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FLEXURAL-TORSIONAL COUPLED VIBRATION ANALYSIS OF A THIN- WALLED CLOSED SECTION COMPOSITE TIMOSHENKO BEAM BY USING THE DIFFERENTIAL TRANSFORM METHOD Metin O. Kaya and Özge Özdemir Istanbul Technical University, Faculty of Aeronautics and Astronautics, 34469, Maslak, Istanbul, Turkey Abstract. In this study, a new mathematical technique called the Differential Transform Method (DTM) is introduced to analyse the free undamped vibration of an axially loaded, thin-walled closed section composite Timoshenko beam including the material coupling between the bending and torsional modes of deformation, which is usually present in laminated composite beams due to ply orientation. The partial differential equations of motion are derived applying the Hamilton’s principle and solved using DTM. Natural frequencies are calculated, related graphics and the mode shapes are plotted. The effects of the bending-torsion coupling and the axial force are investigated and the results are compared with the studies in literature. 1. Introduction A composite thin-walled beam with length L , cross sectional dimension B and wall thickness d is shown in Fig.1. The geometric dimensions are assumed to be d B so the terms related to the warping stiffness and the warping inertia are small enough to be neglected. The bending motion in the z direction, the torsional rotation about the x axis and the rotation of the cross section due to bending alone are represented by w( x, t ) , ( x, t ) and ( x, t ) , respectively. A constant axial force P acts through the centroid of the cross section which Figure 1. Configuration of an axially coincides with the x axis. P is positive loaded composite Timoshenko beam when it is compressive as in Fig.1. 2. Formulation The governing undamped partial differential equations of motion are derived for the free vibration analysis of the beam model represented by Fig.1. After the application of the Hamilton’s principle, the following equations of motion are obtained as follows I EI kAGw K 0 (1) w Pw kAGw 0 (2) I s PI s K GJ 0 (3) Here, A is the mass per unit length; I s is the polar mass moment of inertia; K and kGA are flexure–torsion coupling rigidity and shear rigidity of the beam, respectively. The boundary conditions at x 0 and x L for Eqs. (1)-(3) are as follows EI k 0 (4) Pw kAGw w 0 (5) PI s K GJ 0 (6) A sinusoidal variation of w( x, t ) , ( x, t ) and ( x, t ) with a circular natural frequency is assumed and the functions are approximated as wx, t W x e it , x, t x e it , x, t x e it (7) The following nondimensional parameters can be used to simplify the equations of motion x , W W , r2 I , * d , 1 * (8) L L AL2 d L Substituting Eqs.(7) and (8) into Eqs.(4)-(5), the dimensionless equations of motion are obtained as follows A1 ** A2 A3W * A4 ** 0 (9) B1W ** B2W B3 * 0 (10) C1 ** C 2 C3 ** 0 (11) where the dimensionless coefficients are L4 r 2 kAGL2 kAGL2 K P A1 1 , A2 2 A3 A4 B1 1 (12) EI EI EI EI kAG L2 2 PI s I s L2 2 K B2 B3 1 C1 1 C2 C3 kAG GJ GJ GJ 3. The Differential Transform Method The differential transform method is a transformation technique based on the Taylor series expansion and is used to obtain analytical solutions of the differential equations. In this method, certain transformation rules are applied and the governing differential equations and the boundary conditions of the system are transformed into a set of algebraic equations in terms of the differential transforms of the original functions and the solution of these algebraic equations gives the desired solution of the problem. A function f x , which is analytic in a domain D, can be represented by a power series with a center at x 0 , any point in D. The differential transform of the function is given by 1 d k f ( x) F k (13) k! dx k x x 0 where f x is the original function and F k is the transformed function. The inverse transformation is defined as f ( x) ( x x0 ) k F k (14) k 0 Combining Eqs. (13) and (14) and expressing f x by a finite series, we get m ( x x0 ) k d k f ( x) f ( x) dx k (15) k 0 k! x x0 Here, the value of m depends on the convergence of the natural frequencies [1]. Theorems that are frequently used in the transformation procedure are introduced in Table 1 and theorems that are used for boundary conditions are introduced in Table 2 [2]. Table 1. Basic theorems of DTM Original DTM Function f x g x hx F k Gk H k f x g x F k Gk f x g xhx F k k Gk l H l l 0 d n g x F k k n! Gk n f x dx n k! 0 if k n f x x n F k k n 1 if k n Table 2. DTM theorems for boundary conditions x0 x 1 Boundary Condition Transformed B.C. Boundary Condition Transformed B.C. df dx (0) 0 F (0) 0 f (1) 0 F (k ) 0 k 0 df kF (k ) 0 df (0) 0 F (1) 0 (1) 0 dx dx k 0 2 d2 f (0) 0 F (2) 0 d f (1) 0 k (k 1) F (k ) 0 dx 2 k 0 dx 2 d3 f d3 f dx 3 (0) 0 F (3) 0 dx3 (1) 0 (k 1)(k 2)kF (k ) 0 k 0 4. Formulation with DTM In the solution step, the differential transform method is applied to Eqs.