EFFECT OF MATERIAL PROPERTIES ON VIBRATIONS OF NONSYMMETRICAL AXIALLY LOADED THIN-WALLED EULER-BERNOULLI BEAMS

Matheatical and Coputational Applications, Vol. 5, No., pp. 96-07, 00. Association for Scientific Research EFFECT OF MATERIAL PROPERTIES ON VIBRATIONS OF NONSYMMETRICAL AXIALLY LOADED THIN-WALLED EULER-BERNOULLI BEAMS Gökhan Altıntaş Departent of Mechanical Engineering Celal Bayar University, 4540 Muradiye, Manisa, Turkey gokhanaltintas@ekanik.net Abstract-This paper focuses on effect of aterial properties on free vibration characteristics of nonsyetrical axially loaded thin-walled Euler-Bernoulli beas. Many scientists studied the vibration of coupled systes but adequate consideration has not been given to detailed and paraetric studies on the effect of aterial properties on vibration properties. The current findings of this study show that there have been quite different effects of aterial properties on the systes without syetry axis than on the systes with syetry axis. Material properties are properties affecting the free vibration behavior. Moreover, effect of axial load on natural frequencies was also investigated in this study. Convergence work revealed that the current atheatical odel siulated the proble with excellent accuracy. Key Words- Effect of Material Properties, Triply Coupled Systes, Coupled Modes, Mode Order Change. INTRODUCTION A nuber of studies dealing with coupled vibrations of thin-walled beas have been developed. Gere and Lin [] and Lin [] provided soe of the first analytical works in this field and obtained the coupled, natural vibration behaviors of unifor, open-section channels using the Rayleigh Ritz ethod. The various bea theories for the solution of beas having coupled torsion and bending were copared by Bishop et al. [3]. Freiberg presented a nuerical ethod for the solution of coupled bea vibrations based on Vlasov theory and derived the dynaic stiffness atrix [4]. He developed eleent ass and stiffness atrices for use in standard finite eleent techniques and evaluated the odal asses. Dokuaci [5] derived the exact analytical expression for the solution of torsional bending equations and the results of this study were later extended by Bishop et al. [6] to include warping, especially significant for thin-walled section beas. The use of this ethod was found to be extreely convenient and effective when the responses of finite, unifor structures to point haronic forces or line haronic loads were deterined [7]. Banerjee and Willias [8] took into account the effect of axial load. Yaan [9] investigated the triply coupled vibrations of open-section channels by wave analysis ethod and the coupled wave nubers, various frequency response curves and the ode shapes were presented for undaped and structurally daped channels. Yaan [0] also developed exact analytical odels for the analysis of non-classical end boundary conditions in purely

G. Altıntaş 97 torsional and doubly coupled vibrations of single-bay channels. The exact ethods have alleviated the difficulties encountered in the consideration of coplex end boundary conditions and were proved to be effective. Tanaka and Berçin [] have derived the governing differential equations of otion and obtained solutions for various boundary conditions using Matheatica. Kollár [] presented the analysis of the natural frequency of coposite beas. Arpaci and Bozdag [3] have extended the approach to triply coupled vibrations of thin-walled beas. Arpaci et al. [4] presented an exact analytical ethod for predicting the undaped natural frequencies of beas with thinwalled open cross-sections having no axis of syetry. Prokic [5 ] analyzed triply coupled vibrations of thin-walled beas with arbitrary open cross-section. Starting fro the Vlasov s theory, the governing differential equations for coupled bending and torsional vibrations were perfored using the principle of virtual displaceents. In the case of a siply supported thinwalled bea, a closed-for solution for the natural frequencies of free haronic vibrations was derived. Prokic [6] analyzed the fivefold coupled vibrations of Tioshenko beas. A dynaic transfer atrix ethod has been presented by Li et al. [7] and the boundary eleent ethod was developed by Sapountzakis and Tsiatas [8]. Recently Orloske et al. [9] and Orloske and Parker [0], studied the stability and vibration characteristics of axially oving beas and Chen and Hsiao [] investigated the coupled vibration induced by the boundary conditions. Hijissen and Horssen [] considered the change of frequencies of a standing cantilevered bea subjected to gravity load. Chen and Hsiao [3] investigated the coupled vibration of thin-walled beas with a generic open section induced by the boundary conditions by using the finite eleent ethod. Nuerical exaples were presented to deonstrate the accuracy of the proposed ethod and to investigate the effects of different pin ends on the coupled vibrations of the thin-wall bea. Vörös [4] has analyzed the free vibration and ode shapes of straight beas where the coupling between the bending and torsion was induced by steady state lateral loads. Majority of the work cited above deal with atheatical solution techniques of bea vibration probles in which coupling interaction take places. There are liited nubers of nuerical studies that investigate how the vibration characteristics are affected by the associated paraeters. For this purpose, in the analyses in which the first ten natural vibration odes are considered, the effect of several paraeters on natural vibration odes of the bea analyzed. The paraeters considered include ass density, elasticity odulus, shear odulus, Poisson ratio. The triply coupling vibration behaviors of aterial properties are investigated with the details that has never been that previously. As such this deterinistic study is envisaged to fill an iportant gap in literature.

