Non-Conservative Stability Analysis of Hauger Types of Columns with Different Boundary Conditions

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1 Proceeings of the Worl Congress on Engineering 08 Vol II WCE 08, July 4-6, 08, onon, U.K. on-conservative Stability nalysis of Hauger Types of Columns with Different Bounary Conitions S.. Fazelzaeh,. Tashakorian, E. Ghavanloo,. moozgar,.i. Friswell bstract Fully intrinsic equations are use to obtain a moel compliant with Euler-Bernoulli assumptions for a beam uner a linearly istribute follower force known as a Hauger column. The avantage of the intrinsic formulation in moeling problems with non-conservative forces is iscusse here. Only intrinsic parameters which are inepenent of the choice of coorinate system has been use an four ifferent bounary conitions were implemente. lso, a comparison between the present stuy an similar stuies with the classical formulation has been evelope. The Generalize Differential Quarature etho is use to numerically analyze the critical loa of the beam. It is well unerstoo that there is a remarkable avantage in terms of convergence using intrinsic equations in comparison with the classical formulation. Inex Terms Differential Quarature etho, Hauger Column, Intrinsic Equations, on-conservative Stability F I. ITRODUCTIO ully intrinsic equations may be applie to a wie variety of applications of beams because of its proper moeling of forces an moments as motion-epenent quantities. lthough mechanical elements such as beams have been the center of interest in recent years an has been analyze in many ifferent configurations, there are some cases that have not been iscusse properly as yet. One of these is the non-conservative stability of beams with linear istribute follower loas which is iscusse in this paper using intrinsic equations. The proceure for eriving the fully intrinsic equations was propose by Hoges []. The fully intrinsic equations inclue a set of first-orer partial ifferential equations of motion. The isplacements an rotations are not presente in these formulations an intrinsic parameters exist in this formulation are inepenent of the choice of coorinate system. Hoges [] presente a systematic erivation of a S.. Fazelzaeh is with the School of echanical Engineering, Shiraz University, Shiraz , Iran (corresponing author phone: ; fax: ; fazelza@ shirazu.ac.ir)... Tashakorian is with the School of echanical Engineering, Shiraz University, Shiraz , Iran ( m.tashakorian@chmail.ir). E. Ghavanloo is with the School of echanical Engineering, Shiraz University, Shiraz , Iran ( ghavanloo@shirazu.ac.ir).. moozgar is with the College of Engineering, Swansea University Bay Campus, Swansea S 8E, Unite Kingom (m.amoozgar@swansea.ac.uk)..i. Friswell is with the College of Engineering, Swansea University Bay Campus, Swansea S 8E, Unite Kingom (m.i.friswell@swansea.ac.uk). geometrically exact generalize Timoshenko theory for initially curve an twiste anisotropic beams. oels for elementary beam vibration an stability with various bounary conitions were aresse by Chang an Hoges []-[4] an Sotoueh an Hoges [5]. The fully intrinsic formulation was use by Patil an Hoges [6] to stuy flight ynamics an aeroelasticity of highly flexible flying wings. This work on flying wings spawne several relate works for HE aircraft base on the fully intrinsic formulation, e.g., Chang an Hoges [7], Chang et al. [8], an Sotoueh et al. [9]. Fully intrinsic equations were also use to preict the aeroelastic behavior of oine-wing aircraft by Sotoueh an Hoges [0]. The fully intrinsic formulation was applie to moel multi-flexible boy ynamics problems by Sotoueh an Hoges []-[]. This work has le to a series of stuies on the effect of engine placement on aeroelastic behavior, passive morphing an boy-freeom flutter of flying wings by aranpour et al. []-[5]. Galerkin approach was presente for approximate solutions of the nonlinear fully intrinsic equations by Patil an lthoff [6]. Palacios [7] use the intrinsic formulation to obtain the nonlinear normal moes of beams. Hesse an Palacios [8, 9] an Wang et al. [0]-[] applie the intrinsic