Aerodynamic Drag Chart For Human Powered Mobility

This sort of thing might come in handy for your investigations. Thanks to Troy Rank, an engineering student at RIT and electric bike tinkerer, for showing me this. The chart lists the aerodynamic drag co-efficient and a host of other performance factors for human powered vehicles. I'm not sure of the true source of all this data but will duly give credit if I find out.

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Modern Bicycles and Cycling Speeds : Any Measurable Relation?

Its not about the bike. Or is it?

Without a shadow of doubt, most of us will say that today's Grand Tours are faster than those of the past. True. For instance, since its inception in 1903 to the 1990's, the Tour de France had seen its winner's average speed increase some 50-55%  as this site will show.

But here's the big question - how much of that speed increase came from bicycle improvements alone? If you don't factor in the contributions from all other things- temperature, course, race tactics, improved training methods, nutrition and doping - what role does bicycle technology alone have to play in higher speeds? Is it significant to be appreciated?

This most entertaining problem is one that maybe analyzed with a technique called multiple regression. This method, a staple in any statistician's arsenal of tools, allows one to estimate the effects of many factors on a single dependent variable, in our case - cycling performance.

For starters, there are a number of independent variables that factor into a favorable cycling performance. I have shown these factors diagrammatically below.

In my opinion, these independent or explanatory variables can be broadly termed into 4 categories :

1. Human Performance Related - Physiology, training, nutrition, medicine and doping
2. Technology Related - Bicycles, fancy apparel etc. We'll disregard other things and consider just bicycles.
3. Race Specific - Course, weather, tactics employed, rules, etc.
4. Random Events (Noise) - Example - a freak crash 2 km from the finish line that injured many riders, a neutralized stage due to the death of an athlete, any day to day variation that cannot be predicted but is present. 

In 100 years of cycling history, innovations have come and gone. Some have stuck through to Grand Tour racing, the list of which is mandated by the final word of the UCI.

To consider the effect of just bicycle technology alone on cycling speeds, a multiple regression analysis has to be performed. You would require lots of data for many years and a handy computer to make some meaning out of it. Unless someone gives me serious money, I won't be diving into such an endeavor.

But recently, Ph.D's Jan Heine and Mark Vande Kamp who write for the magazine Bicycle Quarterly sought to answer this question in their article titled "Are Modern Bicycles Faster? An Analysis of Tour de France Speed". To me, the article appeared to be a logical investigation of why speeds increased in the Tour and whether they could be explained by the latest racing bikes. 


The article had ignited controversy in cycling circles about its apparently "flawed" analysis. I think it will be to everyone's benefit if the strategy of the article's investigations are clarified first and foremost. We'll then explore its conclusions.

Here's the strategy behind the article's investigation :

1. Fundamental assumption : The fundamental assumption that the authors imply, but which is not stated explicitly in the article, is that all modern bicycles and related technology are introduced into the market to strictly increase cycling speeds. With this assumption, they proceed to quantify how much that speed increase is.

2. Eliminate day to day performance variations : They selected the Tour de France as the main race of interest with this notion that multiple stages and over 150 riders will eliminate the influence of day-to-day variations in fitness, weather and other factors on individual performance.

3. Eliminate course specific variations : With the view that courses change "somewhat" in the Tour de France, they selected the Milan-San Remo as a supplement in the analysis as the race has been run on the same course for over a 100 years without change. The race's difficulty has also been consistent since smooth speed curves have been displayed for over a century.

4. Separate human performance improvements from bicycling technology improvements :  This one is tricky so pay attention. The authors wanted another race as a control to compare cycling with. They thought of a race from another branch of endurance sports that had little to do with technology or inconsistent conditions and where performance was mostly limited by the "human factor".

They selected medium distance running, specifically the 5 and 10 km running race from all events worldwide and studied trends in running speeds. The logic? If bicycles have truly become faster, the trend line for cycling speeds in the Tour would deviate from that of human speeds in running by showing step increases. If bicycles have not become faster, the trend lines should closely match each other due to the "human factor" common to both endurance sports.

5. Regression Analysis : Using the data of speeds, a regression analysis was performed on the Tour de France and running speeds for the last 100 years. The "athletic performance" regression lines would show the long term speed trends for both races. This was made into a "Chart 1". "Chart 2" was also made where the authors smoothed TdF and 10 Km running speeds for many years by taking a 5-year running average. These curves were compared to each other and to the long term "athletic performance" regression line in Chart 1.


Summary Of Results :

1.  Co-relation between actual TdF speeds and speeds predicted by the runner's trend line was 0.94. Strong.

2.  Co-relation between actual running speeds and the long term running speed trend line was 0.95. Also strong.

3.  88% of increases in TdF speeds over the last 100 years can be explained by improved athletic performance.

4.  For both running and cycling, there appears to be an unexplained 9-12% that are simply random occurrences seen when athletes compete.

5.  The regression curve (or line fit) for TdF speeds have a shallower slope than that of running indicating that cycling speeds increased at a slower rate. The authors propose that this is due to wind resistance factor in cycling as power demand increases by the cube of velocity. But the non-linearity of aerodynamic resistance is not much, it is instead minimized in the Tour de France and spread over a large group of riders.

