 Namaste. Welcome to the session Mathematical Modelling of Mechanical Elements Translation Motion. At the end of this session, students will be able to describe mathematical equations of basic mechanical elements for translational motion. In control system, the important task is to develop a mathematical model of the process of interest. Once it is done, the analysis part of the system becomes easier. The analysis requires mathematical models of the processes in the system which is obtained through equations and formulae that can predict how the various devices will behave. So, what is mathematical model of a system? So, it is a set of mathematical equations describing the dynamic characteristics of a system. So, let us see mathematical modeling of a system. So, there are many methods to obtain a model of a system. Now, a set of differential equations are used to represent dynamic behavior of a system and these equations are based on the fundamental laws of physical system. For example, if it is a motion of mechanical system, we can use Newton's law of motion to find out the mathematical model equations. And if it is electrical system, then we can apply Kirchhoff's law to find out the equations which are representing mathematical model of a electrical system. So, modeling of a system can be done by input output representation using differential equations in terms of function of time. So, for the mechanical system which is having motion, we have to use Newton's law of motion. So, take a pause here and recall what are the Newton's laws of motion. So, there are three laws of motion. The first law, every object in a state of uniform motion will remain in that state of motion unless an external force acts on it. The second law is force equals mass times acceleration. So, it is given as F is equal to M into A. Third law says for every action, there is an equivalent opposite reaction. So, based on these Newton's laws of motion, we are going to find out mathematical model for mechanical system. So, let us see mechanical system and its basic elements. So, most of the control system contain mechanical or electrical or both types of elements and components. So, in the analysis of mechanical systems, three essential basic elements are mass, spring and dashpot which occurs in various ways. So, first is a mass. So, what is mass? The ideal mass element represents a particle of a mass of a body concentrated at the center of the mass and it has inertia. So, in the mathematical model of mechanical system, the mass is represented by this symbol. The second element is you can see here spring. So, a spring may be having different types like helical compression spring, helical extension spring, conical spring, torsion spring. Then, there are different types of springs. So, 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 force. The spring is represented by this symbol. Again, it is applied by force F and having displacement X. The third element you can observe here which is showing a shock absorber which is nothing but damper or a dashpot. So, friction exists in physical system whenever mechanical surfaces are operated in sliding contact. It has three types viscous, static and coulomb. In many situations, viscous friction dominates over other two types static and coulomb and viscous is generally considered. So, dampers are used to minimize the vibrations to improve the dynamics of the system. So, in the mathematical modeling of mechanical system, damper is represented by this symbol. Now, let us see translation motion in mechanical system. So, a mechanical system in which motion is taking place along the straight line is known as translation motion. These systems are characterized by displacement, linear velocity and linear acceleration. So, when force F is applied to the spring, a restoring force F k is produced, the force F k is given by. So, when we are applying the force to the spring, there will be displacement. Let us consider a displacement here D. So, F k is directly proportional to displacement X. So, the equation F k is given as F k is equal to k into X, whereas F k is force in Newton. So, according to the Newton's law of motion, when you are applying force, there will be equivalent opposite force. So, here F k is nothing but reactive force generated by the spring. Here k is stiffness constant, X is displacement in M. So, F is equal to F k. So, we can write here as F k is k into X. So, the equation becomes F is equal to k into X. So, this is the equation for spring. Now, let us consider spring as displacement from both ends. So, as shown in diagram, the spring has displacement X 1 and X 2, then let us write equation for the force. So, according to the equation, F k is equal to k into X 1 minus X 2 as there are two displacements and according to the Newton's third law of motion, F is equal to F k. So, F becomes F is equal to k into X 1 minus X 2. Now, let us consider the second element damper or a dash pot. So, when force F is applied to the damper, it produces a reactive force F B, which is proportional to the velocity. Now, in this diagram, you can see here, the force is applied to the damper and therefore, it will be proportional to velocity. So, the reactive force is represented by F B is proportional to dx by dt. So, F B is equal to B into dx by dt, where B is viscous friction coefficient of a damper and according to Newton's third law of motion, F is equal to F B. So, F becomes F is equal to B into dx by dt. Now, let us consider when dash pot has displacement at both ends as shown in figure. So, the equation becomes here F B is equal to B into dx 1 by dt minus dx 2 by dt. So, again F is equal to F B. So, F becomes B into dx 1 by dt minus dx 2 by dt. Now, the third element is mass. So, when a force F is applied to a mechanical body of mass M, displacement takes place and a reactive force F M is generated, which is proportional to the acceleration. In this diagram, you can see a mass is forced by a person. So, when you are applying a force here, there will be acceleration. So, F M is proportional to acceleration and the same can be written as F M is equal to M into A. So, F M is equal to M into d square x by dt square and as we know that Newton's third law of motion F is equal to F M. So, hence F is equal to M into d square x by dt square. And suppose there are two displacements for a mass, then the equation will be same that is F is equal to M into d square x by dt square. Now, for a mathematical modeling of a mechanical system, one has to draw free body diagram. So, to obtain the mathematical model of a mechanical system, it is necessary to draw a free body diagram including the various forces acting on it. Now, let us see the procedure to draw free body diagram. The first step is to assume the direction of displacement of the mass as positive direction. In the next step, we are going to find out all the forces with the directions acting on the mass. And in the third step, we are using Newton's law of motion to express all the forces in terms of displacement or velocity of the mass. Now, let us have one example here. So, draw free body diagram for given mechanical translation system. So, step number n is considered the mass alone. So, from the given diagram, we are considering mass alone with the force F of t and displacement x. Then in the next step, we are drawing the free body with all the forces. So, here the forces acting on a mass are nothing, but due to the damper spring and mass itself. So, reactive forces are F m which is because of mass F b which is reactive force generated by the damper and F k which is reactive force generated by the spring. And in the next step, we are representing all the forces in the terms of displacement using Newton's law of motion in the form of equation. So, here the equation F of t becomes m into d square x by dt square plus b into dx by dt plus kx. So, in this way, we can find out mathematical modeling of a translational motion in mechanical system. These are references. Thank you.