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ADb-Atlf 387 COMPUITATION OF NATURAL FREQUENCIES OF PLANAR LATTICE I/1

STRUCTUME(U) WER CAIU*IDOE NA J H HILLIM ET AL

ft MAR 87 RFOSR-TR-7-IDW F49620-85-C-S148

UNCLASSIFIED F/0 22/2 N

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MICROCOPY RESOLUTION TEST CHARI

BATOAL BUREAUJ OF STANDARDS 1963-A

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UNCLASSIFIED

SCPsvCLASSIFICATION AD-A 185 387m"F

AD-A 85

37
' iON PAGE A~poT~fiURIV CL"IFIATIN LJ I Iso. %STPICTIVI[ MARKCINGS

U N CLPO A S UI TIE D S F CA IO

* 2.SECURITY C.LASSIFICATION AU? '43. OISTRIOUTIONiAVai.AulLTy OF REPORT

APPROVED FOR PUBLIC RELEASE;

21Lo c. OELASIF ICAT I O4OOWN GRNA Oiftt*6A u DISTRIBUTION UNLIMITED

SPERFORMING ORGANIZATION PIEPW NUMERSI LA .MONITORING ORGANIZATION REPORT NUMBER(S) _______ ______ _ _______

AFOSt. TX. Q ~ 1 0

6aL NAME Oft PERFORMING ORGANIZATION StL OFFICE SYMBOL 7.. NAME OF MONITORING OAGAN1IZATION 0

WEA /C.

6c. ADDRESS oCity. State and ZIP Codes 7b. ADDRESS (City. State and ZIP Coda)

P..Box 260v, MTBranch

£ Cambridge, MA 02139 .

~S.NAME OF FUNOING/SPONSORIING 8b. OFFICE SYMBOL B. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER

ORGANIZATION AIR FORCE OFFICE (it applicabill 0

SOF SCIENTIFIC RESEARCH AFOSR/NA F49620-85- Ql4

1cS. ADD~RESS iCity. State and ZIP Code) 10 SOURCE OF FUNDING NOS ______

PROGRAM PROJECT TASK WORK UNIT

B LLiN ,AFt, DC 20332=~$ ~ ELEMENT NO. NO. NO. 11O. J iiTTLE 'Incugae "cunry CLdeeIfIcataan, COMPUTATION OF NATUR 61102F 2302 B1 FREQUENCIES OF PLANAR LATTICE STRUCTURE-UNCLASSkFIED ________________ * 12. PERSONAL AUTI.IOR IS)

James H. Williams, Jr. and Raymond J. Nagem

-Ar ON

3.. YPE ~ REORT13. TIECVRG14. DATE OF REPORT ,Yr.. .Wo.. D3: 5.PCECON

TECHNICAL Fetoul Sent 85 roLMar 87 1987, March, 1 65

16. SUPPLEM-INTARly NOTATION

17 COSATI COOES IB. SUBJECT TERMS lcontin.. on teuerse of neesary and identify by locle Rumba"

~'E O GOUPSUB ORLARGE SPACE STRUCTURES TRANSFER MATRICES * NATURAL FREQUENCIES '19. ABSTRACT #Con tsnhe on oyuers. it naceue17 and Idennty by bloCa numberCP, Transfer matrices and joint coupling matrices are used to compute natural frequencies of vibration of a five-bay planar lattice structure. The method of #analysis may be applied to general two and three-dimensional lattices. The necessary numerical computations may be performed easily with a personal computer. Numerical results for the first twenty-five nonzero natural frequencies of the five-bay planar lattice structure are given for the case when the members of the lattice are modeled as Bernoulli-Euler beams, and for the case when the members of the lattice are modeled as Timoshenko beams. The maximum difference in the computed natural frequencies of -. the two models occurs in the twenty-fifth mode and is less than one-half of one percent. The natural frequencies obtained here agree within six percent with the natural freqiuencies obtained in a previous analysis using a finite element method * and, an experimental modal analysis. i O. DISTRIGUTION/AVAILAOILITY OF ABSTRACT .121 ABSTRACT SECURITY CL.ASSIFICATION

0nCLASSIFIaDIUNLIMITSO 13 SAM4 AS NOT ZOTIC USERS UNCLASSIFIED

11NAME OF RESPONSIBLE INDIVIDUAL

22a TELEPHONe NUMBER 2c FIESMO

Idnclude .4nre Code)2c aIC

SMO

ANTHONY K. AMOS

202/767-4937

AFOSR/NA

jD FORM 1473, 83 APR EDITION OP I jAN 73 iS OBSOLETE. UNCLASSIFIED -0- SECURITY CLASSIFICATION OF THIS PAGE

AFOSR.T. S 10 0 6

ACKNOWLEDGMENTS

The Air Force Office of Scientific Research (Project Monitor, Dr. Anthony K. Amos) is gratefully acknowledged for its support of this research.

