Dynamic support of instrument components by viscous fluids

Dynamic Support of Instrument Components by Viscous Fluids c.

S. DRAPER, M. FINSTON, W. G. DENHARD and D. GOLDENBERG reason must be carefully considered when high performance is required. This paper is concerned with improving the practical operation of instruments based on floated mechanical receiving components by using hydrodynamic forces for the support of rotating elements in ways that reduce inaccuracy levels.

Introduction

Measurement and control combine to form a segment of modern technology that has been advancing rapidly during the past 30 years. This field is concerned with the theory and practice of receiving, processing and applying information rather than with the handling of material or power. For this reason , the devices involved are usually designed to have the minimum size that is consistent with accuracy and reliability. Equipment of this kind falls into the class called instrumentation, and the activities associated with its conception, design, construction and application belong to the field of instrument engineering. The industry based on the manufacture of measurement and control instruments involves a dollar volume in the range of hundreds of millions each year and is now of major importance for technology. An essential part of the instrumentation business is concerned with the production of receivers to accept physical effects as inputs and to produce numbers as output that are measures of the input levels. Receivers are also very often used to generate indications that represent the input levels without involving numbers. Control elements that receive such indications and deliver corresponding commands to actuators for driving the attached controlled elements account for another considerable part of the instrumentation industry'S activity. The remaining effort in this field deals with recorders, computers and miscellaneous items used in connection with data processing. Accuracy requirements on instrumentation have become increasingly severe during the past 20 years. During the first three decades of this century, measuring instruments with inaccuracy levels in the I per cent range were satisfactory for almost all purposes, and static inaccuracies of 0 ·1 to 00·1 per cent were characteristic of exceptionally high-quality equipment working under favourable environmental conditions. At present, the desired inaccuracy levels of instruments are often less than I part in 100,000 under severe environmental conditions of vibration, acceleration, temperature, pressure, etc. In many situations, rapid and direct indications of the type provided by oscillographs are needed. In an increasing number of other applications, the time integral of the input quantity is the desired output. Many integrating instruments to meet this need use receiving elements that are mechanical in nature and produce outputs that have the forms of forces or torques. Inaccuracies in the output of any mechanical receiver will be introduced by spurious forces or torques. Components of these disturbing effects that are accurately predictable may be determined and effectively eliminated by a compensating mechanism or some means for applying corrections. Uncertainty components of disturbing effects are especially troublesome because they are erratic in ways that make it impossible to do more than estimate averages and probable limits . Uncertainty effects determine the minimum inaccuracies achievable in practice from any instrument and for this

Illustrative Instrument

Inaccuracy-reduction problems for mechanical receiver systems may be described in terms of an illustrative instrument that could be subject to obvious difficulties from frictional uncertainties. The example chosen for this purpose is a receiver for specific force, which is the force acting on each unit mass of matter in the part under consideration due to the instantaneously existing resultant of gravitational fields and accelerations with respect to inertial space. Instruments of this kind must often work under environments that include interfering effects much stronger than the input that should ideally control the output. A very effective method of achieving high performance under these circumstances is to apply flotation in a fluid of proper density for support and to use viscous shear in the same fluid for integration of the specific force input. The essential features of such a specific force integrating receiver are illustrated in F(!{ure I. The mechanical component that serves the function of receiving specific force as its input and delivering torque as its output is hermetically sealed float having a rightcircular cylinder with symmetrical end projections and pivots as its external shape. Internally, the mass of the float is disposed so that the centre of gravity of the float is displaced from the line of pivots, in a plane perpendicular to this line, by an arm of arbitrary length and orientation. The float is enclosed in a cylindrical case that carries bearings for the pivots at either end. A high-density fluid with Newtonian viscosity characteristics (that is, having a coefficient of viscosity independent of the rate of shear) is used to fill the clearance volume between the float and the case. By design, the float is made to displace an amount of fluid equal to its own mass, so that pressure gradients associated with the specific force that exists at any instant substantially support the float without loading the pivots. A signal generator is used to receive the angle of the float with respect to the case about the cylinder axis as its input and to produce a corresponding electrical signal as its output. An electrical torque generator adapted to receive electrical command signals is used to apply corresponding torque components to the float. Omitted from Figure 1 for the sake of simplicity but often included in high-performance receivers are two electromagnetic support units (usually referred to as 'magnetic suspension units'), one at either end of the float . These units receive electrical excitations and apply centralizing forces to the float so that the receiver operates without any physical contact

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DYNAM[C SUPPORT OF INSTRUMENT COMPONENTS BY VISCOUS FLUIDS

~~~~~TO~~~~RAT~ts OF FLOAT

[ (lI , )(OA) "

BEARIN G PIVOT

FLOAT STRUCTURE IS] [ RIGID AND SERVES AS

(bl

(pi

THE TORQUE SU!.WING

FLOAT OUTPUT AX IS

MU.I8ER ORIGIN OF FLOAT TOR QUE SUI.I'.1ItlG MEMBER COORDIN ATES

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( f sm)

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~FICFORCE INTEGRATING RECE IVER

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rC ASE OUTPUT 'XIS

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NOTES : (I) Fo r purposes o f illustra tion , the input receiver is sho wn as a ri gid unbal anced structure pivoted abo ut the lo ngitudina l ax is o f its ex tern a l fo rm of sy mmetry a nd substa ntiall y fl o ated in th e liquid that fills th e clea rance between the float a nd the case. (2) To av o id co mplicat io n in this d iagra m , th e mag netic suspensio n units sho wn in Fig ure J are o mitted .

Figure I . l/Iustrative mechanical input receiver for specific force with viscous-shear illlegration, electrical-signal gel/era/ion, and a torque generator f or accepting command inputs

between the pivots and their bearings. The magnetic suspension units also provide signals that give accurate positional information on the locations of the pivots with respect to their bearings. Operating Principles of an Illustrative Specific Force Integrating Receiver Specific force as an actuating input (that is, one of the physical quantities intended to have primary control of the instrument output) acts at the centre of mass of the float and, because of the arm designed into the system, causes a torque component to act on the rigid torque summing member formed by the float structure*. The combined actions of the pivots, the magnetic suspension units, the fluid within the clearance volume and the cylindrical section of the flo at substantially suppress motion of the float with respect to the case except for rotation about the longitudinal axis of the flo a t. The part played by hydrodynamic effects in achieving this result is the principal subject for discussion in this paper. The specificforce input acting through the pendulous element of the float imposes a torque on the float about the axis of symmetry that is substantially proportional to the component of specific force acting along the float input axis. When no command signal is applied to the torque generator, the torque due to input-axis specific force is absorbed by viscous-shear forces within the thin layer of fluid contained by the cylindrical shell of clearance about the centre portion of the float. The resisting torque due to this viscous-shear action builds up to match the applied torque when the float rotates with respect to the case at an angular velocity that under steady-state conditions is proportional to the applied torque. Because this angular velocity is * A measure of the torque-producing capability of the configuration is its pendulosity, which is the product of the float mass and the centre-of-gravity arm.

proportional to the output torque of the pendulous element, the angular displacement of the float with respect to the case from a given reference represents the time integral of the specific-force component along the input axis of the float. In practice, the specific-force-input torque is usually balanced by a torque from the torque generator in such a way that the float remains close to the orientation with respect to the case for which the signal generator output signal has its null level. This may be accomplished by using the signal generator output as the input for a feedback amplifier system that supplies the command signal for the torque generator. For this mode of operation, the viscous-shear integrator action has the effect of smoothing specific-force-input variations without introducing the uncertainties that accompany the generation of signals . Coordinate systems, concepts, definitions of terms, symbols and the mathematical formulation of performance equations are necessary for proper discussions of specific force integrating receivers. These elements are supplied by the diagrams of Figures 1 to 3 inclusive and Information Summaries 1 and 2. In Figure I , the flo at coordinates selected for use are described . The float output axis [symbol (OA)(fI!)] is chosen along the line of pivots, the flo at centre-of-gravity arm axis [symbol (CGAA)(fI!)] is taken at right angles to the output axis and passing through the centre of gravity, and the float input axis [symbol (IA)(fI!)] is placed so that it forms a right-handed orthogonal system with the other two axes. A corresponding set of axes fixed to the case has the specific force integrating receiver output axis [symbol (OA)(sfir) or simply (OA) when misunderstanding is not likely] identical with the float output axis when the pivots an~ accurately in the centres of their bearings. The specific force integrating receiver arm reference axis [symbol (ARA)(sfir)] is along the float centreof-gravity arm axis when the output axes are identical, and the signal generator output is at its null level. The specific force

273 18

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S. DRAPER, M. FINSTON, W . G. DENHARD AND D. GOLDENBERG

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Figure 2. Functional diagram for the iIIustralive specific f orce integrating receiver of Figure

integrating receiver input axis [symbol (IA)(sfir)J is along the float input axis when the other corresponding axes of the two sets are identical. Because the specific force integrating receiver axes are fixed with respect to the case, they are also referred to as simply case axes, for the sake of convenience (see Figure I). From the standpoint of instrument operation, the angle from the specific force integrating receiver arm reference axis to the float centre-of-gravity arm axis is the essential input to the signal generator. This angle is the output-axis float angle [symbol A(f1t)(OA) or A(f1t)J and directly controls the specific force integrating receiver output signal.

