Complicated Math for an Obvious Solution

Recently, I've been thinking of ways to maximize my average speed during road rides. I had previously heard that it's most efficient to have a steady power output up and down hills, but since one spends more time going uphill than downhill, it seems that to go faster, one should expend more energy on the uphills. My question was how much more energy?

To find the answer, I set up an equation to calculate total energy expenditure over a symmetrical hill, of slope alpha, including the work done against and by gravity along with wind resistance, using the average speeds uphill and downhill.

E = [work uphill] + [work downhill]
E = [m*g*sin(alpha)*d + 0.5*C*p*A*d*v1^2] + [m*g*sin(-alpha)*d + 0.5*C*p*A*d*v2^2]

where:
E = energy
m = mass
g = gravity
alpha = slope of hill
d = length of one side of the hill (hypotenuse)
C = drag coefficient
p = density of air
A = frontal area as related to drag coefficient
v1 = average uphill velocity
v2 = average downhill velocity

I condensed the constants/knowns and split up the velocities into distance/time to come with the generic differential equation below.

C1 = C2[1/x^2 + 1/y^2]

where:
C1 = Energy
C2 = 0.5*C*p*A*d^3
x = time uphill
y = time downhill

Solving for y;

y = [C2*x^2 / (C1*x^2 - C2)]^1/2

I then plugged this into Matlab using somewhat reasonable values for energy and distance and a range of x-values from 210 to 1400. I plotted x vs y, or time uphill vs time downhill (figure 1). I added the arrays to find average velocities, then found the maximum value in the new array of averages, and finally the ratio of time uphill to time downhill for the maximum average velocity.

Figure 1: Time uphill vs time downhill

The maximum average velocity occurs at the apex of this curve. It turns out that the golden ratio, is 1. For the maximum average velocity, you should spend the same amount of time uphill as you do downhill, so uphill velocity = downhill velocity. Perhaps this should have been obvious from the start. To answer the original question, nearly all of one's energy should be devoted to riding uphill to obtain the fastest average speed.

Even though the answer isn't revolutionary, there are a couple conclusions to be drawn from this exercise.

If you're trying to go as fast a possible, put significantly more effort into the uphills. The problem with this is that it is highly inefficient and not sustainable for long. It's much more efficient to have a steady power output. To find this optimum, breaches the realm of biomechanics, and I would guess, would be very difficult to truly quantify and would involve extensive testing for each individual. Knowing the terrain and your own limits and how quickly you can recover will get you a long way though.

Another conclusion is that it's more worthwhile to improve your weaknesses than to improve your strengths: spend less time going slow. In mountain biking, bike selection can be critical in this regard. For example, I'm great at riding uphill, hardtail or full suspension doesn't make much difference. But I lose a lot of time on the downhills, so full suspension it is. I also get tired accelerating out of corners, so I went with 27.5 inch wheels rather than 29. Also, full suspension 29ers are a lot of bike for someone my size and can be difficult to maneuver quickly; however, there are situations where the stability of a 29er would be advantageous (like the fast, flowy trails in Colorado).

There are other fun things to be done with this equation such as finding the ratio of one's power output that goes into overcoming wind resistance compared to riding up hills of various slopes. But I'll leave that for another time.

Gravel Racing

 

Last February and March I competed in my first gravel races. The Rock Crusher in Ackerman and the Bentonia Burner in, well Bentonia where each about 100 miles and roughly 80% gravel. I DNFed in Ackerman. The first 50 mile lap took over 4 hours (partially due to a wrong turn and an extra 6 miles), and I simply wasn't prepared to ride 9 hours that day. Also, the field only consisted of about 8 riders and got completely strung out in the first 5 miles, so it just didn't feel much like a race. After the first lap I opted for an additional 25 mile loop just to get in some extra miles. But overall, I had a pretty good time and that's all I was really after. Bentonia was a few weeks later, and I was prepared to redeem myself. It was a single loop with no real option to bail out. I got pretty tired and hot after about 80 miles. I knew I needed to eat more but wasn't all that hungry due to (I think) eating too much peanut butter and other slow digesting foods. Next time I'll opt for more quick energy during the race. But finish I did, reasonably exhausted after 6.5ish hours.

