Statistical Analysis of Go/NoGo Gages

Background and Theory of Go/NoGo Gages

The use of Go/NoGo gages is quite prevalent in any type of manufacturing environment. Essentially Go/NoGo gages are representations of the respective maximum and least material conditions (MMC/LMC) of a feature. These physical representations of the MMC/LMC are then mated with the feature to assess its quality.

For example, if you think about a simple plug gage, the Go side must fit into the hole to pass. If the hole is too small (exceeds MMC) the Go gage will not fit and the part will fail. Conversely, the NoGo gage must not fit in the hole to pass. If the hole is too large (exceeds LMC) the NoGo gage will fit and the part will fail.

In theory, this principle seems quite straightforward. However the rose colored glasses are shattered when it is revealed that the gages, just as the features they intend to assess, are imperfect. Thus the gages themselves must then have tolerances. This tolerance, known as a gagemaker’s tolerance, is typically around 10% of the feature tolerance. The 10% is then split evenly among the Go/NoGo gages. For example, on a 2.000+/-.005 dimension, the gagemaker tolerance would be .001, and the tolerance for each the Go and NoGo gage would be .0005.

The only remaining question is the bias of the tolerance. The typical bias selected is the most conservative; i.e. it is more conservative to risk rejecting good product than to risk accepting bad product. This will result in the Go gage tolerance range being just within the feature tolerance range, and the NoGo gage tolerance range being just outside the feature tolerance range.

So just as your product has a yield, the gages themselves will also have a yield expressed as False Rejections/Total Rejections. Obviously if the gage is your only way to assess the feature, you will never identify any False Rejections, so a 3rd party measurement some orders of magnitude more accurate than the gage would be needed to make this determination.

Problem Statement

Gages are typically designed/selected based on this 10% rule. I haven’t really found any background on where this originated, or if there is any statistical theory justifying it. From what I have found, it is more of an empirical finding that the 10% rule maximizes yield while keeping gage costs manageable.

What I am interested in is the statistical theory behind this gaging process. In other words, is it possible to analytically calculate the probability of these False Rejections? If this were possible, one could better justify their gage design/selection, or possibly improve yield or reduce gage costs for a given application with knowledge about the distribution(s).

Assumptions

For this analysis, I have made the following assumptions.

• All random variables are normally distributed.

• The means of the gages’ random variables are at the center of their respective tolerance ranges.

  • I’m using this assumption in lieu of having any actual information on the distributions of gages.

• All gages are made with 6Sigma manufacturing processes.

  • I’m using this assumption in lieu of having any actual information on the distributions of gages.

Analysis

Let A be a random variable describing the dimension of the feature. Let G be a random variable describing the dimension of the Go gage. Let N be a random variable describing the dimension of the NoGo gage.

For an internal feature, the Go gage passes when it fits inside the feature, i.e. A>G. The NoGo gage passes when it does not fit inside the feature, i.e. A<N. Thus we obtain the following:

P(Pass_Go) = P(A>G)

P(Pass_NoGo) = P(A<N)

The probability P(A>G) may be obtained by double integrating the joint PDF, F_AG(a,g). Because the feature dimension and the gage dimension are independent, the joint PDF may be expressed as the product of the individual PDFs.

F_AG(a,g) = F_A(a) * F_G(g)

Next we must double integrate this expression. Before integrating, we must understand the space we are integrating in. This joint PDF occupies (a,g) space. Each point on the (a,g) plane represents a combination of the dimension of the feature (a) and the dimension of the gage (g). We are only interested in the case where a>g, and both a>0 and g>0. If a is taken to be the horizontal axis and g the vertical axis, this corresponds to the area under the curve a = g (Fig. 1).

Figure 1

Matlab Script

I wrote a simple Matlab script to visualize the joint PDF. The next step is determining how to evaluate the double integral over this shaded region. I don’t think there will be an analytical solution, so barring that option the next logical alternative is numerical integration. However I’m not sure how numerical integration works over semi-infinite regions such as this one.

Github Link

How to Weigh Your Head: Pt. 1

Background

During my time at Boston University, I was a Teaching Assistant for the Product Design course required for all mechanical engineering majors. In contrast to our other engineering courses which taught scientific principles, this course focused on exposing students to structured ideation methods, materials/design selection, functional analysis and decomposition, and how to generate effective engineering problem statements.