formulation for moel orer reuction, consistent linearization, an moeling of nonlinear aeroservoelasticity for flexible aircraft. Using fully intrinsic equations, Khaneh asei an Ovesy []-[] investigate the static, large eflection of beams uner both conservative an nonconservative loas. They use the Chebyshev collocation metho to numerically solve the ifferential equations. In another stuy, moozgar an Shahveri [4] stuie a similar problem by using the generalize ifferential quarature (GDQ) metho. Recently, ynamic instability of beams uner tip follower forces was stuie by the same group [5] within the framework of geometrically exact, fully intrinsic equations. ore recently, the geometrically exact, fully intrinsic nonlinear beam theory has been utilize to moel the ynamic stability of initially twiste beams subecte to istribute follower forces [6]. on-conservative forces occur in a vast range of applications in mechanical an civil engineering as well as aeronautics engineering an aeroelasticity. ost researchers assume a uniformly istribute follower force which is a proper moel for flui friction an similar problems. etaile review of follower force research can be foun in aranpour et al. [6] an Fazelzaeh et al. [7]-[8]. Because of ifficulties of applying a variable follower loa on a eformable beam, the linearly istribute follower ISB: ISS: (Print); ISS: (Online) WCE 08

2 Proceeings of the Worl Congress on Engineering 08 Vol II WCE 08, July 4-6, 08, onon, U.K. problem known as the Hauger problem has receive less interest. The only research that consiere such a problem was publishe by arzani et al. [8] in which the stability of the Hauger column was stuie using the classical formulation an the GDQ metho. ll of the research in follower force problems an the Hauger column has consiere only one bounary conition. Using intrinsic equations arbitrary follower forces can be applie an so the Hauger column can be moele much more easily than in the classical formulation. In this paper, the implementation of a linearly istribute follower loa on the beam will be iscusse. lso the effects of ifferent bounary conitions will be consiere. II. THEORETIC FOUDTIOS. Fully Intrinsic Equations set of equations has been propose by Hoges an evelope by him an his colleagues uring past ecae, name fully intrinsic which are inepenent of the choice of coorinate systems an by consiering special constitutive rules will provie a geometrically exact analysis of beams with any arbitrary shape of cross section an curvatures. Fig. illustrates the configuration of the beam an reference frames which are use in the intrinsic formulation. Here is the uneforme reference line an is the eforme reference line of the beam. lso, at every point of both the eforme an uneforme reference lines there is a reference frame for which one of the uneforme reference lines is time-inepenent an enote by b i (i,,), an is calle the uneforme reference frame. In a similar manner, a eforme reference frame is consiere at every point of the eforme reference line an enote by B i (i,,). ll intrinsic parameters are measure in the eforme reference frame which makes it easy to apply non-conservative forces. ccoring to the formulation propose by Hoges [], the three imensional ifferential equations of motion in intrinsic parameters can be written as: F KF f P ΩP () Ω K e γ F m H H VP where F is a vector of cross sectional forces measure in the eforme basis an is unknown. Similarly, is the vector of cross sectional moments, P is the linear momentum an H is the angular momentum per unit length. There are also kinematical intrinsic parameters, which are: generalize linear velocity V an angular velocity Ω, generalize strain γ an curvatures κ which all is measure in eforme reference frame. Furthermore, f an m are the istribute force an moment vectors. In equation () there is another parameter which inicates the curvature of the beam in the eforme state an enote by K, which can be written in the form of uneforme beam curvature k an generalize curvature κ as: K k κ. Here F is the space erivative of the cross sectional force with respect to x, the path