98 Effect of Material Properties on Vibrations of Euler-Bernoulli Beas. TRIPLY COUPLING EQUATIONS Figure Bea without Syetry Axis A unifor straight thin-walled bea of length L is shown in Fig.. The cross section of the bea is nonsyetrical. In the right handed Cartesian coordinate syste shown in Figure, the x axis is assued to coincide with the elastic axis (i.e. loci of the shear center of the cross section of the thin-walled bea). The y and z axes are the principal axes of inertia passing through the shear center. The ass axis (i.e. loci of the centroid of the cross section of the thin-walled bea) is separated fro y and z axes by the distance ζ and η respectively. The bending translation in the z direction, the bending translation in the y direction and the torsional rotation about the x axis of the shear center are denoted by v, w and ψ respectively, where x and t denote distance fro the origin and tie respectively. A constant copression axial force P is assued to act through the centroid of the cross section of the thin-walled bea. The governing equations of otion of nonsyetrical axially loaded thin-walled Euler-Bernoulli bea eleent including warping effect can be written as following three coupled differential equations, which can be derived by using d Alebert s principle ( ηψ ) ( ζψ ) IV II II EI v + P v + µ v&& µηψ&& = 0 y IV II II EI w + P w + µ w&& + µζψ&& = 0 z II II II II IV GJψ P( Iψ / µ ηv + ζ w ) + µη v&& µζ w&& EΓ ψ = 0 ( ) = ( ) ( ) = ( ) ( ) =Ψ( ) v x, t V x sinωt w x, t W x sinωt ψ x, t x sinωt s () () V, W and Ψ are the aplitudes of the sinusoidally varying bending translations and torsional rotation, respectively. ( ) ( ) IV II II EI V + P V η Ψ + µ V&& µηψ= && 0 y IV II II EI W + P W ζ Ψ + µ W&& + µζψ= && 0 z II II II II IV GJΨ P( I Ψ / µ ηv + ζ W ) + µη V&& µζ W&& EΓΨ = 0 s (3)

G. Altıntaş 99 E is Young s odulus of elasticity and G is the shear odulus of the aterial, EI y and EI z are the bending stiffnesses of the thin-walled bea about the centriodal principal axes, which are parallel to the y and z axes, respectively. GJ and EΓ are torsional stiffness and warping stiffness of the thin-walled bea, respectively. µ is ass of the thin-walled bea per unit length, I s is polar ass oent of inertia per unit length bea cross section about x axis, superscript pries and dots denote the differentiation with respect to coordinate x and tie t respectively. And notation ω shows circular frequency. 3. NUMERICAL METHOD In the finite difference ethod (FDM), one can replace the differential operators d and by difference operator. To obtain finite difference expressions for derivatives, y= f x a given interval by an interpolating polynoial we approxiate the function ( ) Θ(x) and accept Θ ( x), Θ ( x), Θ ( x),... in place of f ( x), f ( x), f ( x),... It is evident that a better polynoial approxiation of the original function f ( x) at the so-called pivotal point yields better finite difference expressions and therefore iproves accuracy. The siplest way of obtaining usable finite difference expressions for the first y f x f x and second derivatives of a function = ( ) at a point is by substituting for ( ) a second-order parabola through a nuber of equally placed points, as shown in Fig. This collocating polynoial using intervals between points can be expressed by u ui+ ui forward difference x u u ui backward difference x u ui+ ui forward difference x Figure Basic Finite Difference Approxiations y+ y y+ y+ y Θ = +. x y+ y y+ y+ y x Θ= y+. x+. (4)