6.  Over the last 20 years, TdF speed increase trends parallel that of runners' speeds. Technology has had minor roles to play in these achievements according to the logic in the analyses (no step increases were observed).

7.  There were steeper speed increases in the TdF in between 1926-1940 than running speeds during that time. The early 1920's saw periods of low performance and the authors propose that World War I had depleted the pool of cycling champions taking part.

The late 1920's, however, showed a marked speed increase was not observed in the Milan San Remo which got the authors to conclude that something particular to the TdF caused these increases. They propose the radical shortening of stage distances as a possible reason.

There were pronounced speed increases in the 1930's that corresponded well with the significant, revolutionary and long term changes introduced on racing bikes such as lightweight steel frames with thinwall tubing. The authors state that of all advances, lightweight steel frames had the most pronounced effect on Tour speeds. These speed increases were also observed in the Milan San Remo in the 30's as well, indicating that this was a sport-wide phenomena.

8.  Since 1947, speed increases in cycling, relative to runner's speeds, came during times when cycling technology did not even change. The late 1950's saw a jump in cycling speeds but nothing significant was invented or innovated in bicycles during that time, since the introduction of Compagnolo's rear derailleur in 1951. Since speed increase came at a time when technology was stagnant, the logical conclusion is that speed increase cannot be explained by technology. The authors state that other reasons, like the paving of roads, may have been primarily responsible.

9.  In the early 1980's, TdF speeds increased between 1981-1982 without a rational reason and then dipped down without an explainable reason as well. Between 1985-1990, time trial bikes, such as those used by Greg Lemond in his 1989 Time Trial did increase stage speeds but the time trial stages were too short to influence overall speed of the entire Tour. Moreover, the bikes used in mass-start races "evolved little" during this period, wrote the authors.

10.  From 1999-2009, lots of things in bicycles evolved - from index shifting, to rear cassettes, increased gearing, aerodynamic wheels and ceramic bearings. Sure, the speeds of the Tour de France saw an almost linear increase as well. But what the authors found was that the long term trend of running speeds tracked this increase in cycling speeds very closely indicating that almost all these improvements can be tracked to physiological factors common to both running and cycling.

Since 2005, speeds started to drop below the predicted trends, possibly indicating that strict doping controls are responsible for the lower speeds. Speeds decreased 3.5% from their peak, while running speeds decreased only 1.8%. This shows that something not common to both sports have influenced the speeds in cycling.

By now, you must be tired with all this information overload. So let's take the justifications provided by the authors for speed fluctuations and plot it on a chart for the last 100 years. I did it below for you :

Conclusions :

The authors wrote that there is no evidence that advances in cycling technology since WWII led to faster racing speeds. There is no systematic co-relation between the two.  Some speed increases came during times when athletic performance as a whole were increasing. Others came at times when bicycle technology and innovation were stagnant.  The only period where bicycling technology led to a pronounced speed increase was during the 1930's with the introduction of lightweight steel frames. Bottom-line of this whole affair is as follows, quoted from the article :

"It is tempting to look over the Tour de France speed curve and pick [technology] factors that appear to have caused increases or decreases in speeds. [...] However, when taken in the context of all the data, these specific examples don't add up to a compelling case that bicycle technology increased Tour de France speeds. Neither of them stand up to close scrutiny.  [...] Across the whole timeframe of the last 100 years, even radical changes like the introduction of the derailleurs did not alter the trend of Tour de France speeds. Clearly, the larger pattern suggests that bicycle technology has had little, if any, effect on racing speeds, especially in recent decades."

Critique & Suggestions :

1. Choice of control : Why was medium distance running chosen as a control and not ...say, the marathon? I don't know. The authors don't provide an explanation for this deliberation, although they suggest that the medium distance races do not see much "influence of technology". So does the marathon see influence of technology then? I don't know. You would think not. Long distance running, to me, is the purest form of endurance sports. It would be interesting to see if marathon running speeds closely followed all the trends of cycling speeds for the past 100 years.

2. Choice of race : One will agree that are simply too many variables in the Tour de France to make a valid relation between one aspect, such as cycling technology, and another aspect, cycling speeds. Why not extend the research to a solo performance such as the hour record where variability is reduced even further? Or a time trial? Or a sprint? In a past post, I revealed details of a study that found that between 1980 and 1990 before UCI regulations came about, 60% of cycling hour records in the discipline were solely due to engineering. The authors may want to catch up with that.

3. What to investigate : The authors start off the article by asking the question - how much faster are the lastest racing bikes compared to classic machines? But it seems that throughout the article, they tend to look at small innovations across years such as the rear derailleur, or increased gearing, or thin walled tubing to see if they made an effect on the "overall" speeds of each year's Tour. How could does a tiny component translate to anything appreciable in the overall speeds across successive years? Rear derailleurs or improved front brakes alone don't make any appreciable change to Tour de France speeds across successive years.

