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 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 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?  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.


not warned. And there's no rule that says that you can be suspended for life for this. You can't be suspended. So he knows what he could do. He's a lawyer. He's a smart lawyer. He's the guy who did the last CBA that the players hate now. He's the guy who got in the luxury tax thresholds that is actually holding down salaries a little bit. He's a smart guy when it comes to this stuff. He knows what he can win and what he can't win. He knew he couldn't win suspending players because there's no written rule. And it is heartening what Tony Clark said earlier, Don, where maybe it'll be different. Maybe that'll be part of the CBA, where they will be able to suspend players for cheating. Yeah, so this doesn't have to happen again. But the whole warning thing, like the players weren't properly notified, I knew that the teams got the letter. Why wouldn't the players know? They might have known, but they were not ever officially told. But it didn't matter anyway because if it's true from what the caller said that the memo said that it would be the owners and the management that would get suspended, not the players, then what would it matter anyway? Why would I even bother heating the warning? Another would happen to me. It's still odd that the players can still freely roam knowing that nothing can happen to them. There's like a diplomatic immunity that they can now work with going forward, that they cannot be touched. But they have a great union. I know, Michael, that's how that's but that they're being protected by their union. But the thing is, is that still there's this attitude of the players can get away with it because they've got this great union. So that means I can still go out and cheat in 2020, knowing what's going to happen to me. I can't get suspended. The only guy that could end up getting suspended, I guess right now is their new general manager and Dusty Baker. Yeah. Yeah, they can cheat again. Stop me from doing dusty bakers on a one year contract. Just make sure Dusty doesn't know about it. Do it behind his back, but nothing can happen to me. And also, if you're the players, you've already had your reputation and resume completely soiled. You're never getting it back. So cheat more. Win again because the best thing that could happen to the Astros is to win this year. Just like the best thing to happen to the Patriots after spygate was to go 18 and 0. And then after deflate gate to win three of the next five Superholds. That's the great deal. I just remember having that conversation when Melchie Cabrera got suspended. He ended up losing out of a World Series title in 2012 and 2012 with the Giants. Lost a batting title. And what was our conversation? He doesn't care because you know what? He got a $16 million a year contract from the Toronto Blue Jays. The second he became a free agent, but he does care a little bit because he would have had a contract of about $16 million. But that's the only reason he would have cared about it. He cared about losing the batting title or not being a part of a World Series team. We all just assume, well, that's the punishment. You don't get to go to the Hall of Fame. That's the punishment you want. You ended up winning a championship. I mean, we're assuming that the punishment is they've been shamed into this. I wonder how much shame they really feel. They're aggravated about it, but they're circling the wagons now. They're probably going to use it now as a as a motivating factor to go out and win in 2020 and mark my words. If they win in 2020, the narrative is going to be the Astros overcame the controversy and came together and won a title. And exactly what would the punishment be then? And Dusty Baker should use that as the motivational tool. Oh, I'm sure he will be.  They think you stink. There's no way you could win without cheating. Let's show them that he doesn't have to give a long speech. And I do believe, though, Don, that they're embarrassed because your reputation is important. They're embarrassed. But they have a chance to change the narrative of the reputation, Michael. We all thought that Alex would never be able to get up. He got up. I'm telling you, if the Astros continue to win titles, it'll go the same way as the Patriots went. I guess they didn't need the spy gate. They didn't need the deflate gate. And guess what? They didn't need the signs. Look, they obviously weren't cheating in 2020. And guess what? They win another title again, and then maybe win in 21. Unfortunately, they have one year to do it because the team is going to be broken apart after that shouldn't be how you get punished. Okay, is so so if I if I break curfew of like a mark over Jalen break curfew, be like, I can't take your iPad away. I can't make you sit in your room for the week.  Marco and Jalen, you have to live with the shame of coming in late and disrespecting mom and dad. I can hear the giggles. It hasn't happened for 15 years, Peter. I can hear the giggles in the future as they go up to play with their iPads and go into their room and turn on the television set. The giggles of I got away with it. But Shane is a powerful. Yeah, but it's in the eye of the beholder, Michael. I mean, if my father ever said looked at me, you know, he really disappointed me. That would crush me. That would be a big deal to me. You really just even if he didn't yell at me, while you really disappointed me and your mother, that would that would crush me. Okay, well, if their parents were calling to do that, that might mean something. This isn't their parents. Now it's their peers. They're not respected. Here's our parents, though. Yeah, but they're not respected by Aaron judge and Mike trout. Come on, that's a big deal. It's something. It's something. But when people get arrested for a crime, they don't go, Hey, it's enough that the community thinks lowly. They put you in jail. You know that I think that they should be punished. But to say that, it's nothing. But to say it's nothing is wrong. It's not no, it's not nothing. It's just not an appropriate response. I need more on top until five years from now. For some reason, Al Tuve and judge are on the same team and they're hugging and kissing each other and forgiven. Well, that would happen. All right. So this will eventually the time does heal all wounds. We may never forget about this in our lifetimes. But time will heal. We'll get up from this a couple of years from now. We won't. Was that 16 or 17 PM? Was it the 17 championship or the team that went to the World Series in 29? It's fresh in our minds because it's all we've talked about for the better part of a week. But how are we going to feel about a three years from now, five years from now, 10 years from now, 100 years from now? Shame really? Peter made a great point. Yeah, the shame of committing that crime. You don't have to go to jail. We listen, we look at and by the way, I mean, yeah, I know of times when that's happened where someone didn't get in trouble. They should have gotten in trouble. And as a result of not getting in trouble, they really didn't learn the lesson.


