So, in this lecture, we are going to see how to obtain response of a single degree of freedom system subject to harmonic loading, we will first start with undamped system and then find out the solution of the differential equation and then we are going to look into damped system. So, let us get started. So, till now, basically what we have studied on this response of single degree of freedom system, let me draw that chart. So, let us say we have a single degree of freedom system. Now, in terms of the response of a single degree of freedom system, we could have free vibration or we could have forced vibration. Now, in free vibration, it can be characterized or categorized as undamped free vibration and then there would be a damped free vibration. And for the forced vibration, it can be categorized based on what kind of force that is being applied on the single degree of freedom system. So, we could have for example, harmonic or periodic, harmonic or periodic excitation, alright. And then we could have say arbitrary excitation, for example, it could be a pulse, ok. So, pulse excitation like sine pulse or triangular pulse or step function like that or we could also have random excitation. Example could be seismic excitation like you know and Wendell others, ok. Now, we have already studied the free vibration of damped and undamped systems, ok. So, today what we are going to start, we are going to see how to obtain the response of a single degree of freedom system subject to forced vibration and more specifically in this chapter, we are going to focus on harmonic or periodic excitation, ok. So, let us start with that. Now, in terms of periodic excitation, any excitation can be characterized as periodic if or any function can be characterized as periodic. If I can write as f of t is equal to t plus some integral number times t naught, where t naught is the period of function f, ok. So, and like you know j is in the integer, ok. So, it could be I mean it could be from minus infinity to plus infinity all the integers. So, let us say I write it as minus 3, minus 2, 1, 0, 1, 2, 3 and so on, ok. So, if any system can be written like this, it would be characterized as periodic function, ok. And harmonic functions are, ok, functions which can be written as some constant times sin or cos, ok. So, basically if I have functions like something times sin, ok or something times cos, ok. These are called harmonic functions. Now, as you would have noticed, all harmonic functions are periodic. However, not all the periodic functions are harmonic, ok. Now, we will start with harmonic excitation. So, one might ask a question that what is the uses of studying the response of a single degree of freedom system subject to harmonic excitation. In reality, forces are not always harmonic and in reality, forces are not always single degree of freedom system. But the idea is that we are going to start with simple understanding, ok. And then we are going to build on that platform to understand more complicated systems subject to more complex excitations, ok. So, basically the idea is that if you understand the response of single degree of freedom system to harmonic excitation, it would assist you in understanding or provide you insight how the system would behave to other type of excitation, ok. So, I mean if you know like you know Fourier transformation, if you remember from your mathematics class, you know that any periodic function, ok. So, let us say I have any function f of t that I have written above, ok. If it is a periodic function, it can be written as sum of many harmonic excitation, ok. So, we are going to start with harmonic excitation. And once we understand the behavior subject to harmonic excitation, it would also give us insight into understanding to any any type of periodic functions, ok. So, we know that any periodic function f of t can be expressed as sum of these terms, right. This is from your mathematics class, ok. So, I can write this as simply, ok. So, if you understand the response to harmonic excitation, ok, then there are basically, if the excitation reality is being applied as in harmonic excitation, for example, a few examples could be like in a machine foundation. So, you could have machine foundation where the load is applied as you know harmonic excitation or you could have any unbalanced rotatory load. So, any unbalanced and we are going to model that actually in this chapter, these can be, these can be directly modeled as harmonic excitation, ok. And even if it is a periodic excitation, but not harmonic, I could still obtain the response as sum of several harmonic excitation. So, if we understand the response to simple harmonic excitation, ok, then I would be, I would be equipped with like you know knowledge to interpret basically response to any other periodic function as well. And like you know same goes for let us say earthquake as well. So, earthquake in reality might be a random excitation, ok. So, earthquake like you know it is composed of several frequencies, ok. And like you know you will see that later when you will get into that. If the power spectral density, ok, there is a term called power spectral density which basically defines how much of energy is actually so situated at each of these frequencies. So, let us say this is my ground excitation and I am representing it with some random function which is neither periodic, neither harmonic. Even this has basically distribution of these frequencies here, ok. And depending upon which frequency is predominant, I could have, ok, like this up to infinity, ok. And this is basically power spectral density and this is frequency. So, if I understand the response of a single degree of freedom system to individual excitation frequency, right. And then it would even help me in basically analyzing or interpreting the behavior of the structure subject to earthquake excitation as well, ok. So, I hope this provides a background that why we are doing like you know response to harmonic excitation first and how to, how it would be useful in subsequent chapters, ok. So, in terms of harmonic excitation, we are first going to start with an undamped system and then we are going to go to a damped system, ok. So, first let us do undamped harmonic excitation. So, if it is undamped, I know that my damping term in the equation of motion would be 0 for the single degree of freedom system. So, we know that our equation of motion is what the general equation of motion is this. Now, if the damping is 0, I can simply delete this term and then I would be left with, ok, k is equal to p t. Now, any harmonic excitation p t can be represented as either sin omega t or cos omega t, ok. So, first we would be doing sin omega t and the response subject to cos omega t would also not differ by much, ok. So, this excitation, which I am representing as, ok, p t equal to p naught times sin omega t, p naught is the force amplitude of the force that is being applied, ok. So, basically it is the, whatever the force that you are applying, this is the peak value of that force, ok. So, this is p naught is the peak value or amplitude of the applied force and note here, now I have an additional frequency omega, ok. And this is different from the natural frequency of the system which was omega n, right. And we saw that my omega n for this system, right, I could simply get it as under root k by m, correct. Now, omega is different from this omega n, omega is basically the applied frequency of the force, ok, of the harmonic force that is there and it is called excitation frequency or the forcing frequency. So, we would be referring it further as either excitation frequency or forcing frequency, ok. And if you plot this function p naught sin omega t, you know that it would look like a simply sinusoidal function, ok, which let us say starts here, ok. So, I can plot it like that. So, this is my applied force, ok. And this is, sorry, this is time here, ok. This is the forcing time period or excitation time period which I can write it as 2 pi by omega, ok. So, now basically my differential equation becomes, ok, my differential equation becomes p naught sin omega t. And now I need to solve this differential equation. So, we are going to employ the same techniques that we have used in previous chapters to solve this second order linear differential equation, ok. So, how did we solve it? Basically, we said that the total response can be represented as sum of particular solution and then sum of complementary solution. And this complementary solution had two unknown constant, ok. So, this particular solution is any unique solution that can satisfy this equation, right. And complementary solution was the general solution to the homogeneous part of this equation. So, if you, in this equation here, if you set the right hand side equal to 0 and then whatever the solution you got, that was basically the complementary homogeneous solution, ok. And it had two unknown constants and those unknown constants were determined using initial conditions, ok. So, let us assume our initial conditions are given in terms of u 0 and u dot 0. So, initial displacement and velocity, ok. Now, let us see how do we solve this. So, basically in terms of particular solution here, I would write this as, let us say I want to get any unique solution that satisfy this equation, alright. Now, what I see here on the left hand side of this differential equation, I have a term which is u of t or up of t and then a double differential of term, right. And then sum of that is actually equal to some sine function times some constant, ok. Now, let us say if I assume this u p of t as some constant times sine omega t, knowing that the sum of term u p of t and then the double differentiation of that term would be again some constant time sine t, right. Keeping that in mind, I have selected this as particular solution. The only thing that need to be found out here that what is the value of c. So, if I substitute this here, I can write it as this m omega square c u p of t or let me just write instead of u p, here it would be sine of omega t, ok. And that should be, sorry, there is another term here k times c sine omega t and that should be equal to p naught of sine omega t. So, if we compare the coefficient of sine omega t, right, we get as the value of c as p naught divided by k minus m omega square and I can further rearrange the terms here to get p naught by k, ok. And this would be omega square divided by k by m, ok.  