# A Variable-Length Rational Finite Element Based on the Absolute Nodal Coordinate Formulation

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## Abstract

**:**

## 1. Introduction

## 2. RANCF Finite Element

## 3. ALE-RANCF Finite Element

## 4. Element Length Control

#### 4.1. Element Segmentation

#### 4.2. Element Merging

## 5. Sliding Joint Model

## 6. Numerical Examples

#### 6.1. A Falling Beam with a Sliding Lumped Mass

#### 6.2. A Suspended Beam with a Sliding Lumped Mass

#### 6.3. A Suspended Semicircular Beam with a Sliding Lumped Mass

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ALE | Arbitrary Lagrange–Euler |

ANCF | Absolute Nodal Coordinate Formulation |

RANCF | Rational Absolute Nodal Coordinate Formulation |

CAD | Computer-Aided Design |

CAA | Computer-Aided Analysis |

NURBS | Nonuniform Rational B-Splines |

ICADA | Integration of Computer-Aided Design and Analysis |

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**Figure 2.**Arbitrary Lagrange–Euler absolute nodal coordinate formulation (ALE-RANCF) beam element model (${\mathrm{X}}_{1}$ and ${\mathrm{X}}_{2}$ are axes of the global coordinate system).

**Figure 12.**Configurations of (

**a**) ALE-RANCF and (

**b**) ALE-ANCF model before and after element length-control processes. LC is short for length-control.

**Figure 14.**Simulation results of ALE-RANCF and ALE-ANCF model, including (

**a**) model configurations, (

**b**) length of beam, (

**c**) change velocity of beam length, and (

**d**) system energies.

**Figure 16.**Simulation results of dynamic model, including (

**a**) model configurations, (

**b**) length of the beam, (

**c**) trajectory of the sliding node, and (

**d**) system energies.

**Table 1.**Statistical results of simulation time of ALE-RANCF and the ALE-ANCF models. LC is short for length-control.

Number of Elements | LC Times | ALE-RANCF | ALE-ANCF | ||
---|---|---|---|---|---|

Motion Time(s) | LC Time(s) | Motion Time(s) | LC Time(s) | ||

5 | 3 | 59.9 | 16.1 | 34.6 | / |

10 | 9 | 97.1 | 69.1 | 58.3 | / |

20 | 18 | 169.2 | 157.0 | 103.1 | / |

30 | 26 | 292.4 | 238.3 | 192.4 | / |

40 | 35 | 478.0 | 378.2 | 325.0 | / |

Number of Elements | Size of Set | Maximum | Minimum | Deviation $\left(\mathit{\sigma}\right)$ |
---|---|---|---|---|

5 | 6 | 1.32 | 0.44 | 0.27 |

10 | 24 | 1.18 | 0.12 | 0.29 |

20 | 57 | 1.15 | 0.73 | 0.06 |

30 | 81 | 1.16 | 0.95 | 0.03 |

40 | 108 | 1.07 | 0.72 | 0.04 |

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**MDPI and ACS Style**

Ding, Z.; Ouyang, B.
A Variable-Length Rational Finite Element Based on the Absolute Nodal Coordinate Formulation. *Machines* **2022**, *10*, 174.
https://doi.org/10.3390/machines10030174

**AMA Style**

Ding Z, Ouyang B.
A Variable-Length Rational Finite Element Based on the Absolute Nodal Coordinate Formulation. *Machines*. 2022; 10(3):174.
https://doi.org/10.3390/machines10030174

**Chicago/Turabian Style**

Ding, Zhishen, and Bin Ouyang.
2022. "A Variable-Length Rational Finite Element Based on the Absolute Nodal Coordinate Formulation" *Machines* 10, no. 3: 174.
https://doi.org/10.3390/machines10030174