 Namaste! Welcome to the session Mathematical Modeling of Mechanical Elements for Rotational Motion. At the end of this session students will be able to describe mathematical equation of basic mechanical elements for rotational motion. These are contents of today's session where we are going to see mechanical system, its basic elements, the rotational motion and mathematical modeling of rotational motion. Now here you can see a picture which gives you idea about mechanical system where you can see most of the parts are rotational. In the second picture it will give you brief idea about what are the different elements can be there in the mechanical system. So, here you can see gear system then there is a damping, inertia, a shaft through which rotating parts are rotating. Now one more picture gives you another view where you can see few more components of rotational motion or rotational elements in mechanical system such as shaft torsional spring, then flywheel rotational inertia and again fluid coupling damping. Now let us see mechanical system and basic elements. Most of the control systems contain mechanical or electrical or both types of elements and components. In the last video session we have seen translation motion in mechanical system and different basic elements. So, take a pause here and recall what are the basic mechanical elements. So, I think you have recalled all the basic elements which were mass, spring, damper. In the analysis of rotational mechanical system three essential basic elements are moment of inertia J of mass, stiffness constant K of the spring and rotational friction coefficient B of dashpot which occurs in various ways. So, this is the first element can be found in rotational mechanical system that is mass which may be different types of motors or gear systems. So, mass the moment of inertia of the system is assumed to be concentrated at the center of the gravity of the body. So, in case of rotational motion the mass is represented by J with angular displacement theta and torque T. The second element is spring the elastic deformation of a body is represented by the ideal element known as spring. It stores energy during the variation of its shape due to elastic deformation resulting from the application of a torque. So, these are different rotary springs can be found in rotational motion and this is how spring is represented in rotational motion with angular displacement and torque. The third basic element is damper also known as dashpot. So, damping occurs whenever a body moves through a fluid and dampers are used to minimize the vibrations to improve the dynamics of the system. So, these are different rotary dampers can be found in rotational motion of a mechanical system and in rotational motion the damper is represented by this diagram with angular displacement theta and torque T. Now, let us see rotational motion in mechanical system. Mechanical rotational systems are obviously similar to transnational system except that torque and angular displacement are considered for force and linear displacement respectively. So, let us see the first basic element which is spring. So, when torque T is applied to the spring it is get twisted by an angle theta. The spring will produce an opposite torque Tk proportional to angular displacement and the torque Tk is given by Tk proportional to angular displacement theta where Tk is equal to k into theta whereas k is stiffness constant produced by the spring. So, according to the Newton's law of motion every force has opposite force. So, similarly every torque will produce the opposite torque. So, T becomes T is equal to Tk. Hence, T is equal to k into theta. So, this is the equation for spring in the rotational motion. Now, let us consider the spring has displacement on both ends as shown in figure. So, it has angular displacement theta 1 and angular displacement theta 2. Then the equation Tk becomes Tk is equal to k into theta 1 minus theta 2 and according to Newton's law of motion T is equal to Tk. So, T equal to k into theta 1 minus theta 2. Now, the second element is damper or a dash pot. The dash pot is represented like this in rotational motion with angular displacement theta and torque T. So, when torque T is applied to the damper it is opposed by the damping torque Tb which is proportional to the angular velocity. So, Tb proportional to the angular velocity denoted by omega. So, Tb is equal to b into d theta by dt whereas b is a viscous friction coefficient of damper and again by applying Newton's law of motion T is equal to Tb, hence T becomes b into d theta by dt. So, this is the equation for damper in the rotational motion. Now, when the dash pot has angular displacement at both ends as shown in this diagram, then equation becomes Tb is equal to b into omega 1 minus omega 2. So, Tb is equal to b into d theta 1 by dt minus d theta 2 by dt and again applying Newton's law of motion T is equal to Tb, hence T is equal to b into d theta 1 by dt minus d theta 2 by dt. So, this is the equation for the damper if it has two displacements at both ends. Now, the third element is mass. So, when a torque T is applied to a mechanical body it produces an angular acceleration as shown in diagram. The reactive torque Tj is equal to the product of J and angular acceleration. So, the equation is given as Tj is equal to J into d omega by dt is also written as J into d square theta by dt square. So, torque Tj is in Newton meter whereas, J is moment of inertia in kg meter square, theta is angular displacement in radians, omega which is angular velocity d theta by dt in radian per second, then d square theta by dt square is angular acceleration denoted by alpha in radian per second square. Again by applying Newton's law of motion T is equal to Tj, hence T is equal to J d omega by dt or you can write T is equal to J into d square theta by dt square. Now, let us see mathematical modeling of rotational motion. So, free body diagram is essential part to find out the method mathematical model. So, to obtain the mathematical model of a mechanical system it is necessary to draw a free body diagram including various strokes acting on it. Now, let us see the procedure to draw free body diagram. Assume the direction of rotation of the mass as positive direction in the first step. The second step says that find all the tokes with directions acting on the mass and third step is to use Newton's law of motion to express all the tokes in terms of angular displacement or angular velocity of inertia element or you can also use angular acceleration. Now, let us have example here. So, draw a free body diagram for given mechanical rotational system. So, this is the diagram where we have all the basic elements. So, according to the steps the first step says that consider the torque alone. So, we are considering torque alone, then we have to draw the free body with all the tokes. So, in the diagram you can see the torque applied and angular displacement theta, then opposite torques produced by spring denoted as T k, T b produced by tamper and T j produced by mass. Then in the next step we have to represent all the tokes in terms of angular displacement using Newton's law of motion. So, here we can write the equation as T is equal to j into d square theta by dt square for the mass plus p d theta by dt for the tamper plus k theta for the spring. So, this is how we can find out the mathematical model for rotational motion in mechanical system. These are references. Thank you.