Ch15 - 2

we have so far state what's called simple harmonic oscillation in which the only net force exuded on the object is a linearly restoring force such as the force from a spring we know that force is conservative which means that the total mechanical energy of the system is conserved and we study that before. In reality though when you consider any simple harmonic system such as you know a pendulum, the pendulum is not going to swing forever right if you pull it to the side and let go of it, its going to swing for a number of cycles and amplitude is going get smaller and smaller eventually its going to stop. the same thing happens to a block attaches to a spring after several seconds of oscillation it will slow down eventually stopping. So what's going on here? because in the real world there is always some resistance force friction, that is that will take energy away take mechanical energy away from the oscillation system and that energy of course is turned into heat and sound and other things so we must if we want to consider real-life situation at a resistance force and the resultant oscillation is no longer simple harmonic oscillation rather its called damped oscillation or simply damp oscillation there are many different forms of resistance force, for example, if you consider a solid object going through a fluid such as air and water, one way to express the resistance force is something like a constant times speed squared the speed squared is a quadratic form and that will add a lot complications to our differential equation in fact, is going to make our differential equation non linear so we not going to deal with that, we're going to deal with the simplest type of resistance force as far as the solving equation is consider and that type of force is called linear resistance force in words you do get a resistance force okay fr standing for resistance, okay it is negative meaning its opposite to the direction of motion and it is linearly proportional to velocity negative v with a force constant in front of it b which we called b okay again this is not the only type of resistance force but as you will see in solving the equation since f is directly proportional to v to the first power is a linear resistance force that makes our calculation a lot simpler so with that new force added in let's look at what happened in x direction okay what is the net force in x direction well of course I have negative kx that could be a real spring on a effected spring anyhow that is linear restoring force responsible for the oscillation but on top of that you also need add to this term the resistance force minus bv, v is really just dx/dt so that the net force what does that equal to, that equal mass times acceleration, so mass times acceleration which is the second derivative. My question is with this new force add how does that change the oscillation we can look at this from math point of view and physics point of view uh from physics point of view you know when you add this force okay this force always resists motion because you see is always against the velocity so when object moves forward this force is backward and when object moves backward this force is forward it's always slows, tends to slow down a motion and therefore with this additional force you're not going to move not as fast as before you lose kinetic energy and that kinetic energy lost is not turned into potential energy such as in the case of this term rather is turn into heat and sounds therefore you going to lose mechanical energy overall and we know the mechanical energy of a simple harmonic oscillator at a certain moment, when this amplitude is a is always 1/2 ka^2 right as the mechanical energy is lower over time by this resistance force we know the amplitude has to decrease as well okay so if I compare uh our new oscillation with his damping force with the previous simple harmonic oscillation by doing a plot here is T and here is x ok without the damping force you know the amplitude is fixed because the energy is also fixed is like this the amplitude does not change with time as what happens to this new term added well energy is going to decrease over time and therefore you have a situation like this right you have reduced amplitude over time okay in word the amplitude taped down like this, that is the first thing we have to consider okay the amplitude is no longer a constant but rather it decreases over time that's one thing, another thing is what happened to the period of the oscillation okay in words does this term slow down or speed up the oscillation well of course this term always resists motion therefore tend to slow down the motion therefore you can imagine when you add this resistance force ah the oscillator going to take more time then before to complete one cycle becauseit was opposed by this additional force and therefore the period tends to get longer or the frequency tends to get lower ok so again these are the two new things form physical consideration when you add this new term and that is first of all amplitude would decay and secondly pureed would get longer or the frequency will be lower that from physics consideration mathematically let's look at this equation indeed you have here extra first power, here is dx/dt to the first power here the second derivative of x respect to time there is no such thing as x squared for example or sin (x) cos(x) times whatever so every term is proportional to x to the first power so that's good news because that means this is a linear differential equation which is rather simple too solve mathematically okay the idea is this, you have x, you have dx/dt and you have the second derivative and when you move this term to the other side you find this three terms add up to 0.So what does that tells us, it tells us that you were looking for solution x as a function of time ok x function of time, such that this function itself and its first derivative and second derivative should assume the same function of form it should have the same dependency on time that is why this term, this term, this term you add them together they could cancel out right see if this depends on e