Hello everyone, in today's class we are going to continue our discussion from the previous classes in which we basically learned how to find out the response of a multi-degree of freedom system subject to external force. And we used method what is called modal superposition analysis. So we are going to continue discussion of on modal superposition analysis and we are also going to discuss how we can apply modal superposition analysis to seismic analysis of structures and also learned about the response spectrum method which gives us directly the peak response of multi-degree of freedom system. So let us get started. We have talked a lot about modal decomposition of modal decomposition, basically diagonalization of mass matrix, diagonalization of stiffness matrix. Now let us look at the forces. So what I mean to say when I apply any force vector to a structure, the force would also distribute itself among different modes and we said that if we have force Pt which is defined as like P1t and then P2t and so on. The way to find out P and t, I need to be clear about this, this is small p here, P and t is basically the vector or not the vector, the excitation force in the nth mode. So effective force in the nth mode. How do we get that? We simply get that by multiplying like others with transpose of the mode shape times the P vector here, this is this. And this would give me P and t. Now when I take P and t, like any other vectors, like my vector ut here, I know that my force would also be distributed in different modes. So I will have effective forces at different degrees of freedom in different modes. And that's where the concept of expansion of this forces comes into picture that what would be the expansion of my vector P to different modes. And let us see how do we do that. So basically what I am saying, let us say I have a force Pt for which, so if I have something like this, let us say at different degrees of freedom, and all of these have same time variation, same time variation. Then I can write down my force vector P as a vector that represents the spatial distribution of that force times the same time variation Pt. I am writing it like that because it will later help me in the analysis and interpretation of the result. And we will see how do we do that, how do we get the distribution. So basically, let us say I have something like that, this vector what I wish to do for different mode, I want to get the spatial distribution as something like this, okay. So instead of saying like this, let us say some distribution is there, okay. And I want to see and so on. So let us say this is my vector S here. I want to find out a vector S1 plus a vector S2 and so on for each mode. So this is let us say mode 1 and this is mode 2 here, okay. I have a force vector which I have written as Pt and if the Pt has the same time variation at all degrees of freedom, I can write it something like this S which represents the spatial distribution basically if this is 0.1, 0.2, okay. This is let us say 0.15, this is 0.6, okay times the Pt which is the time variation. And like you know why we are doing this because if you remember the seismic excitation has a property something like this where all the degrees of freedom are subjected to the same time variation of the force which is the ug double dot t, okay. So we will utilize this knowledge there. So just keep that in mind, okay. So basically I want to write this as let us say summation r equal to 1 to n and this is SR, okay. So sum of multiple modes, okay. Now this can be written as a modal participation vector which we write it as gamma of r, okay.  So basically what I am saying that my SR or SR here is basically nothing but this quantity here, okay. And a question might ask that how can I directly writing this equal to this much? Well if you remember to get the modal coordinates q1, q2 we had written, okay. We had written again mass or like you know this vector here, phi times qr summation r equal to 1 to n, okay. The idea is that if I write it in terms of multiplication with some mode shape and multiply with the transpose of another mode shape, I can use the condition of orthogonality and I can find out these vectors here or these factors that we have defined, okay. So I am going to do exactly that. I am going to multiply, okay, this as s here and this r equal to 1 to n, okay.  Now as you know this quantity would always be 0 except when r equal to n, okay. And if that is the case I am just only term that is going to survive here would be when r is equal to n. So I am going to write this as phi n times s is equal to this phi n t mass and phi n. And this is nothing but the diagonalized mass element mn, okay. So phi gamma n can be written as s times mn. So remember if I have a vector s which is given remember force is given the external force is given and if I can write down that external force let us say pt equal to a spatial distribution vector s times some time variation, okay let us say small pt. So this is also given I know my mass vector I know my stiffness vector I want to write this as sum of vectors spatial for each mode so that I can see how much is the contribution being made by the vector force vector in each mode. So s can be found out as so remember once you find out this gamma n I can get my spatial distribution in nth mode as how would I get that just get write this expression here gamma n times mass times phi n, okay. And then we can get the total distribution or expansion like this. So once we know the gamma n then Sn we can also obtain like this and this would be the spatial distribution, okay. Now let us see what are the properties of this or what are the physical interpretation of this Sn, okay. Now you know that when you will apply this force external force pt, okay a structure can vibrate in any of its modes, right. Now when a structure in vibrating in each of its modes can I say in each mode it would have some inertial forces associated with each degree of freedom. What Sn basically represents it represents the distribution of those inertial forces associated with the application of the applied force, okay. So again I will repeat it if you apply external forces pt on a multi-degree of freedom system, okay. Let us say something like this, okay. In each mode the structure will vibrate, okay. And when the structure is vibrating it would have inertial forces associated with each degree of freedom, okay. What this Sn represents is basically distribution of those inertial forces in each mode, okay. And when you multiply with the time variation of these force it will give you basically the total external force that is being applied on that mode, okay. But Sn just represents the inertial force distribution in different modes. We did expansion of our mode shapes basically different modes using the displacement vector. Now this is the force that is effective force in each of the modes, okay. And we can do this for earthquake forces and doesn't have to be earthquake forces as long as the time variation at each degree of freedom is same and you are able to represent your force vector as some spatial distribution vector times, okay. Same time variation, okay. Let us say this is not QT, let us say this