Understanding Damped Harmonic Oscillators

Hello friends, this is Mayuri Singh and you are welcome to our channel Let's Study India. So in today's video lecture, we are going to talk about damped harmonic oscillator. So if we talk about any simple particle and say that any particle is simple harmonic oscillator, when it will execute the harmonic motion. So, what is said that if any particle is at its mean position and if we slightly displace the particle from that mean position, then across that mean position, the particle will execute the to and fro motion. and similarly oscillations are called simple harmonic oscillation or motion is called simple harmonic motion. So, what we have seen is that if we talk about an ideal case and in an ideal case if a particle is executing simple harmonic motion, then in the amplitude of that oscillation, we do not get to see any kind of changes. That is, let's assume that if we talk about an ideal simple harmonic motion, then a particle is executing an oscillation. and if we talk about that certain oscillation then with respect to time whatever amplitude of oscillations will be what will happen? will be constant so when amplitude becomes constant then what will be said? such oscillations are known as free oscillations so the particle was oscillating but with time in oscillations there are no changes in any way. Or we can say that the amplitude of these oscillations has become constant. Then we know this certain oscillation as free oscillations. Now what happens is that why these amplitudes become constant? So what was seen that whenever there is no loss in the energy of the system, or we can say that the total energy of the system will become constant, there will be no changes in it, then the oscillations will be constant. the amplitude of oscillations will be constant. So, what was said that whenever a particle is executing a free oscillation, then in that certain case, our system is possessing a constant energy or we can say that energy is not losing in that system, then it is executing free oscillations. So, this was about free oscillation, which we get to see in an ideal case. So, in the ideal case, we saw that the system is executing a free oscillation. But if we talk about the actual case instead of the ideal case, then in the actual case, somewhat... but the amplitude starts changing by not being constant. That is, did you get to see that if we consider a simple pendulum case, then a simple pendulum, we have to... as soon as the particle is displaced from its mean position as soon as the particle is displaced from its mean position by presenting it in a medium then the friction force acts opposite to the motion of the particle and to overcome that friction force, the particle will do some work and as soon as the particle does work, the mechanical energy of the particle will start changing and this is called as the particle shift the mechanical energy of the particle starts decreasing. And the mechanical energy of the particle decreases, that is, the energy that the system is possessing, changes are seen in that. And that too in decreasing order, the energy was decreasing, and due to this, it was seen that the amplitude started decreasing. Then in this case, it is said that as the amplitude of oscillation keeps decreasing with respect to time, the energy of that type of We know oscillation as damped oscillation. That is, if our particle is executing a simple harmonic motion and with time, the amplitude of oscillations The amplitude of oscillation will not be constant and will change with time And the change will also decrease with time Then what is said that our oscillation It is a damped oscillation. Damped oscillation is being executed. That is with respect to time, our body will come to rest. So what was said? That the particle was at our mean position. in a certain medium the particle was in the mean position we displaced it now it will execute a simple harmonic motion across its mean position it will execute a simple oscillation but after some time this oscillation or we can say this particle is coming on rest and the reason for coming on this rest is that we have a frictional force so what is said that frictional due to which the particle comes into rest after some time. And how does this frictional force act? This frictional force acts opposite to the direction of motion. That is, the direction in which the particle is moving, this frictional force will be applied in the opposite direction. And because of that, the body has to do some work. The body has done some work, its mechanical energy The mechanical energy of the particle will decrease. And the mechanical energy of the particle will decrease. Because of this, the amplitude that was obtained in this case, it started decreasing. And what was called oscillation in this way? Dammed oscillation was called. Or a system that possesses and executes this kind of oscillation, we know it as Dammed Harmonic Oscillator. Dammed Harmonic Oscillator So what kind of oscillator was Dammed Harmonic Oscillator? With respect to time, if changes start to occur in the amplitude of oscillation and what can be said that the oscillation comes on rest with time then this kind of oscillation we will call it damped oscillation and a system which is executing this type of oscillation will be called damped harmonic oscillator. So in this lecture we are going to talk about differential equation and solution of this damped harmonic oscillator. So let's see differential equation of damped harmonic oscillator. Differential equation of Damned Harmonic Oscillator So what we saw is that if we talk about Damned Harmonic Oscillator So on this certain oscillator two types of forces act So two forces Act on the particle. Now, which two forces were these? So, the first force was our restoring force. Why? Because whenever we talk about simple harmonic motion, then it is obvious that whenever the