4. Details of each stage : The exact details of each stage were not investigated by the authors. It would be interesting to see how many flat stages and how many mountain stages each Tour so far consisted of and how gravity would play a role in changing outcomes. Data may be tricky to find. Now keep in mind that we do have data for the speeds, distances, number of entrants and number of finishers in each of the Tours. Perhaps blending all this information into one graph for different eras of cycling may lend some insight.

For illustration, lets take the Hinault Era (1978-1985). I plotted speed (kph), % of entrants who finished the race, and number of stages with respect to the years and the distances involved. Check this out :

You may be able to come to some kind of understanding about what was going on in those 7 years. For instance, during 1980-1982, speeds increased drastically. It is also interesting to see that between 1980-1981, the number of participants who finished the race had also risen and the distance in Km of the race had fallen, although the number of stages were increased from 22 to 24. It would be interesting to superimpose the percentage of km in uphill roads and downhill roads on this graph for those years. It would also be interesting to see how the "Badger's" temperamental tactics and pace control influenced the speeds in those years.

5. Successive yearly investigations vs leaping : Based on the initial question posed, it would be more meaningful to take a vintage racing bike and a modern racing bike and compare the two.  Hypothetically, a 1903 racer traveling across a period of 100 years into the future to ride the Tour de France on a 2010 race bike with a lighter frame and aerodynamic wheels should be faster. Similarly, a group of 1903 racers climbing a 2 hour long Alpine climb on 40 pound steel bikes would be slower than the same group of racers riding on flyweight machines of similar sizes in similar terrain. It is basic physics.

Investigating this issue year by year, where all riders would have access to the same bikes and the same technology won't show you clearly how cycling technology is improving overall Tour speeds, if they do at all. Besides, some modern equipment and technical wear don't always serve to increase speed solely. Some of them have intangible benefits as comfort and so on. That is an advantage when you stay seated in the saddle for 90 hours of racing.


Overall, I don't think this is as bad of a study as many people think. Besides, it was published in a popular magazine to open up a forum for discussion. It is not a rigorous scientific white paper.

I do agree with one thing that studies like this discover time and time again - that majority of racing performance is related to the human body.

Racing is never a level playing field, no matter what race it is or how much you want to complain - be it the Berlin Marathon, or the Tour de France, the 24 Hours of Le Mans or the Baja 1000. There's always those few individuals genetically gifted or blessed with the finances and talent needed to win.

Then there are those who cheat to win. They may have the talent, but they want to boost it with some extra energy from extraneous sources, illegal by all rules.

Kamp's and Heine's study corresponds with several people's observations that cycling speeds have been coming down since 2005 due to doping regulations. In July during Tour time, I had done my own analysis of this Stage 17 power to weight ratios and my approximate figure of 6 W/kg agreed with other people's observations, among them the Science of Sport bloggers (see their article).

In the end, we may never know exactly what portion of those early TdF speed increases were "fabricated" through cheating. How much came from Amphetamine use, or alcohol, or narcotics, steroids, growth hormones, EPO and blood transfusions or using mechanical devices? Food for thought?

What do you think? Come discuss this article and its implications!

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How The Copenhagen Wheel Works

One of the prime points behind the marketing of Copenhagen Wheel was its regenerative braking capability. Back in January, being the curious cat that I was, I wondered whether the design would be practical for Copenhagen where the outlook for 2012 is in establishing biking superhighways and such.

The question posed by me was how many stops it would take a cyclist to recover at least 10% of the total battery capacity in the wheel. My argument was that if it was a good amount, then it may make sense in crowded city riding conditions. With biking superhighways, number of stops may decrease so would regenerative braking be a much sought after feature then?

Recently, Christine Outram of MIT's Senseable Lab wrote to me clarifying the working of their design. From her, I learned that the wheel they showed at the Climate Conference was actually prototype number 2, with an off-the-shelf motor selected due to their time constraints. After the Conference, the team began their prototype 3 design phase to custom design the hub motor in order to make it smaller, lighter and more efficient. Through custom building of parts, she claimed that the hub would end up at about 2/3rds of the current size and weighing 3.5 kg.

I also picked up on some other important things from her email. For those who want to know how the wheel really works during pedaling, read on.

1) Using the bicycle : When cycling, the rider can choose one of three motor assist modes : low, medium and high. Each mode represents how much the motor is supplementing the cyclist’s cycling input (eg: effort x1, x2, x3) as calculated through our torque sensor.

There exists a sleeve-style mechanical torque sensor to measure the amount of effort the rider exerts as well as the speed and direction that of applied pressure. The readings from the torque sensor direct the motor on how much to supplement the rider, based on the motor assist mode the rider has chosen. Similarly, when the rider wants a workout or to recharge the batteries, they may choose a low, medium or high ‘exercise’ mode. Here, the rider is working against motor eg: effort x (-1), effort x (-2), effort x (-3).

2) Battery life in motor assist mode : Christine provided me some sample calculations to calculate the battery life. Typically, a flat city commute of 2 km will make a 70 kg cyclist expend 31 Watts of power. This distance can be covered in 8 minutes.

