J. Cent. South Univ. (2018) 25: 208-217

DOI: https://doi.org/10.1007/s11771-018-3730-y

A recursive formulation based on corotational frame for flexible planar beams with large displacement

WU Tan-hui(吴坛辉), LIU Zhu-yong(刘铸永), HONG Jia-zhen(洪嘉振)

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University,Shanghai 200240, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: A forward recursive formulation based on corotational frame is proposed for flexible planar beams with large displacement. The traditional recursive formulation has been successfully used for flexible mutibody dynamics to improve the computational efficiency based on floating frame, in which the assumption of small strain and deflection is adopted. The proposed recursive formulation could be used for large displacement problems based on the corotational frame. It means that the recursive scheme is used not only for adjacent bodies but also for adjacent beam elements. The nodal relative rotation coordinates of the planar beam are used to obtain equations with minimal generalized coordinates in present formulation. The proposed formulation is different from absolute nodal coordinate formulation and the geometrically exact beam formulation in which the absolute coordinates are used. The recursive scheme and minimal set of dynamic equations lead to a high computational efficiency in numerical integration. Numerical examples are carried out to demonstrate the accuracy and validity of this formulation. For all of the examples, the results of the present formulation are in good agreement with results obtained using commercial software and the published results. Moreover, it is shown that the present formulation is more efficient than the formulation in ANSYS based on GEBF.

Key words: recursive formulation; multibody dynamics; large displacement beam; corotational frame

Cite this article as: WU Tan-hui, LIU Zhu-yong, HONG Jia-zhen. A recursive formulation based on corotational frame for flexible planar beams with large displacement [J]. Journal of Central South University, 2018, 25(1): 208–217. DOI: https://doi.org/10.1007/s11771-018-3730-y.

1 Introduction

Flexible beams are widely used in flexible multibody systems, such as large deployable space structures and wind turbines. The nonlinear dynamic behavior of these systems with large displacement and large rotation is rather complicated. A large number of papers about beams based on kinds of approaches have been published [1–6]. The dynamic modeling formulations of multibody systems can be developed by three types of frame approaches: the floating frame, the corotational frame and the inertial frame [7].

The floating frame approach was proposed in the late 1960s, which is commonly used for elastic bodies undergoing small deformation. This approach is extended from rigid multibody dynamics by superimposed elastic deformations on rigid motion. It is efficient for small and moderate elastic deformation problem [7].

Recursive formulations are usually used in conjunction with the floating reference frame and relative coordinates [7]. The formulation was first applied to open-loop rigid multibody systems by CHACE [8], and to open-loop flexible multibody systems by BOOK [9], CHANGIZI et al [10], and KIM et al [11] and BAE et al [12]. Then it was extended to closed-loop flexible multibody systems by adding cut-joint constraints to the equations of motion [13, 14]. The computational efficient of recursive formulation can be increased by two folds. On one hand, the recursive solution algorithm is O(N) [15, 16], where N is the number of degrees of freedom, it means that the computational time grows only linearly with N. On the other hand, a minimal set of dynamic equations is obtained.

The corotational frame approach was proposed by ARGYRIS [17]. In this approach, corotational frames are defined for each element. The motion of element is superimposing by element rigid motion and element elastic deformation. It is used for flexible body undergoing large displacement and small deformation. Based on this approach, several kinds of beam elements were proposed by BELYTSCHKO [2, 18].

To consider large displacement, the inertial frame approach is commonly applied. Two kinds of formulations are suggested among them. One is the absolute nodal coordinate formulation (ANCF), which was proposed by SHANBANA [19, 20]. The other one is the geometrically exact beam formulation (GEBF) which was proposed by SIMO [3, 21, 22]. Both of them could be used to solve the nonlinear beam problem.

In ANCF, nodal displacements and slopes are selected as generalized coordinates. Without the rotation of nodes, this method does not require the parameterization and interpolation of the finite rotation. This feature causes the mass matrix to become constant. Also the strain is objectively treated well in this method. Based on this formulation, a number of beam elements have been developed [23–28].

In GEBF, nodal displacement and rotation coordinates are employed as generalized coordinates. The attentions are focused on the treatments of finite rotation [29–32], and many methods for the interpolation of rotation were developed in the past. For instance, interpolation of rotation tensor [33, 34], interpolation of Euler parameters [29, 35], and interpolation of relative rotation [32] were developed by different researchers. As pointed out by BAUCHAU [36], the interpolation of relative rotation parameter is the most accurate.

