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In the lastfew classes we have been discussing about the modeling techniques employed forengineering system design, what kind of engineering methods can be used at various stagesof system design. We discussed about the data modeling, process modeling and behaviormodeling and today we will talk more about modeling of physical systems especially thedynamics of physical systems.Whenever we make some product or whenever we design a product we need to look at theperformance of the product even before we really make it. So, in order to make sure thatwhatever the product we are designing should work as per the requirement; we need to createsome models and then test it and ensure that it actually performs the way we want in somecases it may be a simple just to check the shape and it is weight and other features, but insome other cases it may be the performance parameters like the acceleration time or thedeceleration time or the time to reach a particular state or what will be the maximumattainable parameter we can achieve.So, these are the things which we need to analyze using models. So, the modeling andsimulation basically the system modeling and simulation looks at the possibility ofdeveloping models for the physical systems and testing them through simulations to studytheir behavior we can actually use various kinds of modeling techniques you can use thephysical modeling or iconic model where you actually we make some small prototypes. Andthen text them or we can actually do some experiments simple experiments and then studythe behavior, or we can actually create some mathematical models and apply some nonmathematical techniques to simulate it and study the behavior.So, there are various ways of doing this modeling and simulation. We will look at some ofthese methods and then see how this can be applied for system engineering of course, thefocus of this course is not to go into the details of these modeling techniques, but basically to
tell you that these are the modeling techniques existing and what are the basic principlesapplied for such modeling methods.So, we will be briefly going through these methods, but not into the details or in depth studywill not be carried out, if you want to know more about those methods you can always referto some other courses which are readily available or methods available on the web or thereare some video courses also available. So, you can use any of these resources and learn moreabout these methods.(Refer Slide Time: 03:01)
So, let us look at the methods existing or you need to see why we actually we need themodels and what kind of models can be used.So, here the uses of models in engineering are basically to seek answers to some of thequestions. Will the system as designed will work. So, we are designing some system and weneed to see whether it will really work or not and which of the two system designs is thebetter suppose you have many designs or alternatives are existing. So, which one will reallywork and do I adequately understand the system, and as well as what kind of trade off I canhave. So, these are the basic purposes of making the model. So, we need to check whether thesystem will work or whether the understood the system completely or whether we can have achoice between two models or whether we can have a trade off in some of the aspects.
(Refer Slide Time: 03:54)
So, this can be studied using engineering models. And coming to the methods of modeling wehave various methods of modeling the system or the physical systems. So, you can go for aheuristic modeling basically depending on our on heuristics and our own intuition, andimagination we can actually create the models and then test them to see the behavior. And theother one is known as the mathematical modeling in mathematical modeling we can actuallywrite down the mathematical equations corresponding to the physical model and then studythe behavior.Another method is known as physical system modeling or bond graph modeling. So, we cansee the physical system modeling method is bond graph is one of the methods of physicalsystem modeling, and then this bond graph method we normally what do is look at thephysical system and develop the corresponding model which actually have a one to onecorrespondence with the model and that is easy to understand. And this method is used formulti domain systems when you have a system with various disciplines like a mechanicalengineering, electronics communication software.So, for such systems we can use the physical system based modeling what bond graph is oneof the methods. Then there are other methods like dimensional analysis the numericalmodeling you might a heard about finite difference and finite element methods and these arethe modeling techniques we can employee for modeling of the system, and then once youhave 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 asfrequency domain analysis to find out the behavior.So, the model is basically converted to a simulation model the physical system model or themathematical model will be converted to a simulation model and using the simulation modelwe can predict the performance in time domain or in frequency domain and then see whatkind of parameters we need to modify in order to get the desired performance. So, that is theuse of modeling and simulation in system engineering. So, this is normally used when we gofor the actual design of the system. So, initial design stages like a functional decomposition,and the design of architecture we do not go for this kind of analysis, but when we really makethis design the components and the sub systems we need to check the performance of thesecomponents and sub systems and we go for this modeling and simulation and then analyzethe performance.Let us go through one or two modeling methods to show how this can be used for ourapplication.(Refer Slide Time: 06:22)