of football fans welcome inside our Fandals studio here in Los Angeles and Ed we are nearly halfway through the season. It is week eight people I don't know how that happened but we're here to bet it. You're watching Fandals Sportsbooks more ways to win. I'm Lisa Kearney alongside our analytics expert Ed Egros here with me. The rest of the guys will join us soon to break down the biggest games of the week hand out best bets and of course give their updated Super Bowl picks. You're gonna want to stay tuned for that. It is go time. So make sure you download the Fandals Sportsbook app sign up for your new account plug in the promo code moreways1000 to get your risk free bet up to one thousand bucks. All right time to kick things off with these six in one buccaneers at the four and two Saints. Yes NFC South we're calling on you the box coming off a thirty eight to three win against Chicago last week Tom Brady playing some of the best football of his career leads the NFL in passing yards and touchdowns. He is thrilling to watch right now super dialed in as for the Saints they won in Seattle on Monday night with Alvin Camaro racking up one hundred seventy nine total yards on 30 touches both of those season highs. The box five and a half point road favorites in this divisional match up here Dave you're joining us from Kindle in this week. Hello my friend. Good to have you Dave and Ed you guys are the ones betting this one and Dave I'm coming to you first. Well you know who Tom Brady is not good against the Saints. As a matter of fact last year Tom Brady through five interceptions against the Saints total score in those two games where the Saints won them both sixty two to twenty six is a better team this year for Tom Brady and the Bucks. I don't know they won the Super Bowl last year but I think that Sean Payton has the keys to Tom Brady. I'm going to take the Saints in this one. We knew last week they were looking ahead over the Seahawks they squeaked that one out it was ugly but they were looking forward to going back home and beating Tom Brady like they always do. I'm taking the Bucks. I mean I'm sorry I'm taking the Saints and I'm taking the Saints not just to cover Ed to win. See that's what you needed Dave just a little bit of fresh air out in Lixington enjoy the bluegrass enjoy the horsies out there that's what you needed to get this one right. It's one of those things where a lot of people are probably wondering wow defending Super Bowl champs the Bucks are an absolute machine right now. Why would they only be five and a point to the Saints. Well the good reason is James Winston is outstanding when it comes to short throws. He has the highest passer rating among any quarterback when throws take two and a half seconds or less. He's also been outstanding in the play action game especially throwing to his running back Alvin Camara. The defense for Tampa Bay blitzes a lot. How do you beat the blitz with quick passes and that's exactly what the Saints are going to do. And this has kind of been the recipe over the last year and change. And it's why the Saints have been so successful. I don't know if I can necessarily recommend a money line pick here but definitely I think the Saints can cover five and a half. All right there's the five and a half the Saints money line though plus two 10 right there if you want a little value and we got to get a right take on this match up and we're going to go to go to you Cole for your money line money makers coming up a little later in the show right now. You're joining us from Chicago to talk about this match up. Which team do you have winning and by how many points. Well at least it's going to be a reunion in the Pelican state. The Boudin brothers they are back of course Mark Ingram dealt back to the Saints and he's hot on the heels of Deuce McAllister becoming the Saints all time leading Russia but we have to look past that and focus on the Pharoah of course Tom Brady and Tampa because the best ever do it has a 7 to 1 touchdown the interception ratio and he's on pace to throw for his second 5,000 yard season of his career and even with the extra game if my math serves me 55 25 when it comes to his final total yardage and you might want to check me on the math there but either way Tom Brady is good so are the Tampa Bay Buccaneers that Saints Kryptonite it's out the window Tampa and they win blowout fashion 32 to 14 here. Okay there it is you heard it that was Cole and if you roll with Brady and the Bucks and give those points a winning $50 bet means you're going to collect more than $93 it's right there on the Fandall Sportsbook app. Now to the AFC South and the five and two Titans who have won three in a row including back to back wins against the Chiefs and Bills Tennessee now in Indy to take on the 3 and 4 Colts. This is a rematch of a week three game that one in Tennessee Titans won that one 25 to 16 despite those three turnovers. Now the Titans one and a half point road underdogs in this one. I love this