Comput~'s & St""CIUl'~1 Vol. 49. No.3. pp. 3~IO, 1993Printed in Great Britain.

0045-7949/93 $6.00 + 0.00<t> 1993 Pergamon Press Ltd

FINITE ELEMENT VIBRATION ANALYSIS OF A HELICALLYWOUND TUBULAR AND LAMINATED

COMPOSITE MATERIAL BEAM

c. I. CHEN, V. H. MUCINO and E. J. BARBERO

Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown,WV 26505, U.S.A.

(Received 14 July 1992)

Abstract-Finite element stiffness and consistent mass matrices are derived for helically wound,symmetrical composite tubes. The tubular element is considered to have constant cross-section and smalldeformations restricted to a plane. Each node has three degrees of freedom: axial and transversedisplacement and rotation (slope of transverse displacement). Shell theory and lamination theory are usedto formulate element stiffness matrices. The stiffness and mass matrices derived from the helically woundtubular composite material are reduced to symmetrically laminated composite beam. The free vibrationand natural frequency are investigated for five different materials: steel, aluminum, carbon/N5280,Kevlar-49/epoxy and graphite/epoxy composites and various layup configurations. One application of arotating flexible beam is investigated. The dynamic model of the flexible rotating beam includes thecoupled effect between the rigid body motion and the flexible motion. The inverse dynamic simulationis performed by a prescribed driving torque in the numerical simulation. The influence of flexibility onrigid body motion are presented and discussed. From the numerical results, the composite materialstrongly possesses the lower power consumption and the passive control in damping the vibration of thestructure.

INTRODUcnON

The dynamic modeling of structures with beam orrod elements has been well developed in the past,especially the area of dynamic analysis of multibodymechanism systems, such as four-bar mechanism,crank--slider mechanism and robotic manipulator.The rigid body assumption in the analysis of multi­body systems is not adequate due to high operationspeeds, large external load and high precision require­ments. The analysis of flexible links in the multibodysystems has been developed in the past twodecades [1-5]. Generally speaking, research in thisarea can be divided into two categories: The firsttreats the elastic links of the mechanical system as acontinuous system [6-10]. The equations of motionfor these continuous systems are derived with the helpof certain simplifying assumptions and solved toobtain the system response. In the second category,the elastic links of the system are modeled as discretesystems via finite element formulations [II-IS]. Theadvantages of the finite element formulations arethat they provide an easier and systematic modelingtechnique for complex mechanical systems. Thedrawback of the second method is the requirementof substantial computer time simulation due to themany degrees of freedoms in the system. However,modal synthesis can be utilized to reduce the degreesof freedom. In addition, the accuracy of the dynamicresponse depends on the selection of the modaldegrees of freedoms [16, 17]. Furthermore, most of

the work in this area considered flexible bodiesmade of isotropic materials. Recently, an alternativephilosophy has been proposed for the design offlexible multibody systems which requires themembers to be fabricated with advanced compositematerials [18-24]. Generally speaking, compositematerials possess much higher strength-to-weightratios and stiffness-to-weight ratios than metals. Con­sequently, the systems experience much smallerdeflections when subject to load.

Thompson [18, 19] used an elastic continuousmodel for a fibrous composite material to predict theglobal features of the dynamic response of a four­bar mechanism fabricated from a graphite/epoxymaterial. The results of the study demonstratehow vibrational activity in high-speed machinerycan be reduced by fabricating members from acomposite laminate rather than commercial metals.Shabana [22] presented a finite element scheme forstudying the dynamic response of crank-slider andPeucellier mechanism with components manufac­tured from orthotropic material. They concluded thatcomposite materials can be treated as an effectivepassive control system. D'Acquisto [23] developed afinite element model for the preliminary design of ahelically wound tubular composite structure whichcan be used as links of high-speed mechanisms.Krishnamurthy et al. [24] investigated a dynamicmodel for single-link robdtic arms from or~hotropic

composite materials. Chen and Yang [25] presented afinite element model for anisotropic symmetrically

399

400 c. I. CHEN et al.

