basically comes from the common sense of ah the users the designers ah think

about the all possibilities and then make some simple models and then create a ah scenario

work actually they want to solve the problem and get a solution so that kind of modeling

is known as ah the heuristic modeling for example if you have a large cupboard ah

to be moved out of the room and you have a small door to be there to take it out then

we need to look at ok how do you actually take this cupboard out of the room may not

be possible to simply take it as it is so you may have to tilt it ah some site or you

need to make some particular angle and somebody has to move forward and then take a turn and

then the other person has to come and take a turn and then tilt the table or the cupboard

in a particular direction to take it out so those kind of things cannot be modelled

or simulated using mathematical methods we need to apply some common sense technique

and do this so the same principle applies for system engineering also we can actually

do some common sense modeling and sold many of the problems which we really faced on the

ground when really implementing the system so there kind of methods are known as heuristic

modeling so this kind of modeling the physical modeling

provides a grasp of the problem that cannot be achieved by any other techniques so if

you apply this common sense methods or the heuristic modeling technique we can actually

ah get the grasp of the problem were other methods cannot be applied and crude models

can be easily develop from basic materials at minimum cost so in order to do this we

can actually create some crude models and using basic materials and ah very low cost

and a good bit of caution needs to be applied to any conclusions reached

so whenever we do this kind of heuristic model you cannot make any generalized conclusions

that particular solution may be applicable only to that situation so bit of cost is needed

when we do the this kind of modeling and make some conclusions out of this so that is the

first method of heuristic modeling then we can actually do different types of mathematical

modeling also so a mathematical model can be defined as a construct that comprises an

abstract representation of a real system models are constructed by people often for the purpose

of system design so mathematical models are abstractions of the physical systems when

you have a physical system we try to abstract into a model using mathematical methods and

their constructed by people often for the purpose of system design and then computational

simulation a mathematical model implemented in a digital computer

so once you have a mathematical model we can implement in a digital computer and ah get

the simulations done so the computational simulation is basically a mathematical model

converted to a the use in a computer and whether you can actually get the output of course

the law of nature a fundamental understanding of causality in a physical system often expressed

in mathematical form or as an algorithm executable by a computer so whenever we make the model

we need to follow the fundamental understanding of the causality in physical systems so what

causes the a particular system to behave in a particular way so that actually comes from

the physics behind the dynamics or the system behavior

so we need to look at those fundamental issues and then only create the mathematical model

and use them so let us ah see how the mathematical model can be used in engineering system design

so most of you may be knowing how to create mathematical models so we take a an existing

system or a mechanical system and then find out the abstract ah nature of that system

and then create a mathematical model and then see how to write the mathematical equations

corresponding to the various ah performance parameters of the system and then write down

the equations and later on you convert the those equations into a form were the it can

be accepted in a digital computer and simulations can be done

so we will just take ah one simple example a spring mass damper system and i will ah

show you how to model this system using the mathematical methods so let us take a very

simple ah case of a spring mass damper system so we will have a spring and a damper so this

can be representation of any mechanical ah system so if you have a a spring mass damper

system like this and a force is applied here a force of f is applied and if this is the

stiffness parameter of the spring and this is the damping parameter of the dashboards

you can actually represent the displacement of this by parameter x so if you want to find

out the displacement of the mass and you apply a force and what will be the dynamic nature

of the response we can actually model this using mathematical relationships and then

convert that into a a simulation model now if you want to know this relationship then

we can write down the equations of ah motion in terms of the force balance so you have

f first force f is equal to m x double dot plus b x dot plus k x

where x is the displacement of mass and ah m is the mass of the body and b is the damping

parameter and k is the stiffness this is the time domain relationship so this is x double

dot stand for the acceleration x dot stands for the velocity and x displa for the displacement

in order to sold this ah and to find out the relationship for x so we want to find out

the displacement x in terms of time domain so x t need to be found out we can actually

change this to the laplace domain to solve it so if you do this then he will be getting

it as x s that is the laplace domain s f s over m s square plus b s plus k so the s stands

for in the laplace domain so f of s is the force in the laplace domain m is the mass

and b and k now we can actually solve this for using ah

standard mathematical formulas or you can actually put this in the x omega is equal

to in the frequency domain omega n square divided by s square plus two it zeta omega

n plus omega n square where omega n is given as the natural frequencies given as square

root of root k by m so we can actually get this as in the this can actually be converted

into a time domain and then you can actually you can solve this and then converted into

time domain we can actually ah use any standard ah simulation softwares to simulate this behavior

this equation can be easily solved i can actually simulate it

similarly in the time frequency domain also we can get the output using this relationship