So, that I am going to substitute here to get this as omega by omega n square, ok. So, the particular solution I have obtained as this term here, all right. Now, the second thing that remains is the complementary solution which is nothing but solution to this equation here. And that we saw in previous chapters can be directly written as some constant times cos of omega n t plus another constant times sine of omega n t. So, I can write the total solution as particular solution plus the complementary solution, ok. And this can be written as this is the particular solution and then a cos omega n t plus b sine omega n t. Now, I have two unknown constants here a and b which can be found out by substituting at t equal to 0 initial displacement at the initial velocity. So, you will need to differentiate it and then substitute again t equal to 0. If you do that, you will see after substituting these conditions u equal to u of 0 and then u dot of t equal to u of 0 at t equal to 0, you obtain the final solution as u of t should be equal to u of 0 cos omega n t plus u dot of 0 and then this term here sine omega n t and then I have the particular solution which is sine omega t, ok. So, this is the final solution for undamped vibration subject to harmonic excitation which is P naught sine omega t, ok. Now, what you see here that one thing I am important thing to note here we have in there is one not two frequencies, right. Let me just write it here again. We have two frequencies. First frequency is basically the natural frequency of the system and then the second frequency is actually the applied frequency of the force or the excitation frequency, ok. So, one question might arise if you try to plot this system at what frequency would the system vibrate? Would it vibrate at omega n t or would it vibrate at omega or the applied frequency? You know and then the question also becomes that what do these frequency actually mean? What is the what is the physical significance of this frequency? So, let us draw each term that we have here, ok and then see how do they look like, ok. So, let me say I am trying to draw here the displacement. I am not going to say if it is the total displacement of if it is like one of these displacement. Let me first just say that it is a displacement quantity, ok, right.  Then I am going to plot this term here, ok. So, this is by let me just call it u 1. Let me call it as u 2, ok. So, if you plot u 1 what you are going to see? It vibrates at its own frequency omega n which is the natural frequency of the system and let us say it looks like something like this with some initial conditions. Remember that I do not have any damping. So, amplitude is going to remain constant here, ok and it will keep on vibrating like this with time period equal to 2 pi by omega n. Now, I have another frequency, another term the sin omega theta, ok and let us see let us plot that and see what do we get, ok. So, let us say it looks like something like this. Now, which one would look like which plot for example, you know you might argue that why it is why the first u 1 looks like this and why not the other way? Well, it depends on the relative ratio of the frequency omega by omega n, ok. So, for just this case just for demonstration I am assuming that let us say it looks like this.  So, these are the two components, ok of the response that we get subject to the harmonic excitation of single degree of freedom system, ok. It is important to note here this u 1 is basically. So, we can say that two components of the vibration response the component 1 is basically vibrates at omega n of the natural frequency of the system, ok and u 2 basically it vibrates with the whatever the applied frequency is, ok. So, it vibrates at the applied frequency, ok. This response is called force response or steady state response and we are going to come back to that why do we call it like that or steady state response and this is called transient response, ok.  So, if I try to again draw the total response which is u equal to u 1 plus u 2, ok it would look like something like this, ok. It is going to look like or let me just first draw it then we are going to. So, it is going to look like something like this, this is not to scale of course, ok. You can perhaps plot this kind of graph using MATLAB. So, this is the total response, ok. So, this is u 1 plus u 2 state transient response plus steady state response which is basically some of these two response. So, what it is basically doing? It is taking the transient response which is u 1 and then it is vibrating along this steady state response. So, the sum of two function would look like this. This is basically the total response of the system, ok. So, there are certain characteristics of each of these response, ok. If you look at the transient response or let me just write down the expression for ut again the total force or sorry the total displacement, ok.  So, this is sin of omega t. So, as I said this is the transient response and this is the forced or more commonly it is called steady state response. And these terms are better explained when we will get into like you know damped forced vibration. However, note here the transient response depends on initial conditions, right. It depends on u of 0 it depends of u dot of 0. However, my force or steady state response is it does not depend on initial condition, ok. And that you can easily imagine. For example, consider let us say or not like that let us draw the spring mass representation, ok. If you apply a force which is harmonic force, right, whether there is initial condition or not, whether there is no initial velocity, ok and no initial displacement is still as long as you have applied force, ok. The body would have some response, ok and that is the steady state response, ok due to this force, all right. Now, for the transient response it depends on the initial condition. However, even if the initial conditions are 0 this force P naught would still provide some response to the system that would contribute or that would oscillate at the frequency omega n. And how much of that response would be the total response contribute to the total response? Well, it depends it would depend on the frequency omega by omega n which we would see later, ok. Now, in reality what happens? So, if you look at here you know the transient response it seems that it would remain constant, ok and it would not decrease with time. So, if you look at here let me go back to this graph here. So, if you look at u 1 the amplitude actually remains constant, ok and it does not change with time. However, in reality all the system would have certain amount of damping and what that damping actually does although not evident from this undamped the equation from the undamped of an undamped vibration. The damping would reduce this transient response to 0 after sufficient amount of time. However, my force response of the steady response due to applied force would still remain and that is why it is called transient. Transient means something that is between a state or a changing state, ok. So, initially system was let us say it was like you know with some initial condition, ok and then finally it achieves a steady state which is the force response, ok. So, in between whatever the response that system has is called the transient response which actually dyes out in a damped system, but you cannot evidently see here. So, when we will do the damped system we will see that this response actually dyes down. Now, in dynamics we are mostly concerned in like in most of the cases whatever is the force steady state response, force response or the steady state response. So, what we have here because in reality all the system would have some damping, ok and the effect of applied force is measured in terms of steady state response. So, we are going to focus right now we are going to neglect this not neglect this, but we are going to turn our focus now to steady state response, ok because as we will see for the damped system from there that that their mathematical equation for the ut this transient response actually dyes down, ok. So, let us look at the steady state response, ok and I am going to write is this although as ut, but remember that I am only considering the steady state response part of this, ok because this is what would be important. However, you have to keep in mind initially the system would have some transient response and steady state response, ok, alright. So, look let us look at what is the steady state response I can write my steady state response as this term times sin of omega t, ok. Now, if you remember from this equation of motion let us say this was the original equation of motion correct. Let us say I ignore the dynamic effect in the system. So, I basically ignore this term here, ok. If I ignore this term here can I say the static displacement subject to a force p which is varying with time, ok would be u of t equal to p of t by k which is nothing but p naught by k times psi omega t, ok. Now, you have to understand what do I mean by this, ok. So, instead of saying u of t let me just write it as u of s t saying this is the static displacement, ok. It simply means that there is no mass in the system. So, that is why there is no dynamic effect and if I am applying a sinusoidal force, ok, the deformation is basically whatever the force is applied divided by k, ok. Now, it was very simple if simply a force let us say if a spring is given to you and set like you know force p naught is applied you would simply find out that the displacement is p naught by the spring constant k here. But if the force is varying with time then you at each and every instant your displacement would be simply p t by k if no dynamic effect is considered, ok. So, this is static displacement here is basically, ok whatever the force you are applying the sinusoidal force here if that is applied without any dynamic effect of the mass then what is the displacement, ok. So, it is different from the ramp loading in which we said like you know we are applying a force p naught and the final displacement was basically p naught by k if this was large enough with respect to the time period of the system, ok. In this sinusoidal variation of the force we are saying that mass term is 0. So, the resultant displacement is and basically force divided by k. Now, the significance of this is that if there was no mass in the system and if I have applied a sinusoidal load the peak displacement of that is u of s t naught equal to p naught by k and it is the same p naught by k that I have here. So, I am going to write my u of t or the dynamic displacement as static displacement times this term that I have here times sin omega t. So, you need to understand what is this u of s t times naught means, ok. u s t naught is basically the maximum value of static displacement, ok, for a time varying load, ok. Otherwise you would argue like you know what do you mean by the maximum displacement of static load because if it is a static load whatever the load does is apply that is the divided by k is the maximum load. However, I am saying that if you apply a load which is varying with time and there is no mass then at each instant of time you can divide that load by the displays or the stiffness k to get the displacement at time t but that is still the static displacement because there is a mass in the system and this u s t of naught represents that peak static displacement subject to the sinusoidal or the harmonic excitation, ok. So, I hope that is clear to you, alright. Now, if I have this term here, ok, let us see what does it mean. For example, you see here that the term omega by omega n or the frequency ratio, ok, of the applied frequency divided by the frequency the natural frequency of the system, ok. Depending upon the value of omega by omega n by value of u