(I)

QI6 =(QII - QI2 - 2Q66)sin 6 cos] 6

where t/J is the polar coordinate. [Q](k) is the trans­formed reduced stiffness matrix and is given by

for a particular layer k in local coordinate can beexpressed as

laminated beam including the effect of shear defor­mation and torsional deformation.

The main objective of this study is to fonnulate themass and the stiffness matrices of laminate tubularbeam element to conduct the dynamic analysis ofcomposite laminate beam structure with future appli­cation to the flexible multibody systems. The massand stiffness matrices of a tubular beam derivedin this paper are also reduced to fiber-reinforcedorthotropic material with a symmetrically laminatedcomposite rectangular cross-section. Validation ofthe element with· the existing analytical solution fora rectangular cross-section is presented. The freevibration of natural frequency is analyzed in terms offive materials: steel,aluminum, carbon/N5280 com­posite, Kevlar-49/epoxy and graphite/epoxy com­posite. Various layups of each composite material areconsidered. A flexible rotating beam is consideredas one application based on the matrices developedin this. study. The coupled effect between therigid body motion and the ftexiblemotion is included.The inverse dynamic simulation is performed by aprescribed .driving torque in the numerical simu­lation. The influence of flexibility on rigid bodymotion is presented and discussed. It can be shownthat the composite material strongly possesses thepassive control in damping the vibration of thestructure.

LAMINATE SHELL THEORY

where 6 is the angle from the x -axis to the I-axis andQij are the so-called reduced stiffnesses and can beexpressed in terms of engineering constants [26]where "Ix. = 2lx4J" Neglecting the high order terms, the

In a laminated on composite material, a lamina isa single layer with thickness h, with a unidirectionalor woven fiber orientation at angle 8 from alocal coordinate· axis as shown in Fig. 1. A laminateis a stack of laminae with various orientationsof principal material directions in the laminae. Incombining shell theory with lamination theory,the assumption is made that each composite laminacan be analyzed as a thin cylindrical shell andelastic deformations are assumed to be small withinthe linear range. The relation of stress and strain

(2)

x

Fig. 1. Schematic of the k th ply in the tubular element.

Finite element analysis of composite tubes 401

Fig. 2. Laminated helically wound beam configuration.

kinematic equations of a cylindrical element can beexpressed by [27]

where the membrane strains in the reference surfaceof the shell are

(7)

(6)

(5)

flal2 ( z)Nx= uxl+-dz

-la/2 a

fla

/2 ( z)M x = Ux 1 +- z dz.-"/2 a

where a is the average radius of the cylinder; u, v, andware the deformations associated with x, t/J and raxes, respectively. The resultant force N and momentM are obtained by integration of the stresses in eachlamina through the laminae thickness as shown inFig. 2. For example

(4)

, (3)

ouox

I (OV )~ ot/J +w

I ou ov--+­aocjJ ax

£~ +zXx

£~+Z(l-;)x~

[I +~}~.+2z[I- ~]X%~

and the changes in curvature and twist of the refer­ence surface are

Substituting eqn (1) into (6) and (7) and integratingthrough the laminae thickness, the final result can bewritten in a compact form as

(8)

·rD66

Xx

D66

Dl1B l1 +­a

B66

B)2AJ2 +­

a

8 16A 16 +­

a

BlIAIJ +­Q

DIlBII +­a

8)2

D J6BI6+­a

8 16

Nx

N.Nx•

N.x

M x

M.M x•

M.x

402

where

N

Aij= L Q~~(h(k)-h(k-1)k=1

B - ~ 1 ?i(k)(h 2 h2 )ij - i..J 2\!ij <k) - (k - I)

k=1

C. I. CHEN et af.

~ the nondimensional parameter defined by ~ = xl/.The generalized coordinates e l , e2' e3' e4,eS and e6

represent elastic deformations at -the nodal points.The deformation at an arbitrary point in the elementis approximated in terms of el' e2' e3' e4, es, and e6and the shape functions as

