The dynamic analysis of a linear viscoelastic planar elliptical beam

Abstract

Keywords: Elliptical planar beam; viscoelastic material; dynamic analysis; impulsive load; mixed finite element method.

1. Introduction

Referring to the developing technology, curved beam elements are widely used in mechanical, civil, mechatronics and aerospace engineering as a structural construction member. They are used in advanced structures such as bridges, railways, aircrafts, turbine blades, connector elements and space vehicles. These needs are required to a wide range of curved element (elliptic, parabola, catenary, cycloid, and circle) having curvature range along the arc length with different section shapes and sizes. Due to internal friction, viscous effects are important on behaviour under dynamic and impulsive type loads etc, of some materials, such as polymers. Different viscoelastic mechanical models, such as the well-known Standard, Kelvin and Maxwell model etc, can simulate the viscous effects.

Corresponding Author: e-mail: ermism@itu.edu.tr, tel: (212) 285 65 52

M. Ermis, A. Kutlu, N. Eratlı, M.H. Omurtag / Sigma J Eng & Nat Sci 9 (2), 157-168, 2018

When elastic material is considered, there are numerous studies about static and dynamic analysis of elastic space (straight, curved planar and 3D) beams in the literature. Some of these works related to circular / non-circular planar curves can be given as follow: Static analysis is considered in [1] and [2]. Free vibration analysis are investigated in [3]-[9]. Dynamic analysis are studied without damping effect in [10]-[12], and, with damping effect in [13], respectively. When viscoelastic material is considered, there are many studies for linear viscoelastic straight beam in [14]- [20], whereas the studies about linear viscoelastic planar curved beams are limited for circular beam in [21]-[23], and, for non-circular beams in [24]. In this study, the dynamic analysis of a linear viscoelastic planar elliptical beam having three different cross-sections (circular cross-section, two different elliptically oriented cross-sections) are examined via mixed finite element method (MFEM) based on Timoshenko beam theory. By using the linear shape functions, the exact nodal arc length and curvature values are inserted into the finite element matrix [25]. The exact analytical expression of arc length and curvature of an elliptical plane curve is derived by using the formulation given in [26]. The analysis is performed in Laplace space by using the correspondence principle [27] and the results are transformed back to time domain numerically by Modified Durbin algorithm [28]-[30]. The mixed finite element algorithm used in this study was verified by Eratlı et al. 2014 [31], also used in [32] and [33]. Through the analysis, the viscoelastic model exhibits standard type of distortional behaviour while having elastic Poisson’s ratio. The linear viscoelastic analysis of planar elliptical beam subjected to isosceles triangular and right triangular impulsive type of uniformly distributed load is performed. Beam is fixed at both ends. As far as the knowledge of the authors, this study is a new contribution to the literature with some benchmark examples.

Figure 1. Planar elliptical beam having different cross-sections subjected to vertical distributed impulsive type dynamic load

2. Functional In Laplace Space And Mixed FE Formulation

Space Curve in Frenet Frame: A space curve is described by a position vector R( s) , where s is the arc length parameter of the rod axis. The unit vectors of the Frenet Coordinate system can be given as: t ( s) = R , s , n(s) = R, ss / R, ss , b(s) = t(s)  n(s) where t , n and b are the tangential, normal and binormal unit vectors, respectively. The differentiation with respect to arc length s is denoted by comma (e.g. dR / ds  R,s ). The differential relations between the unit vectors of

Frenet c oordinate are t(s), s =  (s) n(s) , n(s), s =  (s) b(s)   (s) t (s) , b(s), s =  (s) n(s) where

 (s)  R, ss and  (s)  b(s)  n(s), s represent the curvature and the torsion of the space curve [34], respectively. By choosing the position vector R( )  x( ), y( ), z( ) in Cartesian coordinate system as a parametric function of horizontal angle  , the differential relation between arc length s and the horizontal angle  can be defined as s,  c( )  R, with c( ) being the gradient of arc length [26]. Plane Curve: A planar curve is described by a position vector r ( ) , the analytical expressions for the curvature and the arc length in terms of position vector becomes  p ( )  r, ss , c p ( )  r, , s,p  c p ( ) . We note that the superscript “ p ” is used to denote that these

