no. ri!v, mo.date 11 january 1971 bibliography atm-934, "structural analysis report, laser...
TRANSCRIPT
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LRRR 300 Corner Array Dynamic Analysis
NO. RI!V, MO.
ATM-936
PAGE _j__ OF
DATE 11 January 1971
This A TM presents the results of the structural dynamics analysis performed on the Lunar Ranging Retro Reflector 300 Corner Array.
CONTENTS:
Paragraph
1.0 2.0 3. 0 4.0
Prepared by:
M. Goran
G. B. Min
Ka>~ K. Wadleigh
Summary Description of Dynamic Models Computed Responses Dynamic Loads
Approved by:
D. Dewhirst
(1)
(2)
(3)
(4)
HO. Ri!V. MO.
ATM-936 LRRR 300 Corner Array
Dynamic Analysis PAGI! ii 01"
DATE 11 January 1971
Bibliography
ATM-934, "Structural Analysis Report, Laser Ranging Retro-Reflector, Bendix Aerospace Systems Division, Ann Arbor, Michigan, January 13, 1971
"Vibration Tests of Single Retro-Reflector Assemblies", Arthur D. Little, Inc., Cambridge, Massachusetts, December 31, 1970
Leonard Meirovitch, Analytical Methods in Vibrations, The MacMillan Company, New York, 1967
AL 410900, "Exhibit B, Design and Performance Specification for the Retro -Reflector Array Panels,"Bendix Aerospace Systems Division, Ann Arbor, Michigan, November 1, 1970
LRRR 300 Corner Array Dynan1ic Analysis
MO.
PAGE i jj
REV. HO.
OF
DATE 11 January 197
1.0
1.1
Summary
Motivation of Analysis
The vibrations Analysis of the Lunar Ranging Retro Reflector 300 Corner Array (LRRR 300) was performt>d to answer four basic questions.
These questions are:
l. What loads are imposed on the LRRR 300 structure during qualification and acceptance testing, during launch and boost and during lunar landing? Are the loads a.ssumP:d l)y
the stress group adequate to ensure structural reliability?
2. What is the worst vibration environment seen by any single corner reflector mount? Does this vibration environment impose loads on any corner reflector mount that will cause either the reflector or the mount to fail?
3. What vibration test spectrum should be used at the small array attachment fittings when the small array is separately tested?
4. If the experiment is flown with the upper array removed (the off loaded condition) will the structure and the corner reflector mounts survive?
To answer the four above questions, it was necessary to analyze three different mathematical models of the LRRR 3 00 structure. One math model predicted vibration response perpendicular to the plane of the two arrays. The second model predicted vibration response in the plane of the arrays. The final model was constructed to find vibration response of the experiment in the off loaded condition.
l. 2 Summary of Conclusions Drawn from Analysis
The analysis that was performed to answer the questions of paragraph l. 1 and the reasoning behind the answers make up the body of this report. The four paragraphs that follow give very brief summaries of the answers arrived at.
LRRR 300 Corner Array Dynamic Analysis
NO. RI!V, MO.
PAGI! i v OF
DATE 11 January 1971
1. With the two arrays connected, vibration responses were such that the dynamic loads and deflections seen in the array structure were small (see Table 4. 3.1). The dynamic environment therefore produced low stress levels in the array structure. The stress levels seen in the array structure are discussed in the"Structural Analysis Report" (1). In addition, analysis showed that the reaction loads carried through the LM interface fittings were well under the loads that the fittings were designed to carry (see Table 4. 3. 2 and (1) ) •
2. The analysis also showed that, when the experiment is subjected to its worst vibration environment (launch and boost qualification test), the worst 3 sigma response at a corner reflector mount in the array plane was 40 g. The worst 3 sigma response at a corner reflector mount perpendicular to the array plane was 48 g. Single corner reflector mounts have been subjected to sinusoidal vibration levels in excess of 50 g peak for considerable periods of time (2).
3. Vibration levels on the lower array at thf! attachments of the upper array were calculated both in the array plane and perpendicular to it. Using these calculated vihration levels, test levels for the upper array are proposed which are envelopes of the calculated responses. These envelope test levels are shown in Figures 3.1. 7 and 3. 2. 7 of this report.
4. Analysis of the lower array in the "off loaded condition" showed that acceleration responses on the lower array are reduced by removing the upper array. Thus removing the upper array reduces loads seen in the lower array structure and reduces the acceleration levels seen by the corner reflector mounts.
HO. Rll!V. NO.
9%
LRRR 300 Corn<'r Array Dynamic Analysis PAGE 1 OF 113
DATE 11 January 1971
z. 0 Description of Dynamic Models
2. 1 Dynamic Model Perpendicular to the Array Plane•
The dynamic model of the structun' perpendicular to t:h<' array was a lump('d mass type model. The large array was modeled with 23 lumped lnasst•s and the small array was modeled with 8 lumped masses. Each lump<~d mass point was given one degree of freedom, motion perpendicular to the array plane. With 1 constraint, this gave a total of 30 degrees of freedom for the model. The locations of the lumped mass points on the low<' r array are shown in Figure 2. 1. 1 and on the upper array in Figure 2. 1. 2.
The structure consists of the corner r<~flcctor support cavity sb"udure with a network of cross beams below. The upp<~r array is about half as large as the lower array and is attached over one side of th<~ lower array at four points as shown in figure 2. 1. 3.
The stiffness of the arrays between lumped mass points was evaluated by developing an equivalent beam stiffness for the cavity support structure and the lower ribs. The structure of each array was then modeled as a grid of beams, and a stiffness matrix for each separate array was found using Bendix Program No. 0313, Rigid Grid Analysis _!:>y _!)ire ct ~~~~e_p.J: _Meth_od. The interface fittings, the hinges, and the clevis fittings were assumed to be very stiff compared to the arrays in this direction. The fittings were modeled as rigid links. The stiffness matrices found by this method are listed in Fi gurc 2. l. 4.
The mass matrix for the whole system was then cvaluatNL ed mass points wert~ chosen as shown on Figure 2. 1. 1 and 2. 1. 2. ed at each location is shown in Table 2. l. l.
Thirty one lumpThe mass lump-
Again refering to Figun~s 2.1.1 and 2..1. 2, it is S<'<'n !:hat mass locations 3 and 16 are repeated on both arrays. This bPcaus<' the two arrays art~ connect('d by hinges at these points and the upper and 1 ower arrays havv th<· s aiTit' 1noti on. The clevis locations on both arrays arP also connected, but no mass is lumped at the clevis locations on the upper array.
The stiffness matrix for the upper array was calculated with the four attachments between the arrays fixed in a plane. Since a plane is determined by 3 points, the displac{!ment of any one of the 4 attachment fittings is determined by the other 3. Therefore 30 independent coordinates describe the system. The dependent coordinate chosen here is the displacement of the mass lumped at the rear clevis fitting (shown clotted). The mass matrix for the independent system was calculated
HINGE.
1
6
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RIB LOcATION
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11 Jan
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1971 #936
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>n Weight (Lbs)
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1. 344
2.74 1. 37 2. 714 2. 714 2.74 2.688 2.74 1. 37 2. 714 2. 714 1. 37 2. 74 2.688 2.74 1. 37 2, 714 2, 714 1. 37 I. 37 1. 344
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-" ~---- .......... ··----..... -~.~--.· ·~~-~-:~ ----- -~~:- -------~~--~~.,.,..~--~.~ .. ~,=·-,. ·-~~--r--~---· ~---- --·- ...... -·-·- -·-·- -·~~- -- ~--··."- __ .~ ."_··_·;;-1rfl{i ... is tt
MASS LUMPED FOR LOWER ARRAY
Attached Weight
(Lbs)
.45 + .5
.45+.5+.3
.1
.2 1. 0
1.0
• 2 1. 0
• 375 • 5 +. 3
1. 25
Total Weight
(Lbs)
2.32
2.594
2.84 1. 57 3. 714 2. 714 2.74 2.688 2.74 1. 37 3. 714 2. 714 1. 37 2.74 2.688 2.74 1. 57 3. 714 2. 714 1. 37 1. 745 2.144
2.62
Mass Lumped
A~b~~::~fn In
• 00601
.00672
• 00735 • 00406 . 00962 .00703 .'00709 . 00696 • 00709 . 00354 • 00962 • 00703 • 00354 • 00709 • 00696 .00709 • 00406 .00962 . 00703 • 00354 • 00452 • 00555
• 00678
Table 2.11 Lumped Masses
Remark (For Attached Weight
(1/2) Leg Hinges (1/8) Leveling Leg Asay (1/2) Leg l-Unges (1/S)Leveling Leg Aaay
+(1/2) Tie Pta, Back Rest (1/2) Array Hinges (1/4) Leveling Leg Assy
(1/4) Leveling Leg Assy
(1/2) Array Hinges (1/4) Leveling Leg Auy
(1/2) Pins r. Name P1 Handle & Handling Socket+ (1/2) Tie Pt. Suncompaso Aasy.
Mass Synthesized Joint Mass
Location Numbering
2 23 4 3** 6 24 8 25
10 26 12 27 14 28 16 16** 18 29 20 30
Weight* On Each
Location
1. 5 2.02 2.02 4.55 2.02 2.02 4.ll 2.02 2.02 1. 94
MASS LUMPED FOR UPPER ARRAY
Attached Weight
(Lbs)
.1
.2
.1
.2
.1
.1
Total Weight On Each
Location
Mass Lumped
(Lbs. Sec2) In
1.6 2,22 2.12 4.55 2.02 2.02 4.ll 2.02 2.02 1. 94
• 00414 • 00575 .00549 .Oll78 .00523 • 00523 • 01064 . 00523 • 00523 • 00502
*Weight on each location= 24.2 x~ = 24. 2 x Ai At 237.402
**Mass of stations 3,16 will be added to the stations 3.16 of the Lower Array
***Mass at this location is transformed to locations 3, 16, and 21
Remark (For Attached Weight"
(1/4) Tie Points (1/Z) llingee (1/4) Tie Points
(1/2) Hinges (1/4) Tie Points (1/4)Tie Points
:E~'D WLtiU cr-suOQ
~ (1)
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...... ~ -.ow -J ,_.
LRRR 300 Corner Array Dynamic Analysis
NO.
936
PAGE 10
RI!V. NO.
OJI 113
DATE 11 January 1971
by making use of the fact that both the independent and dependent systems must have the same kinetic energy if all lumped masses have the same velocity in each system.
The only elements in the mass matrix affected by the constraint will involve the coordinates at the attachment points. These elements are calculated below. For the unconstrained system:
Where:
M. = the mass lumped at the th coordinate location, 1
U. = the velocity of the mass lumped at the i th coordinate location 1
( )d = the displacement or velocity of the dependent coordinate
But since the 4 mass points are constrained to move in a plane:
Where the constants are determined by geometry of the plane.
Substituting (2) into (1) givc~s:
K. E.=~ [ ( M 21 + Md) u 212
- 2.796 Mdu 21 u 16
+ ( M 1 6
+ 1. 9 5 44 M d) U 2
2 + 2. 7 9 6 M d U 21 U 3
- 3. 4088 Md U 16 U 3 + ( M 3 + !. 9544 Md ) u 3 21
Expressed in matrix form:
( 1)
(2)
(3)
-l.398Md (M16
+ 1.9544Md) -1.9544Md u16
(4)
1. 398Md - 1. 9544Md (M3 + 1. 9544Md) u 3
LRRR 300 Corner Array Dynamic Analysis
HO.
936
PAGE ll
REV. HO.
OF 113
DATE 11 January 1971
By definition of a mass matrix, the elements of the 3 x 3 matrix in {4) are the non':rivial elements of the mass matrix in the independent coordinah~s. These are placed in the independent 1 [Mj matrix as shown bdow
rM21, 21 M21, 16 M21 3 u 21 '
K. E. = t [ u 21' u 1 6' u 3] M 16, 21 M16, 16 M u (5)
16, 3 16
l M3, 21 M3 16 M3 3 u3 ' '
The remainder of the independent mass matrix is are the masses lumped at the appropriate points. shown in Figure 2. 1. 5.
a diagonal matrix whose elements The independent mass matrix is
Next the two stiffness matric<'s calculated for the large and small arrays were used to calculate a stiffness matrix for the whole system. The stiffness for the upper array was calculated relative to a plane through its four attachments to the lower array. It is necessary to find the stiffness of the whole system relative to ground. To do this, the stiffness matrix for the upper array was transformed using the transformation matrix relating displacements relative to the plane and displacements relative to ground. This was accomplished as follows:
Let: q., {i = 1-22) =displacement of the i th lumped mass point ob. the lower array relative to ground
q,, (i = 23-30) = displacement of the 1. th lumped mass point oh the upper array relative to ground
u .• (i = 23-30) = displac<>ment relative to ground of the i th p6Int on the plane through any 3 of the 4 points attaching the two arrays, u. has the same dimt~nsions relative to the attachment potnts as the i th lumped masH
u,, (i = 23-30) = displacement of the i th lumped mass point oh the upper array relative to the above plane
u., {i = 1-22) = displacement of the i th lumped mass point oh the lower array relative to ground
These coordinate definitions are shown in Figure 2. I. 6.
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11 Jan
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1971 #936
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HO. RI!V. HO.
936
'erospace JStems Divialon
LRRR 300 Corner Array Dynamic AnalysiH PAGI! 16 OF 113
DATI! 11 Januat·y 1971
But since the u ., are displacements of points on a rigid plane, tht>y can be expressed i.n term~l Jr the absolute displacements of the 3 ind<'pendE'nt coordinates determining the plane. In Figure 2. I. 1 above, q
3, q 16, and q
21 wcr<'
chosen as the independent coordinates ddPrmining th{~ plane tnrough tlie attachment fittings. We can then write:
u r 23
u r 24
u r 30
Where:
=
=
[c]
The(C]matrix is determined by the geometry of the lumped mass locations on the plane.
P., (i = 23-30) =the force at the i th coordinate l
q21
ql6 (6)
q-:l J
(7)
[K }= the stiffness matrix calculated for the upper array relative to the plane u through its attachment points.
By definition:
u. = q. u . 1 1 r 1
(i = 23-30) (8)
Substituting (8) into {7),
(9)
To balanc<' thPS<' for<'<'s, fhPr(' nntst b<• equal and opposit<' forcPs on thE:> rigid planP.
LRRR 300 Corner Array Dynamic Analysis
+
Expressing these two sets as one,
p23
[Ku J [Ku] P3o
---
p r 23
p; 23 [-K~ LKu] '-
HO.
PAGE
DATE
q. 1
u . rt
..J
This equation allows calculation of dynamic loads on the st ructurc.
RE!V. HO
936
]7 OF .....ll,;:
11 Janua:ry 1
( 1 0)
( ll)
This set of equations is also used to combine the two stiffness matrices to calculate a stiffness matrix for the whole system.
The coordinate transform can be written using (6)
~ [13] ( 12)
The Ll3 J matrix is shown in Figure 2. 1. 7. It is seen that the [13] n1atrix is simply a unit diagonal matrix with the [C] matrix of equation ( 6) tacked on the bottom.