Heuristic modeling as I mentioned it is a common sense or minimum cost physical modeling.So, here we do not go for any detailed modeling of the system we apply some common senseor make some 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 tell you a case study where this kind of minimum cost physical modeling can be usedthis was a real situation or a real project where actually some common sense modeling wasapplied to get the work done without going for a detailed analysis we mentioned about theatlas missile project in one of the lectures. So, this was a project managed by us air force andcorps of engineers for developing a series of family of missiles for US air force. So, thismissiles need to have some installation support; so the install missile and supportingequipment in a vertical underground concrete silo.So, the supporting equipment need to be inserted or to be placed on a underground concretesilo missile to be lowered to the silo through open doors at the ground. So, once you have aconcrete silo we need to lower the missile to the concrete silo through a open door at theground one of the propellant lines that is prefabricated piping sections could not bemaneuvered into the propellant systems shaft. So, there was a propellant system shaft whichactually should carry the propellant lines, but they were not able to maneuver this propellantsystem 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 intothe silo because the propellant lines were actually obstructing the moment of the systemshaft, but they want to do the complete section then it would I mean cost around 300,000 usdollars for 70 sites. So, that was the problem if they could not do this to they could notmaneuver this particular shaft into the silo then there were to redo the complete section andfor all the sites it may cost around the 300,000 dollars.So, they could not do it for many days and they were trying what to do with this particularproblem and then this heuristic modeling method came to their help they created a verysimple model of the pipeline as well as the shaft and then tried on the ground many ways ofinserting this into the silo. So, they maneuvered this shaft under various orientations andvarious positions, and they would find that there is one position where actually they can bringthis into the shaft or the shaft can be brought inside the silo.So, this was tested using very simple prototype very low cost model and they found out themethod to insert it and then they simply applied the same principle and then they couldmaneuver this shaft in to the silo and could solve the problem. So, this kind of modelingbasically comes from the common sense of the users the designers think about the allpossibilities and then make some simple models and then create a scenario work actually they
want to solve the problem and get a solution. So, that kind of modeling is known as theheuristic modeling.For example if you have a large cupboard to be moved out of the room and you have a smalldoor to be there to take it out then we need to look at how do you actually take this cupboardout of the room, may not be possible to simply take it as it is. So, you may have to tilt it somesite or you need to make some particular angle and somebody has to move forward and thentake a turn and then the other person has to come and take a turn and then tilt the table or thecupboard in a particular direction to take it out.So, those kinds of things cannot be modelled or simulated using mathematical methods weneed to apply some common sense technique and do this. So, the same principle applies forsystem engineering also. We can actually do some common sense modeling and sold many ofthe problems which we really faced on the ground when really implementing the system. So,there kind of methods are known as heuristic modeling.(Refer Slide Time: 10:37)
So, this kind of modeling the physical modeling provides a grasp of the problem that cannotbe achieved by any other techniques. So, if you apply this common sense methods or theheuristic modeling technique we can actually get the grasp of the problem were othermethods cannot be applied. And crude models can be easily develop from basic materials atminimum cost. So, in order to do this we can actually create some crude models and using
basic materials and very low cost and a good bit of caution needs to be applied to anyconclusions reached.So, whenever we do this kind of heuristic model you cannot make any generalizedconclusions that particular solution may be applicable only to that situation. So, bit of cost isneeded when we do the this kind of modeling and make some conclusions out of this. So, thatis the first method of heuristic modeling.(Refer Slide Time: 11:26)
Then we can actually do different types of mathematical modeling also. So, a mathematicalmodel can be defined as a construct that comprises an abstract representation of a real system,models are constructed by people of10 for the purpose of system design. So, mathematicalmodels are abstractions of the physical systems when you have a physical system we try toabstract into a model using mathematical methods and their constructed by people often forthe purpose of system design and then computational simulation a mathematical modelimplemented in a digital computer.So, once you have a mathematical model we can implement in a digital computer and get thesimulations done. So, the computational simulation is basically a mathematical modelconverted to use in a computer and whether you can actually get the output. Of course the lawof nature a fundamental understanding of causality in a physical system often expressed inmathematical form or as an algorithm executable by a computer. So, whenever we make themodel 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 comesfrom the physics behind the dynamics or the system behavior.So, we need to look at those fundamental issues and then only create the mathematical modeland use them.(Refer Slide Time: 12:42)
So, let us 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 existingsystem or a mechanical system and then find out the abstract nature of that system and thencreate a mathematical model and then see how to write the mathematical equationscorresponding to the various performance parameters of the system, and then write down theequations and later on you convert the those equations into a form were the it can be acceptedin a digital computer and simulations can be done.So, we will just take one simple example a spring mass damper system and I will show youhow to model this system using the mathematical methods. So, let us take a very simple caseof a spring mass damper system.
