 We are considering problem of modeling vehicle structure interactions using finite element method, this will be the concluding lecture on structural stability analysis, we will begin a new topic towards the end of this lecture on finite element model updating. So what we have done in the study of time varying systems and stability is we considered flow case theory for periodically time varying systems, these coefficients help us to determine boundaries of stability for response of dynamical systems with period time varying periodic coefficients. In certain class of problems that is statically loaded structures, in certain class of problems a dynamic analysis is needed to resolve the questions on stability and this we discussed in the previous lecture and we have derived the governing partial differential equation for beam moving oscillator systems and we started discussing weighted integral and weak formulations to develop FE models starting from governing partial differential equation. So we will continue with this and now consider the finite element analysis of vehicle structure interactions, so this to quickly recall the supporting structure is modeled as a Euler-Bernoulli beam with these parameters and the moving vehicle is modeled as a single degree freedom oscillator, MU is the unsprung mass, MS is the sprung mass, KV and CV are the vehicle spring and damping coefficients. This type of problems are currently gaining importance study of this type of problems because of development of high speed trains and mobile aircraft launchers and in even in numerically controlled machines where the tools move on the job at fairly high speed to enhance productivity. So when we talk about vehicle structure interactions the scope of the problem need not really be confined to only highway bridges or railway bridges, it can encompass a broader class of problems and the kind of formulation that we are going to discuss is in principle applicable to this wide range of problems. This equation we have derived in the previous lectures, so the vehicle enters the bridge span at T equal to 0 and exits at T exit, it moves with an acceleration A and velocity V which remain time which do not change when the vehicle is on the bridge, and we have one equation for the vehicle, a single degree freedom oscillator it has terms containing the bridge response and the beam response itself is governed by a Euler-Bernoulli beam equation and the wheel force has contribution from weight of the vehicle and the vehicle spring stiffness damper and the inertial effects due to the unsprung mass, we are using total derivatives here because the wheel moves on a deflected profile of the beam and it will induce Coriolis forces, and once the vehicle exits the bridge we consider only the free vibration of the beam assuming that our interest is basically on the beam and not so much on the vehicle. So we discretize the beam into a set of capital N elements, and for an Nth element the coordinates are XN-1 and XN and the degrees of freedom we model this element with two-noded Euler-Bernoulli beam element with two degrees of freedom per node, U1, U2, U3, U4 are displacements, P1, P2, P3, P4 are the corresponding stress resultants. Now LN is the length of the beam and TN-1 is a time at which the vehicle enters the Nth element and at TN it exits the vehicle, sorry the vehicle exits the Nth element. So we are now considering what happens during the time period when vehicle enters this element and leaves this element. So the equation of motion while the vehicle is on the Nth element as follows, this is oscillator equation, so we use an indicator function to indicate that this force is valid only when the element is carrying the load, so the vehicle exists on the element over the time period TN-2 TN, and indicator function takes a value of 1 when T is in this range otherwise it is 0, and this is the wheel force, again it is multiplied by an indicator function and these are the terms that we have already discussed. Now we denote by Y naught of T the deflection of the supporting structure under the wheel load which is Y of X naught, T where X naught is VT plus half AT square, we must note that X and X naught are calculated from the left end of the beam. Now we make a coordinate transformation, this is the Nth element of length LN, we introduce a coordinate XI as defined through this X minus, actually XN minus 1, and by this arrangement the XI would lie between 0 to LN. Now X naught is the position of the wheel in the local coordinate system, so this is XN minus 1 is some addition of all the length elements up to this point, and this XN is the length that includes this LN as well. Now the equation of motion while the vehicle is on the Nth element we have already written, now this prime here is now with respect to XI, so this is the only difference we have introduced the XI coordinate system which is specific to the Nth element. Now we write the weighted residual form of the beam equation, so we take the all the forcing functions to the left side and multiply by a weight function and integrate over 0 to LN equal to 0, so this W of XI is the weight function. Now to get the weak form we carry out integration by parts so that the demands on continuity on trial functions and weight function can be equally distributed, so this we have seen in the previous lecture, so this leads to this equation where the first upon doing integration by parts twice I get this equation. Now this is our weak form of the weak statement of the equation, now we denote by V the shear force and the bending moment EI Y triple prime and EI Y double prime and VN minus 1 is the value of shear force at the left end and VN is the shear force at the right end, similarly we define MN minus 1, MN as the bending moments at XI equal to 0 and LN. The weak form using these notations now take this you know the value is represented by this equation. Now our job is to approximate the field variable in terms of the nodal degrees of freedom and suitable interpolation functions, so this is the representation that we use