Keywords: structural dynamics, Reissner beam, geometrically exact analysis, strain-based finite element analysis, elasticity.

The dynamic behaviour of structural members is of a considerable interest to engineers of several disciplines. In structural engineering, the responce of a structure to the dynamic loading of wind gusts, blast and earthquakes is of great importance, while mechanical engineers are often faced with problems associated with machine-induced vibrations. In the traditional approach to the dynamic analysis of mechanisms and machines, one assumes that the systems are composed from rigid bodies. In modern systems operating at high speed one needs to account for a possibility of the system undergoing severe elastic deformations to the extent that they cannot be ignored. Flexible beams, capable of properly describing large overall motion and strains, become an indispensable part of such a model [3,4,8].

The dynamic analysis of flexible mechanisms has been a subject of many researchers [1,2,3,4,5,8,9]. The most popular approach for this analysis is to develop finite element models. There the mechanism components are modelled by beam elements. Several different approaches have been employed to describe the motion of the beam elements for the dynamic analysis of flexible beams: the inertial approach [4,8,9], the floating approach [2] and the co-rotational approach [1,3]. The advantages and disadvantages of these approaches are well documented in literature [3,4,8].

In the present study, an alternative finite element approach based on an inertial formulation of the non-linear dynamic analysis of elastic flexible beams is proposed. Our finite element procedure is based on Reissner's geometrically exact elastic planar beam [7] whose assumptions are: (i) the Bernoulli hypothesis that the plane cross-sections remain plane during the deformation and (ii) the shape and the size of the cross-section do not change. Reissner's beam theory is capable of considering membrane, bending and shear strains and does not set any restrictions regarding the magnitude of displacements, rotations and strains. The paper introduces a finite element formulation which uses a new strain-based planar beam element for the non-linear dynamic analysis of planar elastic flexible beams. The strain-based finite element is derived by using the modified Hu-Washizu functional [6]. Only the deformation variables are interpolated. The advantages of the proposed element are: (i) no locking and stress oscillations, (ii) high accuracy and (iii) good convergence. In order to find the solution of the system of dynamic equations of motion, an incremental-iterative method based on the Newmark time integration scheme in combination with the Newton-Raphson method is used.

The validity of the present finite element model along with its accuracy, effectiveness and efficiency is proved through numerical examples.

1

M.A. Crisfield, J. Shi, A co-rotational element/time-integration strategy for non-linear dynamics, International Journal for Numerical Methods in Engineering, 37, 1897-1931, 1994. doi:10.1002/nme.1620371108

2

T.R. Kane, P.W. Likins, D.A. Levinson, Spacecraft Dynamics, McGraw-Hill, New York, 1983.

3

K.-M. Hsiao, J.-Y. Jang, Dynamic analysis of planar flexible mechanisms by co-rotational formulation, Computer Methods in Applied Mechanics and Engineering, 87, 1-14, 1991. doi:10.1016/0045-7825(91)90143-T

4

A. Ibrahimbegovic, S. Mamouri, Nonlinear dynamics of flexible beams in planar motion: Formulation and time stepping scheme for stiff problems, Computers and Structures, 70, 1-22, 1999. doi:10.1016/S0045-7949(98)00150-3

5

M. Iura, S.N. Atluri, Dynamic analysis of planar flexible beams with finite rotations by using inertial and rotating frames, Computers and Structures, 55, 453-462, 1995. doi:10.1016/0045-7949(95)98871-M

6

I. Planinc, M. Saje, B. Cas, On local stability condition in planar beam finite element, Structural Engineering and Mechanics, 12, 507-526, 2001.

7

E. Reissner, On one-dimensional finite-strain beam theory: The plane problem, Journal of Applied Mathematics and Physics (ZAMP), 23, 795-804, 1972. doi:10.1007/BF01602645

8

J.C. Simo, L. Vu-Quoc, On the dynamics of flexible beams under large overall motions - the plane case: Part I and Part II, ASME Journal of Applied Mechanics, 53, 849-863, 1986.

9

N. Stander, E. Stein, An energy-conserving planar finite beam element for dynamics of flexible mechanisms, Engineering Computations, 13, 60-85, 1996. doi:10.1108/02644409610128418

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