variable of the reference line is the time erivative of the linear momentum, an an P so on. In equation (), an all of this paper, the tile notation is use to simplify the cross prouct of vectors. Given a vector K with three elements, K K, K, K 0 K K K K 0 K K K 0 () Equation () provies equilibrium conitions for a three imensional beam with intrinsic parameters, but cannot be solve in isolation an nees relations between the kinematical parameters. Here these relations for the kinematical intrinsic parameters are []: V KV e γ Ω γ () Ω KΩ κ Equations () an () simultaneously provie a set of fully intrinsic equations for the beam. ow we can simplify the three imensional equations to make a beam moel consistent with Euler-Bernoulli beam theory. In orer to achieve this, all of the intrinsic parameters are set to zero except V, Ω,,, F, F,, P an κ. Equations () an () F γf H V Ω γω γ Ω κ ISB: ISS: (Print); ISS: (Online) T tile notation will give: can then be written in simplifie scalar form as: F κ F P Ω f F κ F P Fig.. General configuration of the beam, the reference lines an the reference coorinate systems., the (4) where f is the istribute axial follower loa acting on the reference line of the beam. ssuming an homogenous, isotropic beam with the mass centroi coincient with the reference line an the principle axis of the cross section, the constitutive relations of the beam can be written as: WCE 08

3 Proceeings of the Worl Congress on Engineering 08 Vol II WCE 08, July 4-6, 08, onon, U.K. P μv κ H i Ω F γ E F γ G Substituting equation (5) into (4) results in a set of intrinsic equations with respect to only V,Ω, F,F,. Thus, F F μvω f F F μv (6) F F F iω E F F V Ω Ω E G Ω For a linear varying istribution of axial force, known as Hauger column, where the magnitue is at the root an zero at the free en (Fig. ), the istribute axial force may be written as: f x (7) If the non-linear terms in the first of equations (6) is neglecte, then a ecouple orinary ifferential equation is create which can be solve analytically for given bounary conitions Thus F x (8) F 0 an the solution is: F ( x ) (9) (5) Substituting equation (9) into other equations of (6) an ignoring shear effects ( 0 ) an rotary inertia G E (i =0), results in four couple first-orer ifferential equations which can be solve for the stability analysis of the problem. These equations can also be written in non-imensional form to enable an easier assessment of the numerical performance. Thus V & & F ( ) V F (0) One can also write the non-imensional parameters as: F F V Ω p 0 x t V Ω μ μ ξ μ 4 4 τ B. umerical Discretization In orer to make a numerical stability analysis of a Hauger column, the Generalize Differential Quarature etho (GDQ) has been use. In this metho, the space erivatives of a function is approximate by a series of weighting coefficients multiplie by gri point values of that function, as propose by Bellman an his colleagues [0]- [] in the early 970s. There are several approaches to compute the weighting coefficients values. Here, we will use Shu s general approach [], which is an efficient an reliable metho an has been use many times recently. The first erivative in this metho, can be written as: f x where () xi i f x x () () () i are the values of the weighting coefficients, f are the values of the function at the gri points an is the number of gri points. larger number of gri points gives more accurate results, although this also makes the computation more ifficult an the calculation time longer. Fig.. Reference coorinate system an loa moel for a Hauger column in the uneforme an eforme states. ISB: ISS: (Print); ISS: (Online) WCE 08

