Bond Graph Modeling of Dynamic Systems l Modeling Techniques l Alison
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Module 1: Bondgraph Modelling & Decision Making

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Bond Graph Modeling of Dynamic Systems

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today we will discuss about the Bond Graph Method of System Modeling. Inthe last class, briefly explained the bond graph methods what is the importance of this methodand how this method can be used for modeling of dynamic systems. Today, we will go in tothe details of the modeling techniques; what are the important elements in bond graph andhow do we develop a bond graph model for a dynamic system and then how can it be usedfor simulating the dynamic behavior.So, as I explained in the previous few lectures we need to model the behavior of systemsunder various stages of design in order to understand the behavior of the system how weresponds to particular inputs what kind of controls need to be incorporated and what kind oftuning is to be provided. In order to get a desired the outputs we need to use the modelingmethods and bond graph is one of the methods which can be easily implemented for systemdesign as I explained earlier, bond graph method is basically used for multidisciplinarysystems where you have multiple domains like mechanical systems electronic systemselectrical systems software computers and hydraulic systems.So, when we have a system with multiple domains or multiple specialization we need to havea common language to model them and bond graph provides you this common languagewhich actually across the domain, we can use the same kind of elements to model the systemthat is one of the advantages of using bond graph methods.
(Refer Slide Time: 01:57)
So, as I explained in the previous class, this is actually discuss about the exchange of powerbetween 2 parts of a system. So, whenever there is a dynamic behavior of a system there is anexchange of power between elements. So, we are trying to model this power transfer ormovement of power or transaction of power and that is why sometimes it is known as powerbond graph method also and here the flow of power is represent by a bonds a bond isrepresent by a half arrow. So, that direction is represented by the arrow and flow isrepresented by bonds and as I mentioned effort and flow are the 2 components of power.(Refer Slide Time: 02:38)
And this diagram, I explained the difference between the normal or conventional way ofmodeling and the bond graph modeling. So, here basically we do not go for the differentialequations instead we directly go for the bond graph models or direct from the physicalsystem, we develop a an engineering model of system and then convert that in to a bondgraph and then we can use the standards software for simulation. So, that is the maindifference in bond graph and the classical approach.So, we will actually jump through this differential equation, we do not go through thedifferential equations we will bypass this because creation of equations and the blockdiagrams will be carried out by the software itself.(Refer Slide Time: 03:25)
So, we have standard softwares to do the job again coming to the generalized variables forthe system we have 2 important variables one is known as the effort the other one is the flow.Effort is normally represented by e and flow is represented by f they are time dependentvariable.So, we will write that as e (t ) and f (t) . So, these 2 variables actually represents the flowof power the power is obtained by multiplying this effort and flow. So, that actually definesthe power transfer between 2 elements the other 2 variables are a momentum denoted by p tp(t) and the displacement denoted by q t q(t) . So, we can see here the momentum isrepresented by p and displacement by q and they are not independent the basically p is anintegral of efforts.
So, p is obtained byp=∫e ⋅dtandq=∫f ⋅dtthat is integration of f will give you the displacement integration of effort will give you themomentum for example, if you have f is the velocity then integration of velocity gives youthe displacement. So, that is the relationship between the variables f and q and e and p and thepower transfer isPower=e (t )∗f (t)(Refer Slide Time: 04:44)
So, these are 4 generalized variables used in modeling to explain the energy flow. So, theenergy flow is represented by as I explained by a bond. So, they are known as directedharpoons. So, the energy flows are represented as directed harpoons the 2 adjugate variablesthat is the effort and flow are annotated above and below. So, the e is actually above the bondand f is below the bond or the other way you can say that the hook was the harpoon alwayspoints the left and the term above refers to the side with the hook.
So, when this is harpoon and this is the hook and this is the above and this is below. So, whenit is in the vertical direction then e will be on the left side and f will be on the right side.(Refer Slide Time: 05:31)
So, this is the normal convention of representing effort and flow for various domain formultiple domains this effort and flow can be defined for example, for electrical effort isvoltage and flow is current for translational motion force and velocity rotational motiontorque can angular velocity hydraulic pressure and then volumetric flow chemical is chemicalpotential and molar flow thermodynamics is temperature and entropy flow. So, we can seehere for various domains we can still have the effort and flow parameters whether it iselectrical system or hydraulic system or a chemical system you can actually represent theeffort and flow irrespective of the domain.So, we do not need to worry about the domain because effort and flow is always the samerepresentation can be used to represent a voltage or a pressure. So, e will be the common forvoltage or pressure. So, even if there is a change in the domain still the e and f therepresentation remains the same. So, that is the advantage of using bond graph.
