AN EFFICIENT MODELING TECHNIQUE FOR VIRRATION ANALYSIS OF COMPOSITE AND COMPLEX STRUCTURES

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1 AN EFFICIENT MODELING TECHNIQUE FOR VIRRATION ANALYSIS OF COMPOSITE AND COMPLEX STRUCTURES Giuseppe Miccoli CEMOTER, National Research Council of Italy Via Canal Bianco, CASSANA - FERRAP.A Italy, Anil Jacob and Daryoush Alla& QRDC, Inc. P.O. Box 562, Excelsior, MN , David Tarnowski Hutchinson Technical College, Hutch&o, MN ABSTRACT A new modeling approach based on the combination of component-mode synthesis and the receptance methods (CMS- R) is presented. Even though the CMS-R technique can be used for various struchlral analysis, it is most suitable for vibration analysis. For demonstration purposes, an aircraft Wmp is modeled and the results are compared with a classical finite-element sub-struchlring method. The results have demonstrated that the proposed CMS-R method is a efficient approach for vibration analysis of composite and complex stnumres. In the CMS-R model, the large mass and stiffness mabices of the conventional ti ite element models are replaced by the vibration characteristics of sub-structures. The system degrees of freedom are drastically reduced since the components are coupled only along their interface boundaries. Such formulation results in a huge reduction in the size of the model, significant improvement in accuracy of the natural frequencies and mode shapes of the structwe. NOMENCLATURE a = length of plate and beam stiffeners b =width of plate h =height of the plate qi = interaction forces x = transverse displacement A = cross sectional area D = flexural rigidity of the plate (Eh3/12[1-p ]) E = Young s modulus F, = harmonic interface forces between sub-sbuctwes I = moment of inertia L = the length of the beam N = modal mass [R] = receptance matrix U,,,(x) = beam mode shapes W,(x,y) = plate mode shapes as = receptances of the stiffener a p, = reoeptances of the plate structwe & = receptances of the stiffener b y, = receptances of the stiffener c us = weptances of the stiffener d I, = receptances of the wing structlxe v = modal participation factors p = density fl= Poisson s ratio w, = natural frequencies of plate a,,,, o,, natural frequencies of beams oq = natural frequencies of composite sbuctwe 1. INTRODUCTION The development of mathematical models that require less memory, are m re accurate, and have higher computational speed has played a important role in vibration analysis of composite and complex structures in recent decades. On one hand, the finite element method has been improved by developing super-elements, and on the other hand, substmcturing methods have been introduced as a alternative method in such applications. The sub-structuring techniques have been tncorporated in many general pupwe finite element codes used for the analysis of large stn~ctwes. Some of these fmite element models have up to 750,000 degrees of freedom which results in very costly analysis. The ever increasing complexity of structures demand wnstant tiprovement on existing methods and the introduction of better approaches. In order to deal with the increase in analytical demands, substructuring methods have taken three general approaches. The methods of component-mode synthesis (CMS), branch Also Program Manager HTC. * Now employed at QRDC, Inc. 338