NOTICE

This document was prepared under the sponsorship of the Air Force. Neither the US Government nor any person acting on behalf of the US Government assumes any liability resulting from the use of the information contained in this document. This notice is intended to cover WEA as well. -2-

INTRODUCTION

In a previous series of papers 11-5], matrix methods for linear dynamic analyses of lattice structures are developed. A lattice structure, in this context, is defined to be an idealized network of one-dimensional members which are connected by joints. In this paper, transfer matrices and joint coupling matrices are used to compute the natural frequencies of vibration of a five-bay planar lattice structure. The method of analysis is applicable to general two and three-dimensional lattices. The necessary numerical computations may be performed easily using a personal computer. Numerical results for the first twenty-five nonzero natural frequencies of the five-bay lattice structure are given for the case when the members of the lattice are modeled as Bernoulli-Euler beams, and for the case when the lattice members are modeled as Timoshenko beams. The results obtained here are compared with the results of a previous analysis [6] using a finite element method and an experimental modal analysis. A short dis- cussion of the results and of some potential applications of the type of analysis presented here is given. Fo S .!%

3 ...• ..

" ...i %' ~i .. " "' ' .. ;? .. K.

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ANALYSIS

Lattice Structure and Model %

The lattice structure considered here is shown in Fig. 1. The lattice is machined from a single piece of aluminum, and contains no welds or fasteners. The model used here to analyze the lattice structure of Fig. 1 is shown in Fig. 2. The structure is modeled as an idealized lattice of one- dimensional members which are connected by joints. It is assumed that the members and joints can move only in the plane of the structure, and that all motions are small. The joints of the lattice model are labeled 1 through 12 as shown in Fig. 2. It is assumed that each joint is rigid, and that each connection between a joint and a beam is rigid. It is also assumed that each joint is massless and has no spatial extent. The assumptions that the joints are massless and have no spatial extent are made only for convenience; some *comments about the analysis of joints with mass and/or spatial extent are given below. It is assumed that all members of the lattice model of Fig. 2 are identical, and that each member can extend (and contract) axially and bend flexurally. It is also assumed that the axial and flexural motions are uncoupled. Two different member models are used. In the first model, hereafter called the Bernoulli-Euler beam model, the lattice members are modeled as classical longitudinal rods for axial motions and as Bernoulli- Euler beams for flexural motions. In the second model, hereafter called

0 the Timoshenko beam model, the lattice members are modeled as classical

longitudinal rods for axial motions and as Timoshenko beams for flexural -4- .° -a ...... motions. In both models, the stace vector at any point x of a lattice member is of the form u(x,t)

V(X, t)

Z((x~t)

z(x,) xt ) (i)

N(x,t)

V(x't)i

F(x,t)

where u(x,t) is the longitudinal displacement of the member, v(x,t) is the transverse displacement of the member, i(x,t) is the rotation of the member, M(x,t) is the bending moment in the member, V(x,t) is the shear force in the member, F(x,t) is the axial force in the member, x is a spatial coordi- nate which extends along the length of the member and t is time. The com- ponents of the state vector and the sign convention adopted here for the components of the state vector are shown in Fig. 3. Local coordinate directions x i (i 1,2, .. 16) are assigned to the lattice members as 'p shown in Fig. 2. OS.

Joint Coupling Matrix Relationships

The Fourier transforms z and z

2 of the state vectors z I and z 2 shown in Fig. 2 are related by an equation of the form [3] z

0 -(2)

55
where Bl(w) is the joint coupling matrix of joint 1. Eqn. (2) is the joint =1m coupling matrix relationship for joint 1. Joint coupling matrix relation- ships for joints 2 through 12 can be written in a similar manner as

B=() 0 (3)

-4 -5 B 5 (w) z =0 (4) -7- 3 S6.

B (W) z 0(5)

-4 1 5

12(w) z1 =0

(6) -3 -l4

B() z =0 (7)

615
-16 -7 -7(~ -18 =0(8) z2 1 9[f

F20 a.

B89() z 21 0 (9)

-8 ~--21 26

A 9, )-z

2 4051

0 z 2 -23 z I

B I 0 ( W)1 z 2 7

= 0 ( 1 _B 9 z0 (12) a. -243 7a z:bl -281 -'P;' ,%' ,IG. k i" ''.;-,X " :'" "" " v_:i ," >- "-"v v-.?v £31

