26 November 2002

ERRATA SHEET(Change Notice)

C 1 TO: TECHNICAL REPORT ARCCB-TR-02015

MAIN BATTLE TANK FLEXIBLEGUN TUBE DISTURBANCE MODEL:

THREE-SEGMENT MODEL

By

Henry J. Sneck

Please remove the cover from the above publicationand insert new cover enclosed. The report title has been corrected.

ZZZ

U.S. ARMY ARMAMENT RESEARCH,DEVELOPMENT AND ENGINEERING CENTER

CLOSE COMBAT ARMAMENTS CENTER

BENET LABORATORIES

"WATERVLIET,. NY 12189-4000

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1. AGENCY USE ONLY (Leave 2. REPORT DATE 3. REPORT TYPE AND DATES COVEREDBlank) October 2002 Final

4. TITLE AND SUBTITLE 5. FUNDING NUMBERSMAIN BATTLE TANK FLEXIBLE GUN TUBE DISTURBANCE MODEL: Sales Order No. 2WHR-BVTHREE-SEGMENT MODEL

6. AUTHORSHenry J. Sneck

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORTU.S. Army ARDEC NUMBERBenet Laboratories, AMSTA-AR-CCB-O ARCCB-TR-02015Watervliet, NY 12189-4000

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING / MONITORING AGENCYU.S. Army ARDEC REPORT NUMBERClose Combat Armaments CenterPicatinny Arsenal, NJ 07806-5000

11. SUPPLEMENTARY NOTESPresented at the 1 0 h U.S. Army Gun Dynamics Symposium, Austin, TX, 23-26 April 2001.Published in proceedings of the symposium.

12a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODEApproved for public release; distribution unlimited.

13. ABSTRACT (Maximum 200 words)A rational approach to disturbance rejection is proposed and applied to a simple three degree-of-freedom flexible gun tube model using feedforwardand feedback compensation. The first two natural frequencies of the pin-free and cantilever tube are matched by adjusting the dimensions of therigid segments and the stiffness of the torsional springs that join them. It was found that, contrary to the previously analyzed two degree-of-freedomsegment model, the muzzle-end segment could be stabilized by the proper choice of transfer functions and elevation driveline response. Theanalysis serves to establish the requirements for the transfer functions and stabilizing actuator systems.

14. SUBJECT TERMS 15. NUMBER OF PAGESGun Tube, Flexible, Stabilization 22

16. PRICE CODE

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACT ULUNCLASSIFIED UNCLASSIFIED UNCLASSIFIED

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TABLE OF CONTENTSPage

INTRODUCTION.....................................................................................1..

EQU A TION S O F M O TION ......................................................................................................... 1

FEEDBACK AND FEEDFORWARD CONTROL ............................................................... 3

D ISTUR BAN CE REJECTION ............................................................................................... 5

IM PLEM EN TA TION .......................................................................................................... 6

TU BE M O D EL PA RA M ETER S ............................................................................................ 7

CON CLU SION S ........................................................................................................................... 13

REFEREN CES ............................................................................................................................. 14

A PPEN D IX A ............................................................................................................................... 15

AP PEN D IX B ............................................................................................................................... 15

A PPEND IX C ............................................................................................................................... 15

APPEND IX D ............................................................................................................................... 16

APPEND IX E ............................................................................................................................... 17

AP PEND IX F ................................................................................................................................ 17

AP PEND IX G ............................................................................................................................... 18

N O M EN CLA TU RE ..................................................................................................................... 19

LIST OF ILLUSTRATIONS

1. G un generic m odel ......................................................................................................... 1

2. Block diagram of gun tube system ................................................................................ 4

3. Bode plots of G, Gu, ...................................................................................... ...... 8

4. Bode plots of G, G 12 .................................................................................................... 9

5. Bode plots of G, G22...................................................... . . . .. . . . . . . . . .. .. . . . . . . .. .. . . . . . . . . . . . . . . . . . . . 9

