AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UM-39 UNIVERSITY OF MICHIGAN MEMORVAINDUM NO. 39 MOMENTS ABOUT TUE AXES OF A SIMPLIFD FIGEHT TABILE GIMBAL STRUCTURE Project X'-794 -.. - USAF Contract W33-038- ac1i4222 Prepared by: James'H. Br6wnm' Re-se'arch Associate Approved by: D. McDonald Supervisor Controls and Instrumentation Group 1 December 1949

AERONAUTICAL RESE ARCH CENTER - UNIVERSITY OF MICHIGAN UMM-539 ABSTRACT The moments about the gimbal axes of a simplified flight table gimbal structure are developed in terms of the Euler angles and their derivatives. From the equations developed, it can be determined how much torque must be produced by the motors turning the flight table gimbal structure about its axes.

AERONAUTICAL RESEARPCH CENTER - UNIVERSITY OF MICHIGAN UMM- 39 The flight table gimbal structure is a physical device to simulate the angular motion of a craft in flight as it maneuvers. The flight table gimbal structure consists of a platform mounted in a gimbaling arrangement to allow complete angular motion about one point which remains fixed. Figure 1 is a schematic drawing of such a gimbaling arrangement. Referring to figure 1, the outer gimbal axis is fixed in space insofar as the operation of the flight table system is concerned. The middle gimbal axis will always move in the same fixed plane, called the reference plane, perpendicular to the outer gimbal axis. If the coordinates, xf, yf, and zf are fixed with respect to the flight table platform, with Zf along the inner gimbal axis, then the middle gimbal axis will lie along the line of intersection of the xf - yf plane and the fixed reference plane. Rotations about each of the three gimbal axes will generate three independent angles, AL, AN, and Ap, where: AL = the longitudinal angle, the angle between the xf-axis and the middle gimbal axis; = the nodal angle, the angle between the zf-axis and the outer gimbal axis (Zo); Ap = the polar angle, the angle between a line fixed in the reference plane and the middle gimbal axis. These three angles are the Euler angles (Fig. 3) and the three axes about which they may be generated are called the Euler axes. In order to simulate the angular motion of a craft, three motors are used to cause appropriate turning of the gimbals about their axes. The motor which turns the inner gimbal is mounted so that the reaction is between the middle gimbal and the flight table. The motor which turns the middle gimbal reacts between the outer and middle gimbals; and the outer gimbal is turned by a motor reacting between the outer gimbal and inertial space. It is desired to find the moment about each Of these axes in terms of the Euler angles, angular velocities, and angular accelerations. Page 1

,AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN MM- 39 FLIGHT SIMULATOR GIMBAL ARRANGEMENT Ap do,Zo A, |-POLAR AXIS MIDDLE GIMBAL AXIS FLIGHT TABLE PLATFORM f NODAL AXIS.iINNER GIMBAL AXIS NODAL AXIS —Sa N, Xm.Xo.' LONGITUDINAL AXIS | OUTER GIMBAL AXIS FIG. I Ap+ Zszo POLAR AXIS FLIGHT TABLE PLATFORM MIDDLE GIMBAL AXIS (WITH EQUIPMENT) NODAL Zf INNER GIMBAL AXIS NODAL AXIS, ztz AA N, Xm,Xo. LONGITUDINAL AXIS SOUTER GIMBAL AXIS FIG. 2 Page 2

0 z AN (POLAR AXIS) Z (NODAL ANGLE) (LONGITUDINAL\A AXIS. [ROLL= rI]) (YAW = CD XO~ ~~~~~~~~~~I..oAL X LONGITUDINAL (PITCH (POLAR ANGLE) NODE LINE 0 (NODAL AXIS)'?J Fig.. 3, -GEOMETRIC REPRESENTATION OF MISSILE IN INERTIAL SPACE