The Question

The professor, Greg Blonder, posed a question to the class during one lecture as an exercise:

How could you determine the weight of someone’s head, without removing it?

This question seemed underwhelming at first, but the as the groups discussed their ideas, they began to realize a solution was not trivial. Their solutions roughly fell into two categories:

1. F = ma, f = (k/m)^(1/2)

   These involved resting the person’s head on a scale or measuring acceleration response to a known force. The main problems with these approaches are that one must make many assumptions for the reaction forces and moments/damping ratios/spring rates in a person’s neck. Additionally, some of these procedures may be harmful/intrusive to the subject, as they involved vibrating the subject by their neck, whacking their head with an instrumented hammer, rapidly rotating the subject, and administering muscle relaxants to the neck muscles. However depending on your definition of success, these could be interpreted as viable solutions. The question was open ended…

2. Fluid Displacement/Buoyancy

   These methods generally involved submerging a person in water up to their neck, or holding them upside down so only their head is underwater. All of these methods fall victim to the common fallacy that buoyant force informs the weight/mass of the submerged object. This is of course not true; the buoyant force is simply equal to the weight of the water displaced and does not depend on the mass of the submerged object. The displacement measurement from this method informs the volume of the head, and combined with density one can obtain mass. However one must either devise a way to measure the average density of someone’s head, or make an assumption. Again depending on the definition of success, assuming the average density and simply measuring volume could be a viable solution for someone’s application.

When the professor revealed his solution, it became clear that class had overlooked one key idea: remote sensing. With remote sensing techniques, it is possible to measure properties of matter without making physical contact. These technologies work by analyzing the EM or acoustic signals propagated by the test article to inform its properties. This problem is in a way already solved by the various medical imaging technologies such as CT scans, MRI, and ultrasound. Once the subject is scanned, the only remaining question is debating where the head ends and the neck begins. But I believe that is outside the scope of this exercise.

Professor Blonder stated that it would be theoretically possible to apply the principles of remote sensing/medical scanners to a more practical, ‘backyard’ solution. His solution, informed by the principles of CT scanning, was to ‘scan’ the subject by placing them on a balance beam at various positions w.r.t the fulcrum, and recording the mass of the weight required to balance the beam at each position. The resulting system of equations could be solved to determine a mass per length vs. length profile of the subject, and by integrating/summing over the length corresponding to the location of the head, the mass of the head may be determined. This is as much detail as he gave. He also stated that one could perform multiple scans at various angles and transform the data into an 2D density map. This is of course the working principle of a CT scanner.

The Project

I found myself still thinking about this exercise years after graduating, and decided to try and pick up where the professor left off. The goal of this project is to assess the feasibility of the method described above for determining the mass of a person’s head. I’ve done some preliminary research on this topic (mostly reading about the principles of Computed Tomography and writing way too many mass balance equations) and have identified some key objectives/stepping stones:

1. Solving the System of Equations for a Single Scan

   This method begins by modeling the human body as a rectangle divided into n equally sized slices of uniform density. Because the density of each slice is uniform, its CG is at it’s midpoint, thus the CG location for each slice is known. The mass/density of each slice is still unknown. This model makes sense if you imagine the limiting case as n approaches infinity; the slices are so thin that their density will essentially be uniform. The ‘body’ is then placed on the balance beam at a known position and a known mass is added to the opposite side at a known position to balance the moments. Summing the moments on each side of the beam, we obtain an equation with n unknowns (the masses of each slice). Thus if you repeat this measurement n times at n unique positions, we end up with a system of n equations with n unknowns. So it seems the problem is solved.

   However upon further thought, I recalled my linear algebra professor’s musings on linear independence and matrix invertability. Reexamining the model, it occurred to me that the equations generated may all be linearly dependent, as the initial equation is just shifted incrementally to create the subsequent equations. The first task will be to determine whether or not this is true. If the equations are linearly dependent, then the matrix cannot be inverted, and Ax = b cannot be solved. Based on a simple hand calc, the method works for a 2x2 matrix, but I’m not sure about larger matrices. Looks like I will have to dust off my linear algebra textbook.

