So, this is time to give you forth to keys that make you understand the oscillation period and frequency. As I demonstrated by whistling, we will give you a graph that combines these two oscillations. Then we'll ask you how we express this combined oscillation with two different frequencies. Let me then go to next question. So, we talked about oscillation. That means something move back and forth. Let's concentrate what it means by move. Move back and forth or move back and forth with the respect to equilibrium. If something has to move, what governs the movement? Suppose I have this string. If I hit it, this movement move back and forth or you can see there's something that's move back and forth. What governs this motion? Let me say what law governs the motion move back and forth? You know the answer. The law that governs move back and forth is a famous Newton's Second Law. Newton's Second Law essentially, every motion, but we are concentrating the motion that move back and forth oscillation or vibration. Then let me try to express Newton's Second Law graphically because Newton's Second Law essentially say, any force acting on, for simplistic case, particle will move the particle m in certain direction with acceleration. This graphical expression of Newton's Law essentially expresses the relation between forces and the motion. Another important observation we can see from this Newton's Second Law on the graphical expression of Newton's Second Law, is that the expression in terms of vectors. What is vector? Vector has direction and magnitude. What we need to express direction and magnitude is a coordinate of course, right? So, in this general expression we need coordinate like X and Y expressing the magnitude and direction in both direction or generally we need X, Y, Z coordinate to express forces acting on the particle in any direction as well as the motion of particle in any three dimensional direction. Now, let's think about what would be the simplest case of vibration? Vibrating something particle in X and Y, Z direction. That's general three dimensional vibration. Then the next question is what would be the simplest vibration? Because if you understand the simplest case, we can understand more complicated case. In other words, we can understand the vibration in three dimension. So, let's start with the vibration in only one direction. In other words, let's study about the vibration that only allows in one direction. In other words, that vibration has single degree of freedom system. So let's think about the vibration of a particle that has mass m but experiencing force on one direction, say its magnitude is f. The resulting motion would be graphically if I write, express it graphically that would be m acceleration. But using this coordinate xt, we can write x double dot. Where x double dot meaning that taking derivative actually with respect to time. But as we can realize, this graphical expression between force and motion may not introduce vibration because this force is only this force and this motion. Therefore, this just describes something moving in this direction. So if it has to oscillate, in others words move back and forth, then due to the motion in this direction there must be the force that resist this motion. That, therefore, we have to invite the some force that resists the motion in this direction. So if I introduce this force denoting by F reduced, we have a graphical expression of single degree of freedom system vibrating in acceleration would look like this. Okay. There are many forces that oppose the motion of particle in this direction. One very typical force belong to this force would be the force that is proportional to the displacement. In other words, when I push this and move this direction with one unit over here, then this one, oppose the motion of this direction. So if I write this graphical expression of vibration single degree of freedom system expressed using one coordinate xt, mathematically, then it can be written like this: F, that has the same direction of this coordinate, so that has to be plus; and there is a force kx, the force resists the motion but proportional to the displacement x, but opposed to the direction this to the excitation force over here and that direction is reversed direction of this coordinates, so I have to put the minus, okay? That has to be equal to mx double dot. because the direction of this vector is the same as the direction of this coordinate, that has to be plus. I showed very logical straight forward way to apply Newton's second law, inviting the coordinate. Somebody highlight of apply Newton's second law is force that you have to follow the sign convention of coordinate because Newton's second law expresses the relation between force and motion in backed up form and the vector form, vector has magnitude interaction, magnitude and direction has to follow the coordinate system you employed. So this is equation number one. In the beginning of this lecture, I said compare with the other vibration causal we will use rather simple mathematical expression or I said we will not use complicated mathematical expression to explain vibration. Rather than using complicated mathematical expression, we use a simple mathematical expression that possesses every essential vibration concept, so that you can use a new understanding about vibration into practical problem. So this is fundamental starting, essential, mathematical, simplest expression of vibration. I actually tried to make you to feel the vibration or see the vibration, but just assumed this mathematical expression. Okay. Then how to do it? We have to look at very carefully this mathematical expression. Okay. X