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LRRR 300 Corner Array Dynamic Analysis
936
PAGI! OP' 113
DATE 11 January 1971
Then the stiffness matrix for tht> whole system can b(' calculatNl:
K s =
[KL] [Ku] 0
0 [-KJ
The lKsl matrix is shown in Figure 2. 1. 8.
[~] ( 13)
With the system completely described, solution of the eigenvalue problem for the system was accomplished to find the mode shapes and frequencies of the model. These are listed in Figure 2. 1. 9. The mode shapes and frequencies were then used to calculate the system's frequency and time response through the method of modal analysis. Viscous modal damping of 10 percent of critical damping was assumed for all modes in calculating rc sponses. Experience with previous retroreflector arrays has shown this assumption to give good agrcpment between analysis and test results.
2. 2 Dynamic Model In Plane of Array
The dynamic model predicting the structure's re sponsc to inputs in the plane of the array is shown in Figure 2. 2. 1. In this model the arrays are assumed to be rigid in comparison to the fittings connecting the experiment to the LM and the fittings connecting the two arrays. Each array is considered a rigid body with 3 degrees of freedom as shown in Figure 2. 2. 1. All flexibility is attributed to the 8 fittings shown as springs in the Figure. The rear LM interface fittings are seen to have been attributed no stiffness in the X coordinat(' direction. This is because they are free to slide over pins in this direction.
Figure 2. 2. 2 shows plan views of each array defining parameters and showing pertinent dimensions. Given this, a stiffness matrix for each array can be evaluated from the definitions of the various influence codficicnts assuming both arrays fixed to ground. Figure 2. z. 3 shows both stiffness matrices. To find a stiffness n1.atrix for the whole system r<>lativ(' to ground the same procedure was followed as was used for the out-of-Plane model.
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EIGE~VALUE= 0.19t51D-C5 FREQUENCY= 0.113530 03
F17qre rZ.I. 9
#OlJE SHAPES I FR£QU£#CI~S MODE 2
EIGENVALUE= 0.551080-06 FREQUENCY= 0.214390 03
MODE 3
EIGENVALUE= 0.286430-06 FREQUENCY= 0.297380 03
. EIGEhVECTOR ·c·--~-- EIGENVECTOR - ---- -- EIGENVECTOR -0.305810-01 -0.178040 00 -0.115130 00
0.232460 CQ 0.100000 01 - _._.. - -0.297980 00
0.13 52 90 00 -0.186540 00 -0.229490 00 o. 761820-01 - ~- -~- --·-· --- -- -0.234060 00 ~---· -- ._.. --~--- ,_,
-0.367360 00 0.111900 00 o. 806310-01 -0.244~10 00 o. 203520 cc -- .......... --- ~-- ~- -o. 26 7440 oo ·- ,_.__,. -····-·'Y·-~
-0.53lS9C 00 0.206530 co -0.5!:>1500-01 -0.48294[. 00 0.248260 00 -•ro • ......... •-·r• • .. -..--.- ~-
0.398960 00 -- _._ .. .,. -~.._... . .,..... ... --
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-0.3762~C 00 0. 227760 oc -O.HS'-180 00 -0.673760 00
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0.334530 cc 0.159020 00 -0.467380 00 ~· 0,.267360 co . -- .... - --~~ - --
0.533690 00 -- -0.323261i 00 0. 3322ED CO -0.194 .. 00 00 -0.600021i 00
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0.258630-0l -·--·. -0.449870 00 o. 3';2430 00 0.260020 00 -0.32838[. 00 .. ~--- . 0.<;85600-01 ...... - ·--·· -~-. ·-- .. -~. - -0.166030 00 -0.453950 00 o. 567130 00 --().347410-01 -o. 30751C oo
0.570880 00 - -- -~ ....... ·----~ -- - -~ --0.101900 00 ---··-"
-0.21357C 00 0.1226l:D 00 0.190460 00 -0.141440 00 0.395100 00 - • - ,..,....- ~- ••• ¥;-- -- •
0.857690-02 -0 .38C62C-Ol 0.812260 oc 0.345050-01 0. 3&113C-01 0.3S5C'30 00 -- . . --· -...... -~ -~- -· o. 32 8680-01 . - ~ - -.- . -
-0.975490-04 0. 512 fJ 70 0 c -0.938230-01 . 0.145890 00 O. S2032D 00 ' • - - - I --
-0.611670-02 ..
0.625390 00 0.565G50 OC -0.109440 00 0.304580 00 0. 947190 oc - -' ... --
-0.159730 00 -0.38358C 00 0.100000 01 -0.194030-01 0.100000 01 "" -~- . -0.~467CO 00
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0.192140 00 . 0.900600 00 0.109210-01 0.473580 00 F -- ~- .. 0.459360 00 - --·-. , ____ -. -..... -
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MODE 4
EIGENVALUE= 0.192320-06 FREQUENCY= 0.362920 03
E I GENVEt TOR -0.149140 00
C.l 000 00 01 · 0.397010 00
-0.144620 00 -0.191930 00
0.264200 00 -0.282700 00 -0.287240 00
O. 74S52D 00 -0.408640-01 -0.440690 00 -o. 4433 6D oo
0.292510 00 -0.300720 00 -o. 50s37o oo
0.395960 00 -o. 842510-0l -0.385790 00 -C.310600 00 -0.656830-01 -0.218280 00 -0.187630 00
0.895830-01 -0.865270-01
o. 311330 00 0.306840 00 0.131340 00 0.302760 00
-0.509210-01 o. 758330-01 .
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lODE 5
:!GENVALUE• 0.129650-06 REQUENCY• 0.442010 C3
:IGENVECTOR 0.276750 cc
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0.17291D 00 0.404160 00 0.243830 oc o. 317300 00
- . ·- .. 0 • 3 8 3 61 D 0 0 0.404410 oc 0.161950 oc 0.22327D 00 o. 342960 co 0. 496990 00
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0.218890-01 0.2801CD 00
-0.144720-Cl -0.3398'+0 00 -o. 211s~o cc
0.252160 00 -0.405210 co -o.Bneo oo -0.355420 00 -o. ll818D-01
0.941220-01 0.410420 00
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-
Fir;. ~./. 9 ( ccm 'f MODE 6
EIGENVALUE= 0.104060-06 fREQUENCY= 0.493330 03
EIGENVECTOR
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O.l 1tl38D 00 0.240490 00 o. 46 092D 00 0.207470 00 0.114470-01 0.328680 00 o. 844590-01
-0.293750-01 0.123430 00 0.103510 00
-o. 22ono-o2 -0.125700 00
. -0.113290 00
o.2938o0-02 -o. 37 5oao-o1 -0.357010 00 -0.111500 00 -O.B6C280-01 -0.832830-01 -0.12 5250 00 -0.177110 00 -0.835800-01
0.1COOOO 01 0.612700 00 0.279240-01
-0.3!t353D 00 -0.415950 00 -0.478130 00 -0.470990 00 -0.472360 00
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MODE 1
EIGENVAlUE= 0.814650-07 FREQUENCY• 0.557610 03
EIGENVECTOR -0.447260-01 -0.402800 00
0.292990 00 -0.190010 00 -o. 2'70 740-01 -0.420270-01 -0.301200 00
o.33751D 00 0.34566D 00
-o. 381180 oo -0.550310-01
0.10000D 01 -0.15234D 00
-o. 353970 oo 0.418920 00 0.14454D 00
-o. 320 l4D oo -0.492830-01
o.55039D 00 -0.152640 00 -0.107350 00 -0.163620-01
0.83573D-01 -0.448430-01
0.310870 00 0.810900 00
-0.40047[1 00 0.159280 00
-0.29S6&D 00 -0.321580 00
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MODE 8
EIGENVALUE= 0.57610D-07 FREQUENCY= 0.663090 03
EIGENVECTOR -0.252800 00 -0.210500 00
0.260780 00 -o. 2 87970 oo -0.104870 00
o. 66089D-Ol -o. 305140 oo
----·- -- 0.100220 00 0.584170 00
-0.145990 00 ..... - - -0.14555D 00
o. 53355D 00
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-0.160340 00 0.176050 00 0.316270 00 0.436830-01 0.420240-02 0.293820 00 o. 8 34 740-01 ().10319D 00 C.55635D-01 0.566230 00 0.686510-01
-0.35l90D 00 -0.857530 00
0.21741D co -0.200470 00
0.212710 00 0.100000 01
4
4
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MODE 9
EIGENVALUE= 0.40305D-C7 FR'fQUENCY= O. 792760 03
EIGENVECTOR . . O.lOOOOD 01
o. 274860 00 -o. 658560-01
0.67274D 00 - 0 • 7 c 8 6 4D- 0 1
0.12293;) (10 0.282200-02
- 0 • 9 5 2 J 50- 0 1 -0.153030-01 -0.255160 00 -0.185920 00
0.104570 00 -0 .. 290820 00 -·o. 334soo oo
; 0. 40 4 5 00- c 1 0.750040-01
-0.352740 00 O.l2CC40 00 o. 839400-01
-0.186140 00 0.250800 00 0.368880 00
-o. so1no- 01 -o. soo 1 oo-o1 -0.180190 00
0.14334D 00 -0.438CSO-Ol
o. 2815 70-01 0.229080 00 0.549950 00
A? .2'.1' (cant) MODE 10
EIGENVALUE= 0.286550-07 FREQUENCY= 0.940200 03
EIGENVECTOR -0.448860 00
0.224050-01 o. 829130-01
-0.188470 00 -0.984850-01
0.110950 00 o. 694640-02
-0.256680 00 0.136080 00 0.129300 00 0.749270-01
-0.382400 00 -0.764320-01
0.186220 00 0.175260 00 0.691590-01
-0.454820 00 0.626410 00
-0.119510 00 -0.835490 00 -0.135110 00
o. 10 0000 01 o. 227330-01
-o. 546710-01 0.670750-01
-0.124600 00 O.H286D-01 0.17 331 D-Ol
-0.101360 00 -0.113290 00
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MODE 11
EIGENVALUE= 0.202550-07 FREQUENCY= 0.111830 04
EIGENVECTOR
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-0.266200 00 -0.54929C-01 0. 849 89C-Ol
-0.93814C-Ol 0.376540 00
-0.185960 00 0.36522C 00 0.279970 00
-0.220990 00 -0.108820 00
0.421980 00 -0.549040 00 -o. 354 73D oo
0.231640-01 -0.13715D 00 -0. 117380-01 -0.321510 00 -0.109210 00 -0.394390 00 -0.296290 00 -0.689030-01 .-o. 363880 oo
0.298600 00 -0.301670-01
0.135120 00 0.25651D 00
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MODE 12
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EIGENVALUE= 0.161680-07 FREQUENCY: 0.125170 04
EIGENVECTOR
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0.624240 00 0.453960-01
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-0.362220-01 -0.116860 00 -0.435020 00 -0.208390 00 -C.l21170 00 -0.134760 00 -0.261260 00
0.105360 00 -C.212600 00
o. 691820-01 c. 234920 00 C.l60770 00 0.224220 00 o. 336320 00 0.282900 00
-0.657590-01 o. 5215&0 00
-0.101170 00 o. 2 74360 00 O. 365 ObO 00 0.166640 00 C.100000 01
-0.1d4390 00 0.418220-01
-0.71911D 00 -o. 887oso oo
O.•H't07D 00
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ODE l3
iGENVALUE: O.l59750~C7
REQUENtY= 0.125920 C4
IGENVECTOR 0. 216180 00
-0.326510-02 -0.793110-02 -0.400290-01
0.921220-01 . -0.215520 00
o. 552960-02 0 • 871970- 01 0.434C60-0l
-0.190760 00 o'.l67320 oo
':" 0 ~ 2 51 8 50 0 c -0.333380-01
0.561230-01 o. 583830-02 0.168660 00 O. 449CW-Cl 0.926280-01
-O.l753CD 00 0.156420 00 o. 1!84980-02 0. 55737D-Ol 0.100000 01
-0.261640 oc -0.188710 00
0.177190-01 o. 285720-01 0.2()8100 00 0.156510 00
•'· ~0.466640 00 -- .. ~- ' . . . ..
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MODE l4
EIGENVALUE= 0.132760-07
FREQUENCY= 0.138130 04
EIGENVECTOR
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... _.... -- ·- --
-0.290000 00 o. 801160-01
-0.347920 00 0.150250 00
-0.374550 00 0.124900 00 0.196990-01
-0.543620 00 -0.738920 00
0.452400 00 -0.361850 00 o. 501390 00 0.144690-01 0.351780-01 {).204900 00
-o. 217030 oo -o. 942900-02 -0.853180-01
0.584280 00 -0.312560 00 -0.614250-01 -0.273570 00
0.10COOO 01 -0.236400 00
0.125190 00 0.209210 00
-0.183860 00 0.163050 00
-0.363580-01 0.363520 00
"
MODE 15
EIGENVALUE= 0.126800-07
~ '· ~·
'
FREQUENCY= 0.141340 04
EIGENVECTOR
~.,
0.411730 00 O.lo336D 00 0.433620 00
-0.141770 00 -0.587300 00 -0.379570-01 -o .406730 oo -0.892170 00
0.193240 00 -C.304100 00
0.439910 00 -0.674580 00 -0.359880 00
0.640530 00 0.1000QO 01
-0.718660-01 0.804000-01 0.627020 00 0.2!;35CO 00 o. 276850 00
-0.378'13[) 00 -0.890320 00 -o. 56H·30 oo -0.834460-01 -0.124240 00 -o.378550 00 -0.303710 00 o. 599590 00
-O.H628D 00 0.175580 00
.. , ...
·-
. ..4..
MODE 16
EIGENVALUE= Oe10913D-07
FREQUE~CY= 0.152350 04
EIGENVECTOR 0.345480 00
-o. 847740-ot 0.157920 00
-0.923910-01 0.404890-01
-0.111510 00 -0.225740 00
0.427090 00 -0.468180 00
C.123660 00 -C.232170 00
0.42927::> 00 c. 884010-01 0.170940 00
-o. 506940 oo -o. 393360 oo
o. 378450 00 -0.169680 00 -0.205050 00
0.375010 00 -o. 507530 oo
G. 869470 00 -0.581000-01 -0.156280 00
0.138320 00 -o. 665330 oo -o. 721950 oo
0.100000 01 -0.53291D 00
0.423610 oo
• .?'-~
~:::::1:1· 1..\lYI» O'I»(JQ
::s ~ t:: N {» . 1-i ...0
'-< 0 .....~ ...0--J..... 1..\l
;;·
ODE 17 MODE 18 FA 2.;.e; cc<MI) ·
tf MODE 19
:IGENVALUE= 0.942480-CB =REQUENCY= 0.163940 04
:IGENVECTOR -0.551000 oc
0 • 4 211 cD- 0 1 0.196030-01 o. 242 890 00 o. 902110-01 0.24965D oc 0.305060 00
-0.16J670 00 o • 2 4 2 4 so- o 1
-0.111520 00 ... O. 2S0390-01 .