2. Applying the Inverse Radon Transform to Produce an Image

   Assuming the matrix is invertible and the system of equations can be solved, I will write a Matlab or Python script to scan a virtual object at various angles using the scan method developed in objective #1. These scans will then be lined up to construct a sinogram. Each scan represents one column of a sinogram. The Inverse Radon Transform (IRT) will then applied to the sinogram to obtain an image. A filter must be applied in fourier image space to compensate for the center of the image being oversampled.

   I am fairly certain most image processing libraries will have built in functions for the IRT and fourier image space filtering, but I’m not sure about a function to assemble a sinogram from a series of data from multiple scans. I may have to write that function myself.

3. Evaluating the Success of the Solution

   Lastly, I will attempt to estimate uncertainties for a theoretical measurement device, and calculate the theoretical resolution and accuracy of the device.

   Off the top of my head, some of the sources of uncertainty/error will be friction in the fulcrum, position measurement of the body and the balance masses, mass measurement of the balance masses, nonuniform density of the beam and the balance masses, movement of the subject (especially during multiple scans at various angles), and scan angle measurement. There will also be errors from the IRT and image space filter but I’m not yet sure how to quantify those. I suspect it will be quite challenging to figure out how to propagate the base uncertainties of mass and position all the way through the IRT and filtering steps.

   Assuming I’m able to propagate these uncertainties, I will also try to calculate the resolution of such a device by propagating the resolutions of various COTS methods for measuring position and mass. This will let us evaluate the success of this solution and allow for comparison to other solutions.

Stay tuned! More to follow.

Illusionist's Heart Locket Pt. 7

This weekend was our graduation, and with everyone's family in town I took my girlfriend's sister to the machine shop to teach her some machining. She helped make part of the locket I'm making for my girlfriend. We had to cut shallow flat bottomed holes in the barrels for the pictures to go in. Because the barrels have a semicircular geometry, they could not be mounted on the table using a regular vise or any of that hardware. So we made a custom fixture to hold the barrel for the hole cutting operation.

We used a 1/2" ballmill to cut a groove in a square piece of aluminum so that the 1/2" barrel would fit snug in the groove. We then superglued the barrel in the groove and clamped the whole thing in the vise to drill the hole .050" deep with a 3/8" flat end mill. We then heated it up with a propane torch to release the superglue bond and used some acetone to get rid of any residue. Superglue is a very easy way to fix parts that would otherwise require the manufacture of a complex fixture or jig. I don't know if I would trust it for precision machine work though.

We also made some spare parts for our robot on the lasercutter, as we are taking it to ConEdison on Tuesday to present to them and we want to make sure it works! 

Illusionist's Heart Locket Pt. 6

Today I made another piece for the Illusionist's Heart Locket. Now that classes are over I have free time to work on personal projects, which is a relaxing way to end the school year. The purpose of this piece is to punch the pictures that will go into the locket at the correct size. A regular hole punch can't be used as it is much too small. Instead I made a custom hole punch to get the correct size photo.

The Die

The die is a block of 6061 aluminum with a hole drilled and reamed to .376" and a slot cut with a slitting saw at a 15 degree angle. The angle allows the punch to cut through the paper gradually rather than all at once, reducing the force required by the user. 

The Punch

The punch is a cylindrical piece of tool steel turned to .375" on one side and .50" on the other. The .375" side cuts the photo, while the other side is larger to stop the punch from going all the way through and also for the user to grip. Now that I know this works, I can harden the steel. The sharp edge from turning is already starting to degrade even though I've only been cutting paper...

MATLAB Project: Membrane Vibration

I was able to get the membrane vibration to work for a rectangular membrane. The equations are quite different for a circular membrane, so I'll need to write a new function for that. I'm still working on implementing this function into the GUI.

Fundamental Mode

p = 1, q =2

p = 2, q =2

p = 4, q =2

p = 3, q =3

MATLAB Project: String Vibration

We recently covered vibrations of strings in my mechanical vibrations class. I decided to try and make a GUI to animate the vibration of the string. There are lots of options for changing the string properties and simulation parameters, as well as subjecting the string to various end conditions and combinations of initial conditions.