00 Effect of Material Properties on Vibrations of Euler-Bernoulli Beas Since the pivotal point is located at x =0, first derivative of the original function can be approxiated by dy y = [ Θ ( x) ] = y y. dx x ( ) ( ) + Siilarly, the finite difference expression for the second derivative is d y d dy y = ( x) y y y. dx =Θ = + dx dx x ( ) ( ) + (5) (6) A procedure identical to that used above yields the higher-order derivatives 3 d y y 3 = dx x x x = + and = ( ) ( y ) 4 + 4y+ + 6y 4 y + y, The physical eaning of central differences is also shown in Figure. That is, using a second-order parabola for interpolating polynoials, the slope of the chord line fro point x to point x + becoes identical to that of the tangent at point x. We shall thereafter consider only central differences because of their higher accuracy. EI y( V 4V + 6V 4V+ + V+ ) + P ( V V + V+ ) 4 4 + Pη ( ϕ ϕ + ϕ + ) µω V + µηω ϕ = 0 EIz( W 4W + 6W 4W + + W+ ) + P ( W W + W+ ) 4 4 + P ζ ( ϕ ϕ + ϕ + ) µω W µω ζϕ = 0 (9) GJ ( ) ( ) ( y ) 3 + y ( ) ( y ) 3 + y+ y y 4 4 3 d y y y = = dx y 4 4 3 = + ( ) ( y ) 4 + y y PI µ + + + + S ( ϕ ϕ ϕ ) ( ϕ ϕ ϕ ) ( ) ζ ( ) + P η V V + V P W W + W + + + I ω ϕ µηω V + µζω W 4 4 4 S ( ) ( 3 3 y ) 4 + y / + / EΓ( ϕ 4ϕ + 6ϕ 4ϕ + + ϕ+ ) = 0 Nuber of unknown rotations and displaceents were approxiately three ties as the nuber of nodes on discritized bea. Three equations of syste each having its own three different esh layers, and supporting conditions ust be applied considering the esh of layers. (7) (8)

