High-accuracy Mechanical Integration by Shear in Viscous Liquids.

-Integration from the standpoint of instrumentation is provided by any device wherein the time rate of change of an output quantity is accurately proportional to the input quantity. Viscous shear in fluids is a very useful means of realizing integration when mechanical torque acts as the input and the angular velocity of a rotor with respect to a stator is the output. This type of integration is widely used in situations requiring high accuracy for oscillatory inputs with output angles restricted to small magnitudes. The paper outlines the background of viscous-shear integration and develops concepts for describing integrator operation. These concepts are applied to the practically important case of the singledegree-of-freedom integrating gyro unit as an illustration of viscous-shear-integrator operation. The physical dimensions and fluid properties associated with gyro unit integrating components are taken as typical of the problems that may be solved by viscous-shear integrators. The essential problem is that of causing shearing forces in a thin fluid layer between a cylindrical rotor and a cylindrical stator to produce a torque on the rotor with its magnitude closely proportional to the relative angular velocity between the two cylinders. For practical applications, this proportionality should VOL. 45, 1959 ENGINEERING: DRAPER AND FINSTON 529 be at least as accurate as one part in one hundred thousand. Physical properties of a fluid which influence the accuracy obtainable with this configuration are discussed. It is found essential to use a Newtonian fluid in an isothermal state. A detailed mathematical analysis of the fluid motion determines limits for physical dimensions, frequency, and amplitude of rotor motion. Introduction.-Integration, from the standpoint of mechanization, is the process of generating an output that depends on the summation of small increments determined by the input. The information involved may be represented in digital terms or it may be related to continuous variations of physical quantities. In this paper, digital devices are excluded from consideration and attention is concentrated on continuous-variable instruments that fall into the class of analoguetype' integrators. Mechanical integrators of this kind may depend on geometrical changes among rigid members in the fashion commonly used in ball-and-disk integrators, or the principle of torque summation by a rigid member may be applied to generate the output as a mechanical displacement. A number of other instrument-design schemesb are available for integrators, but for present purposes they will be disregarded. The discussion will emphasize special types of torque-summation integrators in which shearing action in a thin layer of viscous liquid is used to provide the essential effect. Perfect Integrator.-Figure 1 is the basic functional diagram for any integrating Input Output device of the continuous-variable type Quantity Integrator Quantity that has perfect performance.c The esq(()()int) sential characteristic for perfect performance is that the time rate of change of the The essential performance characteristic for a perfect ance AS tnat the time rate OI cnange OI tnintegrator is output quantity be proportional to the dqior dq S(nt)[q(i,);q(oitq(v) ( input quantity. By definition, the ratio where of the time derivative of the output to the S(int)[q(n); ()l input quantity output quantity rate inputiis the input quantity-output quantity sensitivity of the integrator input is For a perfect integrator, the sensitivity is constant, so rate sensitivity of the integrator. The that important property of a perfect integrafq(out) 2 q(i, dt (2) tor is that this sensitivity is constant. and ti and t2 Illustrative Torque-Summation Integraq(out) q(out)o S(int)[q(n);4(Ut)1) q(in)dt (3) tor. Figure 2, a is a pictorial schematic diagram that shows the essential componFig. 1. Functional diagram and performance equation for ents of an illustrative torque-summation the generalized perfect integrator. integrator. The input quantity is torque supplied by an external operating component that is unaffected by any action of the integrator. This input is applied to the torque summing member, which is also subjected to drag from an integrating element that is essentially a viscous-shear damper. The integrator output is physically the angular position of the torque summing member with respect to the integrator case. This output is indicated by the position of an index fixed to the rotor with respect to a scale fixed to the stator. Figure 2, b is a functional diagram for the integrator of Figure 2, a. In practice, any change in the input torque will be absorbed in three ways: 53O ENGINEERING: DRAPER AND FINSTON PROC. N. A. S. INTEGRATING ELEMENT SCALE OUTPUT AXIS STATOR (S) \(A (ies) S>-t~i'i-ti'alwl*^lxl..>.s](OA) (ea)-,]1 is identical withA151..,] TORQUE SLIMMING H lAfvi