line pony where's your money on this match up. I don't get it. I mean other than Pat McAfee and Colts fans who wants to bet on Indianapolis here it's one of the most confusing lines of the NFL this season. The Titans just had two massive statement wins what now they're going to get down. They're going to play down to the competition on the road. They can really end the AFC South Division race right here and now by sweeping the Colts. The only time the Titans losses when both of their wide receivers were out Dave like recently that was to the Jets. I'm going to take them to win the game outright. I don't get this line at all. Well did you just not make every point to take Tennessee. I mean Tennessee to me is so much the better team right now and you know you at least pointed out those three turnovers and they still won. The Colts had 10 points 16 points in the game. 10 of them came off of the turnovers. So for me this is all about the Titans. We know Derek Henry is on pace right now to break the all time leading rushing record. He wants it and with the win here the Titans would be up three with eight games to play and have the tie break on the Colts essentially with the win here this division is over. All right time now guys to move on. We're taking it to our research here to the next level for that we turn to next gen stats for an analytical perspective on this matchup. The NFL analyzes the location speed and acceleration of every player on every play and creates these various data sets that we get to tap into. So Ed I'm coming to you're our analytics guy. So give us the numbers associated with this Titans Colts matchup that's going to help us bet this game. Going back to Sunday Night Football you could argue Carson Winsons challenge really was more of the weather than it was anything the 49ers were throwing at him. If we take a look at his past chart from that Sunday Night game. Yeah it doesn't exactly pop off the screen just 150 passing yards with a completion percentage over expected of negative point five percent but more than 30 percent of his throws were behind the line of scrimmage and Winsons was patient with his chances when he could go deep and when he did he went to a three and drew three defensive pass interference calls. These results suggest Winsons more comfortable with this Colts infrastructure. He's been there for a few weeks now. He does the most given the conditions and against a Titans defense that doesn't blitz very much and has had trouble stopping the pass. I think Indianapolis Indianapolis can cover those. All right. There it is. Awesome stuff from Ed. Great stuff from next next next gen stats the next level up. Thank you guys. Now you have to do you though. Right. You can fade and go ahead. Take the Titans getting those points and winning fifty dollar bet means you're going to collect more than ninety five dollars. And remember you can make your first bet on the app risk free. Just sign up for a new Fandall Sportsbook account using the promo code. It's going to be on your show on the screen all show long. It's more ways one thousand you're going to get a risk free bet up to one thousand bucks. Again that promo code more ways one thousand. All right. Now let's get to the Bengals sitting at five and two so much hype around these guys now Cincinnati now gets the one and five jets since he has won four of five including a dominating performance over the Ravens last week. The Bengals hold the number one seed in the AFC right now. Look on the other side the Jets struggles continued last week quarterback rookie Zach rookie quarterback excuse me Zach Wilson lift the game with that knee injury. So we've got back up Mike White taking over throwing for two hundred and two yards one touchdown and those two interceptions at New England. He will make his first career start on Sunday. This one is at MetLife where the Jets are getting 10 and a half. Ed who are you backing in this one. So this is an unenviable position because we're having to do things with a super small sample size Mike White only playing part of a game. And now we have to figure out what he's going to be like for a full game against a really good Bengals team. Well if we look at that performance his completion percentage over expected was negative six and a half percent. Yeah not a lot of throws there but still you want your CPOE to be a good bit better that to have any confidence whatsoever. Joe Burrow was at 3.4 percent. That's a pretty decent number. But what matters a good bit more are throws that are beyond 10 air yards. Joe Burrow is at 46 percent and he's not a great intermediate thrower to begin with. But Mike White is at 38 percent in terms of his completion percentage beyond 10 air yards. This is going to be a really tough tough test for the Jets. And I know the Bengals are on a third straight road game but that's just suppressing the number unnecessarily I think since he can cover the big spread. Sort of a Bengals orange. Not quite but close enough some