t would change, ok. For example, let us consider two extreme cases, ok or the two cases let us say when omega or the applied frequency is smaller than omega n then what will happen this denominator term would be positive and u t would have same sign as sign of t. So, u t vary accordingly to whatever the variation of the right hand term, ok. So, u t vary are same as whatever the value of applied force in the same direction. So, if this is then then u t and p t are of same sign, ok and basically it means that they are in phase. However, let us consider the second case when omega is greater than omega n then you will see that denominator is negative and then u t, ok, varies as negative of sign omega t which is again negative of p of t. So, in that case omega n omega t are of opposite sign, ok and it is said out of phase, ok. Physically what it means basically that you apply a force in first case if you apply a force to the right, ok, then your system also moves to the right, ok. And if in the second case it means that if you apply the force to the right the system moves to the left and like you know you might have a doubt well how is that physically possible remember that we are not considering here monotonic load, ok. We are considering here a sinusoidal load, ok. So, depending upon the frequency of omega by omega n once the motion is started and like you know providing this vibrational motion here, ok. It might not be in the same direction at any time instant as the applied force, ok. It might move to so if my force is positive it might be moving to the negative direction and that depends on the ratio of omega by omega n, alright. So, let us see if we can formulate some mathematical expression to actually represent my dynamic displacement. So, remember ut here is my dynamic steady state displacement and although in like you know in further lectures or like in a further when we discuss we are going to not I am not going to always refer it as a steady state displacement, but it should be understood, ok. You should understand that I am talking about a steady state displacement but just looking at the expression of ut, ok. So, this is the dynamic steady state displacement, ok. Now can you imagine if I again have a dynamic steady state displacement which is varying as a function of sin of omega t it would again have some peak value. So, it would have some basically amplitude. Can I write my ut as u naught which is the dynamic amplitude or the dynamic displacement amplitude, ok times sin omega t, ok where I write it as u of s t times r d times sin omega t, ok. Now what is the r d here? If you compare this equation here and this equation here, basically r d is this expression 1 minus omega by omega n square, ok. Now if you plot this expression, ok this 1 minus omega by omega n square how does it look like, ok. We talked about the phase. Let us see how does it look like, ok. So, I know that when omega by omega n or the horizontal let me just again draw here, first let me draw here, ok. This is my frequency ratio. So, horizontal direction is basically omega by omega n and the vertical axis is 1 minus omega by omega n square. Now I know that at a very small value of omega by omega n this is equal to 1. So, let us say this is 1 here, this is 2, this is 0 minus 1 minus 2 and so on. So, it starts from 1, ok. I do not know what happens to it as I increase the value of omega by omega n, ok. So, what happens? The value of the ratio r d actually increases because as you increase the omega by omega n, the denominator decreases. So, the overall r d which is 1 inverse of that value would increase. So, it increases something like that. I do not know how it would increase. Other extreme is that by omega by omega n is a very large value. So, it is infinity in that case r d would be equal to 0, ok. So, it will start somewhere from here and as you decrease the value of omega by omega n, then your denominator would actually increase, but in the negative direction, ok. And can I say when omega by omega n is equal to 1, then the value becomes unbounded, ok. So, it becomes unbounded here and unbounded here on the negative side. So, the graph would look like something like this. This is of this specific quantity, ok. And the value actually depends on the frequency ratio, ok. Let us say this is 0, 1, 2 and so on, ok. So, it depends on the frequency ratio omega by omega n, ok. And whether it is positive, this expression or whether it is negative, that also depends on omega by omega. So, what I would do? I would further write this term here or let me just actually delete it here, ok. I do not need that to plot this, ok. I am just trying to plot here u naught, u s t of naught times this expression right here, ok. So, now what I am going to do here?  Now, I am going to use that factor r d times sin omega t and I am adding a new term here called phi, ok. Now, this phi is called the phase angle, ok.  So, I have originally this term, but r d is actually the absolute magnitude of this term here. So, r d would always be positive. This is the mod of this function right here, ok. If that is there, then you would ask me, well, then how does the sin of sin t or omega sin of omega t is actually taken into consideration? Because initially it was coming due to this term. Precisely for that I have this angle phi here. Remember we said that when omega by omega n is less than 1, then it is where it varies, the u t varies as a negative of, sorry, it varies as the same positive as sin of omega t and if it is greater than omega by omega, if it is greater than 1, then u varies as negative of that. So, take to take that sin into consideration, I am writing this term here and r d I have taken magnitude of that and all the sin is now being taken care of this phi or the phase angle, ok. So, this is this term here and the phase angle now I am defining phi is equal to 0 for omega less than omega n and it is equal to omega or it is equal to 180 degree for omega greater than omega n, which makes sense. If you look at it, right, if you look at it here, I would again have the same expression, ok. So, let me first write this as dynamic displacement times sin omega t minus phi. Now, remember that now this is always positive because r d I am considering as magnitude, ok. Now, if omega by omega n is less than 1, which is basically this condition here, my phi is equal to 0. So, u t would become, ok, u naught of sin omega t minus 0, which is sin omega t. However, if omega is greater than omega, then u t would become u naught times sin omega t minus phi, which basically becomes minus of sin omega t. So, I will have a negative sin here, ok. So, this is in phase and this is out of phase and when I say in phase and out of phase is basically the displacement u t with respect to the applied force P of t, ok. Once I have this information, let us plot, ok, let us plot r d and let us plot basically the phi, the phase angle and this then see how do they look like, ok. I know that both of them depends on the frequency ratio.  So, that would mean that this function here would simply invert to the positive side of the x axis, ok. So, let us let us try to draw that. So, I have this r d, ok. So, let me just write it here. So, r d is here, ok. I am plotting r d here and then below this I am also plotting by phase angle, which is phi and both these axis are basically omega by omega n, ok. Value of 0 by 1, let us say 2, 3, 4 and so on, ok. Similarly, 0, 1, 2, 3, 4 and so on and the vertical axis in this case for the phi this is 0 degree, 90 degree, 180 degree and here it is 0, this is the value of 1, then 2, then 3 and so on, ok. So, if I simply take the mod of this function, I again it starts at 1 for a very small value of omega by omega n, ok. So, r d let me just write down the expression anyway for your convenience. So, omega by omega n when it is very small it is equal to 1 and it is unbounded at value of omega by omega n equal to 1, ok. So, it will start at 1 and then it will go like this. And the second case, ok, what happens as you increase the value of omega by omega n it starts from here it will start at value of omega by omega 1 equal to 1 approaching to infinity and then it would decrease as you start and at some point it is going to cross the x axis and then go to 0 for a very large value of omega by omega n, ok. Now, compared to that I have the phase angle, ok. The phase angle which is if omega by omega n is less than equal to 1 it is 0 and if it is greater than 1 it is 180 degree, ok. So, we have talked a lot about r d and phi. Let us see what is the physical significance of these two quantities, ok.  So, if the force had been applied statically whatever the amplitude of that function is let us say p naught by k and if the system had mass and then the force is being applied, ok whatever the dynamic displacement that you get the ratio of both two is defined as r d and it is called response or displacement or deformation response factor, ok. So, it is called deformation response factor, alright and we will see later that there are different type of response factors, ok. Right now we are only dealing with deformation response factor which represents what is the amplification in the response, amplification or reduction whatever happens with respect to the static system, ok. r d is a measure of that amplification or reduction, alright and phi represents that if the force p t is applied, if the force p t is applied, ok. Then with respect to the applied force p t whether my response or the u is in with in phase or whether it is out of phase. So, it represents phase of u t with respect to applied force. So, if my force p t is moving right whether my displacement is also in the right direction or whether it is opposite to the applied force in the left direction. So, the phi represents that, ok. Now this plot of r d or the displacement response factor here it gives me several useful information in terms of dynamic response because that is what we are trying to study here, right. So, basically if I have a system and a harmonic load is applied on that based on the frequency ratio omega by omega n I would see what happens to my r d, ok which is the displacement response factor. So, let us come to this graph here, ok. If the value of omega by omega n is very very small I do not see any amplification, right. My r d is equal to 1 that means u naught is equal to u s t of naught. So, means dynamic is equal to starting. So, let us consider first case that omega by omega n is very small. It means that omega is very small the applied frequency is very small. When the applied frequency is very small it means that it is a slowly varying load, ok. It is a slowly varying load and in that case r d is equal to 1 or dynamic displacement is equal to static displacement, ok. So, if the load is applied very slowly we do not see any amplification in the system with respect to the static displacement, ok and that that should be intuitive to you, is not it. I mean if you are applying load at very very small frequency then the your system would move, ok in tandem with whatever the force you are applying without having the much dynamic effect of the mass and