A consistent corotational formulation for the nonlinear dynamic analysis of sliding beams

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Abstract

This paper presents a consistent corotational formulation for the geometric nonlinear dynamic analysis of 2D sliding beams. Compared with the works of previously published papers, the same cubic shape functions are used to derive the elastic force vector and the inertia force vector. Consequently, the consistency of the element is ensured. The shape functions are used to describe the local displacements to establish an element-independent framework. Moreover, all kinds of standard elements can be embedded within this framework. Therefore, the presented method is more versatile than previous approaches. To consider the shear deformation, the sliding beam (a system of changing mass) is discretized using a fixed number of variable-domain interdependent interpolation elements (IIE). In addition, the nonlinear axial strain and the rotary inertia are also considered in this paper. The nonlinear motion equations are derived by using the extended Hamilton's principle and solved by combining the Newton-Raphson method and the Hilber-Hughes-Taylor (HHT) method. Furthermore, the closed-form expressions of the iterative tangent matrix and the residual force vector are obtained. Three classic examples are given to verify the high accuracy and efficiency of this formulation by comparing the results with those of commercial software and published papers. The simulation results also show that the shear deformation and the rotary inertia cannot be neglected for the large-rotation and high-frequency problem.

Introduction

Moving flexible beams have been widely used in engineering applications, such as spacecraft antenna [1], robot arms [[2], [3], [4]] and space elevators [[5], [6], [7], [8], [9]]. These applications all involve dynamical systems of changing mass. The nonlinear equations of motion of sliding beams were first given by Tabarrok et al. [1] through Newton's Second Law and Euler-Bernoulli beam theory. To obtain the approximate solution, the authors linearized the equations of motion by introducing the small-deformation and inextensibility assumptions. The finite element method [10] as one of the most widely used numerical methods has also been employed to study the dynamic problems of sliding beams. Stylianou and Tabarrok [11] discretized the sliding beam with a fixed number of finite elements, while the element length changes with time. Then, they verified the accuracy of the variable-domain element through some examples. Moreover, they investigated the stability [12] of the time-varying system. Al-Bedoor and Khulief [13] modelled sliding beams by conventional finite elements with time-dependent boundary conditions. They introduced a transition element with variable stiffness to simulate the interaction between the sliding beam and the channel orifice.

To consider the geometric nonlinear effect due to the large deformation of sliding beams, Vu-Quoc and Li [14] studied the dynamics problems based on the geometrically exact theory [[15], [16], [17]]. The beams were analysed taking the shear deformation into account and could undergo large overall motion. The parametric resonance problems were also discussed in the authors' numerical examples. Subsequently, Behdinan et al. [18] provided another derivation of the equations of motion based on the extension of Hamilton's principle [19]. Then, Behdinan and Tabarrok [20] employed Galerkin's method to solve the equations. Based on previous works, Gürgöze and Yüksel [21] investigated a more general system of sliding beams and obtained an approximate solution by the assumed modes method. Behdinan and Tabarrok [22] used the U.L. method [23,24] and corotational method [25] to solve the nonlinear equations of the motion of sliding beams. In the U.L. formulation, variable-domain beam elements and constant-domain stretched beam elements were compared. In the corotational formulation, the sliding channel problem and the sliding beam problem were studied using the variable-domain beam elements. In their numerical examples [26], the authors discussed the differences between the linear analysis and nonlinear analysis of sliding beams. Humer and Irschik [27] studied the static problem of beams that can move relative to their supports. However, previous works did not consider the deformation of the inside part of sliding beams. To overcome this limitation, Humer [28] introduced a new non-material coordinate to derive the sliding beam formulation. Steinbrecher et al. [29] used commercial software, ABAQUS, to simulate the motion of sliding beams and compared the results with those of Humer's [28] method. In recent years, variable-domain elements have been used to investigate the dynamic response of space elevators [[6], [7], [8]].

Regarding studies on the dynamic analysis of many structures, such as delaminated composite shallow shell panels, functionally graded panels and laminated curved sandwich structures [[30], [31], [32], [33]], both the geometric nonlinearity and shear deformation should be taken into account. The high-order shear deformation theory and Green-Lagrange strain were introduced in their formulation. For the sliding beam problem, the most important character is that the length of the beam changes with time. If the length is very long, the beam will be flexible. Then, the geometric nonlinearity (large displacement and large rotation) of the beam should be considered. If the length is very short, the shear deformation of the beam will have direct effects on the vibration frequency and amplitude of the beam. Therefore, the geometric nonlinearity and shear deformation of the beam should both be considered for the dynamic analysis of the sliding beam.