so now for any applied ah force f we can find out the response of this system the displacement

of the system displacement velocity and acceleration can be easily obtained from here so if you

simulate this in the response can be something like this the time response can be obtained

like this depending on the parameter values we will be getting various performance parameter

is like ah raise time then the settling time if all those can be plotted for a given force

you will be getting a a constant displacement here

so this is the transient period so you can find out what is the time taken for the transient

to died on so transient period and then you have a a steady state period so using this

equations you can actually simulate it you can use the software like matlab or a simulink

matlab or simulink can be used for simulating the behavior of this particular system so

that is ah very simple system and ah we are ah showing how the mathematical model can

be generated for that kind of systems when you have a complex system we need to

do is to take the abstract that complex system to a mathematical model and write down the

equations of motion for the system and then simulate it and get the behavior so when i

complex system can be ah model using this kind of methods mathematical modeling techniques

if you have a robotic system or you want to find out the behavior of a ah machine tool

any of these even if you have a a simple motor you want to find out the performance characteristics

what you design and then you want to see what kind of response time it has how much time

it will take to reach a particular value of acceleration all those things can be modelled

and then simulated will get the output using this kind of methods

so that is one way of ah doing mathematical modeling not only this we can actually do

many other ah things also using mathematical methods so i will show you another one were

actually we can model and simulate the traffic flow doing the peak time at a traffic junction

so that also can be modelled ah using mathematical methods or you can if you want to find out

the deflection of a beam we can actually do that if you have a a beam like this a cantilever

beam and you want to find out the deflection suppose you apply a a force over here or you

hang a a mass over here you want to find out what will be the deflection

of this ah beam again we can we have standard ah equations for representing the deflection

so we can actually find out what will be the deflection under various situation depending

on the type of load you can find out the deflection as well as the bending moment bending stress

all those thing can be modelled and then simulated to get the output there are different methods

mathematical modeling is one of the methods to get the outputs

now coming to the a traffic problem we will ah see how that ah this kind of ah traffic

problems also can be model using mathematical method so this is the problem given the traffic

lights at ah road junction are set to operate with a red phase of length hundred seconds

so you have a red phase of hundred seconds so we can put it as r g as hundred seconds

sorry red phase r and then green as sixty seconds so you have a traffic signal which

actually there is a a red ah signal for hundred seconds and ah green phase of for the sixty

seconds and vehicle arrive at the traffic lights on average one every four second

so every four seconds one vehicle is arriving so every four second one vehicle is arriving

and when the lights turn to green the vehicles in the queue leave at a rate of one every

second so this is the arrival every four second one vehicle will be coming and when it is

green every one second one vehicle will be going so we need to model how long the green

signal should be there or if you want to find out ah how much delay will be there for the

vehicle to go from the signal or whether there will be any pile up of the vehicle at the

signal we can actually model it using mathematical methods or we can actually use simple methods

to model it and then predict what will be the waiting time for a vehicle or when the

signal is about to change or how long the vehicle will wait at the signal or what will

be the average waiting period can be model using mathematical methods

so we will see how this can actually be done so now we know that a red and green cycle

will actually have one sixty seconds so we have one sixty seconds for a a red and green

phase so when it change to green there will be sixty seconds when it changes to red it

will be hundred seconds so during the green phase a maximum of sixty vehicles may pass

through the road junction since it can actually go through one second is needed for one vehicle

to go so we can actually have only sixty vehicle passing through during the green phase

so sixty vehicles during green phase whenever the signal changes to green you can have sixty

vehicle passing through ok now forty vehicles will arrive so during the hundred seconds

so one sixty seconds so totally we have one sixty seconds so we can actually except forty

vehicles to arrive during this phase so it will be having forty vehicles arriving during

this phase it should we can have sixty vehicle passing

through the green phase so that means ah there will not be any pileup of the vehicle so during

one green phase all the vehicle can actually pass through the green phase because sixty

can pass through and we have only forty vehicles arriving and ah this can actually be model

either by because its a simple system and its only for one ah case we can actually model

it just by writing down the table and then see how many vehicles will pass without any

waiting or what will be the average waiting time for each vehicle that can actually be

model by writing down the table so we if we assume that two seconds after

that is signal changes to red one vehicle is coming then six second then ten seconds

then fourteen second eighteen seconds so one vehicle will be coming at two second another

will be coming at six ten fourteen eighteen this vehicle which is coming over here it

has to wait till the signal changes to green so it has to wait for around ninety eight

seconds so the vehicle coming at ah two seconds after that is signal changes to red it has

to wait for ninety eight second so that is the delay time for the first vehicle and for