particle... when particle is executing simple harmonic motion then restoring force acts on particle and that force act by force act that force again tries to take the particle to its equilibrium position So, the first force is the restoring force. And how is this restoring force given? F is equal to minus k y. Where minus denotes that the negative sign represents that the direction in which we have displaced the particle, this force is acting in the opposite direction. It is obvious because then only the particle will come to its mean position. So, the restoring force is acting on the particle is given as F is equal to minus KY where K is the restoring force sorry K is called force constant and Y is the displacement that how much the particle is displaced from its mean position. This is about the first force. The second force that acts on the particle in the case of damping is F. Frictional force. Frictional force. If we talk about frictional force, then it is said that frictional force is proportional to the velocity of the particle. So, if we assume that the velocity of the particle is V, then the frictional force will be directly proportional to the velocity. And this frictional force is given as F is equal to minus R V, where R is called a damping constant or frictional constant. So, what is R? Damping constant. and V is our velocity. So, what are the two forces to be seen? First is the restoring force, second is the frictional force. Now, what else can we write this frictional force? This velocity can be written in the term of displacement. So, the first force is this. The second force acting is F is equal to minus R. and we can write F is equal to minus R dy by dt. So, these two forces are acting on our particle. Now, if we talk about the total force acting on our particle, then what will happen? Total force acting on the particle. Total force acting on the particle. What will we do? We will add both the forces. And what are those two forces? One is our frictional force and the other is our force acting on the particle. we got restoring force and these two forces are acting on the particle so we are getting the condition of damping so what we got? minus R dy by dt and what we got? equal to minus ky so this is our equation number Now, what is said is that if we consider that the mass of the particle we have considered is M and it is executing a certain Y displacement, it is Y displacing from its mean position, then in that case, what can we say that we know that F is equal to MA where M is the mass of the particle and A is the acceleration. And what can we write this F? What can we write MA? Second order derivative of time. So, let's assume that this equation is 4. So, from here we have seen that we have two force equations. One is the total force acting on the particle and one is the certain force. Now, from here we can equate these two equations. So, what can we say that from equation 3 and 4, From equation 3 and 4, we will get md2y upon dt2 is equal to minus r dy by dt minus ky. Now, we can write this equation further as M d2y upon dt2 plus R dy by dt plus ky is equals to 0. We can write like this. What can we write here? Or d2y upon dt2 plus r upon m dy by dt plus k upon m y is equals to 0 So this is a certain equation we have made Now what we can do here is that we will replace r upon m with a different quantity and we will replace k upon m with a different quantity So what can we write here? Or d2y upon dt2 plus 2b dy upon dt plus omega square y is equals to 0. Further, we saw that solved this differential equation and lastly we got one resultant differential equation and this certain equation we got this is our differential equation of damped harmonic oscillator and here what we saw where What did 2b get? 2b is equal to r upon m. and omega square is equal to k upon m and this 2b is equal to r upon m what is it telling us r was our damping that is r upon m that is 2b is telling us damping per unit mass per unit displacement and here omega square is equal to k upon m it is telling us restoring force per unit mass per unit displacement so in this way we have differential equation of damped harmonic oscillator Now, here we will see that if this is our oscillator's differential equation, then now we will talk about the solution of this differential equation. So, solution of damped harmonic oscillator. Differential equation of damped harmonic oscillator, we have taken out D2Y. upon dt2 plus 2b dy upon dt plus omega square y is equals to 0. Now, if we want to talk about the solution of this differential equation, then what we can assume is that let the solution be 1. Let the solution be Y is equal to AE to the power alpha t. So, we have assumed that Y is equal to AE to the power alpha t which is the solution of this certain differential equation. Where Y is the displacement. and displacement is the displacement of the particle from the mean position and t is time. Now what is said that if we differentiate this particle with respect to time, then what value will we get? a e to the power alpha t will be alpha e to the power alpha t so dy by dt we got certain quantity if we again differentiate this quantity with respect to t then we will get a value d2y upon dt2 and that will be a alpha square e to the power alpha t now what will be seen here that we can substitute these three values in our differential equation so let's see Substituting these values in differential equation, we will get d2y upon dt2, what is a alpha square e to the power alpha t plus 2b dy by dt a alpha e to the power alpha t plus omega square y is equals to 0. So, in our differential equation, when we So, if we substitute certain values, then our differential equation will be changed like this. Now, if we talk about this certain equation, then we will see some common terms from here. That is, we can take A e to the power alpha t as common. And what will happen to our y here? AE to the power alpha