Another post in a series of “trail science” pages.
jump to turn and switchback calculator below
What’s the best turn design for a trail? Just what the “perfect” design arc and/or in-slope for a trail turn is part personal preference, part science/math. It’s not that this can’t be answered. Design depends on the user groups, safety, and sustainability.
Some turns are just awesome, others, not so much. Just how tight should a directional change be for the best flow? The feel is related to the speed at which runners and riders enter the turn. Speeds can be controlled with slopes preceding the turns to slow them down to the speed you want them to go (more on this later in a separate post), or give them a kick and more g’s with a downhill beforehand.Should a bank (or in-slope) be added to a turn? Do you want banks at all? The bank angles and arc degrees can be determined by plugging in variables and solving for what you need: rg tan ø = v^{2} (r= arc radius g= 9.8 m/s^{2} v=velocity ø= in-slope angle). The tighter the arc, or higher the velocity, the steeper the in-slope should be. Makes common sense, but the equation spells it out. Mass is irrelevant! Friction is not, but masses still cancel: centripetal force, Fc = mv^{2}/r = μ_{s}mg cosø + mg sinø
The First Law of Motion applies: riders will continue in a straight line unless acted upon by a force– the faster you go the harder it is (more force) to turn from that straight line, and the more likely that you’ll slide around a corner. As we see in the equation above (and the turn and switchback calculator below), an in-slope can make turn tightness irrelevant. Some turns are in-sloped a priori, some a posteriori, to make a tight turn work without slipping.
Head tube angles and wheel bases will change how bikes handle turns. Most XC bikes are near 70 degrees and most DH/FR bikes around 65. The latter is more stable, but not great at tight turns because the slacker angle increases the wheelbase. The benefit though is more stability in rough terrain, but harder recovery if knocked off-course— perhaps offset by more suspension travel. Take home lesson here: turns should be more drawn out for FR/DH, but this is obvious as speeds are generally higher as well.
When will a bike slide? This is relative to the tire and friction coefficient, as well as the arc or radius of the turn, whether it is in-sloped (and how much), and again the speed of the bike around the turn.
jump to turn and switchback calculator below
So back to the main questions of turn and switchback design:
Just what “should” that arc be? And if in-sloped or banked, how many degrees?
- The turn radius depends on the speed riders have
- The radius can be very tight: if the in-slope (and less-so the height) is changed
- As a general rule, consider this:
- A 10 mph turn speed (should be slower for blind turns, but there should be no blind turns)
- No bank (no in-slope)
- No sliding/slipping around the turn
- A coefficient of friction of 0.5
- The turn radius should be: ~13 feet
- Why 0.5? This coefficient of static friction changes with soil type/conditions, tire inflation, and tire selection. 0.5 is a ballpark average. If slippery or packed decrease to 0.3 or 0.4, if tacky or loose increase 0.6 or 0.8. You can determine the coefficient of static friction by this experiment: **below
- The Crr, coefficient of rolling resistance, kinetic friction μ_{k}: how to pdf | reprinted below
- At 15 mph with no in-slope and a 0.5 coefficient of friction that radius needs to increase to about 30 ft!
- Add a 10% in-slope to the turn and the 15 mph radius of 30 ft can be cut down to 20 ft or 17 ft at 15% in-slope
- Use the turn/switchback calculator below to play with turn radius and angles (for multi-use trails I’d personally recommend not exceeding 20% in-slopes, 3 feet high banks, or turns less than 9 ft radius. Most bikes will do fine at 16 ft, preferably greater. Although bigger turns can become a drainage, rut, and user conflict problem if lines of sight are poor. Correctly designed turns can control speeds to some degree, and eliminate a user-created demise of the turn.
**Determining the coefficient of static friction
- The angle at which the tire starts to slip on the dirt is directly related to the coefficient.
- Your tire needs to be put on the dirt you use (with same compaction as turn) and then the dirt would have to be tilted until the tire slides.
- You could build a 4 foot arc from 0% up to nearly vertical, and place the bike upright at higher and higher points until it slides
- It will slide when when the component of gravity exceeds the maximum value of the force of static friction.
- The coefficient of static friction is equal to the tangent of the angle at which the tire slides. f_{s} = f_{s max} = μ_{s} N or μ_{s} = F/N or sin(ø)/cos(ø)=tan(ø)=μ_{s}
- A similar method can be used to measure kinetic friction μ_{k}.
- Give the bike a push downward as you increase the angle
- When the bike slides with constant velocity, the tangent of that angle is equal to μ_{k}.