there are two nodes and two degrees of freedom per node therefore we need four generalized coordinates and as before we take cubic polynomials and we obtain equations we substitute into this and we use the weight functions Phi I of XI for 1, 2, 3, 4 and get a set of four equations which lead to the required equation for the nodal degrees of freedom UI of T. Now I write I use capital Phi to denote the this trial functions it is one row and four columns here and at XI equal to 0 it is given by this and at XI equal to LN this is given by this and similarly Phi dash of Phi dash at XI equal to 0 and XI equal to LN are given by this. Now using these relations we can take care of the terms at the boundary and we can show that the terms that is this boundary terms lead to force vector minus VN minus 1, MN minus 1, VN and minus MN transpose because of these you know properties of the trial function. Now we can examine each term one by one and see what is the contribution they make. So this first term MW of psi Y double dot of psi, T and the second term W double prime EI Y double prime lead respectively to a beam mass and stiffness matrices given by this where L is LN is given by this, so this we have seen in the last class. Similarly the third term involving damping terms leads to the beam element damping matrix. Now the evaluation of the other term which is W of psi that the wheel force term which is actually since this is a direct delta function we can quickly do the integration it is given by this, so this requires elaboration. Now what I will do is I will use alpha of T to denote the nodal degrees of freedom U1, U2, U3, U4 transpose, so it is a 4 cross 1 vector, so now F of XI naught, T is given by using the definitions that we have introduced given by this is given by this and now Y naught of T itself is Y of XI naught, T which is phi of XI naught into alpha that is a representation we are using. So this is 1 cross 4, this is 4 cross 1 therefore this is a scalar representation. Now I have to take care of these terms D by DT of Y naught D square by DT square of Y naught, so we can write now D by DT as this is the operator nu plus AT dou by dou psi plus dou by dou T, operating on Y naught it produces this equation and we get this equation that is this acts on Y naught and we get this equation and the second derivative accordingly can also be obtained by repeating this product operation on this, so we get the terms this and this are computed. Now so we have D square Y naught by DT square using this notation we get this representation, so we can rearrange the terms and now see what happens, so this is the phi psi transpose and phi double psi transpose that is required in defining these functions that is elaborated here. Now we can now write the resulting equations, so what we have done is the weight function is taken to be one of the trial functions, there are 4 trial functions and we get 4 equations and these equations are assembled in the matrix form, so the terms involving flexural rigidity will lead to K alpha, inertia will lead to M alpha double dot and damping is C alpha dot, this is a wheel force term, so here of course the indicator function multiplies all the terms so that it is clear that these terms are valid only when the vehicle is on the element under consideration, so these details can be absorbed but what we should notice is this has terms involving alpha, alpha dot and as well as alpha double dot, so these terms can be transferred to the left hand side, so if I transfer all the terms that multiply alpha double dot to the left hand side the mass matrix will have the mass matrix of the beam element plus the contribution from the wheel force, the damping matrix that of the beam element plus the contribution from the wheel force and similarly stiffness of the beam element plus the contribution from the wheel force, this must be equal to the contribution due to the self-weight of the structure and the load transferred from the vehicle. Now the vehicle degree of freedom itself can be written in this form, there is a Y naught here and to D Y naught by DT I will use the representation that we have derived earlier, this is what we get here, so I can write now the mass matrix as M plus M tilde, damping matrix as C plus C tilde and stiffness as K plus K tilde, this tilde related quantities are functions of time and they are described here, so XI naught itself is position of the wheel which changes with time, so it is VT plus alpha D square, so there is a time dependency here also, so all these terms are functions of time, so we get the governing equation for the beam and for the vehicle, now I can now combine the beam degree of freedom with the vehicle degree of freedom and define a combined displacement vector call it a DN, that is alpha of T plus and U of T, FN I write it as the nodal forces as consisting of these quantities as FN, so with these notations the equation now become some M of T into DN double dot plus C of T into DN dot plus some K of T into D dot is equal to some forcing function plus the nodal forces, so this you know is of the form that we have been talking, so this is like, so we have now the time varying mass, stiffness and damping terms, so this is the main feature of this problem, this is the equation at the element level, now further steps involves assembly and imposition of boundary conditions, this I am not going to elaborate because we have done this several times for various problems, so we need not revisit this issue again. Now what is to be noted here is that the, because of the time dependency of the structural matrices these equations although they are linear, the concept of natural coordinates etcetera are not applicable here, so you cannot talk about natural frequency and mode shapes for this type of systems and the only way to tackle these problems quantitatively is to integrate them using one of the integration schemes that we discussed like Newmark beta method or any other method that we discussed earlier. Of course if a series of loads pass on this and if there is some periodicity