G. Altıntaş 0 4. NUMERICAL RESULTS 4.. Convergence Study The coparison of results for the first ten natural frequencies of the axially loaded bea with an axial force of P=790N is also ade by including the effect of warping stiffness. The case of boundary conditions claped on supports applied by taking the three variable functions and derivatives of the functions are zero when pivotal point on supports. For eliinating support issue and for coparing only aterial properties, all results obtained for the systes have sae support conditions as claped-claped. The particular thin-walled bea with a sei-circular cross section is considered. The geoetrical and physical properties of the thin-walled bea and axial load are given below: I = 9.60 I =.770 J =.640 y S 8 4 8 4 8 4 z 6 I = 0.00050kg Γ=.50 L= 0.8 y = 0.055 z = 0 µ = 0.835kg C C 9 9 E= 68.90 N G= 6.50 N P= 790N Table. Convergence Results for the First Ten Modes. Nuber of Mod Mod Mod3 Mod4 Mod5 Mod6 Mod7 Mod8 Mod9 Mod0 Nodes 8 84.50 84.6 378.5 467.9 55. 597.4 749.09 8.5 894.55 989.8 9 86.48 87.87 385.77 485.7 566.8 6.6 79.8 873.45 973.66 7.86 0 87.96 90.5 39.30 498.69 576.8 64.6 84.7 9.7 034.55 8.94 89.5 9.00 395.65 508.40 583.59 634. 849.54 94.87 08.36 99.46 90.3 93.3 399. 55.83 589.09 64.77 869.6 964.86 0.56 364.43 3 90.93 94.34 40.95 5.63 593.34 648.0 885.8 983.0 5.54 47.47 5 9.6 95.78 406.0 59.96 599.36 657.40 90.08 009.43 98. 497.69 7 93.04 96.73 409. 535.5 603.34 664.7 97.07 07.7 30.86 554.3 0 93.95 97.64 4.7 540.90 607.7 67.03 944.3 044.79 64. 60.47 5 94.85 98.48 45.8 545.90 60.7 677.78 96.4 06.3 96.84 654. 30 95.35 98.93 46.96 548.59 6.60 68.58 970.55 070.3 35.5 666.93 40 95.87 99.36 48.67 55.3 64.46 685.44 980.07 078.93 333.73 676.50 50 96. 99.56 49.47 55.43 65.30 687.5 984.53 08.96 34.44 680.54 80 96.38 99.77 40.33 553.7 66.0 689. 989.37 087.8 35.9 684.75 00 96.44 99.8 40.53 554.0 66.4 689.67 990.48 088.7 354.09 685.69 00 96.5 99.89 40.80 554.40 66.68 690.7 99.96 089.57 356.99 686.94 50 96.53 99.89 40.83 554.45 66.7 690.34 99.3 089.73 357.34 687.09 500 96.55 99.90 40.87 554.5 66.75 690.44 99.37 089.93 357.80 687.8 600 96.55 99.9 40.88 554.5 66.76 690.45 99.38 089.95 357.84 687.30 750 96.55 99.9 40.88 554.5 66.76 690.46 99.4 089.97 357.88 687.3 900 96.55 99.9 40.88 554.5 66.76 690.46 99.4 089.98 357.90 687.33 000 96.55 99.9 40.88 554.5 66.77 690.46 99.43 089.98 357.9 687.33 Ref [7] 96.55 99.9 40.89 554.53 66.77 690.47 99.45 090.00 357.95 687.35 The nuerical results are shown in Table copared with those available in the literature [7]. It is shown that the convergence with respect to esh size is precisely good and the frequency paraeter closes its analytical value fro below onotonically. Differences between the results and Ref. [7] are less then %0.. Generally, convergency ratio shows faster approxiation for lower odes than high ones. In this paper, all calculations were done for the beas having 000 nubered esh size.

0 Effect of Material Properties on Vibrations of Euler-Bernoulli Beas 4.. Verification The first ten bending torsion coupled natural frequencies of the unloaded bea (i.e. P=0) are calculated for claped end conditions by excluding and including the effect of warping stiffness. Further coparison of results for the first ten natural frequencies of the axially loaded bea with an axial force of P = 790 N is also ade by excluding and including the effect of warping stiffness. The nuerical results are also shown in Tables and copared with Li et al. [7]. It can be seen fro Table, for the unloaded bea, the present results and the published results are copletely the sae. For the axially loaded bea with an axial force of P = 790 N, the axiu relative error between the present results and the published results is less than 0.%. Table. Natural frequencies of thin-walled bea with Claped Claped end condition Mode Natural Frequency (Hz) Nuber Warping Ignored Warping Included Li et al [7] Present Study Li et al [7] Present Study Li et al [7] Present Study Li et al [7] Present Study P=0 P=0 P=790 P=790 P=0 P=0 P=790 P=790 74.08 74.0873 7.77 7.7774 98.8 98.808 96.55 96.554 0.38 0.389 99.9 99.97 0.38 0.389 99.9 99.99 3 353.58 353.589 349.05 349.0493 45.04 45.04 40.89 40.8875 4 5.0 5.996 55.85 55.8534 557.87 557.873 554.53 554.59 5 557.87 557.879 554.53 554.593 68.09 68.090 66.77 66.7704 6 630.9 630.93 68.89 68.8876 695.63 695.699 690.47 690.4675 7 7. 7.93 703. 703.65 999.3 999.3004 99.45 99.435 8 893.88 893.87 88.69 88.6898 093.66 093.647 090.00 089.989 9 07.90 07.886 059.45 059.44 365.73 365.69 357.95 357.96 0 093.66 093.647 090.00 089.989 688.57 688.555 687.35 687.335 4.3. Analysis Figure 3 Effect of Poisson s Ratio a) For fixed Elasticity odulus (68.9e9 N/ ) b) For fixed Shear odulus (.8e0 N/ )

