Comput~'s & St""CIUl'~1 Vol. 49. No.3. pp. 3~IO, 1993 0045-7949/93 $6.00 + 0.00 Printed in Great Britain.
FINITE ELEMENT VIBRATION ANALYSIS OF A HELICALLY WOUND 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 small deformations restricted to a plane. Each node has three degrees of freedom: axial and transverse displacement and rotation (slope oftransverse displacement). Shell theory and lamination theory are used to formulate element stiffness matrices. The stiffness and mass matrices derived from the helically wound tubular composite material are reduced to symmetrically laminated composite beam. The free vibration and 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 a rotating flexible beam is investigated. The dynamic model of the flexible rotating beam includes the coupled effect between the rigid body motion and the flexible motion. The inverse dynamic simulation is performed by a prescribed driving torque in the numerical simulation. The influence of flexibility on rigid body motion are presented and discussed. From the numerical results, the composite material strongly possesses the lower power consumption and the passive control in damping the vibration of the structure.
INTRODUcnON the work in this area considered flexible bodies made of isotropic materials. Recently, an alternative The dynamic modeling of structures with beam or philosophy has been proposed for the design of rod elements has been well developed in the past, flexible multibody systems which requires the especially the area of dynamic analysis of multibody members to be fabricated with advanced composite mechanism systems, such as four-bar mechanism, materials [18-24]. Generally speaking, composite crank--slider mechanism and robotic manipulator. materials possess much higher strength-to-weight The rigid body assumption in the analysis of multi ratios and stiffness-to-weight ratios than metals. Con body systems is not adequate due to high operation sequently, the systems experience much smaller speeds, large external load and high precision require deflections when subject to load. ments. The analysis of flexible links in the multibody Thompson [18, 19] used an elastic continuous systems has been developed in the past two model for a fibrous composite material to predict the decades [1-5]. Generally speaking, research in this global features of the dynamic response of a four area can be divided into two categories: The first bar mechanism fabricated from a graphite/epoxy treats the elastic links of the mechanical system as a material. The results of the study demonstrate continuous system [6-10]. The equations of motion how vibrational activity in high-speed machinery for these continuous systems are derived with the help can be reduced by fabricating members from a of certain simplifying assumptions and solved to composite laminate rather than commercial metals. obtain the system response. In the second category, Shabana [22] presented a finite element scheme for the elastic links of the system are modeled as discrete studying the dynamic response of crank-slider and systems via finite element formulations [II-IS]. The Peucellier mechanism with components manufac advantages of the finite element formulations are tured from orthotropic material. They concluded that that they provide an easier and systematic modeling composite materials can be treated as an effective technique for complex mechanical systems. The passive control system. D'Acquisto [23] developed a drawback of the second method is the requirement finite element model for the preliminary design of a of substantial computer time simulation due to the helically wound tubular composite structure which many degrees of freedoms in the system. However, can be used as links of high-speed mechanisms. modal synthesis can be utilized to reduce the degrees Krishnamurthy et al. [24] investigated a dynamic of freedom. In addition, the accuracy of the dynamic model for single-link robdtic arms from or~hotropic response depends on the selection of the modal composite materials. Chen and Yang [25] presented a degrees of freedoms [16, 17]. Furthermore, most of finite element model for anisotropic symmetrically
399 400 c. I. CHEN et al. laminated beam including the effect of shear defor for a particular layer k in local coordinate can be mation and torsional deformation. expressed as The main objective of this study is to fonnulate the mass and the stiffness matrices of laminate tubular beam element to conduct the dynamic analysis of composite laminate beam structure with future appli (I) cation to the flexible multibody systems. The mass and stiffness matrices of a tubular beam derived in this paper are also reduced to fiber-reinforced where t/J is the polar coordinate. [Q](k) is the trans orthotropic material with a symmetrically laminated formed reduced stiffness matrix and is given by composite rectangular cross-section. Validation of the element with· the existing analytical solution for a rectangular cross-section is presented. The free vibration of natural frequency is analyzed in terms of five materials: steel,aluminum, carbon/N5280 com posite, Kevlar-49/epoxy and graphite/epoxy com posite. Various layups ofeach composite material are considered. A flexible rotating beam is considered as one application based on the matrices developed in this. study. The coupled effect between the rigid body motion and the ftexiblemotion is included. The inverse dynamic simulation is performed by a prescribed .driving torque in the numerical simu lation. The influence of flexibility on rigid body motion is presented and discussed. It can be shown QI6 (QII that the composite material strongly possesses the = - QI2 - 2Q66)sin 6 cos] 6 passive control in damping the vibration of the structure.