6. Bode plots ofd ............................................................. . . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7. Bode plots ofd12 .................................................................. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 11

8. Bode plots ofd 3 ............................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . .. . . . . . . 11

9. Bode plots of d21 ............................................................ . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . .. . . .. .. . . . . 12

10. Bode plots of d3j ............................................................................................................... 12

ii

INTRODUCTION

Modem tank cannons are long, relatively, thin, beam-like hollow cylinders. Theiraccuracy is, in part, determined by their flexibility, especially under dynamic loading. Verysmall deflections and rotations of the muzzle end can have a significant influence on theaccuracy of the shot at long ranges. Muzzle motions induced by firing are inevitable, anddifficult to control because the time scale of firing is of the order of milliseconds.

Another source of muzzle motion is the ground-induced motion of the vehicle. Thesemotions, transmitted through the trunnions and gun actuators, can be quite large and havefrequencies comparable to the natural frequencies of the tube. The time scales of thesedisturbances depend on the tank speed and on the nature of the terrain. They are typically of theorder of seconds or longer. Sensing and actuation to control the influence of vehicle motion onthe muzzle response might be possible, given these relatively long time scales. This raises twoquestions. First, is it possible to reject some, or all ground motion disturbance from the muzzlemotion? In a previous paper (ref 1) it was suggested that not all of the disturbance could berejected. Second, if the more comprehensive model used here indicates that all of thedisturbance can be rejected, what is the required control strategy?

During a discussion with Dr. Purdy, author of Reference 2, he suggested that the fidelityof his two-segment flexible model, documented in Reference 1, was inadequate. Herecommended that the tube should be divided into at least three segments, with interveningtorsional springs and dampers. The author is indebted to Dr. Purdy for this suggestion since thisreport is the result of his recommendation.

EQUATIONS OF MOTION

Figure 1 shows the generic model of the tube and the various quantities that determine itsdynamic behavior.

Y

hull

Figure 1. Gun generic model.

11

A free body analysis of this model yields the classical dynamic equation

[in] {}+[c]{}+[k] {(}-{I} (1)

The elements of mass, damping, and stiffness 3x3 matrices are shown in Appendix A.

Purdy (ref 3) has shown that the tube motion can be adequately modeled if the segmentedmodel matches the pinned-free and cantilever frequencies of the mounted tube. Matching isaccomplished by adjusting the size of the rigid segments and the stiffness of their connectingtorsional springs. The 2x2 matrices for the cantilever mode are outlined in Appendix B.

Transformation of equation (1) into the frequency domain will allow its incorporationinto the control strategy. Taking the Laplace transform of equation (1) yields

Ix,(s) F. (s)l

[a] o2(s)[ = 2 , (s) (2)

Theelmets f a]and[ 0{2(S) jjSO,(S)13o(s)) oP(S)J

The elements of [a] and [1] are listed in Appendix C. This equation relates the responsevector on the left to the disturbance vector on the right. These vectors also contain the actuatorforce, F,, and the actuator displacement, x,, in addition to disturbances (s2y,(s), sOp(s), Op(s)) and

the responses 01(s), 02(s), •3(s).I. x(SY] F,,(s)1,() = 1 ci ,(s) [ (3)

02(S)j det[a] S'~sp(S)Jo.3(s)J i OP(S) J

where [C] is the transpose of the numerators of the cofactors of [a]. The elements of [C] and det

[a] are listed in Appendix D.

The final step in the preparation of the dynamic equations is to perform the operation1

det[a] [c][I]= [B] (4)

where the elements of [B] can be found in Appendix E.

2

The result of these straightforward, but laborious manipulations, is an equation for theresponse to the disturbance in terms of the properties of the model contained in [B], i.e.,IX (s), [ F(s) 1

o,(s)L [] s 2y,(s) (5)92(S)r I so()(s)03(s)J ( o,(s) J

Of course, it was known at the outset that equation (1) could be put into this form. Thissection merely provides the details of how this transformation is performed, and documents theintermediate steps and their components.