the second vehicle it will be ninety five because one second will be for the passing

of the vehicle and then will be having ninety two seconds waiting then will be having eight

nine and so on so we will be having in the total delayed

periods where the vehicle is coming at ah one thirty forth seconds you can see if you

write down the table you will see that one thirty forth second this has turn to green

and all the vehicle have pass so this vehicle will be having a zero seconds delay similarly

till one fifty eight in the last vehicle also pass without waiting at the signal so this

many vehicles so we will be having about ah seven vehicles passing through the signal

without have any delay those zero delay will be there and if you take the average delay

we can actually find out the average delay of the vehicle and you can calculate the average

delay from this one so that is one way of doing it but this again

it is writing about making the table and then calculating it now how do we actually convert

that into a mathematical equation and then do it or generalize it for any kind of traffic

problem so here we are assuming that four is fixed and one is fixed but if it is not

fixed then how do we model it you can actually use the mathematical equations and then model

this particular delay so we need to find out the average delay in this case the average

delay will be total delay so we have the total delay of the vehicle divided by forty vehicle

so we have forty vehicles coming during the phase so there will be sigma delay by forty

so that will be the delay so in this case you can see that the delay will be around

forty one point two five seconds that it is the average delay of the vehicle now if you

want to model this using ah mathematical equations we can actually the parameters and then create

a a mathematical model for it and then use that one for simulation of the system under

various situations so whenever they r g or r v changes so we

can actually model it easily without creating a table like this so in this case you can

actually see that r v is the phase o red phase so length of red phase

and similarly g v is the length of green phase and then a is the time between arrivals and

d is the time between departure the same as what we saw there r v was hundred

and ah g v was sixty and a was four and d was one

now if you want to find out the average delay average delay is one over n sigma delay that

is the total delay divided by n where n is the number of vehicles number of vehicles

will be equal to r v plus g v divided by a so this is r v r v plus g v divided by a will

give you the number of vehicles and the sigma delay over n will give you the average delay

now if you want to find out the sigma delay the sigma delay can be calculated s n is equal

to one over two n a plus l so this is it will be the relationship for

average delay where a is the first term in this that table what we saw in the previous

case that is a minus d and l will be the last term which is ah r v minus one now n can be

obtained as r v minus one divided by a minus d so using this relationship we can write

s n as r v minus one divided by two a minus d multiplied

by so this stands for the n and then a minus d plus r v minus one

that is the delay and if you simplify this you will be getting the s n as equal to a

multiplied by r v minus one divided by two r v plus g v multiplied by one plus r v minus

one divided by a minus d so this actually shows that any scenario any

dynamic scenario can be easily model using mathematical relationships and that can be

used for modeling that particular scenario and predicting the behavior and depending

on the ah requirement we can actually modify the parameter or we can think of changing

the system parameters in order to suit the requirements

so that is the advantage of using mathematical model for this kind of applications so in

the system design we will see many scenarios like this where we need to find out the ah

behavior the dynamic behavior of that system at a particular situations and then we need

to modify the system designs so mathematical tools will be really helpful for such situations

let us see some other methods also so we have few other methods also for ah modeling so

when ah analytical solutions are not available we go for a method called a finite difference

method or finite element methods so finite different method gives a point wise approximation

to the exact solution of a partial differential equation

again ah these are the two methods which can be easily use for analysis of the structures

especially when you want to find out the load acting and then the stress levels or you want

to find out the temperature distribution so this kind of application there are wide application

for this kind methods like finite element method and finite difference methods i am

not going to the details of these methods because you can find many resources for ah

learning about ah this methods but just want to tell you that ah these methods can be easily

employed for ah system design and whenever the requirement is there we can apply these

methods to do the analysis just to show you how it works in the finite

element methods the domain can be analytically model or approximated by replacing it with

an assemblage of discrete elements so that is what we an we have have an assemblage of

discrete elements to represent the system this has some examples or ah methods if you

have a structure like this physical system you can actually create a mathematical model

by idealization and then use different kinds of ah elements so in ah finite element method

we use various elements one d one dimensional element two dimensional elements or three

d elements can be used for modeling there are various ah softwares available ah for

doing this so we can actually ah conversion of the physical system to a mathematical model

is the first task once you have this then actually you can use ah standard software

for simulating it and finding out the behavior ah these are some other examples this finite

element methods are widely used in design activities so for the in especially in the

case of aircraft design missiles and all so basically when you do the machine you will

be getting a structure like this you will be having different elements and we can find

out the stress or temperature or any other parameter using from this elements and then

you can plot it to find out how it will be ah affected by various parameters

so this is a finite element model of an aircraft and the last system i would like to discuss