2 mode synthesis, and component-mode substitution, were fust intmduced by Hurty [I 1, Gladwell [2] and Bentield and Hruda [3], respectively. Since the release of Hurty s work, new strategies to improve the accuracy and computational speed of CMS sub-structuring methods have been developed. Summaries of these various strategies have been published by Hou [4] Melrovitch [5] and Craig 161. Among the strategies developed, the benefits of utilizing fixed- or free-interface methods have been argued. Hwty s fixed-interface component mode method, reworked by Craig and Bampton [7], has been shown to produce accurate results. The method, however, requires that information on interface displacements be preserved in the final assembly with the inclusion of constraint modes. The free-interface component mode methods do not require the interface displacement information since they are based on unconstrained modes only. The accuracy of these results can be improved, as shown by MacNeal [S] and Rubin [9], with the addition of residual modes. The free-interface methods lend themselves to experimental verifications and applications since the unconstrained modes can be tested using a limited number of degrees of freedom. CMS methods have been utilized in the analysis and synthesis of large or complex structures, rotor systems and in the redesign of these types of structures through sensitivity analysis [lo]. The benefits of CMS sub-structuring, as cited above, can be significantly increased, however, through other modeling methods. The method adopted in this paper, component mode synthesis via receptance (CMS-R), takes advantage of the reduced system degrees of freedom, reduced computer memory requirement, increased computational speed, and improved accuracy. It also utilizes efficient interface models, has compatibility with other dynamic models, facilitates component substitution and, in some cases, simplifies design equations. The CMS-R approach, as described by Bishop and Johnson [ll] and Soedel [12] is derived in detail below. The method utilizes the dynamic characteristics of the components, couples them through the receptance interface model, and expresses the result of the complete structure in terms only of the component natural frequencies, mode shapes, and modal damping, which make the structural vibration signature. This method has been applied to many types of structures including, beams, rings, plates, shells, and assemblies with many attachments [ In this paper, to demonstrate the advantages of CM&R method, the following problem was selected: a simplified model of an aircraft wing having two side sti&ners and two cross stiffeners (Figure 1). Comparison was made between the natural frequencies and mode shapes obtained by CMS-R and the FEM. Figure 2 shows the CMS-R model of the aircratl wing. The plate is considered as a host struch~re, with four beam attachments as four sub-structures. To formulate the problem using the CMS-R method, a two stage approach was used. In Stage I, two side stifleners (a and b) were connected to the plate through line interaction forces, and the modal parameters were calculated. In the Stage II, the system analyzed in Stage 1 was used as the host stnwhw, and two cross stiffeners were the sub-structures. 2. STAGE I FORMULATION The plate with two side stiffaers as shown in Figure 3, is represented in the CMS-R sub-structure model by Figure 4. The equations of motion of the system are given in terms of its receptances as shown by Equations l-4: Applying the force and displacement continuity conditions, Equations 1 to 4 can be written in matrix form as: For a non-trivial solution, equating 1R/ to zero gives the characteristic equation of the system whose roots provide the natural frequencies of the composite system The receptances of the individual components (host and sub-structures) of the system are formulated in detail in the following sections. 2.1 PLATE RECEPTANCES Though a cantilever plate is a more realistic model for an aircraft wing-structure, for simplicity, the wing is modeled as a simply-supported plate. The boundary conditions of this model may be relaxed for other applications. The natural frequencies and mode shapes of the simply supported plate are given by Equations 7 and 8. (5) 339

3 0,.2 - n2[d+2(;)*] ;nt,n-1.2,3... (7) The plate modal mass is defmed as N+ Z(xy)&dy - $ (9) The applied inter&ion load is given by Equation 10, where F,, and FQ are the interface forces ate, and %. The forcing function is given by Equation 11: 2.2 BEAM RJKEPTANCES The nahrral frequencies and mode shapes of a simply supported beam are given by Equations 19 and 20 mn BI Yu - -jd- PAL g(x) = ial (19) (20) (12) The modal mass of the beam is given by The modal participation factors are determined by the Equation 13. N-! U,(x)& - ; (21) The applied interaction loads on the two beams are The mode shapes of the host struchue (Plate) are given by Equation 14:. 1 Evaluating the integral gives zero unless p,=p2- bcm;y &yql The host receptances, defined as tbe ratio of the displacement at a point i over the applied force at a point j in the absence of all other forces, are. thus given by where F,, and F, are the interface forces at a, and b,. The general forcing functions on the two beams are given by L1 L1 F,W=+@&W> PANo F,(O=+~,~&)~ G-No (23) Equation 23 can be written as F&O = -!L FE,. 2phN (I F,(t) = ~ F, 2phN (24) 340