that is why your dynamic displacement is equal to the static displacement, ok. Now, in the second case when omega by omega n is very very large that means a rapidly applying force or rapidly varying force. In that case r d is equal to 0 or that means your displacement dynamic displacement is equal to 0 it means it implies that for a rapidly varying force, ok you do not see any response in the system, ok. So, if you try to imagine this basically what do you have system on which you are applying a very like you know fast moving load this system would not even actually respond to this, ok that is what basically it means the displacement would be equal to 0. Now, if third case omega by omega n, ok so these were the two extreme cases if it is greater than equal to root 2, ok which basically corresponds to let us say this point here root 2, ok. Then my r d starts to become smaller than 1, ok my r d is so that I do not get any amplification in the response, ok. So, the dynamic behavior actually does not lead to amplification but actually leads to reduction in the response and as we will see later this is very useful in some design of some systems. For example, let us say I have a system, ok a static system a building just sitting there would I not be happy that if I apply some force or the frequency on my structure the response become actually less than the response it would have if the load had been applied statically I would be very happy, right. So, that depends on the applied frequency and this actually property is utilized in a vibration isolation, ok we will come back at later. So, my dynamic for r d omega by omega n greater than 1 dynamic is actually less than static, ok and of course then the case where omega by omega n is equal to 1 then r d approaches to infinity, ok. So, the response becomes unbounded, ok the response becomes unbounded, all right. So, these are the four typical cases which you might experience in real life based on how the frequency of the applied force varies with respect to frequency of the system and that decides what is the response of the system, ok. Now, we talked about this, this case here when omega by omega n is equal to 1 the response becomes unbounded we are not going to define the frequency or the applied frequency which is called resonance frequency, ok or the frequency is actually resonant frequency this phenomena, ok. So, this would be the forcing frequency or excitation frequency, forcing or excitation frequency at which the r d or the displacement response factor, ok becomes maximum, ok. So, whatever the value of applied frequency at which it becomes maximum is called the resonant frequency and this phenomena is actually called resonance. Many of you might be aware with this, ok this phenomena here. Now, in this case what happens so the resonance is basically maximum happens when omega by omega n as we saw here is actually equal to 1. So, applied frequency is actually equal to the natural frequency of the system so my r d become maximum. Now, when this happens at this point actually the solution that we have obtained for ut is not valid anymore, ok and we actually obtain a new solution, ok. So, basically what I am saying in this case the solution that we had obtained for our equation of motion this is not valid anymore, ok only at this point, ok it is valid it was valid at all other point. So, we developed a new solution, ok when this happens, ok for this case what do we do basically remember our particular solution initially we had assumed c times sin omega t. Now, omega is equal to omega n so this is c sin omega n t, ok and our homogeneous solution or the complementary solution was a sin omega n t plus b sin sorry first term should be cos here. So, let me just write it again, ok. Now, if you look at this carefully this is not a unique solution anymore because the sin omega n t term because your omega is now equal to omega n is a part of this term here initially because they were different omega was not equal to omega n, ok this was a unique solution but now because I have an omega same omega n term here sin omega n term and then it is part of this complementary solution. So, for these type of cases, ok what we do the solution is obtained by assuming u of p t equal to c t the time constant t, ok as this and if you substitute this in this equation you will obtain as u of particular solution as and you can like you know do that calculation yourself can obtain that as cos omega n t, ok and the total solution you can write it as a cos omega n t plus b sin omega n t p naught omega n t cos omega n t. So, you can see with omega is equal to omega n the solution does not suddenly become or the response does not suddenly become unbounded, ok. If you solve this or if you obtain the response for the initial condition u of 0 and u dot of 0 you will get some expression, ok and what I want to do let us say for both these parameters equal to 0 you can get the response u t as minus p naught by k cos omega n t and sin omega n t, right and if you try to plot this function and you can use any of the softwares like in order you can use MATLAB to plot it it would look like something like this, ok it would look like some let me just plot it here it would be easier to do so. So, my response does become unbounded but I cannot represent my solution using the previous equation I have to derive the new solution for omega by omega n equal to 1, ok because of the reason that we just discussed here now that is not a particular solution is not a unique solution anymore and what happens over the time the amplitude keep on increasing if you plot it the amplitude will keep on increasing and it would become unbounded it does not become unbounded suddenly. Now try to imagine this scenario in reality, ok in reality assuming that it is an undamped system which is in self like you know assumption that might not be true all the system would have some amount of damping but let us say I do have a system which has very small value of damping that can be neglected, ok even given that fact what will happen as the real system will start to vibrate and start the response start to become unbounded what will happen if the displacement exceeds a certain value for example if it is a brittle structure like concrete it will start to break, right concrete will start to get damaged and it will start to break at certain value so you do not get actually infinite response but what happens for the brittle system as the concrete gets damaged its stiffness changes so your omega n actually changes, ok so although initially you had applied a response sin omega t for which initially omega, ok so applied frequency become initial value of omega n t it started the resonance the system starts to get damaged so this omega n changes, ok so let us say this is omega n, ok after sometime f so now this omega is different from the omega n the natural frequency of the system so basically this system would come out of the resonance because the frequency of the system itself like you know changes due to unbounded deformation once it starts to develop other solution like in other situation could be if it is not brittle but let us say it is very ductile like for example steel bailing or some steel structure then it would yield at certain point, ok it won't so again the stiffness would change, ok and the forcing frequency again would not be equal to the applied frequency, ok and then this basically solution or it would come out of resonance and again we can go back to the original solution, ok so I hope this discussion gave you some idea what happens in an undamped harmonic motion of a single degree of freedom system where basically undamped system is an assumption in reality all the system would have some damping so in next lecture we are going to see what happens when the system has damping we are going to look into mathematical formulation and then the physical interpretation of the result, ok so this lecture we are going to conclude it here, all right thank you


Welcome back. It's the Friday frenzy edition of the breakfast and it's time for us to look at the headlines. And we begin with the PANCH newspaper. With National Assembly leadership, APC moves to pacify agreed aspirants. The rider there, Stop Agitation Party will revisit zoning formula at Damu Beg's aspirants. Details of that is on page two of the PANCH newspaper. Right on top of the PANCH newspaper, you have Buhari or Shibajo, governors get 651 million dollar hardship allowance. You have details of that on page 17. And debt servicing heats 112 million dollars monthly. And that's according to the CBN, details of that is on page 18 of the PANCH newspaper. Okay, that's the much I'll be taking from the PANCH newspaper. Okay. We move from the PANCH to where are we going right now? The Guardian. Okay. We'll move to the Guardian now. The leading headline there is underutilization debts Hubble 4.12 billion dollar real rehabilitation project. We also have another story here which says National Assembly leadership aspirants adamant despite APC's promise to revisit zoning. Okay. Let's reclaim stolen mandate PDP's lost glory article tax party members. That story is on the Guardian as well. Then we have intrigues protest as immigration boss gets controversial tenure extension. Nigeria's oil output heats record low as Angola takes the lead is also there on the Guardian newspaper. And I think that's the much we can take on the Guardian this morning. Okay. From the Guardian newspaper we'll move to Daily Independent. The Daily Independent newspaper is the next one we'll be taking. And it leads with Nigeria's 77 trillionaire debt calamity of highest proportion. That's Pengasin. Over there on the front page you also have Buhari appoints AIG Garba Umar as advisor on security. Page six is where you have details of that appointment. 