(9)-(11). Here we quit using the bar symbol on , W , and instead, we use , W , . A1 k 2k 1 k 2 A2 k A3 k 1W k 1 A4 k 2k 1 k 2 0 (16) B1 k 2k 1W k 2 B2W k B3 k 1 k 1 0 (17) C1 k 2k 1 k 2 C 2 k C3 k 2k 1 k 2 0 (18) Applying DTM to Eqs. (4)–(6), the boundary conditions are given as follows at 0 0 W 0 0 0 (19) at 1 k 1 k 1 A4 k 1 k 1 0 (20) B1 k 1wk 1 k 0 (21) C1 k 1 k 1 C3 k 1 k 1 0 (22) 5. Results and Discussion In order to validate the computed results, an illustrative example, taken from Ref [3], is solved and the results are compared with the ones in the same reference. Additionally, the mode shapes of the beam are plotted. Variation of the first five natural frequencies (coupled and uncoupled) of the above example with respect to the axial force is introduced in Table 3 and compared with the results of Ref. [3] and [4]. Here, it is noticed that the natural frequencies decrease as the axial force varies from tension P 7.5 to compression P 7.5 . Additionally,it is seen that the coupled natural frequencies are lower than the uncoupled ones. However, the fourth natural frequency becomes less when the bending-torsion coupling is ignored. Table3. Natural frequencies with respect to the axial force Natural Frequencies r 2 0.00002322 P 7.5 P0 P 7.5 Present Ref. [4] Present§ Ref.[3]§ Present Present§ Ref.[3]§ Present Ref.[4] Present§ Ref.[3]§ 40.975 40.97 37.106 37.1 35.283 30.747 30.75 28.064 28.06 21.987 21.99 224.259 224.25 197.672 197.7 217.341 189.779 189.8 210.162 210.16 181.495 181.5 598.668 598.66 525.665 525.6 592.626 518.791 518.8 586.519 586.51 511.818 511.9 647.595 647.59 648.495 648.6 647.411 648.269 648.3 647.228 647.22 648.047 648 1125.71 1125.71 992.878 - 1119.85 986.199 - 1113.950 1113.95 979.473 - § Natural frequencies with coupling The effects of the axial force P and the Timoshenko effect, r , on the first four natural frequencies are introduced in Figs. 2(a-d). When Fig. 2 is examined, it is noticed that the natural frequencies decrease with the increasing rotary inertia parameter, r , because Timoshenko effect decreases the natural frequencies and this effect is more dominant on higher modes as expected. Additionally, since the fourth mode is torsion, the Timoshenko effect makes a slight change in the fourth natural frequency value. 40.00 200.00 2nd Natural Frequency (Hz) 1st Natural Frequency (Hz) 36.00 196.00 32.00 192.00 28.00 188.00 24.00 184.00 20.00 180.00 -8.00 -4.00 0.00 4.00 8.00 -8.00 -4.00 0.00 4.00 8.00 528.00 Force (N) 4th Natural Frequency (Hz) 648.50 Force (N) 3rd Natural Frequency (Hz) 524.00 648.40 520.00 648.30 516.00 648.20 512.00 648.10 508.00 648.00 -8.00 -4.00 0.00 4.00 8.00 -8.00 -4.00 0.00 4.00 8.00 Force (N) Force (N) Figure 2. Effect of the Timoshenko effect on the first four natural frequencies ( , Timoshenko ; , Euler) Mode shapes of the considered beam under the effect of the compressive axial force ( P 7.5 ) are introduced with bending-torsion coupling in Figs. 5(a-d). When these figures are considered, it can be noticed that the first three normal modes are bending modes while the fourth normal mode is the fundamental torsion mode. 1 1 0.75 Second Mode Shapes First Mode Shapes 0.75 0.5 0.5 w 0.25 0.25 0 w 0 0.25 0.5 0.25 0.5 0.75 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 ξ ξ 1 1 Fourth Mode Shapes 0.75 Third Mode Shapes 0.5 0.75 0.25 w 0.5 0 0.25 0.25 0.5 0 0.75 w 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 ξ ξ Figure 5. The first four normal mode shapes of the composite beam with bending-torsion coupling ( ,; ,; ,) References S.H. Ho and C.K. Chen, Analysis of General Elastically End Restrained Non-Uniform Beams Using Differential Transform, Applied Mathematical Modeling 22 (1998) 219-234 Özdemir Ö, Kaya MO, Flapwise Bending Vibration Analysis of a Rotating Tapered Cantilevered Bernoulli-Euler Beam by Differential Transform Method, Journal of Sound and Vibration (In Press). J.R. Banerjee, Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method, Computers and Structures 69 (1998) 197-208 J. Li, R. Shen, H. Hua and X. Jin, Bending-torsional coupled vibration of axially loaded composite Timoshenko thin-walled beam with closed cross-section, Composite Structures, 64 (2004) 23-35