So if the motor assist is set to times two, the same commute mentioned above can be done in 4 minutes. In this case, the battery can be expected to last 8 hours on a single charge.

The same commute but with a 0.01 grade slope (gentle up-slope, 1 in 100), which is common in cities, requires 70Watts of power. This is a more likely scenario to use the motor assist mode. If set to times two, the batteries will last just over 3 hours and 50 minutes.

If the motor assist is set to times two, a rider can climb a 0.05 grade slope such as in a hilly city, like San Francisco, with the same level of effort and at the same speed as he or she would typically apply on a gentler 0.01 slope.

3) Battery charging in exercise mode : Let's consider the same commute distance. If the exercise mode is set to times two (the same commute will require twice the effort), the rider must provide 140Watts, and can burn 6.7 kJ (16 food calories) in his or her 8-minute commute. Additionally, if on exercise mode times two, discharged batteries will be fully charged in 5 hours and 30 minutes.

Note that the above points talk about the charging and discharging of the battery in pedaling mode, and not regeneration through braking. I hope we can get further clarification about that aspect from the team.


Christine stated the goals of their design as follows, in her own words :

"1. Providing ease of usability. Most other e-bikes on the market have bulky battery packs, are hard wired and are not able to be retrofitted in an existing bike.

2. Creating an elegant design where all components can be packed into one hub.

3. Augmenting the bicycle with small amounts of technology to encourage people to cycle - hence the social and data networking aspect. It may not be a motivator for everyone, but the electronics will continue to get cheaper, and if it convinces some people to ride a bike, then this is great.

4. Pushing the research agenda of the lab, which includes understanding how distributed sensing in cities can help us understand a city's infrastructure, how it can influence and promote sustainable behavior and how we can contribute to the design and development of sustainable places. This research agenda is also about imagining a future, 10-15 years from now, when these technologies will be much improved and fully integrated into our lives. Through this vision, we attempt to create leapfrog urban demos. in the present day. Hence, the construction and demonstration of the Copenhagen Wheel, which was made possible through bringing together from a variety of backgrounds including architecture, urban planning, computer science, network analysis, interaction designers, sociologists and mechanical and electric engineers."

Good stuff?

ADDITIONAL READING :

Practicalities Concerning The Copenhagen Wheel
Boston : From MIT, A Quantum Leap In Bike Mechanics

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Crank Brothers Candy C Pedal Failure

A reader who doesn't wish to be named sent me an email with this image. What is shown here is apparently a right side Crank Brothers Candy 'C' pedal that failed on an indoor Revmaster stationary bike. Good for him it was indoors, and the fact that the bike is built with safety features for holding onto something when falling in embarrassment.


The user estimated that he would have put a little more than 100 hours of time on these pedals, all during home use at fairly low pedaling intensities. He is a fairly big person, at 6'2" and weighing 217 pounds.

When it broke, he had been standing on the pedals for want of some variation and was putting forth moderate effort. But he remarked that despite cycling since the 1980's, this is the first pedal he has ever broken. He has other Crank Brothers pedals in stock for outdoor riding but this event has diminished his confidence in those as well. Natural.

The pedal features a chromoly spindle. I do not know its geometrical specifications. Curious enough, I visited the Crank Brothers website (its very well developed actually, so Kudos to them!) and obtained a screen-shot of the spindle from one of their servicing videos. The failure, as I see it, happened just where the cross section started to step down :




Loads & Stresses : The spindle acts like a simplistic cantilever beam with the concentrated reaction force R and pedal force F as I show below.


Then, for a solid spindle acting like a beam, maximum shear stress at the neutral axis would be about :


where V is shear force and A is the cross sectional area of the round section.

By looking at the length of the spindle above compared to its depth (L>>d), one might be able to say that what would be more concerning than the shear stress itself is the spindle's bending stress at one or two critical locations (sites where part sees high stress risers). The designer would then multiply the concentration factors necessary for these stress risers to the equation for maximum bending stress at the spindle's outer fiber given by :


where c = outer radius of section, M = bending moment & I = moment of inertia

High Cycle Fatigue (HCF) : Since the user told me that he uses this moderately at home, the correct technique to analyze the spindle would be in HCF, in a mode of fluctuating loads. Here, the stresses seen would be much below the yield of the spindle material but the loading would gradually accumulate until failure. This happens because of local yielding at critically stressed locations.

To generate a rough idea of fluctuating load on the spindle, I made some assumptions. Research has shown that a good figure for average pedal force for long periods is about 1/5th of body weight. If this is the average, then an approx. max force could perhaps be 75% of body weight. So for a 217 lb person riding moderately like the individual here, a possible range of forces could be as follows :

Fig.1 : Peak forces are seen roughly when the crank is forward, 90-100 degrees past top dead center. These peak forces are a resultant of effective force (radial) and ineffective forces (tangential). For specific illustration, mean component of foot load, Fm, is 43.4 lb and in tension. The alternating component is given by Fa = (Fmax - Fmin) /2 = 119.4 lb. Mean and alternating components of the reaction force Rm and Ra is same as for the foot loading forces. Time period is given by t = 1/frequency. Note : Blue region for upstroke has been exaggerated whereas in reality, peak magnitude might not be so high.

