 Welcome to this very first lecture on structural dynamics. In today's class, I would be just giving you a brief introduction about this course and that will include the concepts that you would be learning as part of this course. Then I will give you a general idea of what is a static loading and what is a dynamic loading and what are the key differences between dynamic and static loading in terms of their effects on a structure. Then I would be discussing the problem statement that you would encounter as part of this course, so what problem to solve and then different solution methods that we could potentially employ to solve though that problem statement. So let us get started. In this very first lecture of structural dynamics, we are going to discuss about this course. What are the different concepts that we would be going through this course? What are the objective of this course? What is basically a simple difference between a dynamic system and a static system and then I would be presenting you the problem statement for this course that we would be studying. So without getting much delay, let us go through this course. If you remember from your undergraduate knowledge, you used to find out forces and deflection or different response quantities subject to a load say p0. So let us say a load p0 is being applied here on a beam of length L and the modulus of flexure EI. Now remember never at a point at any point of time you were told how the load p0 is actually being applied. So it was just said that a static load of p0 is being applied on this beam and then you needed to find out the response quantities such as forces and deflection. So my first question to you, can you tell me if I have to ask you how is this load p0 is actually applied or let us say in general how a static load is applied. So what I mean to say if you have to apply a static load of p0 on a structure, what would be the time variation of it? So that is my first question. My second question to you is what is the deflection of this beam subject to this point load p0. Now you know it from your undergraduate studies that if the load static load p0 is being applied, you calculated the deflection at the midpoint as the maximum deflection and the value was pLp0Lq by 48EI. Now the question is, would this load or would the deflection be always p0Lq by 48EI, okay? So that is my second question, okay? So what I am showing you here, the question is how the static load p0 is applied and I am showing you four options. If you had to represent the time variation of this load p for which the maximum value is p0, how would you describe this static load? Okay, so I have shown you four options, okay? So you need to select one of these options. If you would like, you can scan this barcode and open up a form which would show you all the option and then you can select the correct answer and then it would show you whether the selected answer is correct or not, okay? So let us discuss these four options. So in the first option, I have load p0, okay, which is being applied like this, a constant load p0. But do you think it is physically possible to apply a load like that? Like it suddenly goes to p0, okay? We will discuss about that. In the second figure, you can see I have a maximum load of p0 and then it decreases slightly so that it goes to a value of 0 over a certain duration. In third one, I have a load p0, okay, which starts at certain time, t1, it goes to p0, it is maintained over a certain duration of time, then it again goes to 0. Then in the third one, I have a load which is gradually being applied so that it reaches to a maximum value of p0 after a time duration. Let us say we will call this TR or the rise time, okay? So which one do you think can be used to represent this static load that you have studied till now, okay? If you think about it, the only load that could be or the role representation could be used is actually this ramp load here because if you try to apply this load or this load or this load here, you would see from the principle of dynamics that you would study later, they could never have a static load application through any of these representation, however, if you consider the fourth option here, depending upon the time value of TR, you might be able to apply a static load, okay? And we will come to that, what do you mean by static load and dynamic load, okay? How do you characterize actually, mathematically how do you characterize static load and dynamic load, okay? So this was the first thing you heard about, you used to do analysis subject to certain kind of load, let us say p0 is equal to 10 kilo Newton or 20 kilo Newton used to find out like, you know, the response, okay? But now you need to understand how the load p0 is actually applied. Now, let us come to our second question, okay? So my second question is, out of these four options or the four load representations that I have shown here, which one would give you a deflection that is most likely to be the value p0L cube by 48EI, okay? So you have to select one of these options depending upon the plot of p versus T that I have shown here, that would give you a value that is closest to p0L cube by 48EI. Now again, you can scan this barcode using your phone either through an app or through your camera, okay? And then you can submit the answer and see the correct answer right here. Now let us discuss each of these options that have been provided here, okay? So the basic difference between these three plots, A, B, C, what do you find the basic difference? It is basically the rise time that you see. So you can see TR is an intermediate, this first option has an intermediate rise time. The second option is the fastest rise or I can say like the smallest rise time, okay? And third is the largest rise time, okay? So what I mean by this is that this load is being applied fastest, this slowest and this is somewhere intermediate between these two, okay? Now I can simply discard this fourth option because this would never produce a value of deflection that is equal to p0L cube by 48EI, okay? So my, the computation is between these three options here, sorry, okay? So what you would see that out of these three, the loading that has the largest rise time would give you a load which is closest to being a static load and