(Refer Slide Time: 13:33)
So, we will have a spring and a damper. So, this can be representation of any mechanicalsystem. So, if you have a spring mass damper system like this and a force is applied here, aforce of F is applied and if this is the stiffness parameter of the spring and this is the dampingparameter of the dashboards.You can actually represent the displacement of this by parameter x. So, if you want to findout the displacement of the mass and you apply a force and what will be the dynamic natureof the response we can actually model this using mathematical relationships and then convertthat into a simulation model now if you want to know this relationship then we can writedown the equations of motion in terms of the force balance. So, you haveF=M ́x+b ́x+kxWhere x is the displacement of mass and M is the mass of the body and b is the dampingparameter and k is the stiffness, this is the time domain relationship. So, this is ́x stand forthe acceleration ́x stands for the velocity and x displacement for the displacement; inorder to sollve this and to find out the relationship for x.
(Refer Slide Time: 15:12)
So, we want to find out the displacement x in terms of time domain. So, x (t ) need to befound out we can actually change this to the Laplace domain to solve it. So, if you do thisthen he will be getting it asX (S)=F(S)M S2+bS+k
So, the S stands for in the Laplace domain. So, F( S) is the force in the Laplace domain Mis the mass and b and k.Now, we can actually solve this for using standard mathematical formulas or you can actuallyput this in the
X (ω)=ωn2S2+2 ε ωn+ωn2
where ωn is given asωn=√kMSo, we can actually get this as in, this can actually be converted into a time domain and thenyou can actually you can solve this and then converted into time domain. We can actually use
any standard simulation softwares to simulate this behavior this equation can be easily solvedI can actually simulate it.Similarly, in the time frequency domain also we can get the output using this relationship. So,now, for any applied force F, we can find out the response of this system the displacement ofthe system displacement velocity and acceleration can be easily obtained from here.(Refer Slide Time: 16:55)
So, if you simulate this in the response can be something like this the time response can beobtained like this depending on the parameter values we will be getting various performanceparameter is like rise time, then the settling time, if all those can be plotted for a given forceyou will be getting a constant displacement here.So, this is the transient period. So, you can find out what is the time taken for the transient todie on. So, transient period and then you have a steady state period. So, using this equationsyou can actually simulate it you can use the software like MATLAB or a Simulink. MATLABor simulink can be used for simulating the behavior of this particular system. So, that is verysimple system and we are showing how the mathematical model can be generated for thatkind of systems.When you have a complex system we need to do is to take the abstract that complex systemto a mathematical model and write down the equations of motion for the system and thensimulate it and get the behavior. So, when I complex system can be model using this kind of
methods mathematical modeling techniques. If you have a robotic system or you want to findout the behavior of a machine tool any of these even if you have a simple motor you want tofind out the performance characteristics what you design. And then you want to see what kindof response time it has how much time it will take to reach a particular value of accelerationall those things can be modelled and then simulated will get the output using this kind ofmethods.So, that is one way of doing mathematical modeling not only this we can actually do manyother things also using mathematical methods. So, I will show you another one were actuallywe can model and simulate the traffic flow doing the peak time at a traffic junction. So, thatalso can be modelled using mathematical methods, or you can if you want to find out thedeflection of a beam we can actually do that if you have a beam like this a cantilever beam.(Refer Slide Time: 19:05)
And you want to find out the deflection suppose you apply a force over here or you hang amass over here.You want to find out what will be the deflection of this beam again we can we have standardequations for representing the deflection. So, we can actually find out what will be thedeflection under various situations depending on the type of load you can find out thedeflection as well as the bending moment bending stress all those things can be modelled andthen simulated to get the output. There are different methods mathematical modeling is one ofthe methods to get the outputs