D - ~ 1 n(k)(h 3 h3 )ij - i..J -3. \:!ij <k) - (k - I) • (9)

k-I

For the classical lamination beam theory, we as­sumed that the laminate consists of perfectly bondedlaminae, bonds are infinitesimally thin, and sheardeformation is negligible. The displacements are con­tinuous across lamina boundaries so that no laminacan slip relative to another. Furthermore, only smalldeformations are considered· in the. context of thinplate theory. A line originally straight and perpen­dicular to the middle surface of the laminate isassumed to remain straight and normal when thelaminate is extended and bent. Due to the assumptionofaxisymmetry with respect to the neutral axis of thebeam and neglecting Poisson effects, eqn (8) can besimplified to [23]

el

e2

{u(e, t)} = [N1 0 0 N2 0

~Je3

w(~, t) 0 N'j N4 0 Ns e4

ese6

(11)

where Ni are the shape functions and are given by

{Z:}= {i}, (10)

(12)

where £~ = au/ax, Kx = -C 2W/CX 2 and u and warethe axial transverse deformations, respectively. For arectangular beam, a approaches infinity and eqn (10)reduces to the well known expression for laminatedbeams if all Poisson ratios~re taken to be zero [29].

EQUATION OF MOTION

The equations of motion for a typical element arederived using the Lagrangian function. A typicalelement configuration and coordinate system areshown in Fig. 3~ with I the length of the element, and

z

The kinetic energy of the beam element is given by

where Pm, A, and / are the density, the cross-sectionalarea,_ and the length of the beam, respectively.

The potential energy stored in the laminated beamis given by

xFig. 3. Generalized coordinates of the beam element.

Finite element analysis of composite tubes 403

Table 1. Material properties

CarbonI Kevlar-49/ Graphite/Steel Aluminum N5280 epoxy epoxy

£, (Msi) 30.0 10.6 26.25 11.02 20.0£2 (Msi) 30.0 10.6 1.49 0.798 1.40G12 (Msi) 11.81 3.98 1.04 0.334 0.8

#12 0.27 0.33 0.28 0.34 0.3h (in) 0.25 0.25 0.005 0.005 0.005

£/p (107 in) 10.6 10.6 45.25 20.8 33.33Pm (lb m/in3

) 0.283 0.1 0.058 0.053 0.06

Substituting eqn (10) into (14) yields

v = xal LI(All +~IX::Y

- 2(BII +~I):: ~:~+DII (~:~Y de. (15)

The equations of motion of the element can beobtained by Lagrange formulation as follows:

(16)

where L = T - V is the Lagrangian. Substitutingeqns (13) and (15) into (16), the element equation ofmotion can be obtained as

NUMERICAL RESULTS

Both rectangular and tubular cross-sections areconsidered in this section. The material· propertiesand the geometry parameters are listed in Tables Iand 2, respectively. Two types of laminates areconsidered, a[±tx] angle ply and repeated clusters of[tx/O/-tx]N, where N is the number of clusters andis taken to be 17 in the numerical results presented.All numerical results illustrate the first five modes.Natural frequencies for all material and both cross­sectional shape (rectangular and tubular) in differentlayups are shown in Figs 4-11. It can be concludedthat the natural frequency depends on the layup andthe stiffness-to-weight ratio. The natural frequency ofthe composite material in a certain fiber angle layup

In eqn (17), [M] and [K] are the mass and thestiffness matrices, respectively, and they are givenin the Appendix. The stiffness matrix in eqn (17)can be ·reduced to the case of a rectangular cross­sectional beam element by assuming a approachesinfinity and 2na is replaced by the width of thebeam. In this case, they coincide with the expressions·presented by Cleghorn and Chao [13]. The overallequation of motion is assembled on each elementbasis with adequate boundary conditions whichyields

[M]{q} + [K]{q} = o. (17)

8......- ---,

i Steel0-0 Aluminum....... [01

- -- [22.5JH8 - [45]ei --- [67.5]~ -- [90]c"='~8r:!