One important issue for the simulation of flexible beams is to improve numerical efficiency. In present formulation, a forward recursive procedure has been used in conjunction with the corotational frame and relative nodal rotation. While the recursive formulation is conjunct to the corotational frame, the recursive formulation could be used to solve large displacement problems. And the motion of a flexible component is divided into rigid body motion and natural deformation. This simplifies the calculation of the internal forces, since the stress and strain measures are small.

The bending deformation is usually the dominated deformation in many engineering fields. In present formulation, the large displacement and large rotation are considered while the axial deformation is ignored based on the assumption that the axial deformation is small. The assumptions for Euler-Bernoulli beam are used in this formulation. The generalize coordinates are the relative nodal rotations of beam elements. The radius of beam’s axis curvature can be computed by the derivative of the relative rotation. The accuracy and the efficiency of this formulation are demonstrated by several examples. And the results are compared with the commercial software and the published results.

This work is organized as follows: In Section 1, the background and status of research are reviewed and the purpose of this paper is clarified. Section 2 presents the description of deformation and kinematics for a new finite element formulation. Section 3 presents the dynamic equations of elastic beams undergoing large overall motions using forward recursive approach. In Section 4, numerical simulation results of a static beam, a rotating beam and flexible beam system are presented to demonstrate the validity of the proposed formulation. Finally, conclusions are drawn in Section 5.

2 Deformation description and kinematics of a planar beam element

An Euler-Bernoulli beam is given based on the following assumptions: 1) the cross sections are rigid; 2) the cross sections are perpendicular to the axis; 3) the effects of rotary inertia of the cross sections are neglected [27]. As shown in Figure 1, a two-node beam element is presented. O–XY is the inertial frame; i–x_iy_i is the ith element’s corotational frame which is a tangent frame fixed on the ith node on deformed beam; i–x_0iy_0i is the ith element’s corotational frame fixed on the ith node on undeformed beam, where the subscript 0 denotes the undeformed configuration; α_i is the rotation of the (i+1)th node relative to the ith node in the ith element; l_i is the length of the ith element; s is the arc-length from the ith node to point P. Then, the relative rotation of point P to the ith node can be given by linear interpolation,

                              (1)

And the relative rotation derivate to s can be obtained as

                           (2)

where the prime (′) denotes the derivate to s. Since the beam is subjected to bending deformation, both axial and shear strain are ignored based on the assumption that the axial deformation is small. The curvature κ_i of the ith beam element axis is given by

                           (3)

The point P position derivate to arc-length parameter s can be given in corotational frame:

                        (4)

where the prime denotes the coordinates in corotational frame. The position of point P in corotational frame can be easily obtained as

                         (5)

Substitute Eq. (4) into Eq. (5), we can get

                        (6)

If α′(s)=0, the result of Eq. (6) would be infinity. Thus, consider Eqs. (1) and (2) and Trigonometric function Taylor series, Eq. (6) can be rewritten as

          (7)

While k≥4, an accurate approximation can be obtained for Eq. (6). The translational velocity of point P in corotational frame can be obtained:

   (8)

The translational acceleration of point P in corotational frame can be obtained:

 (9)

The absolute position, velocity and acceleration of point P can be given by:

                             (10)

                       (11)

          (12)

where ω_i is the angular velocity of corotational frame fixed on the ith node; is the relative translational velocity of point P in corotational frame fixed on the ith node. where is a skew symmetric matrix:

                            (13)

where

                             (14)

where A_i is the direction cosine matrix of the ith node. Based on Eqs. (8) and (9), Eqs. (11) and (12) can be rewritten as follows:

                              (15)

                          (16)

where

                     (17)

                        (18)

          (19)

 (20)

       (21)

where is a units matrix; θ_i is the absolute angular rotation of the ith node’s corotational frame.