Both of these races were done on the Inbred single speed, running 38x17 in Ackerman and 38x16 in Bentonia which worked well, along with some wide/flared drop bars, the legendary Brook's saddle, and my home-made frame bag. I think this was a great setup, and yet, I still find myself wanting a gravel/cross bike. Would a gravel bike be easier? No. Would it be more fun? Not really. Would it be faster? Eh, a little, but who cares? Will I eventually get one anyways? Maybe, depends where I live next and how much gravel is nearby. There's a lot of gravel around Oxford, but getting to it requires a lot of paved roads which aren't much fun on a mountain bike, single speed or otherwise. Additionally, gravel isn't much fun on a road bike with 25mm tires. So I would certainly use one here if I had one. But this raises another concern. I currently use each of my bikes roughly equally: since using Strava, I've put 3600 miles on the road bike, 2600 on the full suspension, and 2400 on the single speed. The addition of a gravel bike would probably mean my road bike would gather dust and may only get used for the occasional group ride.

Endurance racing (and racing in general), for me, has never been about beating the other guy but about testing my own limits. It's always a personal challenge to see how far and how fast I can go. And that, I think, is one of the fundamental differences between roadies and mountain bikers. Road racing is about winning; mountain and gravel racing is about finishing as fast as you can. Others may see things differently. In any case, I look forward to the next round of gravel.

A Secret Weapon

Over the last year or so I've observed a pedalling technique used by more advanced riders, and after experimenting with it myself, it's proving to be a valuable asset. It's essentially just out-of-the-saddle pedalling but in an aero position. Basic instructions would be to put your face near your stem, your butt in the air, and pedal, keeping your upper body as low as possible and your weight back. It's also good to have a relatively low cadence for this to work well. The great thing about it is that it's equally applicable to both road and mountain biking.

On the road it makes for a powerful and fast sprint with more power than regular out-of-the-saddle pedalling and far less wind resistance. It's great for keeping speed up short hills, sprinting for city limit signs, and any other time you need to produce a lot of power.

On the trail, where aerodynamics are less of a concern, this position keeps body weight centered which keeps handling and traction in check while allowing maximum power output. I often find myself accelerating in this position without really meaning to.

The massive power produced comes from the full body-weight on the forward pedal and from the rearward hamstring being fully extended - where it is the strongest. The downside is that all the force generated by the rearward hamstring is transferred to the forward quad making this technique inefficient as hell. This technique also requires A LOT of strength. Not just quad strength but core and upper body strength too. It's definitely something that has to be worked up to, but for me it's been well worth it and I've only been at it for a few months now. The good news is that the strength gained from practicing this technique transfers well to seated pedalling.

The Urban Ride

Here in Oxford, the beginning of the fall semester is the most dangerous time to ride. Incoming freshman don't know their way around town and probably aren't used to cyclists on the road. Other students are getting settled into new apartments and creating a lot of traffic. Not to mention the football crowd on home-game weekends. Last fall, I remember thinking I'd surely get hit before the semester was over, but traffic actually got a little better as the semester progressed.

I used to take offense when people would talk bad about cyclists, but after recently observing some other riders, I get it. Riding erratically through intersections in the rain or at dusk, with no lights is a sure-fire way to get injured and piss people off. Seven years of bike commuting, dozens of close-calls (usually my fault), and now one collision (not my fault) have taught me how to minimize risk on the road.

Visibility is most important, and several factors contribute to this.

