The Poor Man's Giro : Amateur Science in a GT Mimicry

Early in May, I set out to do something in the name of science. I'd read about the physical demands of grand tour racing in research papers and wondered what that would translate to for an amateur rider who works 5 days a week in a day job.

The idea was simple. I set out to ride roughly 1/8th the daily distances in the 2019 Giro d'Italia. Each ride tried to capture the intent and spirit of the pro rides.

For example, if one day was the ITT, I'd go out and smash a little ITT of my own. If there was a mountain stage, then I'd go out and do some hill repeats (we do not have mountain passes in Abu Dhabi ! ). If the ride called for a flat stage, I'd go out on a 40km ride and end with a "solo sprint".

The challenge was called "Poor Man's Giro". I even created a little flyer for it and shared it on Twitter with the likes of sports analytics guru Alan Couzens and exercise scientist Stephen Seiler. 

All rides were attempted in the searing heat of Abu Dhabi. The rides were supported by nutrition from Secret Training U.A.E.

Fig 1 : Too poor to be a pro and ride a Grand Tour? The Poor Man's challenge is an answer!

Now, I faced a few challenges which I need to declare before we get started. Namely :

1) In addition to my day job, I also coach a running club and so squeezing in rides everyday became a challenge.

2) I went on vacation round about the 20th pro stage so I ended up completing just 18 "stages".

3) A couple of rides had to be done indoors on a Cybex ergometer.

4) I took one rest day more than necessary. It was inevitable. Too busy to squeeze a ride in one or two occasions.

5) My time trial bike was not fitted with a power meter so power output wasn't captured for two TT's.

95% of the rides were done on a Colnago C40 road bike outfitted with a Powertap powermeter to capture the workload. Daily rides were uploaded into Strava and synced with GC to power the analytics.



Data Results

Below is the data from 18 rides. BikeStress is GC's implementation of Training Peak's TSS when they got rid of the "TSS" trademark from their software. TRIMPS have been calculated most likely using zonal points. IsoPower is GC's implementation of TrainingPeak's NP, again after getting rid of trademarked metrics.

Fig 2 : Ride, workload and stress parameters  from each day's ride of the Poor Man's Giro

To add a little bit of extra science to the investigation, daily HR and HRV related parameters were measured using a Faros ECG device hooked up to a Polar H10 chest strap. Protocol followed was 5 min supine-standing orthostatic format.

All the data was analyzed in Kubios to extract the mathematical nature of sympathetic and parasympathetic function. A self coded script threw the data onto a spreadsheet and automatically plotted the variables.

Fig 3 : Sets of plots showing the HR/HRV related parameters for the duration of Poor Man's Giro

Discussion of Results

We understand from the Grand Tours research done by Sanders et.al that the stress associated with a time trial (TT) as a function of distance is the highest among flat (FLAT), semi-mountaineous (SMT) and mountain stages (MT).

The authors find a typical average TT speed of 36.5 +- 12.9 kph, an average power output of 371 W at 177+/ 10 bpm, TRIMPS of 33 +/ 32 AU and a TSS of 62 +/ 32 AU. That translates to a TRIMPS/km = 3.39 +/- 1.39 and a TSS/km = 3.39 +/ 0.17 AU/km.

The table 2 from their research paper is very instructive of the performance parameters across the spectrum of stages. Borrowed and pasted below for quick reference.

Fig 4 : Typical performance characteristics from Grand Tours from Time Trials (TT), Flats (FLAT), semi-mountaineous (SMT) and mountain stages (MT).  

This can be compared to my own ride characteristics from Fig 2.

Time Trials : Agreeing with the research, the RPE associated with a solo TT is high, around 8.5-9. TRIMP points are 62 vs 58 (mine) which translates to a TRIMPS/km of between 4-5. This is the highest among all rided that I attempted.

Flat Stages : Agreeing with the research, the RPE associated with a flat stage is around 5 (pro =5.8). TRIMP points are 298 vs 94 (mine) which tranlates to around 1/3rd the heart related stress mainly due to the reduction in distance attempted.   This translates to a TRIMPS/km of around 2 (pro = 1.55).  Power output is around 137 W average giving an average TSS/km of 2.9-3 (pro = 1.14). I presume pros show a lesser power related stress per km riding such long stages due to the draft effect.

Mountain Stages : The ride done on May 25 is a perfect example of a flat ride ending with several hill repeats to mimic the feel of climbing a mountain. The TSS/km and TRIMPS/km came out to 3.8 and 3 respectively, compared to the pro stats of 1.97 and 2.1 AU/km. So the stress was a bit greater on my part, and I probably intentionally made it that way when thinking about climbing.

Daily Accumulation Rates : For 3 weeks, the accumulation of stress was as follows :

The sum total of TRIMPS gained over 18 stages = 1937 AU = 108 TRIMPS/day.

The same for TSS (aka BikeStress in GC language) = 1386 AU = 77 TSS/day.

Total workload = 6467 KJ, with an accumulation rate = 359 KJ/day.


