Postgraduate, School of Naval Architecture and Maritime, Zhejiang Ocean University

Postgraduate, School of Naval Architecture and Maritime, Zhejiang Ocean University

2.  Dynamic modelling of an elastic beam using 6-DOF Absolute Nodal Coordinate Formulation (2020.07~2020.10)

Based on the 3-DOF Nodal Position FEM program and the mastering of numerical analysis methods, a Fortran based program using 6-DOF Absolute nodal coordinate formulation (ANCF) was developed which was first proposed by Shabana (1996).

The elastic beam in this study is discretized into ANCF beam elements based on the Euler-Bernoulli beam theory. The effect of the rotary inertia and that of shear strain are neglected. We can deduce the equation of motion of the beam element by generalized D’Alembert principle as follows, $$\boldsymbol{M\dot{V}+KX+CV=F_{ext}}$$

where $\boldsymbol{M}$, $\boldsymbol{K}$, $\boldsymbol{C}$ are the element mass matrix, stiffness matrix and damping matrix, respectively.  $\boldsymbol{K}$, the element stiffness matrix are composed of two parts: the axial part and the bending part. It should be mentioned that the axial strain can be approximated by the average strain value based on the small longitudinal deformation assumption

Then, Equations of motion of the elastic beam can be naturally obtained by assembling the element contributions into the global system. The global system is solved by an implicit symplictic 4th order 2-stage Runge-Kutta Gaussian-Legendre scheme.

To validate the implementation of our code, the results produced by present program were compared with those obtained by Berzeri and Shabana (2000) shown as follows.

##### Case 1. Free-falling of an elastic beam
As shown in Fig. 1, a horizontally orientated elastic beam in the air is arranged initially to have one end connected to a fixed pin (point A) while the other end (point B) is free for induced motion. The relevant parameters are listed in Table. 1 Fig. 1 A free-falling elastic beam in the air

As a solution verification, the present model results are compared with those obtained by Fangfang S. (2020). Before the comparison analysis, the number of elements impacting the dynamic response is investigated. Three different mesh scheme had been applied, and the results are shown in Fig. 2. It is evident that the difference is trivial. In terms of accuracy of the program, the 30 elements mesh scheme is used in the next comparison.

Fig. 3 plots the comparisons of the vertical position of the elastic beam with different element numbers. The red curve represents the result produced by our code. It can be observed that a good match is achieved according to the comparison plot. Fig. 2 The number of elements impact on the dynamic response Fig. 3 Comparisons results between our code and conventional ANCF (Berzeri and Shabana, 2000)

To get a visual results, Fig. 4 shows the time varying configurations of the falling elastic beam in terms of the vertical and horizontal positions at every 0.1 s with g = 9.81 $m/s^{2}$. Fig. 5 is the GIF obtained by MATLAB. Fig. 4 Time varying configurations results at every 0.1 s Fig. 5 GIF of the free-falling motion of the elastic beam
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