Patent GB-754114-A (GB)
Title: Improvements relating to electrical analogue-computing apparatus
Inventor: LIEBMANN GERHARD
IPC: G06G7/46
Filed: 1952-10-02
Abstract: 754,114. Electric analogue calculating apparatus. ASSOCIATED ELECTRICAL INDUSTRIES, Ltd. Sept. 14, 1953 [Oct. 2, 1952], No. 24741/52. Class 37. An electric analogue impedance network, representing the mathematical equivalent of a physical problem capable of representation as a fourth order differential equation in one independent variable, includes a pair of similar impedance chains each of which has a series of junction points corresponding to successive values of such variable and plural resistances successively interconnecting the junction points, the impedance components of the chains and the interconnecting resistances being so scaled that the network represents the equation infinite difference form; and is operable in conjunction with means for feeding analogue currents into the junction points of the first impedance chain and means for observing the resulting voltages derived at corresponding points of the second impedance chain, together with means for iteratively adjusting the input currents until the relations between currents and corresponding voltages satisfy the difference equation at all position points, when such voltages will represent the values of the dependent variable at such points. In apparatus for solution of the differential equation describing the vibrational resonant frequencies of a rotating shaft, where x defines a point on the shaft axis; y=deflection of the shaft from its axis at such point; E =Young's modulus; I= moment of inertia of shaft about its axis; m=weight of the shaft per unit length ; w=angular velocity of shaft; for discrete critical values of w, the differential equation is replaced by its equivalent finite difference equation, for which there is constructed a corresponding analogue impedance network wherein the network voltages represent the displacements y, the network intervals are determined by the positions of the supporting bearings, and quantities E, I, and m are represented by impedances of the network variable in value as functions of x in correspondence with the shaft characteristics. The analogue network (Fig. 1) comprises two similar series resistance chains N 1 , N 2  built up respectively of resistances R 1 <SP>1</SP>, R 1 <SP>11</SP>, &c. and R 2 1, R 2 <SP>11</SP>, &c. interconnected by lateral resistances R 3  between junction points P 1 , P 2 , &c. into a ladder network; the individual meshes of which represent adjacent sections of shaft; the series resistance values being given by R 1 <SP>1</SP>=a 1 #x<SP>1</SP>R 0 ; R 1 <SP>11</SP> = a 1 #x<SP>11</SP>R 0  ; R 2 <SP>1 </SP>= a 2 #x<SP>1</SP>R 0  ; R 2 <SP>11</SP>= a 2 #x<SP>11</SP>R 0 ; where a 1 , a 2  are scaling constants, #x<SP>1</SP>, #x<SP>11</SP> the intervals between network junction points, and R, is a basic constant resistance value. It is shown that such a network solves the finite difference equation equivalent to equation (1) if R 3  is given a value and a current is fed into junction point P 1 , where b is a scaling constant; such that R 3 >>R 2  and R 1 , and where #, y are variables and m and EI represent the average values of the parameters over the interval (#x<SP>1</SP>+ ,#x<SP>11</SP>). The voltages appearing at the junction points P 1  of the N 1  chain then represent the bending moments, and the voltages at junction points P 2  of the N 2  chain represent the displacements y, of the shaft at the corresponding points thereof. For determining the critical angular velocities # 1 , # 2 , &c., currents i 1  are fed in at junction points P 1 , the resultant voltages measured at corresponding points P 2 , and the currents are cyclically and iteratively adjusted until a self-consistent voltage and current distribution is obtained, satisfying equation (3) for the same value of the critical angular velocity # at all points of the network. The residual error current for each junction point given by may be displayed for iterative adjustment by the C.R.O. system described in Specification 754,113. The problem may alternatively be solved by an electrical analogy of the difference equation replacing the fourth order derivatives, wherein for equation (1) in which y is represented by the junction voltages, the suffixes 0, 1, &c. refer to values at points P 0 , P 1 , &c. along the shaft (Fig. 2) separated by intervals h +1 ,  h+2  &c. and which is represented by the analogue circuit shown in Fig. 3 comprising a chain of resistances whose values are respectively given by where c is a scaling constant, R 0  is a reference resistance value, and B 0 =(EI) 0 , B 1 =(EI) 1 , &c. A current i fed in at junction points P 0 , P 1 , &c. is given by and equation (1) is solved by iterative adjustment of i 0  until self consistency is attained as set forth above. Alternatively, resistances R o , o  are inserted between Po, P 1 , &c. and the zero potential point of a value given by where A is a proportionality constant and the resistances are adjusted to resonance by varying A to a value A 0  at which resonance occurs and # 0 =A 0  - ¢. Certain resistances in the above - networks are of negative sign, and may be synthesised by electronic valve or transistor circuits, or resistance networks incorporating signal generators. Alternatively, the network may contain inductances representing positive resistances and capacitances representing negative resistances (Fig. 4, not shown), and may then be energized from a source of alternating voltage of fixed frequency. Fig. 5 shows a section of network for the solution of the equation (1) using the difference equation (7) and comprising positive resistances only, wherein resistance chains N, N<SP>1</SP> are diagonally interconnected by further resistances, and are energized respectively by voltages or currents equal in value but opposite in phase sense, which may be derived, e.g., from a transformer centretapped to the zero potential point and varied by duplicate ganged controls. The resistances R 1 , 0  and R -1,0  in chains N, N<SP>1</SP> &c. and interconnecting resistances R 2 , 0  and R -2 , 0  respectively are of values given by equation (8) and currents i 0  and - i 0  are fed into junction points P 0  and P 0 <SP>1</SP>; such that where α and are constants given by the geometry and construction of the shaft which is the subject of the problem. The iteration process set forth is then applied to solve the problem until the distribution of voltages representing y 0  and currents i 0  is such as to satisfy equation (12) at all points for the same value # of critical speed. Finite deflections at shaft bearings are allowed for in the problem by introducing appropriate constant terms into equation (1), and the apparatus may be applied to the solution of other problems (e.g. bending of shafts under load) reducible to differential equations similar to equation (1).