754,113. Electric analogue calculating apparatus. ASSOCIATED ELECTRICAL INDUSTRIES, Ltd. Aug. 12, 1953 [Aug. 28, 1952], No. 21658/52. Class 37. An electric analogue computing apparatus for solving differential equations representing physical problems comprises a resistance network representing at least part of the equation in finite difference form and having a series of nodal junction points representing successive position points of the difference equation, together with means for feeding currents to the junction points and for measuring the resulting voltages thereon by comparison of such voltages with required voltage values satisfying the problem to be solved, so as to derive error signals from the nodal junction points which are simultaneously displayed in a visual array so that the effect of input adjustments upon the voltage relationships at individual nodal junction points may be determined. Fig. 1 shows a section of a resistor analogue network for solving the 4th order differential equation defining the critical speeds of a whirling shaft where x is a position along the shaft; y is the shaft deflection; E is Young's modulus; I is the moment of inertia about the shaft axis; m is the weight of shaft per unit length; # is the critical speed; which network comprises resistance chains N1, N2 comprising resistances R1, R2 and interconnected by resistances R3; the values of resistances R1, R2, R3 being determined by local values of E, I and m with appropriate scaling factors. Nodal points An, Bn represent the n'th of a series of positional points measured along the shaft axis dividing it into sections corresponding to successive network meshes. Current In from point Cn is fed through resistance R4 into point Bn of chain N1 and a voltage Un measured with reference to the network zero point appears at point An of chain N2 to represent the shaft deflection y. The value of In is adjustable, and for the solution of equation (1) where α is a scaling constant; #xSP1/SP,#xSP11/SP are intervals of x to the left and right of the shaft point represented by An, Bn; #n is a constant bearing reaction at this shaft point, so that the voltage drop Pn across R4 is given by where Un, Pn are measured voltages; bn is a known constant; ao is a constant multiple of #SP2/SP for each network mesh at solution; and during the iterative adjustment of successive networks towards solution where Qn is a residual error-signal. Fig. 2 shows a device for systematic iterative adjustment of the network during which Qn#0 and a#ao for each nodal point, wherein the current In supplied to each nodal point Bn is derived from an alternating source V over an adjusting variable resistance Rx and the series resistance R4, the voltage drop Pn across which is selected by multiple switches S1, S2 to energize one primary of transformer Tr. The voltage Un appearing at each nodal point An is selected by multiple switch S3 ganged with S1, S2 to energize a potentiometer a producing a slider voltage of a Un, and the voltage Un also energizes a potentiometer bn for each nodal point whose slider potential bn Un is selected by multiple switch S4 ganged with S3 to energize a second primary of transformer Tr in. opposition to the voltage a Un so that an error voltage Qn given by equation (4) appears across the secondary for each nodal point of the network selected by the ganged switches in correspondence with different points along the shaft. The series of voltages may be displayed, e.g. on a range of n vibration galvanometers selected in correspondence with the successive outputs of the transformer, or as y-deflections of a long-persistence C.R. oscilloscope wherein the successive Qn traces are shifted along the x axis synchronically with the operation of switches S1 to S4 in selecting successive pairs of network nodal points An, Bn; so that the iterative adjustment of the currents In flowing into each mesh node point so as to reduce Qn to zero in each case is facilitated. Fig. 3 shows a modification in which the several voltages Pn, Un, bn, aUn derived from the successive network sections as shown in Fig. 2, are selected by a 25-way uniselector which is ratchet motor operated from half-wave rectified A.C. to switch at 50 positions per second. Writing E1=Pn, E2= -bn Un, E3= - Un and k2=a0; equation (4) reduces to k2 E3 - E2+E1=Q-(5) which is synthesized by supplying voltages E1 and - E2 from the uniselector contacts to a first subtracting circuit which produces an output voltage - (E1+E2)/(2) and voltage - E3 to a calibrated potentiometer producing an output voltage - kSP2/SPE3/2, which outputs are supplied to a second subtracting circuit to produce an output voltage “(E1+E2 - kSP2/SPE3)= - Q/4. The residual - Q/4 developed for each section of the analogue network and selected by the uniselector is amplified and displayed on a long persistence cathode-ray oscillograph having an x deflection controlled by the same uniselector to increase in successive steps whereby a large number of residuals are displayed together as vertical lines positionally displaced along the x axis in conformity with the positions of the successive nodal points of the network, which lines extend vertically upwards or downwards from the x axis in dependence on the sign of the residual. Spurious voltages arising from switch contact potentials are eliminated by blacking out the C.R. trace for the half-cycle immediately following the instant at which contact is made, so that the spurious voltages are not displayed and the sense of the display represents the sign of each residual error as well as the magnitude. As before, the currents into each network node are iteratively adjusted to reduce each residual voltage to zero for solution of equation (1). The difference circuits may comprise (Fig. 4) a double triode having equal anode resistances and equal cathode bias resistances; the anodes being connected directly to their opposite cathodes by resistances R5, R6. Voltages E1, E2 are applied to respective grids and the difference voltage appears on a sliding tap on resistance R5; adjustable for balance. The resistance analogue network may be D.C. excited, and the uniselector may be operated at such a speed as to act as a chopper; the succeeding amplifier circuits being given an adequate time constant so that the chopped signals are transmitted and are displayed as bright spots positionally displaced with respect to the X-axis of the oscilloscope to give the magnitude and sign of each residual. Specification 684,989 is referred to.
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