4 Proceeings of the Worl Congress on Engineering 08 Vol II WCE 08, July 4-6, 08, onon, U.K. The matrix of weighting coefficients can be compute using Shu s general approach as follows: ( ) i ( ) ii i k, ki xi x x xk ( ) ik k, ki x x k, k k i () The other parameter that etermine the accuracy of the GDQ is the metho of choosing the gri points. Here, for faster convergence an higher accuracy, a non-uniform istribution of gri points has been implemente [], as i x i cos i,..., (4) III. RESUTS D DISCUSSIO Using the Differential Quarature etho, one can turn equation (0) into a system of first orer ifferential equations with respect to time. Inee, the equations of motion can be written in stanar matrix form as: q B q 0 (5) where an B are numerical matrices. In orer to obtain the natural frequencies of beam, harmonic motion is assume an therefore, equation (5) can be rewritten in the form of a stanar eigen-value problem as: B qˆ 0 (6) Where is the eigen-frequency an qˆ is the eigenvector in generalize coorinates. Using a stanar algorithm, the eigen-frequency for ifferent loa parameters can be easily compute. Furthermore, the bounary conition must be efine. For ifferent bounary conitions, the values of the intrinsic parameters shoul be etermine at the bounaries. Four ifferent traitional bounary conitions are stuie in this research an the values of the intrinsic parameters are as follows: Simply supporte beam (S-S) V 0 0 V Clampe-Simply supporte (C-S) V 0 0 V Clampe-Free (C-F) V 0 0 F Clampe (C-C) V 0 0 V The critical loa for the four bounary conitions are given in Table I. The Hopf bifurcation metho was use to etermine the critical conition. lso, in orer to compare the intrinsic results an the classical results, the results of arzani et al. [9] are also presente in Table I. The results are in a goo agreement with those reporte by arzani at al.. While the number of gri points in the classical formulation using the DQ metho is reporte to be 5, but it is reuce to only 7 when using the intrinsic formulation in the same numerical metho. This is because of the first orer erivatives an intrinsic parameters implemente in the intrinsic equations. For more esign applications, a semi-log plot of critical loa versus slenerness ratio is provie. It can be seen that with an increase of the slenerness ratio, the critical loa for all bounary conitions will ecrease. Furthermore, the most critical bounary conition from the point of critical loa is etermine to be the simply supporte beam. lso, it is easy to see that the least critical bounary conition from this point of view is the clampe beam. IV. COCUSIO In this paper, the three imensional fully intrinsic equations of beams have been use to create an intrinsic moel conistent with Euler-Bernoulli beam theory in orer to analyze a column uner a linearly istribute follower force known as a Hauger column. The avantage of the TBE I CRITIC OD FOR DIFFERET BOUDRY CODITIOS USIG ITRISIC D CSSIC FORUTIOS S-S C-S C-F C-C Bounary Conitions Present Ref.[9] Difference (%) Parameter TBE I CHRCTERISTIC PRETERS OF BE Description µ mass per unit length i secon moment of inertia (minimum) E Young s moulus G shear moulus bening stiffness E tensile stiffness G Shear stiffness Fig.. The imensionless critical loa versus the slenerness ratio, when the number of gri point is 7. ISB: ISS: (Print); ISS: (Online) WCE 08

5 Proceeings of the Worl Congress on Engineering 08 Vol II WCE 08, July 4-6, 08, onon, U.K. intrinsic formulation is clearly seen where applying nonconservative loas because of the ease of escribing motionepenent loas in this formulation. Four ifferent bounary conitions have been stuie an compare with each other. comparison between the present formulation an the classical formulation is also presente an the significant avantage of the implementation using the intrinsic formulation was highlighte. For more esign applications, a semi-log plot of critical loa versus slenerness ratio for four bounary conitions is provie. REFERECES [] D.H. Hoges, D.H., mixe variational formulation base on exact intrinsic equations for ynamics of moving beams. Int. J. Solis Struct. 6, 990, pp [] D.H. Hoges, Geometrically Exact, Intrinsic Theory for Dynamics of Curve an Twiste nisotropic Beams, I Journal, 4 (6), 00, pp.-7. [] C.S. Chang, D.H. Hoges, Vibration characteristics of curve beams, J. ech. ater. Struct. 4, 009, pp [4] C.S. Chang, D.H. Hoges, Stability stuies for curve beams, J. ech. ater. Struct. 