LAMINATE SHELL THEORY
In a laminated on composite material, a lamina is a single layer with thickness h, with a unidirectional or woven fiber orientation at angle 8 from a local coordinate· axis as shown in Fig. 1. A laminate is a stack of laminae with various orientations of principal material directions in the laminae. In (2) combining shell theory with lamination theory, the assumption is made that each composite lamina where 6 is the angle from the x -axis to the I-axis and can be analyzed as a thin cylindrical shell and Qij are the so-called reduced stiffnesses and can be elastic deformations are assumed to be small within expressed in terms of engineering constants [26] the linear range. The relation of stress and strain where "Ix. = 2lx4J" Neglecting the high order terms, the
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 be expressed by [27]
£~ +zXx £~+Z(l-;)x~ , (3) [I +~}~.+2z[I- ~]X%~ (5) where a is the average radius of the cylinder; u, v, and where the membrane strains in the reference surface ware the deformations associated with x, t/J and r of the shell are axes, respectively. The resultant force N and moment ou M are obtained by integration of the stresses in each lamina through the laminae thickness as shown in ox Fig. 2. For example
I (OV ) (4) la ~ ot/J +w l2 ( Nx= uxl+-dzz) (6) -la/2 a I f ou ov la --+ /2 ( aocjJ ax M = U 1 z) x x +- z dz. (7) f-"/2 a and the changes in curvature and twist of the refer Substituting eqn (1) into (6) and (7) ence surface are and integrating through the laminae thickness, the final result can be written in a compact form as
BlI B)2 A Dl1 IJ + AJ2 + Bl1 + Q a a
Nx N. 8 16 A 16 + a Nx•
N.x
M x (8) DIl BII + Xx M. a M x• 8)2 M.x D J6 ·r BI6 + a D66
8 16 B 66 D66 402 C. I. CHEN et af. where ~ the nondimensional parameter defined by ~ = xl/.
The generalized coordinates el , e2' e3' e4,eS and e6 N represent elastic deformations at -the nodal points. Aij= L Q~~(h(k)-h(k-1) k=1 The deformation at an arbitrary point in the element is approximated in terms of el' e2' e3' e4, es, and e6 - ~ 1 ?i(k)(h 2 h2 ) and the shape functions as Bij - i..J 2\!ij {Z:}= {i}, (10) (12) £~ 2 2 The kinetic energy of the beam element is given by where = au/ax, Kx = -C W/CX and u and ware the axial transverse deformations, respectively. For a rectangular beam, a approaches infinity and eqn (10) reduces to the well known expression for laminated beams if all Poisson ratios~re taken to be zero [29]. where Pm, A, and / are the density, the cross-sectional area,_ and the length of the beam, respectively. EQUATION OF MOTION The potential energy stored in the laminated beam The equations of motion for a typical element are is given by derived using the Lagrangian function. A typical element configuration and coordinate system are shown in Fig. 3~ with I the length of the element, and z x Fig. 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 1.40 G12 (Msi) 11.81 3.98 1.04 0.334 0.8 #12 0.27 0.33 0.28 0.34 0.3 h (in) 0.25 0.25 0.005 0.005 0.005 £/p (107 in) 10.6 10.6 45.25 20.8 33.33 3 Pm (lb m/in ) 0.283 0.1 0.058 0.053 0.06 Substituting eqn (10) into (14) yields NUMERICAL RESULTS Both rectangular and tubular cross-sections are v = xal LI(All + ~IX::Y considered in this section. The material·properties and the geometry parameters are listed in Tables I and 2, respectively. Two types of laminates are - 2(BII + ~I):: ~:~ + DII (~:~Y de. (15) considered, a[±tx] angle ply and repeated clusters of [tx/O/-tx]N, where N is the number of clusters and The equations of motion of the element can be is taken to be 17 in the numerical results presented. obtained by Lagrange formulation as follows: All numerical results illustrate the first five modes. Natural frequencies for all material and both cross sectional shape (rectangular and tubular) in different (16) layups are shown in Figs 4-11. It can be concluded that the natural frequency depends on the layup and where L = T - V is the Lagrangian. Substituting the stiffness-to-weight ratio. The natural frequency of eqns (13) and (15) into (16), the element equation of the composite material in a certain fiber angle layup motion can be obtained as 8...... - ---, [M]{q} + [K]{q} = o. (17) i Steel 0-0 Aluminum ...... [01 In eqn (17), [M] and [K] are the mass and the - -- [22.5J H8 - [45] stiffness matrices, respectively, and they are given ei --- [67.5] in the Appendix. The stiffness matrix in eqn (17) ~ -- [90] can be ·reduced to the case of a rectangular cross c "=' sectional beam element by assuming a approaches ~8 infinity and 2na is replaced by the width of the r:! beam. In this case, they coincide with the expressions· presented by Cleghorn and Chao [13]. The overall equation of motion is assembled on each element 3 basis with adequate boundary conditions which Mode yields Fig. 4. Frequency variation offiber angle for carbon/N5280 composite material, with a rectangular cross-section. 8...... - ----, where [MG ] and [KG] are the global mass and stiffness I .....-e [OJ matrices, respectively, and {qG} is the global general - (22.5J -- [45J ized coordinates. The free vibration of the system --- (67.5) with harmonic frequency co, eqn (18) becomes H8 -- (90) =-§ ::::::: ~~~76J!452l·5] ~ •.••.•• [67.5/0/-67.5] ; [90/0/-90) =' ~8 Table 2. Geometry parameters &:- Rectangular Tubular Length (in) 48.00 48.00 Width (in) 0.75 ·f3 Thickness (in) 0.25 0.25 Mode Outer diameter (in) 0.75 Fig. 5. Frequency of carbon/N5208 composite material, in Inner diameter (in) 0.25 a different layup, with a rectangular cross-section. 404 C. I. CHEN et OJ. ~-r------' 1....------, § ! [OJ •.••-•• Steel £22.5J D-et 'HI CI Aluminum (45J ...... [0] (67.5] (22.5) £901 [45] (22.5/0/-22.5] [67.5] " (45/0/-45J [90] (67.5/9/-67.5) £90/0/-90) 3 3 Mode Mode Fig. 6. Frequency variation of fiber angle for Kevlar-49/ Fig. 9. Frequency of graphite/epoxy composite material, in epoxy composite material, with a rectangular cross-section. a different layup, with a rectangular cross-section. i-----[o-]------,8 .-r------~ ! •.••.•• Steel [22.5] o-o.ca. Aluminum [45] ...... , [0] [67.5) £22.51 (90) [45] [22.5/0/-22.5] (67.5J [45/0/-45) (90l [67.5/0/-67.5] [90/0/-90l . .'$ ...... - ...... 8.w~~~:::~~~.-r-r-r~r-T"'"'I"*"f'~ 8.--~:;;;:;::;:;:::;:;~,.-,-.,.."T"'T"T-r"T""1r-r-r"T"'T"T1""T""11"""rT"-r-rT-rT'"~ «:> I 3 «:> I 3 Mode Mode Fig. 7. Frequency ofKevlar-49/epoxy composite material, in Fig. 10. Frequency variation of fiber angle for car· a different layup, with a rectangular cross-section. bon/NS280 composite material, with a tubular cross·section. 1 8 ,------. -....,...-.---£0-]------•.••••• Steel [22.5) O-G ••• Aluminum ...... -e [Ol [45] £22.5J [67.5] (45] [90] (67.5] [22.5/0/-22.5J [90J [45/01-45] [67.5/0/-67.51 (90/0/-901 ~_~~;:;::;:~-r-r-r-r-r"T"'T"T-r-r-r-r-r-rT'T-r-r-r--T-r-.,...,--r...,...,....,...I ~~~~~==~"""""""',.