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Differential Analyzer — Mechanical Calculus

Computing · c. 1931

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LabelsExact math

Differential Analyzer — Mechanical Calculus

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Step advances x by one-240th of the run. The inspector’s Step moves one mechanical transfer at a time.

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Teaching reconstruction · dimensions and linkages simplified

Follow the motion: motor → disk → integrating wheel → torque amplifier → bus shafts → pen. Open gear frames expose the bevel pairs. Axial paint marks rotate with every transmission shaft.

How the model works

The solid trace is generated from the numerical shaft state of the modeled mechanism and displayed at the output-table pen position. Toggle 'Exact math' to compare it with a dashed analytical or high-accuracy reference solution.

y″ = −yx = 0.00

x0.00y0.00

y′0.00

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Integrator I accumulates −y to obtain y′, and integrator II accumulates y′ to obtain y. A reversing gear returns −y to the first integrator, closing the loop for y″ = −y; the modeled output table plots the resulting sinusoid.

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c. 1931 · Computing

Differential Analyzer — Mechanical Calculus

Four modeled wheel-and-disk integrators connect shaft variables to numerical solutions and plotted traces.

A differential analyzer is an analog computer that represents variables as continuously changing mechanical quantities. Vannevar Bush and colleagues developed an influential machine at MIT around 1931 using wheel-and-disk integrators, interconnected shafts, gearing, and torque amplifiers. Earlier wheel-and-disk integration work by James Thomson and William Thomson helped establish the underlying method.

For an ideal wheel-and-disk integrator, a small disk rotation combined with the signed radial position of the contact wheel gives an output increment proportional to r dx. This exhibit advances numerical shaft-state variables according to that relationship and maps them onto animated disks, wheels, carriages, shafts, and a pen plot. The visual contact, friction, torque amplification, backlash, slip, and structural compliance are not physically solved.

Preset configurations illustrate sinusoidal, exponential, damped-oscillation, nonlinear-pendulum, and drawn-curve integration problems. A solid trace comes from the model’s numerical state; an optional dashed trace is an analytical or high-accuracy reference. Agreement is therefore a way to inspect numerical behavior, not evidence that a reconstructed brass machine has produced an exact physical solution.

Connecting integrator outputs and feedback paths demonstrates how an analog configuration can encode a differential equation. The exhibit is inspired by historical differential analyzers but is not a dimensional reconstruction of Bush’s MIT installation, and its programs simplify the scaling, range limits, errors, and operator adjustments of physical machines.

Specifications

Principle: Idealized wheel-and-disk relation: dφ ∝ r dx
Representation: Variables stored as numerical shaft-state angles
Comparison: Modeled trace versus analytical or high-accuracy reference
Programs: Oscillation, exponential, pendulum, and drawn-curve examples
Scope: Kinematic visualization; no contact, torque, or error physics

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Differential Analyzer — Mechanical Calculus

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Input table — draw f(x)

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