Undamped Free Vibrations
In the previous lessons, we learned the elements of a dynamic system, we learned about mass, stiffness and damping. We also learned how to formulate the equation of motion of a dynamic system. So, now, onwards we will solve the equation of motion and we will explore the responses of structures under various time varying loads. So, in this lesson we will be talking about free vibrations.(Refer Slide Time: 00:53)So, what is a free vibration? Before defining it let us see a cantilever under free vibration. So, what was happening here? The cantilever was in equilibrium and I disturbed it; so, it started vibrating and the vibration amplitude reduces with time and the cantilever eventually goes back to its equilibrium position. These types of vibration is called as free vibration. So, what is it;? so, free vibration is when the structure is disturbed from its static equilibrium position and when no external dynamic excitation is present. So, here there was no external force present during the vibration, the structure vibrated on its own after I gave an initial disturbance.So, in free vibration the structure is free to vibrate on its own; that means, no external force is there to maintain that vibration. These type of vibrations are called as free vibration. So, this is the equation of motion of a dynamic system we have learned it in the previous lessons. So, in the case of free vibrations this force is 0. So, the equation will be mx double dot plus cx dot plus kx is equal to 0, it is a homogeneous differential equation. So, here we know that m indicates the mass of the structure, c is the damping coefficient and k is the stiffness of the structure.So, now, how this structure is vibrating there is no force acting on it so, how is it vibrating? So, when the structure is disturbed initially we are in fact, giving it some initial conditions; that mean, we are giving it an initial displacement or an initial velocity or both. So, when we are giving this initial displacement or velocity the structure is getting some energy; that means, when we are giving a initial displacement, we are actually giving that structure a potential energy or strain energy and when an initial velocity is given to the structure that is a kinetic energy.
So, we know how to calculate how a kinetic energy and velocity are connected it is like half mv square right. We will come to the energy calculations later, but at this point we have to understand that whenever we are giving an initial velocity (
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