, 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

$$M\ddot q + vC_1\dot q + \left[ gK_0 + v^2K_2 \right] q = f$$

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

$$\det\left( M\lambda^2 + vC_1\lambda + gK_0 + v^2K_2 \right) = 0$$

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.