quantities are written for a plane curve. For a planar elliptical curve shown in Fig.1a, x( )  Rmax cos , y( )  Rmin sin  where Rmin and Rmax are the minimum and maximum radius, respectively. Viscoelastic Material Model: In this study, the viscoelastic material exhibits the standard type of distortional behaviour while having elastic Poisson’s ratio, and the complex shear modulus can be expressed as given in [35] and [36]: G  G 1   G rG z  / 1   rG z  ;  G  Gg / G  1 where  rG is the retardation time, G is the equilibrium value of shear modulus and Gg is the instantaneous value of relaxation function associated with shear modulus. Field Equations: The field equations for a space rod are based on Timoshenko beam theory. The associated functional in Laplace transformed space is given by: I  y   [u, T,s ]  [t  Ω, T]  [M,s , Ω]  12 [C M, M]  12 [C T, T]  12  A z 2 u, u 

   (1) ˆ ˆ ˆ ˆ T]  [Ω, M]   12  z  Ω, Ω  q, u  m, Ω [(T  T), u]  [(M  M), Ω]  [ u,    2

In Eq. (1), the Laplace transformed variables are denoted by the over bars; z is the Laplace transformation parameter; u (ut , un , ub ) , Ω ( t ,  n ,  b) , T (Tt , Tn , Tb ) and M (M t , M n , M b ) are the displacement, rotation, force and moments vectors, respectively (given in terms of Frenet q and m are the distributed external force and moment vectors, frame components). respectively. C and C are the compliance matrices.  is the density of homogenous material,

A is the area of cross-section and I ( It , I n , I b ) is the moment of inertia vector. In the mixed finite element formulation, a two noded curvilinear element is used to discretize the domain of the planar curved beam. The derivation procedure of a functional Eq. (1), exists in [31], is based on Gâteaux differentiation and potential operator concept [37]-[39] .

3. Numerical Examples

Common parameters: Through the analysis, geometric parameters of elliptical beam and crosssections are taken as follows: the minimum radius of elliptical beam to the maximum radius of elliptical beam ratio Rmin / Rmax  0.5 where Rmax  1m ; a circular and two elliptical cross-section having different orientations are used in the analysis. The orientations of the two different elliptical cross-sections are as shown in Fig.1b. The abbreviations "ellipse_n" and "ellipse_b" are used to denote, respectively, the elliptical cross-sections having major axis oriented horizontal and vertical. The cross sectional areas of the circular and the two elliptical sections are taken the

M. Ermis, A. Kutlu, N. Eratlı, M.H. Omurtag / Sigma J Eng & Nat Sci 9 (2), 157-168, 2018

same. The dimensions of the elliptical and circular cross-sections are a  5cm , b  2.5cm and r  3.53553cm . The beam is fixed at both ends. Torsional moment of inertia for an elliptical

cross-section is calculated by using the equation I t   a3b3 / a 2  b2 [40]. A convergence and verification example for elastic analysis: A convergence analysis is performed over the natural frequencies of the planar elliptical beam having the circular, elliptical_n and elliptical_b cross-sections and verified with the solutions of SAP2000 in [41]. Viscoelastic analysis: The material parameters are the shear modulus G  7 105 Pa , Poisson’s ratio   0.3 , the density of material   7850kg / m3 , the retardation time  rG  0.05s and the ratio  G  1.5 . By using the parameters  rG ,  G and G , the complex shear modulus G is determined. The quasi-static and dynamic responses of the beam which is subjected to a vertical distributed dynamic load q  qz (t ) are investigated within 0  t  100s . Isosceles triangular and right triangular impulsive type loads are used (Fig.1c). The area of the forced vibration zone of these loads is kept constant, the duration of load is tload  40s and the maximum intensity of distributed load is qo  103 N / m . The analyses are carried out in Laplace transform space and the results are transformed back to time domain numerically by Modified Durbin algorithm [28][30]. For the inverse Laplace transformation algorithm, N  211 and aT  6 are used [31]. The vertical displacements ubM (t ) at the midpoint M, the rotations Cn (t ) at point C, the forces TbA (t ) and the moments M tA (t ) and M nA (t ) at point A of the elliptical beam (Fig.1a) are investigated within time interval 0  t  100s . For points A, C and M see Fig. 1a. A convergence example for viscoelastic analysis: A convergence analysis of the planar elliptical beam having circular cross-section is carried out for 16, 24, 32, 40 and 48 elements. The first extremum peak values of the time histories of ubM (t ) , Cn (t ) , TbA (t ) , M tA (t ) and M nA (t ) for the isosceles triangular and the right triangular impulsive loads are tabulated in Tables 1-2, respectively. The results ( ubM (t ) , Cn (t ) , TbA (t ) , M tA (t ) and M nA (t ) ) obtained by using 40 elements are normalized with respect to the results of 48 elements and percent differences are also given in Tables 1-2. In the following examples, 40 elements are employed. Table 1. The convergence analysis for the isosceles triangular impulsive load. ( ne : number of element) ne