is some small pt which is the time variation then we can interpret results like this, okay. And like you know we can prove that mathematically as well. Let us say inertial force in my nth mode would be what? In my nth mode the inertial force let us say MIN here would be mass matrix times the acceleration of each degree of freedom in that nth mode, right. So let us say this is the nth mode. The inertial force is mass times acceleration which is UNT, okay. Now UNT is what? Remember UNT is phi n times QNT. So all I need to do is that take the double differentiation of the time function here. So mass times this is phi n here times Q and double t. Now if you compare this remember if you compare this to p, the time variation of the force capital P equal to phi times pt, okay not phi times sorry, S times pt. This is the time variation, this is the time variation. What is S here? S is basically this quantity, okay and this quantity is what? This is the spatial distribution associated with the inertial forces that I have, okay. So I hope this is clear to you. Now what happens that if you have a force, if you are able to find out Sn vector which is the spatial distribution of force, Sn times the time variation pt of this force, okay would give you the response in the nth mode, response of all degrees of freedom in nth mode, okay. And there is no other mode that is going to contribute to this response because now this is the effective force that you have found out assuming that this is how the total force is going to be distributed among different modes. If this is clear now let us look back at our equation of motion in terms of this S, okay. If you remember our equation of motion was mn times Q and double dot t plus Sn times Qn dot t plus Kn times Q and t and this was equal to P and t here, okay. Now I can go ahead and I can divide it by mn so that this would be Qn t and this becomes 2 zeta omega n zeta n sorry plus omega n square Q and t. Now remember P and t I can write as zeta n times the p vector, right. So when I multiply with this mn this is nothing but phi t and pt if it is can be if it can be written as a spatial distribution some time variation I can write it like this, okay the time variation and I am dividing it with by mn, okay. Now what is this quantity here is it not gamma n that we have already derived so our equation of motion can be written as zeta n omega n plus omega n square Q and t is equal to gamma n times the time variation pt, okay. So this is my equation of motion for each degree of freedom where this is a factor gamma n is basically a factor which is a modal participation factor so I will write it anyway this is called modal participation factor, okay. So this factor basically says that of the given force how much is the total force that is going to be applied in the particular in particular mode, okay and this is what basically the excitation force becomes what you are used to do you can solve this equation remember we have done nothing new we are just considering a special case, okay so this is just one of the special case for which it can be written like that, okay so let us say this is a special case, okay and if this is special case is still not there you can still write down the same thing pt as some vector and do the same thing and multiply and like you know solve it, okay it is not that it has to be like this, okay. So using this the modal participation factor we have now written down our equation of motion in this form here which is basically the equation of motion of a single degree of freedom system subject to excitation pt multiplied with gamma n to account for the distribution of the force in its nth mode you know I mean let us say many times what happens, okay you have or you would have the solution of this equation available, okay and I will give you some example, okay let us say I have a building, okay let us say the same two-story building and let us say this is being applied to a blast load or a triangular pulse let us say so this is the load and you know the time variation of this load so it is being applied with the same variation pt here and here similarly instead of this I could have pulse load or I could have like you know ground motion which is something like this, okay for all these type of motion we have a standard result available from the previous chapters, right in which we solved the equation of motion this, right you solve the equation of motion not q let me first write down in terms of u, u equal to pt where pt could be one of these forces here. So equation solution to this one is available which is basically response of a single degree of freedom system to any of these pulses. Now if I know the time variation of these forces are to be the same and I know the solution and the same thing is now being applied to a multi-degree of freedom system what do I need to do to consider response of each mode I just need to multiply whatever response I get here by gamma. So think it in like you know two step first step is to get the time variation of the response, okay for our case it is qt, okay these responses are available using standard results. So solution to this one is solution to this SDUF system is available. Problem is now I have a multiple degree of freedom system, okay. So solution to this is available how do I utilize this for a multiple degree of freedom system. Now I know that multiple degree of freedom system can be represented using n number of if n is the degree of freedom single degree of freedom system, okay. So to utilize this result all I need to do is to multiply response of this using this to get for that particular, okay. So you will also see in the book it might be written as d here instead of u to avoid any confusion, okay. So this is d dot and this is d here. So this is available if you want to find out q and t for a multiple degree of freedom system it would be nothing but this factor accounting for the distribution of the total force in the nth mode time the standard results that you available here. Let us say this is d and t you obtain. Now generalize this to df system instead of zeta sorry instead of gamma n we put gamma equivalent, okay. But here it is for that particular mode, okay. So qn we can get it like this, okay. And if I have to get the contribution of the nth mode to the total displacement at each degree of freedom all I need to do is remember this would be sorry this would be phi n times q and t, right. Now q and t I can write it as zeta times dn. So this I can write it as phi n times dn t, okay. Similarly if I have to find out equivalent static force in the nth mode, okay. So if I have to find out let us say equivalent static force in the nth mode I can find this out simply as writing this as k times u and t, k is here, u and t is my I can write this expression here it would becomes zeta n times phi n times dn t, okay. And if I want I can further write down using eigen value equation this as omega n squared times m times phi n. So this I can write down as m phi n and there would be omega n squared term here as well. So let me just write that times dn here, okay.  