-0.125520 oc .J().337C10 00
··~--0.29476[}00 ~~- ~
•. 0.166780 00 0.457730-01
-o. 744310-01 -0.314370-01
EIGENVALUE= 0.775340-08 FREQUENCY= 0.180750 04
EtGENVECTOR -o. 11011 o-02 -o. 22 6100-01
0.536570-01 -o. 7ll59D-03 -o. 68101 D-Ol
0.17 7100 00 -0.162450 00
0.122270 00 0.451250-01
-o. 300840-o2 -o. 79uoo-o3
o. 494150-01 -0.131100 00
0.123250 00 -o. ·7513oo-o1 -o. 82 5940-0l
Oo482010-03 -. 0.576140-01
0.182580 co 0.100000 01
. -0.164370 00 -·>." • ~- __ :~ .. ~:·~-- .: :J 0.152500-0l ..
-0.144740-01 0.258380 00 O. 382330-C2
-o. 7832 5o-ot O. 635520-Cl
-o .127150 oc -o. 530590-Cl
0.57716D-Ol ··· ·. -o. 576220-ol
· -o. 3C68 ao--o1
..
-0.104380-01 -0.439220-01 -0.107580-01 -0.724860 00
0.419660 00 .. -o. 212970 oo -0.427700 00 -0.215780 00
0.100000 01 ·- --- --·- .. -0.13850D-Ol ,"
I
EIGENVALUE= 0.709120-08 FREQUENCY= 0.189000 04
EIGENVECTOR 0.2~170D 00 0.11802[; 00 0.856100-01
-o. 199 590 oo 0.524530 00
-0.990200 00 0.89885D 00
-0.642530 00 0.161490 00
-0.405690-01 0.6~7270-01
-0.440470 00 0.908810 00
-0.757110 00 0.448970 00
-0.623020-02 0.808740-01
-0.376890 00 0.100000 01
-0.406700 00
0.176370 00 0.209730 00
-0.157020 00 0.83367U-02 0.254480 00
-0.464670 00 -0.590590 00
0.122a90 00 o.43a66D oo
...;.0.131000 00
.. .. '":
· ........
MODE 20
EIGENVALUE= 0.578760-08 FREQUENCY= 0.209200 04
EIGENVECTOR -0.393310 00 -o. 309890-01 -o. 80496D oo c. 342880 00
-o. 4827oo-o1 0.316550 00
-0.235380 00 0.311650 00 o. 694070 00
-0.252730 00 -o. 53oooo-o1 -0.115490 00
(,.241630 00 -0.282100 00
o. 723560-Cl 0.916980-01
-o. 663 060-01 0.447400 00
- ·-- · ·"~ · -o. 5 95960 oo
•
-0.389680 00 --~=- ::,:;-;::.-.:. 0.100000 01
-c.. 500770 00 0.603260-01 0.833120 00 C.292290 00
-0.797380 00 -0.961410 00
0.452200 00 -0.157490 00 -OeZ72180 00
• "#= ::: 'U --.o I» VJ t....t ()Q 0' I» ~
::s s:: w .1» 0
"1 0 '<....,_ ..............
-.o ....... -JW ......
!.<
~ri§'t:,''~~~ m~'H~t•~~~~"~"'-~ ,-- =,;;,;;,.;;;,~~~p1i'i:· ~~0.i:i• :.· •_:::~s~r~-·i~~~
JOE 21
iGENVALUE= 0.4S7850-C8 REQUENCY= 0.225560 C4
IGENVECTOR
'-
0.29B30 OC -o .143950-01 · -0.623350 00 -0.:!~1200 co
0.839620-01 o. :>00650 00
-0.327860 00 0.205460 00 0.1GOOOO 01 \,./"" o. 403430-01 0.234540 00
-g. t>91 900 00 O. 5572 BD 00
.-~. 468710 00 0.49o33D 00
-0.988860 00 -0.1<;0130 00 -0.873730-01 -O.l317CO 00
0.359880 00 -0.41~540 00
0.133270 00 0.232790 co
-0.42B30 00 -o. 45 8190-c 1
0.3657CO 00 0.315710 00
-o. 36oozo-o1 -0.146490 00
0.311510 00
P; 2./., (ccrnt!) MODE 22
EIGENVALUE= 0.437240-08 FREQUENCY= 0.240690 04
EIGENVECTOR -o. as 95 9 o oo
0.208680-01 0.140670 00 0.100000 01 ......--
-0. 9b449D-01 -0.243250 00 -0.172610 00
0.101640 00 -0.919340-01 -0.863980 00
o. 49 5030-01 -0.136730 00
0.382320 00 -0.136~70 00
0.184020 00 -o. 246650 oo . 0.830280 00 0.145710-01
-0.251310 00 -o. 96 0690 oo -o. 1113 oo oo
0.478180-01 0.101390-01
-o. 93 0590 oo 0.127090-01. 0.541310-01 0.607o-30 00
-0.176940-01 -0.506120 00
0.830610-01
--... - ,_ -. _... 'O.w_.. ....... ,. ... •,•~- ,-,_.,
. ; .
MODE 23
EIGENVALUE= 0.408890-08 FREQUENCY= 0.248900 04
EIGENVECTOR 0.198210-01 0.276570-01
-o. 256630 oo -0.622070-01
0.209980 00 0.926680-01 0.15647C 00
-0.206550 00 0.100000 011../'
-0.225470 00 0.558210-01 0.639900-01
-0.528840 00 0.500860 00
-0.415910 00 -0.163240 00
0.247300 00 -0.227400 00 0.946410 00
-o. 3ooaao oo 0.662200-01 0.102110 00 0.507090-01 0.6133l:-0-02 0.154710-01
-o. 673810-03 -0.18370C-01
0.231080-01 -0.422320-01
o.21641D-Ol
.. .. "':.
.. ..,...
MODE 24
EIGENVALUE= 0.386240-08 fREQUENCY= 0.256090 04
EIGENVECTOR 0.298100-01
-o. 399240-02 -o. 957550-01 -0.156400-01 -0.122520 00 -0.607440 00
0.557280 00 -o. 30854D oo
0.861640 00 0.147470 00
-0.234210 00 C.100COO 01 V 0.521070-01 0.136210 00
-0.319060 00 -0.1602 O!J 00 -o. 396500 oo
0.493090 00 -0.795720 00
0.3611:>80 00 0.2!36'380 00
-0.265910 00 0.108720-01
-o. 63003D oo 0.618690-01
-0.995290-01 o. 345350 00 0.482960-01
-o. 440460 oo 0.13 7960-02
$
:
~,{.<t> •· ~-- ;.~f '(
"11:::1-'ITf
-r:> ...... ~ 4
~ t...tOQ 0'!»(1)
g w I» ......
~s. ......::::: -r:>~ -.J .,....
::r.~~:'\~'=--~-~'
ODE 25 MODE 26 F~7 c.f.gJ Cconl)
MODE 27
IGENVALUE= 0.373840-(8 REQUE~CY= 0.260300 04
IGENVECTOR 0.32361D 00
-0.791210-02 -0.281390-02 -0.425510 co
0.598250 00 0.584040 00
-0.137910 00 0.224810 00
-6.646320 00 • o-.233260 00 -0.345030 00 -0.645C80 00 -0.262440 00 -0.3045EO 00
0.475700-02 0.355460 00 0.426820-01
EIGENVALUE= 0.308010-08 FREQUENCY= 0.286770 04
EIGENVECTOR -o. 207620 oo -0.102000-02
0.694730-01 o. 33 3730 00
0.22'i760 00 -0.356320 00 -o. 302670 oo
0.338010 00 -0.148440-01
0.534250-02 -o. 361780 oo -0.487350 00
0.402950 00 o. 266560 00
-0.371850 00 -0.11 1860 00 -0.376870 00
EIGENVALUE= 0.286350-08 FREQUENCY= 0.297420 04
-. . - . EIGENVECTOR 0.798280-01 0. 70909[;-02 0.769970-01
-o .156980 oo
. / 0. 100 (100 01 ../ 0. 271260 00 0.183880 00
-0.263050 00 -o. 196 550-01 -0.174030 00 -0.7'i5460 00 0.313340 00
-0.182&50 00 -0.146450 00
0.290120 00 • - ..... -0. 78'i65D 00
0.585990 00 0.100000 01 0.368760 00
v
0.219400 00 -o • 1so 11 o o o
0.403260 00 0.628700 00
-0.561310 00
,.
-g: ~~6~6g g~ tl'- .. »~·--·- .... -0.251510 00 . -·- ··- -····· -o. 3 ss1 80-ot -0.96-6560 00
0.149380 00 -0.375100 00
0.5050•80 00 O.l18C60 00
-o. 821150 oo. -0.155720 00
..
•
-0.277390 00 -0.216330 co
O.l05Y8D-02 o. 78 740D-01
-o. 215850-01 o. 743460-01
-0.301770-01 -0.232600-01 0.10~970 00 0.379910-01. "'
-O.l6791Q. 00 -0.5~4160 00 -o. 187180 oo 0. 163830-01 0.231200 00
-0.532500-01 0.140050 00
-0.971920-01 -o. 439140-0l
0.222l0D 00 0.62908D-Ol
.. .. •.
,.~ .....
MODE 28
EIGENVALUE= 0.232460-08 FREQUENCY• 0.330100 04
EIGENVECTOR -0.987320-01 -0.260110-01 ·~
o. 528280-01 0.281240 00
-0.184040 00 - o. 910660 00 -0.199760 00
0.653090 00 / 0.100000 01 . ./ 0.871410 00
-0.829440 00 -0.838470 00 -0.919350 00 -0.144540 co
0.806920 00 o. 539810-01 0.303510 00
-o. 317230 oo -0.216300 co -0.139190 00 -0.203130-02
0.70430J-Ol -().51 7790-02
0.372360-02 -0.119860-02
.. -0.279560-02 -0.151710-Cl -0.182900-02
O.l3031tD-02 '..:o.~t7605o-oz
·.~
- ~ -;--~~-
~ =~~:: ....... '1:1 ...0 ....... 11> Wt..tOQ 0"11>~
~ ~ l.oJ II> N
"'' 0 '< ..... .............. '-C),_. -J l,.o.) .......
I
i
f
MODE 29
EIGE~VALUE= 0.199520-Ce fREQUENCY= 0.356310 C4
E IGEi..VEC TOR -o .134120 oo
o. 291900-01 -0.707240-01
o.26747D oo· O.lOOCOD 01 V'. -·
-0.148640 00 -0.801400 00 -0.488370 00 -0.435540-Cl
0.486940-01 -0.253220-01
0.260530 00 ~ 0.2128(:0 co • 0 .7474<;0 00
0.424050 oc 0.927380-01
-0.429380 00 -0.757t!80 co
~- . -o. 477680 co 0.172810 00
. ·- ~··· 0.350970 00 0.110920 00
-0.55597D-02 -0.632530-Cl
O.l97C7D-01 -o. 5368BD-Ol 0. 215510-01
O·el5853D- 01 .;.0.70547D-01 -0.264120-01
~
MODE 3 0 r-;? 2· 1. ::J (cG1.)2t)
EIGENVALUE= 0.143370-08 FREQUENCY= 0.420320 04
EIGENVECTOR -o. nsno-ot
0 • 22 505 D-Ol 0.594600-0l 0.208880 00 o. 450620 00
-0.332140 00 -o. 874140 oo -0.47b960 00
O. 24 C07D CO 0.606110 00 O.lCCOOO 01 '· ·' 0.423980 oo·
-0.358140 00 -0.906040 00 -0.~20150 00 -0.109030-01
o. 27 2140 00 0.456210 00 O.ll676u 00
-0.814800-01 -0.139480 00 -o. 72C6BO-o1 -o. 32 40oD-03
0.144870-01 -o. 54739o-o2
0.133050-01 -o. 38 73oo-o2
-0.451300-02 0.179150-01 0.598940-02
; .
,,,~··
... , ·r ~ ;.,...,, 'l'" ·;
. '
.. "';
~, .....
ct>
'-P
'• ... ~.... ...... ..~ ·- .... -~...-~ ....
--~---~~ ... __ _, .... -.--.-. $7~···-··--"""!": !;'!""
. <-- ._,.......,_..~-. ~~- • _<.....,.-
__ ... -··-·_...,.. ....-..-~--~-~-~--~;- "~~ .. ~ .,
_, >'• ·''·-·~~--- ... -.,..-~ -. ........... ::......s. ...... ·-:· ......... ~-
• ~ .. --..-~~!!"?'-9'""'"•_....-.~-""-----c~--'ffi-'Y ~ .. ~.--·_..,.- -·-~ ..... ~~-·
,....,-. :.....:- - ,.__.~'"""'"-·- ---···· .. . ;.,.. -"~--
---,--· _. -• ~r--....-----;-~---
............ _ . ..,_ __ ,. ___ • ..,.,-4,._.,_'"""' ____ ~- ........ - .•.• ~--- -~---- ....
: ~,. .. -.
'*= ..... '"d -.o ..- IU· \.1.) C-t OQ c- IU ~
~ . ~ Vol
IU IJo)
l'f 0 ·'< .....
..... -...0 ..... -J\.1.) .....
I
I
L M INTERF"AC~ Fl TT/NG ST/~AI.CS5
LOW£R ARRAY
UPPER ARRAY
EQUIVALENT Ct. EVIS
S7!FFNESS
/~
IN PLANE DYNAMIC MODEL Figure 2. 2.1
~ ....... 1::1 ::a-Ill "Wy(JQ
a-.. f.J lb ;j ,... w Pi *" ., 0 '< ..... ............ -.!),.... --.] w ......
~ ·- . --· ,f•~'\\ ,1,~"""~_..,........__._,...,.~ .... -.~
SMALL ARRAY I 1<)(2
~.,l--
7.3,
k. YJ..
2179
LARGE ARRAY
5.61-l l<j,t
~
I 0 1&1 I I s
lAB') ~1 1/.98
I<JJ
~- •I l<tJ
l<x3 f-----IU1 13.25 ~
TOP VIEWS OF ARRAYS
Figure 2. 2. 2
==i!::::::1:J ...oyll> VJ '" (JQ 0'.-~
::l VJ
~ \11
~ 0 '< ......