the first peak of the dynamic behaviour of elliptical beam having circular cross-section ubM  103 Cn  106 TbA  104 M tA  102 M nA  102 (mm) (rad) (N) (N.mm) (N.mm) -47.65 61.81 -12.10 -29.71 44.05 -47.34 60.86 -12.09 -29.94 43.25 -47.24 60.54 -12.09 -30.01 43.00 -47.19 60.38 -12.09 -30.04 42.88 -47.17 60.25 -12.09 -30.06 42.82 -0.04 -0.22 0.00 0.07 -0.14 M C A A A diff.%  1  40el / 48el  100 ,  : ub ,  n , Tb , M t , M n

Table 2. The convergence analysis for the right triangular impulsive load. ( ne : number of element) ne

the first peak of the dynamic behaviour of elliptical beam having circular cross-section ubM  103 Cn  106 TbA  104 M tA  102 M nA  102 (mm) (rad) (N) (N.mm) (N.mm) -90.02 116.57 -19.67 -54.96 78.62 -89.51 114.82 -19.68 -55.33 77.10 -89.33 114.23 -19.68 -55.45 76.63 -89.24 113.96 -19.68 -55.50 76.42 -89.20 113.82 -19.68 -55.53 76.30 -0.04 -0.13 0.00 0.05 -0.16 diff.%  1  40el / 48el  100 ,  : ubM , Cn , TbA , M tA , M nA

The effect of the dynamic impulsive load: The time history of ubM (t ) of the planar elliptical beam having circular cross-section are given for isosceles triangular and right triangular impulsive type loads in Fig.2. The dynamic behaviour of the viscoelastic elliptical beam dissipates within the sampling time interval and approaches to the quasi-static case (see Fig.2). Under the same damping parameters, the behaviour of the viscoelastic case is oscillating close around the quasi-static case results for the impulsive isosceles triangular loading whereas it is not the case for impulsive right triangular case. The first extremum peak of ubM (t ) belonging to the isosceles triangular impulsive load case are normalized with respect to the first extremum peak of ubM (t ) belonging to the right triangular impulsive load for dynamic and quasi static cases, and, these percent differences are 47.12%, and -0.89%, respectively. The effect of the cross-section: The dynamic behaviour of viscoelastic elliptical beams having three different cross-sectional geometries are compared with each other for isosceles triangular and right triangular impulsive type loads. For this purpose, circular, ellipse_n and ellipse_b crosssections are considered. The time histories of ubM (t ) , Cn (t ) , TbA (t ) , M tA (t ) and M nA (t ) are given in Figs.3 and 4. For points A, C and M see Fig. 1a. It is seen that; the time histories are significantly affected from the shape of the cross-section. Figs.3 and 4 may be discussed within the two time intervals namely, 0s  t  40s and 40s  t  100s . In this case:

For the forced vibration zone ( 0s  t  40s ), the first extremum peaks of ubM (t ) , Cn (t ) ,

T (t ) , M tA (t ) and M nA (t ) of the isosceles triangular impulsive load are normalized with respect to the first extremum peaks of ubM (t ) , Cn (t ) , TbA (t ) , M tA (t ) and M nA (t ) of the right triangular impulsive load for each type of cross-sections: - In elips_b cross-section: The percent reductions in the amplitude of ubM (t ) , Cn (t ) ,

TbA (t ) , M tA (t ) and M nA (t ) are 47.1%, 47.0%, 38.6%, 45.9%, 43.9%, respectively.

