, 3 min read
Steve Jobs on Bicycles
1. Energy efficiency. Steve Jobs was impressed by the efficiency of the human riding a bike.
Steve Jobs has cited this comparison multiple times during his life. Probably he was referring to this chart in Scientific American:
Bicycles are very efficient, see The science of bicycles by Chris Woodford.
More on the efficiency is here: stackexchange: Does a human on a bicycle travel more efficiently than any other species?.
2. Differential equation. Below statements and figures are coped from Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review by J.P Meijaard, Jim M Papadopoulos, Andy Ruina, A.L Schwab. A local copy is here. Also see English Wikipedia Bicycle and motorcycle dynamics from where I copy the eigenvalue part.
Model parameters are given in below table.
| parameter | symbol | value for benchmark |
|---|---|---|
| wheel base | w | 1.02 m |
| trail | c | 0.08 m |
| steer axis tilt ($\pi/2$−head angle) | $\lambda$ | $\pi/10 \hbox{rad} (90^\circ–72^\circ)$ |
| gravity | g | $9.81 \hbox{N} \hbox{kg}^{−1}$ |
| forward speed | v | various m s-1 |
| rear wheel R | ||
| radius | $r_R$ | 0.3 m |
| mass | $m_R$ | 2 kg |
| mass moments of interia | $\left(I_{Rxx},I_{Ryy}\right)$ | (0.0603, 0.12) kg m2 |
| rear body and frame assembly B | ||
| position center of mass | $\left(x_B,z_B\right)$ | (0.3, -0.9) m |
| mass | $m_B$ | 85 kg |
| mass miments of interia | $\begin{pmatrix}I_{Bxx}&0&I_{Bxz}\cr 0&I_{Byy}&0\cr I_{Bxz}&0&I_{Bzz}\cr\end{pmatrix}$ | $\begin{pmatrix} 9.2&0&2.4\cr 0&11&0\cr 2.4&0&2.8\cr\end{pmatrix}$ kg m2 |
| front handlebar and fork assembly H | ||
| position center of mass | $\left(x_H,z_H\right)$ | (0.9, -0.7) m |
| mass | $m_H$ | 4 kg |
| mass moments of inertia | $\begin{pmatrix}I_{Hxx}&0&I_{Hxz}\cr 0&I_{Hyy}&0\cr I_{Hxz}&0&I_{Hzz}\cr\end{pmatrix}$ | $\begin{pmatrix}0.05892&0&-0.00756\cr 0&0.06&0\cr -0.00756&0&0.078\cr\end{pmatrix}$ kg m2 |
| front wheel F | ||
| radius | $r_F$ | 0.35 m |
| mass | $m_F$ | 3 kg |
| mass moments of inertia | $\left(I_{Fxx},I_{Fyy}\right)$ | (0.1405, 0.28) kg m2 |
The dynamic can be represented by a single fourth-order linearized ordinary differential equation or two coupled second-order differential equations, i.e., in matrix form
where the time-varying quantities are $q = \left[\phi, \delta\right]^T$ and $f = \left[T_\phi, T_\delta\right]^T$. The constant entries in matrices $M, C_1, K_0$ and $K_2$ are defined in terms of the 25 design parameters.
The eigenvalues of
are now of interest. If the real parts of all eigenvalues are negative, the bike is self-stable. When the imaginary parts of any eigenvalues are non-zero, the bike exhibits oscillation. The eigenvalues are point symmetric about the origin and so any bike design with a self-stable region in forward speeds will not be self-stable going backwards at the same speed.