## The Differential Analyzer of Vannevar Bush is a machine for solving (2007)

by
Dylan Leeman

by
Dylan Leeman

@MISC{Leeman07thedifferential,

author = {Dylan Leeman},

title = {The Differential Analyzer of Vannevar Bush is a machine for solving},

year = {2007}

}

differential equations with reasonable boundary conditions. The original differential analyzer at MIT was capable of solving ”ordinary differential equations of any order up to the sixth, and with any amount of complexity within reason. ” (Bush, 451) An example of an equation that can be solved using the differential analyzer follows (a machine configuration for the equation will be presented in section 2): d2x dx + k + g = 0 (1) dt2 dt An analog machine like the differential analyzer is uniquely suited for the solution of such equations, since the solutions involve the integration of continuous functions. In fact, the machine is actually configured as a mechanical representation of the equation to be modelled. The machine consists of a number of bus shafts, which can provide the input or accept the output of a number of functional units. Functional units are attached to the bus shafts by the use of spiral gear boxes, which allows the machine to be specialized for a wide variety of equations. The machine is ”programmed ” by applying appropriate interconnections of bus shafts, functional units, and input and output tables. 2 Example Machine Configuration Recall equation one (1) from above; rearranging the equation yields: dx = − k dt dx + g dt (2) dt The schematic in FIG. 1 follows more naturally from this arrangement. Note for the input to that the output of the integrator labelled II provides dx dt integrator II: � k dx dt + g �. The output also drives the input to integrator I, which has as its output the function x. Here the constant g is provided using an input table, so that it can be easily changed, while the constant multiplicative factor k has been introduced using a spur gear box. The output table has been set to record the dependent variable x and its derivative as functions of the independent variable t. 1 Figure 1. It was customary in schematics such as these, which represent the basic conceptual layout of the machine, to omit indications of sign and relative scales. In fact, a left-hand spiral gear box should connect the output of II with the input of I, to accurately reflect the equation. 3

differential analyzer vannevar bush bus shaft functional unit output table analog machine constant multiplicative factor appropriate interconnection continuous function input table original differential analyzer spur gear box equation yield relative scale example machine configuration recall equation differential equation dependent variable left-hand spiral gear box basic conceptual layout dx dt dx dt dx dt integrator ii independent variable d2x dx machine configuration ordinary differential equation spiral gear box mechanical representation reasonable boundary condition wide variety

Developed at and hosted by The College of Information Sciences and Technology

© 2007-2016 The Pennsylvania State University