 Welcome to the session Damped Free Vibration, myself Mr. Chetanji Konapure, Assistant Professor, Civil Engineering Department, Walter Institute of Technology. These are the learning outcomes. At the end of this session, students will be able to idealize the structure as a damped free vibration system. Students can draw the free body diagram of the system and they can write the equation of motion for the damped free vibration system. Now, let us see how damping is involved in the system. Basically, damping is the resistance to the velocity of the system when the system is under motion. So, the resistance which is offered by some phenomena against the velocity that is called as damping. How practically we can identify whether the damping is present or not? The amplitude of vibration after every cycle will reduce and the successive amplitudes are smaller than earlier amplitude. So, this is basically because of damping which is present in the system. So, the decay in the successive amplitude is observed because of the damping present in the system. Now, what are the examples for the damping? In general, friction with air is also damping which is very minor damping which we can say. Next in case of buildings, the buildings when they are subjected to vibration, the interaction between infill wall and frame is nothing but the damping or this interaction will generate the damping in the building. Now, for every material the intermolecular friction that is also generating the damping. Now, opening and closing of the cracks in the structures in the masonry walls or material of the structure, this requires lot of energy to be dissipated. So, that is also one sort of damping. In case of steel structure, bracing systems in the steel structures which are employed, these are also plays an important role to create the damping in the system. Now, basically because of damping, the substantial amount of energy is dissipated and that is present in every example which is given above. Now, this simple example I will explain you, that is the example of simple pendulum. Now, this is free vibration system with the damping, force is not existing during the vibration of the system. Now, the pendulum, now let us see some initial displacement is given and the pendulum will start to oscillate. This is neutral position. Now, why this pendulum will come at rest until and unless any obstacle is there. But without any obstacle, without any stopping phenomena, it will come to the neutral position, how it is possible, this is only because of friction with the air. So, the amplitude of this pendulum will reduce after every cycle, means the energy is dissipated and finally it will come at neutral position. This entirely depends upon how much friction is happening with the air. So, this phenomena of friction with the air is damping in this system. Now, let us see what is the idealization of the structure, how the structure is idealized as a damped free vibration system. Now, every real physical system is a continuous system having distributed mass and elasticity because the structure is also made with some small elements, small particles, molecules of the materials. Then every element is a part of that structure, that is why it is a distributed mass and elasticity is a continuum system. Now, these are the assumptions made for idealization of the system. First assumption is total mass of a story is concentrated at its floor level. So, this is lumped mass system means the entire mass of the story is concentrated at cg of the floor. Next is girders in the building are infinitely rigid. Third is effect of axial forces are neglected. Now, the real example is shown here, this is a single story frame with infill wall. The entire mass of the story is lumped at the floor level, it is shown here m. Now, this building when it vibrates, this will be the reform shape of the column, why is the displacement at the top of the girder. The single story frame is converted into the lumped mass system. Now, the k is representing which is a stiffness and it is representing these two columns. m is the mass and why is the displacement of the system and this entire real physical practical example is now idealized as lumped mass system. Now, what is the conceptual model for this? The conceptual model is actually this is also called as wagon wheel model. This box or this wagon is representing the mass means the mass of the story. K is the stiffness which is represented by the spring. Now, this infill wall, this infill wall which creates a damping in the system and that is represented by this piston dashpot arrangement. C is the damping coefficient. Now, we can see here this figure in this figure 2, the single story building idealized as a single degree of freedom system. Now, what type of damping is considered in the structural dynamics? There are various types. One is viscous damping, then coulombs damping, then negative damping. But in structural dynamics, viscous type of damping is considered. Viscous damping means which is proportional to the velocity of the system. Now, let us see this wagon wheel model is ready. Now, we have to solve this wagon wheel model and we have to arrive to the mathematical equation. Now, isolate this body from all point of contacts. This is first step means isolate this K, isolate this C from this body. Mark the force set at respective point of contacts. Now, this is the first that is inertia force, second is elastic force and now one more force or resistance is there that is a damping force. Now, why is the displacement of the mass? You can see here this is the y, y dot is the velocity of the mass that is dy by dt. Then y double dot is acceleration mass means y d2y by dt square. The second derivative of displacement with respect to time. Now, this is free body diagram of the mass you can see here. Now, the quantification of other forces are shown here. How damping force is quantified that is important. So, you just write down the expression for the damping force, take the pause the video and write down the explanation for damping force or expression. Let us see now damping force as I explained which is proportional to the velocity of the body. C is the damping coefficient and it is proportional to the velocity as velocity will increase, damping force or resistance will increase as the velocity of the system is lower, damping force generated will be also lower. So, damping force is equal to C y dot this is the quantification for damping force. Now, as I have explained in free vibration inertia force always acts opposite to the direction of motion. So, direction of motion towards right inertia force is acting towards left. Now, force of inertia is proportional to the acceleration. So, that is why it is m y double dot elastic force which is the resistance against the deformation it is proportional to the displacement of body that is k y. Now, this you can see here these are three resistances and the dynamic equilibrium of the body is maintained. Let us apply summation effects is equal to 0. When summation effects is equal to 0 is applied to this body the dynamic equilibrium of the body is considered. So, all forces are towards left that is why the equation is now minus m y double dot minus k y minus C y dot is equal to 0. Now, why this 0 is there? This is the equilibrium condition that is why it is 0, but if you observe practically this free body diagram no force or any excitation is present on right hand side of the body that is why the 0 is here. Little smaller rearrangement we can do here m y double dot plus k y is plus C y dot is equal to 0 this is the formulation after equilibrium equation. Now, again a small arrangement I am making here rearrangement m y double dot plus C y dot plus k y is equal to 0. So, you can see here this is inertia force this is damping force this is elastic force. So, summation of all these three is equal to 0 this is the conceptual understanding of this equation. Now, this is mathematical equation. Mathematical equation then we can see here it is a second order equation the order of the equation in this entire differential equation is m y double dot y double dot double dot means it is a second order equation. Now, right hand side of this equation is 0. So, the type of differential equation it is second order linear homogeneous equation. So, a single story building with infill wall that is modeled with this equation which is second order linear homogeneous equation. These are the references for these two three this session these are the three references. Thank you.