is the resulting motion, and the resulting motion expresses and it appears in right hand side and left hand side as well as the right hand side. Therefore, is not physically acceptable way to express the vibration. So move this one over here then that will look like F equal to mx double dot plus kx. This mathematically expression, If I translate it into physical domain then I can say excitation force vibrate something that has a mass m and a k into extraction. Okay, that is physically makes sense. I mean, physically sensible explanation. But often we write this mathematical expression in this way, mx double dot plus kx equal to F. That will change with respect to time. We call this mathematical expression asked differential equation second order that has constant coefficient. Okay. Now, we are going to investigate what physics, what simple vibration concept this mathematical expression has. That's our next objective. We have very simplest, but possesses every physical phenomena vibration. Mathematical expression can be written as mx double dot kx is equal to f of t. So, now we are trying to see and understand the vibration of this kind. For example, vibration of this. Okay. Interesting. So, if I pull this one down, this will vibrate like that. If I hit this very slightly, only providing velocity not displacement, it will look like. What's the difference? I wrote f of t. There are many different kind of f of t's. We got this f of t or this f of t. There are many different f of t. What would be the simplest case of f of t. That would be, I like simplest case because every simplest case shows us every fundamentals. So, simplest case of mathematical expression of this. That I say this is equation number one. That will be mx double dot plus kx equal to zero. No force. Solution. There are no force, no response. But that's the trivial case. No force, no motion except rotation. Okay. If I pull this one, unit displacement and then let it go, it oscillates like that. Even though there is no force but I excited this one by giving initial displacement, this would oscillate. Okay. I just to give it at a time t equal zero initial displacement and let it go, it will oscillate. During the oscillation there is no force. Okay. There's a force because of gravity. Okay. In this example, the gravitational force acting on this particle will be m multiplied by g, g is gravitational acceleration, is balanced by the force acting on this spring. Okay. So, the gravitational force acting on this particle is already expressed in this demonstration. So, the excitation, unit excitation pulling down this mass. In this instance, there is initial displacement that it will auction like that. What is the response look like? That is the displacement. We can intuitively find that the response would be with respect to time locally to one unit displacement because I gave him one unit displacement. In this case, the spring expresses the restoring force or the spring force acting on that particle. This is m and I'm using the coordinate toward this direction but measured in amplitude over here. Okay. I'm using coordinate measuring the displacement starting at this point. I give unit displacement and let it go. Therefore, it look like this. I didn't solve this equation. But using simple math, simple demonstration, I can see without any question that displacement will look like that. This is as we learned before is period. One over period, reciprocal of the period, it physically means how often you oscillate compared with unit time. If it oscillate 1,000 times with respect to unit time one second that is one kilohertz, in that case. So, one of t has to measure something frequency. Okay. Therefore that is related with frequency. That usually we denote f. That is then in measured hertz and hertz measured as I mentioned briefly before, hertz measures frequency and measure how often it oscillate with respect to unit time. So, 1,000 hertz meaning again oscillate. The mass is oscillating with respect to time, 1,000 times. Okay. So, I can see there's this relation. Now xt. If I express this xt mathematically that it has amplitude one. It look like cosine. All right. It's not sine, cosine ft. But this one has to be measured in terms of radian per second. So, I have to put two pi and two pi f is equal omega that measures frequency in radian per second. Sometimes it's headache to understand two radian per second rather than frequency. But radian per second simply meaning how often you rotate with respect to time. So, if you rotate one second, one circle. That is, has a period of one second frequency of one but you rotate to two pi. So, back here you have a frequency one and you rotate circle within one second, two pi. So, is now tangible measure. If you rotate this twice a second, then this two pi and the frequency is in this case two hertz. So, two pi multiplied by two. We have investigated the solution of this simple mathematical expression. In other words, there is no force but initial displacement or initial velocity is possible for having. If you have initial displacement. Say one centimeter like, I gave one centimeter initial displacement, let it go to oscillate. The solution look like X of t is equal to one centimeter and it has to be cosine two Pi ft and this is in terms of radian per second, that is, omega. Okay. This a has to satisfy this, because this solution obtained very intuitively. Therefore, we can argue that this intuitively obtained or experimentally obtained, the solution has to satisfy this equation that governs the motion of a single degree of freedom vibration system. So, I have to differentiate to twice of this, that will provide me two Pi f, two