Lecture 19: The Leung Accelerometer

Next: Room at the Bottom

You're in a car doing 50 kph. You approach a cross street. You have a green light, but as you reach the intersection, a truck pursued by a police cruiser crosses in front of you. There's no time to swerve, no time to brake, and you run head-on into the side of the truck. Almost instantaneously, the car's airbags inflate and hold you securely in your seat. As the car comes to a stop, you realise with surprise that you're alive and unharmed.

How does the airbag know it's time to inflate? It doesn't inflate when you step on the brakes or when you slam a door. It's set up to inflate when the car undergoes a very rapid acceleration or deceleration -- an acceleration too high to be caused by anything short of a high-velocity impact. So we need some method of measuring acceleration. It needs to be fast, cheap, and accurate enough to distinguish braking from crashing.

In the first of the two lectures on this topic, Dr Albert Leung, inventor of the hot-air accelerometer, gave a power-point presentation on the development of his invention. This presentation is given here as a .ppt attachment:

The Hot-Air Accelerometer.

For some applications, an accelerometer can be built consisting of a `proof mass' suspended by a low-friction pivot. The motion of this proof mass can be detected by letting it, or part of the pivot, close an electrical circuit. To inflate an airbag, though, we need both a very fast response and a high threshold. These requirements are difficult to realise simulateneously.

Currently, airbags are inflated on a signal from an accelerometer that looks like a pair of intermeshed combs. The two combs have a certain electrical capacitance that changes when the comb teeth are bent under an imposed acceleration. The only problem with this device is that it requires high tolerances, and is therefore somewhat expensive. (Since all new cars have airbags, and about ten million cars are sold every year, saving twenty-five cents per car adds up to an annual saving of two and a half million dollars.)

In thinking about accelerometer design, it's helpful to get an intuitive grasp of how accelerations behave. Curiously enough, the theory of General Relativity can be applied here. In deriving that theory, Einstein noted that it's impossible for an experimenter in an elevator to tell if the elevator is resting on the surface of Earth or acclerating upwards at 9.81 metres per second per second. Thus, an accelerometer is also a gravity detector. This is useful mainly as an aid to imagination: we've had more experience of being in a gravity field than of being uniformly accelerated. To illustrate this, suppose you're in a car, holding a helium balloon on a string. Suddenly the brakes are applied. Does the balloon move towards the front of the car or the back?

It is at this point that Professor Leung enters. His design for an accelerometer doesn't look like anything seen before. He has three thin wires stretched across a cavity. The middle wire gets hot, and he measures the resistance of the other two wires. His idea is that a bubble of hot air will form around the middle wire; then, when the apparatus is accelerated, the bubble of hot air will be displaced sideways, and the relative temperatures of the two side wires will change -- one will get hotter, the other will get colder.

Albert asked me if this would work. I did a simple analysis and decided it wouldn't, because of the `sticky air' problem I described last week -- the hot air bubble will move very slowly, because viscous forces are strong, and it will cool quickly, because its mass is small and its surface area is high. He went ahead and built it anyway.

The techniques he used to build it were described by Ash in last week's lecture. Briefly, he used techniques similar to those used in chip manufacture -- laying down thin films of substances impervious to a solvent, then dissolving away the unprotected silicon. For acceleration, he used the Earth's gravity. This can be adjusted experimentally by rotating the device from the horizontal to the vertical. His results showed that it did work. The only question was, why? Granted, he had a working device and he had an explanation, but that doesn't prove that the device works for the reasons given in the explanation. How do we know that it's not just the bending of the struts under acceleration, changing their resistance?

One way to test that is to try running with the heater off. Once the heater is off, the accelerometer stops working. But maybe the heat just softens the wires enough to bend, and that's why we only get a signal with the heater on.

At this point I started to think about the problem again. I still couldn't see anything wrong with my analysis, but the device was working. I thought about buoyancy-driven flows. It's well known that they occur -- here's a textbook illustration showing a Schlieren photograph of isotherms around a heated horizontal cylinder -- but the smaller the scale of the apparatus, the harder it should be for a flow to get started. Wasn't Albert's accelerometer too small?

Here's a stick of incense. Its tip diameter is about 400 microns, it's at a few hundred degrees, and clearly there's a buoyancy-driven flow carrying the smoke upwards. If we put detectors above and below the burning tip, obviously the one above would get hotter. So buoyancy-driven flow does occur at close to the scale of the accelerometer.