AE:RONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-39 The following right-hand orthogonal coordinate systems are used: Fixed-: Xo, Yo, Zo with the Zo-axis along the outer gimbal axis. Flight Table: xf, yf, zf with the zf-axis along the inner gimbal axis. Middle Gimbal: xm, Ym Zm with xm along the middle gimbal axis and Zm along zf. Outer Gimbal: xo, yo, Zo with xo along the middle gimbal axis and Zo along Zo. Unit triads, such as exf, eyf, ezf, are placed along each of the coordinate axes systems. Since the moment of a system is the time rate of change of the angular momentum of the system, the angular momentum of each of the components, inner gimbal, mid-dle gimbal, and outer gimbal, may be found and the time rates of change of these angular moments will give the desired moment. The coordinate system x0o, yo, zo contains two of the Euler axes, viz., xo (N) and zo (P); therefore, all vector quantities are referred to this coordinate system. Angular momentum is defined as: =h e +h e +h e (1) xoxo x o yo zo zo h =J w -P w - P w xo xo xo zo yo yo zo h =-P w +J w -P w (2) yo zo yo yo xo zo h =-P w -P w +J w zo yo xo xo yo zo zo where: J = moment of inertia P = product of inertia Pxo ~= miYoZ Pyo = E mizoXo Page 4

AERON'AUTICAL RERSEI ARCH CENTER - UNIVERSITY OF MICHIGAN UM,- 39 Pzo = mixoYo ) (3) w = angular velocity Angular Momentum of Flight Table In order to simplify the resulting equations, it is assumed in the following discussion that the moment of inertia of the inner gimbal about any line in the xf-yf plane is the same as the moment of inertia about any other line in the same plane. It will be noted that figure 1 does not indicate this fact. Figure 1 shows only the inner gimbal platform. By proper placing of the craft control system components on the platform, the above assumption can be approximated in actual practice. For purposes of discussion, the inner gimbal, including the platform and its equipment, could be considered as a cylinder whose polar axis is zf (Fig. 2). Thus Xf, yf, and Zf become the principal axes of inertia of the inner gimbal. Since the yo-zo plane is a plane of symmetry for the flight table, the products of inertia, Pfyo and Pfzo are zero. Thus the angular momentum of the flight table becomes: fxo f fxofxo ) (cf. table of yho fyo fyo fxo fzo definitions) h =J w -P w fzo fzo fzo fxo fyo In terms of the moments of inertia about the flight table axes: Jfxo = Jxf f= J ) 2 2 Because o the assumed symmetry of the flight table, the products of in Because of the assumed symmetry of the flight table, the products of inertia with respect to xf, yf, and Zf, which would normally appear in the expressions for Jf and Jf in equations (5), are zero. The flight table is considered to have an angular velocity about each of its Euler axes; viz., AL, AN, and Ae l Page 5

AERONAUTICAL RESEA ARCH CENTER, UNIVERSITY OF MICHIGAN UMM-39 Therefore: w A fxo N - - Asin A (6) fyo L N 0 w = A + A cos A ) fzo P L N where the dot indicates the time derivative. If the transformation between the Xo-Yo-zo and the Xf-yf-zf coordinate systems is given by: e e e ) xo yo zo "xf 11 12 13 ef | 21L m22 23 (7) ) 5zf m31 m m ) zf m3.1 32 M 33 then the product of inertia, Pf, is given by: -P J m m +J m m +J m m(8) fxo xf 12 13 yf 22 23 zf 32 33 (Ref. 1, p. 123) Substituting the values for the m's from Table A into equation (8): -Pfxo = (Jxf - zf) sin AN cos AN (9) Substituting equations (5), (6), and (9) into equations (4): h =J A fxo xf N h =(J - Jzf) Ap sin A cos,A - J A sin A ) (10) fyo xf zf P N N zfL N h =(J sin2A + J cos2 A ) A +J A cos A fzo yf N zf N P zf L N Angular Momentum of Middle Gimbal The yo-Zo plane is a plane of symmetry for the middle gimbal. Therefore, equations (4) apply to the middle gimbal with suitable change of subscript. The angular velocity of the middle gimbal is: Page 6

