Shock and Vibration

Volume 2019, Article ID 9065365, 10 pages

https://doi.org/10.1155/2019/9065365

## Geometrically Nonlinear Analysis for Elastic Beam Using Point Interpolation Meshless Method

^{1}Key Laboratory of Unmanned Aerial Vehicle Technology, Nanjing University of Aeronautics and Astronautics, Ministry of Industry and Information Technology, Nanjing 210016, China^{2}State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Correspondence should be addressed to Cheng He; moc.361@yragnehceh

Received 18 December 2018; Revised 23 February 2019; Accepted 24 February 2019; Published 15 May 2019

Academic Editor: Mohammad Rafiee

Copyright © 2019 Cheng He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The intrinsic beam theory, as one of the exact beam formulas, is quite suitable to describe large deformation of the flexible curved beam and has been widely used in many engineering applications. Owing to the advantages of the intrinsic beam theory, the resulted equations are expressed in first-order partial differential form with second-order nonlinear terms. In order to solve the intrinsic beam equations in a relative simple way, in this paper, the point interpolation meshless method was employed to obtain the discretization equations of motion. Different from those equations by using the finite element method, only the differential of the shape functions are needed to form the final discrete equations. Thus, the present method does not need integration process for all elements during each time step. The proposed method has been demonstrated by a numerical example, and results show that this method is highly efficient in treating this type of problem with good accuracy.

#### 1. Introduction

Many structures can be modeled as beams in structural mechanics, such as bridge trusses, helicopter rotor blades, DNA/protein molecules, carbon nanotubes, and many others. How to precisely describe the dynamic response of beams is of great interest to many researchers [1–3].

As an extension of the classical Kirchhoff–Love rod model, the equations deduced by Reissner in 1972 is regarded as one of the earliest works on geometrically exact beam and have been extended to two/three-dimensional static and dynamic large deformation problems of beam by scholars [4]. Then, the intrinsic equation was proposed for static large displacement space beam by introducing the generalized strain of the reference line [5]. For nonlinear numerical analysis, the second-order nonlinearity is the most ideal case, and the intrinsic beam theory meets this requirement. However, the basic variables in the intrinsic equations were all converted into displacement form when the solution procedure was conducted, and that makes it lack many advantages of the intrinsic formulation. Therefore, Hodges [6] presented a nonlinear intrinsic formulation for the dynamics of initially curved and twisted moving beams. Based on the intrinsic motion and constitutive equations, Hodges expressed the equations in a compact matrix form by only six generalized strain variables. Different to early works, the presented intrinsic formulation does not require reformulation while solving the equations. Thus, the intrinsic beam theory proposed by Hodges was widely used in many industries [7–11]. To evaluate different geometrically nonlinear beam theories, Bauchau et al. [12] compared the accuracy of several commonly used beam models by some standard problems. Pai [13] used a truly geometrically exact displacement-based beam theory to compare with and reveal problems of other geometrically nonlinear beam theories in the literature. In general, the main challenges and problems in geometrically exact beam modeling are (1) the problems of singularity due to the use of three independent rotation variables to describe the cross-sectional rotations, (2) the shear locking problem due to reducing the beam theory’s order by the bending-shear rotation variable, and (3) the accuracy problems caused by numerical discretization.

Meanwhile, corresponding numerical methods were also proposed to solve the static and dynamic problems of the developed geometrically exact beam equations [14, 15]. Borri and Mantegazza [16] firstly presented equations of motion in the intrinsic form for dynamics of beams. The equations developed in their work are suitable to describe dynamic behavior of the initially curved and twisted beam, such as the helicopter rotor blades. Cardona and Geradin [17] developed another finite element method for the Reissner beam element, where the rotation vector appears as a dependent variable, and the singular problem of complete Lagrange formula could be avoided when the rotation angle is close to 2Pi and its multiple. In addition, Karlson and Leamy [18] developed a shooting method for computing solutions to nonlinear intrinsic beam equations. As no iteration is needed in the computing procedure, the presented approach may find application in model-based control for practical three-dimensional problems, such as the control of manipulators utilized in endoscopic surgeries and the control of spacecraft with robotic arms and long cables. Khaneh Masjedi and Ovesy [19] and Khaneh Masjedi and Maheri [20] presented a Chebyshev collocation method for the static and free vibration analyses of the geometrically exact beams with fully intrinsic formulation. Le et al. [21] presented a corotational beam element for the dynamic analysis of 3D flexible frames. Wang et al. [22] developed a novel nonlinear aeroelastic model for large wind turbine blades by combining blade element momentum theory and mixed-form formulation of geometrically exact beam theory. Considering that the beam-shaped structures have been widely used in micro- and nano-electromechanical systems, Karparvarfard et al. [23] developed a new geometrically nonlinear elastic size-dependent Euler–Bernoulli beam formulation in the framework of the second strain theory and the corresponding end conditions were also given. Kurka et al. [24] proposed a simplified numerical model of a long flexible beam with variable cross section. Jeong and Yoo [25] proposed a nonlinear modeling method to conduct the static and dynamic analyses of a flexible beam using the in-extensible beam assumption. Lenci et al. [26] compared the free vibrations of Timoshenko beams with mechanical or geometric curvature definitions. The results of different curvature definitions are different for extensible beams. Ahmed and Rhali [27] discussed the nonlinear transverse vibrations of Bernoulli–Euler beams with general boundary conditions based on the extended Hamilton’s principle.

In the literatures demonstrated above, there are two main methods for solutions to dynamic problem of large deformation beam: nonlinear finite element methods and finite difference method. However, both methods require a preprocessing step where the domain is discretized, and the accuracy of the results depend on the points size, which means the more the points, the lower the computational efficiency. In this paper, the discretization formulations of the intrinsic beam model are presented by using the point interpolation meshless method. Low-order interpolation functions are used to obtain the shape functions. Since the governing equations of the intrinsic beam are in the first-order form, it is easy to get the discrete form of the intrinsic continua beam directly by substituting the shape functions into the governing partial differential equations. Then, only differential of the shape functions will appear in the final discrete first-order equations. Furthermore, no integration means no need of the Gauss quadrature, which will speed up the whole solving process.

#### 2. Intrinsic Continua Formulation for Beams

##### 2.1. Geometric Relationship of the Deformed and Undeformed Beam

Consider an initially twisted and curved beam as shown in Figure 1. The initial shape of the beam can be described by the undeformed configuration , and the configuration of the deformed beam is described by . is the reference configuration, which is the common configuration of the strait beam generally.