 In the last six lectures we discussed about the response of single degree of freedom and multi-degree of freedom system for a specified ground motion. Now this analysis is useful for checking the structures which have already been designed for earthquake and for cases where an earthquake is specified which is likely to occur in future. This is also used many a time for performing a non-linear analysis of the structures in time domain. However most of the structures are to be designed for future earthquake and since the future earthquake is not known therefore we try to model it or represent it in different ways. Two different forms of modeling or representing earthquake were discussed in the input. There we talked of the response spectrum of earthquake and the power spectral density function of earthquake. Response spectrum of earthquake has a particular shape and that has been verified from the past earthquakes and we propose that the future earthquakes that will take place that would have a similar kind of shape in so far as the response spectrum is concerned. And from that we developed what is known as the design response spectrum and this design response spectrum is widely used for designing the structures for earthquake. So that becomes a very good input for analyzing the structures for future earthquake and we will take up the response spectrum method of analysis of the structures subsequently. In this sequence of lecture we will discuss the other input that is the power spectral density function of earthquake that will be applying to structures for analyzing the structures considering earthquake as a random process. That is we assume that the future earthquake is a random process it is not known therefore it is quite rational to model it as a random process rather than a deterministic process. When we model the earthquake as a random process we discussed in connection with the input or earthquake inputs that we describe it in the form of the power spectral density function of the earthquake. And we also describe what is known as a coherence function this coherence function takes care of the phase lag or the time lag between the excitations that occur at different supports which are at specially long distance. So with the help of this coherence function and the power spectral density function one can analyze the structures for random ground motion rather we should say that the ground motions are modeled as a random process. And the technique by which we analyze the structure with the power spectral density function as input and the coherence function in support is known as the spectral analysis. Since the entire analysis is done in frequency domain it is better called a frequency domain spectral analysis. It is a very popular method for finding seismic response of structures for ground motions modeled as a random process. The analysis requires the knowledge of random vibration analysis which forms a special subject of its own and therefore we will not be able to cover this entire random vibration analysis over here. We will take up only a portion of the random vibration analysis which is known as the spectral analysis and using that spectral analysis we will try to obtain the response of the structures for future earthquake modeled as a random process. The spectral analysis is quite elegant simple to understand and provides us a very good estimate of the peak response and the root mean square response of the structure. However the spectral analysis when we are using we have to have some simple concepts before we explain the spectral analysis technique. The two most important concepts that are required for understanding the spectral analysis are the concept of the power spectral density function of earthquake or may be for any random process it may be the response x t of a structure it may be a bending moment of the structure it may be a shear force of the structure since the input is random process therefore they also would constitute a random process. So the concept of power spectral density function not only of the earthquake ground motion but also the power spectral density function of displacement power spectral density function of acceleration power spectral density function of bending moments they must be well understood then comes the coherence function whenever we have got two random processes for example if we have a displacement of a structure x 1 t at the say top of a frame and say x 3 t at the first floor level between these two random processes there will be a lack of correlation or there is a phase gap or a time lag. So because of this phase lag or the time lag or the lack of correlation we have a cross power spectral density function defined for these two random processes and these the concept of this cross power spectral density function with the help of a coherence function that also we introduced in discussing the seismic inputs. So let us say look at these concepts again starting with the definition of the power spectral density function and some other concepts that will be required somewhat in the rigor of the random vibration analysis. However we will not go through all the diggers of the random vibration analysis over here we will simplify our spectral analysis technique by assuming the earthquake process not only to be stationary but also ergodic that is ergodicity is assumed in order to simplify the solution process and in understanding the concepts that we are now will be talking of and you already have undergone the concept of the ergodicity in connection with the seismic input to structures. Now let us take a collection of records of the ground motion and they are available at a particular site which is prone to earthquake. So from the past records one can collect as many number of ground motion records as possible. Ideally if we wish to define a random process we must have infinite number of records and each record should be of infinite duration. However in reality we cannot have a infinite number of the records nor we can have a infinite duration of the ground motion or for that matter any other time history. So we have always a finite number of collection of the ground records and the duration of the earthquake is generally of the order of 30 seconds 40 seconds or so on. So whatever data we