AJERON. AUTICAL RESE;kARCH CENTER - UNIVERSITY. OF MICHIGAN UMM- 39 w =A mxo N w =0 ) (11) myo w =A mzo P In terms of the moments of inertia of the middle gimbal about its coordinate axes: J =J mxo Xm J = (J cos2A + J sin2A ) myo ym N zm N J = (J s A +J co2 (12)A) mzo ym N zm N -P =(J ym-J )sin A cosA mxo yM zm N N Therefore: h =J A mxo xm N hmyo (Jm -J A sin A cos A (13) myo ym zm P N N h = (J sin2 A + J cos2A) A mzo ym N zm N P Angular Momentum of Outer Gimbal The outer gimbal rotates about a fixed axis. Also it is symmetric to the Xo-Yo and yo-zo planes. Therefore, the outer gimbal has an angular momentum of: h =J A e (14) o zo P zo The total angular momentum of the flight simulator is: xo ( xf J xm ANN hy =(Jy +m Zf - J AsinAcosA - J A sinA ) (15) yo yf yinzn P N N zf N h = [(Jfg )sin2AN+(J fzm)cOsA +Jzo ]A zf A osN zo yn N~ ~zf Lz N zog P L Page 7

AAERONAUTICAIL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-39 Moments The moment of the flight simulator system is given by: 6h = h + W X Xh (Ref. 2, p. 345) (16) Where: 8h *- ~ * ~ -- -=h = e h e + h e (17) 6t xo xo yo yo zo zo W=Ae (18) P zo therefore: xo (J + J A + J AA sin A -(J +J f J - sin A cos A yf yM zf za P N N (f J+ -J -ymJ) A sin A cos A J AsinA +[(J )+J )cos2AN +(J f+ J )sin2AN]AA ) - [(Jr + J )sin2AN + (JZf + Jw)cOAN]ANAp ) (19) +(J~f + J );iAA cos A xf MN p zfLN M = [(J + J )siN~JC +os2 + J]p ) +2(Jg Jz-Jf -J)AzA sin A cos A + J A cos A - J A A sin A zf L N zf L N N If ML, MN, M, are the moments along the flight table, middle gimbal, and outer gimbal axes, respectively, then: M =M) zN 10 )MN z z o sMzo ) (20) ML =Me (yL ( N yo AN eZo) ) -. -.-Page 8 )

AE;RONAUTICAL RESE:ARCH CENTER - UNIVERSITY OF MICHIGAN UMM-39 Therefore: 1 zf"L + (Jzf zm zo N Mt-J A +(J +J +J )A-cosA - (J -J +J +J )A sinA xm ym zf zm N P N MN (Jxf +Jm) A + J ALA sin A N xf XMi N zfLP N _ f +J _ Jf - J )2 sin A cos A ) (21) f ym zf zm P N N = [I(J-f + Jm) sin2AN + (Jzf + Jz)cos2 AN + J ]A ) + 2(J + J - J )A A sin A- cos A) yf ym zf zm N P N N + J A cos A -J A A sin A zf L N zf L N N Equations (21) give the total moments about the three Euler axes. However, the useful contribution of the motor turning the inner gimbal is that which changes the angular momentum of the inner gimbal. The inner gimbal motor will need to work only against the inertia of the inner gimbal. Therefore, the moment of inertia terms of ML in equations (21) not containing the letter "f" in the subscript can be dropped to give: M = Jzf + J zfAp cos A - J zfANA sin A (22) tzfL zfP N zfNP N which gives the moment or torque the inner gimbal motor must produce. The motor turning the flight simulator about the middle gimbal axis must not only turn the middle gimbal mass, but also the inner gimbal mass. Therefore, the moment equation will contain moments of inertia for the inner gimbal and also the middle gimbal. Examining the value of MN in equations (21) it will be found that only the moments of inertia of the inner gimbal and middle gimbal appear; therefore, M = MN (23) Likewise: M' = M (24) P P The prime superscripts indicate the value the motors must actually produce to do the turning of the respective gimbals. Page 9