of those stats you mentioned on Joe Burrow go a long way into why he leads the league right now in yards per pass completion at 13.4. He is on a roll. And look this is the type of game that a good team normally would look past a bad team. But don't forget the Bengals have been a bad team for so long. They know what it's like to be in these positions and have good teams come in and thrash them. This is a time for them now to be able to do that and beat up on a bad team. That's exactly what they're going to do. I like the Bengals to win big. There it is giving 10 and a half on the road. So the jets are home this week. You just heard the guys bet it which means the party returns to New Jersey. And of course the Fandals Sportsbook at the Meadowlands is the place to be. The Fandals Sportsbook opens seven days a week by the way. Come watch wager and win with your favorite teams and players. The sportsbook located right across the parking lot from MetLife Stadium. So it is super easy to get to on game day. Just hop that free round trip shuttle from Lot G. And if you don't have a parking pass it's all good. We're going to have your back there to head to Red's Restaurant in Carl's Sat. Catch the $10 shuttle to MetLife. Then head to Lot G for your free ride to the Fandals Sportsbook. Mass transit of course available as well. Sunday is going to be absolutely hoppin. But hey a heads up and make sure you mark your calendars for this. Get yourself to the Fandals Sportsbook there at the Meadowlands for the 2021 Breeders Crown taking place at the Meadowlands racetrack this Friday and Saturday. Get your friends come out for an awesome time at the Meadowlands and the Fandals Sportsbook. Watch, wager and win and have tons of fun with us. This is your personal invitation to get there. All right. Time now to get to prime time. We're starting with Sunday Night Football. The five and one Cowboys are in Minnesota to take on the three and three Vikings. Both teams coming off their bye week. We've got a well rested crowd coming in. Cowboys offense has been absolutely rolling first in points, first in yards per game. And then you look at the Vikings. Court of Accord Cousins has quietly been having a pretty good season. Ranking in the top 10 in yards passing per game. Touchdown passes and completion percentage as well. Here we are. The Cowboys giving one and a half on the road. Dave, you're on the road. Here I come to you. Who do you have in this Sunday Night Matchup?  They both are are going to play well, but that's not the case for the Minnesota Vikings. They're three and eight against the spread following a bye week. And this is not the type of team the Cowboys that you want to play with a little bit of rust. By the way, the flip side of that, the Cowboys are 12 and four against the spread after the bye. Here are the facts here. The Vikings defense give up 128 rush yards a game. The Cowboys run the ball more than anybody. Actually third most a game over 32 times averaging, you know, tons of yards to carry plus the Vikings give up five yards to carry. They have just too much run game, not to factor what they can do in the air. Look, last year, I met the Vikings did beat the Cowboys, but that was with Andy Dalton. They are not going to beat back this year. I like the Cowboys. I don't, you know, there's a stat out there. Curt Cousins in primetime games. He's terrible. The Dallas has been a different team away from Jerry's world.  They beat the Chargers on a last second field goal. They only scored 20 points and they needed overtime to beat Mac Jones and the Patriots offense in Foxboro, Minnesota, Dallas and Cookback healthy. I think this is going to be a game that shows you the Vikings are playoff caliber that they're ready to go back to the postseason. They're going to win this game outright. All right, you guys, let's get to the bright lights on Monday night now. Great job there on Sunday. Here we turn to the two and five Giants. They are in Kansas City to face off against the three and four Kansas City Chiefs. Chiefs are in a bad place. I got to be honest, offensively and defensively. Now you get a hot bothered about this coming off an embarrassing 27 to three loss in Tennessee. Patrick Mahomes streak of the touchdown passes in 31 straight regular season games came to an end. It's the ninth longest streak in league history. Meanwhile, the Giants are coming off their best defensive effort of the season, giving up just three points in a 22 point win over the Panthers. Guys, Giants are getting 10 on the road at Arrowhead here. Pony, who you got in this matchup? Well, this is another one of these lines that looks like it was made for last year's Chiefs team. But when you really peel back the layers of the Giants, you realize that all of their weapons on offense are hurt. Saquon, Barkley, Kenny Galladay, Kaderius, Tony, you know, Sterling Shepherd as a hamstring injury. Evan Ingram might get traded. He could get held out of this game for the deadline. I mean, they're