10th National Assembly reps caucus vows to review APC zoning arrangement. Page seven is where you find details of that nine metal lads on thunderstorm activities in northern states. That will be found on page seven. Supreme Court of Johnson and the county's case to September 14th. Now that's much I'll be taken from the Daily Independent. Okay. We'll take the next paper. And the next paper is Daily Trust. Daily Trust leads with uncertainty over multi-billionaire Abuja Airport runway project. And the writers are criticisms trail fresh 3.4 billionaire consultancy fee. Stakeholders flay delay in project execution and 65 billionaire budgeted in five years. Another story we have not where's deserves the Senate presidency. That is according to Al-Haji Abdulaziz Yari who is also vying for the position. National Assembly leadership aspirants. Okay, leadership aspirants not consulted before zoning. We also have dango terefinery to cut fuel import evils of subsidy shortly. That is Penn Gassen. Tribunal adjoins at TICU APM's motions to May 18. Okay, those are the headlines from Daily Trust newspaper. And those are the four dailies will be taking a look at this morning and we've been joined by our guest. Jide Johnson, senior lecturer at Nigerian Institute of Journalism. He's joining us from Lagos. Good morning, Mr. Jide Johnson. Good morning. It's a pleasure to be with you. Good morning to our viewers all over the world. Thank you for having me. Unfortunately, I couldn't join through. It's your means and the power is you. It's a national problem. Isn't that unfortunate? Would have loved to see your face this morning. Not just a still picture. Yeah, I remember a national event that was hard one time in the stadium and the light went off. And it was like, okay. And then there was another one held in Emo state. The governor was addressing the people and the light went off in government house. You know, if you could be going off only when these people are talking to be very good. All right, let's start with the head, the punch newspaper. And right on top, I want to start with this very one that I'm sure every Nigerian would be interested in looking at. And that's Buhari or Shibat, your governors get 651 million Naira hardship allowance. They actually decorated for Nigerians. Due to their inability to provide critical direction to both the economy, political and the social life of Nigerians. So if there is a justification of their failure in paying themselves hardship allowance issues that they've left untold hardship on Nigeria. But it's unfortunate that we found ourselves where we found ourselves whereby these people can just allocate whatever money they want to themselves. Without any, any, any shintilla of, of, of, of respect for public opinion, any, for due process and the rest of the way. Why will you justify this hardship? Are they going to achieve the living government houses to provide them with opportunities to buy whichever car. They want to buy for themselves, they live in opulence. It took this with worth. One of the stories we read where a governor allocated two billion Naira for himself, his wife, and then the deputy as parting gift, official cash for themselves. So you begin to wonder, do we even have critical institutions? What is the National Assembly doing? We are the civil society. We are these accountability institutions that hold government accountable. What's even the role of the citizens here in all of these things? And I don't know how the citizens are not right by this. And at the same time, they still go ahead to cast their vote for people that are imposed on told hardship on them. And because they know they can get away with this, they've tried it many times. They keep doing it. What can anyone tell me? What type of ashen is the government still going through? Is the government still going through? They are collecting on behalf of the people. Since they are the leaders. They have gone through, they have asked just the two to take. They went to England to treat the two to take.  One of the performers said it was a civil society. Said the president started his tenure with a problem and it is ending it with two problems. Fantastic. So he was not able to hear Nigerian very well when he started. And when he's leaving, he's not able to talk to Nigerians. So it's rather unfortunate because I see no justification for this. And I think that there's a need for lawyers, civil society lawyers to take class action. Class actions against these governors. Because they have been able to let the judiciary interpret whether they indeed have access to this money or not. Let this matter be taken to court and be challenged. And if there is a rule for them to reform this money, then they have to reform it back to the public office. Because it's unbelievable. It's not a 51 million. Yeah, well this brings us back to the issue of the independence of the legislative arm of government. So let's talk about the National Assembly leadership, which is bringing up some really strong questions up in the air. APC moves to pacify a grieved aspirant. With the right of there, stop agitation, party will revisit zoning formula. That's Adamu Beggin aspirant of the APC. Talk to us on this. We would have been partisan and stating the facts and stating the ideas. I think when it comes to independence of the legislature, I think at the national level, PDP has said that at the APC, with respect to independence of the legislature from the executive, all you need to do is just to go back and look at the records. Compared to what we have. If you compare the Eighth Assembly of... Bukola Saraki? Bukola Saraki with the Ninth Assembly of Lawa and that approves anything. We are talking about death. In all of the stories, you know stories about death. Everything the president presented before the Ninth Assembly, they approved. And the APC is trying to make the National Assembly to be a rubber stamp. Now the Ninth Assembly has not been inaugurated. Members have not been inaugurated. And then the party came out and said, okay, we have loaned this office with the principal offices to this particular area. Not only did they zone it to a particular area, they zone it to specific... Specific individuals elected in the Senate. And now what kind of independence do you expect from such body? At least if you want to do that, you can zone it and then you continue with your consultation and in the process of your consultation you say, okay, within your caucus, you tell your caucus member, this is the person we have nominated and this is the person we want you to work for. Not for the party to quickly come out and release a statement. And this is the first time I've seen this type of thing. That's never happened in the history of this country before. That the party will just come out and will release a list. Okay, we have zoned these offices and terms we have zoned to this list is to this various individual. And that's why you see that they have many systems. Both at the as-of-rep level and at the Senate level, you see the as-of-rep members that have gone to see... Two days ago, the as-of-rep members went to see the national... The as-of-rep for the office of the as-of-rep went to see the national chairman. Yesterday, those of the Senate went to see the national chairman. And I think that the party needs to consult white and the party needs to involve critical stakeholders. There's no... There's no crime in zoning and there's no crime in identifying who you want to take the leadership position. But there are better ways in doing it in democracy, not with a fear. If democracy is based on consultation, it's based on the principle of free and fair processes. So, if you want to do that as a party, you consult those that you need to consult one to just oppose this with the letter. The governor of... The governor of Undustry to release some days ago questioning the rationale behind the party coming up with the zoning arrangement and then without getting the governor's info. So, I think that the ATC should not start on this note. I think the matter should be revisited. And in revisiting the matter, explain to the critical stakeholders why you have taken decision. I think there's need for you to make amendments. You make some amendments. However, I think there is not even the responsibility of the party. By virtue of the constitution, provisions of the constitution, it is members of the National Assembly that will elect their presiding officers. In this state, it's just the grand norm. There's no law that says that it must come from the majority. For example, if, on the day of the inauguration, the members decide to elect someone for the minority. Just like what we have in the night in the gate assembly when Dogara emerged, so the opposition can decide to work for someone who is against the interests of the party that has the majority, whether a single majority, because this does not even have supermajority in order to maintain independence of the legislature. But what we have seen over time is the executive trying to control our homes in the legislature. And I think that it's the party for the independence of the legislature beyond the selfish interest of those aspiring for these offices. Beyond that, because if you ask me, I'll ask you all of these gladiators that are parading themselves to be the speaker or to be the selling person. What have they done since they've been in the National Assembly? That is a very valid question. I've asked you, what has the speakership of Bajabia Mila done for Ligos State as a whole? What has the speakership done for Ligos State? Let me talk about the Senatorial District. What has the speakership of Bajabia Mila done for Ligos Central as a whole? What has it done? For the Southwest, because it got the speakership on the basis of the fact that the speakership was done to the Southwest. And I want anybody to point to me the impact of Bajabia Mila in Ligos State, the impact of Bajabia Mila in Nekiti State, the impact of the Gajabia Mila in Oshu State, the impact of... I'm not too sure he even visited those states. So when these people parade themselves on the basis of zoning that they want this thing to be zoned to their geopolitical zone, it is not for the interest of the people. It's for their own selfish interest. Because I ask people this, Bajabia Mila, for example, I'm from Ligos State and that's why I'm being partially... Bajabia Mila, for example, has been a minority leader in the... He was minority leader in the Seventh Assembly. He was majority leader in the Eighth Assembly. He's the speaker in the Ninth Assembly. So there are two federal constituencies in Ligos. There are 24 federal constituencies in Ligos. Anybody should tell me what he has done. He has not done anything for my own constituency. I know. I know that for sure. So what people talk about is just about their selfish interest. And I love this Michael Jackson song.  That's why the Kuala Kip is obsessed with the Seventh and Fifty-One Minions as hardship allowance. When you and I are going through hardship, you can look at how hot the weather is. I don't have like that. That's why you can't see my face. That's why you can only hear my voice. And yet, they are giving themselves hardship allowance. They are collecting on behalf of the people they are pretending over. They have newspaper allowance and I don't know how many of them buy newspapers nowadays when they have their tabs and all that and given free airtime by some service providers and all that. But that's a story for another day. Another worrisome thing on the headlines is the fact that underutilization debts have hobbled the $4.12 billion real rehabilitation project. Especially now that they are milling the possibility of removing subsidy and talking about buying buses, talking about giving 5,000 to the people to ameliorate the sufferings that they are going to suffer. That's the hardship allowance for the people. 5,000 for everybody. 50 million households out of the 200 million people that are in Nigeria. They are targeting 80 million people and 50 million households. Now the real project that could have been one of those things that will aid people in movement from one point to the other is being truncated by the fact that there's too much debt and then underutilization. So I'd like to have your comments on that. When you see these people, they don't change.  