Figure 1 Deformation description of ith beam element

3 Forward recursive formulation of flexible beam dynamics

The dynamic equation of a beam element can be given as

               (22)

where ρ_A is the cross section density, E is the elastic modulus of the material, I_z is the inertial moment of the cross section, and are applied external forces and moments loads. Consider Eqs. (15)–(19), Eq. (22) can be rewritten as

              (23)

In Eq. (23), the generalize mass matrix M_i, the generalize inertia forces w_i, the generalize external forces the generalize elastic forcesare given as follows:

                      (24)

                      (25)

   (26)

           (27)

In order to obtain the single beam dynamics equations, a forward recursive construction approach is used. The floating frame of flexible beam is attached to the corotational frame of the 1st node. So the equation of frame motion can be written by recursive propagation. As shown in Figure 2, the absolute position, velocity and acceleration of the (i+1)th node can be obtained as follows:

                             (28)

                       (29)

           (30)

where v_i is the relative translational velocity of the (i+1)th node in corotational frame fixed on the ith node. The rotation velocity and rotation acceleration can be obtained as

                            (31)

                            (32)

Figure 2 Recursive relation from ith element to (i+1)th element

Then the generalized velocity of the (i+1)th element recursion equation is obtained as

                    (33)

where

,            (34)

where The generalized acceleration of the (i+1)th node recursion equation is obtained as

                (35)

with

                           (36)

where

            (37)

wheredefines matrix as a units matrix. Then the generalized velocity of the (i+1)th element can be obtained by recursion propagation as follows:

                 (38)

          (39)

where

                          (40)

              (41)

           (42)

The dynamic equations for the jth beam with n elements in a multibody system can be given as follows:

(43)

where the superscript j denotes the jth beam, x^j presents the generalized coordinates:

                   (44)

M^j presents the generalized mass matrix:

                         (45)

f^j presents the generalized force matrix:

              (46)

where

                     (47)

                     (48)

                      (49)

is a n-dimensional vector with all elements being 1. The dynamic equations of the jth beam can be rewritten as

                              (50)

where

                         (51)

                 (52)

Since the single beam dynamic equations are obtained from recursive propagation. The dynamic equations of N beams’ multibody system with tree structure can be obtained directly from recursive propagation [37] as follows.

                                 (53)

And the closed loops multibody system can be transformed into open loops by using cut-off joint. The Lagrange multipliers are introduced to express the constraints at the cut point [37]. The dynamic equations of system can be written as

                            (54)

                             (55)

where λ is the Lagrange multipliers. Equation (55) is the constraint equation, Θ_y is the matrix of constraint gradients, and the generalized-method for constrained mechanical systems is used [38].

4 Numerical examples

In the previous section, the system dynamic equations are developed. In this section, numerical examples of a static beam, a dynamical beam and a multi-beam mechanism are calculated to verify the validity, convergence and accuracy of proposed formulation. The results using proposed formulation are compared with the results using the commercial software ANSYS or the published ANCF result. The results using ANSYS are given by using element beam 188 which is developed from GEBF theory.

In Example 1, displacements are calculated for a clamped beam with three different case static loads applied to its free end. The material, geometric properties of beam and load case are shown in Figure 3. Several static load cases are considered and the comparison between ANSYS solutions and results using present model is carried out. The results of beam meshed by 30 elements for both methods are shown in Figure 3. As shown in Figure 3, the results are in good agreement for every load case. When the moment applied to the tip of beam is 2226.6 N·m, the beam deformed to a circle and the displacement of beam’s tip is 0. The displacement relative error of beam’s tip is nearly 10^–15 for a discretization with 5 elements using proposed formulation.

Figure 3 Static deformation of clamped beam

Relative errors are computed with respect to a reference solution obtained by using ANSYS with 200 elements. The displacement relative error using different elements for load F_x=–350 N and F_y=100 N is shown in Figure 4 and for load F_x=–700 N and F_y=100 N is shown in Figure 5. As shown in the two figures, the accuracy of proposed formulation is better than ANSYS with the same element number. It means that for the same accuracy less element number is used for proposed formulation. Moreover, the degrees of freedom using the proposed formulation are less than those by using ANSYS while the same element number is adopted. For nonlinear finite element formulation, the degree of freedom of a planar Euler-Bernoulli node is 3 in general. In this example, 30 elements with 33 degrees of freedom are used in the proposed formulation while 30 elements with 93 degrees of freedom are used in classical planar beam element.

