 In the last lecture, we are talking about the modal spectral analysis. The modal spectral analysis is performed taking help of the mode shapes and frequencies of the structures as we do in the case of the deterministic analysis. The whole idea again is to convert the problem into a set of uncoupled single degree of fit on system and in that case one can write down the power spectral density function relationship for a single degree of fit on system and then the responses obtained from there are then combined with the help of the relationship that exists between the displacements in the structural coordinate system and the generalized displacements through the mode shape matrix. So the essence of the modal spectral analysis is that first we decouple the equation of motion and once we decouple the equation of motion then we obtain for each decoupled equation of motion the frequency domain equation which is shown in equation 4.97A and 4.97B, 4.97A and then from there one can write down and P i omega is written in this way and if R if it is a multi supported excitation case then R is the matrix of the coefficients which is used and J i t is the ith mode shape transpose of that M bar i is the ith modal mass. So once we are able to describe P i omega that is the ith generalized load then using that one can write down S z i z j that is the cross power spectral density function between the two modal displacements z i and z j as h i into h j star then j i t M r S x double dot g R t M t and j j and both i and j vary from 1 to R where R is the number of mode shapes that we consider. So using this expression one can write down the cross power spectral density function between any two modal coordinates and then with the help of this we can make the matrix S z z and the diagonal terms of course will be S z i z i and i varying from 1 to R then after we have obtained S z z then we can obtain the power spectral density function matrix of displacements in the structural coordinate using this formula it is coming from the above relationship that is x is equal to phi z and if it holds good then x x x will be equal to phi S z z phi t. Many a time we do not take all the modes into consideration for example here we are considering the modes 1 to R R is less than the number of degrees of freedom. So in that case the phi matrix will be equal to M into R where M is the number of degrees of freedom and R is the mode shapes that are considered. So using this matrix this truncated matrix of the mode shape one can obtain the power spectral density function of the responses in structural coordinate by considering not all modes but only in limited number of modes. An example problem is solved here to illustrate the modal spectral analysis technique. This problem is the problem that we had solved for the cable state bridge where we want the power spectral density function of the degree of freedom 1 that is the left hand tower top and degree of freedom it will not be 2 it will be 3 that is the center of deck and it is assumed that a uniform time lag of 5 second between the supports exist that is it is a multi support excitation case. So for this problem the K matrix is a 3 by 3 K matrix that is the 2 corresponding to the 2 displacements at the tower top and a vertical displacement at the center of the deck and this is the mass matrix corresponding to these degrees of freedom and this is the R matrix that is the coefficient matrix that is the displacements that occur at the non support degrees of freedom due to unit displacement at the supports at the 4 supports so that constitutes the R matrix. So with the help of that we try to obtain the power spectral density function matrix of the degree of freedom 1 and degree of freedom not to 3 and we proceed with the equations that we described before now in the case of this multi support excitation what we will require is the definition of the power spectral density function sx double dot g and the power spectral density function sx double dot g will be written in terms of the coherence function or the cross correlation function between 2 supports so between the first and the second support it will be minus 5 omega divided by 2 pi because the time lag is 5 second between the first and the third it will be minus 10 omega divided by 2 pi because the time lag is 10 second and between the first and the third support it will be minus 15 omega by 2 pi defining row 1 row 2 row 3 in this particular fashion one can write down the power spectral density function matrix for the excitation as in this particular form that is 1 by 1 row 1 row 2 row 3 so on and they they successively because between 2 and 3 support it will be the row 1 that will come into picture so once we have written down this power spectral density function then we can use this for finding out the power spectral density function of the load vector that we will see later sx double dot g over here is a single it is not a matrix is a single quantity the power spectral density function of say here we have taken the L central earthquake so this is the power spectral density function of the L central earthquake as sx double dot g in fact is multiplied with all of them so therefore we have taken it out the h1 and h2 we written with the help of the first frequency and the second frequency first frequency is 2.86 therefore it will be omega n square minus omega square that is the first term and the second term is i 2 eta omega n into omega so eta is 5 percent therefore it is 0.05 and omega 1 is 2.86 therefore we in place of omega n we write down 2.86 and we have the complex frequency response function for the first mode similarly one can write down the complex frequency response function for the second mode only thing that we do is that we replace the first frequency by the second frequency 5.85 and we get the frequency response function for the second mode similarly one can write down the frequency response functions for the third mode and taking the third frequency once we have obtained these h1 h2 h3 then one can plug in these frequency response function into this equation that is hi and hj star that now is known for any two modes and the jit and jj they are also known that is the mode shapes for any two modes and sx double dot g matrix has been