I hope to expand this simulation to animate 2-D membranes as well. It would be very cool to simulate how a drum head or the body of a violin vibrates in MATLAB.

MATLAB Four-Bar Linkage Design

The Product Design class I am a TA for had an assignment to design a four-bar mechanism to move a lid on a shipping container up and over so that it rested on the top of the container during loading. There was extra credit awarded to those who could analytically arrive at a design using MATLAB (or some other language). I made my own solution for fun and to check student's solutions. I tried to get it to plot each snapshot (in magenta) but it was not correct and I didn't have time to figure it out. 

So, here's how to design a typical four-bar linkage:

1. Define the desired locations of your coupler. You define two or three locations you want the coupler to travel through. My code only works with three locations.

2. Find the locus of points that is equidistant from where the crank connects to the coupler in each position. If two coupler positions are defined, the locus will be an infinite line (think of the intersection of two circles; connecting the two points forms a line). If three coupler positions are defined, the locus collapses to a single point. This point also happens to be the circumcenter of the triangle formed by the three vertices the bearing must be equidistant from. The bearing must be placed on a point on the locus. If the locus is a single point, your job is done. If the locus is a line, you must select a point on that line to be your bearing location.

3. Repeat step 2 for the points where the rocker connects to the coupler.

4. Once you have the bearing locations for the crank and the rocker, you can easily calculate the lengths of each link.

Illusionist's Locket Pt. 5

Quick update or my locket project. The pin is complete, and to spec. I decided to try using a corner rounding tool to give the locket more refined appearance. They are notoriously difficult to setup in the CNC machine, so I did a test run on a plastic part and on 2 aluminum parts to make sure I got it right before machining into my expensive copper. I also tested the drilling operations on the barrel hole. I originally made this feature with a ballnose mill on the CNC, but the finish was very poor and I would have to ream it anyway to improve the finish. So I decided I might as well drill and ream it from the get-go. 

To do this, I simply clamped two pieces together in the vice, using a parallel bar over the top of the vice to keep the faces normal to the axis of the mill spindle. Then I used an edge finder to locate the hole locations and drilled, reamed and countersunk the holes. 

After confirming all of these things could be done, I decided I was ready to begin making the final product from my copper plate. The feeds and speeds for copper are very slow, and it took almost 2 hours just to make these four little parts. The quality of two of the parts was acceptable, however the other two had gouges in them from the .25" end mill. I think that it was caused by two of the parts being mirrors of the other. For some reason, when coming around the parts the other way, the end mill got sucked into the part, causing those gouges. I may need to remake two of those parts. 

Illusionist's Locket Pt. 4

I did some work on the locket today. The first thing I did was I marked the sides of the magnets with a sharpie so I could distinguish between the N and S poles. To do this, I just stacked them all up and sharpie-d the tops one by one. 

The next thing I did was I heat treated a pin I turned from O1 tool steel. I used the furnace to gradually heat it to 1450F and let it soak for 15min. I then quenched it in oil. The subsequent hardness test shows a hardness of almost C59 Rockwell. It started out as B96. The spec sheet says it can be hardened up to C64, but I am satisfied with C59. This just means I don't have to temper it back down again. Next I'm going to turn down the diameter a bit more with a tungsten-carbide tool and finish it with a grindstone on the lathe. I would use a cylindrical grinding method, but we don't really have that capability at the shop. 

Experimenting With Copper Patinas

When debating what material to make the locket out of, I learned of an interesting technique used by metal artists/jewelers to give their work pieces interesting colors. By heating the work, you are speeding up the chemical corrosion of the surface of the copper. You can also submerge the work in chemical solutions to achieve the desired reaction. Using a torch requires less overhead, so I'm going to see if I can get this to work. 

The main problem with flame patination is creating/speeding up unwanted chemical reactions that may produce visual effects that interfere with the one you want. I'm still doing research to try and understand exactly what the red patina is, and how it's different than the black flaky firescale, which, as I understand, is also a type of oxidation reaction. 

I tried making a traditional boric acid flux. Typically, the powdered boric acid crystals are dissolved in denatured alcohol or methanol to form a saturated solution. The work is then coated in this solution, and when the alcohol evaporates/burns off, a thin layer of boric acid crystals remains on the surface of the work. I didn't realize that boric acid was different than borax, and so the flux wasn't very effective, and heating the borax turned it into a weak brittle glass. Boric acid can be easily made from borax simply by reacting it with hydrochloric acid, so I may try that tomorrow. 

