Recumbent Bicycle Design by Charles Meredith Brown - November 2015
Steering and Ride Roughness Formula "If I have seen farther than others, it is because I have copied off the test papers of giants."
   
Sections:
Cornering ability
Steering feel
Handlebars
Ride roughness formula

 

Three laws of bicycle motion
Short drive train designs
Wheels and rolling resistance
Also see: Bicycle frame design, Air drag formula and Putting it all together

In this section I'm combining the results of my experiments with what I've been able to find in books. Let's imagine you have a test bike with two identical wheels. Let's also imagine this bike is adjustable so you can vary the percentage of weight on each wheel, with all other things being identical.

Cornering ability

For the first try let's have, say, only a little weight on the front wheel. In hard cornering you find the front wheel slips out easily, and once it starts to slide, it's almost impossible to recover before you go down. (Incidentally, if you want to do this sort of testing, as far as I can tell bikes skid the same way on gravel as on concrete, only at a slower speed. If your feet aren't fastened to the pedals, you can usually get a foot down in time.)

Well, that was exciting. Now let's try the opposite condition, with only a small portion of the weight on the back wheel, lots on the front. Now the back wheel slides out easily. but now it's much easier to control. You see, the wheel doing the steering now has a good grip on the road, making it easy to keep things rubber-side down. The two steering conditions we have just experienced are called 'oversteer' and 'understeer' in automotive texts, but in the older stuff I used to read, the terms were used interchangeably!

For the best overall cornering, on this bike with identical wheels you would want between 50 and 60 percent of the weight on the front wheel. This way both tires are doing about the best they can, but at the limits of cornering power, the back wheel starts to slip just a bit before the front one does. This gives you a little warning and makes it much easier to control at the edge of adhesion. And if you say, ďIím not a racer, this doesnít affect me!Ē, think about this the next time youíre bombing down a hill through blind curves and something unexpected jumps out in front of you. Better-steering bikes are safer.

How do you figure wheel loadings for bikes with different size wheels? It's easy! As far as I've ever been able to tell, with tires of the same tread and width, desired load varies directly with wheel diameter. For example, let's say you want your bike to skid the same as a bike with 50/50 weight distribution with identical wheels. However, your wheels are different sizes with, let us say, one wheel half the diameter of the other. You want the little wheel carrying half the load you have on the big one (i.e.: 1/3 the total weight). For example, let us say you want the cornering characteristics of a bike that has two equal wheels and has 50- 60% of the weight on the front, but your bike has a 406 front wheel, and 700 back. You would want about 41-49% of the weight up front.

Physicists will tell you that coefficient of friction is independent of surface area. However drag racers, who live in the real world, have found that wider tires give more traction for accelerating their vehicles from a stop. You might think, therefore, if one wheel is slipping on the turns before the other wheel, a wider tire on the slipping wheel might help.

Actually, NO! Surprisingly, here a narrower tire can help bring things back into balance! When you are rolling along in a straight line, the middle of the tire contact patch and the midline of the wheel are both pointed in the same direction. When cornering, this is no longer the case, the side force twists the tire contact patches of both tires. The difference between the direction these patches are pointing and the way the wheel is pointing is called the íslip angleí. As cornering force increases, slip angle increases until it reaches a point of maximum cornering force. Beyond this point, the tire will break free and slide. The maximum cornering the bike can achieve will be if it is balanced so that both tires meet their limit at about the same time. Having the rear wheel skid slightly before the front gives up a tiny bit of cornering power, but gain in control and ďfeelĒ is more than worth it.

Remember to think in terms of balance. Tire width can be used to help tune the steering. For example, most LWB bikes don't have enough weight on the front wheel. A narrower front tire can help get it to stick. In practice, however, I find sometimes a lighter tire reduces the directional steadiness of the bike because there is less gyroscopic effect on the front wheel. If this is the case for you, a wider tire in back might be your answer. Most SWB lowracers have the opposite problem- too much weight on the front tire. Here the front tire should be wider than the back, which can reduce your rolling resistance too. In practice, I find it takes a lot of width to make a significant difference. Wider tires make the bike more controllable, as well as a heck of a lot nicer to ride, on rough, gravel, or dirt roads.

All the above is fairly unimportant for just puttering around. The side forces going around the same corner vary at the square of speed. Skid characteristics become much more important at higher speeds. All through this section I've assumed you had similar tires on both ends of the bike, which I recommend. íWay back in the early í80ís, the only narrow, high-pressure 20Ē tire I could find had a knobby tread. The roads I was riding on often had sand washed over them. The back tire would grip best on clean concrete, the front tire on the sand, which was odd!

Long wheelbase bikes seem to go into skids more slowly, giving more time to react. A favorite trick of mine is to put most or all of the braking on the back wheel. When the bike just starts to skid, front wheel first, the natural reaction is to slam on the brakes. The back wheel thus now has both a braking as well as a cornering force applied to it, so it starts to skid too, more than the front. With the reduced speed and thus reduced cornering force, it could start to recover. I think that on a LWB bike, having the back brake much stronger than the front brake is by far the safest way to stop it. I donít like strong brakes on the front of LWBís. Locking up the front wheel first will cause a certain crash.