FEEDBACK AND FEEDFORWARD CONTROL

The portion of the response due to the applied actuating force is

x,(s)= _B1,0 j(s) B31 F,=[G,]F, (6)

03(s))J [ B41 j1where [Gp] is the "plant" transfer function.

The portion of the response due to the disturbance is

X, (s) B2 B13 B14 1fs2y(s)2°s(s) B4 s OP (s) J (7)

02(S) B2 B33 B34 op (S)J103S) -B2 B43 B4J "

where [Gd] is the disturbance transfer function and {D} is the disturbance vector.

Figure 2 is a block diagram of the gun tube system with a gain Gc, feedback H,feedforward Gda, and reference signal R. Because R is a scalar, the feedforward transfer functionis a row vector, i.e.,

[GJ]=[G,, G,2 G,I] (8)

3

{D}

R 4- FC +4.. G

I 2-

Figure 2. Block diagram of gun tube system.

Assuming that the tube rotations at the trunnions and the muzzle can be sensed, thefeedback transfer function is also a row vector, i.e.,

[H] =[ G22 0 G24] (9)

Now referring to Figure 2, the response to the disturbance for the controlled system is

IGd]-L[Gp J[GC] [GCd]D} =DJI] + [GP j [GC ] [H] {OD} (1Oa)

or in abbreviated notation

[d] {D}= [q] {OD} (10b)

The final step in these manipulations is to solve for the response to the disturbance, whichis

{OD}=[q]' [d]{D}= •1 [r]T [d]{D} (10c)det[q]

where [r] is the matrix of the cofactors of [q].

The matrices [r]T and [d] are given in Appendix F. It is interesting to note that [d]contains only the feedforward transfer functions GI, G12, and G1s, while [r] and det[q] containonly the feedback transfer functions, G22, and G24.

4

DISTURBANCE REJECTION

Referring to Appendix F, the expanded version of equation (10c) is

X/S r~i, , r2, O r41 [di, d12 d13 ir Y(S01(s) _ 1 0 r22 0 r42 Hd21 d22 d23 so )(1d02 (s) e4q] 0 r23 r33 r3 d, d 2 d (,d)

1 3(s)) r2 r 0 r 44 d4, d42 d43 L 9p(S)

where

det[q]= I + B2 G G22 + B4, G G24 (11)

To completely remove the effect of the disturbances on 0 3(S)D requires that

r24 d21 + r44 d4, = 0 (12a)

r24 d22 + r44 d42 = 0 (12b)

r24 d 23 + r44 d 43 =0 (12c)

anddet[q] 0 0 (12d)

One way to accomplish this is to let G22 = 0 so that

r24 = G-B41,G G22 = 0 (13)

and then choose

d 4 1 = B 4 2 -B 41 G 1G = 0 (14a)

d 42 =B 43 -B 41 Gc G12 = 0 (14b)

d43 =B44 -B 41 GG13 =0 (14c)

so that

det[q]= 1 + B41Gc G24 = r1, = r22 =r 33 (14d)

5

and

r =4 r2, = r23 = r 24 = 0 (15)

The effect of this choice on the disturbance transfer function is

[r,, 0 0 r4, [d,, d12 d1311 [r]T [d]=~ 0 r22 0 r42 id 2, d22 d23 (16

det[q] r,= 0 0 r.3 r., d3, d32 d33

0 0 0 r.4 0 0 0

An alternative strategy is to let r44 = 1 + B21 G, G22 = 0, and then choose d2, =d22 =d2 =

0 so that det[q] = B41 G, G24 = rH = r33 . However, in the end the resulting disturbance transferfunction is the same as created by equation (16).

Since the fourth column of [r]T is eliminated by the matrix multiplication and det[q] = rl

= r22 = r33, det[q] will be eliminated from the transformation.