Daily HR and HRV related fatigue : The days after the hardest rides (TT's and MTs) on 11th, 18th  and 28th May respectively show significant drops in time related HRV parameters such as rMSSD and conversely  high supine resting pulses. 

Although all these parameters showed cyclical variations day in and day out, one standout feature was the steady rise in chronic HRV and the steady drop in chronic resting heart rates over the course of 18 days (chronic = long term).

Infact, the drop in resting heart rate, when compared to similar data from the beginning of year show the difference very clearly. The long term difference seems to be a decrease of around 5 beats/min compared to the period prior to starting this mini challenge.

Fig 5 : Highlighted section showing the supine resting heart rate (daily acute and chronic over 7 days) compared with data from March 2019. 

Conclusion

Keeping with the spirit of amateur scientific investigation, an 18 day grand tour was mimicked during the period of the 2019 Giro d'Italia. Despite the limitations of a decreased work load, the aim of trying and matching atleast 1/8th the distance was more or less accomplished.

From the data. I conclude that heart related fitness parameters improved during those days, which shows the effect of a 108 TRIMPS/day and 77 TSS/day loading pattern. However, the data doesn't show the "delayed" effect of improvement that must have come +1 or +2 weeks after the 3 week training was concluded.

I hope to expand on this research during the period of the Tour de France. If you wish to join me in a Poor Man's TDF, please join !  Let's learn together. I can be found on Twitter.

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The G-G Diagram Applied To Bicycle Racing

In the midst of our world of possibilities for data analysis in cycling performance, I was wondering if we could consider extending our scope a bit to other things.

As we speak, more models of GPS systems, power meters and heart rate monitors are being designed and released with are basically just different answers to the same old questions : "what's my speed", "what's my power", "what's my heart rate".

But what about other fundamental questions and problems imposed by bike racing? How could we explore and quantify those parameters?

As an illustration, one topic for discussion rarely brought to the table in our daily techno babble is that of acceleration and its impact on safety. Acceleration is an elementary concept in physics and we all know how important it can be towards team race strategies we see today in cycling. Is it getting the importance it may deserve from an analysis standpoint? I'm not sure.

The question is : Could an acceleration analysis be helpful to the racing cyclist and how?

Consider a human-bike system in an Individual Time Trial, a bike race against the clock on a circuit of predetermined length and design. Often in a race of professional caliber such as this, the standard deviation of the data set of results from the top 5 placers is mere seconds.

While time trials are usually steady power output races, if we considered a very technical race course with lots of curves and S-bends, we could say that a majority of the racer's effort is concentrated towards finding the optimum line between the bends while controlling the bike's speed through braking and acceleration. Miscalculations here could cost seconds and a possible podium spot.

Consider two racers, A and B, who chose different paths around a section of this hypothetical race course. Let's also say that they were riding bicycles of the same design, with the same handling qualities.

Fig 1 : Paths of two cyclists in a section of an ITT course. COG = center of gravity.

Who emerged faster at the right end of this section? Racer A or B?

Of course, we wouldn't know because we don't have enough information, you would say. The information missing here is that of the cyclists' acceleration.

To maintain a curved path on the ground, any vehicle, be it a bicycle or a Formula 1 race car, must be moving sideways as well as forward. Hence, there are two components to the bicycle's acceleration. They are :

1. Lateral or centripetal acceleration (LA), whose vector points towards the center of curvature of the road. A bicycle turns because of applied lateral tire forces. LA is affected by tire-road friction characteristics, angle of lean, square of the speed of the bike and radius of the curve, R. If lean is too large (i.e. rider tilts too much into the circle), centripetal force will be too much and the bike will start turning into a circle with radius smaller than R. If lean is not large enough, there won't be sufficient force to keep the bike on the circle and the bike will veer off, turning in a circle with radius larger than R.

2. Tangential or longitudinal acceleration (TA), whose vector points in the fore-aft direction of the bike rider. It is decided by pedal torque, aerodynamic drag forces, and traction limit of the tires.

If m is mass of bicycle-rider system, v its speed along the course, R the radius of curvature of the curve, t is time and g is the acceleration due to earth's gravity, then the above two are defined mathematically in terms of g-force as :

The vectors of these components and their resultant roughly look like this :

Fig 2 : The tangential and lateral vectors of acceleration represented graphically. A reversal of lateral acceleration vector (purple) signifies reversal of direction while a reversal in tangential acceleration vector (red) signified reversal of speed with time.

If we attached perpendicularly oriented accelerometers at the center of mass of the rider-bike system for the 2 riders in Fig 1 , and if we captured lateral acceleration (LA) and tangential acceleration (TA) through a data recording system, the data points when plotted on a graph could look like this (shown just for illustration, not to be taken for granted).

