 In the previous lecture we considered questions about imperfection sensitive structures, we also derived the outline the derivation of geometric stiffness for a plate element and considered the form of the function U for a Timoshenko beam element, 3D beam element. We will quickly recall some of the details, so for a 3D beam element the question that we asked is it simply that the computational effort increases with no new phenomena kicking in or are there any new questions that we need to address? It turns out that there will be certain newer terms that we need to take care of, that takes into account coupling between axial forces with bending along two axes, torsion about longitudinal axis and also coupling between torque and bending moment. This model is a simplified model, a more refined model would include other effects. We also derived the geometric stiffness matrix for a thin plate element, we didn't complete the derivation but the basic steps involved in arriving at the geometric stiffness matrix was outlined. We considered to discuss questions on imperfection sensitivity we considered three archetypal mechanical models which basically consisted of rigid link AB loaded in three different, supported in three different waves. These three systems were hypothetically designed to possess same critical load and the contention was that if one were to perform an experiment the critical load although theoretically all the critical load for all the three systems should be exactly equal it will turn out that the beam, this structure will have, the first system will have higher critical load than second one and second one will have higher value than the third one. Why that happens was a question that we were trying to answer and that we approached by considering the load deflection diagram by including large deformations and imperfections. So the bifurcation diagram for the first system had a rising load path whose rate of change dramatically as we came close to the point of bifurcation but this path did not encounter any unstable point, so the structure was able to carry the load even after large loads were applied. Whereas for the second system the load deflection diagram raised and it reached a point beyond which it was no longer able to carry the load and it bifurcated into two unstable paths which were symmetric about this. In presence of imperfection the load deflection path rises up to a point and encounters an unstable branch and thereby limiting the load carrying capacity to these values. So the maximum load that this system can carry is limited by the imperfection that is present, so greater the imperfection lesser the ability to carry that is depicted in this graph for the second system. The third system in which the spring was placed like this again the load deflection path raised up to a point and it bifurcated into a stable path and an unstable path, the unstable path was dropping and in presence of imperfection the load carrying capacity was dictated by the behavior of the unstable branch and there was substantial lowering of load carrying capacity because of imperfection. We also considered problems of buckling of beams on Winkler's foundation, specifically this graph is for a structure which is simply supported and on elastic foundation we found that for certain L by R ratio the half sine wave shape of the mode shape leads to the lowest critical load, but this is not generally true for certain other choices of L by R ratio the full sine wave which corresponds to M equal to 2 becomes a critical load for the system, so this was one special feature associated with beams on elastic foundation. We also studied the problem of a ring and derived the critical load that external load that can be applied axisymmetrically on the ring, we derived the value of that using simple formulation. Now we will continue with that discussion as applied to a circular arch and then we will take up the topic for today's class that is questions about stability analysis using structural dynamic concepts. So this is a circular arch which is loaded as shown here, it's a two-hinged circular arch under uniform pressure, it would not have created bending moment in a pre-buckle state, but what is our critical value of Q, that is what is the value of Q which induces bending in the arch, so this is discussed in the book by Timershankov and Gehry, the simple based on the formulation given there we can write the expression for bending moment and one gets this expression which we already saw for a ring in this form. So the Ghani equation here is as shown here with boundary condition W of 0 and W of 2 alpha equal to 0. Now we define K square as this quantity and write the solution as A cos K theta plus B sine K theta, now imposing the boundary condition W of 0 equal to 0 I get K equal to 0 and W of 2 alpha equal to 0 gives me sine 2K alpha equal to 0 from which I get K to be pi by alpha, so the critical load is given by this and QC turns out to be EI by R cube into this quantity, this is exact result assuming thin cross section is thin. So the exercise that I said is you can select in circular arch like this and use Euler-Bernoulli beam elements and discretize this and formulate the structure elastic and geometric stiffness matrix and evaluate the critical value of Q and compare it with this exact value, okay. So this now we will move on to a new topic, this is dynamic analysis of stability and analysis of time varying systems. Now we will