we will see why, we will see like you know mathematically we will also prove that, okay? But physically you just need to like you know imagine that if you apply a load very slowly then load is actually being applied statically and it does not produce any dynamic effect in the structure, okay? So this one is most likely to give you the value which is p0L cube by 48EI, okay? So now let us come to what basically defines a dynamic system, okay? So as we discussed dynamic system deals with the variation of a state of a system or structure with time, okay? Of course under application of external load, okay? So I will write under application of external load, so load, okay? So again I have shown you that representation, a load representation that is used that can be used to represent a statically applied load of the magnitude p0, okay? Now we saw that a parameter to determine whether a system would behave as a static system or a dynamic system under the action of external load is TR, right? We saw in the last slide depending upon how fast you apply or how slow you apply it might behave like a static system or dynamic system, okay? To demonstrate that I will give you a simple example, okay? So let me show you a spring, okay? Which has a mass m and k and another spring, okay? Again same mass, so both of these have same system properties but now I am applying a load p, okay? Of final magnitude p0, only thing in this one TR or the rise time is much, much greater than TR in the second one. So in this one load is being applied slowly and in this one load is being applied fast, okay? Now can you imagine, okay? Forget about mathematical formulation. Let us just talk about like, you know, have the feel of the system, okay? First develop appreciation from the real life examples then we will get into mathematical formulation. So in this case, as you can see if the system are same and I applied two load of final magnitude p0, however if one is applied slowly what will happen? This spring won't vibrate as much as if you have a load that is being applied suddenly. So in the first case load representation is something like this, in the second case load representation is much more sudden, okay? So I can say TR plays an important role in determining whether the system would behave as a static load or a dynamic load, okay? Now can you imagine some other parameter of the system that would determine whether the system is static or dynamic? For example, again consider a similar example except in this case, let us say mass, let me keep the mass same but I have K1 and K2 and in this case K1 is much smaller than K2. That means this is a flexible system and this is a comparatively rigid system. And now I apply a load which has the same rise time. So both load of final magnitude p0 applied over the same duration. Now can you imagine if it's a flexible system then it would vibrate more and if it's a rigid system it would vibrate less. Of course I am doing a relative comparison between these two. So compared to this system I would see that load p which has which is a ramp load it would lead to more vibration in a flexible system compared to a rigid system, okay? And I would come to technicality later like you know what do I mean by more vibration less vibration in terms of what response parameters but in layman terms just try to understand something like this. You can appreciate this example from a real life let us say a pedestrian bridges that you see for example like you know the Jhula bridges that you see in some villages you know it is also called a suspension bridge. So you have something like you know let us say bridge which is very flexible and supported at the end and then you have river going over like this, okay? You can imagine you try to walk very carefully very slowly here because if you jump it would start vibrating because this is a flexible system. So the dynamic effect would be more however if you have a rigid system like a concrete slab bridge okay which is much more rigid than this you don't need to be that careful as you would like you know in the case of a suspension bridge or like you know just a hanging bridge, okay? So the second parameter that determines whether the system would behave statically or dynamically is the flexibility of the system or more appropriate if you would see later is the time period. So it is mass and k. So these two parameter and if I represent time period as this is actually what determines the second parameter, okay? So depending upon the values of tr and tn we can decide whether the system would behave as a static or dynamic under the action of an external load, okay? So this thing you need to keep in mind, okay? We will come back to this later and we will try to do mathematical the derivation of whether the system would act as a static or dynamic, okay, alright? Now let us come to classification. So we saw that we have a ramp load here, right, okay? And if we increase the value of tr to like in a very large value then we can represent it as a basically a static load, okay? Now let us consider loads that you have been dealing till now which were typically what dead load or live load and a typical dead load included a simple load of let us say kilo newton, okay? And we said that is a static load, okay? Can you tell me why the dead load was considered as a static load? So dead load means what? Self-weight of the structure or anything that is attached to the structure is considered as a dead load. And remember that dead load is not applied suddenly to the structures. If you are building a structure you would do it over a day or like you know over days, over months, you know, so slowly you are going to over concrete and build the structure, okay? So it is applied over day and then you also have live load. Live load could be like you know the habitant or like you know people that are going to occupy that building or things that could be moved, okay? So that would be considered as live load. Even that would be applied slowly, okay? Of course we are not considering live load analysis like a vehicle moving on a bridge or something like that, okay? Live load we are simply considering due to weight of like you know people or like you know things that could be moved during the life of the structure, okay? Now so those are the typical