The shear and bending stresses can then be derived from these alternating and mean loads. A Von Mises stress calculation can be done for both alternating and mean stresses. These V M stresses can then be plotted on a modified Goodman diagram along with the corrected endurance limit and yield strength to check visually where in relation to failure boundaries this pair of stresses lie. The safety factor of the part in service comes specifically due to how this pair of stresses vary with each other.

Since the material is steel with a knee in the S-N diagram, a good estimate for design pedal life for infinite life in steel would be a capability to withstand the maximum stresses shown in the figure above for 2 million cycles and over. A normal 90 RPM cadence equates to 1.5 Hz frequency or 1.5 cycles/second. So a designer can make a pedal spindle with chromoly to sustain itself for the range of life shown below.

2 million cycles / (1.5 cycles/sec) = 1.3 million seconds = 370.37 hrs.
200 million cycles / (1.5 cycles/sec) = 133.3 million seconds = 37037.03 hrs.

If a company has the time and money, they'll design for higher life than 200 million cycles.

The point I'm trying to make here is that given the rough loading curve for one rider above you, the actual pedal spindle would have had to be designed to withstand much higher amplitudes of stress cycles for many different riders across a broad spectrum. This should also arrive after providing for fudge factors to account for geometrical & environmental influences!

However in this case, the spindle locally yielded at lower stresses in a matter of 100 hours of life to failure. This was in a controlled environment. This does not correspond suitably with the range for HCF life calculated. On the surface, it appears the spindle might not have been designed/manufactured properly which brought down the strength of the spindle considerably from its predicted life.

What do you think? Should the pedal spindle be beefed up in size to accommodate heavier people, designed with milder cross-sectional variations or heat treated appropriately to overcome this dangerous scenario? And what are your experiences with this particular pedal?



RELATED RESOURCES :

4130 Normalized Steel Alloy Material Properties
Spindle Shear & A Related Injury
Falling Out of Love With Crank Brother's Pedals
Integrated Sensors In Intelligent Bike Parts
How A Clipless Pedal Works

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52x12 vs 52x11 Gearing : A Look At Chordal Action

If you're the type who likes to gun it down the line in high gears, you may have wondered more than once - what is really better in terms of crankset chainring-rear sprocket size, would it be a 52T x 11T or a 52T x 12T?

At the same RPM, you get a slightly higher top end speed with the 11T at the sacrifice of some torque. But if you're asking this from an efficiency standpoint (ratio of input power over output power), its a slightly tougher nut to crack if you don't have actual measuring equipment.

I won't talk directly about efficiency but I'll talk about something else that you may want to start connecting, perhaps more with equipment durability than efficiency.

A bicycle chain has links, connected by a distance called pitch which is usually 1/2 inch in bicycles. When you're riding your bike at a cadence of 100 RPM, the chain has an average velocity called pitchline velocity. At the sprocket though, some interesting things happen with chain velocity.

For one link in your chain to engage a teeth in your sprocket, the link has to swing about an angle before the roller is seated between tooth. This is called Angle of Articulation, calculated by using the relation : 180/T, where T is the tooth count of the sprocket.


For a 11 tooth sprocket, the angle of articulation is 16.36 degrees, while for a 12 tooth sprocket, it is 15 degrees, a reduction of 8.3%.

Because the chain is turning at these sharp angles at the same time impacting the teeth, the velocity of the chain is not constant, but infact fluctuates between the maximum and a minimum each cycle. The maximum occurs before engagement, the minimum occurs after the link has swung in engagement. This change in velocity is called Chordal Action.

The point is that chordal action results in fluctuations in chain transmission and may be minimized by reducing the angle of articulation, which decreases with increasing sprocket size. For 11T and 12T sprockets, which are about the smallest standard sizes you can find in the bicycles, the angles are tight which results in uneven exit velocities.

But is it something you should worry about? The math is complicated to present here so I just did all the calculations myself elsewhere to look at chordal action. Perhaps you can decide whether it matters or not after looking at the following theoretical numbers.

Say you're riding hard at a cadence of 100 RPM with a 52 front chainring. The RPM at the rear sprocket is multiplied by the gear ratio, which results in 433 RPM with 12 T and 482 RPM 473 RPM with 11 T. With a 12T sprocket, one cycle of tooth engagement sees a 3.4% variation of chain linear speed, whereas an 11T sprocket sees 4.1% change in linear speed for the same because of the tighter angle the link has to swing.

Here are two graphs I generated :

Note : Chain velocity is a function of pitch circle diameter, RPM and articulation angle

What this does for transmission efficiency in bicycling is moot. What I wanted to present before you is the fact that chordal action is real in small sprocket sizes, and it has an effect on your tooth wear and pulsating motions at high RPMs. But it may not matter in low cadence (60 and below) and if you're using wear resistant, hardened sprockets. Feel free to discuss.