consider few topics, the first topic that we will look at is how to perform dynamic analysis of structure in presence of static initial stresses, so this is generalization of beam column study to dynamic problems, then we will discuss what are known as parametrically excited systems and consider questions on stability, I will explain what it means shortly, then we will consider certain situations where we deal with statically loaded structures but it turns out that to correctly infer their stability characteristics you need to perform a dynamic analysis, right, these are known as follower force problems, there is a lot of discussion on that on this question in the existing literature, there are questions on whether this exists in the first place also have been raised, but it poses an interesting question, so we will just take a look at that. And then what we will do is we will consider some time varying systems and see how we can use finite element method and discretize that type of problems, okay, by time varying systems I mean some of the structural matrices will become functions of time, so I will explain all this in due course, this is just a glimpse of what the topics we are going to touch upon. So we will start with the simple problem of a beam which is actually loaded and carries a transverse load F of X, T, this is a beam equation and these are the boundary condition as appropriate for simple boundary conditions. Now the question I would like to ask is what is the influence of presence of axial load and response of this system, if P equal to 0 we already studied this how to carry out this analysis, when P is not equal to 0 what happens, so this is analogous to the beam column example where if we are loading F of X we knew how to analyze the problem with P equal to 0 and by including P what was the effect was the question that we consider. So when it comes to dynamic analysis of this type of problems the question arises what is the role of P in determining the natural frequencies and mode shapes of the system, that has bearing on force response analysis subsequently. So let's first start with consider this equation and I will use prime for a derivative with respect to spatial variable and we will start with the free vibration, undamped free vibration analysis, we will use a method of separation of variables and look for a solution where all points on the structure vibrate harmonically at the same frequency omega, such motions are possible if this phi and omega satisfy this equation. Now we rise to omega T cannot be 0, therefore I will get this equation here and I will rearrange these terms I will write for P by EI alpha square and for M omega square by EI lambda to the power of 4, so this is an eigenvalue problem the eigenvalue being the load P, okay all other parameters are known of course load P and omega both, you know P is given omega is the eigenvalue parameter, if you are doing stability analysis for a given omega of course that will be a different question, the question we are considering is P is given and omega is the eigenvalue, this constitutes an eigenvalue problem, let us seek solution in the form, this is a linear ordinary differential equation with constant coefficients therefore exponential function should satisfy this equation, so we seek the solution in the form of phi naught E raise to Sx, substitute this we get the characteristic equation as shown here and the roots of this characteristic equation are given by this, this is a bi-coderatic equation, so S square is this and consequently S itself is given by roots of this, S1 1, 2 will be plus minus epsilon and this will be plus minus I delta, you can see here that S1 square will be positive whereas S2 square will be negative, therefore roots leading from S2 will be pure imaginary complex conjugate whereas this will be real, so the solution will have sine and cosine hyperbolic as well as sine and cosine functions, hyperbolic terms are associated with epsilon and sine and cosine with delta, now the ABCD are the four constants we have four boundary conditions, so we can run through that calculation and I get the characteristic equation, okay. So the characteristic equation is obtained by demanding that the determinant of this coefficient matrix must be equal to 0 that leads to this characteristic equation, now here delta square plus epsilon square cannot be 0, so that is ruled out, sine H epsilon lambda if you assume that it is 0 then it becomes epsilon will be 0 and consequently lambda will be 0 and that is not acceptable, so this is not acceptable, so the only possibility sine delta L is 0 from which I get delta NL is N pi or delta N is N pi by L, so these are the mode shapes and I substitute into this and I get the expression for natural frequencies, so if I simplify that expression I get natural frequency in terms of axial loads and other parameters of the problem as shown here, P critical is the Euler's buckling load, so natural frequency is obtained as natural frequency when P is 0 and that is omega N naught and multiplied by a non-linear function F of P, now if I plot this it turns out that natural frequency will become 0 when P is P critical, okay, and when of course P equal to 0 you get back to the exact natural frequency of a beam without any axial load, so we can make some remarks, due to presence of P the natural frequencies get lowered, this has important implications in characterizing resonant characteristics of the system because