static loads that you have encountered till now, okay? Now in terms of dynamic load, we can consider earthquake load, wind load, impact load, blast load or tsunami load and why do we consider these are dynamic load? If you consider an earthquake load, can you tell me what is the typical duration or can you think what is the typical duration of an earthquake? Maybe let us say 15 seconds, 30 seconds, 1 minute, something like that, okay? So structure is applied a very heavy horizontal like a very large horizontal road over a very short duration. So this TR is very small, okay? And depending upon the time period of the structure, it could lead to dynamic effect, okay? And then I can also have wind effect and depending upon like you know at what rate the wind is flowing or at what velocity the wind is flowing, I can have very large dynamic effects in the structure. I could have impact which is like a sudden impulsive load or I could also have blast which is also like an impulsive load and then I can also have tsunami load which is like you know sustained load over a little bit larger duration of time than the impact and blast load. So these type of loads are characterized a dynamic load due to their duration over which they are applied to the structure, okay? And the typical time period of the structure, okay? So keep these things in mind, all right? Now what I am going to do, I am going to show you some videos of previous loads or like you know the real life example of different dynamic loads that we have mentioned in the last slide. First I am going to show you basically one of the few images of the Bhuj earthquake that happened in Gujarat and you can see what kind of damage it resulted in. So all these buildings, most of these buildings were designed adequately like you know I can say for vertical loads, you know they were designed for vertical loads but I am not sure whether they were designed appropriately or adequately for horizontal load. So earthquake is in horizontal load due to shaking of ground motion. It leads to horizontal load on the structure and this because of inadequate seismic design it led to like a lot of devastation during the earthquake and loss of precious lives. So this is the example of what dynamic load like an earthquake can do to the structure. Then here is a video of Nepal earthquake. You can see this is a live CCTV video at that time and it is showing like you know you can, I will just show you like in a small clip. You can always go to YouTube and see like you know in detail that all the clips you can see let me run this video so that you can see it here. So you can see the shaking happening and soon you will see like you know the intensity increasing and lot of structures. You can see that there is a structure that just fell ok. And there are like you know 10 different footages at different places. So let me, so you can see another footages here I think the structure already collapsed. So you can see the effect of earthquake on this residential house as well, lot of pots falling around and people coming outside ok. So you can go and search on YouTube like you know different type of earthquake loading and the failures on the structure and you can see like you know what is the effect on different structures around the world like you know during different earthquake ok. Now the second type of dynamic load that we discussed was wind loads. What I am going to show you, this is a suspension bridge known as Tacoma Narrows Bridge. You can see that one morning due to wind effect what is happening is actually what happened. Lot of wind started flowing below the deck and due to pressure differential it started creating torsional motion of this deck ok and which finally led to failure of the suspension bridge precisely the deck of the structure ok. So let me just play this video. Of course there are lot of failure studies which would more accurately describe the failure mechanism and what led to this failure can go ahead and look if you are more interested go ahead and look into those studies ok. So you can see this deck of the central span wobbling vigorously and the amplitude keeps on increasing. So I am going to fast forward this video so that I can show you the final failure video of this one ok. Let me again go here this is showing the front view of actually the funny thing happened one of the dog was actually left in the car and then the owner of the dog actually went ahead and then actually rescued like in on this wobbly deck. You can see that the amplitude is increasing. I am again going to fast forward this video so that so we also have a suspension bridge right. We have a suspension bridge in Bandaravadali ceiling if you have not gone there it is a great engineering structure you can always go there just to have a look and look carefully at the suspension bridge that we have there. So you can see that due to increased basically deflection because of the wind load it finally led to the failure of the deck span ok. So and now like in all these type of structure need to be designed against such wind load so that even if there is a constant energy infusion due to wind then there should be some energy dissipation mechanism and which we will learn later so that it minimizes or it actually like you know truncate or basically it keeps the amplitude within a certain limit even if there is a constant fusion of energy ok ok. Now let us go to the next video not yeah this is crash test video so the third type of dynamic load that we talked about is impact load which is a short duration pulse ok. So this is one of the test that was conducted now if you like you know because of the security threat many of this structure now need to be designed against aircraft impact and this was one of the experimental test done in which a concrete wall actually was hit with a phantom F4 aircraft and then it was seen whether the aircraft would be able to take that load or not ok. So let me just play this video. So what do you think whether this aircraft whether did it breach the wall or not well it just so happens even the load was very high the wall was designed in such a way that there was lot of damage due to this impact however there was not a complete breach and that is what these walls actually designed for heavily reinforced concrete