ponents acting on the rigid structure of the float that serves as the torque summing member. Details of the development of the performance equation for the integrating receiver of Figure 1 are given by Figure 2 and Information Summary 1. This summary starts with mathematical terms that represent the torque components acting on the torque summing member about the output axis and gives the development of an equation for the output signal as a function of the actuating inputs (specific force and torque generator command signals) and various interfering inputs. This performance equation shows the equivalence of the torque generator command signals and specific force along the input axis, and the basic integrating action of the receiver. Static performance depends primarily on static sensitivities and a characteristic time.

Specific Force Integrating Receiver Performance Equation Performance of the illustrative specific force integrating receiver is primarily determined by the balance of torque com-

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DYNAMIC SUPPORT OF INSTRUMENT COMPONENTS BY VISCOUS FLUIDS

Information Summary 1 Factors controlling the primary operating characteristics and the performance equations of an illustrative specific force integrating receiver Actuating torque

The actuating effect (that is, the effect the instrument is intended to receive as the primary input) is due to the specific force (the force per unit mass due to the resultant of gravitational effects and inertia-reaction effects) acting at the centre of mass of the float and producing a torque on the torque summing member about the output axis because the float centre-of-gravity arm, (arm)(eg)(flt), is not zero. NOTE: By definition, the product of the arm length, (arm)(eg)(flt), and the mass of the float (mass)(flt), is the float pendulosity, (Pnd)(flt); i.e. (Pnd)(flt) = (arm)(eg)(llt)(mass)(flt)

(1)

In equation form, the specific-force actuating torque, M(ae)(sf), can be represented by the relationship *

(2)

where M(tsm)(Sf)(OA) is the tor.que applied to the torque summing member about the output axis as a result of the specific force acting at the centre of mass of the float, (POS)[I-(ea)] is the acceleration of the case with respect to inertial space, g is the specific force due to the earth's gravity, and (sf) is the specific force acting at the centre of mass of the float. Inertia-reaction effects about the output axis

The specific-force actuating torque is partially balanced by the inertia reaction of the torque summing member about the output axis. This component of the output-axis balancing torque is given by the relationship M(tsm) (ir)(OA) =

-I(tsm)(Ao)(A'[I-(ea)](oA) + A'[(ea) -(tsm)](AO»

(3)

where J(tsm) (OA) is the moment of inertia of the torque summiQg member about the output axis, A'[(I-ea)](OA) is the angular acceleration of the case with respect to inertial space about the output axis, and A [(ea)-(tsm)](OA) is the angular acceleration of the torque summing member with respect to the case about the output axis. Torque generator (tg) ~

(()A)(tg)

~

~

~ (Sg)(lgl(cmd)

TOrqUe Generator Output Axis M(lg)

Torque Generator Torque Generator Output Torque (tf!,os)

Torque Generator

Output Shaft

Command Signal

Figure I.S. 1-1

(4) where S(tg)[(Sg)(tg)(Cmd); M(tg)] denotes command signal- torque sensitivity of the torque generator. NOTES:

(1) The conventions used for writing the sensitivity symbols used in this paper are as followst: (a) The main symbol S represents sensitivity. (b) The first subscript parenthesis symbol identifies the operating component whose static performance is described by the complete

symbol. (c) The square subscript bracket is the input-output bracket.

Cd) The subscript symbol inside the square bracket nearest the main symbol represents the input quantity to the operating component concerned. (e) The subscript symbol inside the square bracket separated from the input quantity by a semicolon represents the output quantity (2) Details of the torque-generator mechanization are omitted as being unimportant for the purposes of this paper. • The conventions of notation applied here for angles. position vectors, and the like are

( main symbol representing) physical quantity [

symbol for entity)] ( axis. direction, ) subsystem Of ( symbol for) _ ( with which main· reference sY!11bo! Qt;'antity component invo~ved. IS associated where appropnate

See reference I, Volume I, t See Chapter 4 in Volume I and Chapter 28 in Volume HI. reference 1.

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S. DRAPER, M. FINSTON, W. G. DENHARD AND D. GOLDENBERG

Signal generator Csg) A(5~"tin ) Signal Gene~ator

Input Ang le

(Sgt;;J)lcu!) Signal Generator

Output Signal

Figure I.S. 1-2 (5)

(Sg)(Sg)(out) = S(sg)[A; (Sg)cSg)cout)]A (sg)(in) where S(sg)[A. (Sg)CSg)(out)J = angle - signal sensitivity of the signal generator.

NOTE: The potentiometer-slider type of signal generator is shown here for simplicity of presentation; in practice, many other principles are applied in realizing the signal-generator function. Specific force integrating receiver torque-summation equation (sfir)

The behaviour of a specific force integrating receiver is determined by the balance of torques acting on the torque summing member (tsm). To simplify the equations, the torques are written for conditions under which A [(ca)-(fIt)](OA) remains so small that the float centre-of-gravity arm is effectively along the arm reference axis of the case. This situation always exists in practice when high performance is required. The torque-balance equation is as follows: - I(tsm) (OA) [A'[I-(Ca)](OA) + A'[(ca) -(tsm)](OA)]- S(id) [A.; M](OA)A'[(ca)-(tsm)](OA) - S(tg)[(Sg)Ctg)(Cmd); MCtg)]CSg)(tg)(cmd) [inertia-reaction torque] [visc?US-She~r torque from] [torque generator output torque] mtegratmg damper

+ SCpe) [(sf)CIA); MCtSID)](OA)(sf)(lA) input-axis specific force com-] ponent input torque [ ( CIA)(flt) substantially parallel) to (IA)(sfir)

+ [(E)M](tg)(OA) + [(E)Mhsg](OA) torque ] [ generator error torque

signal ] [ generator error torque

+ S(tsms)[(sf)COA); M(tsID)](OA)(sf)(OA) + S(tsmS)[(sf)CARA); Mctsm)](OA)(sf)(ARA) [ [output-axis- arm reference axis result~mt specific force torque about] output aXIS

± [(U)M]Ctg)(OA) ± [(U)M](sg)(OA) torque ] [ generator uncertainty torque

± [( U)Mlcflt)(buoyancy)(OA) ± [( U)M] (fit) (hff) (OA) ± float ] [ buoyancy uncertainty torque

float ] hydrodynamic [ fluid flow uncertainty torque

± [(U)M]Cplvot)(OA)

signal ] [ generator uncertainty torque

pivot ] [ uncertainty torque

[(U)M](msu) = 0

[

± [W)M](id)(OA) integrating] [ damper uncertainty torque (6)

s:~~~~~~n]

unit uncertainty torque

where I(tsm) (OA) is the moment of inertia of the torque summing member about the output axis, and A'[I-(ca)] (OA) is the angular acceleration of the case with respect to inertial space about the output axis. (A is the symbol for' angle'; I is the symbol for' inertial space' as the geometrical reference; (ca) is the symbol for 'case'; (OA) is the symbol for 'output axis'.) NOTE: Two dots above a symbol represent the second derivative with respect to time of the quantity represented by the symbol.