to the power of t an extra deep depend on sin (t), you know it never going to be able to cancel each other out for all time right, so therefore you're looking for the functional dependency ok uh, so now question is what kind of function behaves in such a way that when I take a derivative and when I take a second derivative it goes back to himself. Well the only thing we can think of actually two things we think of one is exponential function the other will be sine cosine function right now we know without this term is a sine cosine function okay what happens to this term okay without his term it would look like this the solution simple harmonic solution very familiar with it now A and then cosine omega t plus initial phase angle phi initial you know that right now with this new term. How should I adjust this OK how should I adjust. Well first of all the amplitude is no longer a constant rather decreases over time now do you expect it to decreased linearly over time well if it decrease linearly over time then it's gonna stop right there a sharp cutoff at some point it just completely stop does that really happens like that.Well see in reality as the amplitude decreases so does the speed the average speed you see this term, the resistance forces proportional to the speed, what does that tell you, it tells initially when you go very fast with large amplitude, this term the resistance force is large which means you can take a lot of energy away per cycle but gradually as you slow down,slow down and slow down the amplitude is so small and speed is a small so that these resistance force automatically decreases becomes its proportional to speed therefore energy is still taken away from system but at much slower rate which means the amplitude is not going to descease linearly rather is going to decay but its gonna keep on going going for a long time you word I can expect the amplitude to decay like that and the amplitude itself would decay like that so this is A as a function of time and this is T is self something like that not a straight line that more reasonable now tell me what do you think this looks like, this curve how about exponential curve huh because mathematical there's good reason because exponential curve and exponential functions such that when you take derivative it goes back to itself right and that's what we want, we want derivatives and second derivative of the functional form as x itself so how about if I do this I said e to the power, negative some sort of constant c times t, now why a negative sign because we know its decay instead of going up exponentially over time so this is what's we called trial solution okay this is our trial solution uh we have a constant c to be determined and we have this new omega to be determined by the way this omega is not the same omega we have before is not square root k over m, rather we expect to be a little bit lower because again we have resistance force that slows down the motion okay so there we go. uh at the next step all we need to do is take this plug this into the equation x substituted by that and you take dx/dt you take the second derivative. now it's not hard to do with little tedious because after all it the product exponential and cosine function you take derivative, one derivative you get two terms and you take two derivative you get more term coming out but is doable okay am going leave out the derivation mathematically its just a little tedious that all but when you plug it in and you compare the two sides and make them equal you'll find that this indeed solve the equation okay this trial solution indeed solve the equation, ah with two conditions condition number one, what is c equal to c must equal to b over 2m, that one thing so that the decay constancy c and another thing is what is omega equal to omega must be equal to the square root of k over m, minus something, what is that something, b over 2m squared ok if c equal this omega equals that its turns out that this solution indeed solve the equation I mean you don't have to take one for, you can, you can pluck c in like this and omega like in and you just you take one, two derivatives see if the two sides match they'll definitely will so here is our solution here's our solution, Ae to the power (b/2m) t and cosine omega t plus phi initial now let's look at the physics of c and omega, now first of all look at omega,omega indeed is less than the square root of k/m that will be the frequency angular frequency of free oscillation without any damping right we know that with damping you could have a lower value then that because damping force slows it down that one thing but I don't really mind if damping slows down slows down the motion a little bit but what I really worried about is that we looking at a squared root here and there is a negative sign okay so there is a danger immediately and that is its possible mathematically for the second term to existed the first term and would we be wrong in that well then you wouldn't have a real value for omega anymore you have you know you have a negative value under the square root that is a complex number and that's not all we're looking for so what does that tells us? Its tells us if you want to sustain oscillation okay you want to sustain oscillation ah despite presence of friction of resistance like this you want to sustain oscillation ah you wanna make sure that omega remains a real number: which means k/m must be greater than the next term okay which is b/2m square and if that satisfies and is not satisfied then omega is a magic number and you're not gonna have an oscillation at all ok so we have three possibilities when this first-term is greater than the second term, equal to a second term, or even less than a second term and then can have three different physical situations. so situation number one, that's when k/m is greater than b/2m square so you see you have damping cost on b, but it's not very large it is small enough such that at k/m is greater then that ok now what that tells us is that omega is real there is real damping, okay there is real damping so in this case damping I'm sorry there is real oscillation