• Location in the Lane
• Sidewalks - Most new riders view this as the safest place, when the truth is precisely the opposite. While sidewalks are usually separated from traffic by a curb, a cyclist is, for all intents and purposes, invisible when riding on one. Drivers turning into and out of cross streets, driveways, and parking lots will not see you. If you find yourself forced to ride on one, it's important to check for traffic in every direction at every intersection.
• Bike Lanes - You may think these are better than sidewalks, but realize that they are not built for the safety of cyclists (at least in America). They are built for the convenience of drivers - to get you out of their way. Aside from being littered with debris, man-holes, roadkill, sharp objects, and parked cars, a cyclist is only slightly more visible here. I should also take this time to mention that bike lanes are directional, and it's a bad idea to ride against traffic (known as salmoning). Cross traffic turning right will not see you.
• Where is a cyclist most visible? In the middle of the lane. Unfortunately, it's not always practical to ride in the center of the lane. As a general rule, the further right you are, the less visible. Riding as far right as possible also encourages drivers to try to squeeze passed you which is neither fun nor safe. Laws usually state to ride as far right as is practical. My go-to spot is where the right tire of a car would be. It's far enough left to be visible and to encourage drivers to use the other lane to pass but not so far as to make it difficult to pass. It also gives me room to maneuver around roadkill and pot holes.  When making a left turn, I'll move to slightly left of center to further discourage passing. If riding in the left lane of a four lane road, I'll also ride left of center to allow cars to safely pass in the right lane. When in doubt, take the lane. 
• The exception to this is waiting at stoplights (see below).
  
• Speed
  What does speed have to do with visibility? A lot of drivers may see you but assume you're travelling at walking speed. It doesn't matter if you're going uphill or downhill since many drivers seem to have poor understanding of the local topography (and distance for that matter). I can't count how many drivers have pulled out in front of me (or next to me) not realizing how fast I was going (+25mph downhill). Mostly this is just something to be aware of as a cyclist, although I've found that a front blinky light can catch their attention long enough to gauge your speed.
  
• Lights and Reflectors
  Reflectors may be unfashionable on sport bikes, but for commuters they're well worth the extra weight. They cannot, however, replace a good set of lights. While essential for night riding, I also use lights whenever the weather is anything less than perfect. High-viz clothing/backpacks/bags is also a good idea if you feel the need.

Laws
While following the rules of the road to a T might be considered the absolute safest (it's not), one of the biggest advantages of cycling is not being imprisoned by the grid. Hopping curbs, riding sidewalks (with caution), and taking shortcuts otherwise unavailable to cars often makes cycling a faster and more convenient way to get around. I'm not advocating to blindly disregard traffic laws but to question the laws and find the logic behind them. Often the reasons behind the laws don't make sense for cyclists. In other cases, following the law isn't enough, and in a few cases, following the law is dangerous. Cyclists need to be vigilant and conscious of certain situations.

• Take stop signs for example. Drivers are trapped behind blindspots (A-pillars), dirty windshields, and darkly tinted windows which limit visibility. They're also deaf to the outside world with closed windows and/or loud music, hence the need to stop at intersections and check for other cars. Cyclists on the other hand, have no blind spots in front of them and can hear cars approaching from all directions. Completely stopping at an obviously empty intersection is ridiculous. If cars are present, of course, I'll wait my turn. The same logic can be applied to stoplights, especially since sensored lights are rarely triggered by bikes. When it comes to waiting in line at stop lights, I'm a bit torn. On one hand I feel I should wait my turn; on the other, I think waiting in line is for people who chose their mode of transportation poorly. Recent experience* has taught me that coming to a complete stop can actually be quite dangerous as it means you're less visible to other road users. If you must wait at a stop light, it's best to move out of the lane, if possible, unless your completely certain that the vehicle behind you is aware of your presence.
• Passing on the right, or undertaking, is dangerous regardless of whether or not you're actually in a bike lane. This is usually tempting when traffic is backed up, but caution must be exercised. Drivers turning right won't see you and neither will unloading passengers, which brings me to the next point.
• The Door Zone is the space adjacent to parallel parked cars where open car doors would prevent riding. Give parked cars a wide berth to avoid a surprise door opening to your face.
• Other street-side parking. Diagonal street-side parking is hardly any better. I usually ride mid-lane in this situation, but when a car backs up into traffic from this position, there's really no good evasive option. You can brake and risk being run-over from behind or you can lane-split and simultaneously try to avoid oncoming traffic and the reversing car. Given enough time I'll move right and brake, but usually lane-splitting seems the safest. I've learned to look for reverse lights, but a lot of newer cars are wired up so that these lights come on even when the car isn't actually in reverse such as when doors are open or when the car is unlocked remotely.
• Drivers making a right turn will sometimes cut you off which usually isn't a big deal if you're paying attention; it's just annoying. Recently, however, I had a very close encounter where if I had braked half a second later, I would've found myself under a Tahoe. Luckily I escaped unscathed, and as the driver was pulling into a parking lot I decided to confront him about it since I usually don't get the opportunity. He claimed to have not seen me, but there's no excuse for that. If his eyes had been on the road, he would've seen me.
• Getting passed by cars. People can't be trusted to use good judgment, so use your lane position to force them to. Narrow lanes, blind curves and hills, and riding at high speed require riding further left to prevent or discourage passing. Conversely, wide roads, good visibility, and low speeds allow you to ride further right to let cars easily pass.
• Round-abouts can generally be viewed as miniature Nascar ovals. They're easy enough to get through - yield, be predictable, and stay in your lane, just don't expect anyone else to. It's unfortunate that the round-abouts on Old Taylor Rd are designed so poorly. The exit ramps from the highway split into three lanes before entering the round-about which often leaves a car in the middle lane stopped, yielding to traffic, while a car in the right lane blindly blows through the intersection at a lethal speed. Always keep an eye on approaching cars.