A'[(ca)-(tsm)] (OA) is the angular acceleration of the torque summing member (tsm) with respect to the case about the output axis; S(id)[A; M] (OA) is the angular velocity- output-axis torque sensitivity of the integrating damper: (id) is the subsystem-identification symbol for the' integrating damper'; S(id)[A; M](OA) ~ C(id)(OA), the viscous-shear damping coefficient of the integrating damper about the output axis. NOTE: The symbol

~

is used to indicate equivalence between the two quantities on either side of the symbol.

A'[(ca)-(tsm)](OA) is the angular velocity of the torque summing member about the output axis with respect to the case; S(tg)[(Sg)(tg)(Cmd\; MCtgl ] is the command signal - torque sensitivity of the torque generator; (Sg) (tg)(cmd) is the torque generator command signal; S(pe)[(Sf)(IA); Mctsm)](OA) is the input-axis specific force component - output-axis torque sensitivity of the pendulous element (element formed by the mass of the torque summing member located at the centre-of-gravity arm distance from the output axis); (SO(IA) is the specific force component acting along the input axis; S(tsms)[(sf)COA); Mctsm)](OA) is the output-axis specific force component-output-axis torque sensitivity of the torque summing member system (this system includes the pendulous element, the float and all attached parts, the fluid, pivots, torque generator, signal generator, and all other elements that may contribute to output-axis torque as a result of specific force acting along the output axis); (sf)(OA) is the specific-force component acting along the output axis; S(tsmS)[(sf)CARA); M(tsm)J(OA) is the arm-reference-axis specific force component-output-axis torque sensitivity of the torque summing member system; (sf)(ARA) is the specific-force component acting along the arm reference axis; [(E)Mlctg)(OA) is the error-producing torque about the output axis due to operation of the torque generator [by definition, 'error', which is represented by the symbol (E) immediately before the symbol for the quantity in which the error exists, identifies the component of the quantity that is consistent and predictable in the sense that it may be determined, so that its effect may be eliminated by compensation or correction]; [(E)M](sg)(OA) is the error-producing torque about the output axis due to operation of the signal generator; [( U)M](tg)(OA) is the uncertainty torque about the output axis due to operation of the torque generator [by definition, 'uncertainty', which is represented by the symbol (U) immediately before the symbol for the quantity in which the uncertainty exists, identifies the component of the quantity that is erratic and unpredictable except by the methods of statistics]; [( U)M] (sg)(OA) is the uncertainty torque about the output

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DYNAMIC SUPPORT OF INSTRUMENT COMPONENTS BY VISCOUS FLUIDS

axis due to operation of the signal generator; [(U)M](piYot)(OA) is the uncertainty torque about the output axis due to operation oLthe pivots; [( U)Ml(id) (OA) is the uncertainty torque about the output axis due to operation of the integrating damper; [(U)M](flt)(buoyancy)(OA) is the uncertainty torque about the output axis due to the action of buoyancy on the float; [(U)M](flt)(hff) (OA) is the uncertainty torque about the output axis due to hydrodynamic-fluid-flow effects, other than integrating-damper operation in the fluid that fills the clearance volume; [( U)M](msu)(OA) is the uncertainty torque about the output axis due to operation of the magnetic suspension units,

Specific force illlegrating recl'icer performance equation (sfir) The performance equation for the specific force integrating receiver is formed by carrying out the following operations on the terms of the torque-summation equation: (I) Rc-arrange terms, leaving only those containing A[(ca)-(tsm)](OA) and A'[(ca)-(tsm)](OA) on the LHS of the equality sign. (2) Multiply each term by the signal generator sensitivity, in order to replace the torque summing member output angle of the LHS by the signal generator output signal (Sg)(Sg)(out). which is also the output signal of the specific force integrating receiver, (Sg)(sfir). (3) Reduce sensitivities to overall system terms by combining component sensitivities to form (sfir), specific force integrating receiver, sensitivities. (4) Reduce the dynamic-performance parameter determined by the torque summing member moment of inertia and integrating damper sensitivity to a c//{{raclerislic lime. (5) For simplicity, combine the error terms and uncertainty terms into resultant error and uncertainty terms. The resulting performance is (CT)(Sfir)(Sg)(sfir) + (Sg)(sfir)

S(sfir) [(sf)(lA); (s'g) I( SO(IA) - S(sfir) [(Sg)(cmd); (Sg)](Sg)(cmd) + S(sfir) [M; (Sg) I [(E)M] (rsIt) (OA)

± S(sfir) [M; In operator form, that is, with (") ( Sg)(sfir)

~

(s'g) 1[( U )AJl (rslt) (OA)

(7)

d(dt--> p, the performance equation may be written as

. [ (Sg)(Cmd»)· ] S(sfir)[M; (s!;)] S(Sfir)[(sf)(IA); (Sg)] p[I+(CT)(sfir)P] (sf}(IA) 1-(SR)(sflr)[(Sg)(cmd); (sf)(lA)] (sf)(IA) + p[l+(CT)(sfir)P] [(E)M](rslt)(OA)

+ S(sfir)[M;

(S'g)1

[CU)M]

- p[l + (CT)(sfir)P] .

(8)

(rslt) (OA)

In the performance equations, (CT)(sfir)

l(tsm) (OA)

S(id) [A; M](OA)

= characteristic time of the specific force integrating receiver

S(pe) [(sf)(IA); M(tsm)](OA)S(sg)[A; (S!;)] S(id)[A; M](OA)

(9)

(10)

= specific force- signal rate sensitivity of the specific force integrating receiver NUTF: Under conditions of practice, the signal generator sensitivity, S(sg){A; (Sg)]. is constant, so that S(sg)[A; (Sg)] = S(Sg)[A; (Sg)] = S(Sg)[A: (S~)].

S(tg) [(Sg)(tg)(cmd); M(tg)]SCsg)[A: CSg)] S(id) [A; M](OA)

(I I)

= command signal- signal rate sensitivity of the specific force integrating receiver

S(sfir)[M; (Sg)]

S(Sg)[A; (s'g)] S(id)[A; M](OA)

(12)

torque-signal rate sensitivity of the specific force integrating receiver S(sfir)[(Sg)(cmd); (Sg)] (13)

S(sfir) [(sf) (lA); (s'g)]

=

command signal- input-axis specific force sensitivity ratio of the specific force integrating receiver

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C. S. DRAPER , M. F1NSTON, W. G. DEN HARD AND D. GOLDENBERG

Complete Torque Balance for the Specific Force Receiver Performance of the mechanical system associated with the float would be ideal for the purposes of receiving specific force if the only torque components acting about the output axis were linear outputs from the pendulous element, the integrating damper, the torque generator and the material of the float reacting to angular acceleration. This type of behaviour is represented by the corresponding terms in the torque-summation equation of Information Summary I. In practice, the primary elements may not have ideal performance and other parts may be imperfect to some extent. The result is that the torque summing member is subjected to torques that are not predictably related to the specific force input. For example, the specific force that is effective on the torque summing member is not effectively identical with the specific force component along the input axis fixed to the case of the receiver if the float

account so that they do not have an intolerably great effect on instrument accuracy. On the other hand, torque uncertainties set limits to performance that cannot be improved unless the uncertainty components are reduced. It is true that averaging procedures and filtering methods may improve results to some extent, but they almost always involve time lags that are intolerably great. Except for special cases, the only effective measures for increasing instrument performance are those that reduce uncertainty torque components. Information Summary 2 gives expressions in generalized vector terms for each of the torque sources associated with the torque summing member of Figures I and 3. The actions of the units associated with the actuating inputs and the output signal have already been discussed. It remains to consider the uncertainty torque components introduced by other parts of the instrument. These torque components are represented by

POINT OF FLOAT

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I!A) (flt } ~ / f LOAT INPUT AXIS

t o"rm l{ cb )

CENTRE.,DF, BUOYANCY ARI:.