associated with these functions then one can use flow case theory and determine stability conditions for the vehicle, for the bridge vehicle system, that is of course one possibility if we are interested in qualitative behavior, if only one vehicle passes on the bridge this time dependency is of a transient nature and there is a questions about stability will not be, you know, steady state, stability of steady state solutions are not discussed for this type of problems. Now as I already said what we have idealized the supporting structure as a Euler-Bernoulli beam and the moving oscillator as a, moving vehicle as a single degree freedom system. Now in practice however the supporting structure can be very complex, this is one of the simpler forms of a bridge, the bridge structures are lot more complicated than this, and this is a moving train, you know it's a series of elastic systems having mass and stiffness and damping characteristics, the thing is when do we need to do a vehicle structure interaction problem, that's a question that we can consider, typically if the mass of the moving vehicle is comparable to the mass of the supporting structure, and if the speed of the vehicle is high, and if the natural frequency of the vehicle and the bridge are comparable, there are various conditions under which one can think of expecting that vehicle structure interactions would be important. In this type of problems for example in railway bridge a plate girder bridge, a single span plate girder bridge typically could weigh about 15 to 20 tons, whereas a locomotive like this can weigh more than 100 tons, 110 to 120 tons, so when this type of vehicle passes on bridge the mass of the vehicle becomes predominantly higher than that of the supporting structure, and vehicle structure interaction would be significantly important. Now to illustrate the formulation that we have developed, a simple exercise has been done, this one of our students has developed a software based on this formulation, this is a data for the beam structure, a single span beam, this beam structure is more like a laboratory model than a realistic real life structure, so this structure has been discretized with 12 elements, and there are 25 degrees of freedom in this model, and a numeric beta implicit scheme has been used to integrate with these details of integration, and some of the results are shown here, so this is mid span deflection as a function of the position of the load, if you see here there is one result that corresponds to analysis of structure under static loads, that is a traditional influence line type of studies that we would do. The second graph that is the moving force diagram takes into account the weight of the vehicle moving on the bridge system, the full line is a complete coupled dynamic analysis, and these results are specifically designed to show the reasonableness of the algorithm developed and the calculations performed. So this is actually the mid span displacement as a function of the position of the load, and this is actually the same response now plotted as a function of time, so this is just an illustration of the formulation that is developed. Now there are several studies on structures under moving vehicles, the book by Fryba is one of the classical books in this field, there's another book, recent book by Yang and others on vehicle bridge interaction dynamics, these two books contain quite useful information for those who wish to study this subject further. Now as an exercise I would suggest that the problem of a stack under biaxial earthquake ground motion we have formulated in the previous lecture, the exercise is to develop a finite element model for this structure using the weak formulation, so that means you start with a given partial differential equation, include the parametric excitation terms and of course the time dependent boundary support motion, and then discretize it using say Euler-Bernoulli beam formulation, and just as you have done for vehicle structure interaction problem the exercise is to develop a model for finite element model for this system, here again you will see that the structural matrices would be time dependent. Now before we leave the subject of structural stability analysis there are couple of instances where stability questions arise, I just would like to mention because they have considerable interest both from application point of view and from the point of view of understanding the collage of different kinds of problems that arise in stability analysis. So this is a problem of vibration of a pipe that is conveying fluid, a fluid is moving in this pipe with a velocity V, the fluid this is a pipe cross section, area of flow is A and these are the diameters, outer and inner diameters, the fluid density is rho and P is the pressure under which it is flowing and V is the velocity of the flow. Now the thing is that what we should model here is the fluid mass experiences centrifugal acceleration as it flows through the deflecting pipe, so this is the issue that we have to additionally handle other than the stiffness and inertia of the pipe and mass of the fluid that is flowing through. So this book by Blevins on flow induced vibration contain many examples of fluid structure interaction problems especially vibration problems of piping and other systems, so this is this illustration is picked up from details provided in this book. Without getting into the fluid mechanics arguments I will present the governing field equation and explain what the terms mean, so we have EIY4 rho A VY double dot, 2 rho A VY dot prime and M plus rho A Y double dot equal to 0, so this is a boundary condition for a simply supported beam condition. This is a flexural rigidity terms coming from flexure of the beam, this is the inertial term, M is a mass per unit length of the beam and this is mass per unit length of the fluid, so these two terms are the new terms, if there was no flow we would write this equation, I mean omit these two terms and write the equation. Now what is this rho A VY double prime, prime is DY