A Centrifugal Pendulum is a rotating regulator in which a mass swings on a pivot or arm while orbiting a vertical shaft, so centripetal force — not gravity alone — governs its period. The orbiting bob is the key component: as shaft speed rises, the bob lifts to a larger radius and the system finds a stable cone angle that locks rotation rate to a fixed frequency. We use it to smooth shaft speed in clocks, engines and turbine drivetrains, replacing the tick-tock impulse of an escapement with continuous silent rotation. A well-built unit holds shaft speed within ±0.5% across normal load swings.
Centrifugal Pendulum Interactive Calculator
Vary cone height, arm length, bob mass, and gravity to see the pendulum period, governing speed, cone angle, and centripetal load.
Equation Used
The governing period depends on cone height h and gravity g: shorter h gives a faster regulating speed. Arm length L sets the displayed cone angle and orbit radius, while bob mass m scales the centripetal force but not the period.
FIRGELLI Automations - Interactive Mechanism Calculators.
- Rigid arm or cord with negligible pivot friction.
- Stable conical motion with constant vertical cone height h.
- Bob mass does not affect period, but it affects centripetal force.
- Geometry output clamps h/L to a physical range for display.
The Centrifugal Pendulum in Action
A Centrifugal Pendulum is a conical pendulum forced to rotate around a vertical axis. The bob doesn't swing back and forth — it orbits. Gravity pulls the mass straight down, centripetal force pulls it inward toward the shaft, and the resultant tension in the suspension arm sets a cone half-angle θ. Increase the shaft speed and the bob flies outward, lifting into a wider cone. Slow the shaft and the bob drops back. The geometry self-balances at one specific ω for any given height of the cone — that's the regulating action.
The period depends only on the vertical height h of the cone, not on the bob mass and not directly on the arm length. T = 2π × √(h / g). That's the same isochronous trick a normal pendulum uses, just rotated 90°. If you machine the suspension pivot sloppy — say more than 0.02 mm radial play in a small horological build — the bob wobbles off-axis, h fluctuates and the rate drifts. If the arm flexes under centripetal load, h shortens dynamically as speed rises, and the regulator becomes non-isochronous: it speeds up under load instead of holding rate. That's the failure mode behind most botched rotary-clock builds.
In engine and powertrain use the same physics drives a centrifugal governor — Watt's flyball governor on a steam engine, or the modern centrifugal pendulum absorber bolted to a dual-mass flywheel in a downsized turbocharged engine. The bob's natural frequency tracks shaft speed, so it cancels torsional vibration at every RPM, not just one tuned resonance. That order-tuned behaviour is why every major automaker now puts pendulum absorbers in their CVT and dry-clutch drivetrains.
Key Components
- Orbiting Bob: The mass that traces the cone. Typical horological bobs run 50–500 g; automotive pendulum-absorber masses run 200–800 g per pair. Mass cancels out of the period equation but sets the energy storage, so a heavier bob resists short-term load disturbances more strongly.
- Suspension Arm or Cord: Carries the bob and defines the cone geometry. Must be stiff in tension but free to pivot — flexure errors above about 0.5° at the upper pivot translate directly into rate variation. Hardened steel pivots or knife-edge suspensions keep friction below 1% of bob weight.
- Vertical Drive Shaft: Spins the whole assembly. Runout must stay under 0.01 mm TIR for precision horological work, otherwise the bob's effective h modulates once per revolution and you get a periodic rate error you can hear as a beat.
- Upper Pivot or Gimbal: The point about which the bob orbits. A spherical or two-axis gimbal lets the cone half-angle change freely with speed. Stiction here is the single biggest enemy — any breakaway torque above roughly 0.5% of bob-weight × arm-length kills isochronism at low speeds.
- Drive or Loading Mechanism: In a clock, a small remontoire or weight-driven train inputs energy to overcome air drag and pivot friction. In a governor, the engine itself provides the drive, and the bob's outward travel mechanically actuates a fuel or steam valve through a sleeve and bell-crank linkage.
- Output Linkage (governor variants only): Translates bob radius into a control action. On a Watt governor the sleeve lifts as the bob flies out, closing the throttle. Linkage backlash above 0.1 mm causes hunting — the speed oscillates around setpoint instead of settling.
Industries That Rely on the Centrifugal Pendulum
The Centrifugal Pendulum shows up wherever a rotating shaft needs its speed held steady or its torsional vibration cancelled. Clockmakers chased it for two centuries because a silent continuous regulator beats a ticking escapement for astronomical work. Engineers reached for it on every steam engine ever built. Today it's quietly inside almost every modern downsized engine drivetrain — you've driven dozens of them without knowing.
- Horology: The Reuge and Jaeger-LeCoultre Atmos rotating-pendulum movements use a torsional-conical pendulum variant for near-silent timekeeping. The 19th-century Bilgram cone clock used a true centrifugal pendulum running at constant cone angle to drive an astronomical dial.