G. Altıntaş 03 One can see the effect of Poisson s ratio on the first ten natural frequencies in Figure 3. Curves in Figure 3(a) and Figure 3(b) were plotted for fixed values of odulus of elasticity E=68.9e9 N/ and shear odulus G=.8e0 N/ respectively. It is obviously seen that the Poisson s ratio can change the frequencies of odes and additionally it can change the order of the odes as fundaental ode. It is well known that the effect of aterial odulus on natural frequencies of syetric beas in classical vibration probles which analyze only transversal odes or only longitudinal odes or etc. The effect of Poisson s ratio on natural frequencies is onotonic for all odes in classical vibration probles. But, in the beas without syetry axis, the effect of ratio of odulus is ore iportant than the odulus, since different odulus have different degrees of effects on different vibration types. As a result, the Poisson s ratio which deonstrates a kind of ratio between elasticity odulus and shear odulus has critical iportance for coupled vibration probles. Figure 4. Effect of Elasticity Modulus for Fixed Poisson s Ratio (0.3)

04 Effect of Material Properties on Vibrations of Euler-Bernoulli Beas Figure 4 shows the effect of odulus of elasticity on natural frequencies for fixed value of Poisson s ratio (ν=0.3). When the value of elasticity odulus increased, so did natural frequencies. It is iportant to point in this graphic that there was bigger variation range of elasticity odulus used in Figure 4 than Figure 3, but there was no apparent change for ode sequence in the range of consideration. Figure 5. Effect of Axial Force on Natural Frequencies Figure 5 shows the effect of axial force on the first ten odes. All odes decreased by increasing axial force onotonically.

G. Altıntaş 05 Figure 6. Effect of Mass Density on Natural Frequencies (P=0) Easily, one can see the iportant effects of ass density on natural frequency in Figure 6. It can be seen fro Figure 6 that natural frequency values of odes are inversely proportional with the variation of the ass density. For saller values of ass density, the natural frequencies had high sensitivity. 5. CONCLUSION The present study was designed to deterine the effect of aterial properties on natural vibrations of beas without syetry axis. A odel based on finite difference technique was iproved and verification analysis showed that the ethod could be used to siulate the syste precisely. Effect of shear and elasticity odulus, ass density, Poisson s ratio was individually investigated in certain ranges. The natural frequency values of odes are inversely proportional with the variation of the ass density for all ode types. Frequencies had high sensitivity to variation of the ass density for saller values. Poisson ratio is a ore iportant paraeter than aterial odulus for beas without syetry axis. At constant Poisson s ratio, the effect of aterial odulus is expectable on beas with or without syetry axis. There are onotonic effects on variations of

06 Effect of Material Properties on Vibrations of Euler-Bernoulli Beas natural frequencies. But, the variation of Poisson s ratio can change the variation ratio of different type of odes. This event can be interpreted by the explanation of that every odulus has different effect level on different ode types in coupling probles. Effect of axial load was also investigated in this study. The effects of axial load on natural frequency were found to be onotonic. There was reverse relationship between the value of axial load and natural frequencies. In the range of investigation, axial load caused considerable changes on natural frequencies. In this respect axial load could be a tuning paraeter for natural frequencies in practical applications. Although these findings enhance our understanding of effect of aterial properties on natural frequencies of coupled beas without syetry axis, this research has thrown up any questions in need of further investigation. For practical usage of beas without syetry axis, paraetric analysis is a necessity instead of single point analysis and all calculations and analysis ust be done as precisely as one could. 5. REFERENCES. J. M. Gere and Y. K. Lin, Coupled Vibrations Of Thin-Walled Beas Of Open Cross- Section. Journal of Applied Mechanics, Transactions of the Aerican Society of Mechanical Engineers, 80, 373-378, 958.. Y. K. Lin, Coupled Vibrations Of Restrained Thin-Walled Beas. Journal of Applied Mechanics, Transactions of the Aerican Society of Mechanical Engineers, 8, 739-740, 960. 3. R. E. D. Bishop, W. G. Price and Z. Xi-Cheng, A Note On The Dynaical Behaviour of Unifor Beas Having Open Channel Section. Journal of Sound and Vibration, 99, 55-67, 985. 4. P. O. Freiberg, Bea Eleent Matrices Derived Fro Vlasov's Theory of Open Thin-Walled Elastic Beas. International Journal for Nuerical Methods in Engineering,, 05-8, 985. 5. E. Dokuaci, An Exact Solution For Coupled Bending And Torsion Vibrations of Unifor Beas Having Single Cross-Sectional Syetry. Journal of Sound and Vibration, 9, 443-449, 987. 6. R.E.D. Bishop, S.M., Cannon, and S., Miao, On Coupled Bending And Torsional Vibration Of Unifor Beas. Journal of Sound and Vibration, 3, 457-464, 989. 7. D. J. Mead and Y. Yaan, The Haronic Response Of Unifor Beas On Multiple Linear Supports: A Flexural Wave Analysis. Journal of Sound And Vibration, 4, 465-484, 990. 8. J.R., Banerjee, F.W., Willias, Coupled Bending-Torsional Dynaic Stiffness Matrix of An Axially Loaded Tioshenko Bea Eleent. International Journal of Solids and Structures, 3, 749-76, 994. 9. Y.Yaan, Vibration of Open-Section Channels: A Coupled Flexural And Torsional Wave Analysis. Journal of Sound and Vibration, 04():3-58, 997. 0. Y. Yaan, Analytical Modelling of Coupled Vibrations of Elastically Supported Channels. Proceedings of the 6th International Conference on Recent Advances in Structural Dynaics, university of Southapton, England, 997.