-...0 ...... -JVJ -
KJil
Kxz
KYl
Lower Array
kll
FIGURE: 2 .2,3
STI FFNt.SS MATRICES
.!N PLAN£ MODEL Upper Array
\2 \.KLl = l~l kzz ~'] k;n
LKu] .. lll
kZl
k31 kzz
k13] kzJ
k31 k3Z k33
= 8. 00x105
KYZ = 8. OOxl05
8. 77x105 5
= Ky3 = 9. 08x10 (see Fig. 2. 2, 2)
= 8. 85x105
K Y4 = 9. 53xl05
ku = Kxl + Kxz
l<u = kzl = 0
k31 = kl3 = 2 2
-12. 94 Kx1 + 13.25 Kxz
~<2z - KYl + Ky2 + KY3 + KY4 z 2
k23 = k32 = 10.89 (KYl +KYZ) - ll. 98 (Ky3 +Ky4 l
k33 2 2 2 2 = ll. 98 {Ky
3+ Ky
4) + 10.89 {KYl+Kyzl + 12.94 Kxl+l3. 25 l<xz
l.68rlo6
o -1.99rl07J
[KJ = 0 3. 55xl06 -6,65x10
7
7 7 7. 53:xl05
1. 99xl0 -6. 65x10
KXl =
Kx2 =
KX3 =
[Ku]
k3Z k33
1. 5Zxl04 3
Kyl = l. 76x10
3. 22xl02 4 Ky
2 = 6.17x10 (see Fig. 2. 2. Z)
1. 7lxl04 Ky3
= 3. 61x10 4
~l = ZKXl + KX2 + KX3
kzl \z = o
k22 = 2KY1 + Kyz + Ky3
k32
k33
k31
2 2 2 kz3 = 7. 39 KYl + 10. 89 KYZ - 8.19 KYl alO. 9KY3
2 2 2 10.89{Ky
2+Ky
3) t KYl(7.39 + 8.19)
2. +5. 452 {ZKXl + ~2 + KX 3)
2 2 = ~J = 5, 45 (Kx2 t KxJ) - 5, 45 (2 X Kxl)
(
4. 78xl04 0
0 1. 01x105
-3. BBxlOS 3. 02x106
-3. 88x10J
3, 02x106
1.18xl07
::;;::~~ ....OC-.f~ W~CIQ 0'- ;j ro
s:: w ~ 0'-'i 0 '< 1-+,
.............. ....a ...... -.] w .......
I I LRRR 300 Corner Array
Dynamic Analysis
NO.
PAGE
REV. NO.
936
37 OF 113
DATE 11 January 197]
The coordinate transformation <·rnploy<·d is:
r l
0
!3 0
= -l
I 0
l 0
Where
xy, YL, 8 L L
X ' ru y - ' ru
xz, y ...,, El G 2
0 ru
XL x1
YL yl
OL [!3 J 01
X xz r y y2 r
0 02 ru
0 0 0 0 0
l 0 0 0 0
0 l 0 0 0
0 -7.49 1 0 0
1 -7' 49 0 1 0 - .i
0 -1 0 0 1
Displacements of the lower array relative to ground
Displacements of the upper array relative to the lower array
= Generalized coordinates of the lower array relative to ground
= CX<~I1era1izC'd coorclinates of th<~ upper array relati.vP to ground
The mass matrix is diagonal; made up of the mass or moment of inertia of the appropriate array,
I NO. RI!V. NO.
! LRRR 300 Corner Array Dynamic Analysis PAGE
936
38 OF 113
I DATE 11 January 1971
=
\Vhc rc
0
I 0
I l_
M u
M u
I u
--01555 0
M u
t u
. 1555
150 0 . 07 2(,
. 07 2(,
0 -~. 27
- Mass of the upper array in lb 2
2 -- Mass of the lower array in lb S<'C
= Polar moment of inertia of the lower array> lb sec
2in.
= Polar 1nom0nt of inertia of th<' upper array, 11 2. j_J sec 11:.
The resulting eigenvalue' prob1('m could th<•n b<· solvPd giving th<> modE' shapes and natural frequencies of the systcnJ.. Thvs<· ar<· shown in Figurv 2. 2. 4. Modal ana.lysis was usC>d to calculat<~ rcspontH' of t:h<· syst<'m in tht• freqtH•ncy domain. In calculating rc spans cs, l Oo/o of critical vis co us damping was used on all but modf' 4 which was <:cssigned 20o/o of critical damping. The value of lOo/o has be<'n shown by cxperirnent to be typical of such structur('s. Mode 4 was assigned 20o/o of critical damping since thi.s mode will cause the rear LM interface fittings to slide on their pins and introduce more damping to the structure.
,,_,,_..,
,.., ~~ .
i<ODE 1
EIGENVALUE= O.l4B91D-05 FREQUENCY• 0.130420 03
UG!Ehi'\!ECTDR 0.142400··01
-0.272800-01
I! ODE 5
-o .159190·· 02 0.100000 01 J.l0822D 00
-o. 74693D··02
EIGENVAlUE= 0.36C880-C1 FREQUENCY= 0.837800 03
E IGENV.EC.TOR
~
o. 323630-01 -0.453510 00 -0.556840-01 -0.135140 00
0.100000 01 0.49501[)-01
::.-':~~·<'---:;•· '1"'~~Jr"~~-
_, ·--~""'·""' :.'!>\< .... ~
- .. ,~ ... ··-r-· . ... ~-.~~:!'> ... -.·.- ~ --~
-..&"· : ... .,......,.. ·:.;_-~-,.,-,~-~'4'*"'-----:-- .;- --- .. - ...
, ___ -~r or.-." •
. ..,-.
' .. ,~ ' . ' ·-. ~- . .;\
~ .. ~·-. -~-..-,.. .. ".""':f.~f;"?"".,..-~"'-•. - ... ·-<•·;:;--:·:"":-- ····- ~,., ,. -: -' _.· '"'"'~--, -~~ .... ~....=.~ ~-·'-··<"•l
- .... t-. . --~··, .. - ••. ~ '~· ... ' .......
"" -~·•,
• --~ j.,
\ /clJUi-e 2.2.1
MODE 2
EIGENVALUE• 0.614790-0& FREQUENCY2 0.202980 03
.. E !GEN\IECTOR
MODE 6
0.291050 00 0.185880-0l
-o. 3lB4D-oz -0.279390 00
0.100000 01 -0.424190 00
EIGENVAlUE= 0.974580-08 FREQUENCY= 0.161220 04
EIGENVECTOR 0.279790 00 0.100000 01
-0.927260-01 0.353710-0l
-0.126110 00 o. 635890-02
HODE 3
EIGENVALUE= 0.155360-06 FREQUENCY• 0.403790 03
EIGENVECTOR 0.234270 00 0.504050 00 0.579021}-01 -0.466940~01
·=••·v>•<O~ •A •<·~··~-.,~·- OolOOOOD Ol
0 .. 525260-0l
~
MODE 4
EIGE~VALUE= C.8J034D-D7 FREQUENC~~ Oo552320 03
E i!GfNVEG Tmt O.lOOCOD Ol
-0.261260 00 0.433440-02
-0.553880-02 -D.309ZBD 00
0Gl39150-0l
'
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NO. RI!V. NO.
LRRR 300 Corner Array Dynamic Analysis PAGI! 4Q OF 113
DATE 11 January 197
2." 3 Out-of-Plane Modd of Off-Loaded ConcHiion
To model the experiment with the upper array rt•moved, a lumped mass rnodcl was again ernployed. Masses W<'r<' lumped at: 23 locations as shown in Figure 2. 3. l. These are the same locations as used forth<' modt·l of the lower array with the arrays conn<~cit•d. The Rtiffn<·ss matrix calculated for t.h(~ lower array alone could then be employed di n·ctl y. Since the attachments art' no longer constrained to move in a plane by the upper array, the degree of freedom eliminated by the constraint can no longer be eliminait•d. The mass matrix becomes a diagonal matrix whose elements are simply the mass lumped at each· node, Mass and stiffness matrices for this model arc shown in Figure 2. 3. 2.
Solving the eigenvalue problem again gave mode shapes and frequencies which were used to calculate the frequency response of the system. The mode shapes and frequencies arc shown in Figure 2. 3. 3. Again lOo/o of critical damping was used in calculating the system 1 s response.
2. 4 Vibration Environment
The qualification and acceptance vibration environments to be imposed on the LRRH 300 corner array in tests is that of the LM quadrant III. This then is the environrnent that was input to the base of the three models. The environment was taken from Exhibit B-l
1Design and Performance Specification for the Laser Ranging
H.etro-Reflector Experiment (300 Corner Array)lNovember 1, 1970. This environment is listed in Table 2. 4. l and the qualification levels shown in Figure 2. 4. 1.
The responses that were calculated here are responses to the qualification environment. Since the acceptance environment is less severe, the qualification environm.ent then determines the dynarnic loads on the structure.
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- . ~· -- . '---
-O.l96200CE 05-C.LtE750COE C6-0.4874000E 06-0.2056000E 05-0.7152000E 05 0.5179000E 06-0.71550COE 05
ROW 23-0.l595000E 04-0.l1550COE 04 O.l4640COE 04-0.l942000E 04 0.779lOOOE 04-0.2021000E 05 0.52leOCCE 04-0.l32'tOOOE 05 -o.UHOOOE 05 0.2423000E 04-0.3472000E 05 0.9818000E 05-0.l30d000E .05-0.l6llOOOE 05 0.11640COE 06 Ool205000E 06
~--~ .. - --·.
O. 4314000E 04 ,_0. ~~260GO_E_ 05-:-_o_. 5l270()0E 06-~._2406000E _05 _ O.J709000E 05_.-o. Jl550()0E 05 _9• 53650001; 06 ________ , . ___ , ____ . __ ,.,.._ ___ __.,__~-- ----. . ··- .... -.:.~-" ~ .
------ ------------------· ----------- --~--- -------- . --- -- ----·--
----------------------------- ---------- ---------------------- .. ---
- - ____________ .. _____ --- ------· ----~ -·· --- -· ~~
··-- -- ------.....,...- ---- "';;"'-! --
-----...----... --------------- ---""'~-,._-----~------~ ·--- ---·-
' "*= J;;._1:J ... 'l:) I» - - ···~ W·y OQ --- -- ------~-~==-~=~~---~:~~ -~-- ~~ ~-2'1 ;
-- ------------~----- ------- ·-- -------- --
-----:-·---- ------·· ·----~~ ·-~---- ------- -· -·-- -·-------------------------· ----- --·-----·-----· -- -----
DE l
IGENVM .. UE= 0.836740-C~
l!"QUENCV= 0.113990 03
lGENVf.CTOR -0.902040-0l -·0.137620 00
0. 715840 00 0.95291D-Ol o. 668160-0l 0.226290 00 0. 217880 00 o. 337130 00 0.551690 00 0.213280 00 0. 4516 CD 00 o. 587760 00 Q. 6203 80 00 0. 427980 00
; 0. 705390 00 0. 6223 50 00 0.118090 00 0.745940 00 0.799650 00 o. 264680 00 0.4990c;() 00
-·------- -- - 0.100000 01 . 0.506390 00
-~
r• "04
F;'JCire 2.3.3
MOD E. SHAPES f FRE:QUENClES tmDE 2
EIGENVALUE• D.5L34BD-D& FREQUENCY• 0.2221JD DJ
EH:OEI\PIECH.IR -------·-·- -0.2:)1)()41) 00
-0.250660 00 0.100000 Dl -0.208~60 00 -0.308480 00
0.182630-01 -0.354't80 00 ~0.173720 00
0.295760 00 -0.409200 00 -0.287150 00 0.241220-02 0.43156[) 00
-0.318310 00 -0.166070 00
0.115170 00 -0.200550 00 ..:.0.222420 00 -0.926350-0l
0.136650 00 -0.106110 0() -0.189910 00 -0.8~8370-01
..
1400E 3
EIGENVALUE= 0,163890-06 FREQUENCY= 0.393140 03
EIGENVECTOR
·!., t-.
0. 311090-03 -0.911580-01
0.948920 00 0.502340 00 0.166560 00
-0.523310-0l 0.592230 00 0. 524490-01
-0.179520 00 0.100000 01 0.324100 00
-0.200820 00 -0.457530 00
0.548200 00 -0.672800-01 -0.428320 00
0.500210 00 0.899430-02
-0.360430 00 -0.314940 00 -0.123520 00 -0.391330 00 -o. 272410 oo
_.,. ____ ..,___--~-~ --~"· -~~-·~--.__, -~ _ . ..,_ ____ ..,._-......
.. . '- ..
.. "';
'
MODE 4
EIGENVALUE• 0.139640-06 FRE~UENCY• 0.425900 03
E!GENVEC TOR ··O.t~QB5D 00 ··0.332940 00
0.175240 00 -0.485120-01 -o. 560150 oo -o. 395330 oo -o. 30401+D oo -o. 5 azz1.o oo ~0.693180 00 -o. 3363 ao-01 - o. 2 80490 00 -0.596620 00 - o. 902 000 00
0.632160-01 -o. 796120-0l -o. 555960 oo
0.148050-01 0.468290 00 O.Z06770 00
-0.470650 00 -- ---· - -·----- 0.551430 00
---·-· -------- 0.100000 01 O.'t55't8D 00
. -.,.----..
. .
'if,
. ..
:::;;::~-''1:!4' -.o ...... lll-w t:....t OQ -
"' Ill ~ s ~ Ill "' ~ 0
·'< ..... ,_. ,_. -.o ,_.
''-.J w ......
0 DE 5 MODE 6 Ajttre R.]. 3 (cern-!)
MODE 1
IGENVALUE~ 0.610940-01 ~f.QUENClf::: 0.643910 C3
EIGENVALUE• 0.445040-07 .fRH1UENC1i"' 0.15'>r+3D !Bl
IGENVECJOR EIGEN\i'EClOR -0.366120 00 -0.234320 00 -0.416740 00
0.477440 00 -0.418940 00 -0.134810 00
0.151060 00 -0.461560 00 0. 231130 00 0.992200 00
-o. 255690 o o -0.207050 00
0.100000 01 0.311150 00
-0.326630 00 '0.338420 00
, .. ·"0.521920 00 -o. 591860-01 -o. 643240-01
0.550540 00
----·- ·-----y~ --·<>-- ,_
•,
0.194520-01 0.259980-01 -- .. ·--- ·---0.675080-0Z"
- ( ,;_J
o.1ooooo en 0. 70&0'1\J OIJ 0.366830 00 0.227020 00 o. 575690 00
~O.Zl5550 00 0.193440 00
-0.288850 00 -0.118320 00
0.281250 00 -0.421260 00 -0.364380 00
0.361680 00 -0.259760 00 -O.ft24800 00
0.110620 00 o. 571950-01
-0.235620 00 0.227300 00 0.217970 00
0.130720-01 0.532100 00 0.554920 00
._
....... ~ .. -~_ ... _,._ -.................... __ .. .,...,. ...... ,.,.. ...... ____. .:•.
..
.. ~
EIGENVAlUE= 0.289680-01 fREQUENCY= 0.935110 03
EI GEN\1 ECTOR
.;,
0.424010 00 0.304570 00 0 ollll40-02
-0.189370 00 0.103960 00 0.669650-01
-::!.241470 00 -0.823060-01
0.245280 00 -0.165570 00 -0.168030 00 -o. 873360-0l
0.384830 00 0.159880 00
-0.136620 00 -0.166580 00
0.2l293D-01 0.603880 00
-0.580480 00 0.117070 00
0.100000 01 0.206620 00
-0.9.57960 00
( .