now defined so we use you put here the sx double dot g matrix r matrix I have shown you already so you have the r matrix and the m matrix is known therefore s j di zj that is the cross power spectral density function between any two modal coordinates can be obtained with the help of the quantities that we have just now shown. Now when all the elements of the power spectral density function of s z z are obtained in this particular fashion then we construct the s z z matrix and then obtain the power spectral density function of sxx so using this particular technique we have obtained the values of the power spectral density function of the first displacement and the third displacement and the rms value for the first displacement that is the at the top of the left hand tower that is in the modal analysis is get 0.021 and from the direct analysis we obtain 0.021 so they are same because we have taken all the three modes into consideration similarly for degree of freedom it will not be 2 for degree of freedom 3 the modal analysis get 0.015 and the direct analysis also get 0.015 the shape of the power spectral density function for degree of freedom 1 that is shown over here and we can see that the power spectral density function is picking almost at the first frequency of the structure these shows the cross power spectral density function between the any two displacements that is between u1 and u2 and or rather u3 it will be u3 and this cross power spectral density function the real part is this and the imaginary part is this as we know that the cross power spectral density functions is generally complex in character therefore it has a real part and imaginary part and these shows the power spectral density function of the top of the left tower and these shows the power spectral density function at the center of the deck next we come to the spectral analysis in state space so when we perform a state space analysis then the equation that we use is szz and szz is equal to h sfgfg h star t now for this h is the frequency response function for the state space equation and you know now the what is the frequency response function for the state space equation so it is written as i cap that is a identity matrix of size 2n by 2n and omega is a frequency minus a a matrix that is again 2n by 2n and if you recall the equation in the state space is equal to z dot is equal to az plus fg a is a is of dimension 2n by 2n z is of dimension 2n into 1 and fg also is a vector of 2n by 1 first n values are zeros and second n values second set of n values are rx double dot g that will and show you later now what we have to obtain is that sfgfg matrix and once we are able to find out the sfgfg matrix that is a power spectral density function matrix of vector fg then we can plug in that matrix over here and h omega can be computed from this equation because a matrix is explicitly known and by inverting this matrix we can get the frf or the frequency response function matrix h so let us see how we obtain the sfgfg so this is the fg in the vector form that is the first n values are zeros and then we have minus rx double dot g so it is of sine 2n into 1 and z is defined as x and x dot that is the states of the system defined by the displacement and the velocity the sfgfg matrix obviously would be 0000 rsx double dot g rt because here this last term is equal to rsx double dot g so power spectral density function corresponding to this will be rsx double dot g into rt where xx double dot g is again the power spectral density function matrix of the excitation and now if this is partially correlated that is the there is a time lag between the excitations then say for the case of 3 supports you will have the sx double dot g matrix defined as 1 rho 1 rho 2 1 rho 1 and 1 and on this side it will be symmetric and sx double dot g will be a single quantity that is the power spectral density function of the L centre earthquake in this particular case and rho 1 and rho 2 depends upon the time lag so between the first 2 supports if the time lag is 5 second then rho 1 will be exponential of minus 5 omega by 2 pi that is what we have shown before and between the first and the third support it will be rho 2 and it will be equal to minus 10 omega divided by 2 pi exponential of that so that is how one can calculate sx double dot g matrix and one can plug in this into this expression and obtain the value of the sfg sfg and once sfg sfg is obtained then using the equation that we have described before one can get the values of sj sj now sj sj in fact contains sxx sx dot xx dot sx dot x and sxx dot because z is defined like this therefore from the sj z matrix which will be of 2n by 2n size this will be n by n size this will be n by n size so we will get 2n by 2n matrix so we can choose the appropriate terms from these matrix to obtain the power spectral density function of any response quantity in the structural coordinate that is x so and if we were wanting to know the velocity power spectral density function of velocity then we can select the appropriate terms from this matrix it is to be noted that if I add up sx dot x and xx dot if I add up these 2 then the addition would be equal to 0 because we know that the cross power spectral density function between the displacement and velocity the summation of these 2 quantities they turn out to be 0 that you have proved before. So in the state space formulation one can also obtain the response in this particular way we have solved a problem here this problem is the problem in which the top and the first floor these 2 displacements are considered and the power spectral density functions for these 2 displacements are the responses that we look for so here is the A matrix for the system and this A matrix we have generated before while solving the same problem for deterministic ground motion and then using the formulation that we just described we obtained the rms value of the degree of freedom 1 that is for the top floor displacement and degree of freedom 4 that is a first floor displacements. So we obtain this by 3 methods in which this one is the modal spectral analysis that we just described before this is the direct analysis in which we do not use the mode superposition technique but to take the entire K, M and