get with the help of that we try to look at a property of the random process. Now one important thing that I mentioned in connection with the seismic input is that whenever we are talking of a random process then we are not only interested in only one record but an ensemble of records and I also mentioned that anything irregular is not random. In order to describe a random process you must have a number of earthquake records or number of time histories of the particular variable. Also these kinds of variables are called a parameter random variable. For example xt over here is a parameter random variable t is the parameter and the meaning of the parameter random variable is that at any instant of time t xt itself is a random variable. So if we look at this figure we see that at time t 1 the xt 1 can assume different values and if we have a number of such records then xt 1 itself is a random variable which can assume any value depending upon the number of records that you have got and so these values we can get if I draw a line across this ensemble. Similarly xt 2 is a random variable and it can assume any value depending upon the number of records that we have and we can in the same fashion define xt 3, xt 4, xt 5 all of them will be the random variables and they are these random variables are connected by a function of time t therefore it is called a parameter random variable. Now if I consider the value of xt 1 over the ensemble then if I average them that means take an average of all the values that xt 1 can take across the ensemble then we will get an ensemble average at time t 1 similarly we can get an ensemble of average at time t 2, t 3, t 4 so on. Now if it is found that these ensemble averages are the same or more or less are the same then we can say that this ensemble average is independent of the time shift. Time shift means if we define t 2 by t 1 by tau designating the time shift then whatever be the time shift the ensemble average will remain the same that means the ensemble average that will work out at xt 1 and say at xt 5 both of them will have the same value. So we phrase it in this fashion that the ensemble average is independent of the time shift similarly one can obtain the variance. So these variance would be described as the xt 1 values across the ensemble and the average value of the ensemble at t 1. So we deduct this from xt 1 and take a square of that and expectation means the averaging of this quantity so this is called the variance that we have also discussed before. So if we find that the variance of the ensemble at t 1, at t 2, at t 3 and t 4 if all of them are more or less the same then we can call this also as a process having a variance as invariant of the time shift. Now for our analysis generally we define a process or a random process as a stationary process of order 2 or second order stationary process if we find that the mean ensemble mean and the ensemble variance they are independent of the time shift. And we generally for engineering purposes at least for structural engineering calculations we satisfy our self with the second order stationarity however if one has to really call a process to be a stationary process then it should be an nth order stationary process n being the order that means instead of sigma x square it should be sigma x cube sigma x 4 sigma x 5 higher the number higher would be the order of the stationarity. Then if we consider a single sample out of it then for that single sample we can work out a variance for the sample. So these variance of the sample will be simply the values at different time minus the average value and we take a square of that and one by t and just integrating the square of this quantity that gives the sample variance. In an energetic process we find that sigma x square that is the ensemble variance is equal to the variance of any sample time history. That means if I take out any sample time history from this ensemble this will also give is the same variance across time. Now if you have such an ideal process then that process is called an ergodic process. It is really hard to get an exactly an ergodic process in reality but for solving many problems in random vibration we assume the property of ergodicity because the assumption of ergodicity simplifies the analysis procedure. So in our case also we will assume that the process the earthquake process or the future earthquake is not only stationary but it is also ergodic. Now once we assume that then we will find that many of the concepts that will be subsequently describing they become simpler to define however those concepts can also be defined in the rigor of random vibration analysis. So however here we will not do that in order to understand in a simple manner we will assume the process to be an ergodic process. This Fourier series and Fourier integral we have discussed before in connection with again seismic input and there we have used that Fourier series for finding out the Fourier spectrum as well as defining the power spectral density function of a ground motion. And so therefore I will not be again repeating it in total but the salient features let me again reiterate over here so that you get a recapitulation. If I take a single time history then it can be broken up into a number of harmonics or in other words if we sum up of the number of harmonics then we get a time history and irregular time history x t and a k and b k they are obtained from this relationship if x t basically is a function which is integrable then one can get the values of a k and b k in the closed form if it is not an integrable function then we go for a numerical integration. In order to perform this numerical integration effectively we now use f f t that is first Fourier transform and in order to get back the time history of the signal or time history of ground motion in this case we can obtain this time history from the frequency components of the time history if we use i f f t. So, the pair of f f t and i f f t that we use for Fourier synthesizing a time history of ground motion now how one can obtain a Fourier integral from the Fourier series that is described over here if we simply substitute 1 by t in the previous equation that is in this equation this 1 by t if it is substituted over here then you will find that