IMPLEMENTATION

This analysis of the three-segment model indicates that model muzzle element can bestabilized by properly selecting the feedforward and feedback transfer functions. This is contraryto the finding for the lower order two-segment model (ref 1). Although the three-segment modelonly approximates the real tube, the results of this model are encouraging with respect to realtubes.

In order to achieve muzzle stabilization, the breech-end of the gun must be actuated.Segments 1 and 2 will also rotate. These motions are determined by equations (14) and (16). Allof the elements of [d] can be written in terms of the elements of [B]. Because [r]T acts like anidentity matrix, the product [r]T[d] is quite simple. If the process is carried a step further, theresult can be put in terms of the elements of [C] with startling results, i.e., d22 = d23 = d32 = d33 =

0, and only d,, = d]2 = d13 = d21, and d3, are nonzero. The surviving elements are

,, = ,, ((C,4 C3, C34 , 3, + (C1 4 C ,, - C44 ),2 (17a)C,4 det[a]c 3 )3 (c,)1)

d~l = 1 ((C,4 C32 - C12 C34)i32 + (C',4 C42 -C, 2 C44 ) 42)( bCH4 det[a]

d, =C ((C,4 C33 - C,3 C34 ) + (C,, C43 - C,2 C44 ) I42) (17c)C,, det[a] CC 4)I (C,7

6

2 (C,4 C21 - C,, C24) '23 (1 7d)d2 C14 det[a]

d1 (C,4 C2, - C,, C24)I 24 (17e)C,4 det[a]

These transfer functions relate the disturbances to the responses. All of the Cs are oforder s4 with the exception of C,, which is of order s 6. Since there is no restraining torsionalspring connecting the tube to the mount in the model, s = 0 is a root of det[a]. Removing thisrigid body factor from det[a] reduces it to order S.

TUBE MODEL PARAMETERS

The feedforward transfer functions depend on the length and mass properties of thesegments, the torsional stiffness of the joining springs, and the torsional damping coefficients.These are chosen so that the actual cantilever and pin-free mode shapes and natural frequenciesare matched as closely as possible (ref 3). To simplify the matching process, it is assumed thatthe damping is negligible. The first estimate of the segment lengths can be obtained by "fitting"the straight-line segments to the mode shapes obtained from a finite element model of the tube orother modal analyses. This fitting is best done by graphically overlaying the segments on plotsof the mode shapes to estimate the segment lengths. The calculation of the mass properties ofthe segments can then be performed and these, along with the modal frequencies, inserted intothe characteristic equations. The characteristic equations will then contain only the torsionalstiffnesses as unknowns. The cantilever and pin-free equations are both quadratic so that thestiffness coefficients can be found directly. The degree of matching is determined by howclosely the cantilever and pin-free stiffnesses agree.

The characteristic equations for the cantilever and pin-free segments are given inAppendix G. Although the pin-free equation appears to be sixth-order, it has a double root thatis zero. The calculations for this trial-and-success process are easily implemented on aspreadsheet.

The XM291 tank gun was chosen for modeling because its mode shapes and frequencieswere available from an existing, validated analytical model. Matching the stiffnesses proved tobe surprisingly easy, requiring only modest adjustments to the first estimates of the segmentlengths. Since all their frequencies (cantilever: 97.4 Hz, 40.35 Hz; pin-free: 25.08 Hz, 81.59 Hz)were inserted into the characteristic equations, they are matched exactly. The torsionalstiffnesses for the pinned-free and cantilever modes were matched within 2% using the lengths 11= 6.0 ft, 2 = 5.5 ft, and 13 = 6.0 ft. From this process the model torsional stiffnesses, k12 =

3.6(106) lb-ft/rad and k23 = 1.69(106) lb-ft/rad were obtained

Dynamic analyses (refs 2, 3) have successfully modeled tube response using proportionaldamping, i.e., [c] = fl[k]. In the case of the XM291, 6= 0.0015 second has been found to bereasonable. A reasonable estimate for trunnion damping is cp = 750 lb-ft-s/rad.