t is equals to 0. So if we take AE to the power alpha t commonly, then what we get? We get alpha square from here. What we get from here? We get 2b alpha. Plus when we take it commonly from here, then what happens? Y square is equal to sorry omega square is equal to 0. So, we saw that we were getting A e to the power alpha t common from these three terms. So, after taking common we changed certain equation in this way. Now, what we saw here? that AE to the power alpha t will not be equal to 0. So, since this is not equal to 0, then what option we have left? That this certain value will be 0. So, therefore, alpha square 2B alpha plus omega square is equal to 0. So, we got a certain quadratic equation. Now, why this value will not be equal to 0? Because that is what displacement is telling us. And displacement is the time when particle till then it will not execute simple harmonic motion. So, this certain value will not be equal to 0. So, obviously, our same quantity of bracket will be equal to 0. So, from here we have gained a quadratic equation for α. So, what can we do with this certain equation? We can get the roots of α. So, what is said that if our quadratic equation is given ax square plus bx plus c is equal to 0. So, yeah. where if we have to find the roots of x then how is it given? minus b plus minus under root b square minus 4ac upon 2a so here if we talk about this certain equation then it is related in the same way therefore we can easily calculate the roots of alpha using this certain equation so what will be our alpha roots? so what we have seen? alpha square plus 2b 2b alpha plus omega square is equal to 0. Now, if we want to find the roots of alpha, then what will happen? What will be the value of minus b? 2b plus minus under root b square, that is 4b square minus 4ac. A value is 1 and c is omega square. Upon what will happen? 2 into a, again a value is 1. What will be the value of alpha from here? Minus 2b. we can take 4 as common from here so under root 4 that is if we take its root then it will be 2 so 2 under root b square minus omega square upon 2. So, what will be the value of alpha? 2 cancel out minus b plus minus under root b square minus omega square. Now, from here we will... we will get two roots of alpha with a plus value and a minus so two possible values of alpha two possible Roots of alpha are, first what will happen? Alpha 1 minus b plus under root b square minus omega square and the second root will be minus b minus under root b square minus omega square. Now, using these two roots, we will see our solution, what is our general solution. So, when we talked in the starting, that let the solution be y is equal to a e to the power alpha t. That is, our solution of differential equation was equal to y is equal to a e to the power alpha t. But after solution we saw that alpha has two roots therefore the general solution so the general solution of differential equation or harmonic oscillator the general solution of damped harmonic oscillator oscillator is given by the y is equals to a1 e to the power alpha 1 t plus a2 e to the power alpha2t so the general solution of the damped harmonic oscillator y is equal to a1e to the power alpha t plus a2e to the power alpha2t now we know the values of alpha1 and alpha2 directly we will substitute the values so y is equal to a1e to the power alpha minus b plus under root b square minus omega square t plus a2e to the power alpha t to e to the power minus b minus under root b square minus omega square t and we can write y a1 exponential minus b plus under root b square minus omega square plus a2 exponential minus b minus under root b square minus omega square so this was our general solution of So, what we get to see from here is that B was equal to R upon M, basically 2B and Omega was K upon M. So, what we got to see is that, we will get to see 3 cases relative to r and omega. So, we have seen the general solution. Now, what was said here? Relative to r and omega, what did we get? r upon m and what did we get? k upon m. So, when we saw the general solution, what was said? Relative to r and k or r and omega, We will get to see 3 cases. We will get to see 3 cases relative to these. So, if we talk about the general solution, then it was seen that under root b square minus omega square terms we have got things. So, what was said? That whenever the value of this certain b will be more than omega, then we will get a case of heavy damping or over damped case. That is, what was said? That whenever If we get a case where B is greater than omega, if we talk about this general solution, then B can be greater than omega, B can be comparable to omega, and B can be less than omega. So, we got to see three cases relative to B and omega. First, when B is greater than omega, then this is called heavy damping case. Heavy damping case or over damping case. If B is comparable to omega, then we get critical damping cases. This case is called critical damped. And if B value is less than omega, then what is called? under damped condition. So, relative of B and omega, we have to we will get to see 3 cases of this certain general solution which is the condition of heavy damping or over damping critical damped and under damped so in today's part we will talk till here that is differential equation of damped harmonic oscillator and solution of this damped harmonic oscillator and we will cover these 3 cases of this certain topic in our next lecture that is in the second part so in today's part that's it ja Here we have learnt that the damped harmonic oscillator shows oscillations in a certain way. Why it is called damping condition? How many types of forces act on it? In that certain case, we have seen the differential equation and solution of damped harmonic oscillator. Now in the next part of the video, we will study these three cases broadly. So that's all for today's video. See you in our next video. Thank you so much.