really down to very few weapons plus the left tackle. Andrew Thomas is hurt. The Chiefs have beaten themselves a lot this year with their two and O would blow out wins against NFC East teams. I think they make it three for three and they cover the big number. Yeah, Pony, virtual mash unit for Big Blue Sunday at Arrowhead. It's going to be a redemption story for Patrick Mahomes and Kansas City. We know last week a trio of turnovers. Two came from Mahomes along with four sacks and an early exit. Not what you want to see, but the offense. We know it still has plenty of pop second in the league, but they have to get back in stride defensively in the two and five G men. Well, they're just the team to do it against the Big Blue defense. A bottom third in all the leading if they're not going to be able to stop that Kansas City run or the past game because they're going to come into this one and they're going to play angry. I'm going big time cover city for KC 28 to nine. I hope Patrick Mahomes can play because he's my fantasy QV. There you go. And I just have to throw this out there. Patrick Mahomes tied with Zach Wilson with nine interceptions on the season. They lead the NFL, but hey, one of the receivers just catch the ball and that's going off everybody's hands. This is a message to you, Tyree. Tyree kill. I'm not upset about it. I'd still early, but we're going to have to catch the coach. I'm just throwing that out there. All right. You know, I love the numbers. You know, I love the context here. So I'm going to dip back in on those next gen stats and Ed, you use those stats earlier in the show to bet the Titans in the Colts game. Now give us three stats that you're using to give us some value on the Fandall Sportsbook app. Where are those bets going? You're morphing into Eric B. Enemy on me and I don't know how I feel about it. Hey, let's go. Let's go into the more ways to win Time Machine here and go all the way back to week six. Jaguars captured their first one of the season in London. Marvin Jones had a lot to do with it. Here's his route chart from that game. Seven catches out of 10 targets with an expected catch rate over expected of 17.3% largely utilized on the right side and Trevor Lawrence was more than comfortable throwing in his direction. Jones's area chair among all Jags receivers that game was 41.5%. There's still improvement to be made though. Didn't have a lot of yards after the catch. In fact, as Yak over expected was minus two still with a rookie quarterback. Any offensive success suggests it can be sustained especially against a Seattle defense. So there are paths for Jones and Lawrence to have big days. So give me the Jags on the money line. Next, do we need to adjust our priors concerning the Atlanta Falcons? Well, there'll be above 500 with a win over Carolina and squarely in the playoff hunt. Here is how quarterback Matt Ryan has fared this season. You see some key areas of the field where he's above league average and pass a rating. Matt Ryan has faced a lot of defensive pressure that's affected his box score numbers. But when you separate what he's done with a clean pocket, it's actually not too bad. His total EPA is 38.5 which is roughly in the middle of the pack. His clean pass a rating of 97.8 is slightly around the middle of the pack and his completion percentage over expected of negative 0.7% again in the middle. Even after his hot start, Sam Darnold's numbers, they do not come close to Matt Ryan's. Atlanta has the quarterback advantage, so I'm going to back the birds. Finally, Brandon Cooks had a rough outing against the unbeaten Cardinals. He's unhappy for a lot of reasons as seen by his route chart here. There's five receptions for 21 yards, but there are a lot of reasons why this performance was an outlier. His air yard share was still better than 30%. So everyone struggled. Starting quarterback Tyrod Taylor has returned to practice and once he plays he will get Cooks going again. Even in just 10 receptions between the two, their success rate is nearly 67% and lastly Cooks is still eighth in football and total air yards. The Texans will need Cooks to shine to keep it close against the Rams. I know it's a big number, but I think Houston can cover it. There it is. He puts the numbers, does the whole thing, spits it out the models, right? Thank you. Let's take that information now. You can use it to place your bets on the Fandalsports app right now. You can, hey, place a wager on the bets we just talked about. Edge, roll them out so nicely for you or there are hundreds of others available for you right now. Alright, time to speed things up. We're going to hit our betting markets with some quick picks now guys. Shouting out your local teams as we rip through the rest of the week eight slate and guys, here's how it works. I give you 10 seconds. I give you the game. Then you get 10 seconds and you give me your pick. So let's roll. I know I love this one. Three and three Pittsburgh heads to four and three Cleveland, big division game here. Pony, you live in Pittsburgh. You see both of these teams in numerous times. Steelers are getting three and a half. This one usually goes. Big Ben arrested. He's 24 two and one all time against the Browns. He makes it 25 and the Steelers win the game outright. All right, Dave, three and four Denver hosting two and five Washington. These teams have combined for seven straight losses Denver is giving three. I think I've officially seen enough of Taylor Heinecke after sliding before the goal line last week with nobody around to cost me the cover.  