Bohnenberger’s machine supports a spinning sphere in two pivoted gimbals inside a fastening ring. This historical teaching instrument makes rotational inertia and precession visible. Explore a museum-inspired connected mechanism and calculate the ideal response of a uniform spherical rotor below.
Bohnenberger's Machine Interactive Calculator
Change rotor mass, diameter, spin speed and perpendicular torque. The calculator uses solid-sphere inertia and the ideal slow-precession relation; the connected gimbals show how the rotor is supported.
Equation Used
- Uniform solid sphere; actual historical rotors and axle assemblies have different inertias.
- Axes intersect at the rotor centre; mass is balanced about the support.
- Perpendicular torque turns with the spin-axis direction in the depicted horizontal-axis case.
- Slow-precession approximation: Ω must be small compared with ω; a warning appears when Ω/ω exceeds 0.1, a screening threshold rather than a guaranteed error bound.
The numerical precession result is an ideal ratio, not a complete rigid-body simulation. At fast-precession inputs the animation remains visible and identifies the approximation limit.
How the sphere and gimbals work
A rotor axle turns in the inner gimbal. That gimbal pivots inside a second gimbal, which pivots in a fastening ring attached to the stand. The three rotational freedoms are rotor spin and the two gimbal rotations. Bearing axes intersect at the centre; rings must stay joined at their bearing pairs as they move.
The surviving Tübingen instrument documented by the Institute of Navigation has an ivory rotor, brass rings, adjustable steel point bearings and axle drums for a pull cord. The drawing uses these recognizable features without claiming to reproduce its dimensions.
What the animation shows
The stand is fixed. The middle gimbal turns about the vertical bearings; the inner gimbal carries a horizontal rotor axis. An ideal perpendicular torque turns the angular-momentum direction. With zero applied torque, the axis remains fixed while the sphere spins. Equator and meridian markings distinguish sphere rotation from gimbal motion.
Uses of Bohnenberger’s machine
This is a historical physics and astronomy teaching instrument. Its gimbal-supported rotor illustrates angular momentum and precession, principles used in later gyroscopic instruments. The original device and modern teaching gyroscopes differ in rotor construction, drive, bearings and available measurements.
What the calculator computes
For mass m in kilograms and radius r in metres, I = (2/5)mr². Convert entered spin speed n from rpm using ω = 2πn/60, then L = Iω. The ideal perpendicular-torque relation is Ω = τ/L, with τ converted from mN·m to N·m.
This relation follows from dL/dt = τ for constant-magnitude angular momentum turning at right angles to a vertical precession axis. More generally, the geometry introduces an angle between angular momentum and the precession axis. Torque parallel to the spin axis changes spin magnitude; it is not the torque represented here.
The model neglects the angular momentum of the moving gimbals and the transverse motion of the rotor. It is a slow-precession approximation, not a prediction of startup, nutation, spin decay or friction-driven reversal. When Ω approaches ω, the ratio remains calculable but does not establish the physical trajectory.
Worked example using the default inputs
For m = 0.300 kg, diameter 50 mm, spin 1000 rpm and perpendicular torque 1 mN·m:
I = 0.4 × 0.300 × 0.025² = 0.000075 kg·m². The spin speed is 104.72 rad/s, giving L = 0.007854 N·m·s. Therefore Ω = 0.001/0.007854 = 0.12732 rad/s, or 1.216 rpm. One ideal precession revolution takes 49.35 seconds.
Doubling mass or spin speed halves this ideal rate. Doubling diameter increases inertia fourfold and reduces the rate to one quarter. Doubling torque doubles the rate. These dependencies can be checked directly with the sliders.
Historical machine and modern demonstrations
A hand-spun historical machine reveals the support geometry and rotational principles. A motor-driven teaching gyroscope can maintain spin speed and may provide a measured torque arm. Neither is adequately described by the solid-sphere approximation if its rotor is predominantly a rim, disk or hollow shell. Use measured inertia or the correct mass-distribution formula for quantitative comparison.
Questions about the model
Why does zero torque stop precession but not sphere spin?
The ideal model has no bearing losses. With zero external torque, angular momentum keeps its direction and magnitude.
Why can the displayed precession become very fast?
A small rotor or low spin speed gives little angular momentum. Dividing a large torque by that value produces a large ideal rate. The warning identifies when slow-precession assumptions become questionable; the calculator does not solve full rigid-body dynamics.
Is there a universal RPM at which precession reverses?
No. The sign and motion depend on torque, angular momentum, constraints and transient dynamics. The calculator does not predict a reversal threshold from spin speed alone.
How are the rings connected?
Each moving ring turns on an opposed bearing pair. The two gimbal axes and the rotor axle meet at the sphere’s centre.
Why is the spin slowed in the drawing?
Normal rotor speeds are too fast to show clearly on a screen. Rotor markings run 40 times slower; calculated precession uses its actual rate. The drawing is labelled to make the distinction explicit.
References and further reading
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
