Introduction to Exponential Decay (Damped Oscillations)

hi guys dr. Cole breath here with an introduction to exponential decay specifically exponential decay in the context of damped oscillations I want to start off looking at an ideal oscillator and what we mean for an oscillator to be ideal well the main approximation that we make here is that the amplitude of this oscillator is constant and it oscillates from minus a to plus a forever but a more realistic model is that of a damped oscillator which starts off at some maximum amplitude and then the extent of the oscillation dies out over time now I want to compare both the ideal oscillators shown at the top with the damped oscillator and take a look at their displacement as a function of time and for the ideal oscillator we see that the amplitude doesn't change the maximum displacement of the oscillator is constant and time represented by this horizontal dotted line and the amplitude is constant as the oscillation continues on forever but here for the damped oscillator we start off at some maximum value a and our oscillation is contained within this decaying envelope here and over time in an infinitely long time the oscillation amplitude goes to zero mathematically we can look at the simple oscillator and see that its position as a function of time is given by and I want to make a quick note here although I'm showing a simple pendulum here in my animation that supplies to any simple harmonic oscillator so we're going to use the general position coordinate X so the position as a function of time is equal to a constant amplitude a times the oscillating piece the cosine of Omega T plus Phi 0 now for the damped oscillator our amplitude is not constant so when you replace this constant amplitude a with a function that varies in time and it's the maximum displacement of our position that decreases in time so we're going to replace this with a function x max of T which describes the extent of the maximum displacement in time and as I show in another video the solution to the differential equation for a damped oscillator requires that this x max of t function be an exponential decay function so here we have x max of t is equal to a which is our initial amplitude times the natural number e to the power of minus bt divided by 2 m and this describes this envelope here that the oscillator moves within when T is equal to zero we have e to the 0 which is 1 so we have that the maximum displacement at time equals 0 is just equal to a which is what we see here on our plot and as T goes to infinity we have e to the power of minus infinity which is 0 so as time stretches out to infinity our maximum to play some displacement goes to 0 so we want to take this envelope which describes the maximum displacement and we want to combine it with our oscillating piece which is still just the cosine of Omega T plus Phi 0 so instead of having a constant amplitude we're gonna have this decaying amplitude so if we multiply these two pieces together we get the kinematic equation for a damped oscillator X of T is equal to our exponential decay piece a times e to the minus BT over 2m times the oscillating piece which is the cosine of Omega T plus Phi 0 so we have a new parameter here in our exponential decay B which is called the damping constant and on the left here I have animations which show different values of the damping constant the top here we have b equals 0 so if we take B equals 0 in our kinematic equation we always have e to the 0 here for this first term which is e to the 0 is 1 so if this exponential term is 1 we recover a times the cosine of Omega T plus Phi 0 which is just our ideal oscillator our undamped oscillator kinematic equation now the second animation of third I am innate animation showed increase it values of the damping constant here we have a modest value of the damping constant here we have a more extreme value of the damping constant and we can see that the value of B determines how quickly the oscillation dies out couple more notes on the damping constant first of all has kind of unusual units kilograms per second and the value of B depends both on the shape of the oscillator and the medium that the oscillator is within so this is like an engineering parameter that can be measured or in some cases calculated and it depends on both the oscillator and the medium that it's within so the shape of the oscillator for example we could say that if a damping constant for a parachute is going to be much bigger than the damping constant for a bullet right a bullet moves to the air quickly and is not as damped as a parachute which has a lot of damping so the shape of the object certainly affects the damping constant and also the medium that it's in so if we were to take these pendula here and we were to have them oscillate in water they're going to be damped more quickly than the same pendulum damped in air so we have that B for an object and water is bigger than B for an object in air in other words the medium matters now these are just kind of extreme examples and B for one particular oscillator and it's medium can be measured specifically for the system at hand so I have a question here then we know that the damping constant affects how rapidly the oscillation dies out but how long does it take for the velocity of the pendulum Bob to reach zero and of course according to our model the Bob takes an infinitely long time to come to a stop so if we want to describe how quickly the oscillation dies out or how quickly the oscillator loses energy we can't talk about how long it takes to come to a stop because that's an infinitely long time so we need to build up another way of talking about how damped something is or how quickly the oscillator loses energy and to do that I want to look specifically at the energy of an oscillator so we're going to move away from pendulum for a moment and we're gonna look at the energy of a mass on a spring and the main reason for this is we just don't have to worry about gravitational energy and it just simplifies the analysis here so in the case of an undamped spring where we have constant amplitude if we evaluate the energy at the turning point or at the amplitude at the endpoints where the velocity is zero first of all the energy or the position of an undamped spring has a it has a constant amplitude a as we've discussed already and given that the total energy is just equal to 1/2 K times the amplitude square so we evaluated the turning point the turning point is in the same place every time right the turning point is at the amplitude a and so the total energy of an undamped oscillator is equal to 1/2 K a squared now if we compare that to the damp situation where we start off with a maximum amplitude a but the extent of the oscillation dies out over time if we want to calculate the damped oscillator first of all we'll refer to kinematic equation for the position here and we'll see instead of having a constant amplitude we have this amplitude which decays over time and so in other words the maximum displacement of the amplitude reduces over time and our x max of T is given by this exponential piece and if we want to calculate the total energy of this oscillator well the total energy is not constant right the dance swing spring is losing energy to its environment because of the damping because of the friction because of the air resistance whatever is causing this oscillation to decay energy is leaving the oscillator and moving into the environment so the energy