walls are designed for ok alright. So I am going to go to the next slide next type of blast impact sorry dynamic load is a blast load what I am showing you a building before and after bombing this is a building called federal building Alfred P. Mura building in Oklahoma City and it was bombed in 1995 using explosive kept in a van the explosive weights were 1800 kg what it did actually it due to the blast it took out the columns here columns and been here and it led to so first it led to the removal of columns here and then because of the blast pressure shock wave there was an upward pressure on these slabs which they were not designed for ok. So due to this it led to the progressive collapse of the building and finally what you see here is the building partially collapsed so I think lot of fatalities happened around sort of total number goes around 160 people died in this building so now many of the buildings are whether it is there are critical like you know civil building civil building means like you know for civil purposes non-military purposes they are now being designed aircraft buildings are being designed some operation control centers are being designed for blast loads and also military structure defense structure have always been designed for these type of loads ok. Next is a very simple but demonstrate the concept of dynamic is a vibration due to walking I don't know how many of you have taken a course in sorry have you taken like you know as in your curriculum you did NCC as part of your curriculum or either I don't know whether you know about somebody would have told you that when you walk on a bridge whether it is a pedestrian or regular bridge you should not walk in tandem and even like you know forces military or the like you know these they are supposed to walk in a tandem in a periodic fashion in a normal situation however they are advised against walking like that on a bridge because what happens when you walk in a tandem you create a walking load frequency we will see how does that look like walking load ok and that leads to basically a walking load with a certain frequency the frequency of that walking load matches with the frequency of the structure on which you are walking it leads to the failure of building ok. So, they are asked requested to break their steps breaking their steps mean walking randomly so there are multiple frequencies with very small amplitude ok but not a single frequency periodic load with a large amplitude ok alright. So, let us go to the pretext of this course ok so what you see here what is the problem statement ok for this course or what we are trying to do here ok so the problem statement is how to build a structure that can sustain those dynamic loads that we have discussed in previous slides and mitigate their detrimental effect ok so that is a larger goal ok and what is the solution for that well the solution for that is to design structure for dynamic effects of load ok very simple right but it is not as simple as this statement make a sound ok. So, what is required to do that well first I am showing you a structure here this is a like you know I mean you can think of like you know building structure here to obtain and design a structure first what you need to do create a numerical model of this structure ok because you cannot have a numerical model that includes each and every minute detail of any system or a structure ok so you need to create a simplified representation which might be single degree of freedom system or if you need depending upon what is your response quantity of interest you might create a multiple degree of freedom system ok what is the next step then well once you create a simplified model you need to find out what are the stiffness what are the damping properties and then from that what are the time period of the structure what is the damping is basically represented let us say representing it by viscous damping so you need to find that out ok and then you need to find out the dynamic loads that are being applied so in this case let us say I have an earthquake here horizontal earthquake so I will represent it through some like you know time history function ok and then I have a numerical model and these time history function the next step is to find out the dynamic response of this structure so response means what are the forces the base here let us say what are the moments in the members ok what are the displacement here those are the typical response quantities once we have that I can design my individual component or overall structure to accommodate those kind of forces and deflections ok and the final step once I have that is as I said to design the structure to minimize and sustain those dynamic load effects ok now first three steps are analysis steps ok and the fourth step is basically a design step now the scope of this course is to study first three analysis steps so we are not going to do go ahead and go design structure but definitely we are going to study first three steps so structure through this structural dynamics course we are going to analyze system and find out the response ok there are other courses that are there and which we will see subsequently that will discuss how to design the structure once you have the effects of dynamic loads figured out clear ok so that brings us to the context ok so the context is where does this course you know whenever you start a course for the first question you should ask why I am studying this course ok what I am going to achieve through this course ok and how does it fit into a bigger picture ok so let us say if your goal is to become a structural designer or analyst or like you know computational mechanist so you have to first find out what is the utility of those course and how it is going to be useful and how does it fit into the bigger picture now let me draw out the whole curriculum and then show you like you know how does this whole thing actually builds up to work towards a greater goal to equip with you equip you with the skills that are required to work towards designing a structure analyzing a structure ok and finding out or basically building safe and economical structure ok now in your undergraduate you would have studied a lot of subjects now let us see where those subjects actually fits into