Even without using the proper flux, I was able to get some decent purple-ish red colors. I will buff them tomorrow and see how they look, and try the flux as well. Then I'll need to figure out the best way to clear-coat them to protect the patina. 

A Hard Day's Work

I love working with my hands and making things. At the end of the day, I find it very satisfying to be able to look at the things I've made and say "this is what I did today".

What I did:

1. Finished assembling the scaled circuit breaker model for our senior design project. This involved gluing up the sides of the box, and screwing on the wheels and the rollers. We did not have a thin enough wrench to tighten the roller between the two surfaces, so I had to make one.

2. Began prototyping the device to reduce torsion in coils of optical fibers. I used the laser cutter with 1/4" acrylic. I designed the part so that I could prototype it in 1/4" layers, stacking and gluing together different profiles to get the correct geometry of the final part. The part on the bottom is composed of just two layers. The part on the top (with the spiraling slots) is a single layer part.

Isolating Pure Bending in Optical Fibers

One of the grad students from the nanophotonics lab I developed the microbend apparatus for contacted me regarding a question he had. He was trying to isolate bending and torsional loads in optical fibers to investigate their effects. Introducing pure bending in a fiber is difficult because the fibers have very little torsional stiffness. It's easy to eliminate bends to a twisted fiber by simply adding a bit of tension to the ends. But how do you eliminate twist? There is so little stiffness that you cannot rely on the restoring torque to eliminate small twists in the fiber. The current solution is to coil the fiber very loosely (coiling is just one very long continuous bend) whatever twist is inadvertently introduced by the act of coiling will distribute itself along the length of the entire fiber. In contrast, in fiber optic gyroscope winding, because the coils are so tight, the minor twists in the fiber get trapped between turns of the coil, leaving behind localized residual torsions in the fiber. 

The downside to a loose coil is that it is very hard to get each turn in the coil to the same diameter. A constant bending radius is required for accurate measurements. We decided to use an iris diaphragm mechanism to control the diameter of 6 pins, which the fiber will loosely be wrapped around as guides. This way, a very minimal amount of tension could be used to make sure the coils are all the same diameter, and the diameter can be adjusted using the iris mechanism. To go along with this, it would be helpful to have a limited slip fiber clamp and constant force spring loaded spool so the diameter could be adjusted either direction without having to manually pick up slack in the fiber or loosen the coils to reduce the tension. 

One other alternative to this 'loose coil' solution is to actively measure and correct the fiber twist. However, this method seems overly complex. It really depends on how much twist is tolerable. Once that is quantified, we will be able to determine what type of solution is required. I have a feeling the tolerance will not be so tight as to necessitate some type of active system. But I wonder if there is a better solution for an active system? Perhaps if instead of wrapping the fiber around pins, we could wrap it around an expanding cylinder controlled by the iris mechanism. But I have no idea how that could work. Time to start brainstorming...

Illusionist's Locket Pt. 3

I machined the first parts of a prototype today. I used a 3/8" ball-mill to get the central channel for the barrel pivot to rest in. I'm trying to see if there is a better way to do this, as the ballmill leaves a circular edge when I'd rather have a flat face. Also, slight errors in the depth of the ballmill correspond to the channel for the barrel not being cylindrical. There are still many improvements to be made!

Illusionist's Locket Pt. 2

Today I was able to reverse engineer the mechanism that switches the photos in the locket depending on the configuration. The key is having a two-part barrel that the halves pivot about. The two parts of the barrel are not attached, allowing them to separate when the locket is opened. Each half of the barrel is fixed to one of the movable heart pieces, allowing the other heart pieces to rotate around them, and allowing the barrel halves to slide over each other. 

The next step is to incorporate this mechanism into my existing design. I will probably need a few more magnets to hold the locket closed. I will also modify some of the geometry of the barrels and the barrel slots so they are machinable. 

