This compensating pendulum uses a central bob and two smaller side weights carried by bonded steel-and-brass arms. When warmed, the brass underside expands more and bends the arms upward. Raising the side weights can counter some of the timing change caused by the lengthening pendulum rod.
Compound Compensating Pendulum Interactive Calculator
Explore the two bimetallic arms and their side weights. Temperature changes the actual arm geometry; the calculation includes both inertia and gravitational restoring torque.
Equation Used
- Reference bob position 1000 mm, arm root 850 mm, central point mass 1000 g.
- Steel/brass moduli 200/100 GPa; equal layer thickness; uniform thermal curvature.
- Rod and strips massless, no elastic sag; two entered side point masses.
- Small-amplitude physical pendulum with g=9810 mm/s².
Explicit illustrative dimensions and masses. Equal layers, fixed elastic moduli, no sag, point masses and small-amplitude motion.
A central bob and two thermally bending arms
The reconstruction follows Hiscox figure 1148: steel is the upper layer, brass the lower layer, and the two weights W move with the arms attached near the top of the main bob M. It is not a gridiron pendulum with alternating long rods.
The main view shows the calculated thermal shape, with a dashed reference outline at the same swing angle. The detail below shows both material layers and the computed weight rise. Layer strokes are enlarged for visibility; the arm centerlines and weight centers follow the calculation. The pendulum autoplays with a small 3° swing, shown at half speed.
Explore undercompensation and overcompensation
Change the expansion coefficients, temperature, arm length, weight mass or strip thickness. These alter the curved arm geometry or mass distribution and therefore the predicted oscillation period. The calculator reports remaining daily gain or loss; it does not assume perfect compensation.
Thermal geometry and physical-pendulum period
Reference dimensions are 850 mm from suspension to arm root and 1000 mm to the central bob center. The main bob is modeled as a 1000 g point mass. Each arm has reference curvature 0.0006/mm. Entered thickness is shared equally by the two bonded layers. The model uses constant elastic moduli of 200 GPa for steel and 100 GPa for brass.
Let Δα=(αbrass−αsteel)×10⁻⁶. Force and moment equilibrium of an unloaded equal-layer beam give thermal curvature increment Δκ=(16/11)ΔαΔT/t and centerline strain ε=[(2αsteel+αbrass)×10⁻⁶/3+(4/33)Δα]ΔT. Set e=1+ε, a=arm length×e and κ=0.0006/e+Δκ.
Each weight has horizontal distance x=sin(κa)/κ and rise above the arm root h=[1−cos(κa)]/κ. The formulas use their continuous straight-arm limits at κ=0. Its downward coordinate from suspension is y=850(1+αsteel×10⁻⁶ΔT)−h. The bob coordinate is b=1000(1+αsteel×10⁻⁶ΔT).
For each side mass m, inertia I=1000b²+2m(x²+y²), and gravitational first moment H=1000b+2my, in consistent gram–millimeter units. Equivalent length is I/H and period P=2π√[I/(9810H)]. Daily rate is 86400(P₀/P−1); positive means gaining time. P₀ uses the same entered design at zero temperature change.
Useful checks
At zero temperature change, daily thermal rate error and both displacement changes are zero. Increasing arm length increases the vertical displacement caused by a given curvature change. Making the strip thicker reduces thermal curvature. Increasing side weight mass changes inertia and gravitational restoring torque; it does not simply shorten the pendulum by the weight’s rise.
If the entered brass coefficient is below the steel coefficient, warming reverses the bending response. The display retains that result instead of silently replacing your input.
What the estimate excludes
This is an illustrative small-amplitude physical-pendulum model, not dimensions or test data for a particular historical clock. It neglects rod and strip mass, bob size, elastic sag under the weights, suspension compliance, air resistance, escapement impulses, temperature gradients, material hysteresis and amplitude-dependent period. The colored bob outlines illustrate point masses rather than integrating their visible shapes.
The bonded-beam thermal relation assumes equal layer thickness, fixed moduli and small strains. Arm curvature is uniform. Mechanical stress, load-bearing adequacy and real clock accuracy require a fuller design and measurements.
Compensating pendulum questions
Why do the small weights rise when heated?
The brass underside normally expands more than the steel top. The bonded arm bends toward the steel side.
Why can raising a weight change the clock rate?
Both its moment of inertia about the suspension and its contribution to gravitational restoring torque change. The calculator evaluates both.
Does zero displayed rate prove the clock is accurate?
No. It only balances the thermal terms in this simplified model at the entered temperature.
Construction reference
Hiscox, Mechanical Movements, Powers, Devices and Appliances, figure 1148, printed page 283, illustrates the central bob and two steel-top/brass-bottom compensation arms. The dimensions, masses and elastic assumptions here are stated model choices. The beam relation follows zero resultant axial force and bending moment through the two layers; the period follows inertia and gravitational restoring torque about the suspension.
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