" '
.- "':
MODE a
EIGENVALUE• 0.146670-07 fRHlUEi\lCV"' O.Dl.A2D 04
E !GEN'IIEC TOR -0.242210 00 -0.163090 00
0.106510 00 -o. 221480 oo
o. 239960 00 -~~.~-~----- .. ·-0.741620 00
0.478130 00 -0.243510 00 ~c. a4sozo oo -0.918140 00
·-------·· --... 0.706500 00
-o. 714870 oo --- ·--·-- 0.100000 01
..
0.367590-01 0.127540 00 Oo315130 00
-o. 724660 oo 0.213260 00
-0.813390-0l 0.959900 00
-0.923170-01 -0.216290 00 -0.161430 00 ~
"'
.,.
_,;
-- - -··-•n~ . ~. "",..-~.,...::>"llll
., "* ::::: 1:1 '-'> IU -VJ '-! (JQ 0' IU ~
::s c >IlL .su -.J 11
'< 0 . ....,_
-~ := -.J "-'> -
I
z. 3.:5 ( cc-'711) /t'IODE l.l DE 9 1·1DDE 10
GEN\f UE= 0.131780-01 __ EIGENVAlUE~ D.lZ370D-D7 FREQUENCY= O.l431JD 04. EQU(;:"l::::V~~ 0.138640 0~
>ENVl:tWR !:H>EMI!ECHJR ··0. 364100 00 f)" 558850 00 -0. 192830 00 -~ -~--- . _ _ __ 0. 23fH& D 00 -0.165490-01 ,,_, .. , Gol7493[) 00
0.100000 01 ------------ 0·'•47030-0l., 0.712930~01 --- --- -~--" -0.216911) 00 0.120370-0I --------- ·-------- ___ ~0.532520 00 0.698960 00 __ .... ~· _ .. " ~O.<t0584D 00 0.314210-02 _________ .. __ _ _ -0.1•26500 00
-o. l4-574u- 01 _. _ ... __ -o. BZ736D oo 0.216450 00 - -- ~- ---- -- 0.448430 00
-0.181900 00 ' --~ -0.331940 00 0.208880 00 ---~- - --- - 0.506270 00
-0.280110 00 """'"'"- -- -0.771500 00 -0.843260 00 ---- -- ---- ~-- o. 965360-01
0.473370-0l ---~---- --- ' - . 0.640640 00 0.136080 00 --~--~~-~----·- 0.100000 01
-o. 774340 oo _ . ________ . o.4z4soo oo -0.124620 00 ---------------- 0.516580-01
Oo'!l333D-Ol _ ~" __ --· --~-- • .. 0.641.760 00
EIGENVALUE• 0.944010-08 fREQUENCY= 0.163810 04
f.lGfNIJECHJR -0.856200 00 -0.131340 00
0 .1 1<3'•50 00 -0.803430 00
0.456170 00 0.267770 00
-0.193210 00 0.100000 01
-0.746360 00 0.113490 00
-0.963230-01 o. 571090-01
-0.392320 00 0.839410-01
-0.908400 00 0.608160 00 0.924740 00
-0.465180 00 -0.58334D-Ol
MODE 12.
EIGENVALUE= 0.829910-0B FRE~UEN:1~ 0.170700 04
__ EIGENVftiOR -0.197060 00
0.112800 00 O. '•03 SOD- 01 o. 3986'•0-01 0.507i.JOD-Ol o. 214030-·01
-0.111230-CJl 0.113920 00
-·0.155420 00 0. 53844lJ-Ol
-0.865600-01 0.800840-0l
-0.126470 00 ---- -~---·----··- -0.518240-0l
-- - ' - ' --- - - - - - 0. 8 2 6 2 4 0- 0 1 0.160270 00 0.186620-0l 0.534850-01
-o. a5352D-ol -0.117190 00 ----- --·------- ~---- . 0.182 77[) 00 0.723240 00 ------------------ 0.155030 00 a.tta7750 oo .. -·-· ...... _._ ~ ..... ___ -O.lZ301D oo o. 340010 00_ ~--------------- =0.531480 00 --------- -~- - ----
-0.51.5200 00 -- -·· -- _ -· --~- ---~ __ -0., 8Ul9D 00.
~~--7""-~~---~---------------- - ---· -., -l .• ·,
~
J
- ... .t_ ...
-·- ----..:.-----·-"-..-<r·---···- ..,., __ ._~-~~'-.-~----·--- __ .._._ ~
_.
0.227590 00 0.942120 00
-0.~63160.00
·~ .
' . .. "':
~~
; +\
0.100000 01 _-0.721850 00
o.sso1so oo
..
;.
.. ·
. ~
~
--,
. ..
-. .... ..,
- -~:;
.... ----- ..,..._~
"*' ..... 1:! ...0 ..... Ill VJ C-j (JQ 0' Ill ~
::s .;.. ~
-Ill ())
'"I 0 '< ......
. ..... ..... ...0 ..... -JVJ .....
(' ,_
::
"'
li]i:: ,!3 r:'l!JE 14 ;:::(C!/I~(P 9, ~=\-;( ff'/"AA.,J.f
,]'..,... r .... ._.. .,.,r-.._.1 ...,. v~,l')'i'"f ~ .J ' 'NODE 15 I10DE !S
:s~tJV~LU~= OQ80027D-CE E~S~NV;Lu~= 0.552 D·-03 fX·:;ENV;!JJJE:; 0~a'~'2,70:>0=08 ··- .._ E.1;;tl·~":JtiL~;.E~ OQ3~(13·:c
~f.OUt:tJC't~ 0" 177910 04 _______ ·--~Y.f~t::Ot5l: "'if:-;;; O~ ~1!~!.30 !)
GfNVtC"{QR TOR ·0.784580 GO o. 100000 01
0~175690 00 0.299~00 00
0. 750560- 01 -~-=----~~----···· 0. H62 920"0 1 0.161490-02 0.100000 01
"'~-~-~-- o. za3o aJ- 01 ~~---:-.:~~.-~--~:· ::-·-··- ... r_~. '.i6Ul n oo -0.712010-0l 0.471150 00 -o. Z400 m-ol ~~-~:~-~-=~--~~:-~--~-~- ~o. 400490 oo -0.151220 00 0.444270 00
-----·~- · o. 9 4ll3D- o1 ·----~-~--=~~:~~---- · -o. 351~ no oo ·--~-- ... -0.652670-02 ·-----·--------~· .. 0.110640 00
-0.253450-0l.. .-.. -0.528't'iD-Ol -0.839150-02 0.48938D~Ol
~- -·--···- 0.474490-01 --- ---: ~.-.::_ -0.315320 00
- -·--· -· o. 266060-01 ----- ............ -. 0.232830 00 0.128410 00 . ...-~~ .• ~ -0.234100 00
-0.624320-01 0.115320 00 -- ~-- · ·- -o. 86Z62D-02 ·~~~-:-:·. - ·- · _ . -o. B&B56D oo
Q.41749D-Ol 0.3&2010 00
0.572940-01 -0.133400 00 -0.205210 00
0.680910-02 ,. -0.769030-01··-------·-·
.. . .
-0.429250 00 0.953500 00
-0.596080 00 0.830930-01 0.223970 00
. ~
FRECUE~C1u 0.243550 C~
E~ GEN\1 ~C10fo\ ··C!. 156260~01
r-tif:Qi)[NC lf;:• Oo 2 53
r ltl("•r-"[~;..e:·v J?"r'"'O _ ---~--·-- ~lt;~i!:.~'4,1C\_., :UP.
-0<05183lf0 <JO <·0.631 f2D 00 ~o.:n 791D .. D:t
_____ . ..a. 2124'/D=O 1 o. 293430-CH 0.227650 00 09275460-0l
_ ...... _ .. "------- __ , _____ '"O. 2 096 OD U 0
~o. 32ft21D·-Ol -0.524420 00
-··- ... - -·-· o. 587800 00
- ...........
·~ ..
-0.500170 00 o. 716630 00
-o. sso1so-o1 -o. ee 2010-01
0.998900 00 -o .877340 oo
0.100000 Ol -0.882180 00
0.79U80 00 -o. 28 5620-ot
-o. 511530-0l 0.631700 00 0.139970-0l 0.215711>-01 0.1311t6D-Ol
' '
0,908920 00 -0.130'/20 OG
0.176300 00 ----------- ~0.176430 00
O.td999D ,01 .. ------ -0.395670<•01
-0.9!3889[) 00 0.204460 00
-O.U33HO 00 ---.~-~. -0.102760 00
O.ll2'J5D 00 0.-697860-0l 0.166760 00 0.100000 Ol
-0.296920 00 0.159540 00
-0.993310 00 -o. 360490 oo
0 .. 226130 00
:f;~ •
-··---.. ~--' ............. >~~-..... ~-~~--~ •. · ... -,_ ___ ... _,.. __ _ .... - ....... :,_ ·~- --.. ,...
.. ~ .
"':
:;..
. ., ..__.,..
>il.
-----> - .~- ,. .·· ~"1
-~--·--·-,:r
.. A
~~:;
l
r•- -- ·~·-,..-- • ~
•
-.---- -·--~~ ~ .-..r·· ·~
. ..
.,. _,.;'/,
==!l::t='"O '-0 r • Ill .,. (.,.) -.~ 0' Ill ~.
~ ~ ~
Ill "' l'f .-..:: 2, .......... "' ~ -.J (,» .....
·1
ODE 11 ;clQDE 8 .:?."3~3 (
IGEN1ALUE3 0.364770-0E {EQUENtt~ 0.263520 a•
H;tEN'HCTGR 0.77'1140-0l 0.949950-0l 0. 1.09720- Ol
-0.436610 00 ~ 0 • 16 52 ltD 0 0
0.192€>40 00 0.389830 00
-0.326330 00 0.195580 00
-o .62514D-Ol 0.693090-0l 0.551550-01
-O.b20C70 00 -0.222020 00
0.161150 00 -o .1o039o oo
0.300400 00 -0.133700-01
~-----·--"- -<1.173350 00
0'.100000 01 .,. _____ -- _ ':..o .. 92712D-o2 --------
ACi~1~1ALUE= 003119~0-03 FREQU~~CY= 0.28~010 D~
K~KG:;fJ\?EC lOR --o~ 233420·-~o;~
··0. 26 7!190··02 0.166&30-02
-(),fl49350 00 --Oo 350240-0J
Oo27617D-Ol -o. 322920·-oz
0.36361D-Ol -o. 364390-0l
0.100000 Ol ~0.433160~01
-o. 5o466o--oz 0.615870-0l 0.191900·-01 O.BZJOD-01
-o. 533220-ol -0.869580 00 -0.151800-01
0.424150-0l 0.166630-0l O.l20 1t40-0l
o.ao5700-01 0.606040-0l --~~-~~=---~ -- -0.117540-01
_ -;0 ellt,23D-tll •.
1
·.f ; • ,11'
'·
·j.-;_ ,.4.
_. __ ,.,..._·- .. -~.~· --··· ~~·----........ -~,.--~ .. -
"
19
IGC~V~lUEz 0.1CJS2C·OB ~~CVm .2oa~s0 04
EKGEf~4">} CC~fOR
-0, U269D 00 =0 ~ 150~180 00 -Cl.ll998D~Ol
0.255210 00 0.2976!10 00 0.7l 6d8D 00
-Co 198720 {)Q
-o. 90 5lso--o 1 0.183490 00
-O.l.l964D 00 -o .1snso oo -0.797530 00 -0.234520 00 0. 299450 00 0.804520-01
-0.261710 00 -o. 11 2140 oo -0.687520-01
0.100000 01 0.569650 00 0.174610 00
;·lODE ZO
f:.lGE~-!V.0ttUE':"~ 0~276t::;-JD,·OS
fnEQUEN:V= 0.302~?0 04
~ !GIENV>:C !\JR 0.212650 00 0.2377<10 00 0.295930-01
-·0.510lt60 00 ~0.535:.50 00
O.lOOOOD 01 0.&47810 00 0.209310 00
-0.461190 00 -o. 820l8D-02 -0.266870-0L -o. 5707oo oo
0.47?850 00 -0.&33240 00 -0.1 993 50 00
0.47lt290 00 0.508510 00 o. 570310 00 0.127680 00
- o. 976()60 00 -o. 339390 oo
-0.373510 00 --~----.,J"'~-- -0.290250 00
-~ :t.
-0.358090 00 "- ·.
- - ---- -- - --- -~------ --- -----~-. ..._.-.. ----~--- - ~
-. { ~
'
" . '":
........
-I).Z32l90-0l
.. ·
·fj
--~~ -· ..• ·3 -~·. -l "· ~ ... ,.~-~
... ...
-:----:-.: ~
i
";
~
:::;;:: :.::: 1:1 .. "",.Ill 1./..l -. ()Q o--. Ill ro
::1 ~ \Jt Ill 0 '"f 0
'<!......_ ...... _ ""-JW ......
'-"
I:P,Ol!f: ?l MODE 22 T-1JUVC 2.3.3 (c~~J
;;l':CC1'JCH.llE"' 0.231090-CE FREDUCNCY3 0.331080 04
!~G,::N\/t:GHJR
r
0.100700 00 0.111050 00 0.399970-01
-0.213350 oc -o. 37ono oo
0.290100 00 0.100000 01 0.173680 00
-0.759760 00 -0.932380 00 -o. 89lt6 so oo
0.901020 00 0.940500 00 0.912570 00 0.167590 00
-o .a2111D oo
-o. 253500 oo -0.343470 00
0.288060 00
EIGENVALUE~ D.l95BBD-DB fREQUENCY: 0.359610 04
f:!GENVECTOR -0.147260 00 -0.155HD 00
O.l:i564G-Ol. 0.129750 00 0.412790 00 0.100000 01
-o. 268090 oo . -0.350380 00
-0.451:>3&0 00 -o. esa4SO-o2
0.307510-01 -0.644420-01
0.217330 00 0.284540 00 0.823270 00 0.459430 00
-0.135480 00 -0.473790 00 -0.805500 00
0.166000 00 .. -- -· -- --- . -o. 41l10D oo
;.·
0.156330 00 0. 362600-01 -~
_-0.628060-0L_ ..
,<
0.192560 00 0.389600 00 0.117520 00
-~------------- ..
·~~- ---... ~~~ -... -- ~--......·.....,._, ___ .,~
..
-" .
MODE 23
EIGENVALUE= 0.14260~-08 FREQUENCY• 0.421470 04
El GEf\\IJ ECTOR -0.670700-0l -0.689970-01
D.l6648D-Ol. 0. 843490-01 o. 256190 00 0.46'+350 00
-0.351610 00 -o. 90 3560 oo -o. ·H S490 oo
0.228390 00 0.604630 00 0.100000 01 0.422080 00
-0.365030 00 -0.899930 00 -0.515740 00
0.923310-01 0 0 274830 00 0.450590 00 0.114860 00
-0.82346D-Ol -0.158030 00 -0.69862D-01
~:(
~ ,...,. , -·.
··- .----~ ----~-
. '
--:
'!.""'
IS
~-l'
.-
:::;;::~1::1 '>D llJ.' WY(]Q 0' Ill ~
::l ~ (J1 Ill ,_. '"I 0 '< ...... ,__. ,_. '>D ,_. -,Jt.# ,__.