this picture ok so what are the steps required to design a structure as we discussed before well the first step you need to find out forces on a structure right and which course did you study to find out forces on a structure well it was the first course was engineering mechanics right now forces might not be enough right because most of the times failure criteria governed by the stresses not the forces ok so to get the internal stresses and strains you studied solid mechanics ok now after you studied how to get the stresses and strain you also needed to find out how to get internal forces and displacements in a structure and what did you study for that basically a structure analysis course ok and finally when you were equipped with all the analysis courses ok all the analysis courses then you studied specific courses that were for the design of different type of structure so whether it was a steel design course and concrete design course ok so these were at very preliminary level very basic level now at the advanced level in the graduate curriculum let us see what are the courses and how does this course fit into the bigger picture so now what are the steps required now for advanced analysis and design ok so you did basic analysis and design in your undergraduate let us see how do we do what are the steps required for advanced analysis and design so to find out the same the forces on a structure now you do advanced structural mechanics which discuss advanced methods of analysis ok and then you also now study a course which is called advanced solid mechanics to find out stresses and strains in your undergraduates you mostly focused on linear stress and strains however in a graduate you might do linear and non-linear both ok and then to obtain forces or stresses and strain you need some knowledge of numerical methods ok and to learn those numerical methods you take or you are going to take a course which is called numerical methods ok which is part of a curriculum that we have a department also has ok and then till now whether it is undergraduate curriculum or graduate curriculum these courses were focused on static again these courses are focused on static I mean you can do a dynamic like in advanced courses but the first course that you have which deals with the dynamic effect of loads on a structure is this course structural dynamics because most of the loads are actually not static they are dynamic even when you call a load as a static they are still a dynamic load which has a variation with respect to time but applied very slowly ok so I hope the context of this course is clear to you ok so again let me come to the problem statement so what through this course we would be doing first we need to find out what is the load that is being applied on a structure ok so first we need to determine the load history ok the second step is develop mathematical model of a structure so whether it is a building or whether it is a vehicle or whether it is any other type of system I first need to represent that system or a structure through numerical model or a mathematical model ok once I have that with this loading history and the mathematical model of the structure I would set up what is called equation of motion so I would formulate this equation of motion and we would see how to do that for different type of system and you can extend that to a you know very simplified as well as complicated systems as well so I need to first set up this equation of motion and then solve this equation of motion alright ok now the set after this once I have the equation of motion set up I would find out the response quantities ok so obtain numerical response and what are the typical response quantities I am interested is in they could be this could be displacement velocity acceleration and force ok these are the response quantities I need to find out so these are this is the basically the problem statement and basically through different chapters of this course we are going to addressing one or multiple steps at a time ok so keep that this problem statement in mind ok when we will go through different chapters alright now let us look at how do we obtain this load history what I am going to do I am going to present in front of you typical load history that you a structure or a system might encounter in real life so the first one is a harmonic loading which might be a sinusoidal or like you know cosine loading this type of loading you can imagine due to some machines on vibration happening due to machine attached to a floor or like you know other type of harmonic basically machines producing harmonic motions ok the second could be periodic loading a periodic loading might not be harmonic but harmonic loading is a periodic loading so just keep that distinction in mind so periodic loading is something that repeats after a period which is t0 here ok so what I have shown here is basically this is like a representation of a walking load ok of course this is a zoom a uniform walking at like you know at a fixed pace ok but this basically represents a walking load ok then there could be earthquake load which is more like a random excitation ok there could be a wind load which again could be represented by these random spikes which represent the pressure peaks ok there could be impact load which is like a sudden spurt of a load ok for a very small duration of time and there could be a blast load which is like a sudden spike and then decreased and this duration is actually very small ok so I have shown you some of the load history that a structure might experience these are the load history and typical load scenarios that would experience in while obtaining dynamic response of a structure ok so once you have the load history second thing we talked about is mathematical model of the structure and pardon me for the mistake here so in mathematical model of the structure I can represent my structure using a single degree of freedom system and I will tell you like you know what does that mean the degree of freedom in a later slide let us say I have this pergola type of structure where this is like a walkway in which your heavy slab is actually connected or supported through these steel columns ok or you have a water tank in which again a heavy mass is connected through this column