Elliptical Beam ID : 10694 Free Vibration Analysis of a Planar · 2019. 2. 22. · [14]Mehmet H...

Free Vibration Analysis of a PlanarElliptical Beam ID : 10694Merve Ermis1 , Umit N Aribas2, Nihal Eratli3, Mehmet H Omurtag4Department of Civil Engineering, Faculty of Engineering, Istanbul Technical University, Maslak, 34467 Sarıyer/İstanbul, Turkey

1. IntroductionThe curved beam elements are increasingly used in mechanical, civil, mechatronics and aerospace engineering such as bridges, railways, aircrafts, turbine blades, connector elementsand space vehicles referring to the recent requirements in developing technology. These needs are required to a wide range of curved element (elliptic, parabola, catenary, cycloid, andcircle) having curvature range along the arc length with different section shapes and sizes. In the literature, static analysis are presented in the following non-circular curves havingvariable cross-section: semi- in Gimena et al. (2008); in Tufekci et al. (2017). Huang et al. (1998c) are developed an exact solution for in-plane vibration arches having variablecurvature and cross-section. Huang et al. (2000) are studied the linear out-of-plane dynamic responses of non-circular plane curves having variable cross-section by extendingprevious works about uniform curved beams (Huang et al. 1998a) and in-plane dynamic responses (Huang et al. 1998b). Free vibration analysis are investigated in the followingnon-circular plane curves in Oh et al. (1999) and Oh et al. (2000); in Yang et al. (2008); in Shahba et al. (2013); in Luu et al. (2015); horseshoe elliptic in Lee et al. (2016). Thefree vibration analysis of composite laminated and sandwich circular and non-circular beams is studied in Ye et al. (2016). Free vibration and stability analysis of elliptic beam isconsidered in Nieh et.al. (2003). Rajasekaran (2013) solved the static, stability, free and forced vibrations of axially functionally graded tapered circular and non-circular archesby using finite element method. In this study, the free vibration analysis of the planar elliptical Timoshenko beam having three different cross-sections (two different ellipticallyoriented cross sections and circular cross-section) is performed via mixed finite element method. The exact formulation of arc length and curvature of an elliptical plane curve isderived by using the formulation given in Ermis and Omurtag (2017) and the influence of some parameters (e.g. cross-sections, the ratio of the minimum radius of elliptical beamto the maximum radius of elliptical beam, the opening angle) on the natural frequencies of the planar elliptical beam are investigated.

2. Formulation2.1 Field Equations and FunctionalThe field equations for the spatial beams, which arebased on the Timoshenko beam theory and refer to theFrenet coordinate system, are discussed in Omurtag andAköz (1992) and applied to the free vibration problemof the helicoidal bars having non-circular cross-sectionsin Eratlı et al. (2016). Using u = utt + unn + ubb isthe displacement vector, Ω = Ωtt + Ωnn + Ωbb is therotational vector, T = Ttt + Tnn + Tbb is the force vec-tor, M = Mtt +Mnn +Mbb is the moment vector, ρ isthe density of material, A is the area of the cross-section,I = Itt + Inn + Ibb is the moment of inertia vector, Cis the compliance matrix, q and m are the distributedexternal force vector and moment vector, respectively.The field equations can be written in the form.

where the accelerations are denoted by ü = ∂2u/∂t2,Ω̈ = ∂2Ω/∂t2, Equation (1) can be written in opera-tor form as Q = Ly− f , if the operator is potential, theequality 〈dQ(y, ȳ),y∗〉 = 〈dQ(y,y∗), ȳ〉 must be satis-fied (Oden and Reddy 1976).

2.2 Mixed FEM, Free Vibration AnalysisA two-nodded curved element is employed to discretizethe beam domain. The curved element has 2x12 degreesof freedom. Linear shape functions are employed for theinterpolation. The problem of determining the naturalfrequencies of a structural system reduces to the solutionof a standard eigenvalue problem ([K] − ω2[M])[u] = 0where [K] is the system matrix, [M] is the mass matrixfor the entire domain, u is the eigenvector ω is the nat-ural frequency of the system. Hence the explicit form ofstandard eigenvalue problem is mixed in the formulationis

Figure 1. A planar elliptical beam and the types of cross-sections

6. ContactMerve Ermis, Research AssistantIstanbul Technical UniversityEmail: ermism@itu.edu.tr

Website: akademi.itu.edu.tr/ermism/

3. Numerical Examples

E = 210GPaυ = 0.3ρ = 7850kg/m3Rmin/Rmax = 0.25,0.5, 0.75, 0.9999Rmax = 2mΘ = 90o, 180o, 270oa = 6cm b = 3cm r = 4.24264cm

4. ConclusionA parametric study for the planar elliptical Timoshenko beam is carried out via mixed finite element method toinvestigate the influence of some geometric parameters on the natural frequencies of the beam. As a convergencetest, a semi-elliptical beam having circular cross-section for Rmin/Rmax = 0.5 is handled, results of the mixed finiteelement program is compared by the commercial program SAP2000 and an excellent agreement is achieved. Someexamples are solved to investigate the influence of the some parameters (cross-sections: ellipse_n, ellipse_b, circular,the ratio of the minimum radius of elliptical beam to the maximum radius of elliptical beam Rmin/Rmax , the openingangle:Θ) on the free vibration analysis of the planar elliptical beam.

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