is not going to be a constant instead our total energy is going to be a function of time and so here we have our total energy as a function of time is equal to one-half K and instead of using the amplitude a we're going to use X max of T in here because it starts off with the total energy being equal to one-half K a squared but as the maximum displacement decreases the total energy of the system decreases as well so substituting in X Max of T for a we get that the total energy is equal to one-half K times our exponential decay piece squared and we're gonna do a little more analysis on this equation the first thing we want to do is distribute this squared in here so we're gonna get one half ka squared times e to the minus 2b T over 2m and will note that the twos cancel here in this case and now we can rearrange our terms a little bit and we want to analyze this a little bit more so out in front we have 1/2 K a squared these are all constants and this is the piece that changes in time and causes our energy to decrease in time so I want to evaluate that a little bit more this is really where we get into some of the details about exponential functions so the first thing is if we have an exponential function here like e to the minus BM over T B divided by M times T anytime we have an exponential the argument of the exponential or the exponent of this exponential can't have units it doesn't make any sense to talk about eetu the power of minus two kilograms that just doesn't even make sense we don't know how to raise something to the power of a kilogram so what that means is that the argument of the exponential function must be a dimensionless number with no okay so in this case that means that mm M divided by B must have units of time because T divided by this quantity it must be dimensionless we know that T has units of seconds so T divided by this quantity here must be dimensionless so B if T is multiplied by B over m then M over B must have units of seconds in order for this argument to be dimensionless and so we're going to make a definition now we're going to define tau as being equal to M over B and tau is measured in seconds it's a length of time and we're defining tau in this way so now we can rewrite our exponential function using tau which is called the time constant we can rewrite our exponential function is e to the minus T divided by tau so now our energy function becomes even more simplified we have our energy or total energy of our oscillator as a function of time is these constants out in front times e to the minus T over tau and we can rewrite these constants as just the initial energy of the oscillator and this brings us to our totally simplified form that the total energy of the oscillator is the initial energy times the exponential e to the minus T divided by the time constant tau so why do we do this what does this all mean well this is how we're going to talk about how quickly something decays is based on this value of tau so tau combines the damping constant we've already seen how the damping constant affects how quickly the oscillation dies out and it depends on the mass so the larger the mass the longer it takes for the oscillation to die out and the larger the damping constant the more quickly the oscillation dies out so for our example that we've shown here I want to highlight a special point and the special point here is shown on the plot and the special point is where T is equal to tau so after one time constant of time has elapsed and a time constant is equal to M over B and this is evaluatable as a certain number of seconds right and so when that certain number of seconds has elapsed we can evaluate our total energy and we get our total energy is equal to our initial energy times e to the minus T over tau but here we're taking T is equal to tau seconds so after Tao seconds have elapsed our total energy is equal to e 0 times e to the minus 1 e to the minus 1 is equal to 0.37 and so what we can say is that after 1 time constant which is tau seconds has elapsed the oscillators energy is reduced to 37% of its original value and we can follow this sort of procedure again and we can consider what happens when the time is equal to 2 time constants so here more time has elapsed not just tau seconds but now we've gone by 2 tau seconds and our energy is significantly decreased from its initial value of v-0 and if we evaluate this expression when T is equal to 2 tau we get the total energy is equal to our initial energy times e to the minus 2 tau over tau which is equal to e to the minus 2 which is equal to e times our initial energy times 0.135 so in other words after 2 time constants which is 2 tau seconds have elapsed the oscillators energy is reduced to 13.5 percent of its original value so we're turning back to our expression here for the energy we note that it takes one time constant to reduce an oscillators energy to 37% of its original value but we did some manipulation to get to the energy expression and I want to look back at our original kinematic expression for the displacement or the amplitude and the displacement of the amplitude was equal to AE to the minus BT over 2m and here we have M over B is our time constant here so this simplifies to a times e to the minus T divided by 2 tau and so we can summarize this by saying that it takes two time constants to reduce and oscillators amplitude to 37% of its original value so tau is the amount of time it takes to reduce the oscillators energy to 37% of its original value but tau can also show up in our kinematic equation for the position in other words the amplitude also can be written in terms of the time constant tau but in the case of the amplitude it takes two time constants to reduce the oscillators value to thirty-seven percent of its answer thirty-seven percent of its original value for the amplitude and it takes one time constant to reduce the oscillators energy to thirty-seven percent of its original values so I just highlight this because this is a confusing distinction you know we just went through this work to define tau then why on earth does this guy depend on two tau well we often deal with the amplitude when working problems but the reason that tau is defined the way it is because is because of its relationship to the energy of the oscillator so in summary we have that tau which is called the Tomica time constant is equal to M divided by B and the time constant tau is a convenient way to characterize how fast an oscillation decays in seconds so if we just want to know like how damp the system is we can compare the time constants from one system to another and it will tell us how quickly the system decays because we know it takes an infinitely long time for the system to completely stop but we can consider the system completely stopped at least approximately after many time constants that have elapsed for example after time 10 time constants the amplitude of an oscillator is at 0.6% of its original value so the time constant just gives us a sense of how fast the oscillator decays so I just wanted to provide an introduction to exponential decay and this notion of the time constant and how it fits in with the energy and also how exponential decaying oscillators or damped oscillators compare to their ideal counterparts I hope you enjoyed this presentation and I'll see you in the next one

Channel: Christopher Culbreath

Views: 4,694

Rating: 5 out of 5

Keywords:

Id: qgTQO2LqxaM

Length: 14min 58sec (898 seconds)

Published: Sun Apr 16 2017