and in this case the damping is there but is not very strong and we say it's under damping okay that the term that would describe it under damping you have an oscillation even dough the amplitude does decrease with time okay that the first situation, now the second situation is when k/m equals the second term okay so lets write that down as well you have k/m k/m equals b/2m square so the damping constant b large enough so this two terms are equal so what you get as a result we get a result and result omega equals zero right so with omega is zero you don't have the exponential, you don't have the cosine term anymore omega is equal to zero okay this cosine function has no time dependencies so its just the constant and you can just throw it away so the constant you can bring into that constant A so all you have to all you really have just just measure decay function you have ah you know the oscillation doesn't exists any more this is x, this t, its going to look like is gonna look like this okay no ups and downs this type of damping are is large enough so that kills all the oscillation but if b slightly smaller than that you are gonna have oscillation okay so this b barely large enough to kill all oscillation we call it critical damping critical damping okay now third case what if k/m is less than b/2m squared okay so in this case omega is not real alright omega not even real there is no oscillation okay there's no oscillation and so what you have here is just a x and t it just does down like this monotonically it does not oscillation at all and what do we call that we call it over damping versus under damping of course there is too much damping going on that no oscillation can sustain okay so you have three possible cases under damping, critical damping, and over damping. If you want a oscillation of course you want b to be small enough so we can have under damping but critical and even over damping these are not totally useless. Can you give an example of critical or even over damping? Well here a possibility imagine you open up a door okay a lot of this doors such as the door to our classroom they spring loaded right, they're spring loaded you the reason why they spring loaded is because when you open a door when you go in the door going to close by it self you don't have to close the door behind you is going to close by it self but do you want this door after you enter the room do you want to swing back and forth, back and forth, back and forth, many times you don't right you just want it swing nicely and it close itself okay so you think this door as a oscillator do you want the oscillation to be under damping, critical damping, or over damping or you probably want critical or over damping right so what you do is you add enough resistance into the spring so that when the spring when the spring pulls the door back there is no oscillation it just closed by itself another example would be you know the needle of a fine balance okay so you have a balance like this right and this is where the needle would settle but the thing is if the balance very fine and chance are is going to swing to the other side, swing to that side and swing this side and that side and so on its annoying its going to take a long time before its stop so if you add some damping mechanism particular you can use a magnet to introduce a magnetic damping force okay that can hopefully bring this needle into critical or over damping so that ah you know it go nicely and settle immediately after after a little bit amount it nothing it goes into oscillation on and on forever okay so this are the three possibilities. Next lets look at the value of c, ok so the amplitude is a function of time is A added to the constant e to the power negative ct, c is b/m okay so (b/m) t, the damping constant the decay constant here is b/2m, cost years first of all you can verified easily that b/2m has the right dimension what is the dimension of b, well you know the force is given by negative bv so b has the unit of what actually lets look at this way you know kx that the restoring force then you have bv, b dx/dt that the damping force this two obviously have the same these these two obviously have the same dimension right because they're both forces and that tells you want the dimensional of b is okay ah was the dimension of be that's the dimension of kx actually x and that x cancel so the dimension of b would be the dimension of k times the dimension of t right so k is newton per meter ok in words kilograms per meter squared per second squared that k, you multiply this by the dimension of t the second that will give you the SI unit of b so whats the SI unit of b that will be kilograms per second so kilogram per second you divide by m m is mass so kilograms is gone so what is b, what is the unit of b/m that will be one over second you see it nicely cancels t which is in seconds okay indeed this is a dimensionless number the unit's work out really well and secondly take a look at this damping that decay constant here this is telling us that greater the b value the faster it decays right large b value makes a quick decay mix and that makes perfect sense because after all the reason you have a decay in amplitude is because the presence of this damping constant b without the damping force b is equal to zero, there is no damping of course you don't have this convention factor act so obviously damping gets faster when you have it also makes sense that you had a m in the denominator here what does that tells me it tells me that for the same damping constant, the same damping force that is if the object is massive then the decay gonna be slower thats make sense because you if try to slow down very massive object from oscillation is not going to be very effective track if the object he's not massive then you the same on a horse averages you can slow down very quickly so therefore more massive the obligate us more decay so how's in Texas this is the %ah is how we express be rate at which dickey cars so abhorrent act. in the real world there's always a *** a calm down so that if you want to maintain the oscillation you wanna meet opposition only and