-.r"T'"T~r-r-r-rJ 01 3 o I 3 Mode Mode Fig. 8. Frequency variation offiber angle for graphite/epoxy Fig. II. Frequency of carbon/NS208 composite material, it composite material, with a rectangular cross-section. a different layup, with a tubular cross-section. is less than that ofthe steel material. Natural frequen the natural frequencies due to the symmetry 0 cies of carbon/N5280 are less than the steel material the cross-section with respect to neutral axis, a when the fiber angle is greater than 50 0 as shown can be seen from eqn (7). The natural frequencies 0 in Fig. 4. The more 00 layers, the higher natural steel and aluminum beams are the 'same due to tb frequency of the system. The 00 fiber angle in all same stiffness-to-weight ratio of both materiah layers provides the highest natural frequency as However, the equivalent extemalload (load/density shown in Figs 5, 7, 9,11,13 and 15. Carbon/N5280 to the system with aluminum material is greate provides the highest natural frequency due to the than that with steel material. For this reason highest stifTness-to-weight ratio as shown in Figs 16 the aluminurn material system experiences large and 17. The thickness ofthe layers does not influence deformations. II d Finite element analysis of composite tubes 405 ;------~ 1 ...... [0] ...... Steel (22.5] ...... Aluminum - (45] ...... (0] -..... [67.5] (22.5] HI -- [901 [45] :'2 •.••-•• [22.5/0/-22.5] [67.51 ~ •.••-•• (45/~-45] (901 t' •.••.•• [67.5 0/-67.5] ~ ..••.•• (90/ /-901 ='~I rei ~ ~ Mode Mode Fig. 12. Frequency variation of fiber angle for Kevlar-49/ Fig. 1S. Frequency, of graphite/epoxy composite material, epoxy composite material, with a tubular cross-section. in a different layup, with a tubular cross-section. I --'--.-.-.-.-.-(0-]------, ;..,.------. (22.5] -Corbon/N5280 - [45] - Kevlor49/Epoxy [67.5] -.- Grophite/Epoxy [90] •.••.•• Steel •.••.•• (22.5/0/-22.5] 0-.... Aluminum •.••-•• (45/~-45] •.••.•• [67.5 0/-67.5] ...... (90/ /-901 ~ Mode Fig. 13. Frequency of Kevlar-49/epoxy composite material, Fig. 16. Natural frequency of different materials with rec in a different layup, with a tubular cross-section. tangular cross-section. ;.,...... --:------....., ;__------w .-••.•• Steel - Corbon/N5280 0-.... Aluminum • - Kevlor49/Epoxy ...... (0] -- Grophite/Epoxy [22.5) .-••-•• Steel [45J 0-0 ..11 Aluminum [67.5] [90] .. .-.. , ...... - -- ...... •...... ~ . 8~~~~::;;~~r_rT"T'"T"T'T_rr"T'"T"'T"r'f""rT"'1~ 8-Fr:.r-r-.,.....,...... ,..,..,....,...,...,...,1"""r'T'.,..,..T'T""I-r-r~.,.....,.....r-w-r.,....,...,..,....r-r-r..,....,...,....( 01 3 o I S Mode Mode Fig. 14. Frequency variation of fiber angle for graph Fig. 17. Natural frequency of different materials with ite/epoxy composite material, with a tubular cross-section. tubular cross-section. Validation of the present formulation is presented The natural frequency and the mode shape by comparison with the analytical solution for the depend on the boundary condition of the beam. transverse vibration of the beam structure [28]. An The comparison between the analytical and the approximation to the governing equation of a rec finite element solution in this paper is given in tangutar laminated beam in transverse vibration can Table 3 for fixed-free ends. A total of five linear be derived from the traditional isotropic material elements was used to model the length of the problem by substituting Elwith bD•• [28] (see [29] for beam with 0° layup. The material properties and an exact formulation) geometrical parameters are given in Tables I and 2. The numerical solution