Bicycle physics: Keeping balanced....

In summary: R),\quad...$$$$c=(0,0,1,0,R)$$$$\bigl\{(\delta\theta,\delta\psi,\delta\varphi,\delta x,\delta y)\bigr\}=\{(\delta\theta,\delta\psi,\delta\varphi,\delta x,\delta y)\}$$In summary, when a bike is leaning, the gyroscopic precession causes the front wheel to remain in the same direction.
  • #1
Delta2
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So my main question is why we maintain balance easily when the bicycle has some horizontal velocity, but it isn't the same when bicycle is at rest.

The best answer I could find at the web is that when the bicycle wheels rotate, essentially they form gyroscopes and gyroscopes resist the change in their axis of rotation (they tend to keep their axis parallel with respect to the ground).

Is this the answer to my question or not?
 
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  • #2
Delta2 said:
So my main question is why we maintain balance easily when the bicycle has some horizontal velocity, but it isn't the same when bicycle is at rest.

The best answer I could find at the web is that when the bicycle wheels rotate, essentially they form gyroscopes and gyroscopes resist the change in their axis of rotation (they tend to keep their axis parallel with respect to the ground).

Is this the answer to my question or not?
Not really. The gyroscopic precession plays a role in steering the front wheel into the lean, if you don't hold the handles. But it's not the whole story. See videos below.

But if you are steering, then it's you who is steering to straighten up the bicycle. And that doesn't work if the bicycle is not moving.

 
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  • #3
So I see the real explanation is a bit more complex, video 2 mentions 3 factors that combined are what makes the bicycle stable
 
  • #4
I think it would be a good exercise to write down equations of coin's rolling without slipping on a horizontal table and study stability of its straight vertical motion in the linear approximation.

to begin with:

Screenshot from 2021-07-02 08-41-29.png
Screenshot from 2021-07-02 08-41-53.png
 
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  • #5
Delta2 said:
So I see the real explanation is a bit more complex, video 2 mentions 3 factors that combined are what makes the bicycle stable
The gyroscope effect might be relevant to the bicycle itself, but must be fairly negligible if we compare the mass of the wheels to the rider.
 
  • #6
PeroK said:
The gyroscope effect might be relevant to the bicycle itself, but must be fairly negligible if we compare the mass of the wheels to the rider.
According to video 2 the gyroscope effect alone can't explain the stability of the bike (in the absence of a rider).
 
  • #7
PeroK said:
The gyroscope effect might be relevant to the bicycle itself, but must be fairly negligible if we compare the mass of the wheels to the rider.
is not the stability\instability dependent on the velocity?
 
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  • #8
wrobel said:
is not the stability\instability dependent on the velocity?
That's true. More precisely, the gyroscope effect must be negligible at low speeds.
 
  • #9
We could try an experiment.

1) Turn a bike upside down, so that it is resting on its handlebars and seat.

2) See how much effort it takes to tip the bike over.

3) Spin the wheels as fast as possible and repeat step 2).
 
  • #10
PeroK said:
The gyroscope effect might be relevant to the bicycle itself, but must be fairly negligible if we compare the mass of the wheels to the rider.
The gyroscope effect is about steering the front wheel into the lean, not about holding up the entire bike.
 
  • #11
##\boldsymbol{q} = (x,y,\varphi,\psi, \theta)##\begin{align*}

\dfrac{d}{dt} \left( \dfrac{\partial L}{\partial \dot{\boldsymbol{q}}} \right) - \dfrac{\partial L}{\partial \boldsymbol{q}} = \lambda \begin{pmatrix} 1 \\ 0 \\ 0 \\ R\cos{\varphi} \\ 0 \end{pmatrix} + \mu \begin{pmatrix} 0 \\ 1 \\ 0 \\ R\sin{\varphi} \\ 0 \end{pmatrix}

\end{align*}Too tired to evaluate the gradients.
 
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  • #12
Delta2 said:
So my main question is why we maintain balance easily when the bicycle has some horizontal velocity, but it isn't the same when bicycle is at rest.

The best answer I could find at the web is that when the bicycle wheels rotate, essentially they form gyroscopes and gyroscopes resist the change in their axis of rotation (they tend to keep their axis parallel with respect to the ground).

Is this the answer to my question or not?
The gyroscopic effect resists any roll of the wheels, steering movements and from leaned to vertical as well: it is not inducing verticallity of the chassis.

Please, see:
http://ezramagazine.cornell.edu/SUMMER11/ResearchSpotlight.html

The rider must steer in the right direction in order to keep the moving bike from falling over.

 
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  • #13
ergospherical said:
##\boldsymbol{q} = (x,y,\varphi,\psi, \theta)##\begin{align*}

\dfrac{d}{dt} \left( \dfrac{\partial L}{\partial \dot{\boldsymbol{q}}} \right) - \dfrac{\partial L}{\partial \boldsymbol{q}} = \lambda \begin{pmatrix} 1 \\ 0 \\ 0 \\ R\cos{\varphi} \\ 0 \end{pmatrix} + \mu \begin{pmatrix} 0 \\ 1 \\ 0 \\ R\sin{\varphi} \\ 0 \end{pmatrix}

\end{align*}Too tired to evaluate the gradients.
I would use D'Alambert-Lagrange
$$[L]_\theta:=\frac{d}{dt}\frac{\partial L}{\partial \dot\theta}-\frac{\partial L}{\partial \theta},$$
$$[L]_\theta\delta\theta+[L]_\psi\delta\psi+[L]_\varphi\delta\varphi+[L]_x\delta x+[L]_y\delta y=0$$
 
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  • #14
$$\{(\delta\theta,\delta\psi,\delta\varphi,\delta x,\delta y)\}=span(a,b,c)$$
$$a=(1,0,0,0,0),\quad b=(0,0,1,0,0)$$
$$c=(0,1,0,-R\cos\varphi,-R\sin\varphi)$$
 
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  • #15
Hold a yard stick or meter stick vertical in your open hand. Now move your hand so as to try and keep the stick vertical, similar effect when you steer a bike? When you steer a bike you are shifting the bike under your center of mass?
 
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1. How does a bicycle stay balanced?

A bicycle stays balanced through a combination of factors including the rider's body position, the design of the bicycle frame, and the forces of gravity and momentum. When a rider pedals, their body weight shifts slightly to maintain balance, and the design of the frame allows for a stable center of gravity.

2. What are the main forces that affect a bicycle's balance?

The main forces that affect a bicycle's balance are gravity, momentum, and the rider's body weight. Gravity pulls the rider and bicycle downward, while momentum keeps the bicycle moving forward. The rider's body weight also plays a role in shifting the center of gravity and maintaining balance.

3. How does a bicycle turn without losing balance?

A bicycle turns without losing balance through a combination of steering and leaning. When a rider turns the handlebars, the front wheel steers in the desired direction. At the same time, the rider leans their body and the bicycle into the turn, shifting the center of gravity and maintaining balance.

4. Can a bicycle balance itself without a rider?

No, a bicycle cannot balance itself without a rider. The rider's body weight and movements are essential in maintaining balance, and without a rider, the forces of gravity and momentum would cause the bicycle to fall over.

5. How does the speed of a bicycle affect its balance?

The speed of a bicycle affects its balance in several ways. At higher speeds, the forces of momentum and centrifugal force increase, making it easier to maintain balance. However, at lower speeds, the rider must make more adjustments to maintain balance, and a sudden decrease in speed can cause the bicycle to become unstable.

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