resonance is one of the things that we are primarily interested and because of axial loads if natural frequency is going to become less then it's a, you have to take a call on that, in this case it is observed that the mode shapes are not affected by P, in this case means in the case of simply supported beam, this may not be true in general, instead of simply supported beam if you consider an alternative, other alternatives we may not get the same mode shape, at P equal to P critical omega 1 becomes 0, now we already derived the orthogonality relation of vibration mode shapes with respect to mass, this is this but with respect to now stiffness and axial load there will be a modified version of orthogonality relations I will leave this as an exercise for you to verify, now we can do a force response analysis, we will expand the solution in terms of generalized coordinates use orthogonality of the mode shapes and we will get the uncoupled equations in this form, where N runs from 1 to infinity, now as a special case we can consider undamped free vibration and I get A N double dot plus omega N square A N equal to 0, now if P is less than P critical A N of T is given in terms of sine and cosine, that would mean motion would remain bounded for all times and for all initial conditions, because sine omega T and cos omega T are bounded function, but on the other hand if P equal to P critical A N of T will be a linear function increasing in time, so motion grows linearly in time even for small initial conditions, okay. Now on the other hand if P is greater than P critical A N of T becomes you know there are exponential functions with gamma N being this, and motion grows exponentially in time even for small initial conditions, so presence of damping of course what happens is P less than P critical motion decays exponentially in time, but if P is greater than or equal to P critical there won't be any qualitative change in the system behavior, okay. So this roughly captures the effect of axial load on dynamic behavior, this is reasonably straight forward analysis, so one can consider some simple examples and illustrate using finite element concepts, suppose if I consider a property cantilever beam, so the degree of freedom 1, 2, 3, 4, so for the boundary conditions 1, 2, 3 are, for the given boundary condition 1, 2, 3 are 0 and U4 is non-zero that's the only degree of freedom, so this is the mass matrix, this is the elastic stiffness matrix, the geometric stiffness matrix, so you can assemble this equation and you get omega 1 square as given by this, so obviously when P equal to P critical this term inside the bracket will be parenthesis will be equal to 0, okay, this is for a property cantilever, for a simply supported beam again using the same node numbering 1, 2, 3, 4 we see that 1 and 3 are 0 and 2 and 4 are the degrees of freedom, so we can assemble these matrices and obtain a 2 by 2 set of equations governing this system from which you can do an eigenvalue analysis and again see for what value of P the eigenvalues become 0 and so on and so forth. So this I leave it as an exercise for you to pursue but this gives you a flavor of how to proceed, so an exercise is we will consider a property cantilever which is loaded on a spring, supported on a spring as shown here and we have considered this problem earlier for P equal to 0 we have got the, solved this problem earlier in one of the earlier lectures, so if you go back you will get the elastic stiffness matrix and mass matrix for this problem. Now what I want you to do is to formulate the geometric stiffness matrix, so you have to do first a axial stress analysis and then use formulate the geometric stiffness and then analyze the natural frequencies for this system, this is an exercise, repeat this exercise for this problem, okay, you can use the same data, the thing is here there is a 3P and IP load, so the initial axial thrust diagram will be different for this, based on this you have to perform the analysis, these two are exercise problems. Now we will now consider what are known as parametrically excited systems, an archetypal system is a simply supported beam, the axial load P now instead of being a constant it becomes a function of time, okay, so if that happens the governing equation will be wherever there is P I will write P of T and there is a lateral load F of X, T, so this F of X, T we call it as an external excitation and P of T as parametric excitation, because P of T multiplies dou square Y by dou X square and can be interpreted as a parameter of the problem and an excitation appears as a parameter of the problem and therefore it is called parametric excitation. So the question we can ask is how do such systems behave, how do we analyze the response of the systems, what is resonance in such systems and so on and so forth, so let us see some of these questions. So let's assume for sake of discussion that external load is 0, F of X, T is 0 and P of T is a harmonic load, P naught cos omega T, now if omega is 0 critical load is given by pi square EI by L square, it turns out that when omega is not 0 and if P naught is less than P critical that means amplitude of the load is much less than or less than the critical buckling load, but depending on if that happens you would expect that the beam to oscillate only axially, see we expect beam to oscillate predominantly axially, because P is much less than P critical therefore the neighboring, the geometric stiffness effect