In circular cross-section: The percent reductions in the amplitude of ubM (t ) , Cn (t ) ,

T (t ) , M tA (t ) and M nA (t ) are 46.2%, 46.0%, 36.6%, 44.5%, 42.1%, respectively.

In elips_n cros-section: The percent reductions in the amplitude of ubM (t ) , Cn (t ) , TbA (t )

, M tA (t ) and M nA (t ) are 46.1%, 46.1 %, 39.8%, 45.3% , 43.9%, respectively.

M. Ermis, A. Kutlu, N. Eratlı, M.H. Omurtag / Sigma J Eng & Nat Sci 9 (2), 157-168, 2018

For the free vibration zone ( 40s  t  100s ), the dynamic response of ubM (t ) , Cn (t ) ,

TbA (t ) , M tA (t ) and M nA (t ) oscillates around zero along the free vibration zone with decreasing amplitude for both types of impulsive loads. The reduction in the amplitudes of these responses are more influential in the case of isosceles triangular impulsive type of loading. The responses determined for circular and ellipse_b cross-sections damp rapidly when compared with the ellipse_n.

(b) right triangular impulsive load Figure 2. The vertical displacements ubM (t ) of the elliptical beam having circular cross-sections

(e) the moments M nA (t ) Figure 3. Time histories of the planar elliptical beam having cross-sections (circular, elips_n, and elips_b) subjected to isosceles triangular impulsive load.

M. Ermis, A. Kutlu, N. Eratlı, M.H. Omurtag / Sigma J Eng & Nat Sci 9 (2), 157-168, 2018

(e) the moments M nA (t ) Figure 4. Time histories of the planar elliptical beam having cross-sections (circular, elips_n, and elips_b) subjected to right triangular impulsive load. 164

4. Conclusion

The dynamic viscoelastic response of the planar elliptical beam having a circular and two elliptical cross-sections with different orientations are analysed by using mixed FEM. For this purpose, the viscoelastic material behaviour is simulated by using the standard model and viscoelastic properties are accounted using the correspondence principle. The finite element solutions are carried out in the Laplace space. The results obtained in frequency domain are transformed back to time domain using modified Durbin’s algorithm. The effect of the cross-section on the time histories of the variables are:

The vibration periods of ubM (t ) , Cn (t ) , TbA (t ) , M tA (t ) and M nA (t ) for the beam having ellipse_n cross-section is the largest compared to the other two cross-sections.

The first extremum peaks of ubM (t ) , Cn (t ) and M tA (t ) for the beam having ellipse_n cross-section is the largest when compared with those obtained for the other two cross-sections.

The difference between the first extremum of the time histories of TbA (t ) are insignificant for the three cross-sectional geometries used in this study. This judgement is also valid for the time histories of M nA (t ) .

The first extremum peaks of ubM (t ) and Cn (t ) for circular and ellipse_b cross-section are

normalized with respect to the first extremum peaks of ubM (t ) and Cn (t ) of the ellipse_n crosssection. The discussion is as follows: - In isosceles triangular impulsive load: The percent reductions in the amplitudes of ubM (t ) and Cn (t ) are 47.4 % and 47.4% for the beam having circular cross-section, and, 67.8% and 68.5% for the beam having ellipse_b cross-section, respectively. - In right triangular impulsive load: The percent reductions in the amplitudes of ubM (t ) and

Cn (t ) are 46.4% and 46.5% for the beam having circular cross-section, and, 67.8% and 67.6% for the beam having ellipse_b cross-section, respectively. The effect of the impulsive load type on the time histories of the variables are:

 If the first extremum peaks of the time histories of the quasi-static case for the isosceles triangular and right triangular impulsive loads (see Fig.2) are normalized with respect to the first extremum peaks of the corresponding dynamic loading cases, the percent reductions are obtained as 0.39%, 47.79%, respectively.  The results show that, the dynamic behaviour of the linear viscoelastic planar elliptical beam is affected from the form of applied dynamic loading, although the forced vibration areas and the duration of loading are the same. As a result, the type of cross-section and loading to be used in special problems must be carefully selected. As far as the knowledge of the authors, the linear viscoelastic elliptical beam analysis using mixed FEM is an original example.

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