;chfn CENTRE OF HYDRODYNA'.I IC FLOW FORCE

Additional definitions not represented o n the a bove diagram: (a rm)(b)( _) = negati ve-side-of-(OA) bea ring arm

(a rm)(msu) ( _ ) = nega tive-side-of-(OA magnetic suspension unit a rm

(arm)(b)( + ) = positive-side-of-(OA) bearing arm

(arm)(msu)( + ) = positive-side-of-(OA) magnetic suspension unit arm

N OTES : (I) A s uperscript bar, (-), over a symbo l indicates that the qua ntity represented b y the symbo l is a vecto r. (2 ) T o reduce complica tion o f the diag ra m , direct identi fication of the bearing and magnetic suspensio n a rm s a re omitted. (3) M ag netic-suspensio n fo rces a lo ng th e output axis are genera lly negli gible .

Figure 3, Geometrical quantities associated with f orce and torque components acting the torque summing member of a float ed mechanical input receiver

angle is not restricted to very small values . The inaccuracy involved depends on the way in which the direction of the specific force input vector varies with respect to the case. Under circumstances in which the effects of these directional variations are not determined, the corresponding torque components must be classed as uncertainties. Another source of torque that interferes with ideal action may be the torque generator, whose sensitivity may differ somewhat from being constant at the nominal value . When the variations in sensitivity are known from calibration tests, the effects on the input for the torque summing member are predictable and so fall into the class of errors that may be accounted for by the application of proper corrections. Similarly, the signal generator may impose errors in torque and at the same time may introduce torque uncertainties. In general, error components in torque are undesirable, but they may be taken into

01/

terms in the torque-summation equation of Information Summary 2 and are described in that summary by vector-crossproduct terms based on the definitions of Figure 3. For example, the torque on the float due to the bearing located away from the origin of float coordinates on the positive branch of the output axis is given by the cross product of the arm vector from the origin of coordinates in the float to the bearing and the vector force applied to the float by the bearing. This torque will be about an axis at right angles to the float output axis. Cross products similar to that describing the torque from one bearing exist for the other bearing and for the two magnetic suspension units. When the centre of buoyancy is not exactly coincident with the origin of coordinate and is not in the plane containing the centre-of-gravity arm axis and the float input axis, buoyant forces will apply a torque tending to rotate the float output axis

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DYNAMIC SUPPORT OF INSTRUMENT COMPONENTS BY VISCOUS FLUIDS

Information Summary 2 Force components and torque components acting on the torque summ ing member of a floated mechanical input receiver (a) Vector relationships between force applied at a point of a rigid body and the equivalent force at a designated reference point of the body combined with the torque (moment) acting about the reference poillf ('p i Refe renc e P oint for Force and Torque

Appl;cotion

F (rp)

~

( /II' m)

-F

o M om e nt

about {rpj

([pi Force Poin t

F orce Poi nt and Re fere nce P oin t A te Assumed fa be

Fi xed in

0

Rigid Body

Figure I.S. 2-1 (I)

M (rp) = (arm) x F(fP)

(2)

= a vector through (rp) at right angles to the plane determined by (arm) and F(fP)

(b) R esultant force actillg

Oil

the float

Take (pfgs), the point of flo at geometric symmetry (see Figure 3), as the common reference point for summing force and torque components acting on the float , in order to determine the resultant force a nd resultant torque. Then , Resultant fOrCe] [POsitive-side-Of-] (OA) bearing [ acting on float _ at (pfgs) force F(PfgS)(res) F(b)( + )

+

F orce due to specific force] (resultant of gravity and acceleration) acting at the centre of gravity

f

[ Negative-side-Of-] + [Negative-side-Of-] (OA) bearing + [POsitive-side-Of_] (OA) magnetic + (OA) magnetic force

suspension force

suspension force

F(b)(-)

F(ms)( + )

F(ms)(-)

f +

F(mass) (flt)(sl)

J

Buoyancy force (due to pressure gradients set up in fluid around float by action of specific force) acting at the centre of buoyancy F(fluid)(flt)(sf) or F(buOY)

[HYdrody namiC fluid flow force] (resulting from flow in fluid surrounding the + float as re-arra ngements in the clearance volume occur)

(3)

F(hff)

Since point (pfgs) is chosen as the origin of the flo at coordinate system [axes (OA)(flt)' (CGAA)(flt ) and (lA)(flt)], F(Pfgs)(rcs) is identical with F(flt)o, the resultant force acti ng on the float at the origin of float coordinates. (c) Resultallt torque actillg on the float

Take (pfgs), that is, O([lt) , as the reference point about which the resu ltant torque on the float, M(flt) (pfgs) (res) (or M(flt) 0), acts. This resultant torque is made up of components due to the force components of equation 3, the inertia-reaction torque corresponding to the angular acceleration of the float with respect to inertial space, the viscous-shear torque from the action of the integrating damper, and the torque generator output torque, i.e.

Resu.'tant torque .on float] [ Inertia ] _ reactIOn [ actmg about orlgm of float coordmates torque M(flt)(Pfgs)(res) or M (flt )o M (flt )(ir)

+

f

~o:~~~~i~i~~~~~~~~~ torque

+

[1~!~\~;-~~~~~Of~~~~~ ~ + [ ~~~~t~~ef~~~~Ot~~~U~) ] _ _ M(b)( + )

M(b)( - )

[ (--) -] = arm (b)( +) x F (b)( +)

[ ( - -)

=

i"]

arm (b)( - ) x L(b)( - )

1[ ~~~:~;~~s~~~-~:~~?o~) ] fd~e~:~{~rt~en~o~(~:aSW~~~ed 1 +

torque

+

acceleration acting on mass particles of float)

M(ms)( + )

M(ms)(-)

M(mass)(flt)(Sf)

[ = (a rm) (ms)( +) x F(ms)( +)]

[= (arm)(ms)( _ ) x F(ms)( - )]

[ = (mass) (flt)(a rm) (Cg) (fit) x (sO]

279

1380

+

c.

S. DRAPER, M. FINSTON , W. G . DENHARD AND D . GOLDENBERG

Buoyancy torque (torque due to buoyancy forces produced by specific force acting on fluid)

+

+

Viscous-shear torque from integrating damper (torque due to shearing action in fluid within the clearance volume due to relative motion between float and case without re-arrangement in shape of volume)

Mmuid)(flt)(sf) or M(bUOY) [= (mass)(fluid) (dlsplaccd)(arm)(cb) (fit)

M (vs)

x (sf)]

Torque generator output torque

+

r M(tg)

M]A·[ (ca)-(tsm)](OA)]

Hydrodynamic fluid flow torque (torque due to hydrodynamic fluid flow force resulting from flow in fluid re-arrangement of the clearance volume as float moves within case) M(hff) [= (arm)(chff) x F(hff) 1

+

_ [= S(tg) I (Sg)(tgj(cmd); M](Sg)(tg)(cmd)]

away from the case output axis. Individually, these torque components tend to tilt the cylindrical section of the float with respect to its mating section of the case. In addition to these effects, any specific force parallel to the float output axis will act on the centre of mass of the float at the end of its arm to produce a torque tending to rotate the float output axis about an axis at right angles to itself. When bearing forces are zero and the resultant of other torque components tending to rotate the float output axis is not zero, the float is forced to tilt with respect to the case in such a way that fluid is compelled to flow from one part of the clearance volume to another part of this volume. The same kind of action occurs when an imperfectly floated unit is subjected to specific force components acting in the plane determined by the case input axis and the arm reference axis. The residual force causes the float to be displaced with the float output axis parallel to the case output axis. Changes of this kind in float position with respect to the case, combined with any other shifts of the float that cause rearrangements of the clearance volume, must be accompanied by forced flow of the fluid surrounding the float from the places it is being' squeezed out' to the places where the clearance volume is being opened up. Hydrodynamic effects accompanying this fluid flow form a force-and-pressure pattern that acts to resist changes in the float position within the case. The actions involved are quite complex, but at any instant it is possible to represent the integrated effects by a resultant force applied at a centre of hydrodynamic flow force . The hydrodynamic fluid flow force and hydrodynamic fluid flow torque vectors listed in Information Summary 2 are due to this force, which in effect provides the supporting reaction against the resultants of all the other forces and torques that tend to move the float within the case. By proper design of the system and selection of the fluid , hydrodynamic-flow effects may be made to produce forces that provide an effective pivot support for the float within the case. These forces strongly resist bodily displacements of the float with respect to the case, while allowing rotation restrained only by viscous-shear damping torque. It is true that shear in the cylindrical damping clearance does not provide a great amount of hydrodynamic-flow support for movement of the float with respect to the case along the output axis, but support against movement of this kind may be provided by close-clearance plates at the ends of the float and effectively zero-radius axial contacts of the pivots with' end stones' at the outer ends of the bearings. The purpose of this paper is to discuss the theoretical and practical possibilities and limitations of hydrodynamic flow

[= S (id);[A