Fig 3 : This plot shows an example g-g diagram (a composite of friction circles for both wheels) for two bicycle riders on the same section of the race course on the same bicycle. It consists of a forward acceleration, turning and braking regions. Data points for each cyclist is shown in red and blue. A rough boundary envelopes these points for both riders. It must be noted that the g-g envelope/boundary for each of the cyclists is not fixed and depends on the bicycle, maximum tire friction force, human skill level and environmental conditions imposed on tire-road contact. This is the performance envelope for the bicycle-rider system. Outside this safe envelope, racing a bicycle could be dangerous.

This plot is called a g-g diagram. The concept was described extensively by aerospace engineer and race car driver Will Milliken Jr.

The difference in riding techniques of the two riders resulted in two different maneuverability boundaries, which can also be looked at as the maximum potential of the rider-bike system in any technical section of the race course for the race conditions. A given individual can only generate limited g's of acceleration to get up to speed. Theoretical limits of deceleration are on the order of 0.5g for a crouched rider on level ground before a person flies over the handlebars. If they are riding two different bicycles with different handling qualities and tires, the g-g boundaries will be different in this case too.

The ultimate limit of the g-g boundaries is imposed by the acceleration capability of the bicycle, which is primarily determined by the grip between the tire and road surface. This is represented in the g-g diagram by the outer oval shaped boundary.

When a racer sits down and studies his g-g performance, he could get a graphical picture of how he utilized the components of acceleration at specific sites on the course and how his choice of equipment may have cost him.

With appropriate software, answers could be generated to questions such as : What's the range of my accelerations? Which direction did I spend more time cornering to? Did I decelerate too much before the sharp curves? Did I accelerate optimally after the apex? How much emphasis did I place on acceleration and deceleration? Could I have changed the ratios of these accelerations by riding differently and emphasizing various body movements? How would these have affected or improved my course times at the end? What really caused my wipe out at that sharp bend and was it related to the lean angle and speed with which I faced that bend? How does all this change with a change in my bicycle tires? Or bicycle design.

How may this be specifically applied to improve performance? I have some thoughts :

1) The tire and the bicycle could be engineered to widen the g-g boundary as much as possible throughout an expected range of operating conditions (load, surface, temperature) without bringing about potentially dangerous modes of oscillatory motion.

2) The rider could train and improve his skill level to ride and exploit these maximal g-g limits of his machine.

Perhaps knowing the performance envelope of the bike being ridden for specific operating conditions may also empower the cyclist with a feeling for when he can safely take risks to win a race.

Take a look at the following two videos. One shows the capturing of the acceleration vectors on a g-g circle for a remotely controlled toy car. The one below it shows what looks to be a real time friction circle generation from a computer simulation.

Its inspiring to watch these data recording and software applications. Perhaps we could have a neat cyclocomputer in the future that could show the bicycle's g-g diagram in real time, if its practicalities have been established. Maybe cycling commentators will start talking about g-g diagrams and other cool things in future race telecasts. Who knows. What do you think?







ADDITIONAL RESOURCES :

The G-G Diagram
Rate Of Cycling Uphill Explained
Wild Ideas For New Cycling Products Part 1
Wild Ideas For New Cycling Products Part 2

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Psychology Of Dog-Cyclist Encounters

The human instinct to fear when something on four legs chases you down on a pathway is natural. The speed, ferocity and the sound of animal paws smacking the ground as it tries to catch up behind us sends us scrambling for escape responses. Getting caught up in this moment to save your own butt could be all too easy.

What about the dog? Are all cyclist-dog encounters dangerous for riders? This is the interesting question for today.

I have never had very troublesome dog encounters but I did have a few close ones which I managed perfectly fine by just increasing my velocity. That response wasn't hard to gather. I find it somewhat odd when people talk about carrying sticks of dog spray and other ammunition on their backs as if preparing for some kind of surprise ambush like those between highwaymen and money wagons in western movies.

Maybe you'd have a different perspective and we can surely disagree.

In this post, we take a look at the ingredient of dog sprays and then try to understand the mind of a dog during a dog chase. We think we know animals, but we might actually be surprised by how much we don't. What's the source of the dog chase and what's the best course of action from a cyclist? Read on...


DOG SPRAY AND ITS SHU VALUE

Its become a sort of fashion these days to go with the "Point and Spray" method without a furnishing a second thought. Besides giving the user of the spray an inflated sense of security, there's probably some degree of thrill involved in knowing that you'll be spraying nasty crap into someone's eyes. I did wonder a few times whether people stop and think about what this crap really is.

Bite into a piece of hot pepper and one quickly realizes the complex reactions involved in the body to flush out this irritant heat. If that's not enough for you, try some of the interesting hot sauces out there. People make a living through marketing this stuff and gaining notoriety for the "heat in the bottle". The more, the better.

I was once offered a bloody little drop of Dave's Insanity Sauce to try as relish by a friend. Just a tiny drop about 0.5 inches in diameter on the palm of my hand.

How did it go? Well, let's say I was too saucy in my overconfidence to begin with. The moment the wretched stuff hit my tongue, my eyes started watering streams and my throat, mouth and ears felt like they were lit on fire by a propane torch. Wow. Its relieving to just say that it was something I tried, in past tense.