(Refer Slide Time: 06:31)
Now, there are some basic elements in bond graph modeling. So, these are the most basicbuilding blocks for making the bond graph the first element is the inductance or I. So, this isactually represents the I element represents the electrical inductance or mechanical mass orinertia. So, any electrical inductance mass or inertia can actually be represented by anelement I. So, that is shown in the diagram here. So, this is the representation of an I element.So, we have this I and then a hap bond representing the I element.Then another element is C or the capacitance. So, this one as inertial element or this is thecapacity element. So, this is actually for electrical capacitance or mechanical compliance. So,if we have a spring or a electrical capacitance that can actually be represented by a element Cand then R for electrical resistance or mechanical viscous friction here that is the resistanceelement R. So, these are the 3 basic element of bond graph I C and R. So, I represent theinductance or mass or inertia C represents the capacitance or compliance R represents theresistance or friction forces. So, this are the 3 basic elements and then we have some otherelements called TF; TF represents the a transformer.So, whenever there is a transformation of energy from one form to other form or one somerotary motion to a linear motion or you can have a gear box the rotary motion is transform toanother rotary motion with a ratio of power then we can represent it by an element called TF.So, here you can see that TF is e 1 the flow is e 1 and f 1 are the inputs. So, effort one and
flow one then you have effort 2 and flow 2 and this TF M represents the transformer ratio andthe relationship between e 1 and e 2 is given here
e2=1m∗e1
f 1=1m∗f 2that is the relationship between e 1 e 2 and f 1 f 2.So, here e 1 is converted to e 2 and f 1 is converted to f 2 through a transformer you can thinkof a gear box where there is a change in speed as well as change in torque after the before andthe after gear box. So, that relationship can be represented using a element called TF and thegear box ratio can be represented as M the ratio of gears represented by M and therelationship e 2 is 1m∗e1 or we can say e 1 is equal to m∗e 2
And 1m∗f 2 f 2=m∗f 1 and there is another element called gyrator this actuallydifferent from the transformer. So, in this case the relationship is between e 1 and f 2 and f 1and e 2 basically the effort is converted to a flow and this flow is converted to a effort that iswhy we use a gyrator for example, if you have an electric motor. So, we give a current andyou will get it as a torque output. So, the flow parameter is converted to a torque parameter.So, here is a conversion from flow to effort. So, that kind of relationship is represented usinga gyrator element. So, we can see even related to f 2 and f 1 to e 2 and that relationship isgiven as f 2 e 1 over d e 1 and f 1 is 1 over d e 2.
f 2=1d∗e1
f 1=1d∗e2
So, you are relating the f 2 and e 1 and e 2 and f 1. So, that is the relationship for gyrator andthere are 2 other elements source effort and source flow source effort is effort source that isyou provide a effort to the system as a source effort then that is represented by an s e elementfor example, a voltage source voltage source can be represent by a a source effort. So, here
the source is the effort and that is given to the system then you represent that as a effortsource, then you have a source flow or the flow source which is represented by SF where youhave a current source or a flow source like a hydraulic flow or any other flow input, then werepresent it is in SF. So, you have source effort and source flow which represents the sourcesfor the system that is the input sources with the system an effort source and a flow sourcethen you have a gyrator which represents the relationship between the effort and flow andthen the input effort and the output flow similarly input flow and the output efforts. So, thatcan be represented using a gyrator element.And then you have the transformer element which actually relates the efforts that is e 1 and e2 and f 1 and f 2 and we have other power consuming elements I C and R. So, I C and R arethe power consuming elements these are the power transfer elements and this is the sourceeffort and the source flow elements. So, these are the basic elements you get for bond graphmodeling.(Refer Slide Time: 11:39)
We have 2 more elements which actually represents the junctions in the bond graph. So,whenever you have a junction that is whenever you have different sources coming to onepoint and then it gets separated so that can be represented using bond graph junctions. So, wehave 2 kinds of junctions 1 is known as a 0 junction the other one is a 1 junction. So, you cansee here 0 junction and 1 junction. So, 0 junction is known as a common effort junction. So,the 0 junction the; we said this is a common effort junction because all the efforts are equal in