G. Altıntaş 07. M. Tanaka, A.N. Bercin, Free Vibration Solution For Unifor Beas of Nonsyetrical Cross Section Using Matheatica. Coputers and Structures, 7:- 8,999.. P.L. Kollár, Flexural-Torsional Vibration Of Open Section Coposite Beas With Shear Deforation. International Journal of Solids and Structures, 38, 7543-7558,00. 3. A. Arpaci, E. Bozdag, On The Free Vibration Analysis of Thin-Walled Beas With Nonsyetric Open Cross Sections. Coputers and Structures, 80, 69-695,00. 4. A. Arpaci, S.E. Bozdag and E. Sunbuloglu, Triply coupled vibrations of thin-walled open cross-section beas including rotary inertia effects. Journal of Sound and Vibration, 60, 889 900, 003. 5. A. Prokic, On triply coupled vibrations of thin-walled beas with arbitrary crosssection. Journal of Sound and Vibration, 79, 73 737, 005. 6. A. Prokic, On Fivefold Coupled Vibrations of Tioshenko Thin-Walled Beas. Engineering Structures, 8, 54-6,006. 7. J. Li, W. Li, R. Shen, and H. Hua, Coupled Bending and Torsional Vibration of Nonsyetrical Axially Loaded Thin-Walled Bernoulli-Euler Beas. Mechanics Research Counications, 3, 697-7,004. 8. E.J. Sapountzakis, G.C. Tsiatas, Flexural-Torsional Vibrations of Beas by BEM. Coputational Mechanics, 39, 409-47,007. 9. Orloske, K., Leay, M.J., Parker, R.G., Flexural-torsional buckling of isaligned axially oving beas, part I: three-diensional odeling, equilibria, and bifurcations. International Journal of Solids and Structures, 43, 497-43, 006. 0. K. Orloske, R.G. Parker, Flexural-Torsional Buckling of Misaligned Axially Moving Beas, Part II: Vibration and Stability Analysis. International Journal of Solids and Structures, 43, 433-434,006.. H. H. Chen, K.M. Hsiao, Coupled Axial-Torsional Vibration of Thin Walled Z- Section Bea Induced by Boundary Conditions. Thin-Walled Structures, 45, 573-583,007.. J.W. Hijissen, W.T. Horssen, On Transverse Vibration of a Vertical Tioshenko Bea. Journal of Sound and Vibration, 34, 6-79,008. 3. H. H. Chen, K. M. Hsiao, Quadruply Coupled Linear Free Vibrations of Thin- Walled Beas With a Generic Open Section. Engineering Structures, 30, 39-334, 008. 4. G. M. Vörös, On Coupled Bending-Torsional Vibrations of Beas with Initial Loads. Mechanics Research Counications, 36, 603-6, 009.