You'd think this is the hottest stuff around that you had in your mouth but the gurus of spice will wave you off and tell you otherwise. For perspective, inspect the table below. This gives the Scoville Rating, which is basically the piquancy spectrum, for peppers. The unit of measurement is Scoville Heat Unit or SHU for short.

The Scoville Scale of peppers. Click to zoom.

The predominant ingredient in dog spray is oleoresin capsicum (OC). Take your canister and inspect it. It might even say something like "contains capsaicin and capsaicinoids", which is true as the extracts of OC contain capsaicin. Capsaicin causes neurogenic inflammation.

The hotness of OC is directly related to the amount of capsaicin in it, which varies significantly from manufacturer to manufacturer. The more capsaicin content the OC has, the hotter and more effective the spray will be.

While Dave's Insanity Sauce has about 250,000 to 500,000 SHU's, commercial grade self-defense sprays such as dog spray, mace and pepper spray have a minimum of 2 million SHU's and beyond. That's 4 to 8 times the strength of Insanity Sauce.

Now imagine if someone who didn't like you took the same Insanity Sauce and squirted a bit in your eyes. Good luck, my friend. You may see your bum through your head.

When you spray 2 million SHU's or more on someone's face, into their nose and eyes, you bet its going to hurt real bad. Humans could easily get help and get nursed with water in the event this happens. What's a blinded dog to do in the middle of the road? I'm not sure.


PSYCHOLOGY BEHIND THE DOG CHASE

Many times I have wondered what causes a dog to chase a cyclist. What's the motivating factor for ticking a well domesticated animal, sending it scuttling behind something else it spotted on the road? What's the psychology of a dog's mind during this scenario?

I don't profess to be a dog expert. That's why I posed this troubling question to Alexandra Horowitz, a famous professor of psychology and cognitive scientist with Barnard College. She's probably one of the few in the U.S who leads a dog cognition lab which studies dog behavior.

Her recent book, Inside of a Dog: What Dogs See, Smell, and Know, describes recent discoveries of the fields of dog cognition, behavior, and biology in order to better imagine what it is like to be a dog. Just last week, her work was featured in a well written article in Time Magazine titled The Secrets Inside Your Dog's Mind.

Since she understands dogs better than most of us, I asked her to unravel for me the psychology behind the dog chase, from the dog's perspective. She was, in a way, the perfect person to ask because apart from being a scientist, she also happens to be a runner and recreational cyclist.

Ms. Horowitz's best explanation to me went along these lines. Consider the visual system of dogs. The visual system of canids evolved over the course of many years to notice quick movements, like fleeing prey.

As hunters, dogs and their forebears developed a very high sensitivity to motion, dogs became much quicker to notice a small motion in their peripheral vision than we are. This is adaptive for an animal which chases moving prey.

Not all dogs chase bikes, of course. But for those that do, they see the smooth, quick motion of the bike and it triggers their prey instinct to chase the "animal". In our case, the "animal" to the dog is the cyclist. The cyclist is the source of the problem.

She also pointed out to me that this isn't the same as saying dogs see cyclists or runners as "prey" because after all, they never consume you as you dismount. But they do get very excited and their nervous system just riles up for the chase.

What would her approach be as a cyclist? The best way to stop a dog is to simply stop the illusion of the prey, i.e, stop the bike. A dog may still bark and stay riled up, but does this only for a short time, as their nervous energy subsides. It may be impractical to stop if you're on a long ride, but keep in mind that the dog is just excited and can be calmed by stopping the bicycle.

Of course, this is easier said than done as this seems a counter-intuitive step for a majority of us. But since the dog psychology in dog-cyclist encounters makes sense, the response from a cyclist countering exactly that psychology may also make sense.

I also asked her if owners could train their dogs in such a way that they learn not to do what their instinct tells them to do on seeing a cyclist on the road. According to her, the training itself might be intensive, but something like this is certainly possible. A dog can be trained to notice, but not act on these cues. Unless an owner specifically trains their dog to be still when a bike comes by, it is not something dogs with this visual tendency will do on their own.

Ms. Horowitz believes, like I would also like to, that in most case scenarios, there is no pressing need to spray a dog with a canister of a million SHU's. In fact, she believes this could really up the ante and "cause" a secondary response in the animal.

Do you have a dog chase story to share? What are your thoughts on dog behavior? Please join the discussion if you know you have specific experience as a cyclist, dog owner or as researcher involved in animal behavior.




ADDITIONAL READING :

Guide To Chile Heat
Health Hazards Of Pepper Spray
Scoville Scale Chart For Hot Sauce And Hot Peppers
The Secrets Inside Your Dog's Mind (Time Magazine, 21 September 2009)

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Dynamic Stability Of Bicycle Design : Part 4

"Trail can be found by supporting the bike on a flat surface in an upright position for measuring purposes. A centerline is run down through the head tube until it hits the flat surface. A vertical line is then dropped from the front axle until it hits the ground. The distance between these two points on the ground is the trail. The comfort range of trail is 50 to 65 millimeters. Beyond these limits in either direction, it would be considered less desirable."