,AERONAUTICAL RIESE-ARCH CENTER ~UNIVERSITY OF MICHIGAN UMM- 39 TABLE A Transformation between the inner gimbal and middle gimbal coordinate systems and the outer gimbal coordinate system. e e e xo yo zo exfcos A sin A cos A sin A sin A exfL L N L N e -sin A cos A cos A cos A sin A yf L L N L N e 0 - sin A cos A zf N N zm P N N Page 10

AERONAUTICAL RESEARGCH CENTER - UNIVERSITY OF MICHIGAN UMM- 39 TABLE OF DEFINITIONS A = longitudinal angle L AN = nodal angle N A = polar angle P A = angular velocity about L-axis L AN = angular velocity about N-axis N A = angular velocity about P-axis P. ) L Ai~ ) = Euler angular acceleration (not the angular ) acceleration along the Euler axes) P L = longitudinal axis ) ) N = nodal axis ) Euler axes ) P = polar axis ) X, Y, Z = coordinate axes fixed in space xfj yf, zf = coordinate axes fixed in the flight table x, yY, z = coordinate axes fixed in the middle gimbal xM yo z = coordinate axes fixed in the outer gimbal e = unit vector along positive axis as indicated by the subscript h = angular momentum (a vector quantity) Page 11

AERONA.UTICAL RESE:AnCH CENTER - UNIVERSITY OF MICHIGAN..........M. _-39 Table of Definitions (cont'd) h xo ) = component of total angular momentum of flight h table system on the indicated outer gimbal yo ) coordinate axis h ZO h fxo ) - component of the angular momentum of the h flight table only,on the indicated outer fy~o ) gimbal coordinate axis h fzo h mxo ) = component of the angular momentum of the h middle gimbal only, on the indicated outer myo. ho) gimbal coordinate axis h mZoo ~J = moment of inertia ~xf = moment of inertia of flight table about the xf-axis Jxff Jf = moment of inertia of flight table about the yf-axis J~f = moment of inertia of flight table about the zf-axis J ) fxo Jfs ) = moment of inertia of flight table only, about the Y ) indicated outer gimbal coordinate axis Jz fzo JM) ~J ) = moment of inertia of middle gimbal only, about the ) indicated outer gimbal coordinate axis Jfzo )| J ) mx o J = moment of inertia of outer gimbal only, about the z, ois Page 12

AERONAUTICAL RESELARCH CENTER - UNIVERSITY -OF MICHIGAN MM-39 Table of Definitions (cont'd) mij = direction cosine (i, J = 1, 2, 3) M = moment (a vector quantity) M M ) = component of moment on the indicated outer XO Y ) gimbal coordinate axis M M ZO L M ) = moment along indicated Euler axis N M P P = product of inertia Pf = product of inertia of flight table only, with respect to the xo-Yo plane and the xo-zo plane P = product of inertia of middle gimbal only, with respect to the xo-Yo plane and the x0-zo plane w = angular velocity (a vector quantity) w fxo w ) = angular velocity of flight table only, about the fo) indicated outer gimbal coordinate axis w fzo w mxo w ) = angular velocity of middle gimbal only, about the myo yO) indicated outer gimbal coordinate axis w mzo Page 13

AERONAUTICAL RESELARCH CENTER UNIVERSITY -OF MICHIGAN UMM-39 PREERENCES 1. Whittaker, Analytical Dynamics. Dover. 2. Synge and Griffith, Principles of Mechanics. McGraw-Hill. 3. Coe, C.J., Theoretical Mechanics. MacMillan. 4. Brown, J.H., "Use of Transformations Involving Euler Angles", University of Michigan Report No. EMN-26. Page 14

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN MM- 59 DISTRIB3UTION Distribution of this report is made in accordance with ANAF-GM Mailing List No. 9, dated September 1949, to include Part A, Part B, and Part C. Page 15

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