 Welcome to the session Undamped Free Vibration, myself Mr. Chetan G. Kunapur, Assistant Professor Department of Civil Engineering, Walsh Institute of Technology, Sonapur. Let us see the learning outcomes. At the end of the sessions, student will be able to idealize the structure as undamped free vibration system. Then they can draw a free body diagram of the system and student will be able to write the equation of motion for undamped free vibration system. Now let us see what is mean by free vibration system. Now one example is shown here, the example of simple pendulum. So this is in its extensible string and mass is lump. So this is the vertical line which is represented in neutral position of the pendulum. Now whenever the initial displacement is given to this pendulum, it will start to oscillate. So during vibration of this pendulum no force is acting, that is why it is called as free vibration. But for free vibration initial displacement is required to be given. So this is free vibration system. Now what is the idealization of this structure? Now all the structures or real physical system is a continuous system having distributed mass and elasticity. So following are the assumptions for idealization of the building or structure. For a building in a story, total mass of story is concentrated at its floor level. This is the first assumption made, means this is lump mass system. Now girders or beams in a building are infinitely rigid, that is the second assumption made in the idealization and third is effect of axial forces are neglected, it may be in beam or it may be in column, all effects of axial forces are neglected. Now the second assumption girders in a building are infinitely rigid. This is made because axial deformation of girders are not considered here. That is why if girders are infinitely rigid, then there will not be any axial deformation in the girders. Now this is real life structure, this is a single story frame. So in this frame the mass of this story is lumped at floor level, this is the first assumption. Then girders are infinitely rigid, means this girders are infinitely rigid, that is why there will not be any axial contraction or expansion, that is why both the columns will deflect or deform with the same profile. So why is the displacement which will remain same for both the columns? So the total displacement y is shared by these two columns, that is why this second assumption is made. Now third is effect of axial forces are neglected. So axial forces are neglected means the axial deformation of the column that is not considered, that is also neglected, that is why the deflected profile of the column will remain same for both these two columns. Now this single story building is converted into the lumped mass system, the entire mass is now lumped here, k is the stiffness which is representing these two columns, y is the displacement at top of this system. Now this lumped mass system is converted into the conceptual model, this is also called as wagon wheel model. So this mass is wagon wheel, it is connected to the base by means of the spring. So spring is representing the stiffness of columns. So single story frame is lumped mass system and again it is conceptually model in this way. In this figure 2, the single story frame idealized as a single degree of freedom system, mass is only one that is why it is single degree of freedom system. Now let us solve this single degree of freedom system, this is the y displacement, direction of displacement is shown here. Now this model we want to solve, first we have to go for drawing a free body diagram, isolate the body from all point of contact. So point of contact is the spring. So that is isolated from the base, mark the forces at respective point of contact. Now this is the elastic force which is representing this spring, that elastic force is denoted by ky, y is the displacement. So free body diagram is ready, one more force is shown here which is opposite to direction of displacement and this force is inertia force. Now here you can see the different dynamic quantities, first y displacement of the mass, y dot velocity of the mass, y double dot acceleration of mass. Now these terms are used for denoting all these three dynamic parameters. Now how this inertia force is quantified? Because elastic force is quantified ky, inertia force, its quantification what is the meaning of inertia force, just pause the video and write down this expression for this inertia force. Now inertia force always acts opposite to the direction of motion, everybody has got the inertial property, inertia means the resistance change in condition, it may be rest or motion, but any change is happening in the body, the resistance is upward that is called as inertia. So that force of inertia is always opposite to the direction of displacement of motion. Second thing is force of inertia is proportional to the acceleration induced in the body. So that quantification of inertia force is m into y double dot, m is mass of the body and y double dot is the acceleration induced in that body. So we can compute or we can calculate the inertia force by m into y double dot, but very specific definition of inertia force actually it is rate of change of momentum. Now in inertia force expression when it is m y double dot generally in various systems or cases mass is constant, but in some dynamic system dynamic cases mass is also varying and acceleration is also varying in that case this definition is more specific. So next is elastic force. Now the columns are representing the stiffness. Now actually the definition of stiffness is force required to produce unit displacement. Now these two columns are offering the elastic resistance. That elastic resistance is entirely dependent upon the displacement y. So resistance against the deformation is actually elastic resistance or elastic force. So it is quantified as k into y. So elastic force is generated which basically it is dependent or proportional to displacement of the body. Why is the displacement k is the stiffness k into y that is elastic resistance. Now again the dynamic equilibrium of the body is considered in this figure 3B. Inertia force is marked that is m y double dot elastic force is also shown k y. Now we can apply the summation fx is equal to 0 for this dynamic body. Now the dynamic body and the all forces are marked on this dynamic body. So both these forces are towards left. So that is why the sign is minus m y double dot minus k y is equal to 0. So equilibrium of the body in the dynamic is maintained. And the equation is m y double dot plus k y is equal to 0. Now the equation is rearranged and the number is given 1. So this is the differential equation for undamped free vibration system. So even this is mathematical model of the single story building which was idealized as single degree of freedom system. So whenever we observe this equation we can say that the equation is representing mathematically for undamped free vibration system. Now what is the differential equation it is? What is the type of this differential equation? The order of this equation is second you can see here. This acceleration quantity which is present in this equation. So m y double dot this is second order term which is present in this equation you can see here. Next is the elastic resistance summation of inertia force and elastic resistance in the body that is maintaining the equilibrium this is the meaning of this equation. Now right hand side of this equation is 0. That is why the type of differential equation is homogeneous equation. Now the order of the equation is 2 and the right hand side is 0 that is why it is homogeneous. So this differential equation is second order linear homogeneous equation. So the single story frame single story building is conceptually idealized as a wagon wheel model. Then mathematically it is idealized by this equation. So this second order equation homogeneous equation will be solved with the several methods and y displacement is determined for that system which is the displacement of the structure. These are the references which are used for the above session. Thank you.