This was the expression. So can I not write this I can combine this and this term here I can write this as omega n squared times sn times dn, okay. So this is vector here or instead of writing it like this let me combine it sn times omega n squared times dn t, okay. What is this quantity here? This is the acceleration right in the nth mode and this is exactly what I told you when I was discussing the physical interpretation that if you apply a force which will have the distribution s times some time variation, okay sn represents basically the forces in its nth mode, okay the inertial forces in its nth mode due to those applied forces and that becomes clear here. I also told you that if you multiply those forces with the acceleration you would get the equivalent static forces in that particular mode. Then let us look at the most common problem that you might encounter, okay would be earthquake analysis right. So for seismic excitation, okay. Now for seismic excitation what is my applied force vector? This is my applied force p effective is what is my p effective? It is mass matrix times the influence vector times the ground acceleration. Right I hope influence vector you already know how to get influence vector. Basically you apply a unit displacement in the direction of the ground motion and then you see along each degree of freedom due to that unit displacement what is the corresponding displacement and then you form the influence vector. So I had given you example of this, okay if you define this one as u1 and u2 and you have ground motion like this then if you apply a unit displacement ugt equal to 1, okay influence vector is constructed like here it is 1, if the ground displacement is 1 u1 is 1 however u2 is 0, okay. So as simple as that this is my influence vector and you apply multiply with the mass matrix and this is the time variation of force. So basically this is of the form this that we have discussed right.  I can repeat the same procedure and I can show you how to get that. So let us do this let me write down here basically the equation of motion that we get mn times qn double dot cn times qn dot plus kn times qn is now equal to phi n or let me write down p effective n which is in this case what would be the pn here. It would be minus phi nt times p effective which is basically this force here or let me first write down this, okay. So this is pnt and p effective is no minus sign here sorry minus signs comes after I substitute this l times ugt, okay. This is what we have here, okay.  Remember I am just doing the same thing that I have done here, okay. Now pt instead I am writing ugt. It is said my s is m times l if you remember, okay. This thing the numerator actually is defined as ln. This comes from again the equation that you had done in the generalized coordinate system and zeta n or sorry gamma n is basically ln divided by mn where ln is this divided by m. So this whole thing is actually gamma n, okay. And you could have you could have written the same thing without even me having to write all this you know. So basically your expression becomes or expression or the equation that you need to solve for seismic excitation it becomes this the same thing that we have written above becomes gamma n times ugt, okay. So if you are able to solve this equation for a for the nth mode then you can do for all the modes and combine all the modes, okay using this expression here and you can get the displacement as contribution of each mode n equal to 1 to n and this I can write it as phi n times qn here. But now for a special case q and t is represented as especially for ground motion. This can be represented as gamma n times phi n times dnt. My u and t is basically so I have just write down this, okay. Now in addition to that we had also derived my equivalent static force in the nth mode is sn t times the acceleration, okay. So you know the displacement in the nth mode and you know the equivalent static forces in the nth mode, okay. Your job is now basically very easy. You know the equivalent static for let us say this is m3 n, m2n and m1n. These are equivalent static forces, okay. Displacement is known. These equivalent static forces are known. Base shear is very easy to calculate. Storage shear is very easy to calculate. Just consider equilibrium and same goes for the moment as well, okay. For example moment at the base here would be what would be the moment here. Can I not say moment would be simply f1n, this is h1. Let us say this is h2 and the total height is h3. Can I say moment at the base would be f1 times h1, f2n times h2 plus f3n times h3. This would be the total base moment in that structure, okay. Now consider this, okay. Ground motion, this factor represents basically the modal contribution, okay. We know what is the solution to this equation, isn't it? We have the zeta n, omega n, qn plus omega n square qn is equal to ug t, okay. And dt is basically solution to this equation, right. Maybe I should not write sorry. I should not write q and t here. Let us write down dt here. So, this is known to us what is also known to us in fact, okay. What is given to us is response spectra. What is response spectra? It is the peak value of any response quantity as a function of different time period for single degree of freedom system, okay. So, we have this, we have given this from a response spectra perspective. So, let us say and most of the cases our goal of any structural analysis is to find out the peak responses like peak displacements and peak forces, okay. So, I have my un t, I have my fnt, okay. And I have my let us say mnt which is the moment. These are for all the nth mode, okay. So, let us do something. Let me say this is my displacement un t, okay. And this is the time variation. So, as we know, it would look like something like this here, okay. Now, let us say this is first mode instead of saying un, okay. This is the first mode, okay. And the second mode would be something like this, okay. Third mode would be again something like this. Now, different degrees of freedom system. What is response spectra? Let us say this is my response spectra and I write something like this. I mean, if it is d, it looks like something like this here, not like this. Something like this, okay. If it is a, the acceleration, it looks like something like this here, okay. So, this is d is basically what u max if the damping is very small, okay. This is a means my q double dot max. Now, so for different mode we can find out what is the peak response from the response spectra, okay. But let us look at this. For each mode depending upon the time period of this mode, for example, for this one, this is my u1 max. It is happening at this time t1. For this case, let us say this is my u2 max. And for the third case, let us say this is my u3 max. Can you appreciate the fact that if I have n single degree of freedom system, their maximum response would occur at different time. Like for first case, it would occur at this time for the second case at this and the third case at this point, okay. The question becomes, we can combine different mode and we can find out the displacement as a function of time, okay. And the time variation, how do or how can I utilize the response spectra, which is the peak response, so that I can also find out the peak response