September 30, 2014

Roller Coaster Track Design

The East Campus Roller Coaster happened, and was a great success.  Here's some proof.  I didn't get too many pictures of the construction process, since I was busy with the construction itself, so as I wait for other people's photos to trickle in, here are the details on how the shape of the roller coaster's track was designed.

To start off though, a brief back story about the roller coaster tradition and how I ended up being one of the people in charge of this year's 'coaster.

As far as I can tell, the first EC roller coaster, dubbed "8.01: The Ride" was built in 2004 (making this the 10th anniversary coaster).  The last roller coaster, "The Reverse Cowgirl" was in 2010, and had to be taken down prematurely since they weren't permitted.  So since then, EC has avoided roller coasters.

I've been thinking about the roller coaster since rush last year.  Pretty early on I was joined by Jaguar and then Wesley, hall-mates and fellow Mech E. 16's.  During the fall and winter, we started thinking about general track shape and construction method, and as soon as EC's Rush Chairs were elected near the beginning of spring semester 2014, we started talking with them about getting a roller coaster approved.

This post will cover track design.  A construction post will come later.

Let's get started.

Track designing started out like this:



Slightly fleshed out:


Yeah.

So we wanted a loop.

Loops are hard for lots of reason.  First, designing a loop that doesn't exert excessive g-forces at any point is non-trivial.  From a structure design construction standpoint, it's challenging as well, especially given the materials and labor limitations of Rush.   And convincing MIT administration, the city of Cambridge, and some professional structural engineers that our loop would be safe is an even harder problem.

The first of those problems was the easiest to deal with.

Fun Fact:  going into this project, I had never ridden a roller coaster, and didn't have the faintest idea how roller coaster loops were designed, other than that they weren't just circular.  I did a lot of reading about roller coaster loops, and some of the math behind them.

Before looking at any specific loop shapes though, I needed a way to look at the acceleration experienced by a person riding the roller coaster.  More specifically, I cared about the normal force between the roller coaster track and cart at any given point.  This number is important for both the riding sensation as well as design and analysis of the roller coaster structure.

Here's a simple mechanics view of the roller coaster with the cart at three different locations on the track.  I only cared about the forces normal to the track, and didn't particularly care about forces and accelerations in the direction of the track at this point.


The negative sign if inverted above comes from the fact that my code calculated theta using the arctangent of the slope.  Arctangent has a range from -π/2 to π/2, and therefore can't output an angle with a negative cosine.  Yes, atan2 could have fixed this.

I cranked out some MATLAB code which, given a set of points parametricly defining the shape of the track, could calculate the normal force (and G-forces) experienced at all points along the track.

Without going into the code line-by-line, here's how that works.  

First, I define the track as a set of points that looks something like [(x1,y1), (x2,y2), (x3,y3), ... (xn, yn)], where each x and y is a point along the track in meters with the origin on the ground at the start of the roller coaster.

To calculate the normal force, at any point you need the velocity, the radius of curvature of the track, and the angle of the track relative to horizontal.

Velocity at a given point is calculated using energy conservation with change in height from the starting point.  A friction factor is included in the velocity calculation to account for drag and track friction.

Angle at point is approximated by first calculating the slope from point n to point n+1 by (yn+1 - yn)/(xn+1 - xn).  Angle is just tan-1(slope).

Finally, the radius of curvature at any given point n is calculated by finding the circle circumscribing the triangle formed by points n-1, n, and n+1.

This is roughly illustrated in the image below, where the black dots represent the points along the track, the blue dotted line is the approximate slope at a point, and the red circle is the curvature at a point.




So, now that I had a way to analyze different track shapes,  I needed to come up with an exact shape for the roller coaster track.  I chose to use a clothoid loop (generated from part of an euler spiral).  The curvature of a clothoid loop gradually increases until the top of the loop, at which point it decreases again and transitions to flat.  Gradually increasing curvature means no huge spikes in G's experienced by the rider.  

Here's a simulation of a 45 degree slope followed by a loop:


And a 60 degree slope, with friction included this time:

At the bottom of the slope, G's actually go to infinity because of the instantaneous change in slope. 
 As shown, I was able to create reasonably sized loops with reasonable G-forces.

That's pretty cool.  Unfortunately, we had to scrap the loop.  One of the steps to getting the 'coaster approved by MIT and Cambridge was going through MIT's Environment, Health and Safety department (EHS).  Basically, EHS said No Upside-Down People.  Period.

