A clock-driven centrifugal pendulum carries a suspended bob around a horizontal circle. A fine wire hangs from an adjustable eye, while an arm on a vertical spindle supplies the drive. The arrangement in Hiscox figure 1149 adjusts its timing through the suspension height. Its ideal motion is that of a conical pendulum.
Centrifugal Pendulum Interactive Calculator
Adjust vertical cone height, suspension length, bob mass and gravity. Compare revolution period, orbital speed, cone angle and inward force, with a connected suspension and lower clock-drive arm.
Equation Used
- Point-mass bob on a massless, inextensible wire; steady conical orbit.
- Drive only maintains the ideal motion; drag, friction, drive-force disturbance and transient behavior are omitted.
- The suspension length remains constant. The orbit has nonzero radius only when 0 < h < L.
- The whole drawing is scaled to fit; structural details are illustrative.
- The model does not predict clock accuracy or order tuning of an automotive vibration absorber.
This is the clock-driven conical pendulum described in Hiscox figure 1149, not an automotive centrifugal pendulum absorber. The calculation requires height h to be less than suspension length L. Invalid settings keep a geometry preview but provide no circular-orbit results.
A bob moving around a cone
Hiscox figure 1149 shows a bob suspended by a fine wire from an eye. An arm attached to a vertical spindle carries it around, with the spindle driven by the clock movement. The suspension eye can be raised or lowered to adjust timing.
The animation keeps the suspension wire connected to the bob and its length fixed. The bob travels in a horizontal circular orbit, appearing elliptical in the oblique view. The lower arm turns with the bob; the support and suspension eye remain stationary during each revolution.
Gravity acts downward. Wire tension has an upward component balancing weight and an inward component supplying centripetal acceleration. Centripetal force is the name for the net inward force needed for circular motion, not a separate force to add to gravity and tension.
The ideal steady orbit is useful for understanding geometry and period. It does not by itself establish constant clock rate under changes in drive, drag or friction.
Clock regulator rather than vibration absorber
The historical reference places this mechanism among clock and watch devices. A conical pendulum provides continuous circular motion rather than the alternating swing of a conventional pendulum.
A centrifugal pendulum absorber in a rotating drivetrain is a different mechanism: it oscillates relative to a rotating carrier to address torsional vibration. The gravity-based conical-pendulum equation on this page is not a design equation for such an absorber. A torsion pendulum is also distinct; its restoring torque comes from a twisted suspension.
Conical-pendulum equilibrium
With suspension length L and vertical height h, orbit radius is r = √(L² − h²) and angle from vertical is θ = acos(h/L). A nonzero circular orbit requires 0 < h < L.
Let S be wire tension. Vertical balance gives S cosθ = mg, while the inward component is S sinθ = mω²r. Since tanθ = r/h, eliminating S gives ω² = g/h. Thus the full revolution period is T = 2π√(h/g), and orbital speed is N = 60/T rpm.
The inward force is F = mrω² = mgr/h. Mass changes force but cancels from period. At fixed h, changing L changes radius and angle without changing ideal period.
When h = L the orbit collapses to zero radius; when h > L the wire cannot span the requested height. The calculator no longer hides this by clamping the geometry to zero angle while displaying a supposed operating speed.
Example: a 0.5 m suspension
For h = 0.25 m, L = 0.5 m, m = 0.2 kg and g = 9.81 m/s², radius is √(0.5² − 0.25²) = 0.4330 m and θ = 60°.
The revolution period is 2π√(0.25/9.81) = 1.0030 s, giving 59.82 rpm. Inward force is 0.2 × 0.4330 × 9.81/0.25 = 3.398 N.
Increasing bob mass to 0.4 kg doubles inward force but leaves period unchanged. A height of 0.6 m with the same 0.5 m suspension is geometrically impossible and produces a labeled preview rather than an invented orbit.
What this calculation leaves out
The ideal period depends on cone height. If geometry changes with drive conditions, the period changes too. A real clock also has losses and interactions with its drive, so the equation alone does not prove isochronism, settling time or accuracy per day.
The diagram provides a readable model of the suspension and drive arrangement. It does not specify machining tolerances, bearing materials, acceptable torque or an overspeed limit.
Calculator questions
Does the bob swing back and forth?
In the selected steady motion it travels around a horizontal circle. The suspension sweeps a cone, and one full orbit takes the displayed period.
Why does mass change force but not period?
Mass appears in both weight and the required inward force and cancels from the ideal period equation. It remains in the force expression.
Why can changing suspension length leave period unchanged?
If vertical height h stays fixed, changing L changes radius and cone angle. Ideal period still depends on h and gravity. This is not the same as holding angle fixed.
Why are some dimensions marked invalid?
A wire cannot span a vertical height greater than its length. At equal height and length, the circular orbit has zero radius. The preview keeps the wire and drive plane visible and identifies the conflict.
Can this size a flywheel pendulum absorber?
No. An absorber on a rotating carrier has different dynamics and design variables. This calculator models a gravity-based conical pendulum.
References & Further Reading
- Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances (1901), printed page 283, figure 1149: clock-driven centrifugal pendulum with adjustable suspension eye.
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