by DX, dou by dou X, this is a centrifugal force due to the acceleration of fluid through the curvature of the deformed pipe, it produces axial compression and as you have seen presence of axial load reduces the natural frequencies and it causes buckling, so this is the effect that we can expect from this term, this is also an interesting term, this is a force required to rotate the fluid mass with local pipe rotation, this is a Coriolis force, you should notice the mixed derivative here, Y dot prime is actually this is dou square Y by dou X dou T, so this is a mixed derivative term which creates some interesting features in obtaining the solution, it causes an asymmetric distortion of the classical mode shapes, so sine and pi X by L will not, as we will shortly see sine and pi X by L will not be the mode shape for this, exact mode shape for this, capital M is mass per unit length, that includes mass per unit length of the pipe and the flowing fluid. Now we want to now construct a solution for this and for example estimate natural frequency and see how it depends on the flow velocity, that is a question, are there any stability related issues in handling this problem? Now if we start by taking that phi of X is sine omega T and when you substitute here this Y dot prime term will lead to cosine terms, so you will have problem in dealing with that, similarly if you take Y of X, T as phi of X into A sine omega T plus B cos omega T, now to allow for presence of cos omega T term I can include cos omega T terms, so we can avoid this problem by doing this, but if we take phi of X as sine N pi X by L this prime Y dot prime will produce cosine functions, so again that will be problematic in handling because it will be a specially asymmetric terms, it will induce specially asymmetric terms for symmetric mode shapes for 1, 3, 5 and specially symmetric terms for asymmetric mode shapes, so what we do is therefore a mode shape, a single mode shape is assumed in a Fourier series like this, it has sine N pi X by L term with odd indices multiplying sine omega T, and again sine N pi X by L with even number of terms multiplying cos omega T, mind you this is not a modal summation, this is a series representation for a single mode, okay, that should be understood, this summation is not over omega N, now we substitute this into the governing equation, we compute Y Y dot prime and we will have problem here because we have sine N pi X by L and when I compute the mixed derivative there will be cos N pi X by L term, this term will be difficult to handle, so what we do is this cos N pi X by L term itself we will expand in a Fourier series containing sine P pi X by L, okay, now so this can be done, these terms can be evaluated, but the problem is that the cos N pi X by L term will not satisfy the boundary conditions, although it may represent the behavior away from the boundaries, at the boundaries it will not satisfy the prescribed boundary condition, but we will ignore that effect and we will go ahead, because it brings in only sine terms to represent spatial variations and that is helpful for us, so with all this we can go ahead and obtain these equations, for N odd I get this equation, for N even I get this equation, these are infinite number of equations, they are infinite number of terms, so for practical computation we truncate this at capital N number of terms, suppose A bar is A 1, A 2, A N, then I get K N P as this and we can write the eigenvalue problem in terms of K N P as K A bar minus omega square M IA equal to J A bar, I is the identity matrix, M is the total mass. Now for non-trivial solutions the determinant of this equation should be 0, now for illustration what we will do is we will retain only two terms so that we can get some simple terms and examine how the solution behaves, so we will carry out this as details are not provided here, so we introduce two parameters omega capital N, it is not the capital N natural frequency but instead it is the fundamental natural frequency of the pipe in absence of fluid flow, and VC is the critical velocity for buckling, as we saw this term is like a compressive force so we can compute the Euler buckling load corresponding to that and that gives rise to this VC and this is this, now in terms of that I get this equation, so this is a characteristic equation for omega I can solve for it, I get this equation. Now if we examine this we see that there are two roots because we have written two terms and omega 1 and omega 2 are real whenever velocity is less than the critical velocity, and we can approximate this by ignoring the terms involving mass and this is actually we can simplify this and as a first approximation we can get the natural frequency as given by this, so at critical velocity of flow the pipe buckles because the natural frequency goes to 0, so at that time omega is natural frequency would be 0, now how about the mode shapes, we can put omega equal to omega 1 and evaluate the ratio of A2 by A1 and we can show this, and whenever this velocity of flow is less than critical velocity you can show that this ratio A2 by A1 is less than about the small number this indicates that mode shape is predominantly sinusoidal, although there is an asymmetric distortion this seems to produce marginal effect and results from this formulation have been compared in the Blavin's book with experimentally observed data and reasonable comparisons have been found. Now I would like to set an exercise, we will consider this problem and starting with this equation the problem is to develop a finite element model by performing an Eigen value analysis that includes Coriolis terms estimate the critical velocity, that means we will not adopt this representation in solving the problem that is this representation we will discretize deal with a generalized Eigen value problem and examine the relationship between complex valued natural frequencies, mode shapes and the flow velocity and infer without introducing any ad hoc assumptions or you know selecting functions which don't satisfy boundary conditions and so on