Illusionist's Locket

This is a present I plan on making for my girlfriend for Valentine's Day. It is a metal heart made of two halves that are connected with a dowel pin and magnets. When the two halves are arranged in the heart configuration, the magnets attract each other and so the configuration is stable. When you twist the halves apart, the halves take the form of an oval. With the magnets' positions switched, the halves repel each other, allowing the user to separate the halves and gain access to the secret pockets inside each half. The secret pockets can be used to small rings, a small note, etc.

I've also decided to take advantage of the fact that our initials form a palindrome. If our initials are engraved a certain way on the two halves, on one side of the heart configuration the letters read "MD" (her initials) and on the other side the initials read "DM" (my initials). When rotated to the oval configuration, both sides are the same but are upside down relative to each other. So it reads "DM" or "MD" depending on how you look at it. I like to think of it as being symbolic of our partnership together.

Manufacturing should be relatively simple. The two halves can be CNC'd out of bar stock. I made the angle on each half 45 deg so that any standard shop gauge block could be used to mount the middle face parallel to the table of the mill for drilling the holes and milling the slot. I plan to use a 1/8" steel dowel pin for the center pivot shaft which can be turned on the lathe. Adhesive backed neodymium magnets will be used for retention. If I decide to do the engraving I will need to make some sort of fixture or jig for a flip milling operation on the reverse side of each half of the heart. Hopefully I can make a simple jig with some locating pins mounted on the mill table.

Note: This design is inspired by an Instructable I remembered seeing a while back. The author of the Instructable credits the design to a scene from the movie "The Illusionist". As you can see, the locket in the movie has 2 additional hinges that allow the locket to swing open in both the oval and heart configuration. The "illusion" is that when you open the locket in one configuration, one person's image appears. But when you open the locket in the other configuration, the other person's image appears. And the baffling thing is, it's hard to understand how twisting the locket from the heart to the oval configuration doesn't rip the picture in half. This is a *greatly* simplified version where there really is no illusion. I will use this design as a starting point and hopefully be able to design a locket that has the full functionality as seen in the movie.

Bike Brake Light: Part 1

I've been thinking a lot about developing a brake light for a bicycle. The need for such a device has always been obvious to me, but for those of you who may think otherwise, let me explain. Cyclists communicate with drivers and other road users primarily using hand signals. So if you are stopping, you have to let go with one hand and signal. However, if you are doing an emergency stop, it is both difficult and unsafe to signal - both hands need to be on the handlebars to control the bike during an emergency stop. Unfortunately, that's also the most important time when those behind you need to know you're stopping. The situation is essentially a catch-22: if you need to stop quickly, then you should signal to those behind you; but if you signal to those behind you, you won't be able to stop quickly.

In my mind, a brake light is the most effective way for cyclists to signal to other road users. But this isn't a new idea, and this certainly won't be the first to come to fruition. So why don't we see any bikes with brake lights now? From the bike lights I have seen online, there seem to be a few problems with current bike brake lights:

1. Can't properly sense braking
   This mainly applies to those brake lights that use accelerometers to detect the braking. Accelerometers tend to misinterpret bumps and other vibrations as brake applications, causing erroneous lights.

2. Incompatible with certain types of bikes
   Other brake lights use a separate mechanical switch that connects to the caliper or the cable. These may not fit all bikes. Also, they are susceptible to all the wires and the wear and tear of a separate mechanical switch.

3. Not bright enough
   Some of the bike brake lights have only 1-2 small, low power LEDs. What is the use of a brake light if no one can see it?

4. Concern that other users won't understand the signal
   This is the only issue that cannot be solved directly with the device. Perhaps some experiments can be done to test brake lights on unsuspecting motorists/cyclists and see if they understand what the signal means in time to brake. This could end badly...

I will attempt to create a bicycle brake light that solves these problems. Right now I am leaning towards an improved accelerometer sensing brake, that may incorporate another sensor to improve the accuracy of identifying braking periods. But I will also investigate other methods for activating the brake light. Who knows; maybe I'll find some elegant and unobtrusive way to mount a mechanical switch to trigger the brake.