will not be very much pronounced so it will basically oscillate in the axial mode, but for certain values of the driving frequency omega, the driving frequency of parametric excitation, it turns out that the beam can get into large amplitude bending oscillations, even for small values of P naught, this is known as parametric resonance, now we should note that when such oscillations occur omega need not coincide with any of the bending or axial natural frequency of the beam, so what are those frequencies at which the beam can lose stability, which does not necessarily correspond to the natural frequency of the beam. Under this condition we will show shortly that the amplitudes can grow exponentially, conversely on the other hand if P naught is already greater than P critical, so assume that this axial load P naught is more greater than P critical, then we would expect structure to lose its stability, but for certain values of omega, capital omega the structure remains stable for certain values of omega, that means the axial load will have a stabilizing effect on the beam which would otherwise be unstable, okay. So the term parametric excitation one has to carefully interpret, if you consider axial vibration of this beam P of T is an external excitation, but if you consider bending oscillations P of T becomes parametric excitation, so P of T is external excitation for axial oscillations, whereas it is a parametric excitation for bending oscillations, so since our interest in this discussion is basically on bending oscillations and coupling between the influence of axial loads on bending we call P of T as parametric excitation. Now where do we encounter such type of systems, so we can consider a tall stack chimney under say a biaxial earthquake ground motion, suppose this system is subjected to earthquake ground motion which is XG of T and YG of T. Now I want to analyze this problem, let's assume that it's Euler-Bernoulli beam and we'll include the effect of gravity and the vertical acceleration in formulating the deflection of this beam. So due to XG of T the beam would be here at some time T and YG of T is this, so the coordinate of this point at time T will be XG of T, YG of T, this is a deflected profile, this one. So if you consider any element d psi, this is d psi, and the vertical load that acts on it is M d psi is a mass of that element into gravity plus YG double dot of T, because in the event of an earthquake there will be vertical acceleration also, so this load will be acting like this. Now if you consider the bending moment at cross-section A, it will be this load into the arm, what is the arm? This displacement which is what we are calling U, U of psi, T minus U of Y, this distance is Y, this is psi, so the bending moment will be due to this elementary strip is M d psi into G plus YG double dot multiplied by the arm U of psi minus U of Y, and this you have to integrate from, if you want bending moment here you have to integrate from this to H, where H is the height of the tube. So I get Y to H, M into G plus YG double dot U of psi, T minus U of Y to d psi. So using this now I can write the equation for the oscillation of the beam, this is dou square by dou Y square into the bending moment plus M dou square U by dou T square inertia plus C dou U by dou T minus XG dot damping is equal to F of X, now what are the boundary condition? At X equal to 0 U of 0, T is XG of T, because it is applied ground motion, so this ground motion is getting applied here, so that is the boundary condition, and U prime of 0, T is 0, at the top bending moment and shear force are 0, okay. Now you can carry out this integration, and you can show that the governing equation will be EI U double prime and double prime plus these terms plus MU dot double dot plus this is equal to F of XT, now what you should observe is the field variable here, dependent variable U is multiplied by YG of T here, so YG double dot of T appears as a parametric excitation in this problem, and XG of T is now in this formulation is appearing as a boundary condition, so it is a parametrically excited system under time dependent boundary conditions, okay. How do we use finite element method to solve this problem, okay, so that's one question that we can consider as we go along. Now we can do some simple substitution, I can introduce another dependent variable W I define as U minus XG which is a relative displacement, and I substitute here, and for moment I do this the time dependent boundary condition will become time independent, because at Y equal to 0, W of 0, T will be XG minus XG which is 0, so the time dependent term present in the boundary conditions will be removed by this substitution, and in the resulting equation if you now do a kind of a series expansion, this we get now for the modal equation this set of equations, again you should notice due to a presence of parametric excitation the generalized coordinates will not get uncoupled, so this is an approximation resulting from method of weighted residuals, it's not an exact solution to the problem, because the equations are not uncoupled here, because equation for AN has all the other generalized coordinates, so in a matrix form I can put it as MX double dot plus CX dot plus K plus K naught of T into X is equal to Q of T, so this K naught of T is a time varying stiffness matrix for the problem which contains your YG double dot of T, so this is a parametric excitation or you can interpret as a time