so this can be simplified through a single mass and then a single stiffness component represented by this column and this is called single degree of freedom system or we could encounter like you know more complicated system in which I have a very complicated structure and I might decide to simplify at simple beam columns and then applied by loads here so the slab loads and everything can be applied here ok and then horizontal load might be applied here or a vertical load might be applied here depends ok now this is this structure can be further simplified if we assume that the all mass of these loads are rigidly connected to each other and the flexibility is basically represented by these columns so I can further simplify this system through this three degree of freedom system ok now you would ask which is the appropriate representation well it depends on your problem and it depends what are you trying to achieve through that problem ok so for some system even this could be represented by a single degree of freedom system ok if let us say your goal is to find out what is the total base shear and not what is happening in the structure above ok or if your goal is to find out what are the story shears ok then you need to represent it through a shear type building or building like this ok so it depends on what are you trying to find through that that problem ok so coming to what are the different components of a dynamic system you have to think about what happens when a force is applied to deformable body forget about this slide don't look at this slide right now but just think you have a deformable body and then you apply a force force that is varying with time ok so can I say if you apply a force on a deformable body then there would be an acceleration so motion or acceleration let us say motion acceleration and if it's deformable then of course there would be deformation and can you imagine if I apply a load to a structure there would be some sort of energy dissipation so in reality every system every system every structure in reality would dissipate some amount of energy when you apply a load it might be small so in some cases it might be neglected but in for all practical purposes there is always some energy dissipation ok so the third component is energy dissipation ok so considering these points in mind any structure or a dynamic system has three integral properties which are mass which is basically ability to resist acceleration ok then stiffness which is ability to resist deformation and then damping which is the ability to dissipate energy ok so a simplest representation of a dynamic system is a spring mass damper system ok and this is called a spring spring mass damper representation and if this spring mass damper system is acted upon by an external force f t there would be a response which is here displacement ok so I can say the dynamic behavior of any system can be studied by modeling it as a combination of spring mass damper system ok so although I have shown you here a single spring mass damper system let us say even if I have like you know interconnected body I can lump the masses ok I can lump the masses connect them with a spring and a damper like this ok connect them by spring and damper ok and each of them can be connected through each other node ok depending upon what is the connectivity ok and through this any system can be represented by like you know a spring mass damper ok so what I need you to do basically when you are doing this close try visualizing every system what you see around yourself as a representation of a spring mass damper ok and of course if you can tell me that to solve this system I need to represent it with some additional properties then let me know because this I find is the most simplest representation of a dynamic system ok so what I want you to do ask yourself few questions at this point of time ok in this introduction slide how the car shock absorption works ok so you have cars or like you know any other vehicle ok and like you know our Mumbai especially Mumbai is famous for potholes or our Indian roads are famous for potholes and all the cars or any type of vehicles are actually equipped with shock absorption so we need to imagine how would that actually work and if somebody has to design that car shock absorption how would they do it what would be the critical parameter ok then what I want you to ask yourself you guys have travelled by train and you must have seen or the metro you must have seen there are blast or crushed stones like you know below the rails ok what is the function of that one why can't I just have like you know rails on the top of let us say concrete slab ok and then just run vehicle on the top of that so what is the utility of that ok and then you might have also travelled by aeroplane have you ever noticed which wheel the aeroplane any aeroplane lands on whether it is like you know front wheel or the rear wheels so just try to notice that next time and try to find out the reason why the rear wheel or why the front wheel ok and then I need you to ask yourself what will happen if a very fast moving load hits a very stiff structure ok or a very fast load hits a very flexible structure ok these two questions I need to so these questions like I want you to leave I want you to leave with these questions ok keep pondering about these questions because the answers to all these questions would lie or would be answered in future chapters that we study ok so and like these questions are just example always try to imagine whenever you see the systems around you try to visualize why what is happening is happening and what is the mechanism behind that how can I explain that there are so many examples of dynamics around you or the structural dynamics around you that it would become very interesting when you start looking and analyzing those systems in your mind ok so with these questions I would leave you we will come back to again these questions in subsequent chapter and we will see I will explain the basic concepts behind those principles as well as the mathematical derivations of these principles or the questions unanswered to these questions ok so thank you for your attention and let me just leave you with these questions and we will come back to again in next chapter ok thank you