Steering feel

You can do all the above and still have a vile- handling bike! It is also important to keep the wheels in line, any flexibility will degrade the steering, particularly noticeable in a side-to-side maneuver, such as the slalom. This gets worse with increased speed. The frame needs to be torsionally stiff. Virtually any monotube frame will be torsionally stiff enough, multi-tube frames ought to be checked before construction. (Reducing the trail will ease some of the problems with a flexible frame.) The frame and fork should be carefully aligned, there should not be play in the hub bearings, the wheels should be true and the tires pumped up. Iíve measured this flex on some of my own bikes, and found that 1/3 of the total flex is in the back wheel! If you can find a rear hub with flanges farther apart, this should help.

Iíve learned itís important to have good control over how much the bike leans relative to how much the rider leans. On most recumbents this is controlled through the riderís contact at handlebars and seat. On uprights it is mostly through the riderís interactions with seat and pedals, and to a lesser extent handlebars. One of my own projects where this became a problem was a seat that came just a little bit loose; still it was enough to noticeably degrade the steering. Another example were some experiments with lots of extra padding in the seat base to serve as a type of suspension; these did not grip my bottom well enough side-to-side to give good steering control. Perhaps a suspended seat with no side to side motion would give better control? With all the different bikes Iíve built and ridden, in general the ones I thought had the best steering feel were short wheelbase bikes with two same size wheels.

Handlebars

I can offer only a few general impressions on this subject. You tend to get used to whatever handlebar geometry you have, so my opinions have changed with time as I tried various setups. Narrow handlebars are easier to ride one- handed, which is good when signaling for turns, using a frame mounted shifter, occasionally carrying something with the other hand, or if you just want to rest an arm. As for the amount of tiller, on narrow handlebars I think steering is best with the handlebars 6Ē (15 cm) or less behind the steering axis. Others will disagree with me. More tiller is okay with wider handlebars. The steering seems to be better if the section of handlebar you grip is straight rather than curved.

Formula for finding the ride roughness of an unsuspended recumbent bicycle

by Charles Meredith Brown, who is not the slightest bit ashamed of having come up with it:
I've built over 50 different recumbent bicycles since the early 1980's. By subjectively rating each one for ride qualities, and taking notes of the bike's dimensions, I was eventually able to work out a formula for predicting how smoothly a bicycle will ride. All test bikes had tires 25-28mm wide, pumped up to about 100 psi (6.9 bar), eliminating one variable. A rigid frame is assumed (some of my early test bikes had frames that were deliberately flexible, intended to serve as suspension).
Riders of upright bikes usually stand up on the pedals when going over rough pavement, letting the bicycle 'dance' beneath them, so they're not directly comparable.

Three laws of bicycle motion:

Imagine two wheels with equivalent tires, one twice the diameter of the other. Both roll over a small bump. The axle rises the same amount for each wheel, but the bigger wheel takes twice as long to lift up- the ride seems less jarring. It's not only the distance you are moved, it's the time it took to move you that distance as well. For the small irregularities we constantly run over, much smaller than the diameter of the wheel, ride roughness effectively varies as the inverse of diameter.
Another important point- the energy imparted to our bodies varies at the square of the distance we are moved per unit time. (Kinetic energy=1/2mass x velocity squared)

On a bumpy road, the energy required to shake you about like a pair of maracas comes directly from...you. For every action there is an equal and opposite reaction. Energy used to create your own personal earthquake equals energy taken from propulsion, so a smoother riding bike goes faster.

Let's start by figuring how much vibration you are getting from the front wheel. First, measure the distance from the center of gravity to the bottom of the back wheel. In the bike above, it's 92cm (36.4"). Now, divide it by the wheelbase- 132 centimeters (52") in this case. 92 / 132 = 0.70, or you can say 92 is 70% as much as 132. When the front wheel hits a bump 10 mm high, the center of gravity, you, move 7 mm.

This distance moved represents the velocity, and to represent the energy in that motion, we square it. 0.70 x 0.70 = 0.49 . Now we divide it by the outside diameter of the front wheel, in meters. A 406 wheel with a 28 mm wide tire has an outside diameter of 0.467m, so we get 1.05 . Always remember to divide by the opposite wheel from where you measure the bottom of wheel/CG length.
Whew! That's done!