- You would need ramps of varying angles, not arcs for this.
- More simply, you could just attempt to turn on an open flat section of the same dirt at different speeds until you begin to slide. Measure your turn arc radius with a string by moving the center point away from the arc until the arc of the string matches your actual arc on the ground. Then plug that radius into the calculator’s radius as a starting point. Or just do complete circles or donuts and half the diameter.
- The Crr, coefficient of rolling resistance, kinetic friction μ_{k}: how to pdf | reprinted below
Trail Turn & Switchback Calculator |
enter a radius or bank angle below | v_{max} is max velocity without slipping/sliding |
where the angle of bank is θ = ° |
and the coefficient of static friction is μ_{s} = (0.5 is a ballpark average, if slippery decrease to 0.4 or 0.3, if tacky increase 0.6 or 0.7. The coefficient of static friction can be determined using the method** above, or consider the info on arcs below this calculator) |
the maximum speed for the banked turn with this coefficient of friction is
v_{max} = v_{max} = m/s = km/hr = mi/hr. |
and the maximum speed for a flat trail (no bank or in-slope) with this coefficient of friction would be v_{max} = m/s = km/hr = mi/hr. |
Determining a bike’s turning arc, and turn arcs for trails in the real world, sin-calculator:
- You could skip 2, 3, and 4 if you do circles in the dirt (or asphalt), as the donuts you make will give you the arc, or diameter (and radius) of the turn at the speed you make the circles or turn. Try doing faster circles or turns, and the arcs will get bigger. You now have a starting radius minimum. Err on going bigger, and use the calculator if you plan to use banks. FYI, a really tight bike turn (wheelbase 45″) at about 5 mph has a 9 ft radius.
- Determine your bike’s steering wheel angle at full left or right—ride your bike on the street and make some sharp turns at slow speeds…or better yet, try to do a complete circle. Note the position of your bars relative to the top-tube. Measure with a protractor or speed square.
- Subtract the turning angle from 90 and find sine for that angle.
- Divide your wheelbase, in inches, by the sine value from Step 2, and multiply that number by two to get the turning circle diameter, in inches. Divide by 12 to get feet.
more info:
Step-by-step to do a rolldown test:
1. Determine Crrg
A. Drive at a steady low speed, and start timing the moment you take your foot off the electrons.
Stop timing as soon as you come to a stop.
vstart is your starting speed and t is the time it took to roll to a stop.
Do a few tests at different low speeds and take the ratios vstart/t. They should be the same if you are starting slow
enough. This is the value of Crrg. Also try both directions and average (to eliminate any small slope).
B. If you don’t trust your speedometer you can measure roll-down distance d and time t, with
Crrg=2d/t^2 (still at low speeds). I am using mks units on my charts so take care accordingly (velocity vstart in m/s).
C. If you suspect some nonlinear effect near stopping, you can measure the acceleration at slow speeds but away from 0.
For example, if you measure the time to go from 10km/hr to 5km/hr, then
Crrg= deceleration = Δv/t = (10km/hr-5km/hr) / time. Remember to use consistent units. If you also don’t have a reliable speedometer at low speeds you can use measurement of time and distance only. Put three marks on the ground at some distance you can coast at low speeds, eg at 0m, 10m, and
15m. Measure the time t1 to coast from 1st to 2nd mark and the total time t2 to coast (on the same run – use the ‘lap’feature of your stopwatch) from 1st (not 2nd) to 3rd mark, and the corresponding distances x1 and x2.from 1st to 2nd and from 1st to 3rd marks. The acceleration experienced is
Crrg=2/t2-t1(x2/t2-x1/t1)
2. Determine ρCdA/2m
Now do a test with high starting speed (where wind resistance is important, e.g. above 70km/hr, 43 mph).
We have to solve Vstart = sqr root a/b = tan (change in time times sqr root a/b) for the unknown b(my abbreviation for ρCdA/2m), possible in several ways:
A. Use the contour plots below. Each plot is for a different starting velocity (labeled at top, e.g. the first is for a vstart of
110km/hr). Find your value of Crrg on the x-axis, go up till you hit the color for your measured time, and read off the
value of ρCdA/2m on the y-axis Remember these are in mks units; meters/s^2 for Crrg and 1/meters for ρCdA/2m. You might do this at several different high speeds to get a better value. Or,
B. With a calculator, guess and check values of b, or
C. Use the tables at the end (for 70km/hr starting speed only).
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