- Steam and Stationary Engines: James Watt's 1788 flyball governor on the Boulton & Watt rotative steam engine — two iron balls on swinging arms regulating steam admission. Still in use on traction engines and preserved mill engines worldwide.
- Automotive Drivetrain: LuK (Schaeffler) Centrifugal Pendulum Absorber fitted to dual-mass flywheels in VW, BMW, Ford and PSA downsized turbo engines. Cancels 2nd-order torsional vibration in 3- and 4-cylinder engines so the cabin feels like a 6-cylinder.
- Turbomachinery: Pratt & Whitney and Rolls-Royce use centrifugal pendulum absorbers on geared turbofan accessory drives to suppress blade-passing vibration without the temperature limits of an elastomer damper.
- Helicopter Rotor Systems: Sikorsky bifilar pendulum absorbers on the main rotor hub of the UH-60 Black Hawk and S-92 — tuned-mass centrifugal pendulums that cancel N-per-rev vibration directly in the rotating frame.
- Industrial Speed Regulation: Porter governor and Hartnell governor variants on diesel gensets, marine prime movers and old textile-mill line shafts. Still specified on heritage restorations because the response is mechanical, fail-safe and needs no electronics.
The Formula Behind the Centrifugal Pendulum
The core formula gives the rotational period T of a conical pendulum as a function of the cone height h. What you care about as a builder is how T changes across the operating range. At low shaft speed the cone is nearly vertical, h is close to the full arm length L, and the period sits at its maximum — useful for slow astronomical clock drives. Push the speed up and the bob flies outward, h shrinks, and T drops. The sweet spot for most horological designs sits around a 30–45° cone half-angle, where the rate is reasonably insensitive to small disturbances and the bob isn't fighting the suspension limits. Beyond about 60° the system gets twitchy — small changes in drive torque produce large changes in radius, and isochronism falls apart.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| T | Period of one full orbit of the bob | s | s |
| h | Vertical height of the cone (pivot to bob plane) | m | in |
| L | Length of the suspension arm from pivot to bob centre | m | in |
| θ | Cone half-angle measured from the vertical shaft axis | ° | ° |
| g | Gravitational acceleration | 9.81 m/s² | 32.2 ft/s² |
| ω | Angular velocity of the rotating shaft (= 2π / T) | rad/s | rad/s |
Worked Example: Centrifugal Pendulum in a precision astronomical demonstration clock
Your build is a tabletop astronomical demonstration clock for a university physics department. The drive is a 0.6 N·m synchronous motor through a reduction stage, and you want the centrifugal pendulum to orbit at a nominal 60 RPM (1 Hz) so the seconds dial advances cleanly once per orbit. The suspension arm length L is fixed at 250 mm. You need to find the cone half-angle θ that produces the target period, then check what happens at the low and high ends of the operating range you'll see between morning warm-up and steady-state running.
Given
- L = 0.250 m
- Target N = 60 RPM
- g = 9.81 m/s²
Solution
Step 1 — convert the target speed to a period. At nominal 60 RPM:
Step 2 — solve the period equation for the required cone height h:
Step 3 — back out the cone half-angle from the geometry h = L × cos(θ):
That's a very tight cone — the bob sits almost directly under the pivot and the system is barely centrifugally loaded. Useful for a slow astronomical drive but sensitive to any pivot friction.
Step 4 — low-end check at 30 RPM (T = 2.0 s). The required h would be 0.994 m, which is larger than L. Physically impossible — the formula tells you a 250 mm arm cannot run that slow as a centrifugal pendulum. You'd either need a longer arm or accept that 30 RPM is below the mechanism's lower bound for this geometry.
Step 5 — high-end check at 120 RPM (T = 0.5 s):
At 75.6° the bob is flung almost horizontal. Centripetal load on the arm is 4× the bob weight and any arm flex shortens h dynamically, which speeds the orbit up further — a runaway-prone region. Practical upper limit for a 250 mm arm sits around 90–100 RPM (θ ≈ 65°) before isochronism collapses.
Result
The nominal cone half-angle is 6. 1° at 60 RPM with a 250 mm arm — a small angle that locks the period to 1.000 s within the precision of your machining. The low-end test shows the design has no headroom below about 38 RPM (where θ approaches 0°), and the high-end test shows isochronism falls apart above roughly 100 RPM where the arm runs out near 80°. The sweet spot is the 50–80 RPM band. If you measure 1.05 s instead of 1.000 s, the three usual culprits are: (1) upper-pivot stiction above 0.5% of bob weight, which holds the bob at a smaller-than-equilibrium radius, (2) a suspension arm that flexes under centripetal load and shortens h by 1–2 mm at speed, and (3) shaft runout above 0.01 mm TIR introducing once-per-rev modulation that beats against the orbit period.
