Physics behind a Tuned Mass System

[00:00:07]so here is a Saturday beautiful system and I'm just shaking it and the place and as you can see that just behaves like we it we would think it would and it's whenever the forcing frequency is resonating with us in the degree of freedom system that's actually showing too much dynamic amplification but what happens when we add the two nos system and as I attach and shake you will see that there is a significant reduction in the dynamic amplification of the primary structure even though the frequency of the excitation is still pretty much the same and as we move on I am actually now changing the frequency of the base excitation indeed that primary structure has stopped moving and then suddenly the entire system resonates with the external force then it again starts moving but still the dynamic amplification is nowhere close to six so let's take a look at physics that goes behind this so let's take a look at mathematics behind Qun math system not a damper because we'll study about it later on so here in the picture what I have shown as a coupled system with a main mass Capital m which is attached to secondary system with a mass small m and both have a stiffness of capital K and small K respectively so called this entire system as primary structure and the smaller system will be known as a Kuhn mass system vehicle mass we'll discuss about it later on let's first figure out how a two degree of freedom system responds or interact with each other so what we are looking at is a harmonic excitation on the main mass M you can imagine it as a vent force acting on a tall building and we already saw in previous videos how it affects how the wind forces and the world Isis effects a very tall and slender structure say okay you can imagine this force as the exact same thing as an harmonic excitation on the slender building if I draw a free body diagram of the main mass M after attaching it to the secondary mass small m we know that there is an harmonic excitation P naught sine Omega T let's consider it as in positive direction if we try to push any object there is an inertial force that acts in opposite direction that's called MX double dot that's the mass of the system times the acceleration now there was a spring attached behind this mass M capital K so once we pulled us push the system this way spring will be pulling back the mass in this direction so that's why it's K x1 and the other spring that is in front of the capital my son will be compressed and the compression force will be equal to K displacement of relative displacement between the main mass and the secondary mass so from that equilibrium I can write down the equation as MX double dot plus K plus small K x1 minus K x2 equal to the external force acting on the main mass M for the secondary mass there is no external force acting the only thing that's resisting is the secondary spring force that was in compression as we looked before so that compression will cause this mass to push in this direction so that's K x1 minus x2 and the inertial force will be resisting the system's movements so that's M x2 double dot and that equilibrium can be written down and thus far let's say that the harmonic excitation is represented as a 1 sine Omega T for mass Capital m and a 2 sine Omega T for mass more than secondary mass if I differentiate it twice I get this that's a very basic differential equation and similarly for the secondary mass it's X 2 is equal to minus a 2 Omega square sine Omega T what I did is I just replaced all X 1 double dot value into the main equation and in the end I get an equation that looks something like this now we know that in any equation a sinusoidal force so the sinusoid cannot be 0 it varies on the basis of the external force excitation so to satisfy the equilibrium we know that the coefficients of this equation should be equal or else it cannot be satisfied once I make the coefficients equal we come up with this equation for P naught and similarly for my small M we come up with this equation what I'm looking for is the solution of these two equations in terms of the excitation and the amplification of the excitation because of resonance so these are the exact two equations and what we'll be looking at is we want to make it simpler in terms of just frequency natural frequency of the primary structure and natural frequency of the tmd so we are just doing that by replacing the ratio of capital K over capital n with capital Omega and small K over small and with a small Omega and xst is equal to P naught over K you can consider it as a static displacement so instead of a resonating force that varies on the basis of the natural frequency of the force let's say if I just apply a force P naught instead of P naught sine Omega T a structure will displace on the basis of its toughness now that will be known as the static displacement so that term is important because over here we are interested in the amplification of the static displacement because of resonating force and if we want to understand about more about amplification factor you can actually watch our video on resonance and amplification that we posted before so once I do that what I come up with us these two expressions and these two expressions tells me that there is a relationship between a 1 a 2 and a static displacement of the system and from the second equation it's very clear that a 1 will be equal to a 2 times 1 minus Omega square over Omega a square this relationship is very important it tells me that a 1 displaces in some proportion to a 2 it's not just a random excitation anymore so once we solve that for a 1 and a 2 we get an amplification factor that's this remember it's the actual displacement over the static displacement that's called the amplification factor and since the structure is elastic