this junction. So, we have 1, 2, 3, 4, 5 bonds here and these 5 bonds all this bonds are havingsame effort. So, e1=e2=e3=e4=e5So, here you have all the bond all the efforts equal and the flows sum to 0. So, the flowswere coming inside. So, this arrow represents this flow is coming inside. So, 1 and 2 arecoming inside and 3, 4, 5 are going outside. So, whatever the flow is coming inside should beequal to flow going outside that isf 1+f 2=f 3+f 4+f 5that is the flow sum to 0. So, that is the 0 junction. So, you have a common effort and theflows adds to 0.This is something like you have a voltage source many elements are connected in parallel tothat voltage source, then we will be getting same voltage from across all the elements, but thecurrents will be different. So, this actually represent that kind of a scenario then you haveanother junction called one junction the difference between 0 and one is that 0 is a commoneffort junction, but one is a common flow junction here you have the flows are equal thoughthe flows in these 3 bonds that is 11 12 and 13, they are all equal. So,f 11=f 12=f 13So, all the flows are equal here and efforts sum to 0. So,e11+e12=e13So, you can see these 11 and 12 are coming to this junction and 13 is going out.So, you have e11+e12=e13 So, that is the one junction; now with these elements that is0 and 1 junctions and you have the source effort and source flow then you have thetransformer gyrator and I C and R elements. So, these are the only elements you need tomake a bond graph model of any system whether it is electrical system mechanical system orhydraulic system or a combination of this we can model the system using only this elements.So, we have 3 power consuming elements I C and R then we have the transformer andgyrator and then source effort and source flow and 2 kinds of junction 0 and one with thiselements we can actually model the complete system using bond graph methods.
(Refer Slide Time: 14:52)
We will see few examples how do we do that, but before going to that we have one andimportant function to be defined which is known as the causality in the system. So, causalitybasically it actually explains what actually causes the system to perform. So, whether there isa input is effort or a flow in to that system which actually causes the system to behave. So,that is known as the causality. So, every bond defines 2 separate variables the effort and flowand consequently we need 2 equations to compute the values of for these 2 variables.So, we need to have 2 equations for effort and flow it turns out that it is always possible tocompute one of the 2 variables at each side of the bond. So, you can actually compute one ofthese variables at one side of the bond and a vertical bar symbolizes the side where the flowis being computed. So, here you can see that the a vertical bar is shown to show the causality.So, wherever this effect vertical bar is shown it shows that the flow is calculated on that sideand effort is coming on this side. So, here flow is calculated this side and effort on this sidethat is the representation for the vertical bar or the causality of the bond there are mandatorycausality for sources like TF GY and 0 and 1 junctions and there is a desired causality for Cand I elements and free causality for R element I will explain this how do HD give thiscausality shortly.
(Refer Slide Time: 16:16)
So, here is the explanation for causalization of the sources. So, now, we have this Se and Sf.Se is the source efforts and Sf is the source flow. So, the flow has to be computed on the rightside because effort is coming here. So, we need to calculate what is f? So, f is calculatedbased on the effort. So, you have the effort and it is a it is calculated based on what is theeffort coming here calculate f is equal to a function of the effortSimilarly, here source flow; so the flow Sf. So, the causal stroke is here. So, therefore, theflow has to be calculated here and effort is to be calculated here. So, the source computes theflow here the source computes the effort. So, the source computes the effort here and here thesource computes the flow. So, that is how we give the causalities stroke and these are themandatory causality for source effort and source flow. So, the for the source effort you willalways have a causality stroke here and for the source flow you will have a causal stroke hereat the other end or the left side of the bond. So, here the causality of the sources is fixed Seand Sf you cannot change the causality because here the source computes the effort and thenhere the source computes the flow that is why this already fixed.
