The equation becomes a machine

A digital computer simulates a differential equation: it advances a number, step by step, according to a rule. Vannevar Bush’s analyzer did something else. It was assembled into a physical system whose motion obeys the equation, and then it was simply switched on. The historian Larry Owens put it best: its parts did not calculate the equation, they kinetically acted it out.

That is a beautiful idea, and it comes with a bill. If your computer is a physical object, then its errors are physical too. You cannot ask for another decimal place; you have to buy it, with friction, with torque, with machining tolerance, and with time. This playground is that bill, itemised.

One wheel on one disc

Everything rests on a mechanism James Thomson described in 1876. A disc turns through an angle $\Theta$. A small wheel rests on its face at a distance $r$ from the centre. Because the disc’s surface moves faster the further out you go, the wheel accumulates:

Put a carriage on the disc that holds the wheel at $r = k\,V$, drive the disc with the independent variable, and the wheel’s output shaft comes out carrying $\int V\,dx$. That is a mechanical integral. Wire the output of one integrator into the carriage of another, feed the last one back to the first, and you have an oscillator that solves $y'' + 2\zeta\omega y' + \omega^2 y = 0$ by moving.

Why Kelvin’s machine stayed on paper

Lord Kelvin saw all of this in 1876 and wrote down how interconnected integrators could solve differential equations. Nobody built one for fifty-five years, and the reason is a single number. A friction wheel can only transmit as much torque as the contact can hold:

which is a fraction of a newton-metre. Meanwhile the next integrator’s carriage, the gear train and the pen want several times that. So the wheel does not drive them. It slips. Run the Kelvin, 1876 preset and watch: the discs turn, the shafts turn, the drive hums, and the pen draws a perfectly straight line. The machine is computing nothing at all.

What Bush, Hazen and C. W. Nieman added in 1931 was the torque amplifier: a pair of counter-rotating capstans that let the wheel’s feeble signal command a powerful output without supplying the power itself. It multiplies the available torque by $G$, and the slip that remains is

The whole machine hangs on driving that one number below one.

A tenth of a percent is not a small number

Suppose you win. With a good amplifier the creep is tiny, a hundredth of a percent. But creep is not noise sprinkled on the answer; it is a systematic loss on every turn of every wheel, and both integrators suffer it. The loop gain falls to $g^2$, and the machine simply runs at the wrong frequency:

A wrong frequency is a phase error that grows without bound. The pen starts on top of the true curve and drifts steadily out of step with it, and because a sinusoid is brutally sensitive to phase, a drift of one degree is already a percent of pointwise error. Watch the error-accumulation panel: the gap does not settle, it ratchets. This is the deep reason analog machines were used for short transients and not long integrations, and it is what nobody means when they quote an analyzer’s accuracy as a single number.

Why it had to run slowly

A run of the MIT machine took the better part of an hour. That was not because the mechanism was weak. It is because the follow-up servos need real time, perhaps a tenth of a second, to obey an order. Turning the independent variable at $S$ units per second converts that fixed real-time lag into a lag measured in the problem’s own variable:

and inside a feedback loop a lag is not a delay, it is negative damping. To first order the loop’s damping becomes

so once $\tau > 2\zeta / (g\omega)$ the machine has spent all the damping it had, and it starts feeding its own oscillation. Run run it fast: the same machine that was correct to two digits at a crawl now climbs off the paper. Hurrying an analog computer does not make it wrong at the margins. It makes it unstable.

The scaling problem

Before a single shaft turned, an operator spent hours choosing scale factors. It is the only real decision the machine offers, and it is squeezed from both sides. The gear trains have lost motion, so a variable smaller than the dead band simply never arrives. The disc has an edge, so a variable larger than the rim is clipped flat. Every variable has to live in between:

Notice that $k$ appears on both sides. Turning it up buys resolution and spends headroom; turning it down does the reverse. The window between the two walls is the machine’s dynamic range, and it is a property of the metal: $20\log_{10}\!\left(R / \rho\beta\right)$ decibels, fixed the day the gears were cut. Scaling does not widen that window. It only decides where in the window your problem sits.

The scaling panel draws that window directly. There is a plateau, a cliff where the wheel runs off the disc, and a slow decline into the lost motion. Choosing k was choosing a spot on that curve with a slide rule and no way to check your work until the pen had been moving for half an hour.

The equation is the wiring

The simulator here contains no differential equation. Its state is the accumulated rotation of each integrator wheel, and its update rule is the kinematics of a wheel on a disc, a differential gear, a gear train with lost motion, and a servo with lag. The equation lives entirely in the patch: which shaft turns each disc, which shaft each carriage is screwed to, and which change-gear ratios feed the adders. Choose a different equation in the settings and you are not selecting a different formula; you are rewiring the bench.

Four setup sheets ship. The damped oscillator is the classic two-integrator loop. The exponential decay is the smallest possible program: one integrator and one sign-reversing gear, closed on itself. The forced oscillator wires the input table into the adder, and with it the operator’s hand: the cross-hair wanders a fraction of a percent off the forcing curve, and that wander is stirred into the answer. The van der Pol equation is the machine at full stretch: its squares are built by integrators whose discs are geared to $y$ itself, since $\textstyle\int y\,dy = y^2/2$, so the products are integrals too. Multiplication by parts costs two more integrators and pays the creep budget twice, which is why the calibration panel checks that the limit cycle still forgets its initial condition.

What this model leaves out

Four error sources are modeled: microslip, backlash, follow-up lag, and saturation, plus the operator’s tracking error when a patch uses the input table. A real machine also has disc runout, wheel wear, shafts that wind up under load, paper that stretches, and temperature. Each integrator runs at a fixed multiple of one common scale factor, written into the setup sheet; the hours an operator spent deriving those multiples are assumed already spent, and re-scaling mid-run is not represented. The lag is lumped into a single first-order term rather than the second-order transmission it really was.

None of the constants are fitted to a historical accuracy figure. The calibration panel checks the simulated mechanism against the mechanics it claims to obey: the exact solution of the equation it is drawing, the characteristic roots of its own loop, the travel limit of a carriage, and the invariance that makes a scale factor a change of units rather than a change of machine.

Model changelog