of a multi degree of freedom system, okay. And understand the issue here that we have. For a single degree of freedom system, I know my peak response. But for multi degree of freedom system, I do not know my peak response because it is a combination of different single degree of freedom system and different single degree of freedom system will achieve their maximum response at different time. So, and the idea is why do we do the response spectra analysis? Do you remember? We do response spectra analysis so that I do not have to solve the equation of motion every time and find out the response, okay. For example, if I do the response spectra, okay, or as a matter of fact any other force, I do not actually solve this equation, is it? Because I'm only interested in the peak response. I only need to know the TN value. If I know the TN value, I can come here and I can find out the DMAX. I do not actually numerically solve this. And the response spectra, they have been, they are already given, okay. So, for each ground motion, people have prepared response spectra and those are given. So, now I'm considering not the combining dot doing the time history analysis or anything, I just want to get the peak response. And I'm not interested in like, you know, at each time combine different systems then get the time variation of each degree of freedom system and then find out the maximum at what point the maximum would occur and all those things. I want to resort to a very simple procedure in which the maximum response for each degree of freedom can be found out. My only challenge is that they happen at different point of time. And the maximum response UMAX cannot be simply U1MAX, U2MAX plus U3MAX and so on. In fact, we always, sorry, it would be always less than equal to this. This goes not only for displacement but also for all the things that you find out. Remember, now if you talk in terms of UMAX, all the quantities, whether it is equivalent static force or like, you know, moment at the base, all things can now be found out in terms of UMAX for that particular mode, okay. And then we need to come up with a procedure so that we can combine these three or like in all different modes, the maximum response and get somehow UMAX because this is not simply equal to the algebraic sum. So what do we do? There are people have come up, like, you know, there are different methods that are available through which the maximum response of different systems can be combined so that I can get UMAX, you know, and those methods so basically methods of combining peak responses of different modes, peak responses of different modes, okay. So people have come up with different method how to actually combine different modes and get the total maximum response. Now as you can imagine, these methods are approximate.  And let us look at two of those methods where people have actually utilized that and have come up with the methods, okay. So the first method is called SRSS. SRSS method is basically sum root of square, okay, of all the sums. So square root, sorry, square root of sum of squares. So basically in this method, if you have multiple responses, D1, D2, D3 and so on, maximum response can be obtained as D1 square, D2 square, D3 square and so on. Now note that it is not the algebraic sum and it would always be less than the algebraic sum, okay. So people have come up with this method and these tend to be like, you know, produce good results. Of course, there are like, you know, cases where the difference between actual peak response and the peak response after in this method would be little bit different, okay. And you don't know to get into that, although I'll tell you what, if the modes are very closely spaced, like one mode is like, you know, 0.5 second time period, other mode is like, you know, 0.55 second time period, then this method might not work out that well. And there are other methods to do that. The other method like, you know, people employ complete quadratic combination, okay. And this method is little bit mathematically complicated. So in our course, we are only going to focus on this because, you know, I mean, you can have for hand calculation, SRSS is only amenable. If you have a software, you can implement CQC and get the total response, maximum response using individual peak responses using complete quadratic combination method. For seismic excitation, I can go ahead and find out the time variation of the response, okay, by combining different modes that we have done previously. But good thing about the seismic response is that we have available response spectra, okay. So peak responses are available to us. And this would apply for any other method for which the response of a single degree freedom system are applied in terms of some chart. So even like, you know, you could do it for pulse type motion where the peak response is available in terms of the time period and the time duration of the pulse. So if the peak responses are available, we can combine them, the peak responses, but not using conventional method that we have been doing so far because in the previous method, we were combining responses for any time t. And the final response was as a function of that time t. Now, I don't want to solve it in any time t, I don't want to find out d1 t, because I know my d1 max from the response spectra here. If d1 max is known, I can combine the maximum responses and approximately get the total maximum response using one of these methods, okay. So this is the summary of, okay. So this method is also called, you will see in codes response spectrum method. And the response spectrum method is nothing but a special case of modal superposition analysis, okay. Remember modal superposition analysis is for any time t, but for maximum response, we use the response spectra and it is called response spectra method, okay. So it is a special case of modal superposition analysis for peak responses or responses. Let us see one example how we can actually utilize this method, okay. So what we are going to do, let us take the example of what we have done and extend it here for the same time variation, okay. So basically what I want to do here, we had done this method, right, 2M, M, 2K and K, okay. Now, let us say or what I want to do here basically, I want to find out if earthquake force is applied, let us say here, something like this. I want to find out the expansion of the earthquake forces, okay. Remember earthquake forces can be written as what is S here, this is the influence vector and remember earthquake forces are nothing but S times UGT. If I write this as M times influence vector L, okay. Can you find me out what is, now here there are 2 degrees of freedom. So I will have S1 here and I will have S2 here, okay. So get me S1 and S2, okay. I will again rewrite the equation, S1 is gamma N times mass times phi N, okay. And gamma N is phi N times, not phi N, sorry, phi N transpose time the S vector divided by MN, okay. So once you have that, remember my F and T, the equivalent static forces are nothing but S times A and T, the nth mode it would be S N times A and T, okay. Similarly, so you would have F1 and F2. Once you have that, can you also get me the base here V1, V2 at each degree of freedom flowing this method, also try to get the same answer just directly from the displacement, that is from U1T and U2 to T for each mode by multiplying the story displacement with the S stiffnesses and utilizing this expression here. Remember A1T is nothing but omega N square D1T and A2T is nothing but omega 2 square D2T and see if you get the using the equivalent static method, okay. Let us try, okay. Let us see, let us try to get the answer. So my S vector is nothing but mass times L vector. Now influence vector in this would be, what would be the influence vector here? If you apply a unit ground displacement in each degree of freedom, the displacement is 11. So the influence vector would be 1 here. So effectively what you have here is 2M0M times 11, okay, which basically is 2MM, okay. And that makes sense.  