There's no trust fund in the petroleum trust fund which was headed by the current president and I remember when the current president was in a position he said that many companies they are being subsidy, they are just receiving Nigerians that nobody is paying subsidy. However they are paid more subsidy under the administration that they are paid and there is no justification. Some of us have had duties because there are no provisions there are no provisions for petroleum subsidy. Because when it was hired, he claimed that they are no longer being petroleum subsidy and the question was where is the money being used for payment of petroleum subsidy coming from? Isn't there not a professional beer? And we said it was an impeccable offence. So when you look at their pattern and their style, it's the same. What did they do under a batch? They bought buses and he said they would buy buses, they would give people some relief package and they would look. There is a particular word in our local parlance it's called abracadabra and then you know, rexie and then whiskey that is the release of the song by rexie abracadabra. So what we see the more you look the less you see is just the situation when it comes to the oil and gas sector and the situation with which you have had with respect to Angola in Nigeria because our oil output has become low, record low and Angola is the leading oil exporting country in Africa that should be the shape it should be it should be something that anyone in the oil and gas sector in the oil and gas sector should know that it's petroleum. I can see this for you fact the president has failed because the president is the minister of petroleum and the minister of state for that ministry resigned to contest for the APC gubernatorial ticket of bias as state and as we speak is the APC gubernatorial candidate for bias as state so that ministry has a minister that has the every other minister reporting to him and it has no general minister as we speak and presently so where did they get the data how did they come about the data of the 50 million Nigerians that's the question if you situate that with the relief package during the covid or you situate that with that money you just think that probably some people are catching up on innocent Nigeria because there is no there is no idea of accountability on it now all of these programs they have done over time if you break the word accountability into two if you break the word accountable into two you have account and table now they have not watched the accounts of what they have done in the past to table to the table before we talk about giving them the opportunity to go ahead and spend another money so I just can't comprehend and I just can't understand because they think okay you know what and it's just because of the critical sector of the society and the media itself there is no transport line on it I can't imagine someone like Chief Ghanifa making a life today I know the various class suit that would have been instituted against various government at various level with respect to recklessness with respect to I said Nadjira does miss Chief Ghanifa on me now that you've mentioned with respect to financial recklessness and lack of transparency and property on behalf of those who have given the responsibility to manage our public accounts they would just come and band it up just to cut the various amount of money the various kind of projects in the twilight of an administration an administration that is going in less than two weeks in about three weeks time is now awarding different types of projects 3.4 billion as consultants see to construct a runway the money for consultancy is able to build runway as many runway as possible so you wonder are we cost is something really wrong something really wrong somewhere let's move from that to the case of Kano the Supreme Court adjoins Namdi Kano's case to September 14 this keeps happening time and time again even after the court first of all talked about releasing him and the federal government did not obey that law now they are back to court and they are joining this case until now it is September 14 we do not know what the outcome is going to be I would like your comment on that justice dealing is justice denying and why can the Supreme Court assure that it was this president the institution that abducted Namdi Kano here was an abduction and it was this administration that took Namdi Kano to court but somehow it will be the incumbent administration that will edit the baggage created by this current administration I think that the Supreme Court shall have been able to expedite the abduction on this matter and dispense with the case so that justice will be seen to have been done and must have been done if there is any need for him to go to jail for his activity if there is any need for you to send him free you discharge him and you are quitting you see when you delay from now this is May to September for me when I heard that I was in the matter as quickly as possible to expedite action on the matter and as far as I am concerned it is just one way of this present administration trying to ensure that the incumbent administration inherit its packages because the attorney general of the federation and the minister of justice has a role to play in this to ensure that if they have enough evidence to prosecute this guy to convict him they go ahead with it other than delaying it and one court will release him and then you are food dragging on the matter as far as I am concerned it is just it could be anybody because justice delay is just denied issues of this nature should be action should be expedited on the and then justice must be seen to have been done that's my take on that Thank you so much for your time Mr. Ajida Johnson is senior lecturer at the Institute of Journalism Lagos thank you so much for your time and insight on this very important it is a pleasure to be with you thank God it is Friday make sure you have a drink and then relax to recharge yourself on Monday thank you for that advice you too have a nice weekend you are still watching the Friday Frenzy edition of the breakfast on Plus TV Africa we will be back in a minute to take our very first topic hot topic it is stay with us


the time has finally come i'm at hearth and i'm about to get some shoes that i've been waiting to get for months no it's not the ebay dunks and it's not the brooklyn air force ones either wait hold up they got the organ dunks up in here but they don't got my size oh baby come to papa these things go crazy size 13 just for me yes sir let's go baby shout out to hearth always coming through in a clutch for your boy


for Leighton sports card drip and one box of 2019 Panini contenders football hobby this is for Ryan B look Ryan VP contenders land jack regular base doesn't ship in this hurt Warner VP contenders camp is branded Nemo the okayest major league player just decent Jags round numbers Oliver and Williams instead of war wins against replacement player it should be one wins wins above Nemo there's Elijah Holyfield rookie Otto do you know his dad's only got his ear bitten off play he banned it yeah come on right she's done yeah I think it says him on the back 175 Tariq Cohen me Cole Harbin junior rookie the year contenders Carson Wentz VP contenders rookie Otto Keelan Doss Deppin Bush Riley Ridley rookie of the year contenders Keel Harry rookie of the year contenders opening one football I'll take a look on that Tom Brady winning ticket Hines Ward legendary contenders come on I want to see a huge hit in here for you Ryan how about a big Kyler cracked ice or something Josh Allen bro Henderson and David long this is ticket stub rookie Otto to 43 Justice Hill ticket stub rookie Otto it's Jared Goff it's a Marquise Blair and DK Metcalf round numbers two packs to go for Sean Gary rookie ticket Otto Miles Sanders and JJ Arcego Whiteside and last pack rookie Otto Raquel Armstead all right see you Timmy Russell Wilson and that'll do it for your box Ryan appreciate it we'll get these right out to you


What if I told you that Google's engineers would write your service worker for you? I'm Sarah Clark and while I don't have a posse of engineers ready to serve your every whim, I have something almost as good. It's called Workbox. By now you've written your own service worker. It has a list of files to cache, but what happens if you add a file? What happens if you update a file? It may not be easy to update your service worker by hand. Also, what if you don't want to serve everything from the cache? If your content is time-sensitive, you might want to check the network first. You might want different strategies for network files versus user data. It can become rather complicated and Workbox can help. Workbox is a tool written and used by Google engineers that generate service worker code. Once you configure it, it locates the files on disk and builds a list of files to cache. It detects file changes and updates caches automatically. It supports multiple caches. It provides multiple access strategies so you can do the appropriate thing for each file. Workbox integrates into your build system, so the service worker is always up to date. To give you a quick idea of how this works, you add a service worker template to your source directory. The Workbox build tool adds code to this file and writes it in an output folder. You can control this with any build tool or the command line. The service worker you write may be as short as two lines or contain as much custom logic as you need, but you won't have to write the cache maintenance parts. Workbox starts with a template you supply. Let's take a look inside one. First, we call import scripts to load the Workbox library. You can host this on your own site or get it from the Google CDN. This will be cached for you. Now, add the pre-cache and route line with an empty array. Workbox will modify this line when it generates your service worker. If you need any other custom code, it goes at the end. Let's walk through an example of installing Workbox and creating a service worker. We'll do this from the command line. We'll need to install node, then use npm to install the Workbox-cli tool. Then we call the Workbox wizard to create a configuration file. We tell the wizard where our build directory lives and the files we want to cache. We then give locations for the service worker template and configuration file. By default, the wizard's code precaches everything forever. You may want a little more control, such as limiting the number of cached images. If so, you can add a cache rule and a custom cache name. Now we just run the Workbox tool to generate our service worker. That's all. You can register the service worker as usual. Note that we're assuming that everything is being copied into the build directory. The service worker is, as usual, at the root of the site. There's quite a bit more to Workbox, including the different access modes and cache options. There's also the integration with build systems. But for now, go to the Workbox site and try the getting started section to run this from the command line. In the next episodes, we'll look at Workbox in more depth, including how to connect it to your build system. Until then, have fun checking out Workbox.


