 Hello everyone, welcome to this session. I am Priyanka Bansore and today we are going to study about analogous system in that we are going to see about force voltage analogy. These are the learning outcomes of this video lecture. At the end of this session students will be able to define analogy and state the types of analogous system and also students will be able to explain force voltage analogy. These are the contents of this video lecture. First we see about analogous system. We are having two different physical systems with the same mathematical model. Then they will be called as analogous system. In between electrical and mechanical systems there exists a fixed analogy and exists a similarity between the equilibrium equations. Due to this it is possible to draw an electrical system which will behave exactly similar to the given mechanical system. This is called electrical analogous of given mechanical system and vice versa. There are two methods of obtaining electrical analogous networks. First force voltage analogy that is direct analogy and second is force current analogy that is inverse analogy. Consider simple mechanical system as shown in figure one. The mechanical translational system having three elements as spring, mass and friction. M is the mass, B is the friction or damper and spring constant is denoted as k. Due to applied force mass m will displace by an amount x in the direction of force f as shown in figure. Now before moving towards pause this video for few seconds and you have to think and write the mathematical equation for the mechanical system. So according to Newton's law of motion some of forces applied on rigid body or system must be equal to some of forces consumed to produce displacement velocity and acceleration in various elements of system. So the mathematical equation becomes f of t is equal to m into a plus b into v plus k into x of t where a is the acceleration and v is the velocity and x of t is the displacement. So equation becomes f of t is equal to m d square x of t by dt square plus b dx of t by dt plus k x of t. Then taking Laplace of above equation and the equation becomes f of s is equal to m s square x of s plus b s x of s plus k into x of s. So this is the equilibrium equation for the given translation mechanical system. Already we know that the mechanical system has two types. One is the translational and the other is rotational system. So we will see force voltage analogy for translational as well as for rotational system. Now we are moving towards the main point of this video lecture that is force voltage analogy for translation mechanical system. Now we will try to derive analogous electrical network. Consider an electrical network as shown in figure. In this method to the force in mechanical system voltage is assumed to be analogous one means force f is analogous to voltage v. Accordingly we will try to derive other analogous terms. Consider an electrical network here R, L and C are connected in series. According to Kirchhoff's law the equation can be written as v of t is equal to R i of t plus L d i of t by dt plus 1 by c integration of i of t dt. Now taking Laplace of both sides the equation becomes v of s is equal to R i of s plus L s i of s plus 1 by sc i of s. This is the equation of v of s. Now we compare f of s and v of s but here we cannot compare f of s and v of s directly because they are not in the same form. So for that we will use current as the rate of flow of charge. So i of t is equal to dq by dt. Now taking Laplace then the equation becomes i of s is equal to sq of s. i of s is equal to sq of s or q of s is equal to i of s by s. Replacing this term in above equation so final equation of v of s becomes v of s is equal to L s square q of s plus R s q of s plus 1 by c q of s. Now f of s and v of s are in the same form. Now comparing equations for f of s and v of s. From above equations we can say that inductance L is analogous to mass m then resistance R is analogous to friction constant b reciprocal of capacitor that is 1 by c is analogous to spring constant k. Now we have compared translational mechanical and electrical systems here we compare the analogous elements in translation motion. So first force is analogous to voltage v then mass m is analogous to inductance L then friction constant b is analogous to resistance R then spring constant k is analogous to reciprocal of capacitor then displacement x is analogous to charge q and the last is velocity v that is dx by dt is analogous to the current i that is dq by dt. Now let us see mechanical rotational system. In this figure this is the motion about fixed axis in such systems the force gets replaced by a movement about the fixed axis that is called as torque. Then from the figure torque is represented by t inertia is represented by j omega is angular velocity then theta of t is the angular displacement and the alpha is angular acceleration. So according to Newton's law of motion the equation of rotational system is t of t is equal to j alpha plus k theta of t plus b omega. Here alpha is equal to d omega by dt and omega is equal to d theta by dt. So put the values of alpha and omega in the above equation. So the equation becomes t of t is equal to j d square theta of t by dt square plus b d theta by dt plus k theta of t taking Laplace of both sides. So the equation becomes t of s is equal to j s square theta of s plus b s theta of s plus k theta of s. Now this is the equilibrium equation of rotational system. Now let us see force voltage analogy for rotational mechanical system. Now again compare the equations of t of s and v of s. From above equations it is clear that inductance L is analogous to inertia j then resistance R is analogous to torsional friction constant d then reciprocal of capacitor is analogous to spring constant k. Finally we compare the analogous elements of rotational system with electrical system. So first torque t is analogous to voltage v then inertia j is analogous to inductance L then torsional friction constant b is analogous to resistance R torsional spring constant k is analogous to reciprocal of capacitor then angular displacement that is theta is analogous to charge q and the last one angular velocity alpha is analogous to the current I. So in this video lecture we have discussed force voltage analogy for translational as well as for rotational system. These are the references of this video lecture.