And what I want to do is to see how these, because of these earthquake forces, how the inertial forces in each mode is distributed, okay. And let us see how do we do that. To do that, first what I need to do, I need to find out the parameter gamma 1 or before that, let us do this. Let us first find out M1. I think it was 3M by 4, 3M by 2. How about M2? How about K1? I think 3K by 4 and 6K, right. And so we have that, let us say gamma 1 would be phi 1 times T times the mass vector times L divided by M1. So, let us see what do we get? Half 1 mass times L is 2M M, okay. Well, there is a negative sign here. Factor does not have any negative sign. And then M1, M is 3M by 2. So, this one we get as 2M divided by 3M by 2. So, this gives a 4 by 3, yes. Similarly, gamma 2 would be phi 2 is 1 by 3, okay. So, what would be my S1 which is associated with the inertial forces in the first mode? S1 is gamma 1 times mass times phi 1. So, 4 by 3 times 2M 0, 0M times half and 1. And this comes out to be, this would be M and M. So, both would be 4M by 3, okay. Similarly, S2 phi 2M times phi 2. So, this would be 1 by 3, 2M 0, 0M and minus 1 and 1. So, this would be my, this would be minus 1 by 3. So, there would be minus 2M and M. So, this would be 2M by 3 and this would be minus M by 3, correct. So, and remember this is my U1, this is U2, this is U1, this is U2. So, it would be better interpreted in terms of graphical representation. Total mass is 2M and 2M here and M here. And this is my S. Remember S was 2M and M. And if you look at each mode, so this is S1, okay. At mode, at degree of freedom 1, this is 4M by 3 and it is in positive direction, the inertial force, okay. The same mass 4M by 3 in the positive direction. This plus I have for the second case 2M by 3 and in the opposite direction M by 3, okay. And this is S2. And if you look at this, this is just, you know, if you sum this up, 4M by 3 minus M by 3 becomes M. 4M by 3, 2M by 3 becomes 2M, which is this, okay. And if you want to, let us say, you have a earthquake ground motion UGT for which you have found out the acceleration or you could do displacement as well. I will do the second method using displacement. So, let us say, due to this, for each mode you have A1T and A2T, which is basically the solution of this equation except there is no gamma term here. To convert it from that to for each mode, you need to multiply with the gamma. So, basically for each mode, if you remember, what is the equivalent static force? Fn. Fn is nothing but Sn times A and T, right, in short. So, all I need to do, multiply this with A1T for the first mode, A1T for the first mode, A2T for the second mode times A2T for the second mode. And what I have got is basically the equivalent static forces, okay. And to find out the storage here, what do I need to do? Let us consider storage here. If you consider the equilibrium, what would you get? The total storage here. So, this would be your second degree of freedom, first mode. And this would be first degree of freedom, first mode. This would be second degree of freedom, second mode, first degree of freedom, second mode. Okay. So, we can get U21 as 4M pi 3 A1T V11 as, now it will have, if you take a, it will have some of both. So, you will have 4M by 3 A1T plus 4M by 3 A1T, which is basically 8M by 3 A1T. Similarly, so this is corresponding to this one. For the second mode, this is first mode. For the second mode again, V22 is minus M by 3 A2T. Remember, it is just this force. And V12 is sum of both two. So, I will have 2M by 3 A2T plus or let us say minus here, minus M by 3 A2T, which is basically is M by 3 A2T. Now remember, these things we are doing for the A2T, but even if the quantity is A2 max, for that particular mode, at least I would, I can get using the same procedure, right? It's a different story that when I will combine the first and second mode, I will take the SRSS. Okay. But basically the same procedure is what I will follow. I told you that same, okay, the same expressions I could have gotten using the expressions for displacement or, in short, U1T and U2T. What is my U1T? It's gamma 1 times, well, we had written it as phi n times q and t. Remember, once you get used to all these notations and derivations, even if you forget something, you can easily write it. And that's what happens with me as well.  So phi n times gamma n times d and t. Okay, so this is for n. U1 will become, okay, if you want to write down for U1, it will become gamma 1 times phi n, which is half 1 times d1t. Okay, so I'll just write down here as d1t. Okay. And gamma 1 was what? It was 4 by 3, right? So this basically becomes 2 by 3 d1t or I will keep d1 outside and 4 by 3 d1t. Now, what is d1 here? d1 is nothing but a1 divided by omega n square. What is a1? a1 is k by 2 m. Okay, so this would be 2 m by k times a1.   So I will get here as 1 by 3 minus 1 by 3 d2t. So if you want to look at in terms of displacement, what basically it is saying? In the first mode, you have this displacement, remember this is nothing but U11 U21. This is U12 U22. So for the first mode, this is U12, this is U, sorry, not U12, U11, this is U21 plus U2, sorry, again 1 2, U12 and this is U22. Okay, so this is first mode. This is second mode. So let us, so I am discussing the first procedure, not the equivalent static procedure. So in this procedure, I know that the storage stiffness here is 2k, storage stiffness here is k. Okay, 2k, k here. So what would be my V21 and V11? Similarly, V22 and V12 would be, remember V21 that is here would be storage stiffness which is k times the storage drift which is U21 minus U11. So U21 minus U11 would be 4 by 3 minus 2 by 3. So it is k times 2 by 3 d1t. Now I can substitute my d1t equal to 2m by k1. So that would give me 2m by k, it would become 4m by 3 a1t. Okay, and V11 become storage stiffness which is 2k times the storage deformation U21, okay, minus 0. So that would be U2k, 2k times U21 which is, where is U21, 4 by 3 times d1t. Again I will write as d1t as this quantity here to get this one as 8 a1 by 3. Okay, we can compare the results actually, we can look at the results that we have got and see what is the, so you can do for the second mode. I am not going to repeat it for second mode, you can just do it and we will see how to now obtain the responses using peak response quantities of a SDF system or the response spectra. Okay, all right. Thank you.


