Cycloidal cheeks guide a flexible pendulum cord so its bob follows a cycloid. As the swing grows, more cord wraps onto a cheek and the freely hanging part shortens. In the ideal frictionless point-mass model, the oscillation period is independent of amplitude.
Cycloidal Pendulum Movement Interactive Calculator
Watch a flexible cord wrap around cycloidal cheeks while its bob follows a cycloid. Set beat time and swing amplitude, then compare circular-pendulum clock error at two amplitudes.
Equation Used
- Point bob, massless inextensible perfectly flexible cord and frictionless cheeks.
- Uniform gravity g=9.80665m/s²; no escapement impulse, air drag or support motion.
- Swing amplitude is the peak free-cord angle from vertical.
- Circular clock comparison assumes the same length, with rate calibrated at the reference amplitude.
Loss is for an ordinary circular pendulum calibrated at the reference amplitude; negative means gain. The ideal cycloidal period is amplitude-independent.
The cord wraps without changing its total length
The fixed cheek profiles meet at the suspension point. During a right swing, the cord follows the right cheek before leaving it tangentially toward the bob; the left cheek performs the same function on the opposite swing.
The animation draws the wrapped portion in gold and the free portion in blue. Their lengths always add to the original suspension length. The free-cord angle changes nonuniformly because arc displacement, rather than angular displacement, is the harmonic coordinate.
Explore tautochrone geometry
Beat time sets the required suspension length. Amplitude changes the extent of the bob’s path and the amount of wrapped cord while leaving the ideal period unchanged.
The circular-clock comparison is separate: it shows the rate change an ordinary circular pendulum would have between the selected amplitude and a calibration amplitude. The reference setting changes that comparison, not the cycloidal cheek geometry.
Geometry and exact circular-error comparison
For beat time b and gravity g, suspension length L=g(b/π)² and cycloid generating radius a=L/4. The full cycloidal period is 2b.
Using θ as the cycloid parameter, bob position is [a(θ+sin θ), a(1−cos θ)]. The contact point on a cheek is [a(θ−sin θ), a(3+cos θ)]. Free cord is4a cos(θ/2); wrapped cord is4a[1−cos(θ/2)]. Their sum is4a=L.
Arc displacement from the bottom is s=4a sin(θ/2), and bob height is s²/(8a). This gives simple harmonic motion with period4π√(a/g), independent of amplitude.
For a circular pendulum, period is proportional to the complete elliptic integral K(sin(A/2)). The calculator evaluates it through the arithmetic-geometric mean. Daily indicated-time loss after calibration at Aref is86400[1−Kref/KA]. Negative loss means a gain.
A one-second beat
A one-second beat gives a two-second full period and a suspension length of approximately0.993621m. The cycloid generating radius is approximately248.405mm.
For an ordinary circular pendulum, changing amplitude from1degree to4degrees causes approximately24.67seconds per day of indicated-time loss after calibration at1degree. The former15seconds estimate from subtracting squared degree values was not the correct period calculation.
Ideal isochronism and real clock behavior
The mathematical result assumes a perfectly flexible, massless cord, point bob and frictionless contact. Real suspension bending, friction, wear, bob size, air drag and escapement impulses can alter the rate.
The displayed generating radius does not mean the cheek is a circular arc. Its curvature varies. No machining tolerance, service interval or guaranteed clock accuracy is inferred from this geometric model.
Cycloidal pendulum questions
Does the cord stretch?
No. More wrapped cord means less free cord, with a constant total length.
Why can the angle change without changing the ideal period?
The bob’s height is quadratic in arc displacement, producing simple harmonic motion along the cycloidal path.
Does the loss tile describe this cycloidal pendulum?
No. It is the explicitly labeled ordinary circular-pendulum comparison.
What is a beat?
Half a complete oscillation, so a one-second beat corresponds to a two-second full period.
References
Hiscox, Mechanical Movements, movement1146, printed page283, curved frame and flexible pendulum. NIST DLMF19.8, arithmetic-geometric mean evaluation of complete elliptic integrals. NIST DLMF22.19, pendulum application of elliptic functions. The cheek, cord and tautochrone relationships shown here are derived and numerically checked from the stated parameterization.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