Table 2. 4.1
Quad III LM-10 Vibr2.tion Environment Qualification Levels
l. Launch and Boost Phases
a. Random (all axes -duration 1. 0 min/axis)
20-40 hz 40-85 85-110
110-400 400-450 450-1100
1100-2000
b. Sinusoidal Sweep (all axes)
5-12 12-100
Sweep: 5-100-5 hz Rate: 3 oct/min
+12 db/oct 0. 030 g 2/hz +6 db/oct 0. 050 g 2 /hz -6 db/oct 0. 040 g 2 /hz -12 db /oct
. 2 in. da 1. 4 g. peak
c. Sinusoidal Dwell (all axes -duration 10 sec) 6 hz 1. 5 g-peak
2. Lunar Descent Phase
a. Random (all axes -dur;J.tion 12. 5 min)
20-2000 2
0. 005 g /hz
Page 52 of 113 11 January 1971 /!936
NO'l'E! Levels specified are to the GAC interface points in the LM.
,.
.:I.
FiJure 2.1./ QUADDl SINUSOIDAL £NVIRONM£NT
/0 r CI.O I
··I ! I I a.ol
~ I ·~ ~ \!) 1.0
""' :'(: () .c j:;: I
~ I ~ L•J 1
~ . I I
2
·' ~----~~~----~«0------~~~----~~~~~----~ FRE.Q.Ut:NCY fH4'}
OUAD Iff LfB RANDOM ENVIRONMENT
I I I
¢Hl
I
I I I I I I I I I I 1 I I
65 110
I
.Jo I 1100
· 0001~~--~l~o--~z~'o----~~--~lo~o~z~o~o--~soo~~l~on~~o~z~~~~s~~o Frequency (Hz)
Page 53 of 113 11 January 1971 #936
3. 0 Cornputed Responses
LRRR 300 Corner Array Dynamic Analysis
3. 1 Responses of Out of Plane Model of Both Arrays
HO. RIV. HO.
936
PAGI 54 OF 113
DATI 11 January 1971
The response at each lumped mass location of the out-of-plane model of both arrays was calculated to the sine environment and to the random environment. Transmissibility, sine response, and random response at fiV(' location on the experiment are shown in Figure 3. l. 1 through Figure 3. 1. 5. The random re~ sponses shown are responses to the ''Launch & Boost Environment". The "Lunar Desccnt 11 rand01n environment gave responses that were in all cases less severe than responses to the L&B environment. Out-of""Plane responses are summarized in Table 3. L L
The most severe response was found to be at the center of the right edge of srnaH array, lum.ped mass location 27 of Figure 2. 1. 2. The response to the
:..,&B randorn excitation at this location is s4own in Figure 3. 1. 6. This response has a root mean square value of 15. 87 g, giving a 3-sigma peak value of 47. 6 g. Thus, the greatest acceleration perpendicular to the plane of the arrays likely to he seen by a corner reflector mount is 47.6 g. The greatest power spectral density of this response is in the frequency range between 100 and 400Hz. If the corner :mount design can survive sinusoidal vibration well in excess of 47 g peak perpendicular to the array face, then the corner mount design is of sufficient strength in this direction.
Response at the 3 independent attachment points between the arrays deternJ.inc the vibration levels seer: by the upper array. An envelope enclosing the responses at all the attachrrwnts could then be used for testing the upper array by itself. The three responses were superimposed and a suitable envelope) reduced. by a f2ctor of 1. 69 to af,:ceptance level)is shown in Figure 3.1. 7.
3. 2 Pcesponse of the In Plane Model
Responses in the six coordinate directions of the in plane model (coordinate cErection.s shown in Figure 2. 2. 1} are shown in Figure 3. 2. 1 through 3. 2. 6. Response of the angular coordinates e
1 and 0 , are given in 11pseudo g" units
of rad , _l_ . These responses are su~marized in Table 3. 2.1 2 386
sec
The worst vibra":ion level seen in·plane by a corner mount will occur along the edge of one of the two arrays. The response will be the sum of translational tnsponsc of the center of gravity plus thf~ rotational r<'spons~· about the c. g. times ~he distanct' to the <·dg<~ from th<' c. g .. Th<· max·imum r<·spons<·H a.f corn<·r loca-, ions on oo'Lh arra;,rs to <'Xl·itation in th<· X and y dirt·<·tions ar<· Hhown in Tablt• 3. 2. 2.
Page 55 of 113 11 January 1971 #936
FREQUENCY RESPONSE OF LR3 300, 30 D.F.M~DEL
,, FIGURE 3.1./A TRANSMISSIBILITY
LOCATION 2
'1
~~----------~··~~----------------~--------------0')-co-;: f"-'1 W-j L.'?-1
I :::1'-'l
[ (lil I f
(~l
>-a I ~-~-~~--------------+---------------~--~----------~ e-..g m-' _l
r.--;
co -U1 :::r-: U1 ! I-I ('!/).., ::E: ! Ul ru-l :z l 0: i a:QCJ i i- ..-~_l:.,___......,........:::==----~~---l---4r-------+------------l
m-l oo-j ~'--1 tO-'
\ l.r)-1
' ::!'-~ 1
2 3 1.1 5 6 789
FREQUENCY CHZl o'.
Page 56 of 113 ll January 1971
#936
FREQUENCY RESP~NSE ~F LR3 300, 30 D.F.MODfl
FIGURE 3.1.1 B SINE RESP~NSE
:t .;')-! ,..J
~ -i I
;-j ;
I
1 l
, I -4
-l
D-j ., -i
7'~ ;;::: w
1 Ulcr> z
!-! 8 a....(\. en Wo a::O
...-t
I I
_v -1 -!
l _, o-1 j ' I
-! i
~
1 a•
L~CRTI~N 2
_/
2 3 y 5 6 789t cf 2 3 4 56 789 1cf 2 3 4 56 7891 o" FREQUENCY rHZl
Page 57 of 113 11 January 1971
FA E QUE N C Y RES P Cl N S E CJ F LA 3 3 0 6:36 3 0 0 • F • M CJ 0 E L
F I G U R E 3. 1.1 C RRNDCJM VIBARTICJN SPECTRUM
LCJCRTICJN 2
~~--------------~--------------~~------------~ 3 ~ -1
.i }
'~ rue!
5. 6063 G-RMS
% ~.~~ -----------------4----------------~------------~ l.~ -:i Q :l
-1 ~
>- -l !: ~ (f) ! z -;1
W l
...... ! Dbj
' ~~·----------------·-4+4----~+-------4---------------~ _j - -~
0: j rr.. -l r- -~
' u ,, -1
w_~ 1
n -~ (__ i (il i
_,
CCN : W 1 i .,..o: c=; ~.:-----1------+---=--------+--+-4+-----------i f.L J
_:j
=u:~.~ _, !
-!
2 3 ij 5 6 789
FREQUENCY (HZJ o"
Page 58 of 113 11 January 1971
#936
FREQUENCY RESf~NSE ~F LR3 300, 30 D.F.M~DEL
FIGURE 3.1.2.. A TRANSMISSIBILITY
b gB r~...J
....
~ W-i If) -I --' -~ ('f)
b I 2
LCJCRTICJN 9
3 ~ 5 6 7 a~ 11 cf 2 g ~ 5 6 7 891 03
FREQUENCY (HZl
I 2
Page 59 of 113 11 January 1971 #936
FREQUENCY RESP~NSE ~F LR3 300, 30 D.F.MODEL
FIGURE 3.1. 2. /3 SINE RESP~NSE
0') ro r- .
~~ :::f'l ~~ ruJ
Ul 0 ....... 0')-oo-
~~ w en z: e:::::; (.L U1 We a:
I ~-i
I !
ru-J I
ol/ l .,....... ;: .. ,-} 01j co-! f""--1
j w-, L0~i
~-;
:j b
101
L~CATI~N 9
_/
I ! ~
2 3 t! 5 6 7 89 lei 2 3 ll 5 6 7 89 1d' 2 3 ij 5 6 7891 o" Fn,...QUENCY (HZl
Page 60 of 113 11 Janua:r;y 1971 #9.)6
FREQUENCY RESPCJNSE ClF LR3 300, jQ D.f.MCJDEL.
F I G U R E 3. /. 2 C F~ AN 0 CJ M V I BRAT I Cl N SPEC T AU M
LCJCRTICJN 9
-0
~~~------------~--------------~----------~~ 1 -i ""'I '
'
~ ~ ........ l (\Jo l
9. 6986 G-RMS
~ ~-~~ --------------4---------------+-------------~ )J{ -~ {_;') -· .J
l
>- -~ ~~
1- ' 1-i -l U1 I z 1 w .... ! Db!
~~· --------------~~~~-4------~------------~
~ l a: -l 1- -! u ~ !..LJ .
~ j I cc(\1 1
W' l ~~ ~ 4 -----+--------~--------------~~+---------~ a_ 3
J '
rf 2 3 1.! s s 1as1 o' FREQUENCY (HZJ
Page 61 of 113 11 January 1971 #936
FREQUENCY RESP~NSE ~F LR3 300, 30 D.F.M~DEL
FIGURE 31.3 A TRANSMISSIBILITY
LCJCATI~N 16
~~------~------~--------------~-----------------GD-,1 g~ ... -4 w-! !J11
~-~
' (f)~
ru1 >-'"'o I ~~~-~----------------~--------------~--------------~ t--i m...)
1 00-j --" r--~
t-1 w--1 a:) IJ)...J ~--~ I
ff) :::I'..J
r..n I ... ~ ('1")-1 :;E I
I (f1 N-j z l a: l ,.....a .i '-1- 0 ~
~~-~i --------==~~~~~~--~------4---------------~ m-i QO·Jj r-- .. ~l w-i 11)-,
l ::ll'-1
I
1 01 2 3 4 s s 7891 rf 2 3 LA s s 789
FREQUENCY (HZJ
Page 62 of 113 11 January 1971 #936
FREQUENCY RESP~NSE ~F LR3 300, 30 D.F.M~OEL
FIGURE 3/.3 B SINE RESP~NSE
L~CRTI~N 16
m~ ro r--(.£J
Ln
~-~
"'l ! N~
-0 .....-3
~7)···"1 ro-J r-·-i w-l ~n.J ::fl..,! - I
i (\')-1
I ' ! I ~ ' C'\l-i /l !
I .II ..-. '
crH ro-, ~""'-"""i (.01 Ln-i !
1 ! :l'""'l
! !
~l Nl 'ol
J
...
~1 o' 2 3 4 s 6 7891 cf 2 3 ij 56 789 1 o' 2 3 1.! 5 6789 0 .. FREQUENCY (HZJ
........ N :c
... 0
~b
Page 63 of 113 11 January 1971 #936
FREQUENCY RESP~NSE ~F LR3 300, 30 D.F.MOOEL
FIGURE 3./.3 C RRNDCJM VI BRAT I ~N SPECTRUM
LCJCRTI~N 16
6. SlOL! G-RMS
~ ~~--------------+--------------+------------~ ::( (..:>
>-1-....... Cf)
z
_J
a: 0:: 1-u w a.... (J)
./
3 ij 5 6 7 89 1 cf 2 3 4 5 6 7 89 1 Q3
FREQUENCY (HZJ
Page 64 of 113 11 January 1971 #936
FREQUENCY RESP~NSE ~F LR3 300, 30 D.F.MODEL
F I G U R E 3. /. 4 A T RAN S M I S S I B I L I T Y
LOCATION 21
w (f)
'b -0)-
oo-,...._ w-If)-
::1'-
(!")-
N-
... 0 -0)-c.o-r--tO-If)-
::1'-
(!")-
N-
0
z D CL U1 Wo 0::
m-w-,...._ w-If)-
:1'-
(T)-
N·
... b -
Page 65 of ll3 11 January 1971
#936
FREQUENCY AESP~NSE ~F LR3 300, 30 D.F.MOOEL
FIGURE 3/.4 B SINE RESP~NSE
L~CATICJN 21
I j I I
I
v i I
!
! i
101 2 3 y 5 6 7891 cf 2 3 4 5 6 789 o' 2 3 y 5 67891 o" FREQUENCY (HZl
>I--Cf)
z
_J
a: a: 1-u w a... Cf)
... 0
Page 66 of 113 11 January 1971 #936
FREQUENCY RESPONSE ~F LR3 300, 30 D.F.MODEL
FIGURE 3./.GJC RAND~M VIBRATI~N SPECTRUM
LeJCATI~N 21
· 1 0. 74 6CXJ-RMS
Page 6 7 of 113 11 January 1971 #936
FREQUENCY RESPONSE ~F LA3 300. 30 D.F.MOOEL .
F I GUAE 3.1.5 A TRANSMISSIBILITY
L~CATI~N 28 'b
~-~---------------+---------------+--------------~ ......... 0)
....J~
.......,. W· CDt.n ......... (/")::::3" if) ......... (\")
~ if) (\I
z a: a: Cia
~~~---=~=-------~~~----------~~~----------~
rf 2 3 1.1 56 789
FREQUENCY CHll a•
w (f)
z 0 CL (f)
-0')-CD• ,__ CD· tn-::3'-
(T')-
N-
... 0 -0')-CD· ,__ CD• tn-
::::1"-
('I')-
N-
~0 -0>-co-,__ CD-II)-
::r-
(T')-
ru-
... b -
Page 68 of ll3 11 January 1971 #936
FREQUENCY RESPONSE OF LR3 300, 30 D.F.MODEL
FIGURE3.t . .5B SINE RESP~NSE
LOCRTICJN 28
v
o' 2 3 ~ 56 789t cf 2 3 4 567891o' 2 3 4 s s 78-st o" FREQUENCY CHZl
>t-
U1 z
_J
<I a: tu w 0.... en
Page 69 of 113 11 January 1971
r "'~;:E Q U E N C I R t ) P D N 5 E CJ F L R 3 l 0 0 , IF 9 :fo 0 • F • Mel 0 E L
FIGURE 3.;_ sc RRNDQM VIBRATION SPECTRUM
LCICATICJN 28
~ 4 5 6 7891 cf 2 3 I! s 6 1as1 o' FREQUENCY (HZJ
TABLE 3.1. 1
Page 70 of 113 11 January 1971 1/936
SUMMARY OF OUT OF PLANE RESPONSES
Location (See Fig. 2.1.1 & 2.1. 2)
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Maximum Sine Resp. g's peak
1.4 2.6 2.0 1. 85 2.0 2.5 2.5 2.8 2.7 3.2 3.2 2.9 3.2 4.0 3.3 2.0 4.4 4.5 2.0 3.5 5.7 3.5 4.1 6.3 4.4 6.5 6.7 4.3 6.2 ;3. 8
Random Resp. g's RMS
10.9 5.6 5.7 8.2 6.1 8,3
~ 6. 8 6.4 9.7 8.4 7.0 7.2 7.9 7.7 5.8 6.5 8.0 6.8 6.3 9.7
10. 8 8.4 7. 1
11. 9 7.8
14.0 15. 9 6.9
11. 4 8.4
,..._
"' :r: ........