Heat
Riding in the heat and humidity can be brutal, and eventhough it's an often heard excuse for not riding, I see a lot more cyclists in the summer. Truth be told I sweat about the same amount whether walking, riding, or driving - mostly because I like to roll the windows down and refuse to waste gas to cool the car down before going somewhere. My car's interior can easily reach 150 degrees after sitting in the sun, and while the air coming out of the vents is quite cold, the interior will radiate heat for some time contributing to quite a bit of sweat. A couple ways to sweat less when riding are to ride slower and to use a rack to get any cargo off your back. Also, light-weight, loose-fitting clothing and open-toed shoes can be nice. I usually don't really get sweaty until I stop and even then I can be mostly dry within about five minutes of being indoors.

* I started writing this post in early August. Having been hit by a drunk driver recently (while on a highly visible pedicab no-less) I've gained a little more perspective and realized I've gotten too comfortable in traffic. Reading through this now, I find myself asking, 'Is it worth it?', Yes. The necessity of awareness in every moment is what makes me feel alive, what calls me to ride.

A Test of Endurance

62 mile ride with no food after a 5 mile run and 3 hours of sleep.




After riding 35 miles Saturday afternoon and working Saturday night, I found myself unable to sleep due to being stressed out over basically nothing. I finally went to sleep around 3am and woke up just after 6am (about my usual time). Unable to go back to sleep, I got up and went about my business. I took Lucy for a 5 mile run, then had my usual breakfast - bacon, egg, and cheese burrito, all fried in bacon grease of course. I then prepared for a ~60 mile group road ride at noon.

I didn't have any good snacks to bring along, but there was a gas station stop at the halfway point. The first half was a fun, fast ride, and I was feeling pretty good despite a lack of sleep. At the halfway stop, I was still feeling good and decided to make things interesting by not eating anything. I don't think I've ever ridden 3 hours with no food. I've been trying to train my body to run on stored energy rather than what's in my stomach and wanted to see if I'd made any progress. I fully expected to bonk sometime between 2 and 2.5 hours. Not only did I not bonk, but I stayed near the front on some fast, sprint sections near the end.

Once I got home and showered, I ate enough Chinese food to feed about three people! Fully relieved of all stress, or perhaps just too tired to care, I did sleep pretty well that night. Still, work came too early Monday morning.

Tire Pressure

Tire pressure is the fine adjustment element of a bike's suspension. It also plays a part in traction, rolling resistance, and efficiency. While I don't pay much attention to it on my road bike, I like to have it dialled in on the mountain bikes.

A couple months ago two homework problems got me thinking about front vs rear tire pressure and it's effects on contact patch and traction. The first homework problem was for Fluid Mechanics and involved calculating the contact area of a tire given the weight of the vehicle. The second problem was for Dynamics and involved calculating the maximum acceleration of a car with front, rear, and all-wheel drives.

From basic physics, I learned that the force of friction is equal to the friction coefficient times the normal force (f = μN). From the "real" world, however, I know that tread patterns, contact area, and tire pressure all effect the friction between a tire and the ground. It turns out that the formula from physics makes some assumptions that don't hold true for tires as seen here.