Welcome everyone. My name is Jeremy Friedberg and this webinar is entitled Beyond D-Learning Online Teaching Platforms. Over the next 20 minutes I'm going to be talking about the concept of what a platform is and ultimately its implication for the education system specifically for teachers and students. But just to give you a bit of context about myself, I'm a scientist. I've been teaching at the university and high school level for many years and I'm an educational game developer and currently I run a educational gaming studio here in Toronto, Canada called Sponge Lab Interactive. But before I jump in here, let me I want to give you just a context of platform and ultimately what I believe it means and and how it's being used in the education system. But just for analogy here in the computer gaming space and the commercial gaming space, platform is a delivery vehicle. It's a delivery vehicle for communications and content. Now around the world there are a lot of individuals and organizations who make game content, computer games and ultimately they have to get those games out to users and it's the platform that does that. So Xbox, PlayStation, Nintendo Wii, these are these are platforms for delivering game content. The iPhone, computers, Macs, PCs these are all platforms that developers use to create games and get them out to their users. And in order to measure the success of these platforms they're tracking a whole variety of metrics. They're looking at user engagement, duration, scores. They're also looking at things like revenue, where that revenue is being generated, who their demographic is, what are the things that they're doing. And so they track a lot of information, understand ultimately the success of those platforms. The education system has their own platform. It's really very much the school and the system that that school is connected to and it functions kind of the same way. Users engage with that system, their teachers, their students, administrators, parents and ultimately they engage for a very long time and they do a whole variety of different things in it and we look at the success of this platform and we have a different set of metrics that we look at to judge how well the system is working and ultimately how well we're doing in the system. And those things include attendance, they're looking at exam scores, standardized tests, they're looking at graduation rates, the different disciplines they're graduating in and ultimately they are used to understand how well this platform works. Over the past 10 years we've seen a lot of changes to this particular platform. With the advent of e-learning and a whole variety of technologies that have been infiltrated the school system, it's changing and we're in kind of a state of flux and we're trying to understand ultimately where this change is leading us to but what can it really deliver for us? Because there's some pretty powerful things that are happening and that we're really just on the cusp of. To give you a sense of where I'm sort of heading with this, it's a concept called blended learning and it's based on a sort of a core principle that the individual learns in a very unique way. That really there is no standardized way to teach. That every mind is very very different in how it acquires information, how it processes information and what it can do with it. And this concept of personalized learning is quite amazing that we're evolving to a point where we can actually tailor both the approach and the outcomes to a particular individual. And this is really maximizing our ability to understand some very important things that ultimately we want to get out of our education system. And perhaps I'm on my own on this but in terms of evaluating the outcomes of our education system, we really want students who can think critically, can acquire information to be creative about solutions. Critical thinking and creativity, in my estimation, are of the utmost importance that we should be fostering inside the education system and ultimately what a student should get when they leave the system. That's a very personalized trait and it's very very hard to achieve. Now in our current education platform it's very difficult to really personalize things. And I'm generalizing here but the idea of one teacher with 30 students at a limited amount of time per day, it's very difficult to truly understand how a mind works and ultimately how do I craft the approach around it that's tailored specifically to that individual. It's very difficult to do that. But with blended learning there's a whole variety of technologies and systems that are coming together and they're converging that really have the ability to live on this concept of personalized learning. And what it's doing is blending a whole variety of different technologies from mobile to computer systems to the internet. Cloud-based computing allows for this massive storage of data and access to that data at any point and access to content. There seems to be almost no restrictions. Yes, some of it is open and free, others you have to pay for it, but getting at it you can get it whenever you need it. This kind of approach is open 24 hours a day, seven days a week no matter where you are. You just need to be connected to the internet. What's driving all of this is the analytics and the data. Because when you're engaging in all these different systems you're creating so much data that when you analyze it you can actually start to get a really powerful understanding about a user and not just their performance and engagement, but their culture, their learning culture. How do they consume content? When? What works? What doesn't work? When is their mind most optimum in terms of learning? And this analytics is really starting to deliver right across here into this concept of understanding who this user is and what works best for them. Now the one last component to this is game-based learning. I'm going to touch on this a little bit and then I'm going to talk about what I'm going to move into, what the platform this approach, this platform approach is, and some of the issues that are around it, but game-based learning is another component here. And why it's in here amongst all these other things is because game-based learning has the potential to offer something that we've really never had before. It has the potential to both teach and assess the characters of critical thinking and creativity. And game is, it's a very loaded word and it means a whole variety of things to different people, but really a game is built on some very core principles. Whether you're talking about an educational game or a commercial game or playing soccer, they're all based on the core principles. A simulation, a playable simulation with objectives and challenges. Now this is nothing new. We've been doing game-based learning in the computer space for a better part of 40 years. And it kind of works like this. In the late 60s when people started to train pilots, they were interested in the concept of, well, I'm less interested about whether or not they can take off and land the plane. Yes, that's very important skills to have, but what I'm really interested in is when they are at 32,000 feet, something goes wrong. Can they stay calm? Think critically of what's going on and be creative about a solution and do it in three minutes. And so game-based simulators were developed to actually do this. And they both teach and assess those characteristics in one shot. And they do it over long periods of time. And it turns out what actually happens in those types of environments is directly related to what happens with those minds in the real world. There's something very powerful there. And so game-based learning is a crucial important component to this process. And what started in training pilots is really filtered into almost every aspect of professional training. Almost every area of military training, from multiplayer sub-cruise to ground troops, is actually trained through simulation now. Doctors and surgers, oil rig workers, it seems to be across the board because of the power of the data you can get out of it. Now to think about this, there is a game, we think of it as maybe a hook or a really creative way to communicate information, but ultimately a game is a powerful data collection tool. A data collection tool that when you start to analyze it starts to give you a true understanding, a very honest understanding of how this particular mind works and whether or not this mind is similar to other minds. And it's just opening the door to what we can really start to understand and how it's going to augment this system. But I want to talk about one more aspect of game. And this is an important component because there are two fundamental areas of development you're creating, any kind of game, whether it's a game like soccer or a commercial computer game or an educational game, for that matter. There's two fundamental components, there's the simulation and this is where the communication actually happens. This is in soccer, the simulation is the field, the pitch, the players, the ball, the rules, put the ball in the net or it's the alien ship and you have to save humanity from innovation, whatever it is. But the simulation is there to communicate the content. We do this in educational gaming as well. It's the surgical environment that you're simulating that you're doing a routine gallbladder surgery. And so this is where the communication actually happens in the simulation. Another part of game design is the gamification. And this is a very important element to the design. So it's really looking at that user and ultimately, why do I care? Why should I play this game? I'm not just talking about in game, but I'm talking about engagement with any structure. These blended learning solutions, you want to have things to happen. You want to change behaviors, you want to foster engagement and communication. And there needs to be something in there that actually is motivating that user to engage the system, self-reengage as well. That's a tricky thing to do. Now commercial games, you'll see a lot, well, we'll just throw some points in there, give you some badges. And to some degree, to some users, that actually works and works very well. But with others, it doesn't work at all. And so what's very difficult and challenging but educational game design is creating a very delicate balance between intrinsic and extrinsic rewards. You do that right, and you've developed something pretty powerful. But when it comes back to blended learning, we need to create structures within this that actually allow users to want to be in here. And it's very much an exercise in human psychology, but there's elements of social mastery. And a whole variety of things, I'm not going to go into now, but it's a very difficult thing to actually achieve. And it's something that I've been working at for many years to try and understand and design effectively. So bringing this all back to platforms, all these things are happening. And they're all sort of converging. But really, a lot of people are having difficulty trying to understand the impact of this. And ultimately, what do I do with all this? And it's no different. You know, we've been building educational games for quite a long time. When we bring them into school systems, the single biggest problems educators have is not that they don't like the game or think it's great and what it can do. It's just what we do with it. How do I connect it to this textbook that I'm using to this social classroom experience that we're having, the laboratory, the field trip? How do we stitch it all together? And that's really been an element that's been missing, is stitching, bringing it all together. Because you're no longer just at the school, you're everywhere at any time. And that's where the platform comes in. And platforms is the vehicle to actually not only communicate the stuff, but to actually stitch the environment together. I'm just going to flip out of the presentation here into a browser, and I'm going to introduce you to the Spongebob platform. And you can access this at Spongebob.com. But this is probably a better part of four and a half years of research, looking into the nature of platform technology to deliver personalized learning and blended learning experiences. And