4, 009, pp [5] Z. Sotoueh, D.H. Hoges, oeling beams with various bounary conitions using fully intrinsic equations, J. ppl. ech. 78, 0, 000. [6].J. Patil, D.H. Hoges, Flight ynamics of highly flexible flying wings, J. ircraft 4, 006, pp [7] C.S. Chang, D.H. Hoges, Parametric stuies on groun vibration test moeling for highly flexible aircraft, J. ircraft 44, 007, pp [8] C.S. Chang, D.H. Hoges,.J. Patil, Flight ynamics of highly flexible aircraft, J. ircraft 45, 008, pp [9] Z. Sotoueh, D.H. Hoges, C.S. Chang, 00. Valiation stuies for aeroelastic trim an stability of highly flexible aircraft, J. ircraft 47, 00, pp [0] Z. Sotoueh, D.H. Hoges, Incremental metho for structural analysis of oine-wing aircraft, J. ircraft 48, 0, pp [] Z. Sotoueh, D.H. Hoges, Structural ynamics analysis of rotating blaes using fully intrinsic equations; Part I: Formulations, J. m. Helicopter Soc. 58, 0, [] Z. Sotoueh, D.H. Hoges, Structural ynamics analysis of rotating blaes using fully intrinsic equations; Part II: Verification of ual loa path configurations, J. m. Helicopter Soc. 58, 0, 000. [] P. aranpour, D.H. Hoges, R. euhart,. Graybeal, Effect of engine placement on aeroelastic trim an stability of flying-wing aircraft, J. ircraft 50, 0, pp [4] P. aranpour, D.H. Hoges, Passive morphing of flying wing aircraft: Z-shape configuration, J. Flui. Struct. 44, 04, pp.7-0. [5] P.W. Richars, Y. Yao, P. aranpour, R.. Her, D.H. Hoges, Effect of inertial an constitutive properties on boy-freeom flutter of a flying wing, J. ircraft 5, 06, pp [6].J. Patil,. lthoff, Energy-consistent, Galerkin approach for the nonlinear ynamics of beams using intrinsic equations, J. Vib. Control 7, 0, pp [7] R. Palacios, onlinear normal moes in an intrinsic theory of anisotropic beams, J. Soun Vib. 0, 0, pp [8] H. Hesse, R. Palacios, Consistent structural linearisation in flexibleboy ynamics with large rigi-boy motion, Comput. Struct. 0-, 0, pp.-4. [9] H. Hesse, R. Palacios, Reuce-orer aeroelastic moels for ynamics of maneuvering flexible aircraft, I J. 5, 04, pp [0] Y. Wang, R. Palacios,. Wynn, metho for normal-moe-base moel reuction in nonlinear ynamics of slener structures, Comput. Struct. 59, 05, pp [] Y. Wang,. Wynn, R. Palacios, onlinear moal aeroservoelastic analysis framework for flexible aircraft, I J. 54, 06, [] P. Khaneh asei, H.R. Ovesy, Chebyshev collocation metho for static intrinsic equations of geometrically exact beams. Int. J. Solis Struct. 54, 05, 8-9. [] P. Khaneh asei, H.R. Ovesy, arge eflection analysis of geometrically exact spatial beams uner conservative an nonconservative loas using intrinsic equations, cta ech. 6, 05, pp ISB: ISS: (Print); ISS: (Online) [4].R. moozgar, H. Shahveri, nalysis of nonlinear fully intrinsic equations of geometrically exact beams using generalize ifferential quarature metho cta ech. 7, 06, pp [5].R. moozgar, H. Shahveri, Dynamic instability of beams uner tip follower forces using geometrically exact, fully intrinsic equations. at. m. J. Solis an Stru., 06, pp [6] P. aranpour, E. Izapanahiy, S. Rastkar, S.. Fazelzaeh, D.H. Hoges, Geometrically-exact, fully intrinsic analysis of initially twiste beams uner istribute follower forces, 58th I/SCE/HS/SC Structures, Structural Dynamics, an aterials Conference, 07. [7] S.. Fazelzaeh,. Karimi-obanegani, P. aranpour, Dynamic Stability of Pretwiste Cantilever Beams Subecte to Distribute Follower Force, I Journal, 55 (), 07, pp [8]. Karimi-obanegani, S.. Fazelzaeh, E. Ghavanloo, onconservative stability of spinning pretwiste cantilever beams, Journal of Soun an Vibration, 4, 08, pp [9]. arzani, F. Tornabene, E. Viola, onconservative stability problems via generalize ifferential quarature metho, Journal of Soun an Vibration, 5(-), 008, pp [0] R. Bellman, J. Casti, Differential quarature an long-term integration, Journal of athematical nalysis an pplications, 4, 97, pp.5-8. [] R. Bellman, B.G. Kashef, J. Casti. Differential quarature: technique for the rapi solution of nonlinear partial ifferential equations, Journal of Computational Physics, 4(0), 97, pp [] C. Shu, Differential Quarature an Its pplication in Engineering, Springer, onon, 000. [] C. Bert,. alik, Differential Quarature etho in Computational echanics: Review, pplie echanics Reviews, 49 (), 996, pp.-8. WCE 08