Good day, fellow investors! So, as I promised in the last video, today we're going to talk about gold, because with all what's going on with the Fed, the ECB, everybody pumping more and more money into the system, it's very, very interesting to see how gold reacts and what kind of an investing strategy you can apply to gold. We're going to discuss gold prices, the current movement and the long-term movement of gold prices, can gold go to 5,000? I think it can in the future, when? I don't know. Therefore, we will discuss how gold behaves as a hedge and against what it is actually a hedge, can gold be an investment? Can you have gold with a dividend? Yes, you can. You'll see in a bit. Then what's the current situation from a value investing perspective on gold? What's the cost of gold? What's the risk reward? And what are alternatives to investing in gold? Also from a hedging perspective. Let's start on the discussion on the price.  Some were surprised in the comments, but the gold prices actually fell due to the crisis that we now have. Now, why did gold prices fall? Well, usually gold prices, when there is a rush, run to liquidity, gold prices are sold off as any other asset class. So when something goes down in the market, when the market crashes, everything goes down. And that's a given, that's a normal. So you must not take into account the short-term moves to categorize something as an investment. Also, gold is not a hedge against stock price crashing because when stocks price crash, margin calls and everything forced selling, rebalancing, panicking, going to the safe haven of treasuries makes gold go down or not go up immediately. So that's something you have to keep in mind. What gold is a hedge against? It is a hedge against loose monetary policy, loose fiscal policy and lower interest rates. As you can see, since the Fed topped out with increasing interest rates at 2.5%, the beginning of 2019, gold started going up, up and up and especially up when it was clear that the Fed will be forced to lower interest rates because it was clear that higher interest rates are completely detrimental to the economy. So loose monetary policies gold up because gold is supposed to be fixed, relatively fixed in supply. It is a real asset, therefore not real assets, fiat currencies cannot compare to gold. So yes, gold is and should be a hedge. Even if we look at the long-term situation from 2000, the 2000s, when it was clear that there is an issue with the system, with the debt, with the interest rate, gold slowly started going up, then when the Fed started really printing the money, increasing the balance sheet, gold prices anticipated what happened and went from 200 somewhere in 2001 to 1800 in 2011-12. Since then, gold, let's say, peaked in 2012 and still hasn't reached those levels. So those that invested in 2012 are still waiting to break even, also something to note. Another example of how gold prices, when there is run to liquidity, go down 2008, we know what happened in 2008, the crashes, the high volatility and the downward movement of stocks. Similarly, gold prices also went down for a while until it was clear that the Fed will intervene by printing more and more money, which should increase the value, the price of gold in this case, the value of gold is always gold. Now we are currently again in a similar situation. The crisis has arrived, the recession is upcoming, everything like many expect that the Fed offers more than 5 trillion of liquidity, Bank of China pushing money into the system, European Central Bank immediate injection of 135 billion, lower interest rates everywhere, many central banks have lowered interest rates. So here we are and gold should be a long-term hedge against that. And if you ask me the question, will gold go to 5000? I think that somewhere in the future, gold will definitely be at 5000. When? I don't know. How long it will take? Will it take two years? Will it take 20 years? I don't know, but it's likely that due to the policies, both fiscal and monetary, all around the world, due to the conviction that zero interest rates are the best for the economy, at least short-term they are, due to that somewhere sometimes it might explode, we might see inflation, we might see a scenario that we have seen in the 1970s again and then yes, gold as a hedge against inflation might go to 5000, when? I don't know. Now let's see how gold fits the investment categorization as an asset class. How does gold fit your portfolio? Because if gold triples to 5000, it's not said it's the best hedge or investment out there. It's an easy, very beautiful story to tell, huge fiscal deficits, money printing and everything, so buy, buy, buy gold, but especially on YouTube, most of those that sell that story also sell you gold, so they make money on commission. When you buy on yourself, they don't really care where the price of gold goes, so be careful on that. As an investment, let's start with Buffett, we all know he stands on gold, so they dig it out the ground in Africa or someplace, then we melt it down, dig another hole, burn it again and pay people to stand around guarding it. It has no utility and therefore not an investment for Buffett because you don't get anything and we read Buffett's letter to shareholders here on YouTube, so please check that video. Gold doesn't have a dividend, doesn't grow ounce and doesn't reinvest earnings, doesn't compound, which are the key things when it comes to long-term investing and to what Buffett did over the last 60 years, so that's something you have to know when it comes to investing in gold.  There are other and we'll see in a moment. However, if you go to Ray Dalio, you can see that gold can have a yield, can give you a dividend. If you do what he says and he usually says that each portfolio should have an all-weather portfolio, 8% of gold exposure and then you set a fixed percentage of your portfolio in gold, you rebalance after significant moves or 10%. When it goes up to 12%, you sell 2% of your portfolio, bring it down to 10%. So when it goes down to 8%, you bring it back to 10%, rinse and repeat, you get a yield, you get a dividend, you get returns, so you get somehow a return on the asset class, which gold is, but you have to rebalance every few months or every year when it goes up or after 10%, 20%, 30% moves depending on which is your strategy. So that's also something to think about if you want to have gold in your portfolio. When I started this YouTube channel, I was really looking a lot at gold because I'm a value investor and with gold prices, they were there about 1200, 1100, 1300, so much lower than now and I was looking, okay, I'm a value investor, I look for value, I look at the cost, production cost of gold and then I see, okay, what is the risk and reward? When gold was at 1200, I saw a very low risk that it goes lower, yes, it can go to 1000 to 800, but for a short period of time and a very high possibility that it goes higher as it happened because the stability of the system was there, which now isn't then any more financial system, so money printing and everything and gold has jumped. This was my theory 2017-2018 when this channel was in the building because