I'll just go to the footer here. For those who are interested, this platform is very much a research tool. And we've been studying not just game-based learning and how it's used and what works well, but we've been studying the efficacy of using these types of tools and technologies in classrooms. And we have about 250 teacher contributors around the world who participate in these studies and give us a tremendous amount of feedback. And so if anybody's interested in getting involved in that project, please let me know. And we can certainly get you connected. But what this platform is designed to do is a number of things. First and foremost, it is open and free. It is a community-driven platform. Spongebob has developed the technology and it's really, there's a number of things in the background here. One is engine technology to efficiently deploy and deliver very rich content to meet your internet connections. But it's also been designed to be accessible to any device you connect to. And to really look at these issues of people coming from mobile devices, from desktop computers, from everywhere all over the world. And how do we let them integrate seamlessly into one system? The platform itself was designed to integrate with current technologies that are out there. So it integrates with learning management systems. It connects to content management systems. And there are contributing, content contributors that come from all over the world, from individuals, organizations, teachers, and students. It's quite, it's quite interesting to see how we can actually bring people and converge them into one, into one common system. So it's designed as a fully HTML5-compliant site and it connects a whole variety of technologies together. And when everything starts to filter into one place, it now brings us to the core question of how do I start stitching? And so we developed a number of tools. And within this system there is Explore, My List, My Lessons, My Classes, Community, and My Profile. And then I'll, I'll, if I have a few more minutes and I'll talk with the overall overarching game that's sewn into the fabric of the site to foster engagement of behavior change. But on the Explore page, this is very much a tool to help me rapidly find content and the significance of that content and how it connects to other bits of content. Now within here on the Explore tool I can refine by language, subject area, displays, thumbnails, but I get the content broken down by utility. And this is a very important point because everybody's utility is completely different and it even changes throughout a day. So we break content out by graphics and images, games and simulations, animation and video, linked animations of video, shared lesson clusters, and that'll become more apparent after I show you the lessons tool, lesson plans, quizzes, case studies, e-books, e-text publications, and even physical products. So science related products give you a sense of this.  This is an important, an important component that these physical devices are as important as the book, as the game, as the simulation. And all of this, this is still content to us in our perspective. So let's just say for example just to highlight I'm searching for something like animals, there we go, and the system gives me back something that I need and I'll find something here. Here's an animal cell and it gives you an indication of authorship and this is, there's content from all over the place here, it's a graphic, it is vetted. So you get a vetted description of what it is. A tool to find all the related materials within the system very quickly and if you happen to be using a physical resource like a journal or a textbook, it actually starts to connect it directly to it. So if you happen to be using this publisher's book, it goes great with this book and these two chapters. This is important, this is an important point that it's already in a very quick step it's converging a number of different medias and tools together in one step and this is alleviating some of the some of the frustration many teachers have right now because the top resource teachers worldwide are going to for digital teaching tools is Google Images. But you have to wade through all the stuff there to find anything good. Once you've found it, is it valid? Am I allowed to use it? Ultimately how do I use it? So the Explorer tool allows you to find things very very quickly and see their connected significance and everything within the system is intimately stitched together in a number of different ways. But once you've found something you can click on the plus sign and add it to my list and my list as a tool is an aggregator. It lets you find and remember everything that's real important to you and your teaching but lets you then define them as unique clusters of content and these are what we call lessons. It's a unique package of content for a particular utility. And through the my lessons tool, this is when you can start to work with this content. Now I'm not going to have time to go through every feature of the site. There's quite a lot and it's almost an entire other webinar but I'll just give you the highlights of this that when you create a cluster of content you can actually control its status and you can keep it private, you can share it, you can do a number of things with it but you can actually start to manage this cluster, you can rearrange the order of it, you can customize it, put your own content in and around it and even save hard copy versions of this for if you're ever going to be offline. But once you've packaged up this particular content you can present directly with it. So I can be up in front of a group of students, they could be remotely anywhere. We can all be looking at this at the same time or at different time intervals and at any point I can zip through the content and I can pull open what the creator of this content has the notes they put in and I can actually participate in real-time dialogue in and around this content. Some pretty powerful tools on the fly but what's interesting about this presentation window, it looks pretty basic but what's happening in the background is that it is aggregating all types of content into one in single interface. It's a game, simulation, animation, video, images, documents, it all filters into one tool so you don't have to moving into all different places and that's been a real game changer for the users on this platform. This is one simple way to engage with everything. Once you've created those lesson clusters, the My Classes tool allows you to create umbrellas and draw users underneath those umbrellas and then very, very quickly you can create deployed lesson clusters to those users and the system will track an unbelievable amount of data for you and distill that data on a variety of things. But just to hide this, you can create as many of these as you want. So you can create a class of 30, 300, 3,000 or a class of one. So you have a very rapid tool for actually starting to deliver on personalized approaches. Again, this is all open and free and developed by our community. So once you have started to, your users are generating, you're using constant under your umbrella and generating information, the system starts to distill data back for you. Are my students engaged? Let's take a look at this. Well, now I'm breaking it up by students and I can see some variations here but I can quickly play with the data and I can look at the duration of that engagement, I can look at the time of day, I can even look at the location that that engagement was occurring. When it comes to, and I'm just going to select all the content here, just give you a sense of this tool set. I can even see a cross section of what people are, the dialogue that's happening in and around the content itself. But when it comes to games and simulations, we actually pull open a tremendous amount of information and distill it back to you in a variety of ways. So I can actually look at this particular user and it's going to give me information on their success rates within the game compared to their class and even relative success rates between themselves and their classmates, areas where they need to brush up on their understanding but it's also starting to look at their ability to solve problems and think critically and looking at those abilities over time and how it's changing over time. So some very powerful tools right at your fingertips and the system distills that they have done so you can really very quickly understand what's going on. But what's bringing this all together of course is community and there's no one organization that can do it all and this community platform approach is really making this very interesting because we have users from all different types of communities. So I'll just give you a sense here. Another piece of content here. Movement to cellular analysis. This is a video clip and you can see it's developed by a professor at the University of Guelph. Specific attribution in the description but what the site is doing is bringing together a number of different communities through the platform technology. It's bringing together teachers, students, administrators, content developers. It's also bringing in those who create products for education space, devices and tools. But also brings together in a very meaningful way the scientific community within the education community. And this is a great example of a piece of content and I can show you what this looks like. It's a piece of video content that's actually stunning. This is real research that on its own is very difficult to use as a teaching tool. But through a platform like this you can actually bring it together, connect it with other resources and then push it out to a variety of users all over the world and then use data to understand what it's doing, how it's being consumed and what the ultimate end metrics are in terms of the success of this piece of content. And so the platform is really able to bring together a number of different communities. Now I know I'm sort of running out of time here but just to touch on the game layer that's built into the fabric of this system there are credits and experience points awarded for everything users do on here. And those, the experience points they allow users to level up and that unlocks certain social features of the site and your awarded certain achievement, certain things that you do and your credits are ultimately are transient and you can redeem them in the, in under the explore page in the marketplace and actually offsetting the discount coupons on educational product. It's part of that reward system and I understand that is very much an extensive reward but the system has a whole variety of intrinsic rewards built into it and really this is an ongoing research platform to truly understand what is that appropriate balance between the two. But what the purpose of the game layer is it's actually fostering a deeper level of engagement, it's changing behavior. And what it's there to do is to allow users to take a deeper understanding and a deeper level into the tool set, into the data and to really reward the full use of this blended learning approach. And to reward this idea of personalized learning. It's a very interesting concept and really it's about using platform effectively to bring multiple different communities together in a very meaningful and powerful way. So this is kind of what this, what we do at Sponge Lab and this is, this platform is an ongoing project, it's an ongoing and evolving project and we also heavily study this. So if you're interested in getting involved with the platform, certainly have a look at it and it's very easy to find at spineslab.com and give us your feedback. And I believe in the description below here will be some contact information to get a hold of us if you'd like but at that I will, I will wrap things up and just say that things are changing and they're changing rapidly and these tools are incredibly powerful and have the ability to deliver some amazing things for the education system. It's going to take some time to figure out how to make it all work and stitch together but this is what the power of community can do for us, bringing us all together to really affect some amazing change. Thanks very much and we'll connect soon. Take care.