The equations for buoyancy-driven flow are the Navier-Stokes equations, which as I've mentioned are in general insoluble. At micro-scale, the flows are dominated by viscous forces, so some terms in the equations can be dropped. Even the simplified equations looked daunting, though, so I decided to look for a numerical solution, using the finite-element analysis program FLOTRAN.

Before trying to model a problem, you have to decide what you're going to include and what you're going to leave out. For this problem, some fairly drastic assumptions were needed. Most importantly, I assumed that the cavity could be treated as infinitely wide; this made it possible to build a two-dimensional model, which can be solved much faster than a three-dimensional one. (This assumption was subsequently backed up by some order-of-magnitude analysis.)

With these assumptions, and a suitable finite-element mesh, FLOTRAN was able to solve a trial problem. The output showed that a convection current was set up, and as a result, the hot air bubble was displaced and distorted, giving a temperature differential between the detectors. Repeating the experiment with cavities of different size showed that device sensitivity increased superlinearly with size, and also helped determine the best place to put the detectors. Coincidentally, Dr Leung had already got his detectors in more or less the right place.

So at this point, we were convinced that the accelerometer did work on the buoyancy principle. The next need was to determine what factors affected the device sensitivity and response speed, so that the design could be optimised for particular requirements. Unfortunately, there seemed to be a lot of factors that might influence things: the properties of the fluid in the cavity, including specific heat, conductivity, density, expansion coefficient and viscosity; the size of the cavity; the aspect ratio of the cavity; the size and shape of the heater; and the power supplied to the heater Would the device work better using water in the cavity instead of air? Or would it be better to use a light gas, such as helium?

These questions could be investigated by varying each parameter separately and noting the effects. But this would be a lot of work, either experimentally or using FLOTRAN. Fortunately, we were able to use dimensional analysis, the method hinted at in Lecture 3. By a suitable choice of non-dimensional numbers, it can be shown that the ratio of the temperature difference between the detectors to the temperature difference between the heater and the walls is a function of just two dimensionless groups, the Rayleigh number and the ratio of heater size to cavity size, R. Both experiment and simulation indicated that the measured temperature difference increased linearly with Rayleigh number, up to some bounding value where it started to fall off.

The effect of the ratio R on performance was more difficult to assess. At this point I discovered that the simplified problem I'd been modelling had been solved analytically, about twenty-five years ago. The analytic solution confirmed the linear dependence on Rayleigh number, and gave a horrible equation for dependence on R: an equation that takes several pages just to write out. However, using the symbolic mathematics package Maple, it was possible to evaluate and plot this equation.

The Rayleigh number contains the square of gas density. This shows that we can increase the device sensitivity by pressurising the device, a prediction borne out by experiments conducted in Albert's kitchen.

Further studies on the transient behaviour of the device showed that the characteristic response time increases linearly with gas density; thus there's a design trade-off between increasing the sensitivity and increasing the response speed.

Over the last year, I have been trying another method of increasing the density of the working fluid -- use a liquid instead of a gas! Water is about a thousand times denser than air, and device sensitivity is proportional to the square of density. So a water-filled accelerometer might be a million times as sensitive as an air-filled one.

Water is probably not the best liquid to start with, since it will short out the electronics. Instead, we tried using isopropyl alcohol. Putting the physical properties of isopropanol into the expression for device sensitivity, we predicted sensitivity would increase by a factor of about a thousand. Experimental work by Dr Lin Lin, who was at that time working with me on her PhD, confirmed this. However, we also found that the device response time was increased by a factor of about a hundred.

This suggests that we can scale down the device by a factor of 10 -- this will reduce sensitivity by a factor of 1000 and reduce response time by a factor of 100. The resulting alcohol-filled chip should be as sensitive and as fast as an air-filled chip, but be lighter and cheaper.

The biggest market for the device may be the airbag automotive market. However, there are many other possible applications. By adding some software to the accelerometer, we can integrate once to find device velocity, then integrate again to find device position. This could be part of an innovative mouse, or used as part of a position-sensing virtual reality interface. It could also provide inertial guidance for robotic navigation, for example, in one of John Bird's underwater robots.