varying system, so this is a coupled multi degree freedom system with time dependent stiffness matrix, consequently the notion of natural frequencies and mode shapes are not valid for this type of systems, if you ignore K naught of T and if you were to find the natural frequency and mode shapes those set of Eigen solutions will not uncouple this equation, so they are no longer the natural coordinates for this problem, in fact for this type of problems natural coordinates do not exist, normal modes with ground motions equal to 0 would not uncouple the equation of motion. Now we can ask a few questions, how to solve these equations, are there any new phenomenological features associated with the response, what happens to the notion of resonance and what happens to notion of instability, some of these questions can be posed and we can address these questions, but before we do that we will review one more type of problem that is of great interest to civil engineers that is how do structures behave under moving loads, moving loads could be vehicular traffic for example on a bridge, so for sake of discussion we will assume that bridge has been idealized as a single span simply supported beam. We start with a simple model where the vehicle model is represented by a single concentrated force and that force is equal to the weight of the vehicle, so the vehicle is moving with velocity V and acceleration A, so assuming Euler-Bernoulli beam hypothesis we can set up the governing equation of motion, so at T equal to 0 the load enters the span and it will be on this span till T exit, it exits there and after the load leaves the bridge there is no external force on the beam, during the time the load is on the bridge the point of application of the load keeps changing, so the forcing function is expressed in terms of a direct delta function delta of X minus VT minus half A T square, at any time T the position will be VT plus half A T square, so that is what we are writing. Now a solution to this problem can be constructed using a modal expansion, this is an externally driven system, this is not a parametrically exaggerated system, so I will get this finite effects if they are the exact mode shapes, this will uncouple the equation and I will get this equation for T less than T exit this and afterwards it is 0, now for simply supported beam we know that the mode shapes are given by sine N pi X by N, and if we assume that beam is moving with, sorry the vehicle is moving with constant velocity acceleration will be 0 and the simplified equation will be given by this. Now obviously now there is a critical condition that is possible if the natural frequency of the system coincides with the driving frequency here, because mode shapes are sinusoidal in space the driving on generalized coordinates will become harmonic in time, okay, so a harmonic excitation on a single degree freedom system can potentially cause resonance, but you should understand that this excitation is a transient excitation, because this term exists only for the time duration T between 0 and T exit, afterwards it becomes 0, therefore strictly speaking when a single concentrated load passes the beam we cannot talk about resonance, but still a critical condition can be taken to prevail if omega 1 is given by pi V by L, that pi V by L, so this leads to the concept of a critical velocity which is given by pi by L square root EI by M. Now although for a single concentrated load passing on the beam a resonant condition is not possible, but a series of concentrated loads can pass on the beam, that is typical of vehicular traffic on a bridge, and therefore if all the loads were to travel with this velocity the structure would indeed get into resonance, and this velocity would indeed become a critical velocity from the point of view of creating a resonant condition. Now we will improve upon this model, we will assume that vehicle is now represented as a moving mass, not as a moving force, so what is the implication? So as the vehicle travels on the bridge at some point suppose if it is here the mass of the vehicle into gravity is one force that acts on the beam, but since the beam itself is exhilarating, the mass of the vehicle into exhilaration of the beam is another force, but now this mass is rolling on an inclined path, actually the beam is oscillating and this mass is rolling on that, so when I consider exhilaration I should take total derivatives, it's not the ordinary derivative, so if I write for example what is D by DT of Y of X, T this it is dou Y by dou X into dou X by dou T plus dou Y by dou T into this, so the operator D by DT is given by this, so based on that D square Y by DT square the total derivative can be shown to be given by this, so these terms are known as Coriolis terms, because there is a rolling motion and a transverse oscillation which interact producing this Coriolis force. Now what is the consequence of all this in the equation of motion? So again when writing equation of motion the force should now include MAG plus this term, we get this, this is a point of application at time T the position of the wheel will be at distance VT plus half AT square, so this is direct delta of X minus VT minus half AT square, and after the vehicle leaves the bridge of course this term will be 0, now this D square Y by DT square is in fact given by this, so now if you focus our discussion on T between 0 to T exit the equation will be given by this, so if we now expand this