We had very little time to come up with a new track design, so we threw around some ideas including things liked banked curves, but decided on a (relatively) simple straight, multi-humped track design.  More specifically, it's an exponentially decaying cosine function plus an additional exponential decay, with a turn up at the end.  Even more explicitly, the track is the function 2.124e-.055xcos(.7x - .3) + 4.676e-.055x

Here's what that looks like in the G-force simulation:


Not too bad.  ~4.5 G's is pretty high though.  I traced the track curve into a Solidworks sketch as a spline, and then rounded out the curve at the bottom of the first hill to reduce the acceleration there.  I then exported the sketch as a DXF.  I found a MATLAB script that could convert DXF files into a matrix of coordinates.  I used this to pass the modified track shape into my same simulation.  Unfortunately, the process of converting things into splines and DXFs made the curve much less smooth, so the resulting acceleration plot is a bit jagged.  But the peaks were indeed reduced:


If you look carefully, you'll see that, with the friction estimate I used, G-forces actually go just barely negative at the top of the second hill.  Negative G's mean the cart would actually lift off the track at this point.  This turned out to not be the case on the actual ride, unless we gave the cart a really big push at the top.

I planned on taking some measurements to see how accurate the simulation was, but I didn't get around to securing good test equipment in time.  I tried out using a smartphone accelerometer strapped to the cart, but the resulting measurements were an incomprehensible pile of noise.

There's the abridged version of how we came up with the shape of the roller coaster.  Next in the series:  Roller coaster mechanical design, and maybe a glimpse into the whole approval process for this project.

September 3, 2014

Carbon Fiber Scooter

You may be wondering what's been happening with that extremely shiny hub motor I built ages ago.  Well, the vehicle being built for it is nearly complete, but I've been bad at documenting things this summer, so I haven't gotten around to making a post about it.  Until now.  Get ready for a long one.

Since the motor was intended to serve as the drive for some sort of reasonable and practical vehicle, it will be stuck on a kick-scooter shaped object.

Aluminum U channel and rectangular tubing are great ways to make scooter frames with integrated batteries.  Unfortunately, MITERS, did not have any of either of these types of stock in appropriate dimensions for efficiently packing batteries.  However there was some carbon fiber cloth, epoxy, and foam from the same stash I used in my bicycle building adventures.

I quickly laid up a rectangular carbon fiber tube:



I tried out something new for layup.  Instead of vacuum bagging the foam mold, I bent an aluminum sheet metal form, and clamped the carbon-wrapped foam mold into it.

The surface finish on the bottom and sides was excellent directly out of the mold, but not great on top.  It'll all get covered in more carbon anyways though.  The pink foam was melted out with solvents.


To hold the hub motor, I machined some aluminum forks that extend from the carbon fiber tube.  They were thoroughly machined away for weight reduction.


The side in contact with the carbon fiber was wrapped in fiberglass to prevent galvanic corrosion, and they were epoxied in place.  They were additionally screwed in place, using small aluminum plates to distribute the pressure across the carbon fiber, like a big washer.


I also added a rear fender using a segment of a conveniently sized carbon fiber tube I found at a lab cleanout over a year ago.


I machined my own fork from some thick walled aluminum tubing.


The fork is two parts.  The bottom half was thick walled tubing with 1/4" walls and 1" I.D, while the actual steerer tube is 1" O.D tubing.  The two halves of the fork were crammed together with a very tight press fit.  Maybe I'll bother with adding a fastener to prevent the press fit from slipping, but probably I won't.


The head tube assembly for this scooter is by far the most strangely shaped object I've ever tried carbon-fibering.  I started out by cutting a section of carbon fiber tubing from a broken bicycle frame.  I bored out the ends, and epoxied in bearing races for a bicycle headset.  I then cut up a carbon fiber hockey stick handle, and tacked the pieces to the head tube.


The form was vacuum bagged over with move carbon fiber, and then epoxied to the rectangular tube of the scooter frame.  A new and more powerful vacuum pump at MITERS has greatly improved the quality of my vacuum bagging.



Hey, that kind of looks like a scooter

The entire assembly was vacuum bagged with even more carbon fiber over everything, with a few extra layers around the joint holding the head tube.


This was probably the best layup I've done, and took only minimal sanding to get smooth, despite the awkward geometry of the head tube joint.



I made a long split collar to attach the steering column to the steerer tube, and used some bicycle seatpost clamps to hold it all together.  The quick release at the top allows the entire handlebar assembly to be removed for compact storage.


Mike TIGed me some sweet aluminum handlebars:


As far as scooter internals go, the chassis was designed to just barely fit a 12S 2P pack of A123 26650 cells.  For motor controller, I am using one of Shane's FF1.1 controllers.  For the fancy field oriented control algorithm to properly commutate a motor, it needs to know some of the motor's characteristics.

Phase resistance of .6 ohms was measured with a four-wire measurement.  That's kind of high, and it may be worth rewinding to pack in a bit more copper.

Measuring the back EMF took a couple tries to get right, because either I was using the MITERS scope wrong, or the scope was being weird.  I switched to a fancier scope and got it sorted out though.  This was done by scoping two leads, and spinning up the motor by friction driving it with a drill:


Kind of funny looking back EMF.  Fairly trapezoidal but with something 5x the electrical frequency added in.