Centrifugal Pendulum vs Alternatives
The Centrifugal Pendulum competes with the gravity pendulum for clocks and with electronic governors for engine speed control. Each option wins on a different axis. Pick by what your application actually needs — silent continuous rotation, vibration cancellation, or pure timekeeping accuracy.
| Property | Centrifugal Pendulum | Gravity Pendulum + Escapement | Electronic PID Governor |
|---|---|---|---|
| Timekeeping accuracy (per day) | ±2–10 s typical, —0.5 s best | ±0.1 s (Riefler), ±1 s (longcase) | N/A — speed control, not timekeeping |
| Operating speed range | 20–300 RPM practical | Fixed beat (typically 0.5–1 Hz) | 0–20,000 RPM, software-defined |
| Output character | Continuous silent rotation | Discrete tick, audible | Continuous, electronic |
| Vibration suppression | Excellent (order-tuned across RPM) | None | Poor (only as fast as control loop) |
| Sensitivity to friction | High — pivot stiction kills isochronism | Medium — escapement tolerates wear | Low — sensors compensate |
| Component cost (unit basis) | Medium — precision pivots required | Low to medium | High — sensors, ECU, actuator |
| Failure mode | Drifts then stalls — fail-safe | Stops dead — obvious failure | Can fail unpredictably or runaway |
| Best application fit | Rotary clocks, vibration absorbers, mechanical governors | Precision timekeeping, regulators | Modern engines, gensets, turbines |
Frequently Asked Questions About Centrifugal Pendulum
Because the system is running non-isochronously, almost always due to arm flex. Extra drive torque overcomes friction and lets the bob settle at its true equilibrium radius, but if the arm bends outward under centripetal load, h shrinks at higher speed and the period drops with it. The mechanism becomes a positive-feedback loop instead of a regulator.
Check arm stiffness with a simple static test — hang the bob horizontally from the pivot and measure deflection. Anything over about 0.5 mm at full bob weight will show as 1–2% rate sensitivity to drive torque. Switch to a stiffer arm material (hardened steel rod over brass) or triangulate the suspension.
For anything under about 5 kW and below 400 RPM, a Watt flyball governor is the right call — it's the same physics but the linkage geometry directly actuates the throttle valve, which is what you actually need on an engine. A pure centrifugal pendulum gives you a stable rotating reference but no control output unless you bolt a sleeve and bell-crank onto it, at which point you've built a Watt governor anyway.
Use the standalone centrifugal pendulum form when the goal is timekeeping or vibration absorption. Use the Watt/Porter/Hartnell governor form when the goal is speed control of a prime mover.
Hunting means the bob is oscillating between two radii instead of finding equilibrium. Three causes worth checking in order: drive torque ripple from the input gear train, backlash in any output linkage exceeding 0.1 mm, and an under-damped suspension. The first two are mechanical fixes — tighter gear mesh, take-up springs on the linkage. The third needs a small viscous damper or air-vane on the bob.
Rule of thumb: the natural settling time of an undamped conical pendulum is roughly 5–10 orbits. If yours takes 30+ orbits to settle after a disturbance, you have a damping problem, not a stiffness problem.
The bob mass doesn't set the tuning frequency — that's geometry — but it sets how much vibration the absorber can cancel before it saturates. The rule used by Schaeffler and ZF for automotive pendulum absorbers is bob inertia equal to roughly 1–3% of the engine's primary inertia per cylinder order being cancelled. Below 1% the absorber clips at low engine load; above 3% you're paying weight penalty for no benefit.
For a single-cylinder hit-and-miss engine that means roughly 200–400 g per bob on a typical flywheel. For a 1.0L 3-cylinder turbo, production absorbers run 600–800 g total split across two or four bobs.
The classical formula T = 2π × √(h/g) requires gravity along the rotation axis, so on a horizontal shaft or in microgravity the basic conical pendulum doesn't function — there's no restoring force pulling the bob inward. The bifilar pendulum used on helicopter rotors solves this: it uses centripetal force itself as the restoring force, with the bob rolling on two pins so the effective tuning depends only on the rotor RPM, not gravity.
If your application has any axis tilt, use the bifilar variant. Sikorsky and Airbus Helicopters both rely on this on rotor heads where the local gravity vector swings through 360° per revolution.
Almost certainly temperature. The suspension element — usually a thin elgiloy or invar torsion strip — changes stiffness with temperature, and unlike a gravity pendulum a torsional centrifugal pendulum has no pendulum-length compensation built in. A 5°C swing on an uncompensated torsion strip is good for 10–20 s/day of rate change.
Either move the clock to a temperature-stable spot (Atmos units rely on ±1°C ambient stability for their published rate) or fit a bimetallic compensator on the suspension boss. Without one, no amount of pivot work will get you under 10 s/day.
References & Further Reading
- Wikipedia contributors. Conical pendulum. Wikipedia
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