4 The modal participation factors of the two beams are determined by the following equations: The new modal mass of the host structure is given by (35) The mode shapes of the beam stiffeners are given by where W, is the mode shape of the composite sub-struchue given by Equation 14. The applied interaction load is given by Equation 36. Thus, the beam receptances are q, - F,b(x-x,)sin(~) + F&-x&in(~) (36) where F, and FE4 are the interface forces at c and d. The forcing function is then given by Equation 37: F,(t) (37) With the receptances now determined, Equation 6 can now be shed numerically to determine the natural frequencies of the composite struchre. The mode shapes of the system of Stage I (Figure 3) can then determined by utilizing Equation STAGE11 Tlx plate in Stage II, with cross stiffeners as shown in Figure 5, is represented by Figure 6 in the CMS-R sub-stnchwe model. The equations of motion for this system are given by Equations 29-32: The modal participation factors are determined by the following equation: where the subscript s is as-skated with the natural frequencies of the composite struchwe. of Stage I. The displacements of the host structure in Stage II are given by Equation 39: The recqtanoes of the new host struoture are then given by the f&wing Applying the force and displacement conditions, Equations 29 to 32 can again be written in ma&ix form as: Using a derivation simih to Stage I,the beam receptances we (33) (34) The nahual frequencies of the wmplete strwmre can be determined by s&kg Equation 34 numerically. The mode shapes can then be determined. 341

5 4. RESULTS To demonstrate the advantages of the CMS-R method, a plate (144 X24X l/4) with two side beam stiffener (144 X 1 X 1) and two cross st&ners (24 X 1 X 1) were selected. The following side stiffener locations were selected: (i) y, = 2.4, y,=16.8,(ii)y,=7.2,y,=19.2,and (iii)y,=4.8,y,=21.6. AU dimensions are in inches and are measured from the bottom left caner of the plate. The modulus of elasticity, density and Poisson s ratio of the plate and the beams are psi, lb sec*/iuch4 and 0.29, respectively. The results presented in this paper are only for the first step of the formulation (addition of side stiffeners). Ihe results of the addition of cross stiffeners will be presented at the conference. It can be observed from the frequencies in Table 1 that the frequencies of the assembled structure are lower than the natural frequencies ofthe plate. Observing the mode shapes of the assembled shucture (Figure 7), it was oonfinued that the beam does not stiffen the plate, rather, the plate acts like an elastic fouudation for the beams. Ibis follows intuitively if we consider that the beams have four times the thickness of the plate. The frequencies and mode shapes obtained oompare well with those obtained using the Finite Element Method, and will be presented at the conference. In the FEM formulation, the plate was modeled by quadrilateral thin shell element (200 linear elements, 231 nodes, 6 DOF per node) and the beam was modeled by 2-node beam elements ( 20 linear elements, 21 nodes, 6 DOF per node). The computation iu FEM was carried out by means of System Dynamics Module of I-DEAS. The three modal components (plate and the two stil%ners) were defined by modal data representing their natural frequencies and modes of vibration. The matrices were defined in tenus of modal properties and modal DOF s. The components of the fmal system were connected by det?niug displacement relationship between a node on one entity and a node on the other entity. In order to extract the eigenvalues and the eigenvectors of the assembled system, the similarity transformation solver was applied. This solver allows one to chose from four algorithms and calculates the energy disbibutiou for the whole system and for each entity and each modal DOF. 5. CONCLUSIONS It is observed t?om Equations 6 and 14 that solving for the natural Gequencies and mode shape of the plate-stiffener system, only a 2X2 matrix is required. To fmd the natural frequency and mode shapes of the same system using FEM, a 1386X1386 matrix for the plate and a 126X126 matrix for the beam are needed. Thus, memoty savings of the proposed method is significantly better. The natural frequencies appear in the denominator of the receptance and mode shape equations, therefore, the inaccuracy of the higher natural tiequencies of the sub-structures does not significantly effect the calculated vibration characteristics of the assembled stmcture. In FEM, however, the frequency range and accuracy depend on the DOF, generated mesh, and the type of the element used. In the CMS-R method, replacing sub-structures does not require the regeneration of any structural meshes, etc. as in the FEM. Only the mode shapes and natural frequencies of the new sub-structure need to be revised. This feature makes the CMS-R method au excellent tool for analysis of space structures or for sensitivity analysis. Using the dominant terms in Equation 38, simplified equations can be obtained for design and parametric study. Two important distinctions between FEM and CMS-R methods are clear. In CMS-R method, the substructures are connected by a superior line interface, whereas there are only point interfaces in FEM. Also, the formulation of beam and plate mode shapes are exact. Thus, the CMS-R method can be seen as an efficient modeling technique for vibration analysis of composite and complex structures. 16. REFERENCES [l] Hurty, W. C., Vibrations of Structural Systems by Component-Mode Synthesis, Journal of the Engineering Mechanics Division, ASCE, v. 86, Aug. 1960, pp. 5 l-69. [2] Gladwell, G. M. L., Branch Mode Analysis of Vibrating Systems, J. ofsound and Vibration,v. 1, 1964, pp [3] Benfield, W. A. and Hruda, R. F., Vibration Analysis of Structures by Component Mode Substitution, AIAA Journal, v. 9, No. 7, 1971, pp [4] Hou, S. N., Review of Modal Synthesis Techniques and a New Approach, Shock and Vib. Bull., v. 40, 1969, pp [5] Meirovitch, L., Computational Methods in Srrucrural Dynamics, Sijthoff & Noordhoff, 1980, pp [6] Craig, Jr., R. R., A Review of Time-Domaiu and Frequency-Domain Component-Mode Synthesis Methods, J. ofilnalyt. and Exper. Modal Analysti, v. 2, 1987, pp [7] Craig, Jr., R. R. and Bampton, M. C. C., Coupling of Substructures for Dynamic Analyses, AIAA Journal, v. 6, no. 7, 1968,~~ [S] MacNeal, R. H., A Hybrid Method of Component Mode Synthesis, J. of Computer.r and Strucrurcs, v. 1, 197 1, pp [9] Rubin, S., Improved Component-Mode Representation for Structural Dynamic Analysis, AIAA Journal, v. 15 no 8, 342