(Refer Slide Time: 08:39)
And for other elements to fix the causality for the other elements we need to actually find outthe relationship for these elements. So, here you can see this is the R element and for Relement we can actually have a relationship likeu=R∙i
i=uRthat is how you calculate the.So, here u actually represent the effort and i represent the flow it is the voltage and current.So, now, you can actually supply a current to this element and get a voltage drop. So, i isgiven as in input and R is given as the output. So, actually you can calculate u is R multipliedby I that is the relationship for this causal stroke. So, in this case the causal stroke is on theleft side then the relationship isu=R∙i .If the causal stroke is on the right side then the relationship is that the; i is computed u is theinput. So, u giving a voltage to the resistance and then you are getting a current flow output.So,. i=uRSo, here we are calculating i on this side and here we are calculating i on this
side. So, that is i=uR. So, if you are calculating it on this side then this i=uRand here it
is u. So, u=R∙i . So, the causal due of resistor is free.So, depending on the situation again actually provide the causality on this side or this sidejust this free causality for R elements then coming to the C element here actually you cannothave various causalities because you can have only one kind of relationship here you can seethis is at differential relationshipdudt =icSo, normally we can actually write it as the
i=1c∫u⋅dtSo, that is the relationship you can actually find for this. So, in this causality you will begetting an integral relationship for i we should not go for the differential relationship if youhave differential relationship which is difficult for us to visualize the system and therefore,we go for an integral causality and integral causality actually gives us stroke over here whichactually tells us that the effort or the u is 1c∫u⋅ dt . So, we can actually find out the outputusing the integral relationship similarly we can actually for the industrial elements also youcan go for the integral relationship where u is obtained as u=1c∫i⋅dt . So, the flow isobtained as integral of I d 1 over I integral of dt can be obtained. So, that they actually givesyou an integral relationship with this causality.I will explain it little bit more when we take some examples. So, we can understand that thisis the integral causality for C and this is the integral causality for i. So, here actually the icalculation is represented here. So, i=∫uI⋅dt and here u=∫ic⋅dt I that is the flowthat is the integral causality for C and I elements of course, in some cases you may find thatthere are there are becomes a differential causality, but then there are methods to solve thatone by providing some additional elements, but in normal cases we always try to provide aintegral causality for C and I elements that is about the causality of R C and I elements.
(Refer Slide Time: 21:04)
So, this is the relationship for I
f =1I∫edt
something similar to the relationship F=ma . So, here f the flow; flow is equal tof =1I∫edt . So, the flow is calculated here. So, the effort is the input. So, the effort iscoming here and the flow is calculated here f =1I∫edt that is the integral relationship you
are getting and for C elements you can see the
e=1C∫f ⋅dt
is the relationship that is the flow is calculated here. So, effort 1C∫f ⋅ dt dt gives you the
effort relationship. So, e=1C∫f ⋅dt that is the relationship and this integral causality ispreferred when given a choice. So, whenever you have a choice you should try to go for theintegral causality for these elements.
(Refer Slide Time: 21:56)
And the final one is the causality for the junctions 0 and 1 junction as I told you 0 junction isa common effort junction. So, there can be a only one effort coming in to the block to thisjunction and therefore, we can see that this is the effort coming where the flow is calculated.So, this is the effort coming in that one stroke over here and all other bonds will be without acausal stroke on this end then there will be having causal stroke on the other end. So, that isthe causalization of 0 junctions.So, the junctions of type 0 have only one flow equation and therefore, they must have exactlyone causality bar because this there can be only one flow equation we can that is why this isactually one stroke over here and for the one junction junctions of type one have only oneeffort equation. And therefore, they must have exactly n minus one causality bars. So, herewe can see there is only one flow and there is it is a common flow. So, one flow cominginside and then that is common for all the others therefore, will be having causal stroke on allother bonds near to the junction and one without any causal strokes that is the causality forone junction.So, we can see that f 1, f 2, f 3 are equal and e 1 is equal to e 2 plus e 3 because this is inputeffort this is the output. So, e 1 is equal to e 2 plus e 3 that is the causal stroke for causalityassignment for 0 and 1 junction. So, we have fixed causality for source effort and source flowthen we have preferred causality for I and C elements and we have a flexible causality for Relements and we have fixed causality for 0 and 1 junctions.
Now, the last one to be seen is the TF and gy elements. So, the TF and gy it is not on here inthe slide I will explain it to you using the boards. So, we go to the board and then see how dowe actually represent these elements and develop the bond graph model for physical systems.