and reorganize the term we will see that the mass matrix, so the mass term has all these additional terms that is dou Y by dou X, dou Y by dou X, dou T, dou Y, dou X etc., and this is a time varying term, the time varying term here is captured through a direct delta function, but it is still a function of both X and T, so I get a partial differential equation now where the dependent variable is multiplied by a function which is function of both X and T, so therefore this term constitutes the parametric excitation term, so of course the force due to moving, the load due to moving mass, the weight of the vehicle drives as an external excitation that continues to be present, so if I were to we will see how to discretize this, we will visit this problem, we will develop a finite element formulation for this and we will see how the equations can be derived, but as a prelude we can see that if one were to do that the governing equation will be of this form, the mass damping and stiffness matrices will now be functions of time, and of course there are right hand side which is F of T, so again here we can ask several questions, how do we actually formulate this, how do we use finite element method and formulate this equation, again questions on possible phenomenological novelties in system behavior and actual quantification of the response, we want to find out beam displacement strains and stresses, reaction transfer and so on and so forth, we have to analyze this problem, how do we do that and how do we eventually compute stresses and reactions and so on and so on, so bending moments, shear force etc, a further refinement to this model now will include the stiffness characteristics of the vehicle, in the first case we simply considered vehicle as a force, which is weight of the vehicle, in the second case we included the inertial property of the vehicle, now vehicle itself will have elastic properties, there will be suspensions and so on and so forth, so we can model the vehicle now as a single derivative system with spring stiffness KV and a dissipation damper CV, and we will divide the mass into what is known as a unsprung mass and a sprung mass, so this system now enters the beam at T equal to 0 and exits at T exit, and why is the displacement and properties of the beam are EI, MC and L, and MU, MS, KV, CV are the properties of the vehicle, and the degree of freedom associated with the vehicle I take it as U of T, and we will assume that the vehicle is moving with a velocity V and acceleration A, so now the question is as the vehicle traverses the beam, the vibration of this system comprises of vibration of the beam as well as the vibration of vehicle, and they mutually interact, that means the vehicle oscillations will modify the oscillation of the beam, and beam oscillations will modify the oscillation of the vehicle, and this is a problem of vehicle structure interaction. So, we will do two things now, we formulate this problem and see what are the governing equations, and then see how to use finite element method to discretize that, that we will have taken up later. Now we will focus on T lying between 0 and T exit, so the equation for the vehicle that is MSU double dot plus KV into U minus Y here, plus CV into Y minus here, right, and the inertial term will be MS plus MU into U double, and MU we will see how these terms come up, so MSU double dot plus CV U dot into the total derivative, because this mass is rolling on a deflected profile, so again I have to take a total derivative, and this is the force in the spring. The beam itself carries the wheel load, I will call it a wheel load, let us call it as F of X, T, F of X, T and the point of application keeps changing, X minus VT minus half AT square. What this wheel load consists of? The gravity load due to the weight of the vehicle, which is MU plus MS into G, plus the reaction transformed from the spring, which is this, that is spring of the vehicle, and this is the reaction transformed from the damper of the vehicle which is this, minus the inertia due to the unsprung mass, this is this, so we will see, we will consider this problem in some detail in the following lectures, we will discretize this using finite element method, and we will see in that, in course of doing that, that the discretized form of equation will have this generic form, M of T X double dot plus C of T X dot plus K of T X is equal to F of T and certain initial conditions, so here again I get a system where mass damping and stiffness matrices are functions of time, again the notion of natural frequency in mode shapes and uncoupling the equation through natural coordinates, et cetera, are no longer applicable here. Before embarking on qualitative analysis of this type of system, we will mention a few more examples where this type of questions become relevant. Suppose if you consider an arch carrying a load, it is a symmetric circular arch, and P of T if it is periodic, periodic loading applied on symmetrically loaded arch, generally is expected to produce symmetric oscillations of the arch, okay. Under certain conditions however, the arch can undergo asymmetric oscillations of large amplitude, the concept of modes are no longer applicable here because there can be energy transfer from what we consider as generalized coordinates, suppose there is no geometric interactions then the first mode and second mode they are all uncoupled, but because of parametric presence of parametric excitation