Modeled after “Bad Science” by Alistair
B. Fraser
Bad Mechanics: Bicycles balance because their wheels act like gyroscopes.
Here are a couple of examples of this common misconception on the web:
On physlink.com at http://www.physlink.com
Question: Do gyroscopic forces from the wheels make any significant contribution to the rideability of a bicycle?
Answer: Yes, the gyroscopic forces, better known as the angular momentum, of the wheels on a bike allow us to ride a bike.
On The Straight Dope at http://www.straightdope.com
Dear Straight Dope: Why is balancing on an unmoving bicycle so much harder than balancing on a moving bicycle?
SDSTAFF Karen Lingel replies: Because modern bicycles are equipped with a pair of gyroscopic stabilization devices that require the motion of the bike in order to operate. These devices are known as "wheels."
http://www.straightdope.com/mailbag/mangularmo.html
This has since been “revisited”. The new answer is: trail.
The reality is that bicycles stay upright when they are steered to keep the wheels under the center of gravity. This is the same mechanism that keeps runners, skaters, skiers, and snowboarders upright. Lock the handlebars of a bicycle and it is unrideable. Cancel the gyroscopic effect of bicycle wheels by adding counter-rotating wheels, and it is still easily ridable.
For examples of the latter, read "Bicycle Science" by Dr. Klein at http://www.losethetrainingwheels.org/default.aspx?Lev=2&ID=34 or "The Stability of the Bicycle" by David Jones at http://socrates.berkeley.edu/%7Efajans/Teaching/MoreBikeFiles/JonesBikeBW.pdf
The factor that has a lot more to do with how easy or hard a bicycle will be to ride is called ‘trail’ the distance by which the front wheel ground contact point trails behind where a line through the steering axis intersects the ground. The more trail a bike has, the more stable it feels. Bikes with negative trail, while still ridable, feel very unstable.
A curious fact is that most standard bikes are stable even without a rider at various forward speeds. Here is a video demonstration. I have not yet read a comprehensive analysis of inherent bicycle stability due to a combination of mechanical trail and gyroscopic effects on a freely turning front fork and how it interacts with a rider's control system.
So, what exactly is the gyroscopic effect of wheels on the balance of bicycles?
Let us first take a quick look at just what is the gyroscopic effect generated by the spinning wheels. To isolate this effect, suppose that we have a bicycle with the steering locked straight ahead so that it can roll forward, lean to the left and right, but not steer.
Any college level introduction to physics or mechanics will explain that gyroscopic spin rate p, precession rate ω, and moment M are related by M = Iωp where moment is another word for torque or tendency of a force to cause rotation, precession is the phenomenon you see as a spinning top wobbles about the vertical or a football wobbles during an imperfect spiral pass, and I is the moment of inertia of the gyroscope about its spin axis: a measure of how mass is distributed about that axis.[1]
This
is a simplification that ignores the angular moment due to the precession rate
because it is much less than that due to the spin rate. It also skips over the
fact that these are actually vector quantities and that they all act at right
angles to each other and are oriented, by convention, according to the
right-hand rule.
The right-hand rule is a handy trick for keeping track of vector orientation. In this case it works in two ways. First the direction of these vectors is determined by the right hand rule: fingers wrapping in direction of rotation leave the thumb pointing in direction of the corresponding vector. Then the fingers of the right hand initially oriented in the direction of the spin vector and bent in the direction of the moment vector leave the thumb pointing in the direction of the precession vector.
Fortunately,
none of these assumptions are inconsistent with this analysis, and we'll use
the spin rate of the wheels as p and let gravity acting on the bicycle
center of mass provide the initial moment M about the ground contacts.
This equals the weight (mass times gravitational constant) of the bike and
rider times the moment arm: the horizontal distance from the center of mass to
the line connecting the ground-wheel contact points: Mgravity
= mgzcmsinθ
Here we encounter our first problem: that the initial moment and precession vectors are at right angles to each other. If gravity creates a moment about the longitudinal axis of the bicycle, causing it to tip to the side, the spinning wheels induce precession about vertical axes, not in any direction that counters the tipping.
In
fact, the precession about the vertical axis is resisted by friction between
the tires and the road at their points of contact. This frictional force, along
with the moment arms due to the wheelbase introduces a second moment, this time
about the vertical axis, but in the opposite direction. This second moment also
causes precession, again at right angles, and so about the longitudinal axis.
Worst of all, this precession is in the same direction as the moment caused by
gravity. The bicycle just keeps tipping in the direction gravity initially
pulls it.
All of this is easily verified with a toy gyroscope.[2] First suspend the spinning gyroscope from a string about one end of its spin axis or balance it on its edge. Observe how the tipping moment due to gravity causes the gyroscope to precess in a horizontal plane about the vertical. You can verify that the right hand rule predicts the direction of precession. A slightly more sophisticated experiment confirms that M = Iωp as well. Then, prevent this precession with a vertical object such as a pencil. Observe how the gyroscope rapidly tips in the direction that you would expect gravity to make it go in the first place.
That the gyroscopic effect is not necessary for balancing a bicycle has also been proven by direct experimentation. British scientist David Jones built a bicycle with an additional counter rotating front wheel to cancel out exactly any gyroscopic effect. He reports that "it could easily be ridden."[3] Separately, Cornell University Professor Andy Ruina has built and ridden multiple bikes with tiny front wheels, as small as inline skate wheels that have negligible gyroscopic effects.
A more formal verification can be made with Euler's Equations for 3D rigid body motion:
If we set up a right-hand coordinate system such that x and y axes define the ground plane with y in the direction the bicycle is pointing and the z axis is vertical, then we can immediately cancel some terms:
The friction of the wheels on the
ground prevents rotation about the vertical axis so
and
its rate of change 
.
We can simplify things without loss
of generality by supposing that the wheel spin rate is constant so its rate of
change 
also.
That leaves us with just:
Then let the center of mass be
half way between the front and rear wheel and zcm
off the ground. The moment about the y-axis due to gravity on the center
of mass will be 
and the moment about the z-axis
due to tire friction will be equal to or less than 
. So finally:
The moment about the y-axis is due only to gravity, and the bike simply tips over as if the wheels were not spinning at all.
So then, what role if any, do gyroscopic effects play in bicycle balancing, and what exactly does keep a bicycle upright? That is a more complicated question, and the simple answer is that gyroscopic effects are not necessary. A bicycle stays upright in the same way that you can balance a stick upright on the palm of your hand: by moving your hand in the direction that the stick starts to lean.
The more complete answer depends in subtle ways on many geometric parameters of the bicycle such as the steering tube angle (head angle), mechanical trail, front assembly center of mass location, wheelbase, etc.
However, one factor is related to gyroscopic effects: the steering of the front wheel. As we have seen above, the leaning of a bicycle induces precession in the wheels about the vertical axis. The rear wheel is prevented from precessing, but if the rider lets go of the handlebars, the front wheel is free to precess about the approximately vertical steering axis. By following the right-hand rule, we see that as the bicycle leans to the right the front wheel precesses to the right, and as it leans to the left it precesses to the left. This is precisely the correct direction, but not necessarily the right amount, to help keep the bicycle upright.
Depending on the other geometric parameters listed above and the speed at which the bicycle is rolling forward, this can play more or less of a role a bicycle's handling characteristics, the ease at which it can be ridden with no hands, and even the surprising ability for many bicycles to coast down the road without any rider at all.
[1]Fundamentals of Physics by Halliday,
Resnick, and now
[2] Available at most education stores or from Edmond Scientific (www.scientificsonline.com) for $4.95
[3] Physics Today 23(4), 1970, pp34-40.