# BioMEMS Module 2D – Scaling Laws and Analysis in Micro and Nanosystems

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### BioMEMS Module 2D – Scaling Laws and Analysis in Micro and Nanosystems

alright see where we left off here so like I said we’re gonna finish off this chapter today so last time we ended off talking about the thermal domain we’re basically going through the different domains it’s been a week and a half so it’s our last lecture so just to remind you we’re looking at the different energy domains we’re looking at electronic the electronic domain the fluid flow domain the thermal domain we talked a little bit about diffusion and now we’re going to talk about finish up with a mechanic’s domain today what I want you to be able to do in this module is to be able to analyze these different domains with equivalent circuit models let me show you an example of what we did okay in the in the fluid model okay you can model a microfluidic Channel by using an electrical circuit where a current source represents a flow and these different resistors represent hydraulic resistances okay so that’s one way of modeling a fluid system using an electrical circuit in the you know moving ahead I’m not going to go over all this stuff again in the thermal domain we talked about circuits to model the thermal domain okay in this case you have a current source which represents a heat heat being generated in the system if you recall with the micro bolometer you had a membrane and when infrared light hits this membrane it gets it gets hot so that’s represented by this current source and you have a thermal capacitance here and two thermal resistances here just as a reminder what is this thermal capacitance what does it represent physically stored energy that’s correct so if stored thermal energy is the fact that the object is hot okay it’s going to take some energy to increase the temperature of the object and also the energy stored the heat stored in the object can’t go away instantaneously so the temperature of the object can’t attain change instantaneously just like the voltage across a capacitor cannot change instantaneously so it’s modeled by this capacitance these thermal resistances represent the heat flow paths where this pixel is connected to the substrate through these legs heat can flow through the legs okay so from a purely thermal standpoint this is heating up okay and when that when that membrane is heated up heat flows between the hot object to the cold object the substrate is cold so it flows from the membrane into the cold substrate here like this that heat flow can be represented by currents in this electrical circuit model currents flowing through these two resistors basically model heat flow going from the membrane into the substrate this thermal capacitance models the energy storage of the heat storage in this memory the advantages remember of thermal systems is that these thermal capacitances can be very small so basically objects can be heated up and cooled very quickly at the micro scale all right we finished off with the thermal domain here today we’ll talk about the mechanical domain mm-hmm okay and again these concepts I’m showing you are you know they’re very basic concepts that you’ve probably probably learned in in undergraduate mechanics course even in an advanced high school course but it’s it’s useful for us to understand some of these basic concepts especially those coming from different different domains so what we’re going to get at here is that I also want you to be able to analyze a mechanical system using an equivalent circuit model and understand the scaling benefits of mechanical systems just as we’ve understood the scaling benefits of fluid systems thermal systems and so on alright so in the mechanical domain we can just review a few essential concepts mechanical systems with rigid body mechanics were looking at the displacement and velocity and acceleration of some type of object a very basic system can be modeled by a spring and dashpot we’ll get to that in a second but we’re looking at the movement of a mass all right now the different forces that a mass might experience one is acceleration so let’s now look at some of the details of this let’s say that you have let’s say that

you have a mass here we’re gonna draw a Western accelerometer system looks like by the way does anyone know where accelerometers are the most popular uses of accelerometers right now what are accelerometers what do they measure acceleration yeah and do you know where accelerometers are used a lot the biggest in in the 90s the biggest application for accelerometers were automotive airbags for detecting crashes so when the accelerometer detected a large acceleration acceleration that would trigger the airbag to deploy all right great technology for crash safety but does anyone know what the big use of accelerometers are right now every one of you is using one cell phones yeah exactly they’re used in cell phones for when you pick it up when you rotate when you rotate your cell phone your screens turn right their accelerometers built into certain phones for detecting like how much I’m walking like my my Samsung Galaxy has you know that the app that detects how many steps I’ve taken per day accelerometers are and things like fitbit’s you know that also detect motion they’re everywhere so the basic I’m using accelerometers as an example because it a nice example of a miniaturised system which is very quite commonly used the way that an accelerometer would be modeled this is pretty ugly let me redraw this you have a mass okay and that mass is attached to a cantilever the cantilever has a spring constant given by K all right now this whole thing is in some sort of housing okay and this cantilever beam is the one half of it is attached to the housing itself we’re drawing a very simple simplified diagram of this the the mass is actually made of a material that’s that’s either conductive or it has an electrode on the bottom and like this most of the time the mass itself is just conductive all right so let’s say we have a mass that has an electrode on it and then on this side we also have an electrode and this is connected to some type of circuit and the circuit measures capacitance what’s going to happen to the capacitance here fastened in sizzle to epsilon a divided by D this is the distance D the distance between the place right now let’s say we take this whole master so we’re talking about this entire thing here and we subject that to an acceleration what’s going to happen to this mess mohammet it’ll swing that’s correct it’ll swing it’ll swing up and down that’s correct absolutely exactly right when this thing is exposed to acceleration this mass is going to end up either moving closer to this electrode or farther away that changes this distance D okay and that distance D changes the capacitance and you can have a circuit that measures like practice so you can electrode electrically read out the acceleration because the force on this mass is force is equal to MA all right we’ll go back to the equations here but this is this is what the system looks like now accelerometers are used like I said you have accelerometers in your cell phones you have accelerometers and automotive airbags are there everywhere so in this this is a good example because it demonstrates rigid body mechanics and you have a rigid mass in there that’s attached to something that’s holding on to it that that rigid mass is going to be subjected to acceleration F equals MA right when you