effects those so-called modes will get coupled, so a symmetric arch like this which is symmetrically supported and symmetrically loaded can undergo large oscillations, asymmetric oscillations. Similarly, if you consider a ring and if it is subjected to uniform external pressure, external load P of T, if P of T is periodic one generally expect that the ring will undergo axial oscillations, it will be simply be oscillating in this direction, but however under certain conditions the ring can undergo bending oscillations of large amplitude, this again due to parametric effects. Another case is suppose you consider a rectangular beam as shown here loaded symmetrically with a load P of T, again if you apply, if a P of T is periodic one would expect that such a load will create a bending deformation in the plane of the loading, but under certain condition the beam can undergo torsional oscillations of large amplitude, so this oscillations will be excited by this load, so how that happens is another issue that we need to consider. Now I briefly mentioned when we initiated discussion on stability of structures that we are considering stability of state of rest and stability of steady state motions, the parametrically excited systems also arise in a somewhat different context if one is studying stability of steady state motions, this I had already shown once but let us quickly recall, if we have suppose a single degree freedom system with cubic nonlinear stiffness and it is driven harmonically by P cos omega T, suppose if you postulate that as T tends to infinity X of T is periodic with this property and call this as X star of T, now I want to know whether X star of T is stable or not this periodic motion, so what I do I impress upon X star of T a perturbation delta of T which is small, now upon substitution of this equation into this we can get an equation for delta and this equation will have X star square of T which is the solution as a coefficient here, so X star square of T acts as a parametric excitation for this problem, this is known as variational equation, in the variational equation the solution whose stability is being investigated appear as a parametric excitation term, in this type of questions we are not so much interested in the detailed time history of delta of T, we are basically interested in knowing what happens to amplitudes of delta T as T tends to infinity, so we are asking a relatively simpler question on whether delta of T as T tends to infinity amplitude decay or not, so that question of course can be answered by complete solution of this equation, but since the question being asked has a simpler scope they should be simpler ways of answering that question, there is one another class of problems that is of interest in stability analysis, suppose if we consider this we have considered suppose there is a cantilever beam and it is actually loaded and as it deforms suppose if a neighboring equilibrium state is possible the load P continues to act you know what the direction of load P doesn't change because of deformation of the beam, but suppose if you were to assume that load P the point of application of load P also you know remains same, but the direction remains tangential to the deform geometry, okay, if this happens now if we ask the question what is the what would be the value of critical load for this system, okay. Now so in this case line of action of P remains unaltered as beam deforms, static analysis can be used to find P critical that we have done. Now here line of action of P remains tangential to the deformed beam axis, if you were to perform a static analysis as we have done for this case it turns out that for any value of P this structure would be stable, that is somewhat counter-intuitive, a structure cannot obviously carry large values of P, some at some value of P that should be a problem. Now if you were to go back to the basics and ask how do we exactly study the stability, for every increment in value of P you will give a perturbation and see how the oscillations caused by that perturbation decay or not, so the question of stability is being settled through actual analysis of a vibration problem, so if motivated by that if you were to pose this problem as a problem in dynamics and analyze this problem, it turns out that if we include inertial effects there will be a value of P, finite value of P at which the structure loses stability, that a pure static analysis will not be able to capture, so that is one novel element of this type of problems I will briefly touch upon this as we go along, so to summarize what we will do is we will consider basically three types of problems, I am giving a overview of problems that we are going to consider. Now first question we will ask is how to characterize resonances in systems governed by equations of the form M of T X double dot plus C of T X dot plus K of T X equal to 0, when the parametric excitations are periodic, okay, so we talk about resonance in presence of periodic excitations, now the periodic excitations now appear as parameters and then can we talk about resonance. Second question we will consider is how to arrive at finite element models for partial differential equations with time varying coefficients, we saw that in the problem of say vehicle structure interaction the governing partial differential equation had a time varying term, and similarly the chimney under combined horizontal and vertical excitations also had a time varying