So, while I was bored in class one day, I decided to calculate what my front tire pressure needs to be relative to the rear to have equal contact patches and therefore equal traction. One assumption I made is that my weight is centered over the bottom bracket. On my singlespeed this is reasonable as I tend to unweight my handlebars often (corners, bumpy sections, etc). I measured the distances from the center of the bottom bracket to each axle and calculated the fraction of my weight on each wheel. Calculating the weight on each wheel was kind of a round-about way of doing things; all I really needed was the lengths. Alternatively, a more accurate way to do this is to use a bathroom scale to measure the actual weight transmitted through each wheel. From there I found that my front tire pressure should be 68% of the pressure in the rear tire (for the singlespeed; 62% for the full suspension) for an equal contact patch.

The result was a bit surprising. The difference in pressures was much greater than I anticipated. I used to run a couple psi higher in the rear just because it seemed to make sense, but 7psi seemed like a lot. I decided to try it out anyways to see how it rode and ended up really liking it. My method is to find the minimum acceptable front pressure and adjust the rear accordingly. For the singlespeed, this is 14psi front, 21psi rear.

One question that came up is 'Do I want equal traction front and rear?' Another was 'Is there anything in practice that would make this not work?'

For the former, if anything, it's better to have more traction in the front. The rear tire sliding out isn't a big deal, but the front sliding out will put you on the ground in a hurry. Running near equal tire pressures means you're either giving up front wheel traction or increasing rolling resistance in the rear unnecessarily or both. It's noteworthy that I use the same tires front and rear. Front traction can also be increased by using a bigger and/or knobbier tire which could complicate things.

For the latter question, it can be argued that the front wheel is subject to higher impacts since the energy it absorbs is absent when the rear wheel hits the obstacle. This could necessitate a higher front tire pressure relative to what the calculations show. On the rigid singlespeed, this hasn't been an issue. The full suspension is still in testing.

Bump 'N' Grind '17

When I first started racing mountain bikes, I would always get full of nervous excitement waiting at the starting line. My heart would be pounding five minutes before the race even began. In the last year or so this has changed; I'm much more calm and relaxed at the start. I think it's due to experience, getting stronger, and longer races. Racing singlespeed/cat 2 was a sprint from start to finish, but cat 1 races leave plenty of time to figure out who gets to be first. For some reason though, Bump 'N' Grind brought out a little bit of that nervous energy.

The race started out well enough. There was a split early on; out of the ten riders in my age group, four of us started trailing behind. I was having trouble keeping my speed through the corners due to running worn out tires at higher than optimal pressures due to the fast and rocky nature of the Oak Mountain trails. When we hit a short gravel road section, I saw the faster group up ahead and an opportunity to close the gap. Surprisingly, I did manage to catch up to them, but the other three riders behind me didn't follow.

At this point we were only a few miles in, and it started drizzling which slowly increased to a steady rain. The roots got slick in a hurry, and it wasn't long before there was a small river running down the trail. I soon fell behind again, back to the slower group which was beginning to spread out. I made up some ground on a long jeep road section, but lost it again in the technical singletrack. As the mud got deeper and my chance of doing well disappeared, I basically gave up and cruised the last ten miles or so. I finished the 29 mile course in 2:39, dead last not counting two DNFs.

After the race, I was completely covered in mud. My legs had a solid coating and the 'Ole Miss' on my jersey was illegible. The line to the bike washing station was a mile long, and I needed to check on Lucy - my windows had been halfway down the entire time... I had also used my only towel to cover the windshield to keep the sun out. After finding the parking lot mostly empty already and letting Lucy out, I decided to head for the lake across the road. I wasn't the only one with the idea. I went for a swim, shoes, helmet, bike, and all. I air dried, changed clothes, claimed a couple of free beers, watched the awards, got some food, and headed home.

Aside from the mud and crappy tires, there were other things that contributed to my poor result such as a lack of familiarity with the course and a lack of racing technical trails. Oak Mountain isn't the narrow ribbon of singletrack I'm used to. Hopefully next year I'll be better prepared.