So, what we have seen is that if we talk about an ideal case and in an ideal case if a particle is executing simple harmonic motion, then in the amplitude of that oscillation, we do not get to see any kind of changes. That is, let's assume that if we talk about an ideal simple harmonic motion, then a particle is executing an oscillation. and if we talk about that certain oscillation then with respect to time whatever amplitude of oscillations will be what will happen?

will be constant so when amplitude becomes constant then what will be said? such oscillations are known as free oscillations so the particle was oscillating but with time in oscillations there are no changes in any way. Or we can say that the amplitude of these oscillations has become constant. Then we know this certain oscillation as free oscillations.

Now what happens is that why these amplitudes become constant? So what was seen that whenever there is no loss in the energy of the system, or we can say that the total energy of the system will become constant, there will be no changes in it, then the oscillations will be constant. the amplitude of oscillations will be constant. So, what was said that whenever a particle is executing a free oscillation, then in that certain case, our system is possessing a constant energy or we can say that energy is not losing in that system, then it is executing free oscillations. So, this was about free oscillation, which we get to see in an ideal case.

So, in the ideal case, we saw that the system is executing a free oscillation. But if we talk about the actual case instead of the ideal case, then in the actual case, somewhat... but the amplitude starts changing by not being constant. That is, did you get to see that if we consider a simple pendulum case, then a simple pendulum, we have to... as soon as the particle is displaced from its mean position as soon as the particle is displaced from its mean position by presenting it in a medium then the friction force acts opposite to the motion of the particle and to overcome that friction force, the particle will do some work and as soon as the particle does work, the mechanical energy of the particle will start changing and this is called as the particle shift the mechanical energy of the particle starts decreasing.

And the mechanical energy of the particle decreases, that is, the energy that the system is possessing, changes are seen in that. And that too in decreasing order, the energy was decreasing, and due to this, it was seen that the amplitude started decreasing. Then in this case, it is said that as the amplitude of oscillation keeps decreasing with respect to time, the energy of that type of We know oscillation as damped oscillation.

That is, if our particle is executing a simple harmonic motion and with time, the amplitude of oscillations The amplitude of oscillation will not be constant and will change with time And the change will also decrease with time Then what is said that our oscillation It is a damped oscillation. Damped oscillation is being executed. That is with respect to time, our body will come to rest.

So what was said? That the particle was at our mean position. in a certain medium the particle was in the mean position we displaced it now it will execute a simple harmonic motion across its mean position it will execute a simple oscillation but after some time this oscillation or we can say this particle is coming on rest and the reason for coming on this rest is that we have a frictional force so what is said that frictional due to which the particle comes into rest after some time. And how does this frictional force act?

This frictional force acts opposite to the direction of motion. That is, the direction in which the particle is moving, this frictional force will be applied in the opposite direction. And because of that, the body has to do some work. The body has done some work, its mechanical energy The mechanical energy of the particle will decrease. And the mechanical energy of the particle will decrease.

Because of this, the amplitude that was obtained in this case, it started decreasing. And what was called oscillation in this way? Dammed oscillation was called.

Or a system that possesses and executes this kind of oscillation, we know it as Dammed Harmonic Oscillator. Dammed Harmonic Oscillator So what kind of oscillator was Dammed Harmonic Oscillator? With respect to time, if changes start to occur in the amplitude of oscillation and what can be said that the oscillation comes on rest with time then this kind of oscillation we will call it damped oscillation and a system which is executing this type of oscillation will be called damped harmonic oscillator.

So in this lecture we are going to talk about differential equation and solution of this damped harmonic oscillator. So let's see differential equation of damped harmonic oscillator. Differential equation of Damned Harmonic Oscillator So what we saw is that if we talk about Damned Harmonic Oscillator So on this certain oscillator two types of forces act So two forces Act on the particle. Now, which two forces were these? So, the first force was our restoring force.

Why? Because whenever we talk about simple harmonic motion, then it is obvious that whenever the particle... when particle is executing simple harmonic motion then restoring force acts on particle and that force act by force act that force again tries to take the particle to its equilibrium position So, the first force is the restoring force. And how is this restoring force given?

F is equal to minus k y. Where minus denotes that the negative sign represents that the direction in which we have displaced the particle, this force is acting in the opposite direction. It is obvious because then only the particle will come to its mean position.