it becomes delta omega by pi here also you will get delta omega by pi and if we assume that t is tending to infinity then delta omega converts to d omega and the summation sign can be replaced by an integration sign. So, that is what is shown over here x t can be shown to be equal to this particular form 2 times omega is equal to 0 to alpha a omega cos omega t d omega plus b omega sin omega t d omega s like this now one can remove this 2 if we instead of integrating from 0 to infinity if we integrate from minus infinity to plus infinity. Now with this definition of for the x t over here where a omega and b omega they are obtained from this expression that is a k and b k and b k they take the form of a omega and b omega then we try to define a quantity x omega as a omega minus i b omega and if we substitute for a omega and b omega from the equation of a k and b k this a k and b k they become a omega and b omega if you substitute this integrations into this equation and do a algebraic manipulation then x omega turns out to be 1 by 2 pi minus infinity to plus infinity x t to minus infinity to plus infinity to the power i omega t. So that is how the first part of the Fourier integral has is derived that is x omega is given by this integral the second part that is if we wish to find out x t given x omega that is a frequency contents of the ground motion then we use this i f f t that is inverse Fourier transform and it can be easily proved like this that if we take x omega e to the power i omega t d omega this integral if we wish to write down then we have a omega minus i b omega multiplied by cos omega t plus i b omega t that is e to the power i omega t is replaced by cos omega t plus i sin omega t and if and then we also keep d omega over here and when we multiply these two then we have this quantity this quantity that is b omega sin omega t a omega cos omega t and then also these two quantities that is i a omega sin omega t minus i b omega cos omega t d omega. Now this we keep as it is these two and these two in this if we substitute for a omega and b omega by the equation of a k and b k that I have shown you before if we substitute that then we will find that these two terms they cancel finally we have a omega sin omega t d omega plus b omega cos omega t d omega this is the these are the quantities that remain and this is nothing but equal to x t because we have defined these two terms this like this x t has been defined by equation 4.7 b as a omega cos omega t d omega plus b omega sin omega t d omega. So, we can prove that x t becomes equal to the i f f t of x omega and they form what is known as the Fourier integral pair. So, the Fourier integral pair that we use in Fourier synthesizing any particular time history that can be straight away derived from the Fourier series. Then when we try to use f f t and i f f t we perform that integration with the help of a summation sign or we numerically calculate this integration and that format is known as the discrete Fourier transform and inverse discrete Fourier transform and that is what is used in the computer programs for obtaining the Fourier components of any signal or of any time history. The Fourier amplitudes a k and the phase etcetera that has been discussed before. So, I am not going to again repeat that only one important thing that I wish to mention over here is that Fourier integral that we have shown you before that Fourier integral is valid only under this condition that is the absolute value of x t integrated over the entire duration must be less than infinity that means that quantity must be a finite quantity. And in reality with this particular condition is met because our time history records are not in finite they are all finite time history records. Therefore, if I wish to make it infinite that is if you wish to make the integration from minus infinity to plus infinity then we have to add on zeros to that and that way the in reality this particular condition is satisfied. Now, if we are using the discrete Fourier transform then the Fourier integral is expressed with the help of these summation signs and this has also been discussed before in connection with the seismic input. Also we discussed about the Perseval's theorem in order to find out the mean square value. So, mean square value of the process of or x x t a square integration of it from 0 to t divided by 1 by t that gives you the mean square value of a particular time history that can be shown to be equal to the some absolute square absolute value square of the complex quantities that we get after we obtain a f f t of a time history. So, and if they are added together then we get the same mean square value. So, that is what the Perseval's theorem is and how from that we use we obtain the power spectral density function that also we have discussed before that is if we use MATLAB then the whatever values we get as x omega that is the complex number a plus i b then we take n by 2 plus 1 values first n by 2 plus 1 values from that a plus i b quantities. And then we take the absolute square of these complex quantities and we have got n by 2 plus 1 such quantities that we plot against omega then divide each ordinate by d omega. And we get bars small bar diagrams and the small bar diagrams small bars will be again equal to n by 2 plus 1 in number then join the center of those bars in order to get the power spectral density function. So, all these things have been discussed before. So, I do not want to again repeat it only I wish to mention over here that we are able to define the power spectral density function of ground motion in this particular fashion because we assume the process to be an edgodic process. And therefore, we looked at only one single time history record and from there we are constructing its power spectral density function and that power spectral density function represents the entire stationary process because the area under the power spectral density curve is again equal to the mean square value that also we have discussed before. So, since a stationary process is characterized by a unique mean square value therefore, the power spectral density function constructed in this fashion provides a good estimate of the input that will go as an input for the random vibration analysis of the spectral