3Mode

Fig. 4. Frequency variation of fiber angle for carbon/N5280composite material, with a rectangular cross-section.

where [MG ] and [KG] are the global mass and stiffnessmatrices, respectively, and {qG} is the global general­ized coordinates. The free vibration of the systemwith harmonic frequency co, eqn (18) becomes

Table 2. Geometry parameters

Rectangular Tubular

Length (in) 48.00 48.00Width (in) 0.75

Thickness (in) 0.25 0.25Outer diameter (in) 0.75Inner diameter (in) 0.25

8......- ----,I .....-e [OJ

- (22.5J-- [45J--- (67.5)

H8 -- (90)

=-§ ::::::: ~~~76J!452l·5]~ •.••.•• [67.5/0/-67.5]; [90/0/-90)='~8

&:-

·f3Mode

Fig. 5. Frequency of carbon/N5208 composite material, ina different layup, with a rectangular cross-section.

404 C. I. CHEN et OJ.

3

Mode

[OJ£22.5J(45J(67.5]£901(22.5/0/-22.5](45/0/-45J(67.5/9/-67.5)£90/0/-90)

1....--- ------------,!

Fig. 9. Frequency of graphite/epoxy composite material, ina different layup, with a rectangular cross-section.

"

•.••-•• SteelD-et 'HI CI Aluminum........ [0]

(22.5)[45][67.5][90]

~-r-------------------'§

3

Mode

Fig. 6. Frequency variation of fiber angle for Kevlar-49/epoxy composite material, with a rectangular cross-section.

.'$.­.-........................

•.••.•• Steelo-o.ca. Aluminum........, [0]

£22.51[45](67.5J(90l

.-r-----------------~!

8.--~:;;;:;::;:;:::;:;~,.-,-.,.."T"'T"T-r"T""1r-r-r"T"'T"T1""T""11"""rT"-r-rT-rT'"~«:> I 3

Mode

Fig. 10. Frequency variation of fiber angle for car·bon/NS280 composite material, with a tubular cross·section.

8i-----[o-]--------------,[22.5][45][67.5)(90)[22.5/0/ -22.5][45/0/-45)[67.5/0/-67.5][90/0/-90l

8.w~~~:::~~~.-r-r-r~r-T"'"'I"*"f'~«:> I 3

Mode

Fig. 7. Frequency of Kevlar-49/epoxy composite material, ina different layup, with a rectangular cross-section.