So, as I explained we have this I element with a causality like this and then we have this Celement with a preferred integral causality. So, C element and then we have an R element. So,you can have a causality over here or you can have it here depending on the situation andthen we have this source efforts. So, this is the source effort and then we have the source flowand we have these junctions. So, we have a one junction. So, in one junction as I told you it isa common flow junction. So, you can actually have one flow there is only one junction onlyone without a causal stroke and all other will be having causal stroke at this point and thenyou have this 0 junction. So, here can have only one effort and all others will be here.So, this is the one junction and this is the 0 junction and the other 2 elements are thetransformer and gyrator the transformer we can have a causality like this either like this; thatmeans, the effort the flow is calculated here and flow is calculated here or you can have acausality here also; so, any one of this. So, if this stroke is on this side then here also it will behaving on this side. So, you can have either this or you can have this for the causality fortransformer and for gyrator it will be like this can have a causality like this here actually theeffort is converted to flow here that is why you can have it here. So, this is one possibilitycausality and the other one is this one you can have a causality like this also.
So, you can have a causality like this or you can have causality like this for gyrator. So,depending on the situation you will have to identify what kind of causality exists for thatparticular element where it is connected to which junction it is connected to or which elementit is connected to accordingly we have to assign the causality for these elements. So, this arethe basic building blocks for the bond graph modeling. Now let us take a very simpleexample and then see how do we actually develop the bond graph model for such systems formechanical systems or hydraulic electrical systems; let us take the simple example shown inthe in the slide it is basically a mass spring damper system.(Refer Slide Time: 27:18)
So, you can see here this subjected to f force and then this is the displacement x(t) and if youcan assume that there is a reference velocity also xr
(t) of course, this is fixed then it will
be 0; that means, it can be actually represented x(t) and xr
(t) as the reference and this is themass and this is the spring constant k and this is the damping coefficient B. So, this is themechanical system if you want to develop a bond graph model for this we know that if youdo for a mathematical modeling you have to write down theM ́x+Bx ́ +kx=Fwe can write this is equation and then we need to simulate it and then find out the outputs.But in the case of bond graph we do not need to really go for the mathematically equationswe can actually directly write down the bond graph or the we can make the bond graph and
then directly simulate it how do we do this basically we identify there is a force input to thesystem; so will write down that as a source effort. So, we know that a force input is a sourceeffort. So, it is an effort it is given as a input. So, this is a source effort and we have acausality over here. Now for the time being I am assuming that it is 0 there is no velocity overhere that is 0. So, I do not consider that part here now I know that this mass this force isacting on the mass and then this having a velocity displacement x which actually we canwrite ́x is the velocity here and then this k is having a displacement. So, this is actually thisspring is subject to a displacement of x and B subject to a velocity of ́x .So, we are having only one velocity here which is actually the ́x is the velocity. So, wehave a common velocity here which is actually k and B elements they are subjected totherefore, will be having a one junction over here because there is a common velocityjunction now will look at now there is a common velocity here and there is a mass attached tothis. So, we write down we make a bond for the mass that is I element which is equal to Mand then it is connected to k and B the same velocity. So, therefore, we can actually connect itto a C element which actually represents the spring stiffness and then we have a R elementwhich actually represents the damping in the dash board.So, that is all what we need to do we have completed the bond graph model for this systemand what we need to do is to just give the causality assignment we know that I has got a fixedcausality as you can see here I has got a fixed causality and then I has got a causality fixedcausality C has got a fixed causality and then R can be given like this because a one junctioncan have a only one bond without a causal stroke at this point it is a common velocityjunction and this velocity at here is the ́x . ́x is the velocity at this junction. So, wehave these source efforts. So, we know that this is a force coming and then this is a R this isthe effort this is the velocity of this coming. So, here it is. So, this flow; so, we have an effortcoming here.So, if I number this 1, 2, 3 and 4; I can write it as e 1, f 1, e 2, f 2, e 3, f 3 and e 4, f 4, I cancalculate f 2; f 2=1m∫e 2 dt . So, that is the velocity or f 2 here f 2 is nothing but ́x .
So, we have this f 2 which is nothing but the junction velocity ́x .So, ́x is calculated and then all others are because of this now we
have we know that ́xor this f 2,
f 1=f 2=f 3=f 4So, we have all the flows are equal here. So, we have this relationship and now we knowwhat is e 3 we can actually calculate what is the friction forces here e3=1c∫e3dt and
similarly e 4=R⋅f 4
So, that is the relationship here. So, we know that all this effort can be calculated e 3 e 4 andwe can calculate the flow also; that means, once we know this bond graph we can actuallycalculate all the parameters that what is the velocity of this what is the displacement andvelocity of this mass what is the force acting on the spring that is obtained by here this is theforce acting on the spring and what is the force due to friction again we are gettinge 4=R⋅f 4 .

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