and so forth we can examine what the finite element model teaches us, so this is left as an exercise. Now there is one more topic which has been quite widely studied in existing literature, can there be parametric instabilities if the parametric excitations are transient in nature, for example if you have suddenly applied loads or impulsive loads, impulsive axial loads or suddenly applied axial loads, how does the structure behave? Now we can quickly recall a simple model that is undamped single degree freedom system which is subjected to an suddenly applied load, so we can examine how the system respond, so you can see U of T is the step function that means a constant load is applied suddenly at T equal to 0 and let us assume that initial conditions system is addressed when this happens and this is a complementary function plus particular integral and using this prescribed initial conditions we can show that the total solution is given by 1 by K 1 minus cos omega T, 1 by K is actually the static response, if the load were to be not suddenly applied, so the dynamic amplification because we have applied the load suddenly for the undamped case is about 2, okay, so this tells you that for statically loaded systems, actually loaded structures which are actually loaded by static forces, if the force were to be applied suddenly there could be instabilities, so this has been studied I have given 2 references here which have useful information, see for example we have studied this problem of snap through under constant load, so when P of T was P we found that this has the load deflection path and there is a snap through and things like that, now suppose snap through occurs at some P equal to P critical, now if this P instead of applying slowly if I applied suddenly then you can easily imagine that there is room for dynamic amplification of the response because load is applied suddenly, then the response can shoot up, so even for values of P which is less than P critical the structure can lose stability and it can you know escape and it can snap and start oscillating somewhere else, so when that happens the load that you are applying may be less than the static critical load value, because the load has been applied suddenly that happens, so this type of problems have been studied I am not going to discuss this in detail, suddenly applied loads and even impulsive loads that is load acting for a short time you know, so they can also cause buckling, so in general the shell type of structures and other structures if you have impulsive type of loading there will be membrane forces that will be set up due to that transient membrane forces and during that period they can interact with flexural responses and create instability conditions, so that intuitively one can expect that that might happen, but how to characterize this etc is this matter that we can to be studied and I have as I said already there are couple of references that I have suggested for that, so with this we conclude our discussion on stability of structures, so we move on to now a new module that is that addresses studies on existing structures, so before a structure comes into existence the only way we can investigate how the structure might behave is through developing mathematical models, that is what we often do when we design structures and at this time of design the structure would not have come into existence, but once the structure comes into existence then you have the opportunity to measure the response of the structure and you get an experimental approach to study the structure you know as an option. Now even after the structure comes into existence we can continue to use the mathematical modeling, now if we conduct a measurement on the performance of the structure under a given set of loading and observe the structural response and if we were to predict the same structural response for the similar type of loading using a mathematical model it is quite likely that the two results would not match, now the basic question is can we combine these two models, okay, what are the issues that arise when we try to answer this question, so this type of questions are studied in a subject known as finite element model updating, so the question is you make a finite element model for an existing system you have a set of measurements that are available to you on this structure and you can mimic the situations under which you have got these measurements through a mathematical model and the prediction from mathematical model and the observed performance of the structure may or may not match and if there is difference how do you deal with that, can we for example improve upon the finite element model, so that the results from finite element model are reconciled with prediction measurements from experiment, to address this we have to understand that given the advent of sensing technology it has become possible now to instrument structures, so this is a railway bridge structure in which we did a field work where this was instrumented through nearly about 50 to 100 sensors, we had strain gauges, we had LVDT that measured displacement, we had vibrations via strain gauges that again there is a strain gauge, axillary meters, uniaxial, biaxial, etc., so this bridge structure was instrumented and its performance was measured under operating loads, static, dynamic, diagnostic loads where we knew what loads were applying and ambient loads which was due to the prevailing traffic on this line, bridge line, so on and so on. Now the question is after obtaining this data from this existing bridge and suppose if I were to make the finite element model for this and conduct this test numerically on my finite element model I will still be able to predict what should be the readings from these sensors, but those readings will not match with what exactly we have observed in the field, so we can measure several things, this is a test that is done to measure impulse response functions from which we can extract the frequency response function of the system, there are various tests that can be