going and going you must somehow feed energy into it to replace the energy has lost decay ok this system wrist when you force and this dickey for sake desiderio this is a force to do to the year presents a French in order to sustain our solution you must fee urged into as much his new term to drive the %ah sludge so we can happen driven of forced hostage there are different voices they used to tried I'm if you're looking for us simple harmonic expression final lost nation them it's clear of what we want here wouldn't be something really funny back square we want to be just a simple harmonic variation as well okay at City aptitude abyss force is after not and %uh it also the reason to sign a consent function time some lets agents at not times %uh function all-time sign on the city now this oh my god we can be anything I want this is how the young force the Rings all the time and it is not necessarily ewbal to Omega of free oscillation lead is the home it's not happening unscrewed children ride at is not necessary this this league is determined by the driver drugs Akropolis %ah spring an ashy okay depends on how fast I she had has not nothing to do was how you this spring's clubhouse lead by itself so fast course an app equals mass times acceleration mast so this time I have wouldn't want her Solutions Limited more complicated as a matter fact it comes as the two terms okay there is a transient and his I baptized in Galway overtime it only happens for brief moment when be one initiated the ostrich and then it's followed by state-run which does not change over time which which at which which calls on how to now you can understand why did a transient imagine spring and there's a mass right nothing's gonna first which you do is you crack mass used force ok he's touching of course you know the system is startled by tried to happen and he was going on right haha you know how how much force one point how is he doesn't know how fast you want us lead I'm so after a few cycles okay this is nice Aldis Hodge 12 you want to call me down with this amplitude of course and was disarming in frequencies is learns to react okay there is a learning process which may take a few cycles back is was on a transient solution not gonna be mom back wiggle Street into next phase Anders with %uh steady speed solution okay steady-state steady state solutions in a steady state solution d amplitude of the oscillation no longer changes overtime let's eat mathematically one has two kids be because remember a function distract his defeat energy into the system now initial suppose he starts at rest our solution started crest CT castles grungy initial these pretty small you agree he's pretty small okay and a course we knew out his force %ah it's just not possible easily will increase has be- increases this raises for schools increases now %uh this force feed energy into the system this 46 energy away France's initially because motion is not relatively small are you feed more energy into it and last energy beat the system as a result nectar is a nap increase in energy persistently mpg shooter increased up gradually at the MPD increases over time v on average his creature the reasons for schools increases and then the energy taken away by Dez reasons forced also decreases so you have to compete sources this one feat energy even system this wanted immediately from the system at some point the speed reaches the average speed which is such a value that the energy you feat inconsistent per cycle equals to energy has taken away find resistance precise with that I'm in towns need act point that speeches reached is no monger any met change energy of the system i'd which is the amplitude system must be constant that he's steady states so in a steady state solution I have just Anna fixed amplitude of large T and axe please do not confuse this we use ask Sahaj this is not possible how much does the forced harmonic passage first the Akkadian frequency he's not equal to school can work and necessarily property secret to my heart to drop rwanda's ok you graphic she could have passed is costly to past in the crotch she could slow it has to do this slowly okay the system has no choice as to what are Michael Campbell musasa ap is dependent on detractors what doesn't exist ike's to be dry you know the system can have its own reaction I has reaction imagine you know a child us you put a shot past you don't you hung dude on a charge moved toward like this you push back rocker you wait for him to come over and you can push channels forward with great aptitude he comes back pushkin even higher compact pushing higher now if the charts we like this my hand South my kids my keys you don't want to grab his seat and shake a leg is because you know between Seoul is not gonna be very effective in terms of increase in after to the Muslim so that tells you system true he has no choice as to what frequency you want us lead pot don't have a choice whether it lacks the frequency are not my liking the frequency Miami is not you are using a frequency drum for DC which he's close to natural frequency the oscillations system %ah *** like that match you push also my so that pretty much at every moment you are pushing it in the right direction in the system moves this week you push it is when my symbols back you push it back two pieces you always dream party were to desist are you not resist the bush you don't you go on sequences incident witnesses this week you try to push it back unique I I so we knew our applying a driving force with the session same frequency as the scoop you were an and we not gonna natural frequency natural frequency only got mascots cannot just talents heart his natural frequency eavesdropping frequency his calls to africans then what happens is that the system moves in C withdrawn first drums always push the system in the correct direction given it positive work and that dramatically increases decathlete into the %ah solution so that if if we say this is a situation where driving forces you too fast or too slow donkey very large am pretty right and then when a driving force is just right I in frequency you can give it a very large