ADDITIONAL RESOURCES :

Dan Connelly : Drivetrain Losses

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Practicalities Concerning The Copenhagen Wheel

Good morning!

For lack of a super-stimulating post today, I thought I'd immerse myself a bit in the hype surrounding the Copenhagen Wheel before the subject becomes rusty and unwrap this pretty looking present.

So I'm taking it for granted that you folks will have some prior knowledge of this new hybrid e-bike introduced by MIT's Senseable City Lab.

Well, the folks are not actually marketing a bike technically.

They actually demoed a hub-motored wheel during the Climate Summit in December that you can shift around and adapt to different bikes for "city riding", the idea being that it stores energy through regenerative rear wheel braking action and gives it back to the rider when they need a 'boost'. It is sort of a leaner, maybe meaner version of the gas powered Revopower idea that never took shape. Remember that?

An attractive proposition is that this wheel can be monitored by a smart phone via blue tooth and all the electro-mechanical aspects are packaged into one assembly - the wheel itself. And then it measures environmental variables like noise, pollution and so on and throws all that data into your phone for your city riding pleasure.

It is by all means an academic project, funded by the city of Copenhagen, Ducati Energia and the Italian Ministry for the Environment.

Now when I hear of projects like these, I switch off the ads and start looking into specifics. I thought for an institution like MIT, you would expect to see some numbers posted on their website at the least, or a whitepaper indicating weight, wattage requirements, cost, efficiency etc. I certainly haven't found much other than a Java based applet you can sort of play around with.

So here are my questions & concerns for the boys and girls at MIT, which I think I sounded off a month back to Bicycle Design.

What is the energy content of the battery (Wh/kg) and energy use per distance (Wh/km) of this product? Multiplying both will give us an estimate of distance/kg of battery weight and total energy capacity per charge. Knowing cost of electricity, someone can easily calculate the true cost of riding the bike or a bike equipped with the wheel and then compare it with with a bike not equipped with it.

Then comes the question of braking energy recovery. How many stops would a cyclist have to make to recover at least 10% of the total battery capacity back and by how much will their range be extended?

Let me give you a perspective. A 180 lb man with a 20 lb bike traveling at 15 mph has 2026 joules or 0.56 Wh of kinetic energy to shed, in order to come to a stop. Note : These are back of the envelope calculations I just roughly did.

So, with what efficiency will the recovery system be able to capture some of that? 70%? 80%? Let's assume 75% which is a reasonable number. With that, I can get back 0.42 Wh or 1516 joules back, but I can only put this energy back into the battery at a certain charge rate which really depends on the battery's specs.

So for illustration, if battery's max charge rate is 100 joules/sec, 1516 joules will take 15 seconds or more to put back in.

Now realistically, no one takes that long to stop, even though the US CPSC recommends that brakes should stop a cyclist in 15 seconds from 15 to 0 mph.

So realistically, let's say people take about 3-4 seconds every time to stop. Using the 100 J/s charge rate, cyclist can only put in 400 Joules or 0.11 Wh back every stop. So the question again becomes, how many times does a cyclist have to stop to put back 10% of kinetic energy of braking, at 75% efficiency to get the extra amount of miles, given the limitations of the battery?

If its a reasonably good amount, it may make some sense in a crowded urban area like Copenhagen, otherwise not really considering buying costs and life cycle costs. Again, only the average numbers over long riding in the city will give you any solid perspective on the practicality.

In short, without specifics like what I'm calling for, no one should start rooting for this product as a game changer just yet, for that is just silly in my book - a case of inflated yet unwarranted attention that a lot of blog fodder command these days.

Sometime back, I noted that the city of Copenhagen is en route (forgive the pun) to "biking superhighways". With that kind of a future for this city, I expect some of the stopping usually encountered in tight city riding conditions to reduce. So then the question is, will the Copenhagen Wheel even be deemed necessary then?

As usual, I ask too many bothersome but nevertheless important questions. The answers are reluctant to arrive. Or it just might. Right MIT? :)

ADDITIONAL READING :

MIT Press Release : MIT's Big Wheel In Copenhagen
The Revopower Puzzle : Where Is It?

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Thomson Masterpiece Seatpost Failure

In this post, we will explore the mechanics of a simple no-nonsense bicycle seatpost collar. What encourages me you ask?


I was alerted to a Thomson seatpost failure at the Weightweenies forum. The item is a "Masterpiece" brand, the lighter cousin of the utilitarian "Elite". The post severed at the seatpost clamp as an image provided shows. Thomson makes everyone's favorite seat-posts.

As we often have it in such cases, the claim from the poster is that a torque wrench was indeed used to install the product and it was done so by a "pro mechanic". By that, I take it that he knew what he was doing. By that, I meant that he did not give a "few extra" turns than required to tighten the screw.

You may know that I have explored two other Thomson seatpost breakages in detail this year. Those efforts involved a background study of Thomson's seat post design and interviews with the user and the product person over at Thomson. Statistically, in the end we all had to agree to the idea that such failures result more from a common human condition called the "Over-Tightening Syndrome". I have written about it here with the help of Thomson's Dave Parrett.