support for the float against dynamic disturbances due to move ments of the case and changes in specific force. Sources of Output-axis Uncertainty Torque Uncertainty torque components about the float output axis are the factors that limit the high-quality performance of instruments with the general features of the specific force integrating receiver illustrated in Figures 1- 3. Components of this kind may be present because of gas or dirt in the fluid around the float, or because of imperfections in the signal generator or torque generator, but primarily they result from friction generated by motion between surfaces of solid parts that are in intimate contact. Difficulties associated with the fluid must be corrected by care in preparing the materials used and by the use of proper techniques in filling the clearance volume. Uncertainties introduced by signal generators and torque generators can be minimized by design and construction , but also very important is the fact that with units of several types the torque applied to the float varies as the float output axis moves laterally with respect to the case output axis. This source of uncertainty torque may be entirely eliminated if the float is continuously maintained in a position of symmetry with respect to the case, with the float output axis identical with the case output axis. This condition also ensures the absence of any uncertainty torque component due to friction between solid parts, because layers of uniform fluid thickness will ensure that the only friction effects acting are linear inasmuch as they are due solely to the viscous shear that is essential for proper operation of the instrument. It follows from these facts that if the float can be accurately placed in its position of symmetry with respect to the case and continuously maintained in this position , uncertainty torques will be eliminated for all practical purposes. The buoyancy that results from careful flotation will reduce disturbing forces on the float due to gravity, acceleration, vibration and shock to magnitudes that are never more than small fractions of the weight of the float. This small effective mass will be maintained under all conditions because the action of specific force on the fluid is identical with the action of specific force on the material of the float. Under the long-time, constant-direction action of gravity or the short-time influence of high specific force, the residual lack of perfect buoyancy for the float will tend to cause linear shifting and rotation of the float within the case. This kind of motion will be accompanied by uncertainty torques as the pivots are moved away from symmetry with respect to the bearings, and signal and torque generator rotor~ are moved

280

1381

DYNAMIC SUPPORT OF INSTRUMENT COMPONENTS BY VISCOUS FLUIDS

with respect to their stators. Magnetic suspension units placed at either end of the float as shown in Figure 3 act to force the float into its position of symmetry by imposing forces in planes normal to the case output axis that are proportional to the displacement of the pivots from their centred positions. To be practically successful, this centring action must occur without any accompanying torque about the float output axis. Details of the design of such units are beyond the scope of this paper. It is sufficient to remark here that high-performance magnetic suspension units have been made and successfully tested. Buoyancy acts to reduce greatly the effective mass of the float insofar as response to static and dynamic linear specificforce variations is concerned. The magnetic suspension units act to place the float in its position of symmetry with respect to the case and to supply positive information that this condition exists. When no rotational movements of the case occur under either static conditions or linear-acceleration conditions of moderate severity, the continued symmetrical position of the float means that uncertainty torques about the float output axis are substantially eliminated. It remains to examine the circumstances that exist when the case is subjected to angular oscillations or very severe changes in specific force. Any shifting of the float within the case that might exist because of these input changes would necessarily be accompanied by a flow of the fluid within the clearance volume. It is the purpose of the following sections of this paper to show that the hydrodynamic forces associated with flow of this kind tend either to drive the float back towards the position of equilibrium or to make the rate of float motion so small that no significant deviations from symmetry are built up during any of the required periods of operation. Hydrodynamic Support Forces The role of the fluid in the clearance volume as an element in the process of integration and in providing buoyancy has been mentioned earlier and is discussed in detail in reference 2. In the present section, the following question is examined. When the float and case are in relative motion, do the forces and moments arising from the shear and pressure patterns associated with the motion of the viscous fluid introduce uncertainty torques on the float? Ideally, the float and case are exactly concentric, with equal end clearances. Under this condition, the resultant hydrodynamic force on the float vanishes and the only component of torque arises from the uniform shear associated with the rotation of the float about the common float and case output axes. This torque is that which is essential to the process of integration, and in this ideal case no uncertainty torques are produced. In practice, it is to be expected that the float output axis and the case output axis will not be exactly coincident and that the end clearances will not be identical. The typical situation is illustrated in Figures 4-6. In addition to the torque that provides integrations, the variable hydrodynamic shear and pressure patterns arising from fluid motion in the non-symmetrical clearance volume will produce a resultant force and a resultant moment on the float. The question is now whether these will increase or decrease the non-symmetry; that is, what relative motion, if any, develops between the float output axis and the case output axis*? To determine the answer to this question, it is necessary to calculate the actual fluid motion and the resulting shear and pressure distributions. In the present application, the fluid is

* With the problem now stated, it is convenient to change to the notation used in Figures 4 to 10 inclusive and Informaticn Summary 3.

essentially incompressible, so the Navier-Stokes equations of motion are involved. As is well known, these are generally not amenable to analysis; fortunately, however, the fluid properties and geometrical quantities pertinent to the present problem permit substantial simplifications. (z IS COINCIDENT 'filTH

CYLINDER FORMED BY OUTER SURFACE OF FLO",T

"~~:G

FL,UlO

'----+H-.. CYWIDER FORMED BY INNER SURFACE OF CASE

SECTION M

SECTION AA SECTION B8

''',:~ SECTION 8B

SECTION B8

NOTE: The rotational and linear dispiacements shown here are exaggerated for the purposes of illustration.

Figure 4. Changes ill the shape of the fluid regio/l as the float output axis moves with respect to the case ANGLE FROll y AXIS TO LINE OF CENTRES Ar),·(Ic)i

/

LINE OF CfNTRES

(le )

rLOAT RADIUS

R,/It; USE RADIUS

~ru)

INNER SURFACE ___ -

OF CASE

C(co)

centre of case cross_section

Ci:lI~l •

centre of float cross-section

Figure 5. View, normal to case Olltput axis, showing eccellfric crosssection of cylinders formed by outer surface of float and inner surface of case

281

1382

C. S. DRAPER, M. FINSTON, W. G. DENHARD AND D. GOLDENBERG

force and moment about the case are computed using equations 6 and 7 of Information Summary 3. The contributions of the cylinder ends must also be considered. Since the end-clearance-to-radius ratio is very small, most of the flow between the end-plates is not influenced by conditions at the outer edge, that is, in the space between the float radius and case radius. Thus, the recent work of Sir Geoffrey Taylor 4 on flow between slightly canted discs can be used. There, the pressure distribution between the discs when one rotates relative to the other is analysed. Use of the quasisteady condition again allows the study of oscillatory motion here. The pertinent information is presented in the diagrams of Figures 4 and 6 and equations 8 to 14 inclusive of Information Summary 3, from which resultant hydrodynamic forces and torques may be obtained. These hydrodynamic forces and torques are found for fixed relative positions of the float output axis and case output axis. Actually, one axis will be caused to move relative to the other and new hydrodynamic forces will develop. These have not yet yielded to analysis, but some experimental results are available as described in the following section.

First, it has been shown 2 that for typical viscosities and ratios of radial clearance gap to radius the time constant for the fluid motion is about one microsecond. Since oscillations considered as being in the practical range have periods that are four or five orders of magnitude greater than this time interval, the conditions are essentially quasi-steady. This eliminates time-derivative terms. Secondly, the physical constraints at the pivots limit the possible eccentricity of the float within the case to less than 0'1, so that the non-linear terms are negligible. The only terms left are those representing the pressure gradients and viscous friction. r, OARE POLAR COORDINATES

..

,'

.. ...

,'

.

\

\

A

fLOAT OUTPUT AXIS, (O.AJ(flt)

rOA1(co)

..

