# A Space-Time Absolute Nodal Coordinate Formulation Cable Element and the Study of Its Accuracy and Efficiency

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## Abstract

**:**

## 1. Introduction

## 2. Establishment of SAC Element

**F**is the surface nodal force. Besides, the moment can be regarded as a special nodal force that can be calculated as additional work by integrating the moment and torsion angle. The work generated by the moment can be given in tensor form:

## 3. Element Assembly and the Solver Construction

## 4. Simulation and Verification

#### 4.1. Static Simulation

#### 4.2. Double-Ended Velocity Impact Simulation and Verification

#### 4.3. The Efficiency of Space-Time ANCF Method

## 5. The Simulation and Experimental Verification of the Free Flexible Pendulum

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) RC and SC constraints in no-constraint cable. (

**b**) RC and SC constraints in cantilever cable and the constraint conversion process.

**Figure 4.**The transfer of longitude stress waves along the cable. (

**a**) The 3D space-time stress distribution figure discretized by SAC-2 element. (

**b**) The contour line figure of (

**a**). (

**c**) The 3D space-time stress distribution figure discretized by SAC-3 element. (

**d**) The contour line figure of (

**c**).

**Figure 5.**The dynamic simulation of example-2. (

**a**) The spatial state of motion of the left end rope at the center point. (

**b**) The energy conservation in double-ended shock issues.

**Figure 6.**(

**a**) The spending time by calculation in part A and part B of Equation (32). (

**b**) Total spending times in different examples. (

**c**) The average iteration times during calculation in different examples. (

**d**) The residual error in different examples.

**Figure 10.**The comparison between the fitting curve of experiment and the simulation curve in different times.

Element No. | SFC ^{1}-x | SFC ^{1}-t |
---|---|---|

SAC-2 | Equation (4) | Equation (6) |

SAC-3 | Equation (5) | Equation (6) |

^{1}shape function component.

Material Properties | Value |
---|---|

Elastic modulus (GPa) | 3.4 |

Density (kg/m${}^{3}$) | 1380 |

Moment inertia (m${}^{4}$) | $1.25\times {10}^{-13}$ |

Area (m${}^{2}$) | $3\times {10}^{-6}$ |

Element Type and Constraint | 5 Elements | 10 Elements | ||
---|---|---|---|---|

Semicircle | Circle | Semicircle | Circle | |

SAC-2 | 65.31 | −197.3 | 0.400 | 18.30 |

SAC-3 without | ||||

${r}_{xx}$ constraint | 0.172 | $1.380\times {10}^{-3}$ | $-4.622\times {10}^{-4}$ | $2.636\times {10}^{-6}$ |

SAC-3 with | ||||

${r}_{xx}$ constraint | No data | No data | 21.17 | 0.063 |

No. of Examples | Initial Condition of A End | Initial Condition of B End | ||
---|---|---|---|---|

Position Vector | Velocity Vector | Position Vector | Velocity Vector | |

No.1 | $\left[\begin{array}{ccc}-2,& 0,& 0\end{array}\right]$ | $\left[\begin{array}{ccc}10,& 0,& 0\end{array}\right]$ | $\left[\begin{array}{ccc}2,& 0,& 0\end{array}\right]$ | $\left[\begin{array}{ccc}-10,& 0,& 0\end{array}\right]$ |

No.2 | $\left[\begin{array}{ccc}-2,& 0,& 0\end{array}\right]$ | $\left[\begin{array}{ccc}0,& 10,& 0\end{array}\right]$ | $\left[\begin{array}{ccc}2,& 0,& 0\end{array}\right]$ | $\left[\begin{array}{ccc}0,& -10,& 0\end{array}\right]$ |

Example No. | Element Type | Mesh Method ^{1} | Steps | Solver | Integral Points ^{1} |
---|---|---|---|---|---|

No.1 | SAC-2 | 8 × 1 | 400 | N-R | 3 × 3 |

No.2 | SAC-2 | 8 × 1 | 400 | BR1 | 3 × 3 |

No.3 | SAC-2 | 8 × 1 | 400 | N-R | 6 × 3 |

No.4 | SAC-2 | 4 × 1 | 200 | N-R | 3 × 3 |

No.5 | SAC-2 | 4 × 2 | 100 | N-R | 3 × 3 |

No.6 | SAC-2 | 2 × 1 | 100 | N-R | 3 × 3 |

No.7 | SAC-3 | 4 × 1 | 200 | N-R | 5 × 3 |

No.8 | SAC-3 | 4 × 1 | 200 | BR1 | 5 × 3 |

No.9 | SAC-3 | 4 × 1 | 200 | N-R | 5 × 6 |

^{1}The element distribution in Space × Time direction.

Scheme | Grid | ${\mathit{L}}_{1}$ | Order |
---|---|---|---|

Newton-Raphson | 4TEs × 2SEs | 0.0040 | - |

4TEs × 4SEs | 0.0032 | 0.3254 | |

8TEs × 2SEs | $2.66\times {10}^{-4}$ | 3.9099 | |

8TEs × 4SEs | $2.00\times {10}^{-4}$ | 4.2576 | |

Broyden rank 1 | 4TEs×2SEs | 0.0040 | - |

4TEs × 4SEs | 0.0032 | 0.3254 | |

8TEs × 2SEs | $2.66\times {10}^{-4}$ | 3.9099 | |

8TEs × 4SEs | $2.00\times {10}^{-4}$ | 4.2576 |

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

CR | 1.089 | 0.3040 | 1.592 | 1.125 | 1.171 | 1.219 | 1.538 | 0.524 | 1.526 |

Methods | Element No. | ECT ^{1} | Solution Time | Steps | ITC ^{2} | Accuracy |
---|---|---|---|---|---|---|

Longer steps | −− | / | −− | − | / | − |

Higher order | − | ++ | − | / | / | + |

Time direction | ||||||

assembly | / | / | + | −− | / | / |

Quasi-Newton method | / | / | − | / | + | / |

^{1}Spending time in element calculation.

^{2}The iteration loops until convergence.

Material Properties | Value |
---|---|

Elastic modulus (MPa) | 0.91 |

Density (kg/m${}^{3}$) | 1202 |

Moment inertia (m${}^{4}$) | $9.15\times {10}^{-12}$ |

Area (m${}^{2}$) | $1.075\times {10}^{-5}$ |

Time (s) | 0.01 | 0.19 | 0.30 | 0.42 | 0.53 | 0.67 | 0.79 |
---|---|---|---|---|---|---|---|

RMSE (mm) | 0.4145 | 0.3821 | 0.5058 | 0.6869 | 0.6761 | 1.5226 | 0.7383 |

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**MDPI and ACS Style**

Chen, D.; Li, K.; Lu, N.; Lan, P. A Space-Time Absolute Nodal Coordinate Formulation Cable Element and the Study of Its Accuracy and Efficiency. *Machines* **2023**, *11*, 433.
https://doi.org/10.3390/machines11040433

**AMA Style**

Chen D, Li K, Lu N, Lan P. A Space-Time Absolute Nodal Coordinate Formulation Cable Element and the Study of Its Accuracy and Efficiency. *Machines*. 2023; 11(4):433.
https://doi.org/10.3390/machines11040433

**Chicago/Turabian Style**

Chen, Dekun, Kun Li, Nianli Lu, and Peng Lan. 2023. "A Space-Time Absolute Nodal Coordinate Formulation Cable Element and the Study of Its Accuracy and Efficiency" *Machines* 11, no. 4: 433.
https://doi.org/10.3390/machines11040433