 Chetanji Kunapure, Assistant Professor, Department of Civil Engineering, Valchin Institute of Technology, Solapur. These are the learning outcomes. At the end of this session, student will be able to idealize the structure as undamped force vibration system, even the student can draw a free body diagram of that system. And the last outcome is writing the equation of motion for undamped force vibration. Now this is the example which explains to you the force vibration system. This is the example, simple pendulum which is shown in figure one. This mass is subjected to time dependent force FD. So this force is acting during the vibration of system, that's why it is called as force vibration system. Earlier two videos were of free vibration system. Now for force vibration means during the entire vibration period. So time dependent force is acting on that body. Or the time dependent force is leading oscillation or vibration in that body. Now this is idealization of the structure. Now every physical system is a continuous system having distributed mass and elasticity. This is the important statement for idealization for every system we have to adopt this statement. Next is assumptions for idealization of the structure. How these are useful in every system that I will explain to you. Now first assumption is total mass of a story is concentrated at its floor level which is lump mass system. Now second is gardens in a building are infinitely rigid. And third is effect of axial forces are neglected. These three idealizations assumptions are there. Let us see how these are useful. Now the example is there which is used to explain the undamped force vibration. The first example is reciprocating machine exerts dynamic force on supporting structure. Now this frame is the supporting structure. This is reciprocating machine. When it starts it exerts some motion and due to that motion the time dependent force actually exerted on the structure and the structure is subjected to vibration. So this m is mass of this story which is lumped at floor level. This is the first assumption. So the majority of the mass is at the floor level. So that's why the mass is lumped at the floor level so that we can analyze the mass of this entire system. Second these are two columns. These two columns are providing elastic resistances. And why is the displacement which is common for both these two columns. It is possible when the effect of axial forces are neglected or second assumption is there the girder is infinitely rigid. So girder is infinitely rigid then only the wide displacement is common or same for these two columns. And third assumption is there the effect of axial forces are neglected so that the columns length or height will not alter or modify. So these assumptions are made for this purpose so that the displacement will remain same for both the columns. Now this single story frame is idealized as lumped mass system. Let's see what is a lumped mass system. You can see one mass and this K is the stiffness which represents these two columns. M is the mass and when the time dependent force is acting on this mass why is the displacement? So very single sketch is there which explains this entire example of reciprocating machine supporting on the structure. Now this lumped mass system actually one conceptual model is required to solve the problem. So this is the conceptual model. It is also known as wagon wheel model. So this is wagon M which is representing the mass of the body. This is subjected to the time dependent force Ft and it is connected to the fixed base or any support by means of this spring which represents the stiffness of the column. So our structure single story frame is converted into the conceptual model wagon wheel model. So this figure 2 which is a single story frame idealized as a single degree of freedom system. Now I will put one question here. This question is what is the type of force exerted by the reciprocating machine? Now there are four options also. You just see these options. A is static force, B harmonic force, C random force, D is non harmonic force. What is the answer for this? You pause the video, think on that and write on the answer. Now the answer is harmonic force. How it is harmonic force? Let's see. Here the example is shown or this figure sketch is shown here. Whenever this time dependent force is acting on the mass, this mass vibrates both side of the neutral position. Vertical line of this pendulum is the neutral position and this mass is vibrating on both side. Displacements are occurring on both side. That's why it is called as harmonic force or these type of excitations are also known as harmonic excitations. Now this wagon wheel model is solved for further outcome. That is the last which is free body diagram and equation of motion. Now let's isolate the body from all point of contacts. Then this body is isolated. This body is isolated and mark the forces at respective points. So this is inertia force and elastic force. So the spring is removed and at that point the some resistance is marked that is elastic force. And this is why is the direction of motion or displacement? Displacement wise occurred and inertia force is marked which is opposite to the direction of motion. Now this is figure A which is a free body diagram of the mass. Time dependent force is acting towards right Ft. Now the free body diagram is ready. Now we have to write the proper equation of motion which is a mathematical model for this. Why is the displacement of mass? Why dot is velocity of the mass and why double dot is acceleration of the mass? Now this conceptual model wagon wheel model it is solved and the free body diagram is made. Only thing is that inertia force and elastic force quantification is required. After this quantification we can solve this problem. Now next is how this inertia force is marked opposite to the direction of motion. The force of inertia which is a resistance again change in the body. So body is dispersing towards right that is why inertia force is towards left. So what is the quantification for that? Actually inertia force is rate of change of momentum which I have explained in the first video also. So whatever acceleration is induced in that body into the mass of that body that will give you the inertia force. So M y double dot is the inertia force which is marked opposite to the direction of motion. This is possible by applying the element principle the inertia force is marked on this body and the dynamic equation is written here. Now next is elastic force or elastic resistance. So the resistance against the deformation is the elastic force and always it is proportional to the displacement of the body. So that is why the quantification for that is K into Y. K is the stiffness of the body. So stiffness of the body into the displacement that will give you the force which is elastic force. Now these two forces are towards left time driven force is acting towards right. Now Y is the direction of motion or displacement. Now the figure 3B is the dynamic equilibrium of body. So we have to apply the summation fx is equal to 0 by the element principle. Now whenever the summation fx is equal to 0 is applied all horizontal forces are considered. These two forces are towards left. One force external force or excitation is towards right. Now this is minus M y double dot minus K y is equal to time dependent force F t. Or actually this inertia force is negative. Elastic force is negative. F t time dependent force is positive equal to 0. This is the actual mathematical equation. So after rearrangement we will get this equation. So M y double dot plus K y is equal to 0 is the mathematical equation for undamped force vibration. Now what is the importance of this equation? What is the significance of equation? Whenever we observe this M y double dot K y is equal to F t that is undamped force. But the meaning or significance of the equation is for any body in which damping we are not considering then the resistance offered by the body against deformation is elastic resistance. Resistance against the change in the body that is the inertia force. So summation of elastic resistance and summation of inertia force that will counterbalance the effect of external dynamic force. So inertia force plus elastic force both are counterbalancing the effect of this external dynamic force F t. Now after rearrangement this is the final mathematical equation which is equation number one. M y double dot plus K y is equal to F t. This is the mathematical equation of undamped force vibration. Now above equation is differential equation. And this differential equation we have to solve to determine this y which is the displacement basic parameter of this equation. What is the order of this equation? And next thing is what is the type of differential equation it is. Now the order of the equation is second M y double dot. You can see here y double dot actually that is the acceleration quantity but this is the order of equation. Now right hand side of the equation is not zero. Time dependent force is present that is why it is non-homogeneous equation. So this differential equation is second order non-homogeneous equation. When we solve the second order non-homogeneous equation and if we determine y then we can quantify the value of displacement in the undamped force vibration. These are the references which are used for the above session. Thank you.