I~~

PLANE PERPENDICULAR TO THE

INNER SURFACE OF CASE

p ,..,

CASF OUTPuT AXIS

PC) ...,

local

pressure

pressure

01

Experiments on Radial Damping

rim of end clearance

Preliminary experiments have been made on a typical instrument to determine the rate at which the float will move with respect to the case under a known applied force. Although the results are being extended and will be reported later, one piece of information is described briefly, in order to indicate the magnitudes involved. In one of the experiments concerned, the float and case axes were carefulIy aligned and the fluid temperature adjusted so that a known imperfection in buoyancy was introduced. Then the translation of the float under the applied force was measured as a function of time. Treating the problem as a lumpedparameter system gives

ANGLE THROUGH WHICH FLOAT IS ROTATED ABOUT THE x AXIS OF THE CASE (SEE FIG.41 ANCLE BETWEEN END OF FLOAT AND END Of CASE ALONG 0- 0 DIRECTION

Non:: The rota tional displacement a shown here is exaggerated for purposes of illustration.

The thickness h of the thin fluid layer between the end of the float and the end of the case can be expressed as follows in terms of the polar coordinates rand 0, the thickness ho at the centre of the end clearance, and the angle Cf.: h = ho + Cf.r cos 0

h = ho Figure 6. Views

for r = 0

0/ the

(mass)x + CdX + kx

end clearance

In the general case, the fluid can move circumferentially, radially and axially in the region between the float and case cylindrical surfaces. Furthermore, if, in addition to oscillations of the float about its output axis, this axis moves with respect to the case, the fluid region changes shape (see Figure 4). Even with the cited simplifications, these conditions result in an intractable boundary-value problem. This implies the need for further approximation. It can be expected that motions of the float output axis with respect to the case will be small. This is clear from experiments on the sinking motion discussed in the next section. Therefore, it is reasonable to consider such motion of the float output axis to be negligible in calculating shear and pressure forces. With this assumption, only the circumferential velocity is forced by the float oscillation about its output axis, and the other components are of a secondary flow nature and may be neglected. Then the flow at each normal cross-section to the case output axis may be treated as though the section were part of an infinitely long pair of cylinders with parallel but displaced axes (see Figure 5). Wannier 3 has presented a complete solution to this problem. For any assumed orientation of the float with respect to the case, the result is that the force varies along the float output axis in a known manner and the resultant

=

F eapp .)

where x is the displacement of the float output axis with respect to the case, the mass, symbol (mass), involved is that of the float, Cd is the viscous damping coefficient, k is the elastic coefficient of the forces tending to return the float to its initial position, and F(app.) is the applied force. Experimental results showed that

and that Cd was very nearly constant and had a magnitude of about 10 10 dynes/(cm/sec). From equation 6 of Information Summary 3, for any given axes positions, the quasi-steady resultant force normal to the float output axis can be computed. The end-plates do not contribute to this force. Similarly, using equations 7, 13 and 14, the torque about the centre of the float can be found. These can be considered as contributions to the applied force and torque. A step-by-step procedure over small intervals of time can be used to find the subsequent motion of the float output axis. For present purposes, this fairly tedious process is not necessary. Calculation of the hydrodynamic force for anyone case will provide the order of magnitude to be expected. This was done for the case when the axes were displaced but not rotated (that

282

1383

DYNAMIC SUPPORT OF INSTRUMENT COMPONENTS BY VISCOUS FLUIDS

Information Summary 3 Hydrodynamic forces and torques The assumptions on which this analysis of hydrodynamic forces and torques is based are as follows: (a)

Motion is quasi-steady

(b) Flow between cylindrical surfaces is circumferential (c)

Flow in end-clearance spaces is like that between discs

(d) Fluid is incompressible.

Wannier 3 has discussed the problem as governed by assumptions a, band d. Taylor and Saff'man 4 have discussed the problem as governed by assumptions a, c and d. These two problems are considered here as Case 1 and Case 2, respectively. Case 1. Flow between the cylindrical sur/aces For this case, the result is that the hydrodynamic force at any cross-section (see Figure 5) is normal to the line of centres and is given by the relationship3 7T,,!deR(ca) V(flt) F(hf) = -----""'-"'--:=.:;"-:-"'7:'-;---;---(I) , , (d] +s)(dz-s) [R-(ca) + R-(flt)] log (d] _ s)(dz + s) 4sde

where,,! is the fluid viscosity; de = IC(ca) - C(flt) I is the centre displacement = distance between centres of the two cross-sections; C(ca) is the centre of the case cross-section; C(flt) is the centre of the float cross-section; R(ca) is the case radius; R(flt) is the float radius; V(flt) is the peripheral velocity of the float with respect to case, and dJ, dz, s are constants depending on the pertinent dimensions (see reference 3). For the dimensions of interest here (see reference 2), it is easy to show that equation 1 reduces to

~ l7TrI V(flt) ~ I~) z - 4' (Cl)(rad) \(Cl)(rad)

F(hf)

(2)

where (Cl)(rad) = R(ca) - R(f1t) When the axes of the two cylinders are not parallel and the centres are displaced, the line of the centres and the distance between centres of the cross-sections will vary along the length of the cylinder. The angle between the line of the centres and the y axis and the distance between centres are given by the following relationships (see Figure 5): dx (3) tan A[Y-(lc») = d y+z SIn . a z de = (dx)2+(dy+zsin a)2 (4) where A[y-(Ic») is the angle from the y axis to line of centres; dx, dy are the displacement components of the float with respect to the case; z is the distance of the cross-section from the origin of coordinates, and a is the angle of rotation about the x axis. The force per unit length in vector form is -

3

.4

F(hf) =

7T"!

( R (ca) ) 2 1 . V(flt) (Cl)(rad) (Cl)(rad) [(dy+ z SIn a)lx- dx ly]

(5)

where lx, I y are unit vectors in the x and y directions, respectively. The total hydrodynamic force and the total hydrodynamic torque acting on the inner cylinder are the integrals along the length of the cylinder, L of the forces and torques, respectively; i.e.

2: Forces =

3 ( R(ca»)2 JL2 F(hf)dz = -4 7T ,,!V(flt) (-Cl) (-Cl) (dy (rad) (rad) L

-L!Z

2: Torques

=

JL2

- 1 ( R(ca) zlzxF(hf) dz = -2 7T ,,!V(flt) -(Cl) (rad)

lx-dx ly)

)2 L3 sin a -(Cl) ly

(6) (7)

(rad) The displacements dx and dy give rise only to a resultant hydrodynamic force; the rotation a, about lx, produces a resultant hydrodynamic torque about ly. -L.'Z

Case 2. Flow in the end clearances For this case, the shear force produces negligible contributions compared with the pressure distribution in developing forces and moments. Summarized here, with some slight notational changes, are the results of the Taylor-Saffman study4. Expanding the normalized pressure, p', in terms of a (see Figures 4 and 6) gives P' = PI" +pz"z +

...

(8)

where P' = (p - Po)/po; p is the local pressure, and Po is the pressure at the rim of the end clearance. PI = A(r') cos

where rand

e are polar coordinates (see Figure 6) and r' A( r ') -_

e+ B(r') sin e

(9)

= (rl R(ca».

ber] VA ber] r'VA+ bei] VA bei] r'VA MjZ(V A)

-r

ber] ,/ A bei] rVA- bei] VA ber1 r'VA MjZ(VA)

M](x) = IJI(xV=-i)l;

283

1384

i = (-l)t

It turns out that

,

C . S. DRAPER, M. FINSTON, W. G . DENHARD AND D. GOLDENBERG

The parameter A is given by the expression A = 617 R2 (f1t)A'(flt)(OA) Poho Z

(10)

where A(fIt)(OA) is the angul ar velocity of the float about the output axis, ho is defined in Figure 6, and the other quantities are defined earlier in this summary. The quantity A is a dimensionless parameter indicating the relative magnitudes of the pressure gradient induced by viscous shear and the pressure gradient produced by fluid inertia. In the present case (see reference 2), A ~ 1. From equation 9,

L21T PI dll

= 0, so the first-order term does not contribute to the axial force .

