A compensation balance uses two split bimetallic rim sections to move weights inward as temperature rises. Brass on the outside expands more than steel on the inside, bending each free rim end toward the staff. Reducing the balance’s inertia can offset a hairspring that becomes less stiff when warm.
Compensation Balance Thermal Response Calculator
Explore a split brass-and-steel balance wheel. Enter temperature and calibrated thermal responses, then see how the computed weight positions and inertia affect an ideal oscillator’s rate.
f/f₀=√[(k/k₀)/(I/I₀)]; daily rate=86400(f/f₀−1). Inertia is integrated over the two curved rims plus weights and diametric bar.
Positive daily rate means gain. Frequency uses complete oscillations, not beats. Main wheel uses actual modeled thermal dimensions.
- Reference radius 10 mm; two 120° rims of 20 mg each; two 10 mg weights; 10 mg diametric bar.
- Anchor expansion 11 ppm/°C; rim mean expansion 15 ppm/°C; entered curvature response.
- Uniform rim mass per arc length and point compensation weights.
- Ideal torsional frequency; entered linear stiffness response; no escapement or damping.
Stated model dimensions and calibrated curvature response; no certified watch accuracy claim. Spring curve schematic. Thermal displacement inset is 20×.
Two fixed rim roots and two free ends
A diametric bar rotates with the balance staff. Each end anchors one brass-and-steel rim section. Cuts near the opposite ends leave the two rim tips free to bend. Compensation weights move with the rim; their distance from the staff affects rotational inertia.
The balance oscillates under a hairspring’s restoring torque. It does not spin continuously. This animation keeps the outer spring attachment fixed while the staff and inner attachment oscillate. The spring curve is schematic; the balance and rim geometry are calculated at the entered temperature.
The grey dashed outline shows the reference-temperature rim at the same oscillation angle. A separate stationary inset magnifies the selected weight’s thermal displacement by 20, making the small movement visible without exaggerating the main balance geometry.
Explore compensation rather than assume perfect correction
This educational model lets you vary temperature, calibrated bending response, compensation-weight position and hairspring stiffness response. Moving the weights changes how strongly rim bending affects inertia. The rate estimate can improve, undercompensate or overcompensate; it is not forced to zero.
The page previously had no functioning calculator and included an unrelated weighing-scale video. This reconstruction adds a stated geometric model and removes the unrelated content and unsupported historical performance guarantees.
Calculate inertia from the curved rim
The reference wheel radius is 10 mm. Each rim section spans 120° and has mass 20 mg. Each compensation weight is a 10 mg point mass; the diametric bar is a 10 mg slender rod. These are explicit model choices, not measured dimensions of Brown’s balance.
For temperature change ΔT, the anchor radius is a=10(1+11×10⁻⁶ΔT) mm. Rim length is L=10×120π/180×(1+15×10⁻⁶ΔT). Curvature is κ=0.1/(1+15×10⁻⁶ΔT)+C×10⁻⁶ΔT, where C is your entered calibrated response. C is not calculated from unspecified layer thicknesses or elastic moduli.
Measured from a rim root, x(s)=a+[cos(κs)−1]/κ and y(s)=sin(κs)/κ. The opposite half is a 180° rotation. For c=a−1/κ, the mean squared radius of a uniform rim is c²+1/κ²+2c sin(κL)/(κ²L). Each weight lies at s=L×position/100.
Total inertia is the sum of both distributed rims, both point weights and the bar. Let J=I(ΔT)/I(0) and K=1+β×10⁻⁶ΔT, where β is the entered stiffness coefficient. Then f/f₀=√(K/J), and daily rate is 86400(f/f₀−1) seconds/day. Positive means gaining time in this page’s convention.
Read the model’s response
At zero temperature change, the inertia ratio and stiffness ratio are both one. The frequency equals the reference frequency and estimated temperature-induced daily rate is zero.
With a negative stiffness coefficient and no compensating bend, warming tends to slow the oscillator. Increasing inward curvature or placing weights farther along the free rim can reduce inertia more strongly. Too much correction can make the warmed oscillator gain time.
A calibrated model is not a chronometer rating
This model uses uniform curvature, linear expansion, a linear stiffness coefficient and a small-amplitude ideal torsional-oscillator relation. It excludes escapement impulses, damping, amplitude dependence, gravity, poise error, material hysteresis and nonlinear thermal effects.
Real brass-steel compensation balances can retain middle-temperature error even after their hot and cold rates are matched. A displayed daily rate is the result of the entered assumptions, not a prediction of an unidentified watch’s certified accuracy.
Compensation balance questions
Why is the rim split?
The cuts allow each bimetallic section to bend from its fixed root and move its free end and weight radially.
Does brass go inside or outside?
In the illustrated traditional arrangement, brass is outside and steel is inside. Greater brass expansion on warming bends the rim inward.
Why is the main movement small?
The actual thermal deflection is small compared with the wheel radius. Only the separate displacement inset is magnified.
Is one oscillation the same as one beat?
No. One complete oscillation has two half-cycle beats. The frequency control and output use complete oscillations per second.
Why is the rate not always zero?
Compensation requires matching inertia change to spring stiffness change. The model reports the remaining error instead of assuming a perfect match.
Construction and timekeeping references
Brown, movement 319, describes the diametric bar, outer brass/inner steel compound rims, attached weights and inward thermal bending. NIST, The Effect of Materials on Time, discusses temperature effects on mechanical oscillators. Bureau of Standards Circular 392, Testing of Timepieces, explains residual middle-temperature error and temperature testing. Model dimensions, curvature calibration and masses are stated separately above.
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