So, the restoring force is acting on the particle is given as F is equal to minus KY where K is the restoring force sorry K is called force constant and Y is the displacement that how much the particle is displaced from its mean position. This is about the first force. The second force that acts on the particle in the case of damping is F. Frictional force. Frictional force.

If we talk about frictional force, then it is said that frictional force is proportional to the velocity of the particle. So, if we assume that the velocity of the particle is V, then the frictional force will be directly proportional to the velocity. And this frictional force is given as F is equal to minus R V, where R is called a damping constant or frictional constant.

So, what is R? Damping constant. and V is our velocity.

So, what are the two forces to be seen? First is the restoring force, second is the frictional force. Now, what else can we write this frictional force? This velocity can be written in the term of displacement.

So, the first force is this. The second force acting is F is equal to minus R. and we can write F is equal to minus R dy by dt. So, these two forces are acting on our particle. Now, if we talk about the total force acting on our particle, then what will happen?

Total force acting on the particle. Total force acting on the particle. What will we do?

We will add both the forces. And what are those two forces? One is our frictional force and the other is our force acting on the particle.

we got restoring force and these two forces are acting on the particle so we are getting the condition of damping so what we got? minus R dy by dt and what we got? equal to minus ky so this is our equation number Now, what is said is that if we consider that the mass of the particle we have considered is M and it is executing a certain Y displacement, it is Y displacing from its mean position, then in that case, what can we say that we know that F is equal to MA where M is the mass of the particle and A is the acceleration. And what can we write this F?

What can we write MA? Second order derivative of time. So, let's assume that this equation is 4. So, from here we have seen that we have two force equations. One is the total force acting on the particle and one is the certain force. Now, from here we can equate these two equations.

So, what can we say that from equation 3 and 4, From equation 3 and 4, we will get md2y upon dt2 is equal to minus r dy by dt minus ky. Now, we can write this equation further as M d2y upon dt2 plus R dy by dt plus ky is equals to 0. We can write like this. What can we write here?

Or d2y upon dt2 plus r upon m dy by dt plus k upon m y is equals to 0 So this is a certain equation we have made Now what we can do here is that we will replace r upon m with a different quantity and we will replace k upon m with a different quantity So what can we write here? Or d2y upon dt2 plus 2b dy upon dt plus omega square y is equals to 0. Further, we saw that solved this differential equation and lastly we got one resultant differential equation and this certain equation we got this is our differential equation of damped harmonic oscillator and here what we saw where What did 2b get? 2b is equal to r upon m.

and omega square is equal to k upon m and this 2b is equal to r upon m what is it telling us r was our damping that is r upon m that is 2b is telling us damping per unit mass per unit displacement and here omega square is equal to k upon m it is telling us restoring force per unit mass per unit displacement so in this way we have differential equation of damped harmonic oscillator Now, here we will see that if this is our oscillator's differential equation, then now we will talk about the solution of this differential equation. So, solution of damped harmonic oscillator. Differential equation of damped harmonic oscillator, we have taken out D2Y. upon dt2 plus 2b dy upon dt plus omega square y is equals to 0. Now, if we want to talk about the solution of this differential equation, then what we can assume is that let the solution be 1. Let the solution be Y is equal to AE to the power alpha t.

So, we have assumed that Y is equal to AE to the power alpha t which is the solution of this certain differential equation. Where Y is the displacement. and displacement is the displacement of the particle from the mean position and t is time. Now what is said that if we differentiate this particle with respect to time, then what value will we get?

a e to the power alpha t will be alpha e to the power alpha t so dy by dt we got certain quantity if we again differentiate this quantity with respect to t then we will get a value d2y upon dt2 and that will be a alpha square e to the power alpha t now what will be seen here that we can substitute these three values in our differential equation so let's see Substituting these values in differential equation, we will get d2y upon dt2, what is a alpha square e to the power alpha t plus 2b dy by dt a alpha e to the power alpha t plus omega square y is equals to 0. So, in our differential equation, when we So, if we substitute certain values, then our differential equation will be changed like this. Now, if we talk about this certain equation, then we will see some common terms from here. That is, we can take A e to the power alpha t as common.

And what will happen to our y here? AE to the power alpha t is equals to 0. So if we take AE to the power alpha t commonly, then what we get? We get alpha square from here.

What we get from here? We get 2b alpha. Plus when we take it commonly from here, then what happens?

Y square is equal to sorry omega square is equal to 0. So, we saw that we were getting A e to the power alpha t common from these three terms. So, after taking common we changed certain equation in this way. Now, what we saw here? that AE to the power alpha t will not be equal to 0. So, since this is not equal to 0, then what option we have left?