Pz

= - 3

f

r' cos O( cPI/cr') dr' -

In reference 4, it is shown that

~ fil 2

(11)

where (- ) = -1

i21T ( ) dO

27T 0

NORMALIZED PLO TS BASEO ON £OS. ( 13) MC. (1 4) GF IN FO RMATION SUU'MRY 3; SEE SW,IMARY FOR DEF INITIONS OF SYMBOLS USED HER E.

C?

~ u

I

-u '.

-

-D ' :

, ~

°OL---L iO ---~L--~30--~ 40-~~~~50~~)~ O -~8~O--~9~O--~lOO DIMUlSIONLES~

DlMENSIONLESS PARAMETER .\

PI\RAMETER .\

Figure 8. N ormalized hydrodynamic torques acting 011 float produced by pressure distributioll 011 float end-plates

Figure 7. Normalized axial hydrodynamic f orce acting on float end-plates

When the float and case axes are co-planar, it follows that (neglecting higher-order terms) the axial hydrodynamic force on each end can be calculated to be (12)

27T(q2/4)C

F(axial)(hf) =

where C

= q~ .0Cr'jl(r', q) dr'

P;

= 6

J.I" r' dA(r') dr' dr'

[A2(r') + BZ(r')]

and q =

R(ca)

sin a/ho

Figure 7 shows C, the normalized axial hydrodynamic force, plotted as a function of the dimensionless parameter A.

The total axial hydrodynamic force, which acts to equalize the end clearances, is given by the relationship F(axial)(hf)(to tal) =

q1 27T [ T

2

q22 ] CI-T Cz

where ql and Cl pertain to the end with the larger clearance, q2 and C2 pertain to the end with the smaller clearance. The hydrodynamic torques are obtained from PI. Resolved about axes through the point of float geometric symmetry, these torques are MAR = 7Tq

Iol r'2B(r' ) dr';

MNAR = 7Tq

Cr'2A(r') dr'

(13), (14)

.0

where MAR is the moment about the axis of rotation of the angle a, and MNAR is the moment about the norma l to the axis of rotation of the angle a. The results are given graphically in Figure 8. MAR always tends to reduce a, while MNAR tends to skew the axes slightly.

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DYNAMIC SUPPORT OF INSTRUMENT COMPONENTS BY VISCOUS FLUIDS

RADIAL CLEARANCE - RADIUS RAnO (d)(,••I)"R(,.)

Figure 9. Normalized radial hydrodynamic force on /foat as a function of the radial clearance to radius ratio NOTES ,. A GYRO UNIT V,'AS USED FOR THE rURPGSES OF THIS TEST, RATHER THAN A SI~~PLE V!SC:::',jS INTEGRATCR IN ORDER TO TAKE ADVANTAGE OF THE HiGH TCRQ:JfS

OBTAINAflLE ABOUT THE OUTPUT AXiS BY THE ACTION OF THE GYROSCOPIC ELEMENT IN RESPCNSE TO SMA~:'"

OSCILLATIONS ABOUT THE INP'-!T AXIS

O-jC

DURING THIS TEST, THE MAGNETIC

S'JS~ENSION

FORCES

WERE NEGLIGIBLE. THE RADIAL FLOAT.TC·CASE CLEAPA~CE. 'CI\·'od'· ... AS 0-015 CENTII.'.ETER

t

(VERTICAL;

U FCR~~C:O 'JSCI:"'_~ TI':~ AS:::UT I'JDUT AXIS 0,3 [A)..',PLI .. rUDE AT coon (PS f,"PRCXI'.'ATELY THE SAME CSCIl...LATlON c· THE f'LCA, AB:);.)T ThE OUTPUT AXIS

-jP~T,\BL

:~P,;-;-

......... I

_........ _.-., I

I

- · ..... ! ..... ·-·I~.

TI~E

JSCILl..',TION

ACS

,-

/,IA~ESGN~ CO/J~L[T[

I~

36 HCL,RS

'-·-'1 ......

C'~[

,e

PEVCLLTICN A3J:..,'

F.A"TH'S RAT[

I

IN HOURS

---

O~

J~CILLtlTl'J!i

OFF

Figure 10. Float centre displacement/radial clearance ratio along the horizontal axis as a function of time for an experimental gyro unit subjected to an intermittent forced oscillation input about its i11put axis

is, a 0), with the results shown in Figure 9. The shaded area covers values found in typical applications. For an instrument such as that discussed in reference 2 *, the hydrodynamic force, F(hf), is found to be approximately 104 dynes when dc/(Cl)(rad) = 0·1. In practice, if the float output axis moves with respect

* For the viscous-shear integrator discussed in reference 2, the float length was about 10 cm, the float radius was about 3 cm, the radial clearance between float and case was less than 25 x 10- 3 cm, the viscosity was 2,500 cP, and the frequency of float oscillation with respect to the case lay in the region between 0·1 and 100 c/sec.

to the case, F(hf) will, of course, change. For a conservative estimate, this is neglected and F(hf) is treated as constant. Then, since only hydrodynamic uncertainty forces are of concern here, F(app.) = F(hf) and CdX ~

or

x~

104 dynes 10- 6 cm/sec (= 3·6 X 10- 3 cm/h)

If any suspension system is present (that is, k #- 0), this displacement rate will be further reduced.

285

1386

C. S. DRAPER, M. FINSTON, W. G. DENHARD AND D. GOLDENBERG

This surprisingly small value of displacement rate has been confirmed qualitatively by preliminary experimental results presented in Figure 10. Sinking tests were conducted both with and without float oscillations and the results are essentially indistinguishable within the limits of experimental accuracy. At arbitrary angular displacements, the end-plates provide inputs. These have not yet been evaluated, but from equation 13 oflnformation Summary 3 and the pressure plots in reference 4 the results are that the force tends to decrease end-clearance differences and the torques tend to reduce ex and the direction of skewness. Discussion of Results As noted in reference 2, in order to utilize viscous shear as a means of integration, the c1earance-to-radius ratio should be as small as possible and the viscosity as high as possible. From Figure 9, it is obvious that these conditions produce high hydrodynamic forces also. However, they also increase the resistance to float output-axis motion with respect to the case and this appears to be the dominant effect. It should also be noted that high values of TJ also increase the favourable effects of the end-plates (see Figures 7 and 8). Conclusions From the preliminary calculations and experiments described in this paper, certain general conclusions can be drawn concerning the effect of hydrodynamic motion of the float output axis.

In practice, the float oscillates about its axis so that the direction of the resultant hydrodynamic force oscillates in direction and magnitude. This, together with the extremely low rate of motion of the axis itself, implies that, in operation, the drift of the float with respect to the case will be very small. In particular, if the float and case are ideally positioned initially, the hydrodynamic action of the viscous fluid is to resist any change from this condition. This also means, of course, that some means such as electromagnetic suspension is essential to centre the float initially. Finally, the design features needed to provide high-accuracy integration are compatible with the requirement that uncertainty torques be kept negligible. References 1 DRAPER, C. S., McKAY, W. and LEES, S. Instrument Engineering. McGraw-Hill : New York; Vo!. 1. Methods for Describing the Situations of Instrument Engineering, 1952; Vol. 11. Methods for Associating Mathematical Solutions with Common Forms, 1953 ; Vo!. Ill. Applications of the Instrument Engineering Method: Part 1, Measured Systems, 1955; Part 2, Control Systems (in preparation) 2 DRAPER, C. S. and F1NSTON, M. High-accuracy mechanical integration by shear in viscous liquids. Proc. Nat. Acad. Sci. Wash. 45, No. 4 (1959) 528 3 WANN1ER, G . H. A contribution to the hydrodynamics of lubrication. Quart. J. appl. Maths, 8, No. I (1950) I 4 TAYLOR, Sir G. and SAFFMAN, P. G. Effects of compressibility at low Reynolds numbers. J . aero. Sci. 24 (1957) 553

Summary High-performance measurement and control instruments such as integrators, gyroscopic units and accelerometers for modern applications must operate with high accuracy under severe environmental conditions of shock, vibration and acceleration. For many purposes, support of the essential moving parts by fluid pressure gradients acting on fully submerged flotation elements is very satisfactory. By use of properly selected small clearances and Newtonian fluids, erratic friction effects may be eliminated almost completely, leaving only pure viscous-shear forces that not only do not hinder but actually are

essential for high-quality performance. Flotation for any given assembly can never be made perfect, but substantial assistance may be derived from pressure patterns accompanying viscous flow under dynamic conditions. This paper deals with the viscous-flow support of a cylindrical, moving element submerged in a viscous fluid, confined in a cylindrical shell, and subjected to various types of oscillatory and non-oscillatory acceleration inputs. The general theory of viscous-flow support is developed and some preliminary experimental results are reported.