1.\1
~
Figure 3.1. G
Page 71 of 113 11 January lq71
f/9 3 tJ
WORST RESPONSE AT A CORNFR LOCATION LOCATION :27
PERPENDICULAR TO ARRAY
~Or--------------.---------------.--------------~
15.87 G RMS
1
». !::: Vl c q,
Q . 1 -0 (.. ...... v v 0.
V)
L OJ
~ 0 0.. I
Ol ~J------··-·---------1
• 0015~--~,o~--2~0----~so~--~,o-o---~~o-o---1o~o~---,~oo_o ___ 2o~o-o--~cooo
FREQUENCY (Hz}
UPPER ARRAY ACCEtc~TANC£
RANDOM VIBRATION LEI/ELS' Figure 3" l. 7
Z AXIS -PERPENDICULAR TO ARRAY
Page 72 of 113 lJ January 1971 //936
1~--------------------------------------------~
. 1! -:v--107 ?'!"• '\'l
---N I :r. .D'l~l y''/11;!
' I I ~ en I I ......
>--.. I I
~ I
\;') i I c I I Ql
0 .01 I , -t\0107 9j!Hz - , I 0 I.. I I I -u <II I I I Q..
V) i I I t.. Ql I I ~ 0
Q.. ll I
I I I I I I
• 001 I ! I 1 I I
: I : I I I I
I I I I I I 10 110 1'15 22f, 400 (QOO 1400
2t5
• 00015 10 20 50 100 200 500 lOOO 2000 5000
Fre91.1eocy (Hz}
; ..... ,., r" 11'""'""'\:
--' r--1--1 <D-aJ
lfl-1--1
U1 :::1'-lf) 1--1
(\"'.!-
L if) N-z a: a:~ t- ..........
en-a:>-r-~
<D-II)~
:::::1"-
(\"'.!-
N-
-b -1 01
f,~ r-"t ~ ....... <
LBCf-=!T I llN
., ,_ 3
11 .}::: H9
I I 3 4
.~ ! '"' -l '·• . ~ ' :>- \,,.'~ ._~
i?"" !" !_.,.,. \-..~ . ..,_
014
w en z 0 o_ if)
Page 74 of 113 11 January 1971 #936
FREQUENCY RESP~NSE F~R LR3 300,6D.F. M~DEL
FIGURE 3.2./B SINE RESPCJNSE
LCJCRTICJN 1
b X RESPONSE OF LOWER ARRAY -X E~C!TAT!ON ........ 0)-ro-r--<D-U")-
:::1'-
en-
N-
0 ........ 0)-ro-r--<D-If)-
:::1'-
en-
N-
I
~a v ........ 0)-ro-r--<D- I
Ln-
:::1'-
en-
N-
-b ........
01 2 3 4 56 7891 cf 2 3 4 56 789 03 2 3 " 5 6 7 891 o" FREQUENCY (HZ)
Page 75 of 113 ~9~~nuary 1971
FREQUENCY RESPONSE FOR LR3 300,6D.F. MODEL \
N :c ~b
FIGURE 3. 2. JC RRNDCJM VI BART I CJN SPECTRUM
LCJCATIDN 1
X RESPONSE OF LOWER ARRAY -X £%CITATION
9. 3229 G-AMS
~ ~~------------~------------~~----------~ )jC t:)
>t--en z
....J a: a: 1-u w 0... en
"' 'o ~~----~~~~~~~~~~~~~~~~~~~
"'"
, .. :::.
Page 76 o£ 113 11 January 1971 #936
FREQUENCY RESPONSE FOR LA3 300,60.F. M~DEL
F I G U R E 3 2. 2 A T R R N S M I S S I B I L I T I
LCJCRTICJN 4 a )'RESPONSE OF UPPER ARRAY TO% EXCITATION
N
-'a ~~~-.--r-~~~~~~~-+~~~--~~----~~
rf 2 3 I! 5 6 7 89 Q3 2
FREQUENCY (HZl
w (f)
z E)
a... (f)
Page 77 of 113 11 January 1971 #936
FREQUENCY RESP~NSE F~A LA3 300,60.F. M~DEL
F I G U R E 3. 2. 2 . .B S I N E A ESP~ N S E
LCJCRTJ(jN 4
o X RESPONSE OF UPPER ARRAY -Y E{CITATION .....-4
0>-oo-r--(D-
t.f)-
:::1'-
(V")-
C\1-
0 ~
0>-oo-r--CD-Ln-
::1'-
(V")-
_/ C\1-
~b 1/ I ..--. 0')-
I 00-r--CD- l lf)-
::1'-
(V")-
C\1-
-b .....-oi
101 2 3 ll 5 6 7 891 cf . 2 3 t! s 6 7 as'1 03 2 3 y 56 789 o" FREQUENCY (HZ)
>-1---_J
a: a: tu w Q_ (f)
Page 78 o£ 113 11 January 1971 #936
FREQUENCY RESP~NSE F~R LR3 300.60.F. M~DEL
FIGURE 3.2.2 C RRNOLIM VI BRAT I LJN SPECTRUM
LLJCRTION 4
-0
)(RESPONSE OF UPPER ARRAY -X E.XC!TifTION
7.0021 G-RMS
Page 79 of 113 11 January 1971 #936
FREQUENCY RESP~NSE F~R LR3 300,60.F. M~DEL
FIGURE 3.2.3 A TRRNSMISSIBILITI
LeJCPIT I eJN 5
b Y RESPONSE C)F UPPER ARRAY- K ElCITATION
I I 2 3 2 3 4 5 6 7 89 1 03
FREQUENCY (HZl
Page 80 of 113 11 January 1971 #936
FREQUENCY RESP~NSE F~R LR3 300.60.F. MODEL
FIGURE 3.2.3 8 SINE RESP~NSE
LCJCATI(jN 5 a Y RESPONS£ OF UPPER ARRAY- i ElCITATION
0>-co-r--CD-lf)-
::::f'-
(Y')-
C'\1-
0>-co-r--CD-
1-
·-lf)
::::f' w )-U1(Y')
:z: 0 Q...C'\1 J-
(f1
~b ....--1
)-)-
·->-
0> co r(0
1.1) )-
,_ :3'
)-
C'\1 J-
-b
I I
/ 01 2 3 4 5 6 7891 cf 2 3 4 5 6 789 . 0!3 2 3 ~ 56 7891 o"
FREQUENCY (HZ)
Page 81 of 113 11 January 1971 #936
FREQUENCY RESP~NSE F~A LR3 300.60.F. M~Df'
FIGURE 3.2.3 C RRNDCJM VI BRAT I ON SPECTRUM
L~CRTICJN 5
-o Y RESPONSE OF UPPER ARRAY'-% EXCITATION
3. 0928 G-RMS
)-
1--(J)
z W- I o~al
......-f
_j
a: a: 1-u w a_ (f)
a::N W'o ::s::: ...... 0 a_
., \. \::- • .J \ '
Page 82 of 113 11 January 1971 #936
FREQUENCY RESP~NSE FOR LR3 300,60.F. MODEL
F I G U R E 3. Z. 1 A T R R N S M I S S I B I L I T Y
LLJCRTICJN 1
b X RESPONSE OF LOWER ARRAY-Y £A'CITAT!ON
I I
3 4 5 6 7 8 9 1 02 2 3 4 5 6 7 8 9 1 03
' FREQUENCY (HZ) lj " ' 1
( ,\ , . I,
i
>--1-...... (/)
z w 0
_j
a: a: 1-u w CL (J)
a: w 3::: 0 o_
Page 83 of 113 11 January 1971
#936
FREQUENCY RESP~NSE FOR LR3 300.60.F. M~DEL
F I G U R E 3. 2.1 B R R N 0 ~ M V I B R R T I Cl N SPEC T R U M
L~CRTION 1
... o %RESPONSE OF LOWER IIRRAY-Y El'CITATION ----- 3. 7173 G-RMS --
-
0 -------
--I 0 ........ --
~ I ---
N I
0 .........
-
\ ---- M --
-I
0 ........
01 2 3 4 56789to2 2 3 4 5 s 7 s9 1 03 2 3 4 567891 of! FREQUENCY (HZJ
\
Page 84 of 113 11 January 1971 #936
FREQUENCY RESPONSE FOR LR3 300.6D.F. MODEL
FIGURE 3 2.5 A TRANSMISSIBILITY
LCJCATION 2
56789102 2 3 4
FREQUENCY (HZl
w (I)
z CD CL (I)
en-CD-r--CD-Lf)-
::r-
(T)-
N-
... 0 ....... en-CD-r--CD-Lf)-
::r-
(T)-
ru-
~'b ......-1
en-co-r--<D-Lf)-
::::1'-
('~")-
ru-
I 0
, I Page 85 o£ 113 11 January 1971 #936
FREQUENCY RESPONSE FOR LR3 300.60.F. MODEL.~tPT
FIGURE 3.2.5 B SINE RESPCJNSE
LCICRTICIN 2
- ..
I I
I I
------
I
10 1 ..!. .! ·'· .!. .!. ..!. 2 .!.. .!. ·'· .!. .!. ..!, .!...!. 3 2 3 45678910 2 3 45678810 ? It ,I-,!..!,1,.1 3 -;; ;:J 0 I 8 9 .l 01&
FREQUENCY (Hl)
,........ N :c ~0
Page 86 of 113 11 January 1971 11936
FREQUENCY RESPONSE F~R LA3 300.6D.F. M~OEL
F I G U R E 3. 2. 6 A R R N 0 CJ M V I 8 R R T I CJ N S P E C T RUM
LCJCRTI~N 2
Y RESPONSE OF LOWER ARRAY ·-Y OC!TATION
9. 4335 G-RMS
~ ~--------------~--------------~------------~ ;$:
c..::>
>t-....... (f)
z:
_I a: a: tu w o_ (f)
I
~~l-0-1 -2..--' ..._,.3____..4. 5 ld M ~ o2 - £ ~ 4 H 1M 11 o' 0~
FREQUENCY (HZ)
Page 8 7 of 113 ll January 1971
lf.936
FREQUENCY RESP~NSE F~R LR3 300w6D.F. M~DEL
FIG U A E 3. 2.6 A T R R N 5 MISS I B I LIT Y
LLJCATICJN 5
b Y RESPONSE OF UPPEl? ARRAY- Y £XCI !-.4T!ON
FREQUENCY (HZ)
Page 88 of 113 11 January 1971 1#936
FREQUENCY RESP~NSE F~R LR3 300,60.F. MODEL jfp·i
FIGURE 3.2. 6 .B SINE RESPClNSE
LClCRTIClN 5
a Y RESPONS£ OF UPPER ARRAY- V' E.IJC/TA!IOIV' 0)- . <D-.....,.._
w-tn-
::J'-
('f)-
N-:
-0 ........ -· -0)-(X) ...I·
~~ tn:]
I ::J'~1
~j NJ
l -- -------· -!--~
0>-co-r---CD-tn-
I ::J'- I
('f)-
N-
-b ~
01 2 3 ij 56 7891 cf 2 3 ij 5 6 7 891 cf 2 3 1.& 5 6 7891 011,
FAEQUENc·r (HZ)
>t-........ (/')
z
_.J
a: a: tw w a_ (f)
Page 89 of 113 ll,January 1971 #936
FREQUENCY RESP~NSE F~R LA3 300~6DaF. M~DEL
F 1 GURE 32.~ C RRNOC!M VI BRAT I LlN SPECTRUM
LClCATION 5
a Y RESPONSE~ OF UPPER ARRAY - Y EXCITATION
10m 673'-!G-RMS
___)
~----~~-----Jl-----------4~+------------~
(')
'o
1 01 3 4 5 6 7 8 9 1 02 3 4 5 6 7 8 9 "' Q3
FREQUENCY (HZ) \
\ J, ': '\ I ... i '
TABLE 3. 2.1 SUMMARY OF IN PLANE RESPONSES
Excitation Array Coordinate Max Sine Direction Direction Resp. (g's peak)
X 3.28 Upper y . 215
e -2 1. 84 x 10 pseudo g's
X X 1. 57 Lower y . 512
e 1. 91 x 10-3 pseudo g's
X 7, 31 X 10-2
Upper y 1. 64 e 1. 56 x 10-2 pseudo g's
y X 1. 55 X 10-2
Lower y 1. 54 e 6 -3 3. 8 x 10 pseudo g's
.,
Ramdom Resp. (g's RMS)
7.00 3.09
. 235 pseudo g's
9.32 2.84
. 247 pseudo g's
.653 10.67
. 611 pseudo g's
3. 71 9.43
. 859 pseudo g's
::,;::::::::1:1 --o~Pl Wp.lO'Q 0' ::; (b
c "' Ill 0 ~ 0
'< ""' .............. "' ...... -J w ......
~ ~~Divl•lon
Excitation Direction
X
y
Array
Upper
Lower
Upper
Lo"\ver
LRRR 300 Corner Array Dynamic Analysis
Translational Response (.grms)
7. 00
9. 32
10. 67
9.43
TABLE 3. 2. 2
Rotational
Response rad 2
sec 386
. 235
. 248
. ~ 11
. 8.39
Distance To Edge (in)
6: 5
12. 9
10.9
l o. 9
HO. ,, PAGE
DATE
Total~:~
Re sp. (grms)
7. 16
9.82
12. s
13. 3
~:Total Response =/Translational Response 2
+ (Rotational Resp. x Dist. to Edg:e) 2
,
OF
~......, ::::::::: ~ u
~~~ C' ~ (0
:::; '"' -..o $ ....... .,
0 '-<: ...... - ....... -..o--JVJ .......
NO. RI!!V. NO.
...... 4erospace
LRRR 300 Corner Arr;cy Dynamic Analysis
936
PAGE 92 01' 113
· rstems Division DATI! 11 January 1971
This table assumes that rnaximun1 responses in translatiot1 occur at the same frequency as maximum rotation responses, a consf:rvativf: asstunption. The table indicates that the rnaximm11 responses at a corner occurs along the front and rear edges of the large array. Tht~ root nH.~an square response along these edges is 13. 3 grms giving a 3 sigma peak value of 39. <) g. The peak power spectral density of the response occurs at about 400 Hz. Thus a corner mount design that will survive a sinusoidal vibration environment well in excess of 40 g peak in this frequency range is an adequate design.
In order to test the upper array aloneJvibration levels in the plane of the arrays at the attachments between the two arrays were calculated. Since the test set up required the upper array to be tested with trans~ational vibration only, the vibration spectrum on the lower array under the c. g. of the upper array was found. Due to the proximity of the centers of gravity of the two arrays and the relatively low levels of the rotational responses at the lower array c. g. , the response under the c. g. of the small array due to rotation of the large array could be neglected. In order to simplify the tPst set up, the responses due to X and Y excitation were superimposed to give one set of test vibration levels for both the X and Y axes. The test en velopP levels) reduced hy a factor of].(,<) to acceptancP levels> are shown in Figure 3. 2. 7.