done on the structure, like a bridge structure, and these are the typical readings that we get, this is a strain gauge reading, this is an axillary meter reading, and this is readings from LVDTs, so this type of data can emanate from existing structures. Now we need, now the question that we try to answer is having had these measurements can we identify the parameters of the finite element model that we postulate for the given structure. The need for such exercise for example in the context of these bridges arise for example, these bridges might have been designed to carry a specific level of axle loads, the user may like to enhance the axle loads for future use, or have trains that move faster than what they were envisaged at the time of design, heavier vehicles, faster movement, longer trains, so on and so forth, so can these bridges cope up with those increased demands, or should we repair and retrofit these bridges to carry higher level of loads and so on and so forth, to be able to answer that one way is to really apply those enhanced loads and see whether the bridge can take it or not, but in live bridges that is not possible, we can't really apply loads that might cause destruction to these bridges, so the best way is to apply the loads that are permitted on these bridges and then take measurements of the kind that I mentioned and make a finite element model which whose predictions are reconciled with the measurements made by updating the model parameters or details of modeling, and then on the updated finite element model we can apply enhanced loads, make the loads move with faster speeds and so on and so forth, and predict the performance of the structure. Now, so the general framework for this type of problems can be classified as shown here, so we have a system input and output, suppose if we know inputs completely and if the system is represented by a known mathematical model completely and if we want to predict the output that is a problem of response analysis, but it is not always that we deal with that type of problems. Inputs could be partially known, it could be noisy, in the noise itself could be having certain mathematical features, the system could be represented by a known mathematical model, but the model parameters could be partially known, and the system behavior could be linear or nonlinear, the output could be related to state variables through a mathematical model, for example if you measure strain and in the mathematical model you have displacement as the state variable, the relationship between measured strain and displacement is through a strain displacement relation which is could be linear or nonlinear, that is a call that modular has to take, so there is a mathematical modeling issues involved in relating what we measure through the system states of the mathematical model, the measurements would be noisy, the parameter of the model could be partially known, and again the noise characteristic could have complicating features. So with this, within this framework we can classify problems in vibration engineering into several categories, for example if input is given and system is given and if response has to be determined this is a problem in response analysis, this is a forward problem. On the other hand if I know the input and if I know the output may be partially and we are task is to determine the system, that means we have to determine the mathematical model parameters of the system, then this problem is known as system identification, there are various levels of making system identification it could be parametric or non parametric as well, but it pertains to a mathematical model for the system behavior. Similarly if an input is to be determined and system is to be determined and response is given, this is what is known as blind system identification, you only know the response of the system you have not even measured the input nor the, nor you know many things about the system. Now if input is not given and if system is known and the response is known the unknown will be the applied input, so this is a problem of force identification or it is also a problem of measurement, this is a principle on which a sensor would work, in a sensor we know the system characteristics and the response of the sensor, but we will not know what is causing the sensor to produce an output. Now we can generalize this kind of classification, for example if input is given and system has to be selected and on the response I have prescribed bounds, then how do you select parameters, system parameters that's a problem of design, now response analysis is a forward problem, it is a fairly simple problem that is what we have been discussing so far in the course, other problems are more difficult, problem of system identification is an inverse problem, it has its own set of difficulties and some of that hopefully we will be able to see as we go along, so in structural system identification basically these are studies on existing structures and these studies represent combined experimental and analytical studies and these are inverse problems, as I already said given the input and the output our problem is to determine the system parameter. Now if I make a finite element model for an existing structure what could be sources of errors, those which are inherent in finite element method, for example there will be discretization errors and a question of interpolation, then there are solution errors, you carry out integrations, there is a round off on a computer, you will adopt certain algorithms for eigenvalue extraction, you expand response in mode shapes and you may truncate the modes at a given value and there are many computational issues like inverting matrices and so on and so forth. The other issue is modeling of damping, so damping is a very contentious issue, whether damping is linear or nonlinear, viscous or structural proportional or non-proportional or a mixture of all of them, there are many issues there. Now these are some of the things that are inherent in any finite element, whenever you use finite