Now, coming to a traffic problem we will see how that this kind of traffic problems also canbe model using mathematical method. So, this is the problem given the traffic lights at roadjunction are set to operate with a red phase of length hundred seconds. So, you have a redphase of hundred seconds. So, we can put it as r g as hundred seconds sorry red phase r.(Refer Slide Time: 20:04)
And then green as 60 seconds. So, you have a traffic signal which actually there is a redsignal for 100 seconds and green phase of for the 60 seconds and vehicle arrive at the trafficlights on average 1 every 4 second.So, every 4 seconds 1 vehicle is arriving. So, every 4 second 1 vehicle is arriving and whenthe lights turn to green the vehicles in the queue leave at a rate of 1 every second. So, this isthe arrival every 4 second 1 vehicle will be coming and when it is green every 1 second 1vehicle will be going. So, we need to model how long the green signal should be there or ifyou want to find out how much delay will be there for the vehicle to go from the signal orwhether there will be any pile up of the vehicle at the signal we can actually model it usingmathematical methods or we can actually use simple methods to model it and then predictwhat will be the waiting time for a vehicle or when the signal is about to change or how longthe vehicle will wait at the signal or what will be the average waiting period can be modelusing mathematical methods.
(Refer Slide Time: 21:19)
So, we will see how this can actually be done. So, now, we know that a red and green cyclewill actually have 160 seconds. So, we have 160 seconds for a red and green phase. So, whenit change to green there will be 60 seconds when it changes to red it will be 100 seconds. So,during the green phase a maximum of 60 vehicles may pass through the road junction since itcan actually go through 1 second is needed for 1 vehicle to go. So, we can actually have only60 vehicle passing through during the green phase.So, 60 vehicles during green phase whenever the signal changes to green you can have 60vehicle passing through now 40 vehicles will arrive. So, during the 100 seconds; so 160seconds. So, totally we have 160 seconds. So, we can actually except 40 vehicles to arriveduring this phase. So, it will be having 40 vehicles arriving during this phase.It should we can have 60 vehicle passing through the green phase so; that means, there willnot be any pileup of the vehicle. So, during 1 green phase all the vehicle can actually passthrough the green phase because 60 can pass through and we have only 40 vehicles arrivingand this can actually be model either by because it is a simple system and it is only for 1 casewe can actually model it just by writing down the table and then see how many vehicles willpass without any waiting or what will be the average waiting time for each vehicle that canactually be model by writing down the table.So, we if we assume that 2 seconds after that is signal changes to red 1 vehicle is coming then6 second then 10 seconds then 14 second 18 seconds. So, 1 vehicle will be coming at 2
second another will be coming at 6 10 14 18 this vehicle which is coming over here it has towait till the signal changes to green. So, it has to wait for around 98 seconds. So, the vehiclecoming at 2 seconds after that is signal changes to red it has to wait for 98 second. So, that isthe delay time for the first vehicle and for the second vehicle it will be 95 because 1 secondwill be for the passing of the vehicle and then will be having 92 seconds waiting then will behaving 89 and so on.So, we will be having in the total delayed periods, where the vehicle is coming at 134seconds you can see if you write down the table you will see that 134 second this has turn togreen and all the vehicle have pass. So, this vehicle will be having a 0 seconds delay similarlytill 158 in the last vehicle also pass without waiting at the signal; so this many vehicles. So,we will be having about 7 vehicles passing through the signal without have any delay, those 0delay will be there and if you take the average delay we can actually find out the averagedelay 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 thencalculating it now how do we actually convert that into a mathematical equation and then doit or generalize it for any kind of traffic problem. So, here we are assuming that 4 is fixed and1 is fixed, but if it is not fixed then how do we model it you can actually use the mathematicalequations and then model this particular delay.(Refer Slide Time: 25:06)