analysis in particular over here spectral analysis of structures. Next we this also we mentioned before, but again let me reiterate it. If we have got a random variable x t 1 and then we define another random variable x t 2 t 2 defined as t 1 plus tau tau is the time shift between t 1 and t 2 then although there would be the expected value of x t 2 square and expected value of x t 1 square they are same, but they would differ in so far as the phase lag is concerned that is there will be a phase lag between these two time histories and in order to denote these phase lag we use what is known as the autocorrelation function. This autocorrelation function is a function of tau that is the separation time between x t 1 and x t 2 in the ensemble let me show you again in the figure what are those that is this t 1 and t 2 and tau is this distance. So, greater the value of tau more is the time lag between two random variables and autocorrelation between the random variables x t 1 and x t 2 they are a function of tau that is a time separation. So, we write down expected value of x t 1 multiplied by x t 2 in place of t 2 we write t 1 plus tau. So, this is the definition of the r x x tau. So, what we do in obtaining the r x x tau we take a particular time shift and describe two random variables in time multiply the ordinates of these two random variables over the ensemble and then take an average of that that is what is called the r x x tau that is the autocorrelation function and it depends upon the value of tau. If tau is equal to 0 then you can see that the r x x tau simply becomes is equal to r x x tau becomes equal to the expected value of x t square. So, r x x tau becomes equal to the mean of tau. In this fashion we can also define the cross correlation between two random processes say x is a random process and y is another random process then we call r x y tau as a cross correlation function between x and y over a time separation which is called tau. So, and one can define the r x y tau in the same fashion that we have done for r x x tau in this case we take at t 1 at time t 1 on the values of x and for the y process we take the values at t 1 plus tau. So, at that particular time we take the values of y and then multiply them over the ensemble and take an average. So, that is how we define r x y tau and it could be easily shown that r x y minus tau is equal to r y x tau. So, that is a property that can be easily proved. Figures 4.2 to 4.3 they show the plots of r x tau and r x y tau with tau. The figure 4.2 shows the variation of the auto correlation function r x tau with tau and it is seen that the value of the r x tau becomes maximum at tau is equal to 0 as it would be expected and the value of the r x tau at tau is equal to 0 is equal to sigma square plus m square where sigma square is the stand the variance of x and m square is the square of the mean value. For 0 mean process the value of the r x tau at tau is equal to 0 becomes equal to variance or is equal to the mean square value as we can see it from the definition of r x tau. The r x tau or the auto correlation function drastically falls down as tau increases and it is seen that after a value of tau the value of r x tau oscillates about the mean value m. For 0 mean process r x tau is equal to this oscillation takes place at the axis tau axis and this fluctuation is of very small order. So, as the tau increases then there is a lack of correlation between two random variables x t 1 and x t 2 and for large value of tau the correlation between two random variables may be negligible. Similarly, we can see the variation of r x y tau with tau in figure 4.3 unlike the case of the auto correlation function r x y tau does not become maximum at tau is equal to 0, but at some other value of tau that is tau is equal to tau 0 and the maximum value is equal to sigma x multiplied by sigma y plus m x into m y where sigma x is the standard deviation of the random process x and sigma y is the standard deviation for the random process y and m x and m y are the corresponding mean values. After the r x y tau reaches maximum then it again falls down drastically as tau increases and it fluctuates about the value of m x into m y and this fluctuation again is of very small order. So, thus we can see that for large value of tau the correlation between the process x or the stochastic process x and the stochastic process y can be ignored or we can say that that is very becomes very negligible. Now the correlation function and the cross power spectral density functions they form the Fourier transform pair. So, that is what basically we are going to use now what we would say that previously in the case of seismic input we said that the cross power spectral density function is equal to the power spectral density function of the ground motion multiplied by a coherence function. Now this particular thing define the cross power spectral density function between two excitation, but now we will define it slightly in a different way although the meaning remains the same that is the s x x tau can be shown to be equal to the inverse Fourier transform of s x x omega that is the inverse Fourier transform of the power spectral density function of the process and the power spectral density function of the process is the Fourier transform of or f f t of r x x tau. So, given r x x tau one can in principle compute the power spectral density function of the process similarly given the power spectral density function of a random process one can obtain its autocorrelation function. So, in this particular fashion one can also construct the power spectral density function from the ensemble of records. So, if you have an ensemble of records one can obtain the autocorrelation first and then take a Fourier transform of that f f t that will give us the definition or the power spectral density function. We need not then assume ergodicity and try to find out the power spectral density function from a single time history that we discussed before. Similarly if we have r x y tau defined for a two processes then taking a Fourier transform of that or using f f t on this r x y tau we get the cross power spectral density function. So, then we need not again define the cross power spectral density function by stating that cross power spectral