- A quote from Chapter 1 : Frame Geometry, The Paterek Manual for Bicycle Frame builders

Question : Why do we know what we know about the comfortable range of bicycle design parameters and ride desirability? How do we know it? Can such claims be applicable to all bikes with any rider in general? What does science say about these statements?

Continued from Part 4

In the previous post, I presented a mathematical bicycle model to you (validated by research) and a computer program called JBike6 that uses this model to calculate the bicycle's stability eigenvalues. We also explored an important point that this model is, regardless of complexity, still simple in terms of being a riderless model not accounting for the frictional properties of the tires. Hence, whatever results you see in the JBike6 is only so true as long as you consider a riderless bike with other simplifying assumptions established.

Interestingly, Jim Papadopoulos (thanks Jim!) pointed out to me in a comment to the previous post that in terms of ridden bikes, he surmises that JBike6 might apply best to recumbents (where the rider is secured to a seat) with extremely hard tires, ridden no-hands. So its applicability is not lost.

So what is the bottom line of all this mess? What I've been trying to convey to you through this series is that studying a bicycle is a difficult and complex task. The bicycle really is a complex vehicle. Why do we know what we know about the bicycle dynamics, and how do we know it?

Some of us like to think we know bicycles and like to give out general rules of thumb for design so as to get a self-stable bike. Now this could be true for the particular bike design being considered but the point is, it may not be true for different designs and different people. A different bicycle with a differently sized rider can have totally different dynamics.

Hence, it turns out that when someone makes general claims about bicycle design that he thinks he or she knows will work for all bicycles, that's just an unvalidated statement in a true scientific sense. They're what's called an anecdote. Anecdotes come through hearsay or someone's personal experience with building something. However, the state of the art in bicycle science has yet to concretely come out with the unifying principles behind why a rider controlled bicycle, any bicycle, behaves the way it does.

Science has a long way to go before establishing the truth behind general statements about parameter changes and their effect on ride characteristics as applicable to all bicycles of any design. Science also has some ways to go in studying complex modes of motion in bicycles that we talked about in Part 2, particularly the dangerous ones such as high speed wobble that can bring harm and loss of property to the owner.

While I leave you with these thoughts, I'd like to present some research by one of my readers, Jason Moore. Jason is working towards his Phd in Mechanical and Aerospace Engineering at UC Davis. He's currently a Fulbright Visiting Scholar and Researcher at the Bicycle Dynamics Laboratory at Delft University.

Jason and his advisor, Prof. Mont Hubbard, employed the same validated bicycle model we've been talking about and studied the dynamics for the design parameters of an old Schwinn bike he owns. The model was then used with a physical parameter generation algorithm to evaluate the dependence of four important design parameters on the self-stability of a bicycle. These parameters were :

1) Front wheel diameter

2) Head tube angle

3) Trail

4) Wheelbase

In the end, the duo were able to generate interesting results through graphs that showed how changing the above four parameters independent of each other affected bicycle stability in weave and capsize critical velocities.

Their research paper was featured in Engineering of Sport, the journal of the International Sports Engineering Association. The graphs show definite parametric dependence of bicycle stability. Most interestingly, their results disagree with the general claims made by the Paterek (shown at the beginning of the post) about the comfortable limits of trail by showing an increasing stable speed range with increase in trail, provided this increases was kept within reasonable limits.

With Jason's permission, attached below are 8 pages of the paper titled Parametric Study of Bicycle Stability. Please click on them to expand and read. This is also available to read via Google Books Online. See this link.

Finally, if you have any questions about bicycle stability that bothers you, please ask away and I guarantee you'll receive an adequate reply from Jason, Arend Schwab or Jim Papadopoulos, as they all read this blog.

Thanks for sticking along on this journey!

* * *

The Engineering of Sport 7 : Proceedings of the 7th International Sports Engineering Association Conference. Biarritz, France. June 2-6, 2008.

CONNECTED READING :

Dynamic Stability Of Bicycle Design : Part 1
Dynamic Stability Of Bicycle Design : Part 2
Dynamic Stability Of Bicycle Design : Part 3
Dynamic Stability Of Bicycle Design : Part 4
A Bicycle Model To End All Models
"Linearized Dynamics Equations For The Balance & Steer Of a Bicycle" - J. P. Meijaard, Jim M. Papadopoulos, Andy Ruina, A. L. Schwab, 2007
History Of Bicycle Dynamics

* * *

Dynamic Stability Of Bicycle Design : Part 3

Continued from Part 2.

Hello bike nerds! Before you engage yourself in another installment in my series on bicycle stability, have a look at this clip from The Daily Planet shown on Discovery Channel Canada. The individual interviewed is Arend Schwab, a mechanical engineer with Delft University of Technology.