6 1975, pp [lo] Heo, J. H. and Ehmann, K. F., A Method for Substructural Sensitivity Synthesis, J. of Vibration and Acou.s/ics,v. 113, Apr., 1991, pp [ll] Bishop, R. E. D. and Johnson, D. C., TheMechanics o/ Vibration, Cambridge University Press, London, [12] Soedel, W., Vibration of Shells and Plates, Marcel Dekker, Inc., New York., [13] Alla&, D., Jacob, A., and Tamowski, D., Application of Localization to Optimize Service-free Life of Aircraft Fasteners, Third International Congress on Air- and Structure-Borne Sound and Vibration, June 13-15, 1994, Montreal. Canada. [14] Allaei, D, Soedel, W., and Yang, T. Y., Natural frequencies and modes of rings that deviate from perfect axisymmeby, Journal o/sound and Vibration, v. 111, no. 1, 1986, pp [ 151 Allaei, D, Soedel, W., and Yang, T. Y., Vibration analysis of non-axisymmetric tires, Journal of Sound and librarim, v. 122, no. 1, 1988, pp. 1 l-29. [16] Alla&D, Soedel, W., and Yang, T. Y., Eigenvalues of rings with radial spring attachments, Journal ofsound and Vibrarion, Y. 121, no. 3, 1987, pp Table 1. Natural frequencies of simply supported plate and beam (Hz). Table 2. Natural (CMS-R) and resonance (FEM) frequencies of Stage I model (Hz). y, = 2.4 y, = 7.2 y, =4.0 y2 = 16.8 y2= 19.2 y,=21.6 CMS-R i FEM CMS-R CMS-R I I] b a gure 1. Aircraft wing-stiffener system, a and b are sid stiffeners. c and d are cross stiffaers d Figure 2. CMS-R model of the aircraft wing-stiffener system t \ e I 2 I Figure 3. Stage I sub-structure model a and b are beam stiffeners. Figure 4. Stage I CMS-R sub-structure model. 343

7 Figure 5. Stage II sub-sbuchlre model. c and d are cmss stiffeners. igure 6. Stage II CMS-R model Hz d Fieure 7. CMS-R mode shapes of the Stage! model. First four mode shapes (a-d, respectively). Figure 8. FEM mode shapes of the Stage 1 model. First four mode shapes (a-d, respectively)~ 344