more the static displacement the more will be the actual stresses in the buildings and the structural systems so once we have this two equations we want to understand how these two equations affect the system so for that what we have done is we have created a spreadsheet which you can play with we'll be attaching the link to the spreadsheet in the description and you can use that spreadsheet to understand how by varying different frequencies of the TMV as well as main mass and changing the stiffness of the system how the entire amplification factor that gets affected because that's the only thing we are interested in studying them and the beauty of Kuhn mass system is if we select the natural frequency of the secondary system such that the natural frequency of the TMD resonates with the frequency of the exciting force the end result is the main mass remains static it doesn't move at all because all the natural frequency excitation coming from the external force will be absorbed by the natural frequency movement of the tmd so the TMV will move a lot but the main structure will not move at all and that's reflected from this graph you see um there is a resonating frequency for the combined system of tmd and in primary structure and this is the actual displacement of static our primary structure over the static displacement of primary structure and as the Omega over Omega and so the exciting frequency over the natural frequency of the system varies there is a changing dynamic amplification factor of this coupled system and at one point in the entire graph you can see that there is a point where there is no displacement of this the main mass and that's what we call the tuning of the combined system and that tuning is achieved on the basis of understanding the frequency of the exciting force understanding the frequency of the mainmast and actually calibrating our secondary mass so that it actually resonates with the main mass and once we calibrated in this manner we get a system that is known as tuned mass system so let's go ahead and play with our Excel spreadsheet and see how changing the frequencies and changing the mass of the secondary structure affects the overall response of the coupled behavior so um and this spreadsheet what you can play with is four different parameters for a two degree of freedom system the first couple of parameters are the natural frequency of T til mass damper and the second parameter is the natural frequency of the main system primary structure as well as the stiffness of the secondary system or a tuned mass damper and the stiffness of the primary structure I've written down the basic equations that I actually showed before these equations are again the same it's presenting the dynamic amplification of the primary system that's the first equation and the second equation is the dynamic amplification of secondary system now why dynamic amplification is important to study so as we discussed before in an elastic system stress is proportional to strain so if you have more dynamic amplification that means your system is displacing that much more and at resonance without any damping it can go to infinity that means the structure will break so that's what dynamic amplification tells us so more the displacement means more the strain in the system and from the proportion of stress equal to strain more the strain means more will be the stresses that means more loads in the system but there is an ingenious way to reduce these lords's taking advantage of the secondary system and tuning it this word tuna is very important tuning it so that the response or the amplitude of the primary structure reduces close to zero that can be achieved only by varying certain parameters for the tune mass damper such that it kind of resonates with either the exciting force that's the external force P naught sine Omega T or it resonates with both the main structure as well as the exciting force so in case of when a structure oscillates back and forth only at its natural frequency so the excitation is occurring at natural frequency now if we calibrate the toon masked ample to the natural frequency of the structural system just like in case of Taipei 101 what you can achieve is even at very high external functions that are trying to excite the structure the structure will not move at all surprisingly if we damper inside the structure that would be moving so if you are standing on the floor of this structure let's say on 88 story of Taipei 101 you'll not feel any accelerations but when you look at the damper the damper is moving so that's the fascinating thing about unis temper now here let's say if I shift the natural frequency of tune masked ampere they are away from the natural frequency of the structural system or the primary structure you see that it doesn't create that much impact it does to a bit because now the combined system is not resonating at the frequency of primary structure but it's resonating at about 1.25 times the natural frequency of the primary structure and that's only because we have calibrated the TMB in such way and it could be a little bit harmful if we don't understand what will be the frequency of exciting sunshine but for well that's very simple if I just calibrate it and make sure that the frequency of TM D is matching the frequency of the primary system I get a very beautiful band and add the natural frequency of primary structure I see that there is zero displacement the primary structure doesn't move at all but at the same time there is a movement in the tune mass damper and that's fine because Q mass damper is designed for absorbing these loads and it's there so that it provides more human comfort so that building doesn't move and we don't feel any accelerations so that's okay we can account for this displacement and inside the building