*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.

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**

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.

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.

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