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: [email protected]
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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