The deformation of flexible beams has the characteristics of small strain, large displacement and large rotation. For this kind of geometric nonlinear problem, the corotational method has high efficiency and accuracy [34]. Nour-Omid and Rankin [35,36] established the general framework of this method. The key idea of the corotational method is to decompose the motion of the element into rigid body motion and pure deformation. The pure deformational part is measured in the rotational frame. Crisfield et al. [37,38] derived the corotational formulation of 3D beams. To ensure the consistency of the elements for dynamic analysis, Le et al. [34,39] made some improvements to the formulation. Behdinan et al. [22] proposed a corotational formulation [25] for sliding beams. However, their formulation cannot make the element independent because they used the shape functions to describe the global displacements. Furthermore, the rotational kinetic energy was not considered in their formulation because the calculation accuracy may not be high enough for studying high-frequency problems.

Based on the above analysis, this paper presents a consistent corotational formulation for the geometric nonlinear dynamic analysis of 2D sliding beams. The element-independent framework is consistent with the idea of Nour-Omid and Rankin's [35,36] corotational method. Then, many standard beam elements can be embedded within the dynamic framework. The shape functions of interdependent interpolation element (IIEs) [40] are used to derive the kinetic energy and potential energy of elements to consider the shear deformation. The total kinetic energy includes the translational kinetic energy and the rotational kinetic energy. The system of sliding beams is discretized using variable-domain beam elements in space and the HHT method [41,42] in time.

Section snippets

Basic kinematic assumptions of the sliding beam

As shown in Fig. 1, the beam slides along the axis of a rigid channel. The beam can undergo large deformation but we assume that the part inside the channel is non-deformable. The time-varying material configuration can be described asB0={(X,Y)R2|X[0,L(t)],Y[h2,h2]}where L(t) and h are the length and height of the beam outside the channel, respectively. The origin O of the material coordinate system (X,Y) with basis vectors {I,J} is located at the centre of the channel orifice. We can set

Strain energy and kinetic energy of the beam element

The sliding beam can be divided into n elements of equal length l0(t)=L(t)n. The displacements of nodes 1 and 2 of the ith element can be denoted by (u1,w1,θ1)=(u(P),w(P),θ(P))P=((i1)Ln,0)Q and (u2,w2,θ2)=(u(P),w(P),θ(P))P=(iLn,0)Q, respectively (see Fig. 2). For convenience, we ignore the subscript i of variables associated with the ith element. The global displacement vector can be defined byq=[u1w1θ1u2w2θ2]T

The local displacement of the element is measured in the corotational coordinate

Nonlinear equation of motion

Based on the assumption above, the Hamiltonian principle for a system of changing mass [19] can be expressed as [1,43].δt1t2(EKEP)dt+t1t2δWdt=0where the virtual work of the external force isδW=δqTfwhere f denotes the external force vector.

The item δt1t2EPdt in Eq. (32) can be expressed in the global and local systems, i.e.,δt1t2EPdt=t1t2δqTfGdt=t1t2δq˜TfLdtwhere fG denotes the global elastic force vector, and the local elastic force vector fL is defined as in Eq. (20). To obtain the

Numerical algorithms

The nonlinear equations of motion for the discrete system are solved at each time step using the Newton-Raphson method. The time discretization of the system uses the HHT method. Eq. (43) can be expressed in the general formΨ(q,q˙,q¨)=fI(q,q˙,q¨)+fG(q)f=0

Linearization of Eq. (44) at the (k+1)th iteration of step i is performed as follows:Ψ(qik+1,q˙ik+1,q¨ik+1)=Ψ(qik,q˙ik,q¨ik)+(Ψq)ikΔqik+(Ψq˙)ikΔq˙ik+(Ψq¨)ikΔq¨ik=0withq¨ik+1=q¨ik+Δq¨ikq˙ik+1=q˙ik+Δq˙ikqik+1=qik+Δqik

The external force

Numerical examples

In this section, three numerical examples are given to study the problems of beams with fixed lengths under small and large deformations. The accuracy and efficiency of this formulation are verified by comparing the results with those of commercial software and published papers.

Conclusion

In this paper, a consistent corotational formulation of 2D sliding beams was presented. Because the material configuration changes over time, variable-domain IIEs were used to discretize the sliding beam system in space. The shear deformation, nonlinear axial strain and rotary inertia were considered in the formulation. The elastic force vector and inertia force vector were derived through the extension of Hamilton's principle and used the same independent element. Thus, the consistency of the

CRediT authorship contribution statement

Lanfeng Deng: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft. Yahui Zhang: Conceptualization, Validation, Investigation, Resources, Supervision, Project administration.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors wish to acknowledge the financial support from the National Natural Science Foundation of China (11672060).

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