C matrix to obtain the responses and this is the state space solution that we just described and we can see that the rms values obtained by the 3 different methods they are almost the same similarly for degree of freedom 4 that is a first floor displacement the rms values are again quite comparable they are close to each other. So thus we can see that one can obtain the power spectral density function of response of a particular structure by different methods for example one can use the direct method using the second order differential equation and they are the input that we required is M, K and C and once we know the matrices M, K, C then from there one can obtain the frequency response function matrix H and then obtain the power spectral density function matrix of excitation and it depends on whether the excitations are single support excitation or a multi support excitation if it is a multi support excitation then from Sx double dot G that is the power spectral density function of the special earthquake this what to call quantity and the time lag between the supports and using a coherence function one can construct the power spectral density function matrix of the excitations and once it is known then one can straight away obtain the power spectral density function matrix of the responses. In the case of modal analysis in addition to M, C, K matrix we must know also the mode shape matrix and using the mode shape matrix one can decouple the equation of motion into a single degree of freedom equation and then use the relevant expression for finding out the cross power spectral density function between any two generalized coordinates and with the help of those cross power spectral density function matrices defined one can obtain the power spectral density function matrix of the generalized coordinate that is HZZ matrix and in terms of the structural coordinate the power spectral density function is obtained by simply multiplying pre multiplying the SZZ matrix with phi and post multiplying with phi t. So that is the modal analysis technique and for the state space analysis the H matrix that is is different than the H matrix that we obtain for the second order differential equation and once we obtain the H matrix then one has to find out SFG SFG matrix how we obtain SFG SFG matrix that I have described and one can obtain the SZZ matrix that contains also the power spectral density function of velocity along with the power spectral density function of displacements and depending upon the requirement one can choose from the matrix the relevant terms to obtain the power spectral density function of any response quantity. For the previous problem this is the power spectral density function of the response of the top displacement and this is the cross power spectral density function between the top displacement and the first floor displacement the real part of this cross power spectral density function is shown similarly we have a imaginary part for this cross power spectral density function. Now the in the state space one can obtain the responses using a again modal analysis now for that say this is the A matrix in the state space formulation so we write down Z to be is equal to Phi into Q where Phi is the Eigen values and Eigen or other Eigen vectors of the matrix A so this will be of size 2n by m provided we consider the m number of the modes but if we consider all the modes then it will be 2n by 2n matrix. And Phi inverse Phi of course will be an identity matrix of size 2n by 2n so Phi inverse A Phi that is equal to a diagonal matrix of size 2n by 2n and the diagonal terms are the Eigen values of matrix A therefore using this relationship Phi inverse A into Phi we can decouple the equation of motion in this particular form that is Q i dot is equal to lambda i Q i plus f bar G i where the lambda i is the ith Eigen value of the matrix A and Q i is the generalized i h generalized coordinate and a bar G i is the ith generalized load vector or other load not vector so f bar G i is obtained like this Phi inverse f G and ith element of that is f bar G i and once we have these relationship that is the Phi f bar G i is equal to Phi inverse f G then one can find out the power spectral density function of f bar G i simply by finding out S f G S f G and multiply it pre-multiply it be Phi inverse and then post multiplied by Phi inverse t and once we do that we get f bar G i and once we get the f bar G i then Q i omega or Q j omega that means for a particular mode the generalized coordinate in frequency domain is related to the f bar G j through the frequency response function for the jth mode and frequency response function in jth mode is given as this i omega minus lambda j inverse where lambda j is the jth Eigen value of the A matrix so we know H j omega or in other words the frequency response function for any mode and once we know that then we can use simply for a single degree of freedom equation the power spectral density function relationship between the response and the excitation that is S Q is equal to H omega S f bar G j into H omega star this H omega is for the jth mode if this is the jth mode and S Q Q will be the quantity of the power spectral density function of the jth generalized coordinate and one can obtain the S Q i Q j by simply taking the ith mode for this frequency response function and jth mode for the this frequency response function for which we are obtaining the complex conjugate so that is how all the elements of the S Q Q matrix that is written over here can be obtained and once we know that S Q Q matrix then one can obtain the S j j matrix by simply pre and post multiplying by the mode shapes and the transpose of the mode shapes with S Q Q so that is how one can also obtain a modal spectral analysis using the state space formulation now we have seen that there are many ways that we can perform the spectral analysis and these spectral analysis is or can be obtained for a single support excitation and multi support excitation by carefully obtaining the power spectral density function matrix of the excitations whenever we have the multi support excitation case then coherence function must be defined the different methods that you have described can be programmed in MATLAB following a particular