I'm going with Denver. First time before seasons, we've actually used the word turd on the show. Congratulations, Dave. Hey, Cole, you're in Chicago coming to you for this one. The three and four bears hosting those two and four 49ers bears are getting three and a half. Who wins this one? And by how many points? Well, Lisa, it's the underachievers Bowl because Matt Nagy and the coaching staff they continue to approach this. Justin feels offense like an Ikea bookshelf. They just can't put things together. I think the bears, they get it back together against the tough San Francisco defense. They win this one 21 to 17 this week it out. All right. This is your week eight rapid fire picks right now. I just need to talk to you, Cole.  You're back after a couple of weeks off looking well rested nice and tan. Your streak of six straight correct picks still alive, my man. So give me two more underdogs that you're going to win outright this week. Who you got? All right, well, first up the entire season for Detroit. Well, it's been a rough patch. We know about that. We're going to get that one just a second. But Carolina and Atlanta I'm going to have to go with Carolina in this one. The Falcons, they won three of their last four, but it's been versus the G men, the Jets and the Dolphins. They're four and 16 combined the Panthers. However, they dropped four straight. But this is a new week. Every week can get back on the horse. They win this one 24 to 21. Now on the flip side, the one I brought you first, that's Philly at Detroit the entire season. Like I said, well, it's been a rough patch. Dan, Nickab, Campbell and those Detroit Lions. But Philly get this. They scored 20 or more in each of their last four games. The problem is is that there are one in three over those four just a bunch of hollow points for the Eagles. And I haven't seen a whole bunch from these birds. So I'm putting my eggs in the Lions basket. They win this one 20 to 17. All right, go ahead and tell those upset pics. Again, Cole has hit six straight. They're all there on the Fandall Sportsbook app right now for you. Risk free up to 1000 bucks when you plug in the promo code more ways 1000. All you have to do is sign up for your Fandall Sportsbook account and new users do get that risk free bet up to 1000 bucks just by plugging that promo code more ways 1000. And of course, our experts are here to give you their risk free bet of the week. So let's go around the horn here for week eight. And Dave, I'm coming to you first. What's your best bet? I'm going to play a two team parlay. I'm going to take the Saints to beat the box outright and I'm going to buy up the line on the Cowboys. I think they'll win by a touchdown or more. It's going to take the alternate spread there minus six and a half. I'm surprised this page is so big. It seems so easy. Thousand dollars gets me sixty seven hundred. Dave, are you stealing my thunder doing an alternate spread? That's rude. Yes. Yeah, don't do that. Yeah, please don't do that. Alternate spread here for Jacksonville Seattle. I'm going to take Jacksonville minus four and a half. They have that extra week of rest. Meanwhile, Gino Smith on Monday night, his EPA per drop back was negative point one seven against an OK Saints defense. Seattle's completely falling apart and Jacksonville is on an uptick right now. Thousand dollar risk free bet gets me a total of thirty five hundred. Hey, Dave, go have some drinks at the Kentucky Castle right down the road for me. I think a guy bought it for or constructed it for his girlfriend. Then she dumped him.  Go check that baby out. I'm going with two huge underdogs. Two long shots. I'm taking Houston and the points with Tyrod Taylor back Miami with the points. They're actually doing they're losing, but they're more competitive with Tua catching the bills, napping off a buy. Huge payout when you put that two team parlay together. Go ahead, Dave, check that out and then report back to us a little later. We'd love to see how that one goes. Great stuff by you guys. And gamblers, make sure you sign up for your Fandalsports account right now to place your bed. Remember, use that promo code more ways one thousand to get your risk free bet up to one thousand bucks. It's all right there on the Fandalsports book app. Easy, legal, live, take your winnings if you hit and just get your money back. If you don't. And it is not just football on the show. We're bringing