validates that the fonnulation 04W 02 W of the mass and the stiffness matrices developed in bD Il 4 + pA O. (20) ox arr = this 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 Present Node An~lytica1 analysis Analytical analysis Analytical analysis 1 7.34638 7.34661 4.98922 4.98931 6.31053 6.31264 2 46.03922 46.06278 31.26706 31.28269 39.54763 39.56743 3 128.91107 129.37574 87.54862 87.86316 110.73446 111.13215 4 250.91829 255.57972 170.40857 173.57235 215.53852 219.53982 5 417.59052 424.19880 283.60230 288.08681 358.70978 364.38143 APPLICATION theory, the coupled effect neglects such that the forward dynamics analysis can be applied to a specific In this section, a flexible rotating beam is con prescribed motion. However, this simplification can sidered in the application of the matrices developed cause inaccuracy in simulating the dynamic behavior in this study. Figure 18 shows the configuration of the system. In order to maintain the equations of in finite element model of a flexible rotating beam motion with the time variation of the mass matrix system. The finite element method in this type of and the coupled effect, the inverse dynamic procedure problem has been well developed and the approach is performed without any simplifications. The beam can be referred to in [5]. rotates under a prescribed torque in horizontal plane Based on the small linear deformation, the without any other external load. The prescribed equations ofmotion can be expressed in matrix forms torque is given as follows: ~ ~ dl 41 (Ib-inch) 0 1 2 1Ds6 IIIsfJ{Ii} [0 0]{ 8} {F } {Qve} ~ [ m"m mff ijf + 0 K.u: qf = F~f + Q"f' T(/) = 8 (lb-inch) 2 < 1 4 (23) { ~ (21) 8 - 41 (Ib-inch) 4 < 1 6. where 8 is the rigid body motion generalized coordi In this numerical simulation, a total of three linear nate; Clr are the overall elastic generalized coordinates elements were used to model the length of the beam of the nodes; m.o- and Kff are the global mass and with a 0° layup. A small integration time step musl stiffness matrices; Ft6 and F~ are generalized forces; be selected in order to obtain the accurate solution and QIl6 and Q"f are the quadratic velocity vectors for' such type of stiffened system. The pre resulting from differentiating the kinetic energy with dict-eorrector Adams-Moulton algorithm is used respect to time and with respect to the body coordi in this example [30]. However, the carbonfN528C nates. Equation (21) can be-expressed in a concise material cannot converge no matter how small the form time step due to the high stiffness-to weight ratio The material and geometry parameters are given ir. [M]{ij} + [K]{q} = {F~} + {Q,,}. (22) Tables 1 and 2 for a rectangular cross-section and fOJ a length of 60 in. Due to the coupled effect between rigid body Four materials are selected in this simulation: steel motion and elastic deformation, the equations of aluminum, Kevlar-49/epoxy, and graphite/epoxy motion are nonlinear; the nonlinearity being at The major difference relies on the stiffness-to-weigh1 tributed to the time dependency of the mass matrix. ratio of each material. The responses of angulaJ The time-dependent mass matrix contains the gener displacement and acceleration of the isotropic bean alized elastic coordinates. In linear elastodynamics in rigid body motion are shown in Figs 19 and 20 respectively. The corresponding responses of the y composite beam are shown in Figs 22 and 23. It i: (qf}·(q~,~,~) observed that, as shown in Fig. 