capital like these see like this I so what you have here is a simple is this is a simple harmonic variation at a function time so it's like sign on the kitty plus *** but there is a amputee this amplitude does not change with time because remember this for snack pack was the do the same amount work over one cycle positive iniki's the amplitude does not change but it does deep and hard rock britain's okay if you are be destroyed pretty his calls to the National pretty easy he will be March because you doing hun amor and the only got is too fast to slow computers home we cannot the oscillation were not happy March County this he's Phys this is not a representation absolute steady state singe now you can find with act eighties we have to do his party seem to be sick expressed into DC question a force on you know you up to to Drews here you have two hundred he ansel as long you can match it I'm not gonna be market with that at just mathematics my to its liberties by soon dual we wanna know he's the physics and that is what is she of be oMG ok we now the plot be as a function home thirst my free can see that stands out an *** on africans okay national grid system if only got his copies host on cannot the awesome our chapter 22 only has two large too small response the system will be weak you would you get a small smaller County so you can expect something like this ok TC around back at a busy time for his imagine that a pretty greatest amplitude are and when you and yet not one for kids to March too small you have small cavity this phenomena Hitchcock Presents and this is called president speak ok I you control two possibilities let's say you have a car like this units there is a registered but is not pronounced how its theory shallow be situation different situation hacked gonna ICM registry good as the first one but compete is the three pronounced why is there a difference well that depends on a mathematical details of be only we're not which may not get but physically we can understand why there's a difference %uh if you look at these three forces ok out this force simply is responsible for providing the hot africans K of this force is the driving force its frequency on will determine whether this is my large house legends hostage wanna force see that ease the reasons the resistance forces almonds opposite to the direction of abortion all slow things down doesn't care about whether to frequencies to how to know no it doesn't actor care out my country tis too hard to convince africans soon soup is drying something resistance force he's insensitive to freaks okay when B is small then a pause when I don't worry about it too much soon and you happy very pronounced resins peak like this because after all this force this term have homemade in it monica mattos not very large house legends course when B is very large then the system is barred by the resistance force you we can't have much about in next which is %uh which is driving force so mad at the current will not be here would not be very sensitive to frequencies and get a flock okay %uh it is clear pack in many situations we want aftershock response like that now this situation in decisions is system is highly selected hockey fevers this frequency the reprint house in regions almost home africans hockey become application are on the situation well this is very discriminating cars okay if a person with this because the project everything else are can you think of a situation we want %ah the system to be highly selective well why not how how I revealed soon okay martel engine you know our Paris fields all have dreams all have frequencies I'd how do you to you really should have you slept only this particular frequency okay reject of africans Yuma listen to to reduce this isn'ta Dr how you do not simple which you do is you too real soon at you just natural frequency after receiving circuitry of course is natural hostage the IDS you are actually adjusting we cannot now if all my cannot matches my preconceived my particular radio station then be a signal from acquisition witnessed very large response numeracy and neighboring stages with slightly different because you be rejected you see so when you buy review on TV set you want to hear that is highly selective so that you can pick up whatever frequency hundred-channel you wanna listen to watch ok you know I as a junior because and you listen to how much these decent hockey see response is not active from Sol patties to suck *** residence and residents widespread applications in all areas of Physics are he sometimes good things come sometime acting lacking cases medium tuner presence is good we long dresses to be very pronounced on the draft class can also cause disastrous ok for example %ah you happy bridge you have a bridge the bridge be and mechanical structure has gone rat as frequencies it may not have a single frequency become structures hockey you can have several natural freaks without me it means that if evil force GB if actual was crap stockbridge Sheikh Syed was not particular frequency which is close to the natural ricans that's because the bridge to sheik mom because threatens right so when you design a bridge you wanna find a way to counter to predict these natural frequencies you want these natural frequencies to be far away from natural oscillation current on the bridge for example weekend okie in a week bowl %ah on off on off is like a regional rocking math was rocking the bridge but Eve the frequency due to due to the blowing in the wind gets close to natural frequencies the bridge itself back and cost your march ha solution in the bridge second even class to bridge and I'm not just seen at the american Honda City this has actually happened in tax he sees the most exact okay 1940 term weeds asap torsional vibration common Harris bridge him wash %ah this is %ah page for 474 back cause the bridge to collapse when he designed the bridge the young wanna natural frequencies happen to be around frequency which twin in mocks the bridge happen once the statue impact is %uh sup you wanna make sure how bridge you don't is on a bridge to have a frequency data as close to to his ranch