As for this new breakage, it is advisable to let Thomson know so they can test it for signs of extraneous use of force. There's also a good chance they might replace it for the customer on goodwill/warranty.

THE MECHANICS OF A SEATPOST CLAMP

The common cyclist's seatpost clamp may be closely approximated by what they call in engineering circles as a "Split-Ring Clamp Type Shaft Collar" (there are all kinds of weird shapes and designs out there for bicycle seatpost collars but we will limit this discussion to circular). A hex socket cap screw is used to clamp the collar around the seatpost so it constrains it from motion/slippage due to seated weight acting downwards.

When the collar is set with the cap screw, three brilliant things are taking place in your equipment that you probably never heard of :

1) Pre-Loading The Cap Screw : As the cap screw is tightened in the collar, it is stretched to produce the clamping force regardless of the external load. This force, or initial tension, is called a "Pre-load". So without considering the external load, the torque necessary to achieve this pre-load is given by the relation :

Fig 1 : Pre-Load Screw Equation. Courtesy for K table : Bowman Fasteners

The seat post manufacturer or supplier is to select/supply capscrews that have sufficient pre-load capability. They specify diameter, pitch, threads per inch etc of said screw for the particular service. Pre-load force is found from the torque desired.

The pre-load stress is found by dividing pre-load tension over the tensile stress area for the fastener from reference tables. The same process is used to calculate service load stresses, i.e external stress divided over the tensile stress area of the fastener. When you have these two quantities, a pre-load stress and a service load stress, you compare them both to the known proof strength of the said fastener from reference tables. Ideally, they should be lower from the proof strength by a sufficient safety margin because proof strength is the maximum load a screw/fastener/bolt can take before acquiring a permanent set (plastic deformation, no good).

2) Axial Holding Force : When tight, an internal pressure acts on the collar surface due to contact with the seatpost through its internal diameter. There exists an axial holding force from the collar that is proportional to the co-efficient of friction between the collar and seatpost and the torque in the screw. Avoiding lengthy integral based derivations, axial holding force is given by :

Fig 2 : Axial holding force of a seatpost clamp/collar

3) Hoop Tension : A tangential component of stress acts inside the collar at any radius due to the action of internal pressure of the seat post mentioned above. It is called "Hoop Tension". By relating this tangential stress αt to hoop tension, we arrive at the following relation :

Fig 3 : Collar hoop tension

When the collar is set by tightening the cap screw to some torque value, the hoop tension is equal to the screw tension. This is the role of the screw - to induce hoop tension so that it produces clamping for the seatpost.

Armed with that knowledge and the above 3 equations, one can easily find the tension in the screw and the internal pressure of the seat post on the collar. While out of the scope of this blog, you can even calculate the maximum shear stresses in the collar and find out your factor of safety for the design so that it is safe for a rider to use.

You can take this information further and ask - what happens if I screw my cap in too much, vastly exceeding specified torque values? The result could be too much hoop tension, too much clamping force, too much shear stress, too much pre-load. Its a combination of these things and/or manufacturing related defects and the kind of external loadings induced by the weight of the rider and his usage patterns which all set a clear path to failure.

So be cognizant of these things... and why, Happy New Year too! :)

ADDITIONAL RESOURCES :

Shigley's Mechanical Engineering Design
Fastener Tech : The Nuts & Bolts Of It
Bolted Joints (Plymouth University)
Strength Of Threaded Fasteners
Torque Wrench Accuracy Measurement
LH Thomson & Company

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Analysis Of The Bicycle Endo

The endo, short for end-over or end-over-end, is a type of pitch over crash where the cyclist goes over the handlebars, the weight offset of which causes an inertial moment to act about the front wheel resulting in rear portion of the bicycle to flip in the air above and behind him.

Usually, the cyclist, as a sudden reflex action, yanks out their hands or legs at some point to cushion the impending fall and ends up letting go of handlebar control. Meanwhile, the bicycle is bound to fall either sideways, due to its motion about the steering axis or right on top of the cyclist. In the latter scenario, the saddle or even the rear wheel itself could land on the cyclist's body.

Again, an endo is a crash that could cause injury. It is not a bicycle trick. That one has another name. Its called a 'stoppie' or a 'wheelie'. An endo is caused due to strong front wheel braking or when the bicycle hits a curb, a structure more rigid than the wheel itself. Endos may also occur if the front wheel is loose, i.e, if it is not secured properly to the fork dropouts by the quick release skewers.

In this post, we will cover the "endo parameter", study the relationship between braking force and endo parameter on level ground, outline some common reasons for endos, check out a video analysis of an endo and finally study the relationship between gradient of the road and endo parameter through a literature source.


ENDO PARAMETER

Braking a bicycle naturally upsets equilibrium and transfers weight to the front wheel. With a stark increase in the overall braking force, the load on the rear wheel approaches zero, after which the rear wheel will start to lift off the ground. Hard braking may stop the bicycle but Newton's first law reigns supreme as the cyclist's body continues in motion in the headed direction. This rider motion has some momentum. If not self-controlled, the rider will flip over the handlebars and the bicycle will pitch-over as well. What results is the endo.