subject it to acceleration this is a mechanical model okay so when let’s say this mass was connected attached to the housing here and this whole device was accelerated in in the upwards direction okay so in the direction of X the acceleration is happening in the positive x-direction so the first force we’re going to talk about is that the force of the uncertain on this mass is equal to mass times acceleration whatever acceleration was being applied to the entire system that’s the first force on this mess the second force well let’s let’s do this one as the second force the second force is that the the mass is attached to a spring okay in this case you had the mass that was connected to some sort of beam element okay these beam elements can be microfabricated all right this beam element is going to resist the mass removing it acts like a spring now with Springs there’s a very simple way that we can model Springs us by using Hookes law force is equal to KX K is a spring constant of the cantilever and X is the displacement okay you imagine what the spring the more you displace it the more force the spring tries to pull back so that’s a simple law Hookes law K is a spring constant so the more rigid the spring is that yeah the more rigid the spring is the higher the spring constant is going to be all right so you have this force that you have the mass here and then you have this resistor element or this is a spring element and that’s represented by K that represents a force exerted on this mass that’s pulling back on it it’s it’s being accelerated in this direction but the force is pulling back in the negative x-direction the third force is the friction force okay now the friction force in this case in the case of a simple cantilever beam is suppose the mass is being accelerated in one direction now let’s say it’s a it’s going in the upwards directions since that’s the example that we showed now there may be air molecules in here okay so if those air molecules are there the mass has to push aside the air molecules right so the air molecules will actually damp the system that’s called a damping coefficient all right so that’s where this force comes in a frictional force or a damping force that force is equal to be V it turns out that that it scales with the velocity the faster the object is moving through the air the more force is going to be exerted on it it’s like a drag force right you imagine a car moving on moving through a highway faster it’s going to more drag forces there’ll be so that’s your frictional force drag force force is equal to B times B so if we look at all these three forces together we have this three system this simple system here the the frictional forces represented by something called a dashpot that’s what this see is just to clarify this C and this B are the same thing so in this mechanical system we have three forces at work one forces acceleration dependent force equals MA another force is velocity dependent and then another force is displacement dependent you can add all those three things up so the force in the upper X direction the acceleration force ma is equal to negative B V minus KX okay this is the force from the the frictional force and then this is the force from the spring all right and you can express that as a differential equation M times d squared X DT X’s X is the displacement plus B DX DT minus KX equal to 0 DX DT is velocity and then you have this this term here so reminding ourselves from our undergraduate mechanics class that this is just a basic differential equation it’s a second-order equation and it can be modeled by this system here very easy to solve these types of differential equations this particular system will give you if you recall it will give you a resonant frequency so the mass can actually vibrate the resonant frequency is a square root of K divided by M and the damping coefficient is given by B divided by 2 times square root K M now we’ll talk a little bit more about this in a second when you solve this system basically you’re looking you’re either looking for the displacement of the the

that’s a displacement in an under damped system from the moment you tap the beam the beam will start to resonate at its resonant frequency so it’ll just basically vibrate okay and in a perfectly under damped system where there’s no this is if there’s no damping at all right that damping coefficient B that we looked at previously if that’s if that doesn’t exist this thing will vibrate indefinitely it’ll continue forever so this is an example of an under damped system and an overdamped system you know it might just look like this where you displace it at time zero but then in Ischl e goes back to zero displacement and so this would be over damped and in the case of a critically damped system you might have some oscillations that eventually die down and until it rests if you want to look more at physics concepts this is another nice page from hyper physics any questions on this stuff this is really hopefully just a reminder of things that you’ve looked at earlier good question so the question is is a goal to have an under damped system if you want the system to recover from whatever perturbation that you gave it the fastest response the system that returns to its initial state the quickest would be the would be the critically damped system see the over damped system it resists the over damped system would be if if there’s a lot of let’s say if there are a lot of air molecules next to the mass if it was a high pressure system those air molecules would resist that thing from moving at all that the problem with over damped systems is that there’s a lot of energy loss it’s actually resisting those air molecules would resist the mass from moving at all okay now a critically damped system is an under damped system is where there were no air molecules at all in which case we were just Asli indefinitely there are certain situations where you would like the system to oscillate indefinitely in the case of resonators you actually want the system to be resonating with without losses and with at least energy input as possible the critically damped system is important when you want to when you want the system to recover from oscillations as quickly as possible so initially there’s some oscillation with those oscillations died down relatively quickly we’ll see a few examples of the resonant system so here’s an example of a micromechanical system the accelerometer that we just talked about so you assume that in as accelerometer as a proof mass that’s attached to a cantilever beam the beam bends on the proof mass experiences acceleration just like I drew earlier in this case the arrows are sort of opposite here the the acceleration is going down in the spring the spring force and the damping force is going in the optimum going up now these this is a practical device as I said these accelerometers are in your accelerometers are in your mobile phones you know you can see these types of devices here you can also see these STMicroelectronics accelerometers in for airbag deployment as I mentioned earlier some of the systems you know these are examples of what the systems look like I drew a very simple diagram in in the notes here all right but sometimes these devices can actually be a little bit more complicated than that the device that I drew here this only detects acceleration along one axis all right the accelerometers that are in your cell phones and some of the sensors even in the automobiles are multi axis sensors so they’re like sense