itself and the other thing to notice is now there are two exciting frequencies are the resonating frequencies instead of just one first one occurs at somewhere between 0.5 and 1 so that's 0.5 times the natural frequency of primary structure and natural frequency of primary structure and the other resonance peak occurs between 1 to 1.5 as it is visible over here so if we are outside these resonating frequencies then for sure our damper is going to reduce the loads in the primary system but for example let's say the forcing function is occurring at this frequency you know what we can do there is an ingenious way you can just straightaway lock the damper and now the frequency of the system won't be over here it will be somewhere at this point and that will let us move away from the frequency of the exciting function so that's that's how tune math Stamper can help us and with today's computation we can either switch it on switch it off on the basis of the accelerations that the primary structure is experiencing now this you can play with four different parameters understand how stiffness of the damper and stiffness of the system also affect for example if damper doesn't have any stiffness you see this band becomes very very narrow and that's because if damper doesn't have stiffness it's not going to get excited so easily and it's not going to absorb that thing that much load but let's say if it's exactly opposite damper has too much stiffness and I mistrusted doesn't in a slick it's like putting your building on top of like four jelly beans or even gummy bears like the building itself is so soft like gummy bears and on top of that you put a wood block so even if you shake it it's it's not going to do anything because gummy bears are just going to deform so that's why it's important to actually see okay what's the stiffness of the primary structure and what should be the stiffness of that the cue mass damper now the other tab just talks about Omega eight that's the frequency of the damper and mu nu is the ratio of the mass of the humours damper over the mass of the primary structure which is also equal to the stiffness of the Qun mass temper over the stiffness of primary structure if we go back and take a look at the equation that describes the natural frequency of the system it's equal to K over m now because they both are we showed the same over here I can safely say that the tuned mass damper is resonating with the primary structure correct now let's say even if I change this thing you can see that the graph does because forever the qumar stand for a primary structure are resonating with each other so this this graph isn't going to ship with changing in the ratio of the mass of the to mass damper over the mass of the primary structure this band widens so the lower the ratio of the tooth the narrower the band when you can get a point of zero amplitude in the tiny structure as you increase the mass efficient mass standpoint this band becomes wider and this shows the combined natural frequencies so the first frequency is right there natural frequency of combined system is certain frequencies right there and this is a plot of ratio of masses over and on the y-axis it's the exciting frequency over natural frequency of the system you can see as Lou reducers the two natural frequencies become closer and closer to each other so you have a very short window between your temperature now it's very intuitive why does this happen as you increase the mass of the Qun mass damper what happens along with it is it can absorb more energy and at a wider frequency because this is the frequency of combined system so as your tool mass dampers mass increases the total mass of the system increases and as the total mass of system increases it leads to this increase in couple of natural frequencies of systems so the first natural frequency at mu equal to 1 will occur somewhere at like 0.7 or so or 0.6 and the second natural frequency will be somewhere about 1.6 or so so instead of having a resonating force at 1.0 that's omega over capital omega exciting frequency over the frequency of the natural of the primary structure instead of that it shifts wider apart this kind of behaves as a two degree of freedom system or it is a two degree of freedom system but the only thing is both the mass have significant influence on the entire system while if you change it to Nu equal to 0.01 it's the influence of primary structure that is like governing the response of the entire system and you can see that they are approaching closer to one child makes sense so I'm attaching the link where you can download the spreadsheet and play with it so that you can understand how you can tune the damper and how this entire excitation works and if you go back in our video where I showed you the actual structure moving back and forth with a damper there was a point in the video where you notice that the primary structure isn't moving at all and that's because the frequency of excitation was matching the frequency of the tuned mass dampers and it was tamper which was absorbing all the loads and the primary structure was not moving at all so I hope I cleared something some concepts about a tuned mass system over here what we'll be looking at in our next videos when you add a damper what happens to this entire system if you want to go back and read about damping concepts you can watch our video on dynamic amplification and damping and you can clearly see that as you add more damping in the system the peak of resonance reduces because damper is now dissipating so much more energy so till then leave it the spreadsheet redraw a blog post to understand more about resonance and watch our video carefully to understand how it you must ampere and the physics behind it so take care guys I will see you in next week all right you