Modeling For Stability Analysis

The bicycle is a complex system to analyze. In Part 2, we talked a little about modeling the idealized passive rider-bicycle system, simplifying many things through assumptions but still capturing enough detail to go ahead with a reasonable analysis. The full analysis, particularly that involving the derivation of the equations of motion, is beyond the scope of this blog.

But to put it in simple words, what the analysis yields are two coupled second order, non-linear, differential equations in lean and steer. Then, these equations are linearized through a carefully followed algorithm to give us an eigenvalue problem. The eigenvalues gained from the characteristic equation help us assess the stability of the modes of bicycle motion. Eigenvalues are the cool numbers that give us an idea of the stability of the engineered system when the system is disturbed.

Fig 3 : What eigenvalues tell us about stability. To make an engineered system stable, we're all interested in attaining negative real numbers for eigenvalues. Stable motion of a bicycle has negative, real, eigenvalues. Courtesy : University of Michigan, Dynamics And Controls

Many years of research has allowed us to understand the nature of the bicycle's linearized equation of motion (LEOM). The LEOM, expressed in terms of small changes in the lateral degrees of freedom being the rear frame, roll angle ф and the steering angle δ, from upright straight ahead configuration at a forward speed v, looks like this in matrix form :

Fig 4 : LEOM Of A Bicycle

where

M = symmetric mass matrix which gives the kinetic energy of bicycle system at 0 forward speed
C1 = damping matrix, proportional to forward speed v
K0 = first component of stiffness matrix, which when combined with 'g' yields a symmetric quantity proportional to gravitational acceleration and can be used to calculate changes in potential energy
K2 = second component of stiffness matrix, which when combined with the square of forward speed, v, gives a quadratic quantity in forward speed and is due to centrifugal effects.
f = applied forces

Consider each of these matrices as packages and the contents of these packages would be specific combinations of the bicycle's design parameters shown in Fig 2 of Part 2. To know what goes where in these matrices, you need to read this paper from Delft.

Finally, the time varying variables in the LEOM are :

Fig 5

ф = roll angle
δ = steering angle
Tф = action-reaction roll moment between fixed space and rear frame due to external causes like wind or lateral pushing force from behind.
Tδ = action-reaction steering moment, torqued by rider's hands.

Because we're analyzing a passive, uncontrolled bicycle, both these moments are taken to be 0. The characteristic equation is then the determinant of the equation in Fig 4 which gives us the eigenvalues of the problem. Eigenvalues are the exponential part of the solution to the differential equations of motion and as said before, help in stability analysis.

Computer Program To Plot Eigenvalues

Solving all this by hand takes pages of tedious work. If we can program these rules into a computer, it can quickly solve the characteristic equation. The input for the program would be all the bicycle's design parameters. The output would be the eigenvalues. We can even tell the computer to plot them for us as a function of forward speed v, for any particular bicycle configuration that we provide.

That's exactly what JBike6 does. It is a program written in MATLAB, a collaborated work between Delft University of Technology and Cornell. For small values of steer and lean, the program is perfectly accurate. Jim Papadopoulos, a contributor to JBike6, is also the co-author of the book, Bicycling Science (I had interviewed the main author, Prof. David Gordon Wilson from MIT earlier this year).

As the main illustration for today, I pull up an example bike, already provided in the program. It is a Litespeed Ultimate bicycle and Fig 7 shows its design parameters, all values in metric units.

Fig 6 : Litespeed Ultimate

Fig 7 : Litespeed Ultimate's 25+ design parameters entered as inputs to program. Click to zoom in.

After checking these boxes, I hit the calculate button on the upper right hand side. The program solves the linearized eigenvalue problem and gives me 4 generalized eigenvalues. It plots these values on the y-axis as a function of forward speed, v on the x-axis.

Fig 8 : Litespeed Ultimate's eigenvalues vs forward speed. Re = Real, Im = Imaginary

The above plot tells me something about the stability of this bicycle as a function of speed. To understand what's going on, let us remember the information about eigenvalues in Fig 3 and commit it to mind, or go back and refer to it. Now read the plot in Fig 8 slowly from left to right in the order of increasing speed v. We'll take it piece by piece.

Speed Range Of Interest : 0-0.5 m/s (0-1.118 mph)
Motion: Capsize
Nature : Stable, non-oscillatory capsize

What would you expect from a bike standing still or nearly so? It will simply flop over. How do we know this? The fact that the plot yielded large positive eigenvalues or real numbers tells us right away that this is a very unstable motion. Two positive and negative pairs of roots correspond to both falling and uprighting of the bicycle. One pair corresponds to when the steering is turning toward the lean; the other when it is turning opposite to lean. Since there are no imaginary parts of eigenvalues, it tells us that this capsize motion is non-oscillating, like I mentioned in Part 2.

Speed Of Interest : 0.5-1 m/s (1.118-2.237 mph)
Motion: Transition to weave
Nature : Unstable, oscillatory weave with stable capsize

As the forward speed v is increased from 0.5m/s to slightly more, two real eigenvalues (in blue) become identical, coalesce and form a conjugated pair, which is where oscillatory weave motion actually shows its face. In this oscillation, the bicycle sways about the headed direction.