Figure 4 Relative error of tip displacement for load F_y=100 N, F_x=–350 N

Figure 5 Relative error of tip displacement for load F_y=100 N, F_x=–700 N

As shown in Figure 6, Example 2 concerns a simple pendulum, which is connected to ground at one side by special joint and falls freely under gravity. The material and geometric properties of beam are shown in Figure 6. Figure 7 shows the X displacement of the free end of the pendulum versus time. The result of ANCF uses 4 elements with 57 degrees of freedom. ANSYS result uses 10 elements with 63 degrees of freedom. And 10 elements with 12 freedom of degree are used in this example using the proposed formulation.

Figure 6 Pendulum beam

Figure 7 X-coordinate of tip of pendulum

Apparently, the proposed formulation has much less degree of freedom than other formulation. The comparison shows that the present solution is generally agreed with the others.

As shown in Figure 7, the present solution is more agreed with ANSYS, although the theory of each formulation is different. Figure 8 shows the X displacement of the tip of pendulum versus time with different elements using proposed formulation. The results are in good agreement, it means that the convergence of proposed formulation is good for dynamic systems. As shown in Figure 9, the configuration of each time is agreed with ANSYS.

In Example 3, a flexible four bar mechanism is given in Figure 10. The geometric and material parameters of each beam are listed in Table 1. The moment applied to crankshaft is shown as a time function, as shown in Figure 11.

Since the elastic modules of Crankshaft and Follower are large, their dynamic behavior is similar to rigid body’s, both of them are meshed with 3 elements. The coupler is meshed with 10 elements to show its flexibility.

As shown in Figure 12, the results are compared to the results using ANSYS. The results are in good agreement. Figures 13 and 14 show the displacement of Point A and Point B. With the same discretization, it costs 39 s using proposed formulation, however, it costs 66 s using ANSYS. All the three numerical examples show that present formulation can be used for beams with large deformation and rotation of mutibody systems. And the present formulation is an efficient and accurate formulation.

Figure 8 X coordinate of tip using different elements

Figure 9 Configuration compared with ANSYS

Figure 10 Initial position of four bar linkage

Figure 11 Moment applied on crankshaft

Figure 12 Configuration of four bar linkage at each time

Figure 13 Displacement of Point A

Table 1 Parameters of four bar linkage

Figure 14 Displacement of Point B

5 Conclusions

A forward recursive formulation is proposed in conjunction with the corotational frame and relative nodal coordinates. The recursive scheme which can improve numerical simulation efficiency is used for beam elements with large displacement and rotation. It means that the recursive scheme is used not only for adjacent bodies but also for adjacent beam elements. The relative nodal rotation coordinates are used for the generalized coordinate. It is different from ANCF and GEBF, in which absolute coordinates are used. In present formulation, the large displacement and large rotation are considered while the axial deformation is ignored based on the assumption that the axial deformation is small. Numerical examples are carried out to demonstrate the accuracy and validity of this formulation. Furthermore, it is shown that the proposed formulation uses fewer degrees of freedom than those formulations based on absolute coordinates and has higher numerical efficiency. The recursive formulation will be extended from planar beams to spatial beams in future.

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(Edited by FANG Jing-hua)

中文导读

基于共旋坐标的平面大变形梁递推方法

摘要：基于共旋坐标法提出了一种大变形梁递推建模方法，用于解决含有大变形梁的柔性多体系统的动力学问题。传统递推方法与浮动做标法相结合，已应用于柔性多体系统的小变形问题。本文将递推方法与共旋坐标相结合，把传统递推方法中物体之间的递推拓展到单元之间的递推，利用共旋坐标法能高效地解决大变形问题的优点，使得所提的递推方法可应用于梁的大变形问题。递推方法优点为获得的是一个O(N)阶的动力学方程，N为系统的自由度，这意味着动力学方程求解时间随着N线性增长，使得该方法能在解决大体量的问题时具有较高的计算效率。通过静力学以及动力学的算例验证所提方法的正确性与效率。研究结果表明，所提方法在解决平面大变形梁问题上具有较高的精度和效率。

关键词：递推方法；多体动力学；平面大变形梁；共旋坐标系

Foundation item: Projects(11772188,11132007,11202126) supported by the National Natural Science Foundation of China; Project(11ZR1417000) supported by the Natural Science Foundation of Shanghai, China

Received date: 2016-01-28; Accepted date: 2016-04-29

Corresponding author: LIU Zhu-yong, PhD, Associate Professor; Tel: +86–21–34206496; E-mail: zhuyongliu@sjtu.edu.cn