MATLAB Project: Boids, AKA Artificial Swarm Intelligence Simulator

If you've ever seen a flock of starlings flying in seemingly effortless coordination, or a school of fish avoiding some hungry sharks, you know what swarm intelligence is. The mesmerizing patterns make it appear like the flock or school is one (or several) cohesive unit(s) being guided - but it's not. The amazing thing about swarms like these is that there is no 'leader'. These patterns arise naturally as a result of a few simple rules that each swarm member follows. For example, each member in the school of fish wants to get into the center of all the other fish, to avoid the predators. In the flock of starlings, each bird has this centering tendency, but also must maintain some separation from other birds to avoid crashing. The swarm members also tend to travel in the same direction/speed as the other swarm members. For example, if they see one member turn, they'll turn too. 

These are the fundamental rules that govern flocking/schooling behavior in animals. Using MATLAB, I made a simulation of this flocking behavior based on Craig Reynolds' original program, using the three aforementioned rules. In the code, each rule has a weighting term which indicates how important the rule is to the boids. By changing the weights, different types of swarms can be modeled. Fish, for example, tend to stay much closer together than birds. So the weighting factor for the cohesion rule would be greater. Insects, on the other hand, don't always tend towards a center or even in the same direction, so those weighting factors would be very low for insects . This yields much more random motion, as one might expect. 

Many other modifications can be made to this simple code. For example, in this version, each boid sees every other boid. But in reality, birds and fish obstruct each other, and so the perceived size of the swarm by each member would be different depending on the location. By defining a radius around each boid that constitutes the visual range of the boid, one can more accurately model the swarming behavior. Additionally, this reduces the number of calculations drastically, and thus increases the speed and number of boids the simulation can handle. For example, my simulation can only handle about 50 to 100 boids before MATLAB runs very slowly. 

Check out the video below to see what my boids look like! For more information about the actual boids programming, click here.

Update 4/22/15: Added mouse tracking capabilities. Boids now follow the mouse pointer! Apologies for the choppy video; it looks like this took up a good chunk of my memory.

MATLAB Project: Airfoil Generator

I'm taking a class in aerodynamics this semester, and I know we will be working a lot with airfoils. So for fun I decided to write a MATLAB GUI to display any NACA series airfoil. Currently it only works with the 4-series airfoils, but soon I will expand it to work with 5, 6, 7,  8 series airfoils as well.

The program takes inputs of the NACA airfoil number, desired chord length, angle of incidence (AOI), and the wing length. The program then plots each individual airfoil, and lofts the two together in an isometric view. The 'export airfoils' button doesn't work right now, but soon it will export XY coordinates for the airfoils to a text file, so they can be loaded into SolidWorks or another CAD program to make a 3-D model of the wing. 

Download the files to run the program here.

Project: Drafting Bars

Engineers convey ideas through drawings and sketches. When sketching rectangular objects in orthographic or isometric views, it is necessary to be able to draw parallel lines. Additionally, for isometric views, the lines are offset 30˚ from the horizontal. I wanted to make a tool that would help me draw parallel lines and 30˚ angles to be able to make sketches quicker and more accurately.

My initial conceptual idea was to have two bars connected by two rods to form a parallelogram linkage. This way, one bar could 'follow' the original line, while the other [parallel] bar could be extended and a parallel line could be traced along the edge of the bar at any desired distance. 

Making the 30˚ angle was more tricky. My original thought was to have different sized rods that could be swapped out to make different angles, but I thought there was probably a simpler and more elegant way to do it. I eventually decided to add extra holes as alternate positions for the rods. In their alternate positions the rods are no longer parallel, and thus the angle between the bars can change. The freedom of motion of the bars is limited by their interference, and so by adjusting the geometries of the parts, a 30˚ can be obtained when the bars are touching each other. 

The last element of the design was how all the pieces fit together. I decided to use knurled brass knobs that threaded into a nut. The nut head fits into a channel, and the shank of the nut slides in a slot. The rod end fits around the nut, and the length of the rod is contained within a pocket in the bar. Tightening the knob presses the bar against the rod to lock it in place. If the knob is unscrewed completely, the rod end can be moved to a different hole to get the 30˚ angle.

This is the first version of the design. I'm still not sure what materials to use for the different parts. I really like the look of knurled brass knobs and rods, but I'll need to make sure they are strong enough. I originally planned to use 304 Stainless Steel for the bars for longevity and cost, but the locking function depends on the the ability of the bar to deflect under the force of the knob and press against the rod.