8-....,...-.---£0-]-------------

[22.5)[45][67.5][90][22.5/0/-22.5J[45/01-45][67.5/0/-67.51(90/0/-901

~~~~~==~"""""""',.-.r"T'"T~r-r-r-rJo I 3

Mode

Fig. II. Frequency of carbon/NS208 composite material, ita different layup, with a tubular cross-section.

~_~~;:;::;:~-r-r-r-r-r"T"'T"T-r-r-r-r-r-rT'T-r-r-r--T-r-.,...,--r...,...,....,...I01

1,-------------------.•.••••• SteelO-G ••• Aluminum........-e [Ol

£22.5J(45](67.5][90J

3Mode

Fig. 8. Frequency variation of fiber angle for graphite/epoxycomposite material, with a rectangular cross-section.

is less than that of the steel material. Natural frequen­cies of carbon/N5280 are less than the steel materialwhen the fiber angle is greater than 500 as shownin Fig. 4. The more 00 layers, the higher naturalfrequency of the system. The 00 fiber angle in alllayers provides the highest natural frequency asshown in Figs 5, 7, 9,11,13 and 15. Carbon/N5280provides the highest natural frequency due to thehighest stifTness-to-weight ratio as shown in Figs 16and 17. The thickness of the layers does not influence

the natural frequencies due to the symmetry 0

the cross-section with respect to neutral axis, acan be seen from eqn (7). The natural frequencies 0

steel and aluminum beams are the 'same due to tbsame stiffness-to-weight ratio of both materiahHowever, the equivalent extemalload (load/densityto the system with aluminum material is greatethan that with steel material. For this reasonthe aluminurn material system experiences largedeformations.

II d

Finite element analysis of composite tubes 405

;---------------~........ Steel...... Aluminum........ (0]

(22.5][45][67.51(901

~

Mode

Fig. 12. Frequency variation of fiber angle for Kevlar-49/epoxy composite material, with a tubular cross-section.

1 ....... [0](22.5]

- (45]-..... [67.5]

HI -- [901:'2 •.••-•• [22.5/0/-22.5]

~ •.••-•• (45/~-45]t' •.••.•• [67.5 0/-67.5]~ ..••.•• (90/ /-901='~I

rei

~

Mode

Fig. 1S. Frequency, of graphite/epoxy composite material,in a different layup, with a tubular cross-section.

I--'--.-.-.-.-.-(0-]---------------,

(22.5]- [45]

[67.5][90]

•.••.•• (22.5/0/-22.5]•.••-•• (45/~-45]•.••.•• [67.5 0/-67.5]........ (90/ /-901

~

Mode

Fig. 13. Frequency of Kevlar-49/epoxy composite material,in a different layup, with a tubular cross-section.

;..,.-----------------.-Corbon/N5280- Kevlor49/Epoxy-.- Grophite/Epoxy•.••.•• Steel0-.... Aluminum

Fig. 16. Natural frequency of different materials with rec­tangular cross-section.

8~~~~::;;~~r_rT"T'"T"T'T_rr"T'"T"'T"r'f""rT"'1~01 3

Mode

Fig. 14. Frequency variation of fiber angle for graph­ite/epoxy composite material, with a tubular cross-section.

.....---.-...., .......•.......~ .

;__----------------w- Corbon/N5280- Kevlor49/Epoxy-- Grophite/Epoxy.-••-•• Steel0-0 .. 11 Aluminum

8-Fr:.r-r-.,.....,.........,..,..,....,...,...,...,1"""r'T'.,..,..T'T""I-r-r~.,.....,.....r-w-r.,....,...,..,....r-r-r..,....,...,....(o I S

Mode

Fig. 17. Natural frequency of different materials withtubular cross-section.

;.,.......--:-------------.....,.-••.•• Steel0-.... Aluminum •...... (0]

[22.5)[45J[67.5][90]

(20)

Validation of the present formulation is presentedby comparison with the analytical solution for thetransverse vibration of the beam structure [28]. Anapproximation to the governing equation of a rec­tangutar laminated beam in transverse vibration canbe derived from the traditional isotropic materialproblem by substituting Elwith bD•• [28] (see [29] foran exact formulation)

04W 02W

bDIl ox4 + pA arr =O.

The natural frequency and the mode shapedepend on the boundary condition of the beam.The comparison between the analytical and thefinite element solution in this paper is given inTable 3 for fixed-free ends. A total of five linearelements was used to model the length of thebeam with 0° layup. The material properties andgeometrical parameters are given in Tables I and 2.The numerical solution validates that the fonnulationof the mass and the stiffness matrices developed inthis paper.

CAS 49/l-B

406

- -----------------------------------------------------------

C. I. CHEN et al.

Table 3. Comparison results of analytical method and FEM

CarbonfN5280 Kevlar-49/epoxy Graphite/epoxy

Present Present PresentNode An~lytica1 analysis Analytical analysis Analytical analysis