Looking at the amplitude and frequency of the back EMF gave me a torque constant of .31 N-m/A, or 30.6 RPM/V.  I was shooting for 30 RPM/V when I wound the motor, so this worked out surprisingly well.  This gives a no-load top speed of  ~18 mph with a 12S battery pack.

There's not too much left to make this thing rideable.  I built a battery pack for it, but I did not assemble it quite flat enough and it had to be forced into the carbon fiber tube.  I'd like the battery and electronics to be easily removeable, so I need to reassemble the pack to make it a millimeter or so thinner.

Finally, the motor controller needs to be reprogrammed again with the correct motor parameters.  When I did this the first time around, my torque constant was off by a factor of 3 somehow, so the motor didn't commutate particularly well.

In other news, we built a roller coaster, and it was great.  Documentation to follow.

Also, there is now a virtual python-scripted version of the robot arm that draws virtual squares.  The virtual robot arm also can do some virtual gradient descent tuning on its virtual controller, which is pretty cool.  Once I improve that a bit, the real robot arm will have a real controller that doesn't suck.

August 16, 2014

It's Time

Thanks to Zach Both from Formlabs for taking some awesome pictures!



1:60 Scale.  Printed in 15 parts on a Form1+ and glued together with some extra black resin and a laser pointer.

Full-scale construction starting approximately now.

August 7, 2014

Chibi Atomic Jeep, the Power Racing Series, and the Detroit Maker Faire

Summers wouldn't be summers without frantically building something a week before a competition, and this summer is no different.

This year's event was the Power Racing Series at the Detroit Maker Faire.  Charles has been preparing for this event for a while, with the glorious Chibi-Mikuvan, but Dane spearheaded an effort for a brand new vehicle to race, just over a week before the event.  By the combined abilities of MITERS, we got the new cart, dubbed chibi atomic jeep, into a driveable state approximately 30 minutes before beginning our long drive to Detroit.

Here's some of the story of the fabrication of the Atomic Jeep, and its adventures in Detroit.

A frame came first.  Dane quickly welded up the frame out of some thin-walled steel tubing.

Photo credit to Dane.  I did a pretty poor job documenting the build, so most of the pictures are borrowed from other people.


I whipped up a brake disc and caliper mount or the back axle.  The brake disc is retained entirely by clamping force.  Keyways are for sissies.

Photo credit to Dane

The motor of choice was an old mikuvan alternator.  Alternators are pretty cool, because they have 3-phase windings, and a wound claw-pole type rotor powered through brushed slip rings.  When operated as a motor rather than generator, you put a current into the rotor to generate a magnetic field, instead of using permanent magnets.  Then you can commutate it like any other 3-phase motor.  Since you can control current to the rotor, you can on the fly change the torque constant of the motor by increasing or decreasing this current.  This effectively gives the motor a continuously variable transmission.  Start out at low speed with high field current for high torque, and then back off the field current as you speed up, so that you can reach high top speeds.  

We quickly got the alternator spinning using an RC car motor controller, but for the competition wanted something a bit more reliable and controllable.  The Power Racing Series limits you to 1440 watts by means of a fuse, so some form of battery-side current limiting is necessary to prevent you from popping fuses.


I had the old Kelly KBS48121 from my tricycle lying around, so I worked on convincing it to commutate the alternator.  Unfortunately, there was no way to cram hall effect sensors inside the alternator itself.  This meant externally attaching sensors somewhere to the motor shaft. 

I made the exciting discovery that CD drive brushless motors have the same pole count (12) as the alternator.  They also conveniently come with PCBs that have hall effect sensors already mounted in the correct locations.  Unfortunately, the hall sensors weren't the type that work with normal motor controllers, so I popped them off and glued standard halls in their place.  I also laser cut a mount to fix the CD drive motor behind the alternator:


Because of a slight lack of concentricity in my coupling between the alternator back shaft and CD drive motor, I mounted the CD motor on a laser cut delrin flexure, to accommodate the shaft wobbling around.  The delrin flexure was later made unnecessary by removing brushless motor stator and bushing, so that the rotor magnets and hall sensor board were not physically connected to each other.


By now I've memorized the important parts of wiring up Kelly controllers, so I hooked everything up and by some magic the alternator spun:




The cart then got steering and a real go-kart seat (borrowed form CapKart).  Thanks Mike for not letting us accidentally build reverse steering

Photo credit Dane

Dane provided some beautiful A123 Systems battery modules for the kart:

Photo credit Dane

Vice grips make the best steering wheels:


While the rest of the cart fabrication was going on, Rob Reeve was machining gears.  He volunteered to make a two stage herringbone gear reduction for the cart.  Unfortunately due to time constraints it became a helical gearbox rather than herringbone, but the gears are still some of the most beautiful objects to ever be produced on the MITERS bridgeport.