flow chart and using certain relevant inputs and at the end of the book that from which we have we have made all the slides in the end of the chapter of the spectral analysis the flow chart is given with the help of which one can obtain a or write down a program for the spectral analysis of structure by using all methods so let me explain that flow chart here so first what we do is that we consider or we take the structure and obtain the stiffness matrix of the structure considering all the degrees of freedom that is the rotations also are considered and from this matrix we obtain a K matrix which is a condensed stiffness matrix corresponding to the dynamic degrees of freedom which are generated the translations and at that time we store in the program this relationship that exists between the theta degrees of freedom and the displacement degrees of freedom which is related through a matrix A and this matrix A can be easily obtained from the condensation relationship and M is the diagonal generally the diagonal mass matrix but it need not be diagonal it can be also a matrix which is a coupled matrix depending upon the problem which you have seen before now from next we obtain once we obtain the stiffness matrix corresponding to all displacements then we obtain the R matrix and if it is a multi support excitation case then in this K matrix we consider also the degrees of freedom at the supports that is a translational degrees of freedom at the supports in the sway direction and once we have that matrix then from that matrix one can obtain a R matrix which is also stored and this R matrix is of the form of minus K s s inverse that is the inverse of the non support degrees of freedom the stiffness matrix corresponding to that and K s g that is the coupling matrix between the non support degrees of freedom and the support degrees of freedom so after we have obtained for the multi support excitation case this R and it is stored then we have the non support degrees of freedom and corresponding to that the stiffness matrix and the mass matrix that we use for the solution we obtain the phi matrix from these K s s and m and omega the natural frequencies then from this also we can obtain phi R that I have discussed before that is if you these phi is the mode shapes for displacements whereas phi R could be a mode shape matrix or mode shape vector for the response quantity of interest and how to obtain that that we have described before that is if I subject the structure to a force of m omega square where m is the matrix and omega i square is the ith frequency then we the response that we get displacement response that will correspond to phi and the solution for any response quantity that is bending moment shear force taken out from that solution that will give the mode shape coefficient for that response quantity so here depending upon the response quantities that we want we can have a phi R matrix and that can be stored from the frequencies one can obtain alpha and beta coefficient and then construct the required c matrix so once we have that then one can obtain h omega matrix that is required using K m and c if we are not interested in all the response then the reduced frequency response front turn matrix h bar omega that can be obtained then for each mode one can obtain h i omega and for that what will require is the modal mass at the at a particular mode and after we have obtained these two quantities then we can go for different kinds of analysis the first kind of analysis is the direct spectral analysis where h bar omega is the complex frequency response function matrix for the degrees of freedom which are of interest that is we use the reduced complex frequency response function matrix sx double dot g matrix that is constructed with the help of the given power spectral density function of the ground motion and the coherence function from which we obtain rho 1 rho 2 etcetera and that is how sx double dot g matrix is determined and rho ij that must be expressed with the help of an expression for the coherence function taken from the literature which is appropriate for the particular site then one can obtain the sji zj that is the cross power spectral density function between any two generalized displacement and once we know the sji zj then one can obtain sj z matrix from that and if you use that sji z matrix then this particular equation uses the mode shapes to find out the sxx that is the power spectral density function matrix in the structural coordinate. So this constitute this one this one and this one sorry this one that these two constitute the Moodle spectral analysis part of it if we are interested in finding out the power spectral density function matrix for any other response quantities then the displacement response quantities then that can be also obtained by pre and post multiplying sji zj matrix with the phi r matrix phi r matrix is the mode shape matrix for that response quantity that we have stored before. Then if you are wanting to obtain the power spectral density function matrix for absolute displacement which you again we had discussed before we have two kinds of displacement one is a relative displacement with respect to the base other is the total displacement by considering the support displacement also if you are interested in finding out that then what we required is an additional expression which is sx double dot g x that is the cross power spectral density function matrix between the excitations and the displacements and this is also given in the form of an equation which we have explained before. So we obtained that and also keep it in the memory if we are required to obtain the power spectral density function matrix of absolute displacement. Next if we are wanting to go for the state special analysis then a matrix must be first generated from the K, M and C matrix then we obtained the eigen values and eigen vectors of matrix A and obtain the frequency response function matrix using matrix A and an identity matrix I then we obtain sfg sfg matrix that is the power spectral density function matrix of fg vector which consists of 0 and minus rx double dot g and how to construct sfg matrix is that I have described just before. So we obtained that and stored it and then go for a