7

The first attempt to determine the feedforward transfer function using equations (12) and

(13) failed because some of the roots of B41 were positive. This difficulty was eliminated by

using the alternative strategy described above, with the following results:

r44 = 1 + B 21 GC G22 = 0 (18)

GC Gi = B22 (1 9a)B 2,

Gc G12 = 23 (19b)B21

G G13 = B21 (19c)

The det[a] plays no role in these functions because it is canceled by ratioing the B's.

Figures 3 and 4 show the Bode plots of Gc GI and Gc G12. The transfer function Gc G13 is zero

so that Op is not fed forward. The numerators and denominators are all fifth-order polynomials,

so that the high and low frequency gains are bounded.

50

30

S 20 -.

"o10 -

a 0

0.

.50

-100

-150101 10) 10'

Frequency (rad/sec)

Figure 3. Bode plots of G, Gil.

8

52.5

52

51

-o 50.5

s" o

-0.5)0)

.0.5

.1

10" 10. '0' 10,

Frequency (rad/set)

Figure 4. Bode plots of G, G12.

The feedback transfer function, G, G22, is shown in Figure 5. The remaining feedbacktransfer function, G, G24, plays no role in disturbance rejection in this case.

-350

-400 -

S .450-

'E

tM

M -500

• 200

*0.

-150

100 101 102

Frequency (rad/sec)

Figure 4. Bode plots of G• (712.

009

-20

10, 10. 10,

Frequency (rad/sec)

Figure 5. Bode plots of G, (722.

9

The elements of d 22, d 23, d3 2, and d 33 of [d] were found to be identically zero. Theremaining nonzero elements of equations (1Od) and (16) yield the following response equations:

xC ==d,,s 2 y,(s)+ d,2 sO(s)+ d,3 OP(s) (20a)

9, =d 21 s 2 Y, (s) (20b)

02 =d 31s 2 y, (s) (20c)

Figures 6 through 10 show the transfer functions required by the equations above.Figures 9 and 10 show that the effects of the trunnion acceleration on 01 and 02 are highlyattenuated so that large angular displacements of the tube are not required to achievestabilization.

Figures 6 through 8 are quite similar. It appears that the required x, will depend largelyon the trunnion acceleration and pitch rate at very low frequencies. There is a considerableattenuation of the disturbance inputs up to 100 rad/second (-15 Hz) with a return to the lowfrequency levels at 103 rad/second (160 Hz).

2

0- -

-40

. .... ... . .I

€o 100

"0. so

00-

0

- 100 . . . .- . . ..- W. . . . . . . , . .

10"2 10-, 100 101 10, 10, l0e

Frequency (rafts•)

Figure 6. Bode plots of deI.

10

0-

0)0

-80

cc 400

300

200-

100

010 .210 160 10 t10 10 310,

Frequency (rad/sec)

Figure 7. Bode plots of d12.

00

08

CD 200

L I-SO

=-100

0 0

10' 10' 10, 10, 10 103 10

Frequency (rad/sec)

Figure 8. Bode plots of d13.

-100

-120

-140

¢0"0 160

=E -180o.....

IO.

C02

-50

0 100

10": 10" 10" 10e 10• 10, 10e 10'

Frequency (rad/sec)

Figure 9. Bode plots of d2i.

-80

-100

-120

'-140

I-1

r

- . . . .. .. ........ . ... ... . . . .. . . . .. . . . .. . ... . . . .. ... . ....------------

-W0

10" 10 t0- 100 10 10 102 10 10'

Frequency (rad/sec)

Figure 10. Bode plots of d32 .

012

CONCLUSIONS

It appears that the results previously obtained with the two-segment model, (ref 1), led tothe erroneous conclusion that the effects of the disturbance could not be entirely repeated fromthe muzzle angular displacement. The analysis of the three-segment model presented suggeststhat this is possible, at least theoretically. Of course, the unanswered question is, "What wouldbe revealed by a higher-order multi-segmented model, and how many segments are enough?"