I saw gold as a value investment and this was the core of my theory. The cost of mining gold is around 1000 per ounce, so if gold is close to 1000, when you add the debt, the limit resources that miners have, then you say, okay, it will not go that easy below 1000. So my theory was that gold was an investment, a value investment and therefore I made a lot of videos about gold and now you don't see them anymore because the risk reward has changed. I don't know where gold prices will go in the future, but I know that the margin of safety is closer to 1000 than it is to 1500. That's just my value investing perspective. Also, if I look at Barrick's five-year plan, their cost of sales for gold are set at 1000 again, their long-term plan 2021, they still expect gold prices to be at 1200. So there we are, this might last longer, gold might go to 2000-3000, but then it's a rebalancing question, exposure to portfolio, more a speculative exposure to portfolio, more than a value exposure to portfolio. So from a value investment perspective, I don't think gold has a place now in a value portfolio, in an all-weather portfolio, in let's say a speculative part as a hedge portfolio, then gold might have a position if that fits your investment style and preferences. So really see how it fits you and apply the best strategy for you. What are the gold hedge alternatives? We see all this news pumping money, pumping money, pumping money and then everybody says the first thing that comes to your mind, okay, gold is a hedge against inflation, I need to buy gold, is gold a smart investment now and all the questions that I get. However, if I compare gold prices and copper prices over the last 20 something years, since we have had all the bank, all the expansions, balance sheets, low interest rates and everything, the returns as a commodity, commodities, copper is also limited in supply, also gives you a hedge against possible inflation. So it's not that, yes, gold will go up sometimes, somewhere, but you don't know when. So you might want to own also real assets that are also fixed in supply and that also give you protection, plus maybe even yield a return and we have seen what has happened to farmland. So this is Iowa farmland, went from what 1200 in 1990 to 8000 in 2010. So much more, much higher than gold, much better returns or even similar returns, low interest rates push asset prices, real asset prices to the moon and in addition you get a yield growth, you can reinvest earnings, et cetera, et cetera. So there are plenty of alternatives that you might want to consider when it comes to investing in gold and then comparing what gold is as an asset class, as an investment to other options. We recently discussed visa as a stock. So if there is inflation, if things start, if there is more money in the system, visa will process more money, more payments, more transactions, they will make more money, more margins, higher margins, improved earnings, more dividends, more buybacks. So visa will also be a business that has pricing power that will give you a hedge against whatever we might see in the future, also paperless payments, et cetera, et cetera. So there are plenty of things that gold has to be compared to before jumping into gold.  However, this is a message from my stock market research platform on my gold and gold miners section, how a new member came and he says how he invested everything in gold in 2012 and over the last eight years, nothing happened. He recently broke even and he asks whether he should buy businesses and sell gold, but then if it goes up and that is always the same issue with gold. When you invest it, you are speculating on the gold price. I have no idea where gold prices will go.  It's easy to get into the gold buck, let's say, theory that gold prices will go to the moon, but years and years pass and it doesn't happen very often. 25 years of nothing while the index fund was up 15 times from 1980 to 2005, even longer if you consider long-term returns and we are still eight years of nothing and counting from 2011 to 2012. Also from a very, very long perspective, this is something you have to keep in mind. So to create the best strategy of investing in gold, gold returns $1 in 1802 would now be $4.52. So 0.7% is the annualized return per year compared to stocks 6.6%. $1 is now $704,000, $705,000, probably even more because this ends in 2005-06. So $1 is now probably a million and a half something like that compared to gold that might be 20, 25 bucks. So really consider this when investing and how you're approaching it. You need to have a strategy when it comes to investing in gold. What will happen? How does that fit your portfolio? What are other opportunities out there? Nobody knows what will the price of gold be in 12 to 14 months. And yes, you can take advantage of volatility within an all-weather setting, a little bit of trading, rebalancing for yield. But remember, gold doesn't produce anything. Thank you for watching. Check whatever I do in the links in the description below. You have my blog, podcast, book, stock market, research, platform, charity, whatever. Subscribe to this channel. Click that notification bell. And I'm looking forward to your comments and I'll see you in the next video.


Hi, my name is Philip Burke and I'm from Tipperary Institute in Ireland and I'm founder of Computer Games Festival Games Club. I also lecture in a third level college at Tipperary Institute and the subject area that I specialise in is Games Design and Development. Okay, can you tell us about the game you're preparing for this conference and when are you going to announce it and what does it contain? Okay, so what we're doing as part of the actual workshops over the next couple of days is we're actually going to prepare a game around one of your desert animals, a gerbil and also a desert fox. And we're going to have the actual desert fox chasing that gerbil around the desert. And people involved in the actual workshops are going to help me prepare that game. So many teachers want to ask why should they use games and education? So what benefits can it have for them? Games really bring all the subject areas together. Everything from English, Arabic for preparing the actual stories for the actual games, right through to maths, science. Even art is very important, music is very important. These are all the important components of the game. So it covers all the subjects. So how can games impact children, their imagination, their motivation for learning? How can it impact them? I think when they actually have to solve, you know, creating games and creating good games is about actually coming up with innovative ideas. So I think that's the first thing is that it helps them to develop that process of becoming innovators. And also they have to develop their problem-solving skills as well because building games involves lots of things. Everything, as I said, like from storytelling, right through to the actual technology challenges that you might actually face in actually preparing and getting a game released. So you discussed the camera-based controllers and how they're used in games. So for those who didn't attend the session, can you elaborate on