direct analysis in which sjz is straight away obtained by using this power your frequency response function matrix and sfg sfg matrix and the transpose of the complex conjugate of each matrix. Note that sjz matrix contains sxx, sx dot x dot and cross spectral density function between displacement and velocity and velocity and displacement some of them is equal to 0 and depending upon whether you require the power spectral density function of displacement or velocity we can choose the appropriate quantities from these two matrices. So that is the direct state space analysis that is direct state space power spectral density function approach. If you are wanting to find out or use the modal state space analysis in the spectral analysis then we obtain for each mode the complex frequency response function and this will require the knowledge of the eigen value of the particular mode this eigen value is generally a complex quantity. So once we obtain this complex frequency response function for the jth mode then sqyqj that is the cross power spectral density function between any two generalized coordinate can be written in this particular fashion that is hi into aj star and sf bar gi f bar gj. Now this f bar gi and f bar gj is obtained in this particular fashion say f bar g is equal to phi inverse f g and sf bar g sf bar g matrix would be given by phi inverse 0 0 0 rx double dot g rt and how to obtain sx double dot g for a multi support excitation that I have described before in which you have the cross correlation function terms rho 1 rho 2 etc and from this one can choose this f bar g f bar g will be a 2n by 2n matrix and from this matrix one can choose any cross power spectral density function term between two generalized load f bar gi and f bar gj. So that is how one can obtain this term over here and this is of course stored before and this matrix also is stored and once we have the knowledge of sqyqj then we can form the sqq matrix that is the power spectral density function matrix in the generalized coordinate and obtain the power spectral density function matrix in the state space form that is sjz which will be equal to phi sqq and phi t and this sjz matrix will contain both x and x dot. So from that one can obtain all the power spectral density function of displacement velocity and any response quantity of interest. So we see that the frequency domain spectral analysis can be carried out in different ways and the method that one uses depends upon the response quantity of interest and the way one wants to solve the problem it can be obtained in the direct form in which either one can use a state space formulation or one can take the second order differential equation in both cases the c matrix is to be constructed by obtaining alpha and beta using two modes of the structure and once we obtain the c matrix then one can perform a direct spectral analysis. If we are wanting to obtain the spectral analysis in modal coordinates or what we call as the modal spectral analysis then we have to find out the mode shapes and frequencies of the structure and this is usually done for systems which has many degrees of freedom therefore inversion of the H matrix and construction of the H matrix that becomes somewhat tedious and therefore one can go for a modal spectral analysis and we can take only a limited number of modes that is may be the first 5 or 10 modes of the system. And then obtain the first the power spectral density function matrix in generalized coordinate using the complex frequency response function for each mode and constructing a matrix of the generalized forces and for that what one has to obtain is the first one matrix is the sx the whole dot g matrix that is the matrix of excitations if it is a multi support excitation and once we have obtained that then one can find out the power spectral density function or cross power spectral density function between any two supports in the generalized system and with the help of that one can obtain the cross power spectral density function between any two generalized coordinates and from that one can obtain the sqq matrix and once the sqq matrix that is the matrix of the power spectral density function matrix of the generalized coordinates that is known then one can obtain the corresponding power spectral density function matrix in the structural coordinate by using the modal transformation rule that is sxx will be equal to phi into szz into phi t. The state space formulation can be used in many cases where we may be interested in the velocity that is the power spectral density functions of the velocity of the degree of freedom then the state space formulation is advantageous and again in that case one can go for a direct state space analysis and when we obtain the state space analysis in the direct form for the spectral analysis then h omega matrix that is a complex matrix frequency response function matrix that is of different form than the complex frequency function matrix that we use for the second order differential equation here it uses simply the A matrix for obtaining the complex frequency response function matrix for the system. Once it is known then we again use the same formulation to obtain the szz matrix that is the psdf matrix of the state space of the states of the system x and x dot. So in the szz matrix we get sxx as well as sx dot x dot. And depending upon the requirement one can choose any term from these matrices to obtain the power spectral density function of the required response. In obtaining and doing this particular formulation one has to obtain a power spectral density function matrix of f bar g that is the generalized load which consists of the first n terms of that will be 0 and second n terms will be equal to minus rx double dot g and how we can obtain the sf bar g sf bar g that we have described before. Similarly one can also obtain a modal analysis in state space for performing the spectral analysis and in that case we use the mode shapes and frequencies of the matrix A and with the help of that one can obtain the responses in the generalized coordinate q and or in other words we obtain the cross power spectral density function between any to generalized coordinate from that we obtain sqq matrix and then we can obtain szz matrix.