That this certain value will be 0. So, therefore, alpha square 2B alpha plus omega square is equal to 0. So, we got a certain quadratic equation. Now, why this value will not be equal to 0? Because that is what displacement is telling us. And displacement is the time when particle till then it will not execute simple harmonic motion.

So, this certain value will not be equal to 0. So, obviously, our same quantity of bracket will be equal to 0. So, from here we have gained a quadratic equation for α. So, what can we do with this certain equation? We can get the roots of α. So, what is said that if our quadratic equation is given ax square plus bx plus c is equal to 0. So, yeah.

where if we have to find the roots of x then how is it given? minus b plus minus under root b square minus 4ac upon 2a so here if we talk about this certain equation then it is related in the same way therefore we can easily calculate the roots of alpha using this certain equation so what will be our alpha roots? so what we have seen? alpha square plus 2b 2b alpha plus omega square is equal to 0. Now, if we want to find the roots of alpha, then what will happen?

What will be the value of minus b? 2b plus minus under root b square, that is 4b square minus 4ac. A value is 1 and c is omega square.

Upon what will happen? 2 into a, again a value is 1. What will be the value of alpha from here? Minus 2b.

we can take 4 as common from here so under root 4 that is if we take its root then it will be 2 so 2 under root b square minus omega square upon 2. So, what will be the value of alpha? 2 cancel out minus b plus minus under root b square minus omega square. Now, from here we will... we will get two roots of alpha with a plus value and a minus so two possible values of alpha two possible Roots of alpha are, first what will happen? Alpha 1 minus b plus under root b square minus omega square and the second root will be minus b minus under root b square minus omega square.

Now, using these two roots, we will see our solution, what is our general solution. So, when we talked in the starting, that let the solution be y is equal to a e to the power alpha t. That is, our solution of differential equation was equal to y is equal to a e to the power alpha t. But after solution we saw that alpha has two roots therefore the general solution so the general solution of differential equation or harmonic oscillator the general solution of damped harmonic oscillator oscillator is given by the y is equals to a1 e to the power alpha 1 t plus a2 e to the power alpha2t so the general solution of the damped harmonic oscillator y is equal to a1e to the power alpha t plus a2e to the power alpha2t now we know the values of alpha1 and alpha2 directly we will substitute the values so y is equal to a1e to the power alpha minus b plus under root b square minus omega square t plus a2e to the power alpha t to e to the power minus b minus under root b square minus omega square t and we can write y a1 exponential minus b plus under root b square minus omega square plus a2 exponential minus b minus under root b square minus omega square so this was our general solution of So, what we get to see from here is that B was equal to R upon M, basically 2B and Omega was K upon M.

So, what we got to see is that, we will get to see 3 cases relative to r and omega. So, we have seen the general solution. Now, what was said here? Relative to r and omega, what did we get?

r upon m and what did we get? k upon m. So, when we saw the general solution, what was said?

Relative to r and k or r and omega, We will get to see 3 cases. We will get to see 3 cases relative to these. So, if we talk about the general solution, then it was seen that under root b square minus omega square terms we have got things.

So, what was said? That whenever the value of this certain b will be more than omega, then we will get a case of heavy damping or over damped case. That is, what was said? That whenever If we get a case where B is greater than omega, if we talk about this general solution, then B can be greater than omega, B can be comparable to omega, and B can be less than omega. So, we got to see three cases relative to B and omega.

First, when B is greater than omega, then this is called heavy damping case. Heavy damping case or over damping case. If B is comparable to omega, then we get critical damping cases.

This case is called critical damped. And if B value is less than omega, then what is called? under damped condition.

So, relative of B and omega, we have to we will get to see 3 cases of this certain general solution which is the condition of heavy damping or over damping critical damped and under damped so in today's part we will talk till here that is differential equation of damped harmonic oscillator and solution of this damped harmonic oscillator and we will cover these 3 cases of this certain topic in our next lecture that is in the second part so in today's part that's it ja Here we have learnt that the damped harmonic oscillator shows oscillations in a certain way. Why it is called damping condition? How many types of forces act on it?

In that certain case, we have seen the differential equation and solution of damped harmonic oscillator. Now in the next part of the video, we will study these three cases broadly. So that's all for today's video. See you in our next video. Thank you so much.

Transcript for:Understanding Damped Harmonic Oscillators

Transcript for:
Understanding Damped Harmonic Oscillators