times. Differentiate cosine with respect to time, give you two Pi. Then, this cosine will change to sine and giving me minus. Then, if I differentiate cosine two Pi ft, again, with respect to Pi it will go to cosine again and give me another two Pi f. Therefore, I have two Pi f squared. So, that is the solution mx double dot minus m. Then, I have kx. So, that is. So, I have a plus k times cosine two Pi ft, okay? That has to be zero. What do I did, I plot this intuitively or experimentally obtained solution into here, then I got this. Therefore, what I can say is minus m two Pi f squared plus k equal to zero. What it means. Or if I use omega, then it looks like minus m omega squared plus k equal to zero. That says this omega is not independent. It has to be related with m and k. So, I can write omega squared, is equal to k over m. Mathematical expression of this is saying that this oscillate with the frequency of this governed by the single degree of freedom system property which is mass m and k. So, you will see omega over here like this. Omega has to be like that and this omega follows the relation with k and m like that. We call this natural frequency. In other words, if I give initial displacement, it oscillate with the natural frequency of the system. Okay. Wow. That's interesting. That's the interest. That means I know how to measure the natural frequency of a certain single degree of freedom system. For example, you want to imagine natural frequency of your car when car is moving up and down. What do you do? You just push your car using your feet giving unit displacement or some displacement and then let it go, it'll oscillate. If you measure the frequency, then you can measure the natural frequency of your car in this direction. You want to measure the natural frequency of your car in this direction? You push, I don't know whether you can do it, but you push and then let it go, oscillate and measure your frequency, then you can measure natural frequency of your car. Or you can also measure stiffness of your suspension, for example, k if you know the mass of your car. Now, you measure the frequency and if you know the mass of your car, you can measure the suspension stiffness of your car. So, simple expression, but has many practical implication. I prepared a better oscillation, better experimental device. I give initial displacement. Then it oscillates. Okay. I invite coordinate over here. Denoting this is X of t. Okay. Then, I give initial displacement and I draw X of t. Then it look like X of t, X of t and then let it go. Mathematical expression should be equal to one cosine two Pi ft. I plug into this experimentally observed solution into this of an equation, gives to me interesting result that relate with the frequency and stiffness and the mass. Omega is equal to two Pi f. Therefore, I can say the frequency has to be equal to square root k over m, one over two Pi, right? That relates natural frequency and the stiffness and the mass. Now, let's move to another case where we have, mx double dot plus kx equal to zero. Another simple case is that I have this, but I excited this with initial velocity. Look at carefully. The reason why I'm saying I am giving this physical system initial velocity is simply meaning that I am exciting this impact with very shorter distance of time. Therefore, it follows principle of impulse and momentum that gave me the initial displacement. Okay? So mass in the beginning, a thrust, therefore there is no velocity before I excite system and that is zero because it was at rest, and I excite it using f Delta t zero. Very short period, very short time, it oscillates. That is this momentum, right? So, I got f mass times x times 0 and it looks like this. You can see that in this case, the solution is no longer look like this, but it look like this. It look like sine not cosine and I can say the solution in this case look like x of t is equal to sin 2 Pi ft. But having something that is introduced by initial velocity that look like mx dot at time t equals 0, I may say this is A or one if I am expert in making f Delta t1. You can say this is A then. I plug this solution into again mx double dot, plus kx equal to 0 again. Then I can come up with same solution that characterize the vibration of the system that look like f equal to 1 over 2 Pi square root k over m. So generally speaking, I can say the following things. The solution due to initial velocity look like A and the response look like that. If I have initial displacement and the solution cosine 2 Pi ft but I put B over here. So, that will be the solution due to general initial velocity and displacement. So, A measure the contribution of initial displacement. So, measure of the contribution of initial displacement and the B is measure the contribution of initial displacement. Okay. Now, and this solution follows the governing equation zero. What we learned from this selector we found that, the frequency Omega in radians per second has to follow or if I use the frequency f hertz it follows k over m but 1 over 2 Pi square. That we called natural frequency. Okay, in this lecture, we start with the definition of vibration. Vibration can be expressed as the oscillation with respect to equilibrium, and then we invite very simple or simplest vibratory case like this and then we apply the Newton's second law on this vibration system and that expressed the, in terms of mathematical expression and x double dot plus kx equal to F of t. We investigate physical behavior of the simplest case when F of t equal to zero, and then we found that the solution look like this and interestingly it expresses a system's behavior, natural frequency concept.