1 7.34638 7.34661 4.98922 4.98931 6.31053 6.312642 46.03922 46.06278 31.26706 31.28269 39.54763 39.567433 128.91107 129.37574 87.54862 87.86316 110.73446 111.132154 250.91829 255.57972 170.40857 173.57235 215.53852 219.539825 417.59052 424.19880 283.60230 288.08681 358.70978 364.38143

APPLICATION

In this section, a flexible rotating beam is con­sidered in the application of the matrices developedin this study. Figure 18 shows the configurationin finite element model of a flexible rotating beamsystem. The finite element method in this type ofproblem has been well developed and the approachcan be referred to in [5].

Based on the small linear deformation, theequations of motion can be expressed in matrix formsas follows:

theory, the coupled effect neglects such that theforward dynamics analysis can be applied to a specificprescribed motion. However, this simplification cancause inaccuracy in simulating the dynamic behaviorof the system. In order to maintain the equations ofmotion with the time variation of the mass matrixand the coupled effect, the inverse dynamic procedureis performed without any simplifications. The beamrotates under a prescribed torque in horizontal planewithout any other external load. The prescribedtorque is given

[1Ds6 IIIsfJ{Ii} [0 0]{8} {Fdl

} {Qve}m"m mff ijf + 0 K.u: qf = F~f + Q"f'

(21){

41 (Ib-inch) 0 ~ 1 ~ 2T(/) = 8 (lb-inch) 2 < 1 ~ 4

8 - 41 (Ib-inch) 4 < 1 ~ 6.(23)

where 8 is the rigid body motion generalized coordi­nate; Clr are the overall elastic generalized coordinatesof the nodes; m.o- and Kff are the global mass andstiffness matrices; Ft6 and F~ are generalized forces;and QIl6 and Q"f are the quadratic velocity vectorsresulting from differentiating the kinetic energy withrespect to time and with respect to the body coordi­nates. Equation (21) can be-expressed in a conciseform

[M]{ij} + [K]{q} = {F~} + {Q,,}. (22)

Due to the coupled effect between rigid bodymotion and elastic deformation, the equations ofmotion are nonlinear; the nonlinearity being at­tributed to the time dependency of the mass matrix.The time-dependent mass matrix contains the gener­alized elastic coordinates. In linear elastodynamics

y

(qf}·(q~,~,~)

x

Fig. 18. General finite element configuration of flexiblebeam.

In this numerical simulation, a total of three linearelements were used to model the length of the beamwith a 0° layup. A small integration time step muslbe selected in order to obtain the accurate solutionfor' such type of stiffened system. The pre­dict-eorrector Adams-Moulton algorithm is usedin this example [30]. However, the carbonfN528Cmaterial cannot converge no matter how small thetime step due to the high stiffness-to weight ratio­The material and geometry parameters are given ir.Tables 1 and 2 for a rectangular cross-section and fOJ

a length of 60 in.Four materials are selected in this simulation: steel

aluminum, Kevlar-49/epoxy, and graphite/epoxyThe major difference relies on the stiffness-to-weigh1

ratio of each material. The responses of angulaJdisplacement and acceleration of the isotropic beanin rigid body motion are shown in Figs 19 and 20respectively. The corresponding responses of thecomposite beam are shown in Figs 22 and 23. It i:observed that, as shown in Fig. 19, the total angulaJdisplacement of the steel beam is about one and a

half revolutions. On the contrary, the total angula:displacement of the graphite beam is about seve)revolutions. Lower power consumption is require(for the beam with a higher stiffness-to-weight ratio ithe same angular displacement occurs. Although thlsteel and the aluminium have the same stiffness-toweight ratio, the aluminium material has a higheequivalent external load (load/density) so that tbaluminium material experiences larger deflectioncompared to the steel material. The influence 0

Finite element analysis of composite tubes 407

..... Steel (ri,id motion)

........ Steel (nexible motion)

....... Aluminum (rield motion)

........... Aluminum (fiexible motion)

Fig. 19. Angular displacement of the rotating isotropic material beam.

e52 3Time (sec)

......... Steel (rigid motion)

..... Steel (fiexible motion)........ Aluminum (ri,id motion)

......... Aluminum (nexible motion)

~

ucuI)

~ •.0 ..,---------------------;--------,cu~

"-~ 2.0as~

""-'

Fig. 20. Angular aCceleration of the rotating isotropic material beam.

I,

Time (sec)2o

0.5 ---------------------------,

+JCQ)

a-0.5cuU......g..

.!3 -1.0Q

.-..

.E 0.0'-'