Figure 3. Z. 7
UPPER ARRAY ACCEPTANCE RAI\IDOM VIBRATION LEVELS
X I y AXES IN PLANEOFARRAY
lr-------------------------------------------~
.1
'H' ::X: .......
at .oJ., ll"* \!) ·- I .o2.78 cf/Ha
~ .01'5 ~'/He I I
I I 'll.t I I I c: I ~ q, 1o,3tO f'J He
0 I II • 01
I I
II I Cl \,. I I .... ! I \,) I cu I! I a. I CJ)
11 I L I Ql
I' I ~ I 0 I I Q.
I II
I II • 001 I I I
I
I II
I II
Jl I I I
I I I I I I I, I
41 85 liS 2.10 320 600 '/'200 /700
960
Page 93 of lJ 3 l1 January 1971 /1936
• 00015 10 20 50 100 200 500 1000 2000 5000
Frequency (1-12)
NO. Rl!V, NO.
936
LRRR 400 Corner Array Dynamic Analysis PAGE: 94 OF 113
DAtE 11 Jrtnuary 1971
3. 3 Rc sponsc s In Off-Loaded Condition
Response to the LM. Quad III vibration environn1ent for the LRRR in the "off-loaded condition''were calculated perpendicular to tlw plane of the array using the model of section 2. 3. Responses of several locations on this model are shown in Figure 3. 3. 1 - 3. 3. 5. The responses for the experiment in this condition are seen to be slightly reduced compared to outof-plane responses of the lower array with the upper array in place.
In-plane responses of the lower array alone were not calculated. This is justified since in the plane of the arrays the upper array adds no stiffness to the lower array but does mass load the lower array. Since the random excitation spectrum is almost flat in the region of the natural frequencies of both cases, larger loads on the LM interface fittings will be experienced with the upper array connected.
4. 0 Dynamic Loads & Deflections
4. 1 Method of Calculation
Dynamic loads on the structure were calculated using the out-ofplane model of the two connected arrays. The in-plane model gives dynamic loads on the interface fittings, hinges, and clevis fittings. The loads used in designing these fittings were so conservative that any conceivable dynamic load produced by the rlynamic environment will be easily survived. Dynamic loads with the upper array removed will be low('r than with it attached since mass loading induced by the sn1all array is large compared to the amount of stiffness the small array contributes to the lower array ..
The acceleration response at each lumped mass location, the stiffness matrices for each array, and the transformation between coordinates local to each array and coordinates of the whole system determine the dynamic loads on the system.
LKIJ 0
[-Ku] l I [KJ I
Let [K] -- 0
[Ku] J 3H
tKJ t ... - 38 -Where[KL] & (Ku1arc as previously defined
Page 95 of 113 11 January 1971 #936
LRRR 300 IN THE ~FF L~AOEO C~NDITION 23 O.F
FIGURE 3.3./ A TRANSMISSIBILITY
L~CATICJN 3
3 q 56789lcf 2 3 q 567891d'
FREQUENCY £HZl
en (.!)
Pa.ge 96 o£ 113 11 January 1971
#936
LRRR 300 IN THE OFF LOADED C~NOITION 23 0.1
F 1 GURE 3.3. I B SINE RESPCJNSE
-en-CD• ,..._ CD• Ln-:::r-
C')-
N-
... 0 ...... en-CD• ,..._ CD-Ln-:::r-
C')-
N•
-en-CD-,..._ <D-Ln-:::r-
C')-
C\J•
... I 0 -
v
1 o' 2
LOCRTICJN 3
'
_..--/1
3 ai 5 67a9tcf 2 3 ~ 5 6789to' 2 3 4 56 7891 a•. FREQUENCY £HZl
->t-...... (/)
z
_, a: a: tu LLJ a... (/)
Page 97 of 113 11 January 1971
LRAR 300 IN THE ~FF L~ADED CON~f~I~N 23 D.F
F I G U A E 3. 3. I C A AN 0 0 M V I 8 RAT I ~ N SPEC TAU M
L~CATil'JN 3
10. Y934G-AMS
o"
Page 98 o£ 113 11 January 1971 #936
LARA 300 IN THE OFF L~AOED CONDITION 23 O.F
FIGURE 3.3.2. A TRANSMISSIBILITY
LOCATION 8
'b -0)-CD• ,...._ CD• ..,_
~-
('1)-
N•
-0)-CD• ,....._ CD• ..,_
~-
('1)-
N·
-· 0)-(l)-,....._ CD-II)~
:r-
(f)-
N•
... b -
Page 99 of 113 11 January 1971 #936
LRRR 300 IN THE ~FF L~ADED CONOITl~N 23 D.F
F 1 GURE 3 . .3. 2 8 SINE AESP~NSE
L~CATICJN 8
'
~
v
tO' 2 3 4 56 7891 cf 2 3 4 56 7891 o' 2 3 4 5 67891 o' FREQUENCY rH"7t
"" 1:...1
-
->t-...... en z
__. a: a: tu IJ.J a.. en
Page 100 of 113 11 January 1971
LRRR 300 IN THE ~FF l~ADED C~N~~liON 23 D.F
FIGURE 3 3. 2 C RAND~M VI BRAT I ~N SPECTRUM
LCJCATICJN 8
7. 5895 G-RMS
Page 101 of 113 11 January 1971 #936
LRRR 300 IN THE ~FF L~AOED CONDITI~N 23 O.F
FIGURE 3 3.3 A TRANSMISSIBILITY
L~CATI~N 10
o•
'b
Page 102 of 113 · 11 January 1971 #936
LRAR 300 IN THE ~FF L~AOED CONOITI~N 23 D.F
FIGURE 3.3.3 B SINE AESPCJNSE
LCJCATICJN 10
a•
-
->t-~
en :z
....J a: a: tu l&J a... en
Page 1~3 of 113 11: January 1971
LARA 300 IN THE ~FF L~ADEO C~Ndt¥I~N 23 O.f
F I G U R E 3. 3. 3 C RAN 0 ~ M V I BRAT I CJ N SPECTRUM
L~CATI~N 10
8. 1379 G-AMS
o'
Page 104 of 113 11 January 1971 #936
LARA 300 IN THE ~FF L~ADED C~NOITION 23 D~f
F I G U A E 3. 3.1 A TRANS M I S S I B I L I T Y
L~CATION 15
Pag~ 105 of 113 11 January 1971 #936
LARA 300 IN THE ~FF LOADED CONDITION 23 O®F
F I GUAE 3. 3. 4 B SINE RESPONSE
LOCATION 15
-en-CD· ,...._ CD• &1)-
:::1'-
(\')-
N·
0 -en-CD-,..... CD• &n-::1'-
(\')-
N':"
~ ...-4 v en-CD-,...._ w-&n-:::1'-
(1")-
N-
... b -to' 2 3 q 56 7891 £f 2 3 ~ 5 s 7891 o' 2 3 ~ 5 s 78st
FREQUENCY £HZl
-
->t-
...J a: a: tu w a.. en
Page 106 of 113 11 January 1971
#936 LRRR 300 IN THE OFF L~ADED C~NOITION 23 Dsl
FIGURE 3.3. 4 C RANDOM VI BRAT I ON SPECTRUM
LOCATION 15
10. ijlj 71G-RMS
Page 107 o£ 113 11 January 1971 #936
LRRR 300 IN THE ~FF LOADED CONOITI~N 23 O.F
F I G U R E 3 3. 5 A TRANS M I S S I B I L I T Y
LOCATION 22
cf
'b
Page 108 of 113 11 January 1971 #936
LARA 300 IN THE OFF LOADED CONDITION 23 O.F
FIGURE 33 5 B SINE RESPONSE
LCJCATION 22
->..... ...... en z
..J a: a: 1-u L&J a.. en
Page 109 of 113 11 January 1971 #936
LRRR 300 IN THE OFF LOADED CONDITION 23 O.F
FIGURE 3.3.5 c RANDOM Y I BRAT I ON SPECTRUM
LOCATION 22
17. 2065G-RMS
r:f
: : . . ' '
~'t.:c t<;'::.;~ :,< { ' LRRR 300 Corner Array Dynamic Analysis
NO. Rl!:V. NO.
PAGE ll 0 OF 113
DATE 11 January 1971
The K matrix is 38 x· 3 8. As explained in Section 2. I the K matrix
is reduced in si:.r,e to correspond to the ::-;elected 30 coordinates by the transformation
K is shown in Figure 2. I. H. s
[ 13 J = The coordinate transformation matrix as previously defined
The internal loads on the structure and associated displacen1ents can be determined from the following relations:
{P} " 1 r K J -l IK1113 s {Q} (2)
{S} ···· f13l fKsl-l {Q} (3)
Where {P} is the internal load vector, {S} is the displacemPnt vector
The next step is to compute the load vectors, {Q} , resulting from the dynamic environment.
4. 2 Load Vectors
The expression for the load vectors, Q, for the random environrrwnt is
T raJ = r M 1 rAJ [ M] = Mass matrix (Figure 2. l. 5)
[ A ] = A diagonal matrix composed of local rms accelerations as previously computed
NO. R!Y. NO.
: :' ' . ' . ':, \,' ::;_-i ·' •' j~
.LRRR 300 Corner Array Dynamic Analysis
I'AGI 111 113
DATE 11 January 1971
Random Loads
When the {o~ column vectors have been dt)tern1int•d, they are substituted into equations ( 2) and ( 3) to compute the internal loads and displacen1ents on the arrays. For the random case the rms internal loads and displacements arc co1nputed for each mode giving 30 sets of rms values at the 38 coordinate locations. Then for each location the 30 modal rms values are combined by taking the square root of the sum of the squares of the 30 values, thus reducing the number to 30 load and 38 displacement values.
Sine Loading
The load vectors, fo1 , are again formed by equation (4), but real and imaginary steps are computed separately and combined into one load vector for each frequency where a relative maximum sine response at any location occurs.
For the sinusoidal environment case, there is only one {Q} vector per frequency selected and each one will produce a set of internal loads and displacements.
4.3 Discussion
Application of these procedures to the out-of-plane model results in load and deflection sets due to the random excitation shown in Table 4. 3. 1 Loads due to sinusoidal environment arc all less than the 3-sigma random values obtained by multiplying the Table loads by 3, and arc therefore not shown.
To calculate reactions at the LM interface fittings, the dynamic loads at each mass location on the lower array were applied to the "rigid grid" model of the structure as a static load case. The reactions at t:he fittings were calculated by the program. The largest reactions were o1,tained for the random excitation case. These reactions are listed in Te:.;Jle 4. 3. 2.
..
~-·· ROOT MEAN SQUARE--COMBih£0 lOADS
j l ~-- DEFLECTIONS
' r
r::::::~ ,....._.... .. _. . r:-:~-~~--
··,....--· I..
r·· L.~
r:::""·--~
c.~-
j .. ; ..
L
t:.::-:~ ---- -----y.
r----r:· !'..-:~
( --
1 c. 216( 7160-03 2 O.ll935460-C2 3 o.~u~325o-o 4 C.405t5~60-C3
C. 3H3186C-C3 6 C. 74393230-03 1 c. 6e1H42C-C3 8 C.E12Ct7CD-C3 'i c.eH2714C-C3
10 o. 1 c 7133 20-02 ll 0.11128810-02 12 0.99~13290-CJ 13 O.ll16t 530-02 14 C.H37tl5D-02 1~ (.12043430-02 16 (l.~1~!4650-C3 ll C.1~821920-02 18 C. 20C€2250-C2 H C.41ol2C680-03 20 o.uz 25380-02 21 0.2'it24080-C2
-------- 2c C.l42t:4170-02 23 O.l8H5C40-C2 24 c. 339tl2 10-02 2~ O.J<i<;~9940-C2
26 C.33t~3600-C2 27 C.Jt21Hl0-C2 ze C.l89<;C19C-C2 29 c. 3.23 t 11 70-C2 30 O.l59CH50-C2 H C.l€5f2240-02
rat:J{t! 1 '._lf/.3,/
:--~-
c--- ---------------- .34 c.:21639aa-o2 -- ------------33 O.ll5'i8460-02
r-~-- H o. 328H940-C2 35 C.310!2UO-C2
(:::' ___ --- ----;--_ -----------:-- 3e Co1665940D-02 -.-------------31 Co3CCl6760-C2
C::~---_:-- -=·:·----~~-- -~:=--:--=- 38 O.l51l34lD•02 -- --:-:--:·--:---::-_-~_-----
INTERNAL LOADS
0.25247300 02 o.l~922COO 02 0.815210~0 02 0.30545710 02 0.16t>bt:230 02 Oo22411680 02 0.184288~0 02 0.1H76890 02 O.l326t280 02 0.311Ct:3l0 02 0.18938630 02 0.98845470 01 0.21616090 02 O. 20797C30 02 0.15917130 02 0.82.851100 02 0.297U<;t>O 02 0.18552850 02 o.e~848210 01 0.17051580 C2 0.96221020 02 0.22036280 02 O.ll307lb0 C2 0.2,28'1120 02 0.35556460 02 0.28279210 02 o. 32072380 02 0.28391610 02 0.23075630 02
1 0.1625C340 02 0.11307160 02 0.25289120 02 0.35556460 02 0.28279210 02 o. 32072380 02 0.28391610 02 o. 23075630 02
- 0·16250340 02
•
l A
' ..... ·1 . .., _.J
-.a
j .. I-
--..,--.--- ··:.r
l
"'l
•4 ··-· .......... - ·- ••• "7 ~ ~-..-f
~ .... - ..J
-·-·---.......... -. -- ............ ~ _ __..., __ A
;:. ~
c::-::::=.=-.:.:.~~=-=~=-=-==--.===~::_:_._~:=.~==-=·-=--~=:--.. :: .::_- --- ..... ·-- - .. ----, c-.-_:~:-·-:----~-.:-:-- ·-·--c·=- -==--:---:-:- .--::------ ----- -- -------- ___ , - -~
t:=::-:-:~==···:~~~-- -~ --- ·-··~-- -- ----- --- ------------· -- ------·---
r-~- --~- ~~~~=~~H- -~~-=-~~- ~-=-=-:_-=~-~- --------------- ----------- -------------------------- -------------- --.,
j
-- ·--- -- - ~- - --~ -- . - . - -·- .. - -__ _:-:::-·--:--::-:-·-=-:-:-:::---~ -.:1
r-·
,.-
• I
:::;; ..... 1:1 --o,_.l» VJL.t()'Q 0" I» ~
::s ,._ ~ ....... I» N 1-f
r '< o -............ --o,_.. -Jw ,._
Fitting
Rt. Front
Rt. Rear
L. Front
L. Rear
TABLE 4. 3. 2 VERTICAL REACTIONS
AT LM INTERFACE FITTINGS
RMS Load (lb)
142
156
226
144
Peak Load (lb)
426
468
678
432
::jj:::.-.'lj -.o'"""'ll' WLtOQ 0' ll' Cb
l:! .... ~ .... Ill w li 0 '< ...., ........ -.c.-. -..] w ....