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 40 vehicle.So, we have 40 vehicles coming during the phase. So, there will be1N ∑ Delay140∑Delay=41.25 sSo, that will be the delay. So, in this case you can see that the delay will be around 41.25seconds; that it is the average delay of the vehicle. Now if you want to model this usingmathematical equations we can actually (Refer Time: 25:42) the parameters and then create amathematical model for it and then use that one for simulation of the system under varioussituations.So, whenever they rv changes. So, we can actually model it easily without creating a tablelike this.(Refer Slide Time: 26:06)
So, in this case you can actually see that rv is the phase o red phase. So, length of redphase, and similarly gv
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 rv was 100 and gvwas 60 and a was 4 and d was
Now, if you want to find out the average delay average delay is1N ∑ Delaythat is the total delay divided by N, where
N=number of Vehicles=r v+gva
So, this is
rv+gvawill give you the number of vehicles and the
1N ∑ Delay will give you the average delay.(Refer Slide Time: 27:42)
Now, if you want to find out the ∑Delay the sigma delay can be calculated,∑Delay=Sn=12n( A+L)
So, this is it will be the relationship for average delay where A is the first term in this thattable what we saw in the previous case that is a−d and L will be the last term which isrv−1 . Now n can be obtained as
n=r v−1a−dSo, using this relationship we can write
Sn=rv−12 (a−d)[ (a−d)+rv−1]
That is the delay and if you simplify these you will be getting the
Sn=a (rv−1)2 (rv+gv ) [1+(rv−1)(a−d) ]
So, this actually shows that any scenario any dynamic scenario can be easily model usingmathematical relationships and that can be used for modeling that particular scenario andpredicting the behavior and depending on the requirement we can actually modify theparameter or we can think of changing the system parameters in order to suit therequirements.So, that is the advantage of using mathematical model for this kind of applications. So, in thesystem design, we will see many scenarios like this where we need to find out the behaviorthe dynamic behavior of that system at a particular situations and then we need to modify thesystem designs. So, mathematical tools will be really helpful for such situations.
(Refer Slide Time: 30:42)
Let us see some other methods also. So, we have few other methods also for modeling. So,when analytical solutions are not available we go for a method called a finite differencemethod or finite element methods. So, finite different method gives a point wiseapproximation to the exact solution of a partial differential equation.Again these are the two methods which can be easily use for analysis of the structuresespecially when you want to find out the load acting and then the stress levels or you want tofind out the temperature distribution. So, this kind of application there are wide applicationfor this kind methods like finite element method and finite difference methods. I am notgoing to the details of these methods, because you can find many resources for learning aboutthese methods, but just want to tell you that these methods can be easily employed for systemdesign and whenever the requirement is there we can apply these methods to do the analysis.
(Refer Slide Time: 31:42)
Just to show you how it works in the finite element methods the domain can be analyticallymodel or approximated by replacing it with an assemblage of discrete elements. So, that iswhat we have an assemblage of discrete elements to represent the system.(Refer Slide Time: 31:54)
This has some examples or methods if you have a structure like this physical system you canactually create a mathematical model by idealization and then use different kinds of elements.So, in finite element method we use various elements 1D one dimensional element, twodimensional elements or 3D elements can be used for modeling there are various softwares
available for doing this. So, we can actually conversion of the physical system to amathematical model is the first task once you have this then actually you can use standardsoftware for simulating it and finding out the behavior.(Refer Slide Time: 32:32)