you news from the track as well. And it is almost here. The Breeders' Cup World Championships returns in November to where the turf meets the surf in Del Mar, California. The world's best thoroughbred racehorses will compete for championship honors in more than 31 million dollars in prize money. It's all happened in Friday, November 5th and Saturday, November 6th. Tickets available at breederscup.com right now. I will be there. Come out, say hi and enjoy an incredible weekend with us. And for those of you who cannot make it, you can still enjoy. You can watch all 14 championship races on NBC Sports or TVG. TVG, of course, not only your place to bet the Breeders' Cup, but you can bet every race every day right now by downloading the TVG mobile app. And before we get out of here, guys, let's check out some fun futures bets. And for that, we're going to start with who will win the MVP award. Bill's quarterback, Josh Allen, now the favorite there, plus 300, followed by Jack Prescott and Kyler Murray as well, tied with the second shortest odds there on your screen. MVP picks, guys. Let's roll. Make your money, Dave. Start us off. Amazing that Jamar Chase even shows up on that list. But why does he? Because Joe Burrow is on fire. I like Joe Burrow at plus 4,000. If he could cut down on the intercepts, he has eight. He can go about four or five games without throwing a pick. He's going to make a huge run. Well, Jamar Chase is on the board because I'm picking them as my guy with some MVP value here. If you look at EPA among all quarterback wide receiver duos, the number one is Joe Burrow to Jamar Chase. And there's going to be a wide receiver renaissance when it comes to awards like these. It might as well start this year. 75 to one's great value. All right, this is creepy like out of a cheesy Halloween movie. We're giving the Bengals the kiss of death because I'm taking Joe Burrow like Dave. Wow. 40 to one. It's a quarterback award and he's got the best value. But we jinxed him. All right, let's see who he jinxed. On the next one. Cole, go ahead. I'm not going to waver from my original pick. I'm going Kyler Murray. He's thrown for over 225 yards each and every single game this season. 20 touchdowns overall. He's like a combo. Patrick Mahomes, Russell Wilson and Lamar Jackson. So you can take that one to the bank. He's taking home the honor. That's right. My starting quarterback had to give that one up for Richard McManus. There you go. Same drill guys here. But let's get to our Super Bowl picks. Tampa, Buffalo, Arizona. The three favorites here. Dave, where's your money in this in this market? You know, we keep saying where's the value? And I keep picking the Rams because every time they win, their odds never drop. They have the tiebreaker on the box because they beat them already. Unless Arizona runs the table, the Rams are going to host the NSC championship game when they go to Arizona and beat them. I got the Rams. Let's talk process here. You probably want to pick a team that's going to get the number one seed and you probably also want a team that is going to have an easy schedule the rest of the way. Well, you could argue that the Tennessee Titans have the easiest schedule among all division leaders up to this point. And if they beat Indianapolis, but even if they don't, they still have a nice comfortable margin in that division. 19 to 1, great value. Well, I love the logic, Ed, but I'm going to apply it to the Bengals at 40 to 1. The playoffs started today. They'd have the buy and they'd be the number one seed in the AFC and they're still long shot odds. I'm taking senty. I feel like all you guys get your news via the radio and WKRP with all these Cincinnati picks. But I'm going to have to go with the squad I had as my Dark Horse Super Bowl team. The Arizona Cardinals. We know what they've been able to do. Plus they have Kyler Murray. And if I'm not mistaken, there are plus 3,600 at the beginning of the year. So you can bet on Cliff Kingsbury and the fellas. I like it. I like that pick a lot. Cool. Great stuff, guys. How are we already to the end of the show? Well, alas, we are. Game picks, best bets. DFS, we covered it all. It's what we do. Week eight is here. You can bet it right now. Check out all the bets that we talked about on the Fandals Sports Book app right now. New users sign up using that promo code and moreways1000. You're going to get a risk-free bet up to 1,000 bucks. From all of us here at Fandals, from the guys on the screen and everybody behind the scenes, thanks for hanging with us today. We'll see you right back here next week. And good luck with your bets. After an unprecedented year apart, the time has finally come to say welcome back to the prestige and excitement of witnessing history firsthand. This November, beautiful Del Monte will welcome back the world's best thoroughbreds, the brightest stars, and most passionate fans to the Breeders Cup, one experience. That's a breed apart. Tickets on sale now at breederscup.com.