element method you have to make your choices on all these parameters related to this so that you get acceptable results, but whenever an analyst uses finite element method for a same problem not two analysts will come out with the same model, so one has to choose elements to represent the given geometry, then you may omit unimportant details, then modeling of boundary conditions requires careful consideration, most of the important decisions that a modular takes in context of modern finite element method, application of finite element method to structural engineering problems relate to questions on boundary conditions, is the boundary condition on displacement fixed or hinged or sliding or whatever, then modeling of joints, so if there is partial fixity or the joints are flexible, for example in a transmission line tower the joints may not be pinned, there may be partial transmission of movements, and similarly at joints where we think there is a fixity condition it may not be truly fixed, there could be some flexibility, then choice of constitutive laws, we often make assumptions of homogeneity isotropy and so on and so forth, how far they are valid, then numerical values assigned to the model parameters, these are the decisions that analyst makes, analyst has options to make more and more refined models to make you know to deal with each of these issues, but in any given situation some choice will be made, how about in experimental work, so there is data acquisition errors, so there are mass loading effects of the transducers, then the exciter and structure interactions could be there, and how do you support the structure in a model testing, these are in relation to model testing, and there will be measurement noise and how do you know the structure is behaving linearly when you are conducting a test aimed at finding natural frequencies and mode shapes, then there is again in experimental work also the number of sensors is limited, so measurement of limited number of points on the structure and certain degrees of freedom may not be possible to measure like rotation and interior degrees of freedom, the sensing technology is improving with you know many things are becoming possible, but still there are limitations, then limits on frequency range, then signal processing errors, there are leakage, aliasing, windowing, and effect of using discrete Fourier transform and when you measure frequency response function you will have to do averaging across a fixed number of samples that has, that introduces sampling fluctuations, and then after you measure frequency response functions to extract the natural frequencies and mode shapes, you have to use certain algorithms, a simple one is circle fit method, you can use a single FRF or multiple FRF methods so on and so forth, so we are not discussed these issues, but I am summarizing the possible sources when we do model testing, model testing are aimed at finding natural frequencies, mode shapes, participation factors, and model damping through experimental methods. Now the problem of updating, you know the issue is that finite element model are aimed at studies on an idealized mathematical model, and experimental model are studies on actual structure, so it is generally believed that experimental results are more trustworthy than the result based on numerical modeling, the general philosophy of updating is that results from numerical models plus corrections lead to results from experiments, I mean this is over simplified statement, but this is a general philosophy. Now we can do, there are two things, we can do local corrections, that means we can find out where exactly within the finite element model there are errors, we localize errors and correct the errors, it is assumed that the mathematical model consists of discrete locatable errors associated with the physical meaning. Then global corrections, corrections are made in a curve fitting sense, the corrections would not have any specific physical meaning, simply the mathematical model is forced to reconcile with the prediction from experiments. Why experimental models are more acceptable, there are no compromises on constitutive laws, boundary conditions, joint behavior, damping characteristics, and stiffness and mass distributions, and any presence of residual stresses, so we don't make any assumption, they are what they are in an experiment, but whereas in a mathematical model we need to make a model for each one of this, it can be as refined as you wish, but finally a choice has to be still made. Now difficulties associated with locating errors in the theoretical model, we may have insufficient experimental modes, insufficient experimental coordinates, and size and mesh incompatibility of the experiment and finite element models, then experimental random and systematic errors, and absence of damping in FE normal modes and presence of damping in experimental normal, so when you measure natural frequencies using experimental methods it will be always damped natural frequencies and damped normal modes, there is no way you can switch off damping in an experimental work, whereas in finite element model traditionally we use undamped natural frequencies and undamped normal modes and derive natural coordinates from them, so this is just a quick overview of some of the issues that arise when we think of reconciling finite element model predictions with experimental, so what I aim to do in next one or two lectures is to give a glimpse of basic issues related to finite element model updating, the discussion will be focused on mathematical framework for carrying out FE model updating, we specifically discuss what is known as inverse sensitivity methods, and then what are the tools for comparing two models, suppose one experimental model you have and another mathematical model you have, how do we compare the two models, so that takes us into a discussion on model correlation, so some of the matrix used for this we'll discuss, so these two topics we'll try to cover in the following lectures, at this stage we will close this lecture.