A12(W) = 0

(13) g3 2 The locations of the state vectors z. (i = 1,2, ..., 32) are shown in 5- Fig. 2. The joint coupling matrices B. (i = 1,2, ..., 12) may be derived from the general formulas given in [3], or directly from first principles, as is done in Appendix A. The joint coupling matrices are written as a function of radian frequency w because the elements of the joint coupling matrices depend, in general, on frequency. (Note, however, that for the rigid, massless joints considered here, the elements of the joint coupling matrices derived in Appendix A are independent of frequency.) The deriva- tions of the joint coupling matrices in Appendix A are based on the assump- tion that each joint is completely unconstrained. Joint coupling matrices Bi' 2' B and -12 are 6 x 12 matrices, and joint coupling matrices B through B 10 are 9x l8matrices. The right hand side of each of eqns. (2) through (13) is zero because it is assumed that there are no external forces or moments applied to the joints. Eqns. (2) through (13) can be combined into a single equation of the form

B (w)Z = 0 (14)

-r -Cg where B is a global joint coupling matrix given by -8- %5

B, 0 *0

-B1 o • o p 0 B 2

B (w) = (15) S

-0 . 0. 0 B2 and i is a global state vector given by -4 -1 "5 4% * (16) The global joint coupling matrix BG(w) is a 96 x 192 matrix, and the global state vector Z G is a 192 x 1 matrix. Eqn. (16) contains all the information about the dynamics of the joints in the lattice model of Fig. 2 and all the connectivity information (that is, information about which members are connected to which joints), but contains no information about the dynamics of the members of the lattice.

Transfei Matrix Relationships

.1 The Fourier transforms z3 and z1 of the state vectors z and z in .,

Fig. 2 are related by an equation of the form

-9- z =T(W)z (17) -3 where T(w) is the 6 x 6 transfer matrix of the member connecting joints 1 and 2. Eqn. (17) is the transfer matrix relationship for the member con- necting joints 1 and 2. The transfer matrix for the Bernoulli-Euler beam model and the transfer matrix for the Timoshenko beam model are given in Appendix B. The transfer matrix is written as a function of radian frequency w because the elements of T depend, in general, on frequency. Transfer matrix relationships for each of the remaining members in the lattice model of Fig. 2 can be written as

Z 5 T(w)z

2 (8 =T() (19) . -5 T(w)z10(2 z 1 ' -~ z1 (24) -8.

8 1T(w)z (22)

Z11 = T(w)z (23)

1 z20 = T( )i19 (26) i -10 -- ","€'€',; '........ ...... € ........ , Z .,..; .... i'",<'"' ,,-'.'.'- ," .-

T()-- (27)

_ 1 -!W18 z27 = T(w)z 22
(28) -26 = T(w)_ 25
(29) -29 = T (M)z 24
(30) -2 _ (1 -32 -=T )28 (1

Ile z T(w)z

30
(32) The transfer matrices in eqns. (17) through (32) are identical because the *. members in Fig. 2 are identical, and because of the choice of the local coordinate directions. (As discussed in [i], reversing the sense of the local coordinate direction changes the transfer matrix.) Eqns. (17) through (30) can be combined into a single equation of the form z !C(W)z (33) -,kb.2 where the global state vector Z is given by eqn. (16), T (w) is a 92 x 196 -., global transfer matrix given by N %J, ,,A. ' ,%-%,, , ,-11i- p-5"

I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

o 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 T 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 o o o o

0 0 _0 0 1 o 0 o o o o o o o o o

0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 T 0 0 0 0 0 0 0 0 0 0 0

0 0 T 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 o o o o o o o

o o 0 0 0 0 0 T 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 o o o o o o o ,

0 0 0 0 0 T 0 0 0 0 0 0 0 0 0 0.

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 o

0 0 0 0 0 0 0 0 0 T0T 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 T 0 0

0 0 0 0 0 0 0 0 0 0 T T 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 o

0 0 0 0 0 0 0 0 _ 0 0 0 T 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0I0 0

0 0 0 0 0 0 0 0 0 0 0 T0T 0 0

0 0 0 0 0 00 0 0 0 0 0 0 0I1

0 0 0 0 0 00 0 0 0 0 0 T 0 0

0 0 0 0 0 0 0 0 00 0 0 0 TO "

LI 0 0 0 0 0 0 0 0 0 0 1 -

12 - and GIZ is a 96 row vector given by -C/2A Z4 z 6 z -72 -24

£125

-13 z

£252

contains half the state vectors in Z .Eqn. (35) contains all the information about the dynamics of the members in the lattice model of Fig. 2, but contains no information about the joint dynamics and no 4 connectivity information. -13 p e r -FF r r fr % ' .CC 'r'r

Determination of Natural Frequencies

Substitution of eqn. (33) into eqn. (14) gives

B (w)T ()Z 0 (36)

2 - where the product B (w)T (w) is a 96 x 96 matrix. For each value of w, -G -- eqn. (36) is a system of homogeneous linear equations for the components of the state vector Z The values of w for which a nontrivial solution for Z may exist must satisfy the equation r -0G/2 det(B (M)T (M)) = 0 (37)quotesdbs_dbs45.pdfusesText_45

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