Fig. 21. End point displacement of the rotating isotropic beam.

flexibility on the angular displacement is minimal.The flexibility effect on the angular acceleration isimportant for all selected materials.

As in Figs 21 and 24 show the transverse defor­mations are dominated by the angular acceleration.In Fig. 20 the steel beam subjected to a maximum

angular acceleration of 0.8 (rad/sec/sec) experiencesa maximum deformation of 0.28 in as shown inFig. 21. However, as shown in Figs 23 and 24, thecorrespondence of angular 'acceleration and maxi­mum deformation are 4 (rad/sec/sec) and 0.45 in withgraphite/epoxy material. The angular acceleration

C. I. CHEN et ale

~ Graphite \ri1id motion).............. Gra hite fiexible motion).......... Kevfar-49 riCid motion)............ Kevlar-.9 nexible motion)

2 3Time (sec)

Fig. 22. Angular displacement of the rotating composite material beam.

•

53Time (sec)

2

~ Graphite Iri1id motion).......... Gra bite nexible motion)..... Xevfar-49 riCid motion)............ Kevlar-4-9 nexible motion)

~

<>4>~

" 10.0<>4>~

"~as S.O..."-"

d0

::3 0.0as...as....4><>(,) -0.0<~as.....~-10.0

= 0<

Fig. 23. Angular acceleration of the rotating composite material beam.

0.& ~--------------------------..

:s 0.0

"'-'

...,l

; -0.5

at)

~ -1.0.....C.CD

S -1.5

..........- Graphite

........... Kevlar-4-9

2o 3Time (sec)

Fig. 24. End point displacement of the rotating composite beam.

with graphite/epoxy material is five times higher thanthat of steel material but the maximum deformationis about two times greater than that of the steelmaterial. On the other hand, the composite materialcan be subjected to higher inertial forces. This can beattributed to the high performance of the stiffness-to-

weight ratio in composite materials. The maximumdeformation in the linear deformation model is about50% greater than that of geometry nonlinearity de­formation for both materials. Therefore, the correc1mathematical model is indispensable for more accu­rate analysis of realistic flexible structure systems.

Finite element analysis of composite tubes 409

CONCLUSIONS

The free vibration natural frequency analysis of ahelically wound tubular and laminated beam com­posite material has been investigated in this paper.Based on the shell lamination theory and smalldeformation assumption, the mass and stiffness finiteelement matrices of a helically wound tubular be~mhave been derived. Each element has three degrees offreedom: axial, transverse, and rotation. The mass andthe stiffness matrices derived from the helically woundelement are also reduced to the orthotropic fiber­reinforced symmetrical laminated composite beam.These matrices can be used for the dynamic analysisofmultibody flexible systems with composite materiallinks. The natural frequency of the beam elementdepends on the layup and the stiffness-to-weight ratio.

A flexible rotating beam is investigated in thisapplication. The dynamic model of the flexible rotat­ing beam includes the coupled effect between the rigidbody motion and the flexible motion. The equationsof motion of the flexible rotating beam are nonlinearwith time-dependent coefficients and are expressed interms of the elastic generalized coordinates and thelarge angular displacement of the beam. The inversedynamic simulation is performed by a prescribeddriving torque in the numerical simulation. The re­sponses of the transverse deformation and the influ­ence of the flexibility on the rigid body motion areevaluated and discussed. For the composite materialit is shown that it strongly possesses .lower powerconsumption and the passive control in damping thevibration of the structure.

In future work, shear deformation and geometricalnonlinearity will be considered for more accuratemodeling of realistic system such as robotic manipu­lators, rotating blades and mechanisms, built withcomposite materials.

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410 C. I. CHEN et ale

APPENDIX

1 13

0 06

0 0

J.0

13 11/0

9 13/

35 210 70 42011/ L2 13/ 12

0210 105

0520 140

[M] =p",AI 1 10

60 0

30

09 13/

013 III

70 420 35 210

0131 /2

0111 12

420 140 210 105

PI 0p2 PI 0 P2

/ I / /

012fJ3 6{J3

012fJ3 6fJ3

13 12 -13 rfJ2 6fJ3 4P3 fJ2 6fJ3 2fJ3/ p.- I I -12 I

[K] =21ta fJI 0 P2 PI 0fJ2

/ / / /

012fJ3 6fJ3

012fJ3 6fJ3

-13 -r 13 -p.-fJ2 6fJ3 2fJ3 P2 6P3 4fJ3/ r I / -12 /

where

BIIfJl=A II +-

a

DIIfJ2=B II +-

a

fJ3 =DII •"