density function between two support excitations is equal to the power spectral density function of the ground motion multiplied by the coherence function. So, we need not try we need not define in that fashion, but we can define it in this particular way through if we wish to define it through an ensemble that means then we are not no more again using what is known as the ergodicity property. So, the cross power spectral density function and the cross correlation functions and the power spectral density function and the autocorrelation function they form on the Fourier transform pair. So, that is used in many cases when we will be deriving the spectral analysis technique. Another important thing that I should mention over here is that s y x omega that means s x y omega is the cross power spectral density function between x and y. Similarly, one can have a cross power spectral density function defined between y and x. So, that is called s y x omega is nothing but complex conjugate of s x y omega. So, if we know s x y omega you can just take a complex conjugate of that and that would give you the value of s y x omega. This also can be proved and indirect proof to show that the power spectral density function and the autocorrelation function the cross power spectral density function and the cross correlation function they are they form the Fourier transform pair. We can use this indirect relationship that r x x at tau is equal to 0. We know that the value is equal to r x x square that is the mean square value of the process and if we substitute in this Fourier transform the value of tau to be equal to 0 or rather in this equation tau to be equal to 0 then this becomes simply is equal to s x x omega d omega and we know that the area under the power spectral density function curve is equal to the mean square value. So, this is an indirect proof of the Fourier transform pair that r x x and s x x they conform to. Similarly, r x y 0 would be the integration of s x y omega d omega and the integration of this s x y d omega will give you the expected value of x y. So, in this particular way one can indirectly proof that the cross power spectral density function and the power spectral density function they and the autocorrelation function and the cross correlation function form the Fourier transform pairs. Now, the p s d f the power spectral density function forms an ideal input for frequency domain analysis of structures for two reasons. These reasons are firstly a second order random process is you would know that the uniquely defined by its mean square value that is the definition of the second order stationarity which we assume to be sufficient and for our structure analysis. So, therefore, if we have the mean square value known or as a unique value known for the random process then the random process is defined to us. Next since we are wanting to find out carry out an analysis in frequency domain then it is better to have not only the knowledge of mean square value of the process, but how the mean square value is distributed over the frequency and that comes from the definition of the power spectral density function itself because we know that the area under the power spectral density function curve is equal to the mean square value. So, in other words we can say that the power spectral density function is a distribution of the mean square value with frequency. So, in a frequency domain analysis now we expect that at each frequency what is the contribution of the power spectral density function to the mean square value if that goes as an input then we can get an output in a similar fashion that is we can get a power spectral density function ordinate at a particular frequency contributing to the total mean square value of the response. So, with this particular concept in mind we develop the spectral analysis technique or frequency domain spectral analysis technique. How we obtain the power spectral density function curve that I define I described before. So, if you can have the FFTs of the time histories and plot the amplitude squares in close enough range and then divide it by d omega then these are the different bars and you will have n by 2 plus 1 number of bars over here and through the center of these bars if you draw a line that gives you the value of s omega denoting that the area under the curve is equal to the mean square value of the process. So, in this fashion you get the value of the or you can obtain the power spectral density function provided we assume the process to be a ergodic process. So, I am not going to again repeat the same thing that I had repeated discussed before. So, I am not only thing which again I wish to mention over here is that there is a physical meaning which is attached to the power spectral density function that is the area under the power spectral density function curve is equal to the mean square value or alternatively we can say that distribution of the mean square value with frequency is a power spectral density function. So, that gives a physical sense to the spectral analysis. However, we cannot give such a physical interpretation to a cross power spectral density function as we can give it for or the power spectral density function. So, but some idea about the physical significance of the power spectral density function would be clear from this figure if in this simply supported beam if we have got two loads dynamic loads with a phase difference of phi then as phi goes on changing the degree of correlation between p 1 and p 2 they would be changing and the moment this phi is changed you will find the bending moment at the center that goes on changing with the value of phi. So, thus the degree of correlation is very much related to the response of the system. So, that is the physical meaning that we can attach to the cross power spectral density function of two random processes. So, in that case p 1 will be a random process p 2 will be another random process, but having a phase lag and in these two random processes since they are not perfectly correlated they will have a distinct what to call contribution or influence on the response of the system. Therefore, these need to be considered along with the power spectral density function of the two processes of as an input for spectral analysis.