Now let's do the back wheel. This is done just like the front wheel, only reversed. Let's say your center of gravity is 70.8 centimeters above the pavement. The center of gravity is 72.6 cm behind the front axle, so from the Pythagorean theorem, or by drawing a diagram and measuring, you get a center of gravity to bottom of front wheel distance of 101.4 cm. This is 76.8% of the wheelbase. Squared, this is 0.590 . The back wheel is 700 x 28, .0683m outside diameter, so you end up with 0.86 .
Or to put it another way, for the front wheel:

R(f) = (A(r) / Wb) ≤
              D(f)

Where:
A = lever arm from center of gravity to bottom of wheel
D = diameter
f = front
r = rear
R = ride roughness
Wb = wheelbase

Actually, this isn't the whole story, there is a pitching motion caused by the two ends moving over bumps at different times, it would be better if the bike stayed more level. Thus you get the smoothest ride for the wheelbase if C. G. is adjusted so that the front end has a little bit lower number, giving the front end a smoother ride. As a rough rule of thumb I find 90% is about right for a typical SWB; a shorter wheelbase and/or higher speeds increase the weight shift needed to get the best ride. The front and rear wheels are not additive; the rougher end predominates. Just multiply the rear roughness rating by 0.9 and use whichever one, front or back, is higher.

CALCULATED VERSUS SUBJECTIVE RIDE RATINGS:
Here are some of my homebuilt bikes rated both ways. I like rating the ride by how it feels to you, because this way you cover absolutely all the variables involved. The downside to this is that the seat of your pants is not a precision instrument, resulting in considerable scatter in the graph below:

I also want to show the effect of frame weight and stiffness on the ride. Light, stiff frames, like I'm sure you will be building and riding, have a rougher ride. If you want to build a really decent bike, and not a toy, I recommend a ride roughness rating under 1.0 . This is not the setup you want for best cornering! This is why I was playing with front suspension- just a short travel setup on a SWB bike would be the best of both worlds! I'd like to go back to studying suspension when this paper is finished. If anyone has come up with front suspension that does not absorb some of the rider's energy, please let me know.

The formula smiles favorably upon long wheelbases, low seats, and large wheels, as happens in real life. Many other things that affect ride don't show up in this math. Flexible frames, and heavy ones, will smooth the ride somewhat. Wider tires, even pumped up to the same inflation pressure, give a smoother ride. I've had some success with extra padding in the base of the seat. Results were better when the seat was contoured to match the rider's shape, reducing the tendency for your butt to move side to side. Some of my bikes with a more upright rider position seemed to give a smoother ride when the back of the seat didnít go so high up. Struts going from near the back axle to the top of the seat send vibrations straight to your noggin. I now run struts from near the back axle to the lumbar region of the back of the seat.

One simplification of the formula is that it assumes all vehicles are going the same speed- if you modify it for use at different speeds, it predicts that the roughness experienced will vary at the square of speed. Fortunately, something is lost between theory and practice- at speed, the wheels no longer follow every undulation of the pavement, and tires and frames flex more. It is an argument for having at least a front suspension on streamliners.

Short drive train designs

The long drive trains of most recumbents result in more weight, flex, and thus slower climbing than desired. Here are some ideas for bikes with shorter drivetrains:

Prone- I've built one prone, my first HPV ever. To minimize air drag, I had the rider as horizontal as possible.

I quickly learned that to make a practical bike, for use on the road, you want a low bottom bracket to make it easy to get your feet on the pedals, and a raised head for better visibility. You can raise yourself up further if you want for a look around. The result still has less air drag than an old-fashioned upright, and the center of gravity is lower and farther behind the front wheel. In an accident or heavy braking you're less likely to go flying. You don't have so far to fall, either. Note that you may need to strap the rider in for power production.

Front wheel drive (twisting chain)- I've never actually built one, but the design looks promising. My sketch shows a remote steering system, as it allows a shorter drive train with less bend in it, and you can turn further before the front wheel hits your thighs and feet.

Moving bottom bracket front wheel drive- I've built several, including some with hand cranks as well. My best and last was much like the Cruzbike Vendetta. With half the weight on the front wheel, the front wheel would sometimes slip a little when accelerating hard from a stop, but I didnít find it a problem in normal use. This is one SWB with a ride like a LWB. I enjoyed riding mine.


Wheels and rolling resistance

Later on in this article, we will be needing a good approximation of rolling resistance, which I'm getting by swiping it from other people's research. Some people have tested tires by rolling them down smooth floors. This gives a somewhat lower rolling resistance than you get on normal, rougher road surfaces. Tires that are more flexible, of thinner, lighter construction, gives lower rolling resistance, as do higher inflation pressures. Wider tires can offer lower rolling drag and a smoother ride, but at the cost of more air drag and weight. Assuming otherwise identical tires, rolling resistance varies inversely with diameter. Thanks to the work of John Tetz, we know rolling drag goes way up in cold weather. For really low rolling resistance, as David Gordon Wilson keeps reminding us, land speed records could be set on rails.

For finding wheel outside diameter, this formula comes out just about right:
Wheel rim bead seat diameter, plus tire width x 2, plus 5 mm = outside diameter.
With the above in mind, these are the figures we will be using with 28 mm wide tires through the rest of this article:

 
Nominal size Bead seat dia Outside dia Coefficient of rolling resistance
Small 16" 305 366 .00787
Large 16" 349 410 .00703
Small 20" 406 467 .00617
Large 20" 451 512 .00563
650 571 620 .00456
700 622 683 .00422
27" 630 691 .00417

  
  Next: Air drag formula
 

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