On the practical side, it is certain that the transfer functions cannot be duplicatedprecisely. There are four of these that must be implemented with reasonable fidelity to achievethe predicted results of the three-segment model. That number, along with the input signals,indicates the magnitude of this task. While feedforward and feedback control have long beenused in fire control, it is hoped that this report provides some guidance in their use when tubeflexure is a consideration.

13

REFERENCES

1. Sneck, H.J., "An Assessment of Main Battle Tank Flexible Gun Tube DisturbanceRejection," Proceedings of the 9h US. Army Symposium on Gun Dynamics, ARDECTechnical Report ARCCB-SP-99015, (E. Kathe, Ed.), Benet Laboratories, Watervliet, NY,November 1998, pp. 8-1 to 8-18.

2. Purdy, D.J., "Modeling and Simulation of a Weapon Control System for a Main BattleTank," Proceedings of the 8? US. Army Symposium on Gun Dynamics, ARDEC TechnicalReport ARCCB-SP-96032, (G. Pflegl, Ed.), Benet Laboratories, Watervliet, NY, May 1996,pp. 20-1 to 20-19.

3. Purdy, D.J., "An Investigation Into Modeling and Control of Flexible Bodies," Ph.D. Thesis,Cranfield University, England, 1994.

14

APPENDIX A

mi11 =11 + M1 2 +l12(mM2 +M 3)

m22 =12 + M2 + m3l2mil m12 M13 2

[] 21 in2 2 in 23 M 3 3 =13 + m 3 7732

Lm31 m 32 mi33 I = m 21 = m2 11772 + m 31112

m 13 =m 3 1 = msl73

m 23 = m 32 = m 312773

[C] C1[ 2 C12+C23 -C23

0 - C23 C23

[k]= -k12 k,2+k23 -k23

0 - k23 k23

APPENDIX B

[m, I=[r22 m23 [Cc][C= +,+c23 -c]2 [k"] [k-2+k23 k23lm 3 m2 m33 -c23 c23 -k2 k2

APPENDIX C

a il a,2 0 0 a l =kd

[a]= a2, a22 a23 a 24 a,2 = a2, =-kdXtp0 a32 a33 a 34

0 a 42 a 43 a 44 a 24 = a 42 = M13S-

a22 = mis 2 + (C,2 +cp)s+ (kdX•,2 + k12 )

a 23 = a 32 = m 12 S2 -_ c, 2 s-k, 2

a33 = m22s 2 +(c12 +c 23)s +(k,2 +k 23 )

a34 =a 43 = m23s2 -c 23s-k 23

a 4 4 = m 3 3s2 + c 2 3 s+k 2 3

15

122 = ý-(q71M] +lIM2 +l'IM3)

221 230 1 1=1.0 '23 CIP0 22 23 IN'24= kd X p2

0 3 0 014 kd X4 32 = 7 2 M2 + 12 M3 )0 142 0 0

42 = 773M 3

APPENDIX DEC1/ C12 C13 C141[C]= C21 C22 C23 C24

C31 C32 C33 C34

_C41 C42 C43 C44

C,, = a22a33a44 +2a 23a34a24 -(a 24a33a42 + a 43a 34 a22 + a23 a32 a44)

C22 = a,, (a33a44 -a34a43)

C33 =a,,(a2 ,a44 -a 24a42 )

C.4 = a,,(a22a33 - a23 a32 ) -aal

C,2 = C2, = -a21 (a33a44 -a 34a43)

C,3 = C3 1 = a2 , (a 3a44 -a 34a42 )

C,4 = C4, =-a 21 (a3 2a43 --a33a42)

C23 = C 32 = -a,,(a32a44 -a34a42)

C24 = C42 = a,, (a32a43 -a33a42 )

C34 = C,3 = -a,, (a22 a43 - a23a,2 ) + a,2 a2,a4,

det[a] = a,l(a22a33 a44 + a23a34a42 + a 32a 43a 24)