It turns out that while outrageous situations cannot be helped, some factor of safety from bicycle design and rider positioning skill can provide for a cushion against pitch-over tendency in the above mentioned situations.

I'll call my main parameter of interest the pitch-over parameter (or endo parameter for lack of a better word) - A/H - as can be seen in the diagram below :

Fig 1 : Free Body diagram of a bicycle-rider system just at pitch-over. Now you may be able to appreciate from geometry and c.o.g as to why recumbents and tandems are stable in pitch-over. O is the point signifying the contact point of the front wheel with the ground.

TERMS :

W = combined rider-bicycle weight
Wf = normal load on front wheel
Wr = normal load on rear wheel
Ff = braking force at front wheel
Fr = braking force at rear wheel
Fb = braking reaction (mass times deceleration)
L = wheelbase
A = location of center of gravity (c.o.g) aft of front wheel
B = location of c.o.g forward of rear wheel
H = height of c.o.g

The endo parameter, A/H (a ratio), in the combined bicycle-rider system should be large enough to avoid front pitch-over. Obviously the vertical height, H, of the center of gravity (c.o.g) and the location of the c.o.g aft of the front wheel, A, are going to vary with variation in rider's weight, height and sitting position.

This highlights why its important to get a proper bike fit for the type of bicycle you wish to ride. Its not just a question about comfort. Its also a question about safety. Enlarged riders who overwhelm miniature bikes not made for their size will quickly find out what they're doing wrong. All they have to do is hit the front brakes hard and they're right on target to be turned into human projectiles.

Looking at the free body diagram above, we can deduce that the system is in static equilibrium about the front wheel contact point O if the sum of the moments due to all forces about that point is zero. In other words, rotation is just initiated at :

Fig 2 : Endo parameter relationship to braking force. This also gives us an expression for the braking force at the point of the pitch-over.

From this simple relation for level ground, we see that endo parameter is equal to the braking force as a percentage of total weight at just about the initiation of the endo. Braking force is a function of the co-efficient of friction at the tire-road interface.

Before the initiation of pitch-over, the braking force-weight ratio is lesser than the endo parameter. Well after the pitch-over has been initiated, the endo parameter falls lesser than the braking force-weight ratio.

We can now infer that making A/H larger is better for safety. Otherwise, a lesser braking force relative to total weight will be sufficient to initiate pitch-over. How? Simply because the braking force-weight ratio catches up with the endo parameter sooner. Oops.

A/H can be fixed to be greater with good bicycle design and proper fit. It can also be superficially made larger by the cyclist while riding by positioning his body rearward (relative to bottom bracket) as the following picture shows :

Fig 3 : A cyclist ducks and shifts his c.o.g rearward to increase his endo parameter

People succumb to pitch-overs because of other factors too. They may not be skilled enough to increase the endo factor, A/H. They also may not be skilled enough to modulate and may tend to hitting the front brakes really hard without realizing that a front brake can cause more deceleration than a rear brake. For comparison, front brakes generate upto 0.5g's of retarding force whereas rear brakes produce a max of 0.1 or 0.2g's. Note that maximum deceleration is limited by the co-efficient of friction between tire and road and the normal load.

You can notice front braking power for some mountain bikes through a speed vs time graph.

Fig 4 : Speed vs Time graphs of MTB's (Courtesy : Beck Forensics)

Obviously, the graphs show that you can bring a bike to a stop faster using the front brakes alone than the rear brakes. Using both brakes is even better for reducing stopping distance even further.


SEQUENCE OF MOTIONS IN AN ENDO

Beck Forensics did an interesting little video analysis of an endo. The following image as well as the snippet below it is taken from a web sampler of their book Bicycle Collision Investigation. It shows the steps involved in an endo before the crash.

A mountain bicyclist traveling at about 22.5 mph (36.2 kph) applies only the front brake. Once the front wheel is nearly locked, the rear wheel starts to lift up. At about 0.20 seconds, the rider really has no chance to recover. At about 0.33 seconds, he releases the brake and prepares his right hand, and then his left hand for landing. The cones are shaped in 25 foot (7.6 m) intervals and the grade is about -2% (descent). (Courtesy : Beck Forensics)

RELATIONSHIP WITH PERCENTAGE GRADIENT OF GROUND

This section is a little more involved. It uses the same analysis techniques shown above to derive a relationship between the "endo parameter" and braking force-weight ratio with the percentage gradient of the ground. You will see that the chances of an endo are more likely on a descent.

The following literature is from one digest of IHPVA (2001), written by a retired engineer named Frederick Matteson. Click on the series of images to zoom the text. Alternatively, you can also read the paper here.

Enjoy!

Page 1 : Click to zoom

Page 2 : Click to zoom

Page 3 : Click to zoom

ADDITIONAL READING :

Budbrake : Proportional Brake Control For Safer Bike Stops
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

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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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