19, the total angulaJ displacement 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 i the same angular displacement occurs. Although thl steel and the aluminium have the same stiffness-to x weight ratio, the aluminium material has a highe equivalent external load (load/density) so that tb Fig. 18. General finite element configuration of flexible aluminium material experiences larger deflection beam. compared 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. ~ u cu I) ~ •.0 ..,------;------, cu ~ "-~ as 2.0 ~ ""-' ...... Steel (rigid motion) ..... Steel (fiexible motion) ...... Aluminum (ri,id motion) ...... Aluminum (nexible motion) 2 3 5 e Time (sec) Fig. 20. Angular aCceleration of the rotating isotropic material beam. 0.5 ------, .-.. .E 0.0 '-' +J C Q) acu -0.5 ..U ....g.. .!3 -1.0 Q o 2 , I Time (sec) Fig. 21. End point displacement of the rotating isotropic beam. flexibility on the angular displacement is minimal. angular acceleration of 0.8 (rad/sec/sec) experiences The flexibility effect on the angular acceleration is a maximum deformation of 0.28 in as shown in important for all selected materials. Fig. 21. However, as shown in Figs 23 and 24, the As in Figs 21 and 24 show the transverse defor correspondence of angular 'acceleration and maxi mations are dominated by the angular acceleration. mum deformation are 4 (rad/sec/sec) and 0.45 in with In Fig. 20 the steel beam subjected to a maximum graphite/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 3 Time (sec) • Fig. 22. Angular displacement of the rotating composite material beam. ~ <> 4> ~ 10.0 <> "4> ~ ~ "...as S.O "-" d 0 ::3 ...as 0.0 ....as 4> <> (,) -0.0 ~ Graphite Iri1id motion) < ...... Gra bite nexible motion) ~ ..... Xevfar-49 riCid motion) .....as ...... Kevlar-4-9 nexible motion) ~-10.0 0 < 2 3 5 = Time (sec) Fig. 23. Angular acceleration of the rotating composite material beam. 0.& ~------.. :s 0.0 "'-' ...,l ; -0.5 a t) .....~ -1.0 C. CD S -1.5 ...... - Graphite ...... Kevlar-4-9 o 2 3 Time (sec) Fig. 24. End point displacement of the rotating composite beam. with graphite/epoxy material is five times higher than weight ratio in composite materials. The maximum that of steel material but the maximum deformation deformation in the linear deformation model is about is about two times greater than that of the steel 50% greater than that of geometry nonlinearity de material. On the other hand, the composite material formation for both materials. Therefore, the correc1 can be subjected to higher inertial forces. This can be mathematical model is indispensable for more accu attributed to the high performance of the stiffness-to- rate analysis of realistic flexible structure systems. Finite element analysis of composite tubes 409 CONCLUSIONS 8. P. K. C. Wang and Jin-Duo Wei, Variations in a moving flexible robot arm. J. Sound Vibr. 116, 149-160 (1987). The free vibration natural frequency analysis of a 9. V. Masurekar and K. N. 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CHEN et ale APPENDIX 1 1 0 0 0 0 3 6 13 11/ 9 13/ J. 0 0 35 210 70 420 11/ L2 13/ 12 0 0 210 105 520 140 [M] =p",AI 1 1 0 0 0 0 6 3 9 13/ 13 III 0 0 70 420 35 210 131 /2 111 12 0 0 420 140 210 105 PI p2 PI 0 0 P2 / I / / 12fJ3 6{J3 12fJ3 6fJ3 0 13 12 0 -13 r fJ2 6fJ3 4P3 fJ2 6fJ3 2fJ3 / p.- II -12 I [K] 21ta fJI P2 PI fJ2 = 0 0 / // / 12fJ3 6fJ3 12fJ3 6fJ3 0 -13 -r 0 13 -p.- fJ2 6fJ3 2fJ3 P2 6P3 4fJ3 / r I / -12 / where BII fJl=AII +- a DII fJ2=BII +- a " fJ3 =D II • •