- a,, (a24a3,a42 + a34a43a22 + a23a.2a44 ) -a, 2a2, (a33a44 - a34a43)

16

APPENDIX E

"Bl, B12 B13 BM1

[B]B 2 B22 B23 B24

B3, B32 B33 B34

B41 B42 B43 B44

B,, = C,II,, / det[a]

B 2 =(C 21I 22 +C31 32 + C41, 42 )/det[a]

B, 3 = (C 2112 3)/det[a]

B,M = (C,,I,4 + C 2 I24)/det[a]B2, +(C,21,,)/det[a]B22 = (C22 122 + C32, 32 + C42I42) /Idet[a]

B23 = (C 22 23 ) / det[a]

B24 = (C1,2 ,4 + C 21 24 ) / det[a]

B3, = (C3,11 )/det[a]

B32 = (C23, 22 + C33132 + C43, 42 ) / det[a]

B33 = (C 23123 ) /det[a]

B34 = (C,M1, 4 + C23 124 ) /det[a]

B41 = (C,4 ,, ) /det[a]

B42 =(C2 412 2 + C34I32 + C44142 )/det[a]

B43 = (C2412 3)/det[a]

B44 = (C,, 4 + C24 24 )/ det[a]

APPENDIX F

[ 0 r22 0 r42

0" - r2 r3 , r ,43/-0 r"24 0 r.4

17

r,, = det[a] = 1 + B2,GcG22+ B, GcG 24 r33 = r,,

r2, =-B,,GCG22 r4, = -B,,GCG24r22 =1+B41 GcG24 r42 = -B 21GzG24

r23 =-B 31 GcG 22 r4, =-B 3sGcG24

r'24 = -B41GcG22 r44 =J+B21GCG22Ed,, d12 d,31d2, d22 d 2 3 d, =B,2 -B,,GG,,

Li]- d3, d32 d33 d 21 =B 22 -B 21GCG,,

d4, d42 d43 - d3, =B 32 -B 3 ,G G,

d12 =,B3 -1 B1 GcG12 d 4, =B42- B4 , 4 GG,

d 22 =B 23 -B2,GcG12 d13 = B14 -B,,GcG13

d 32 =B.3 -B3,GG,2 d 23 =B 24 - B2 GcG,3d 42 = B 43 - B4,GcG12 d33 =B 34 - B31GG13

d43 =B 44 -B 41 GcG,3

APPENDIX G

t,= LU2 = first two natural frequencies

k 3 -[(ZUI +2 XM22M33 -,..,.,)/(m. + 2mn. +m.)]

+ mi33 , X(,M)22M33 - M,23 M32 )1(M33 + 2m23 + Mi22)= 0

2 =( 2 + XM22M33 - M23,M 32 M 3,,- (m., + 2M23 + M2 2 )k23/ iM33

18

NOMENCLATURE

[a] Dynamic matrix

[B] [C] [I]

[c] Damping matrix

C12, C23 Damping coefficients

cIP Trunnion viscous friction coefficient

[C] Cofactor matrix of [a]

[d] Disturbance input matrix

{D} Disturbance vector

Fc Elevation actuating force

Gc Scalar gain

Gcd Feedforward transfer function vector

Gd Disturbance transfer function

Gp Plant transfer function

G,1 , G12, GJ3 Feedforward transfer function vector components

G22, G24 Feedback transfer function vector components

H Feedback transfer function vector

[1] Forcing function matrix

[k] Stiffness matrix

k12, k23 Stiffness coefficients

kd Drive line stiffness

11, 12, 13 Segment lengths

[m] Mass matrix

19

[q] Disturbance response matrix

Xc Elevation actuator displacement

XtP Distance from trunnions to drive

Y, Vertical displacement of the trunnion

[r] Cofactor matrix of [q]

,6 Proportional damping coefficient

r/, 772, 773 Center of mass coordinates

20

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