what they are? There are companies working on camera-based controllers. The most important one at the moment is a project called Project Natal. And already I believe there are games being prepared for this particular platform. And it's exciting from the point of view of it actually removes the actual need for the typical hand-based controller. And I'm really looking forward to actually seeing Project Natal released at the end of the year. So you discussed the 4D experience that the students went, experienced the 4D dimension. So what is 4D and how is it different from 3D and 2D? Well, for me 4D is basically using all your senses either to interact and play the game or that the actual gameplay uses your senses. An example of that is where my own students this year in preparing their game for this year's game swap is they actually used a neural impulse actuator as the actual game control. So essentially what happened was they were actually using their mind and their thoughts to actually control the character in the game. So also one of the points you discussed is the use of games in mathematics and how they relate it to matrix. So also can you briefly explain that? Everything in math is important from the point of view of games development. We mentioned everything from linear equations to write true to matrix and matrix transforms. Matrix transforms is the most important thing from the point of view of 3D games. And I think in actually if teachers can elaborate a little bit more to students from the point of view of what they're actually covering in math and where it actually relates to real worth situations like developing games. So I think students will actually begin to appreciate more how important mathematics is. Okay, so first student wants to develop his own game and fourth teacher wants to embrace games in education. What advice would you give to each sector or each segment of this? One of the things that we found very successful in Ireland is we as part of the games lab we run an event called the XNA Ireland Challenge. Not only do we involve students from colleges and universities but we also involve second level schools. And I've seen kids from 13 years of age come up with really innovative ideas as regards games and actually implement those. So do everything from preparing the music, preparing the art to actually programming those games. And one of the actual tools that really helps with that is a development environment called XNA Game Studio. So you also discussed the tips for designing a game. And there were really important tips regarding harmony, regarding audio and having kids not to be so involved with the game because they may get tired of it. So can you highlight on these points? It's important when you're actually coming up with your ideas for the games that you actually test those against as many people as possible. The longer you're actually involved in the actual process of creating your own game the less objective you become. So that's why it's really important to bring in new people all the time, look at how they interact with the game that you're actually playing and see that it continues to be fun. So that's one of the key things. And you asked about harmony as well. That is one of the most difficult things to achieve. It's that harmony between the music, between the actual gameplay and even just the balance of the game. Getting that actually right is a challenge and that's where you need feedback from as many people as possible. Also one type of game you discuss is role-playing games. So how are they different from the usual games we know? When I was actually talking about role-playing games I mentioned one in particular where we did a collaboration project with a local theatre who were running a youth outreach program. So what we did was we actually, during that summer workshop we actually built a game, a role-playing game and we also created a script for the game as well. And we combined both gameplay and actual theatre performance. That was pretty unique. I haven't heard of it happening before. Literally we had 30 to 40 people on stage and the audience were actually controlling the next acts and the next scene that would actually happen out on stage. So how important is audio to the game? Can we have games without audio and will they be successful? Audio, I suppose there's a couple of things. There's the music that you hear or the actual audio cues that you hear. I think it's an absolute component from the point of giving feedback to the user. They must understand place and time and one of the ways in which people understand place is from what they hear around them. I think one of the examples that we gave in the workshop today was if a conversation is happening on my right side I'd like to actually hear that in the earphone on the right. And all those cues build up to create that immersive gaming environment. So many students would think that if they create a game they would never compete with multinational expensive games. So how can a student make his game even though it's simple but make it unique? One of the examples that we mentioned in the workshop is what we participated in is an event called Global Game Jam which is run by the International Games Developers Association. And what happens as part of the Global Game Jam over 48 hours you actually build a game. So literally on 5pm on a Friday you get a team for a game. The team this year was actually deception. And then by 5pm on Sunday you actually have to have a finished title. So those type of events really focus on innovative gameplay. And getting as one of the actual keynotes Kyle Gabler mentioned it's getting that idea across to gamers as quickly as possible. So as quickly and easily as possible. So one of the tips you mentioned is for any game creator is to avoid expectations. So what did you mean by that? Sorry could you repeat that question? Yeah one of the tips for designing a game was to avoid expectations. So how can a game designer avoid an expectation? What does that point? Okay I think that's more to do with resetting expectations.  So it's to actually set expectations for the actual size of the team that you have and the time that you have available as well. So funny question you mentioned that Pac-Man is a game that has been played 10 billion times. So why do you think people frequently play the game although it's very simple and there are other competing games with it. So why is this game so popular? I think the people that really get into it. Pac-Man was one of the first games that appealed to both male and female. So that was one of the first thing that happened with Pac-Man. It's just a simple it's a very simple game and it can be actually played by a 3 year old or an 80 year old and they can they can get enjoyment. So that's the first thing. The second thing I think is everybody aspires to get that perfect score which I believe is 3 million, 333,000, 360 something like that. So everybody aspires to get that perfect score by completing all 255 levels on one line. So that's just one of those really rich and classic games. Thank you so much for this insightful interview and we wish you best of luck in the conference. Thank you very much.