Speed Range Of Interest : 1-4.8539 m/s (2.237-10.858 mph)
Motion: Weave+Capsize
Nature : Transition to stable weave, with stable capsize and oscillation

The positive eigenvalues tend to decrease in magnitude, so the motion is tending towards stability. Eigenvalues with imaginary parts lead to oscillation with increasing frequency, and the rate of increase is rapid at first, but then slows. The bike will weave back and forth one or more times before falling over.

Speed Range Of Interest : 4.8539 m/s (10.858 mph)
Motion: Weave + Capsize
Nature : Weave speed critical point, with stable capsize

Weave speed is that speed at which weave does not grow or decay, as can be seen by the eigenvalue crossing 0. Hence, weave speed is 4.8539 m/s. It is stable and forms the lower stability range bound for this bicycle. Eigenvalues corresponding to imaginary parts is oscillating motion. Beyond this point, weave is stable until infinity.

Speed Range Of Interest : 4.8539-7.0994 m/s (10.858-15.880 mph)
Motion: Weave + Capsize
Nature : Asymptotically stable behavior

This speed range is the stable range for the bicycle, as the eigenvalues corresponding to both weave and capsize have no positive numbers. The bike will weave back and forth, less so each time, and eventually roll straight ahead, although not necessarily in the original direction.

Speed Range Of Interest : 7.0994 m/s (15.880 mph)
Motion: Weave + Capsize
Nature : Capsize speed critical point, with stable weave

Capsize speed is that speed at which capsize does not grow or decay, as can be seen by the eigenvalue which is at 0. Hence, capsize speed is 4.8539 m/s. It forms the upper stability range bound for this bicycle. Crossing this point gets us over the stable range.

Speed Range Of Interest : Greater than 7.0994m/s (v>15.880 mph)
Motion: Weave + Capsize
Nature : Stable weave with unstable capsize

Small positive eigenvalue for capsize gets it into unstable mode. Eigenvalues with imaginary parts, but whose real component is much smaller than the positive eigenvalue overwhelms oscillations. The bike slowly leans farther and farther to one side, without oscillation, until it finally falls over.

Thus, stable speed range for an uncontrolled Litespeed Ultimate is between 10.858-15.880 mph, but for all practical purposes, we could say it becomes easily balanced above 2 m/s. One important thing to realize is that capsize instability in these regions is very slow and thus can be easily corrected by a controlling rider. Also note that all this time, we have deliberately avoided talking about wobble. Wobble is complex and cannot be described without analyzing tire dynamics. This is an on-going study in bicycle dynamics.

In the final series to come shortly, I'll show you a cool peice of literature one of my readers has authored from which we might be able to study how changing the parameters in bicycle design affects the modes of bicycle motion. Now you can all take a deep sigh and have a good weekend.




CONNECTED READING :

Dynamic Stability Of Bicycle Design : Part 1
Dynamic Stability Of Bicycle Design : Part 2
Dynamic Stability Of Bicycle Design : Part 3
Dynamic Stability Of Bicycle Design : Part 4
A Bicycle Model To End All Models
"Linearized Dynamics Equations For The Balance & Steer Of a Bicycle" - J. P. Meijaard, Jim M. Papadopoulos, Andy Ruina, A. L. Schwab, 2007
History Of Bicycle Dynamics

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Dynamic Stability Of Bicycle Design : Part 2

Continued from Part 1.

Here, we'll study some fundamental concepts associated with bicycle motion, before we step into play mode. This is necessary for understanding what is to follow in later parts.

A bicycle is a single track vehicle and its dynamics can be studied with or without a rider. For purposes of our discussion here, we consider the human to be passive, rigidly attached to the rear frame of the bicycle, providing no control feedback whatsoever.

Fig 1 : A diagram of the model. Courtesy : Koojiman et.al

Engineers and scientists love to make models to study behaviors of systems. And it turns out that we can go ahead and make a mathematical model (see above) of the bicycle with a rear frame, a front frame with handlebars, and finally two wheels. Certain assumptions are made in the process, such as frictionless revolute joints, non-slipping rolling contacts for tires and knife-edge wheels.

Researches such as Francis Whipple and others have done this for almost a century, hence we stand on the shoulders of giants. The state of the art in bicycle dynamics also adopts the model and the linearized analysis behind it happens to be experimentally verified to justify the assumptions made in creating the model.

After introducing non-holonomic rolling and kinematic constraints, the typical 24 dimensional bicycle can be reduced to 3 to represent its configuration space. These are :

1) The roll rate of the rear frame, ф* (phi dot)
2) The steering rate, δ* (delta dot)
3) The angular rate of the rear wheel relative to the rear frame, θ* (theta dot)

Just imagine how complex analyzing bicycle dynamics in 3 dimensions is , leave alone analyzing it as a 24 dimensional system.

