 Welcome to the session damped force to vibration. This is Mr. Chetan G. Konapure, assistant professor department of civil engineering, Vulture Institute of Technology, Swalapur. Let us see the learning outcomes. At the end of this session, student will be able to idealize the structure as damped force vibration system. Student can draw a free body diagram of the system and even they can write the equation of motion for damped force to vibration system. Now damping which is involved in the system that is actually viscous type of damping and it is the resistance against the velocity of the system. Now there are decay in the amplitude of vibration after every cycle that decay is possible only because of damping involved in the system. Now the amplitude of vibration reduces after every cycle that is only possible because of the damping present in the system and the decay is also because of damping present in the system. These are few examples which are contributing damping in the system. These examples even I have explained in earlier video also. But the main concept in these example is substantial amount of energy is dissipated in every example given above. Let us see one example, simple system. This is forced vibration system. The pendulum with this is a mass which is attached with the inextensible string. Now it has oscillating. Now force is present during the oscillation of the system. This is the first thing which is called as the forced vibration. Presence of the force or oscillation which is leading due to the force or time dependent force which is acting on the mass. Now after every cycle this pendulum escalation or displacement will reduce, reduce, reduce and it will come to the neutral position. So this neutral position is a vertical line. So after several oscillations or cycles the pendulum will return at the neutral position how it is possible? This is possible only because of the damping present in the system. What is the damping in the system? That is friction with the air and this will reduce the amplitude of vibration after every cycle and it will come at neutral position. Now these are the idealization of structures, concept and assumptions. Every physical system is a continuous system having distributed mass and elasticity. Assumption for idealization of structure always these assumptions are important and these are required to be made that the total mass of story is concentrated at its floor level. So which will give us lumped mass system. Gerders in a building are infinitely rigid. Effect of axial forces are neglected. Now these assumptions how these are useful that we will see in the next case. Now damped forced vibration. So actual problem or real physical problem that is given here. This is a single story frame with infill wall and the total mass of this infill wall is lumped at floor level. So that is why this dark portion is shown here. This frame with infill wall is supporting one machine. This machine is reciprocating machine. So the reciprocating type of motion is induced by this machine. Whenever this machine starts or vibrate whenever it works it induces the vibration and due to that the time dependent force is generated which is FT acting on this structure with infill wall. Why is the displacement? Now this single story frame which is supporting reciprocating machine is subjected to the force which is a time dependent force. So this is the problem of forced vibration. Now this single story frame is now modeled as a lumped mass. You can see here this entire mass of the story that is m. It is lumped at the top. K is the stiffness which is representing these two columns. Why is the displacement? Now time dependent force is acting on this mass. This is FT. Now how it is converted into the conceptual model? So the mass that is represented by this wagon wheel this is m. Now this mass on which FT the time dependent force is acting K that is stiffness that is represented by the spring and damping is represented by C. Now what is damping in this case? That is very important. Now this is conceptual model which it is ready now. This is single story frame idealized as a single degree of freedom system and the type of damping considered in structural dynamics even in this example it is viscous type of damping which is proportional to the velocity of the system. Now one question I will put here after this discussion. This question is which is the phenomenon creates major damping in the above structure in last slide which was shown in the last slide. Now there are four options I will show it to you. A and B option, C and D option. Okay now all these three are existing but these are the question is which will create major damping in the above structure? What is the answer? Just pause the video and think on that and write down the answer. The answer is the interaction between in field wall and frame. This creates major damping in the above structure. How it is? Here let us see. Now the sketch of that diagram is again shown here. You can see this is a reciprocating machine which vibrates or which generates the periodic motion and it exerts FT force on this frame. Now the interaction between the in field wall and the frame or friction between in field wall and frame that creates major damping in the structure which will again provide the resistance against the velocity of the motion also. This interaction always and also helps to reduce the amplitude of vibration after every cycle. This is the correct answer. Now let us see this wagon wheel modern. Let us go for the next outcome that is a free body diagram and equation of motion. Now this wagon wheel model which represents the damped forced vibration. Now let us go for the free body diagram. Isolate the body from all point of contacts. So this is isolation. Isolate this spring, isolate this piston dashpot arrangement. Why is the displacement and this mass is now vibrating? Mark the forces at respective point of contact. So first is this elastic resistance elastic force KY. Next is this damping resistance C y dot and whenever the mass or body is under motion one more force is their inertia force that is acting here. Now this is external force FT which is towards right. Why is the displacement of mass y dot is the velocity of mass y double dot is the acceleration of the mass. Now this is actually the free body diagram of this mass. Now here the again equilibrium of the body is considered inertia force acts always opposite to the direction of motion M y double dot elastic resistance is always the direction of displacement like KY. C y dot is the damping resistance damping force which is proportional to the velocity. Elastic force is also proportional to the displacement. Now these three are the resistances against this displacement direction. External force is FT it is towards right. All resistances are towards left. So this figure dynamic explains dynamic equilibrium of the body. Now let us apply summation fx is equal to 0 for this body which is in dynamic mode. Now all these three resistances are towards left that is why negative and FT is actually positive that is on right hand side of this equation. Now small rearrangement is there. So this equation is M y double dot plus KY plus C y dot that is equal to FT. Now this is the mathematical equation. What is the meaning of this mathematical equation? What it actually represents? Whenever a damped force to vibration system is subjected to motion which is because of the time dependent force FT this is external force which is a time dependent force which is dynamic force also. This dynamic force is balanced by the inertia force M y double dot plus elastic force this is elastic resistance plus damping force this is elastic sorry damping resistance. So all these reactive resistances will come into picture whenever any dynamic force is applied on the body and the equilibrium is maintained in dynamic mode also. So summation of inertia force elastic force and damping force or resistances is equal to external dynamic force. Now let us see this equation. Now this equation is little rearranged first is M y double dot plus C y dot plus KY is equal to FT. So this is equation number one which is a differential equation. We can see here the order of this equation is 2 the order of equation M y double dot then next is C y dot term and next is the regular displacement term Y and is equal to some right hand side is present on the equation. What is the type of this equation? So differential equation it is we have to solve this difference equals to determine the displacement Y but what is the type right hand side is present and the order of equation is 2. So this differential equation is second order linear non-homogeneous equation. Now if we want to solve this equation we have to apply all the concepts of second order non-homogeneous equation then we can determine this displacement Y. Now these are the references which are used for this session. Thank you.