is the bond graph methods as i told you ah bond graph is a physical ah system based ah

modeling technique so here we will look at the physical system we dont convert that into

a direct mathematical equation what we do is ah we will try to create a graphical representation

of the physical model and then using a method called bond graph we can actually a represent

the physical system directly and then can be simulated

so this was developed by a person called painter ah long ago and it is ah widely used by ah

professionals and there are few softwares available for using this method to simulate

the particular system when we model the system using bond graph so here in bond graph method

the exchange of power between two parts of a system has been considered so we always

assume that whenever there is a system there will be some exchange of power so the power

interaction between the elements is considered here and thats why sometimes it is known as

power bond graph method also the flow of power is represented by a bond so whenever you have

a an exchange of power he represent that by a bond an effort and flow are the two components

of power so there are two parameters we need to define

to get the power that is the effort and flow the main difference between the classical

approach for modeling a physical system and the bond graph modeling is ah shown here in

this ah diagram as you can see here in the classical approach for modeling ah we start

with the physical system and then we create an engineering model using some mathematical

methods or we convert that into a simplified an abstract ah model and that is the engineering

model and then we write down the differential equations to corresponding to that model and

then we use some standard methods like block diagrams and then simulation language to simulate

it and get the output but in the case of ah bond graph modeling

we dont go through this stages we keep some of this stages we have this physical system

then we create an engineering model using converting of the ah physical system by taking

abstracting that we an engineering model and then we create a bond graph so we actually

we dont write down the differential equations or go for the block diagrams we convert the

engineering model to a bond graph and then this bond graph will be used in the computer

through some software and simulated and will get the output

so that is the difference here in bond graph modeling we dont need to know the differential

equations corresponding to the particular model or we dont really need to worry about

what kind of equation to be used how to solve this equations so all this are done using

the bond graph method the only difficulty is that we need to know how to create the

bond graph for this engineering model now direct equations here only the graphical representation

so we will ah create ah the bond graph from the engineering model and it will be used

for generating the differential equation through the software so the software can actually

do the job of creating the differential equations and then simulating it

so we dont ah really going to this part of this modeling in the bond graph method so

that is the advantage of using bond graph modeling ok so the methods used an bond graph

we need to learn some of the basic ah elements used in bond graph in order to understand

how the bond graph modeling can be used for ah system simulation or how the engineering

systems can be simulated using bond graph there are some basic elements as i told you

there is a parameter called effort and flow and apart from this we have many other generalize

parameters what we do today is to look at those generalize

parameters and then we will ah stop after seeing the parameters and then we will ah

see how to use these parameters in the system modeling in the next class so i will just

explain you about the generalized variables as i told you initially the power variables

are effort and flow so you have a effort variable and the flow variable effort is represented

by e and the flow is represented by f so this is the effort representation e and flow is

denoted as f then we have two other parameters known as

energy variables the energy variables are the momentum and displacement so the momentum

is represented using p and displacement is denoted by q and there is a relationship between

this p q and ah e and f the power flow in the system is given as power is a effort multiplied

by flow so whenever there is a power flow taking place that is actually a product of

the effort and the flow so what whatever maybe the effort and ah flow the power actually

can be calculated ah by taking the product of this effort and flow and this momentum

is basically a integral of the effort so p is obtained by integral e d t and similarly

q is obtained by integral f d t that is the integral of flow gives the displacement and

integral of effort gives the momentum so integral of effort gives the momentum as you can see

here this is if you take the variables ah you can represent them in various domains

as i mentioned ah earlier ah bond graph is basically used for multi domain systems and

you have mechanical systems electrical system hydraulic system and their combined we need

to have a common language to use for modeling so here the effort and flow are common in

all the domains so whatever maybe the domain whether it is ah mechanical hydraulic or electrical

we can use the same variables effort flow momentum and displacement this ah chart shows

the variables effort and flow and different domains so we can see the electrical domain

the effort is voltage and flow is current and voltage into current will give you the

power so multiplication of the effort and flow will give you the power and then translational

the force is in the translational motion the force is the effort and velocity is the flow

and ah rotational motion torque is the effort and ah angular velocity is the flow and in

hydraulic pressure is the effort and the volumetric flow is the flow

similarly for chemical chemical potential and molar flow thermodynamics temperature

and entropy so as we can see here whatever maybe the domain you can still represent the

effort and flow and that actually helps us to have the a common modeling language for

multi domains so i would like to stop here ah today we will see the methods by which

how do we actually use the bond graph for modeling the engineering system and what are

the basic modeling elements used in bond graph we will see all these details in the next

class so till then goodbye to all of you

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