Here are the gears being machine.  The contraption on the left of the mill bed is essentially an indexing head coupled to the mill's X stage.  The liked rotation and translation allow you to machine helices.  Combining this motion with a gear cutter and a funny angle on the head of the Bridgeport allows you to machine helical gears.

Photo Credit Dane
Some of the resulting large gears:

Photo Credit Dane
 The gears were assembled into this glorious gearbox.  The gears were TIG welded to steel shafts by Mike.  The aluminum plates holding everything were waterjetted, and Rob machined the spacers from stainless steel.  I turned the aluminum spacers that hang the motor, as well as the giant steel shaft coupling to connect the alternator to the gearbox's input shaft.


Despite its helical-ness, due to a combination of slight lack of concentricity from welding, low-quality gearbox bearings, and possibly use of the wrong gearcutter on the larger gears, the gearbox was very noisy and had to be run in to spin smoothly.  This was done by slathering the gears in polishing compound and backdriving the gearbox with a giant DC motor coupled to the axle.


The gearbox was attached to the cart and I did some quick test driving a few hours before we planned to leave for Detroit.  And it sucked.  Despite vigorously spooling up on the bench, the alternator failed to produce any real torque when pushing a person.  An electronic shifter was set up for switching the rotor current with a rotary switch and some power resistors, and in the highest torque "gear", the cart would give a brief kick before topping out at around five miles an hour.  In faster "gears", the torque produced was pitiful, and failed to bring the cart to any reasonable speed.

With only a couple of hours to work, we decided to swap the alternator for the go-to MITERS motor, an 80mm RC airplane outrunner, like the trike motor but all black and 180 Kv instead of 130.  This particular specimen used to live on Brickscooter.  Unfortunately, no pictures were taken in the rush to do the conversion.  Performance was much more reasonable after the conversion, even when limiting battery current to 40 amps.

The cart was also fitted with the modified bodywork from a Jeep Power Wheels car.  For the drive, it was ratchet strapped to the roof of a rental jeep.

Photo Credit Bayley

The cart at the Racing Series track on saturday morning:

Photo Credit Dane
Atomic Jeep and Chibi-Mikuvan next to our pit:


3D-printed doge hood ornament and googly eyes:


The MITERS/ MIT Department of Silly Go Karts pit was very busy during and between races:


As soon as we started testing on the track we ran into problems with the Kelly motor controller.  Just as I experienced on the tricycle, before upgrading to the high speed version of the controller, the Kelly would spontaneously cut out.  This usually occurred when quickly going from full throttle to zero, back to full throttle again, and forced the driver to quickly cut power and reset the controller before starting up again.  Still, we qualified with a lap time of around 18 seconds (the fastest were around 15.5), seeding us 7th overall for speed.

After qualifying was a "Moxy Round", during which you are tasked with entertaining the crowd as best you can, and are judged by three small children selected from the audience:  We paraded all the small silly EV's we brought: Dane and Rob on the Atomic Jeep, Mike on the (really) tinykart, me on my trike, and Charles on eNanoHerpyBike.  This procession was followed by bribing the judges with Oreos, which earned us one of the highest moxy scores of the day.

For the first race, a 20 lap sprint, everything went wrong immediately, with the motor controller cutting out constantly.  Still, we somehow managed to qualify in the top half of cars for the next round of racing.  Lots of Kelly setting tweaking and more careful driving made the controller somewhat more reliable.

By the endurance race, we'd gotten good enough preventing the Kelly from cutting out, and resetting it quickly when it did, to perform reasonably well.  That is, until all our harbor freight wheels started exploding.  It turns out the super cheap hand truck tires are pretty terrible in every way for go-kart duty.  Mostly we burned through tires, but in a couple cases, turning forces caused the stamped sheet metal hubs on the front wheels to shear apart, leaving us wheel-less.  We managed to complete 132 laps, for 10th place in the endurance race.

Atomic Jeep, Chibi-Mikuvan, and Nimby Ferarri in the endurance race.

Photo Credit Bayley

Photo Credit Bayley
 There was plenty of down time for triking between races.  Apparently someone on the side was timing some of my laps, and recorded a 12.4 second lap.  2.7 times the power and 1/2 the vehicle weight of most Power Series cars means 25% greater speed (than the fastest cars) around the track.  The trike definitely needs a larger track to really make use of its power and gears.


The future of the Atomic Jeep:
Unclear at the moment.  Numerous improvements could be made:  Tires that don't suck, a high-speed Kelly to eliminate spontaneous cutouts, new gearbox bearings (the current ones got pretty sad from racing), better steering geometry...
Most likely, some of these changes will be implemented in the week or two before the New York Maker Faire.

Lots more pictures and videos can be found on Dane's much more thorough writeup.