Sommaire Les instruments de mesure et de calcul analogique it hautes performances, tels qu'integrateurs, elements gyroscopiques et accelerometres, pour les applications modernes, doivent fonctionner avec une grande precision, dans des conditions ambiantes severes de chocs, vibrations et acceleartions. Dans de nombreux cas la suspension des pieces mobiles essentielles au moyen de gradients de pression de fluide , agissant sur des elements de flottaison submerges, est tres satisfaisante. En utilisant de faibles jeux, convenablement determines, et des flu ides Newtoniens, les effets de frottement erratiques peuvent etre elimines presque compIetement, laissant seulement de pures forces visqueuses laminaires qui, non seulement ne genent pas, mais

sont essentielles pour une performance de haute qualite. La flottaison pour un assemblage quelconque ne peut jamais etre parfaite, mais on peut tirer une aide substantielle de la configuration des pressions qui accompagnent un ecoulement visqueux dans des conditions dynamiques. Le rapport a trait it la suspension par ecoulement visqueux d'un element mobile cylindrique immerge dans un fluide visqueux, enferme dans une boite cylindrique, et soumi it divers types d'accelerations d'entree, oscillatoires et non oscillatoires. On developpe la theorie generale de la suspension it ecoulement fluide et on indique plusieurs resultats experimentaux pn§liminaires.

Zusammenfassung Me13- und Regelgerate fUr erhohte Anforderungen in modernen Anwendungsbereichen, wie z.B. Integratoren, Kreiselgerate und Beschleunigungsmel3gerate, miissen mit hoher Genauigkeit unter erschwerten Bedingungen, wie Stol3belastungen, Schwingungen und Beschleunigungen arbeiten . Bei vielen Anwendungen bringt eine Lagerung der betriebswichtigen bewegten Teile iiber das Druckgefiille einer Fliissigkeit, die auf voll eingetauchte Schwimmer arbeiten, sehr befriedigende Ergebnisse. Durch geeignete Wahl enger Passungen und Newton'scher Fliissigkeiten konnen die Reibungsfehler fast vollig ausgeschaItet werden , so dal3 lediglich die Krafte der Viskositat iibrig bleiben, die keine Beeintrachtigung darstellen und fUr erhohte

Anforderungen sogar notwendig sind. Die schwimmende Aufhangung, kann fUr eine gegebene Anordnung niemals vollkommen sein , jedoch lal3t sich aus der Druckverteilung der viskosen Stromung unter dynamischen Bedingungen eine wesentliche Verbesserung ableiten. Der Beitrag behandelt die Lagerung eines zylindrischen bewegten Teiles in viskoser Stromung. Das Teil ist dabei in eine viskose Fliissigkeit eingetaucht, die sich in einer zylindrischen Buchse befindet, und wird schwingenden und einseitigen Beschleunigungen unterworfen . Fiir die Lagerung in viskoser Stromung wird eine allgemeine Theorie entwickelt und iiber erste Versuche dazu berichtet.

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DYNAMIC SUPPORT OF INSTRUMENT COMPONENTS BY VISCOUS FLUIDS DISCUSSION MASLOV (U.S.S.R.) Integrating gyroscopes, based on the fluid suspension principle described in the report, have very small drift. Does the amount of drift remain constant with time or must the instrument be regulated each time it is used, to keep the drift small ? C. S. DRAPER, in reply. This report describes hydrodynamic suspensions. I can say that gyroscopic instruments differ; some are stable, others are not. Our article in the journal Aviation Week, 6th June 1960 describes gyroscopes with a change of 1 x 10- 3 deg/h. Drift of a gyroscope is a function not only of friction in the supports but also of other factors. This subject is discussed in the following literature: 1 The Floating Integrating Gyro and Its Application to Geometrical Stabilization Problems on Moving Bases. by C. S. Draper, W. Wrigley, and L. R. Grohe. Sherman M. Fairchild Fund Paper No. FF-13. Paper presented at the Annual Meeting of The Institute of Aeronautical Sciences, Jan 27, 1955 2 Instrumentation and Inertial Guidance by C. S. Draper, Proceedings of the Frontiers of Science and Engineering Symposium, 1960. The Institute of Aeronautical Sciences S. M. NIKITIN (US.S.R.) Is it necessary to ensure a certain ratio between the moments of inertia of the float and of the displaced liquid? C. S. DRAPER, in reply. No. A. Yu. ISHLINSKI (U.S.S.R.) Is not the Bingham structure of the liquid a hindrance? C. S. DRAPER, in reply. The fluid was carefully selected to have

Newtonian properties. The fluid chosen is such that its molecules are of the same size. SLOMYANSKI (US.5.R.) What is the size of the gaps and what is the viscosity of the liquid? C. S. DRAPER, ill reply. The article describes an instrument with very small gaps. If a thick film of fluid is formed several actions are possible when the float rotates in the case. With a thickness of 50 /1 the thin film is subject only to shearing stress and all the other deformations are so small that they may be neglected. Moreover, the liquid is very viscous. This problem is discussed in a paper: 'High-Accuracy Mechanical Integration by Shear in Viscous Liquids, by C. S. Draper and Morton Finston. Proc. Nat. Acad. Sci., 45, No. 4, (1959). The gap in the bearings with magnetic suspension is 10 p. With magnetic suspension the shafts carried by the floats are free from contact with the jewels. The magnetic suspensions used in current practice keep the float shafts centered within the jewels with accuracies of 1-2 /1. Work carried out in the Instrumentation Laboratory, of which

the speaker is the Director, is described in a paper, 'A Magnetic Support for Floated Inertial Instruments' by Philip J. Gilinson, Jr., William G. Denhard, and Richard H. Frazier-Sherman M. Fairchild Fund Paper No. FF-27. Paper presented at the Institute of Aeronautical Sciences Meeting on Guidance of Aerospace Vehicles, held in Boston, Massachusetts, on May 25-27, 1960. SLOMYANSKI (US.5.R.) Please describe the advantages and disadvantages of various types of gyroscopes advertised in the American journals. C. S. DRAPER, in reply. Many firms in my country manufacture gyroscopic instruments and try to sell them in general competition with each other. Most of those that apply the principle of flotation use a lubricant type fluid in which hydrocarbonlike molecules have their carbon atoms completely replaced by fluorine atoms. These fluids are chemically inert and are compatible with the materials commonly applied to the construction of instruments. As a matter of practice, the electrical leads carrying alternating current power to the gyro rotor drive motor have been completely immersed in the fluorolube liquid. With this arrangement the electrically generated heat is easily dissipated to the liquid. In earlier gyro unit models the very flexible electrical leads were surrounded by relatively great volumes of fluid. When the units were cooled off during periods of storage, the leads were sometimes damaged by cracks that developed as the liquid solidified. This difficulty has now been eliminated by some firms who have designed grooves for the wires that are so wide that the leads do not touch their sides, but are narrow enough to prevent large cracks from occurring as the fluid solidifies. Other firms have attacked the problems resulting from freezing of gyro flotation fluid by seeking liquids that do not become solid at low temperatures. A. Yu. ISHLINSKI (US.S.R.)

We see from this paper how instruments of very high accuracy must be designed to allow for all the little things that can influence their performance. We know of the immense part which has been played by Dr. Draper in the development of such instruments which are manufactured in the U.S .A. They could not have been developed unless the experiments had been paralleled by important theoretical studies and it is difficult to decide which is the most important. In developing new instruments it is never possible to foresee the difficulties that will be encountered. I recall, in particular, one interesting investigation on the drift of a gyroscope float during conical motion of the base. Moreover, apparently, great difficulties were encountered in developing the magnetic suspension. In essence, this is the idea of Mahomet's tomb! However, it is necessary to try not to set up turning moments.

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