Patent GB-757783-A (GB)
Title: Improvements relating to electrical analogue-computing apparatus
Inventor: LIEBMANN GERHARD
IPC: G06G7/46
Filed: 1952-10-02
Abstract: 757,783. Electric analogue calculating apparatus. ASSOCIATED ELECTRICAL INDUSTRIES, Ltd. Sept. 14, 1953 [Oct. 2, 1952], No. 24740/52. Class 37. An electric analogue computing apparatus for the solution of a physical problem affecting a material body and involving a fourth order differential equation of the general form #<SP>4</SP>F =G, where F is a dependent variable and G is prescribed or determined by such dependent variable, comprises a pair of identical or geometrically similar impedance networks representing the equation in finite difference form and of configuration representing the body limits, having high impedances connecting the several corresponding junction nodes of the network, representing positional points of the body, and the network impedance values chosen to represent the physical properties of the material at each point thereof; there being provided circuit means for feeding currents representing G into the junction nodes of the first network and measuring means for determining the voltages representing F at the junction nodes of the second network, whereby the several currents into the first network are adjustable until the voltage distribution over the second network satisfies the required boundary conditions; the boundary junction nodes being supplied with voltages representing the prevailing boundary conditions of the problem. The apparatus is particularly applicable to the determination of mechanical stresses arising in materials due to forces applied externally or arising internally, due e.g. to thermal expansion or centrifugal forces of rotation. It is shown that such stresses may be derived from the Airey stress function # (x, y) when the 4th order partial differential equation (1) is solved for # and its derivatives for given boundary conditions; f being a function of the independent variables x and y, or a function of #, or a constant, or zero, in accordance with the particular problem to be solved. For such stress problems, Fig. 1 shows two identical or geometrically similar resistance networks N 1 , N 2  whose corresponding mesh nodes P 1 , P 2  are interconnected by resistances R 3 . The corresponding network resistances R 1 , R 2  have the same ratio everywhere within the networks, and are constant throughout or (as in problems of rotational symmetry) are inversely proportional to the distance of nodes P 1 , P 2  from the axis, and also R 3  >> R 1  and R 2  to avoid interaction. It has been shown that in a resistance network N 2  of equal resistances R 2  the difference equation corresponding to the 2nd order partial differential equation (2) is soluble for prescribed boundary conditions of the function #=# (x, y) imposed by the application of appropriate potentials of the boundary nodes of N 2 , and g is represented by the currents I 2  fed into nodes P 2  such that (3) where h=#x=#y; the mesh interval. If currents I 2  are supplied from voltage sources of values # (>>#) at nodes P 1  of network N 1  such that (4) then from equation (2) (5) but since the voltage sources at nodes P 1  are distributed analogously to the solution of equation (2) by network N 1  for its own boundary values, i.e. (6) then from equation (5) operated upon by #<SP>2</SP>, (7) whereby the cascaded resistance network (Fig. 1) solves the 4th order differential equation (1), provided that appropriate boundary conditions for  #  are set up on N 1  and for # on N 2 , and a current I 1 = is fed into the mesh nodes P 1  of network N 1 . The potential distribution at the nodes P 2  of network N 2  gives the required solution  # , from which the stress components and principal and sheer stresses are obtainable in known manner by numerical or graphical differentiation. The sum of principal stresses = #<SP>2</SP> # , and the measurement of the distribution of # at the nodes P 1  of network N 1  gives a check on the evaluation of stress components from function  # . For the solution of the three dimensional equation the networks N 1 , N 2  may be constructed as three dimensional models of the problem, and an additional network may be connected in cascade therewith for the solution of the partial differential equation #<SP>6</SP>#=f. Where boundary values are given only for  #  and ##/#x or ##/#y the equation is iteratively soluble by setting up boundary values for # on network N 2  and arbitrary boundary values on network N 1 ; thereafter comparing the resultant boundary values for ##/#x and ##/#y with the given boundary values and readjusting the boundary values on network N 1  successively until the resultant and prescribed values for ##/#x and ##/#y are coincident. The comparison between voltage values at the network boundaries representing the resultant and prescribed boundary values may be effected by a long-afterglow successive cathoderay display apparatus of the kind described in Specification 754,113. In a modification, the network may be constructed in the form shown in Fig. 2 wherein P 0  represents a network node, P 1  to P 4  its nearest adjacent nodes, P A  to P D  its next nearest adjacent nodes in the x and y directions, and Pa to P d  its nearest adjacent nodes in the diagonal directions. The difference equation equivalent to differential equation (1) is then given by in which # 0 , # 1 , &c. are the values of # at nodes P 0 , P 1 , &c. of the network and h is the mesh interval in the x and y directions. Representing functional values # 0 , # 1 , &c. by analogue voltages, the equation (8) is satisfied by the network of Fig. 2 in which the current to be supplied to point P 0  is given by (9) where R is the resistance value connecting P 0  to P 1 -P 4 . Node P, is connected to node P A -P D  by negative resistances of value - 8R and to nodes P a -P d  by negative resistances of value - 4R, while the remainder of the network is built up in a similar manner within the boundaries of the model. Negative resistances may be synthesised by retro-active valve circuits, transistor circuits, or resistance circuits incorporating signal generators in known manner. Alternatively, positive resistances may be substituted by inductances and negative resistances by capacitances such as to represent the required impedance values at a fixed working frequency (Fig. 3, not shown). In certain vibrational problems wherein f =Kv 0#  in equation (1) where K is a constant and v 0  is the vibration frequency, positive and negative currents are supplied to or withdrawn from the mesh nodes by connecting them respectively to the zero potential point over positive or negative impedances in the form of inductances or capacitances of appropriate values. Arbitrary values may be chosen for which the network is non-resonant, but from equations (9) and (10) it is shown that resonance will occur at a value of negative impedance given by (10) so that by synchronously adjusting the node to ground capacitances of the network until resonance ensues, the value of the resonant frequency and the standing vibrational pattern may be determined from the network. The conditions at the boundary nodes may be determined by successively or simultaneously displaying the voltages thereupon by a cathoderay oscillograph.