As it turns out, the state of the art model of the bicycle has 25 different design parameters, like the real world bicycle. These are shown in the graphic below. Just count the tick marks as you go along.

Fig 2 : Parameters affecting bicycle motion

Here, we get an idea of the different things that affect bicycle design. Its not just trail, or this angle, or that length, or this mass, but a picture bigger than that. Moreover, we can infer that there is an inter dependability among parameters. For example, changing the moment of inertia of your bicycle wheel is likely to change its mass as well.

In terms of their physical significance, single track vehicles possess 3 main modes of motion. There are precise scientific terms for these modes and its important that one doesn't muddle up their definitions and meanings.


1. CAPSIZE

Capsize mode is a non-oscillatory behavior involving both roll and steer and its prominence depends, among others, on the bicycle's speed and deceleration of the bike. The forward speed at which capsize motion neither grows nor decays is called capsize speed.

Basically, the mode tells you when the bicycle will lean over and fall and how easily it will do this. A bicycle without a rider at very low speed is unstable in roll and will simply fall to the ground laterally after moving into a tightening progressive spiral, sort of like a broomstick upon the action of gravity. A rider with some basic skills can easily stabilize this mode.

If a bicycle is rapidly decelerated by locking up the front wheel, it could capsize. In cornering at higher speeds, the ease with which capsize occurs (if you would consider it a rider controlled capsize) determines the cornering maneuverability of the bike. If capsize mode has a lesser time constant (less falling time), you can lean into turns and execute curves a little more correctly. So the lesser the falling time, the more unstable is this capsize mode.

Thus, we see how taking some stability away from the bike affords maneuverability. An overly stable bike is sluggish to control. Its not very responsive.


2. WEAVE

Weave is a complex, oscillatory behavior, 2-3 Hz in frequency, in which the bicycle oscillates or steers sinuously around the axis of the ground in the headed direction. The forward speed at which this oscillatory motion neither grows nor decays is called weave speed. This, although separate from the idea of the high speed shimmy or wobble cyclists always talk about, has a component of wobble in it that it is difficult to say which is which. These two modes are associated with each other in reality, although we like to think of them as separate.

Here's a video demonstration of weave I obtained from the net :

3. WOBBLE

Wobble is an unstable, oscillatory, steering motion. It can also be called steering oscillation. In popular literature, it is called Shimmy, a word that originates from an American dance style in the 1920's.

Here's a video demonstration of wobble. Use it to differentiate from weave.

From observations and anecdotal evidence, it is widely agreed that this mode occurs at some low speeds (see weave above) and also comes into play at some critically high speeds when frequencies are rapid from 5-9 Hz in range. To put things into perspective, a baby being rocked to sleep is at about 1 Hz. 9 Hz or more is rapid and dangerous and can quickly lead to a loss of control unless the rider consciously reverses the negative damping through body movements or braking.

The problem with rider provided damping is that sometimes, high speed wobble can be so quickly induced by some external disturbance that it takes the unassuming rider by surprise. This disturbance could arise from an irregularity on the road, or a bad mass of air, such as the wake turbulence from a box truck passing a cyclist on a descent. This initial condition could become large quickly before the rider can even react appropriately. The self-excited growth of this oscillation could lead to catastrophe.

From decades of detailed studies in motorcycle and airplane wobbles, researchers have agreed that a study of high speed shimmy is one involving the study of the elasticity of the steering head and frame and the complex interactions that come into play at the tire-ground interface.

When someone tells you that your loose bearings are what's causing the shimmy and you are completely sure that there are no loose bearings after periodic inspections, its time to expand your curiosity to the flexibility of the front end of the bike as well as the type and condition of the tyre you're using.

Almost all vehicles have the shimmy problem. It seems to be one of the engineering challenges in transport. A well designed bicycle is one in which the natural frequency of wobble is well above the speeds at which people normally travel on a bicycle. But meeting this is a challenge as bicyclists often like to mix and match different products and components while building bikes. Perhaps it would be wiser on the cyclist's part to keep the idea of a restricted speed range in mind while enjoying high speeds.

At this point, I'd like to shift your attention to the topic of this series. It is bicycle stability. In the next post, we will look at the bicycle parameters of our state of the art model and see how changing the values of the bicycle's parameters as shown in Fig 2 influence dynamic stability.

Keep the cup of your favorite beverage ready. And the rubber side down.



CONNECTED READING :

Dynamic Stability Of Bicycle Design : Part 1
Dynamic Stability Of Bicycle Design : Part 2
Dynamic Stability Of Bicycle Design : Part 3
Dynamic Stability Of Bicycle Design : Part 4
"Bicycle Dynamics", Experiences Blogged By Engineer Jason Moore
"Linearized Dynamics Equations For The Balance & Steer Of a Bicycle" - J. P. Meijaard, Jim M. Papadopoulos, Andy Ruina, A. L. Schwab, 2007
History Of Bicycle Dynamics

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