Whipple–Carvallo benchmark bicycle (Meijaard et al. 2007), linearised lean + steer dynamics. Drag the 3D view to orbit, wheel/pinch to zoom.
Gyro = 1 is a normal wheel, 0 adds a counter-rotating wheel that cancels its spin angular momentum, −1 spins it backwards. Mass and trail edits change the model parameters; the picture only hints at them.
Root locus vs speed (click to set the speed). Green band = self-stable window (all Re<0). weave, capsize, caster.
Lean–steer phase diagram. To turn right (φ>0) a balancing rider first steers left (δ<0): the loop dips into the shaded countersteer quadrant.
The science
A bicycle is a 2-degree-of-freedom system for small motions: roll φ (lean) and steer δ, with forward speed v held constant. The linearised equations of motion of the Whipple model are
M q̈ + v C₁ q̇ + (g K₀ + v² K₂) q = T, q = [φ, δ]ᵀ, T = [0, T_δ]ᵀ
M is the mass matrix, K₀ the gravity stiffness (negative: an inverted pendulum falls), v²K₂ the centrifugal stiffness, v C₁ the gyroscopic/damping-like term. Entries follow from the geometry and mass distribution of the benchmark bicycle (wheelbase 1.02 m, head angle 72°, trail 0.08 m, rider+frame 85 kg). This page re-derives them from the 25 benchmark parameters (Meijaard et al. 2007, Proc. R. Soc. A 463, 1955–1982) and finds the eigenvalues of the 4×4 state matrix at every speed.
- Weave: oscillation of the front end (complex pair). Unstable below the weave speed vw ≈ 4.29 m/s.
- Capsize: slow real fall to one side. Unstable above the capsize speed vc ≈ 6.02 m/s.
- Caster: fast real mode where the steering aligns like a shopping-trolley wheel; always stable for positive trail.
- Between vw and vc all real parts are negative: the riderless bicycle is self-stable. A push is absorbed by the weave oscillation, which dies out.
Trail (caster action) makes the front wheel steer toward the lean (steer into the fall) and shifts the window; at zero trail the window almost closes. Gyroscopic torque of the spinning front wheel also steers into the lean. In the benchmark bicycle cancelling it removes the stable window altogether (try "Jones: cancel front gyro"). That does not mean gyros are needed in general: David Jones (1970, Physics Today) built a bicycle with a counter-rotating front wheel that was still rideable, and Kooijman et al. (Science 2011) showed a bicycle with no gyro and no trail can be self-stable if the masses are placed suitably. The preset "Low-slung, no gyro" is such a case in this model (low body centre of mass, heavy front frame, larger trail). Self-stability needs a combination of mass distribution, trail and gyro; no single one is "the" reason.
Countersteering: at speed, steering toward the desired turn would tip the bicycle out of it. To lean right the rider briefly turns the bars left, the wheels swing out under the mass, the bicycle rolls right, then the bars are steered right to hold the lean. The simulated PD controller (δ_cmd = k_p(φ−φ_ref) + k_d φ̇) does this by itself; the dip in the phase diagram is not scripted.
Steady turn: in a circle of radius R at speed v the lean angle satisfies tanθ = v²/(gR) (resultant of weight and centrifugal force passes through the contact line). The steer angle follows from geometry, δ ≈ w/(R cos λ).
Simplifications: small-angle (linear) model, valid up to roughly 20°; rigid rider; knife-edge wheels with no tyre slip or tyre width; constant speed; no air drag; rider steering torque is an ideal servo. Changing trail with the wheelbase fixed is a parametric change, equivalent to altering the fork offset. The 3D model is an illustration and shifts only partially with the sliders. Lean in the picture is the frame roll φ, which differs slightly from the lean of the centre of mass.
Numerical checks (Experiments): thresholds from this code vs the published benchmark values 4.292 and 6.024 m/s; a time-domain simulation of the riderless bicycle at three speeds against the sign of the largest eigenvalue; and steady-turn lean against tanθ = v²/(gR).
References: Meijaard, Papadopoulos, Ruina, Schwab (2007); Jones (1970) "The stability of the bicycle", Physics Today 23(4); Kooijman, Meijaard, Papadopoulos, Ruina, Schwab, Moore (2011) "A bicycle can be self-stable without gyroscopic or caster effects", Science 332, 339; Whipple (1899); Carvallo (1900).