term, so how do we use finite element method to discretize that, last problem that we will consider is are there any situations in statically loaded systems wherein one needs to use dynamic analysis to infer stability condition, so these three problems we will consider in the following you know discussions, so we will start by considering the first problem, qualitative analysis of parametrically excited systems, that is a topic, suppose if I consider a single degree freedom system U double dot of T plus P 1 of T U dot of T plus P 2 of T U of T equal to 0, if mass is time varying I can divide that term and arrange the equation in this form, that amounts to multiplying this equation by M inverse, now let's assume that there are some nonzero initial conditions and this P 1 and P 2 are periodic functions, now this governing equation is a linear second order ordinary differential equation with time varying coefficient, the equation is still linear, therefore it admits two fundamental solutions, so what are these fundamental solutions we can digress for a minute, suppose if you consider undamped free vibration of a single degree freedom system the governing equation is X double dot plus omega square X equal to 0 and we know the solution is A cos omega T plus B bar sine omega T, I can write this as A X 1 of T plus B X 2 of T where X 1 I will take as cos omega T and X 2 as sine omega T by omega, what is special about these X 1 and X 2? X 1 of 0 is 1, X 1 dot of 0 is 0, similarly X 2 of 0 is 0, X 2 dot of 0 is 1, so that means the initial conditions are 1 0 0 1, if you start with, if you integrate this equation with those two sets of independent initial conditions 1 0 and 0 1 you get two solutions X 1 and X 2 and any solution to the problem can be constructed by superposing those two solutions, these X 1 and X 2 are known as fundamental solutions, for this type of fundamental solutions exist for all linear systems, so the point being made is this equation, this governing equation is a linear second order ordinary differential equation, time varying equation, it admits two fundamental solutions. Now let U 1 and U 2 be the two fundamental solutions, so I can write U of T as C 1 U 1 of T plus C 2 U 2 of T, U of T is any solution to this problem. Now what we will do is, we will consider this equation at time instant T plus capital T, where capital T is a period of P 1 of T and P 2 of T, if you do that I will have U double dot T plus T plus P 1 T plus T plus U dot T plus T plus P 2 of T plus T into this equal to 0, but P 1 and P 2 are periodic, therefore P 1 of T plus T is simply P 1 of T and P 2 of T plus T is P 2 of T, so the equation gets simplified, I get this, so that would mean if U of T is a solution, U of T plus capital T is also a solution, okay, because U of T satisfies this equation and so as so U of T plus T. Now therefore this U of T plus T can be expressed as linear combination of the two fundamental solutions, A 1 1 U 1 of T plus A 1 2 U 2 of T. Now similarly U 2 of T can be written in this form, or I can put this in a matrix form and write U of T plus T is A into U of T. Now we are interested in nature of solutions as T tends to infinity, clearly you can easily see that this behavior will be controlled by this matrix A, okay, and this matrix A is what we should focus on determining. If we now consider the question what happens to U of T as T tends to infinity, this is equivalent to asking the question limit of N tending to infinity U of T plus N T, what happens? Now I can write based on this equation U of T plus T is A U T, U of T plus 2 T is A into U of T plus T that itself is A square U of T, so it follows U of T plus N T is A to the power of N U of T. So the behavior of this quantity U of T plus N T as N tends to infinity is controlled by the behavior of A to the power of N as N tends to infinity. Intuitively we can see at this stage that this would depend on eigenvalues of A, again we will digress, we will consider first a scalar case, suppose I have a map X of T plus T is alpha X of T, therefore X of T plus N T will be alpha N X of T, so as N tends to infinity this function will go to 0 if modulus of alpha is less than 1, and it will go to infinity if modulus alpha is greater than 1, it will be if alpha equal to 1 what happens? The function remains same, so it is periodic with period capital T, if alpha equal to 1. On the other hand if alpha is minus 1, the period will be 2 T, because X 2 will be minus X 1, X 3 will be minus X 2 which is again X 1, so the period will be 2 T. So similar situation can be expected to prevail in analyzing this vector equation, so what we will do is in the next class we will examine this issue how to investigate the behavior of U of T plus N T as N tends to infinity, it will turn out that we will be able to formulate this matrix A in terms of fundamental solutions of the governing equation, and to answer this question we need to integrate the governing equations over one period of the parametric excitations for a set of independent initial conditions, using that information I will be able to answer this question whether the response becomes how the response behaves, so there will be various situations that we will encounter, the response will be periodic with period capital T, with period 2 T and it can decay to 0 and it can exponentially blow off, depending on the eigenvalues of matrix A, so this question we will take up in the next class, we will close this lecture at this point.