# Application of the Recursive Finite Element Approach on 2D Periodic Structures under Harmonic Vibrations

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## Abstract

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## 1. Introduction

## 2. Recursive Method

#### 2.1. Review

#### 2.2. Behavior of One Cell

#### 2.3. Behavior of Two Cells

#### 2.4. General Case

- Calculate the dynamic stiffness matrix for a structure of eight cells, decomposing them into four of two cells each.
- The studied structure is then assembled with a structure of two cells, studied previously.
- The resulting matrix is assembled with a structure of one cell.

## 3. Applications

#### 3.1. Truss Application

#### 3.1.1. Truss under Forced Vibration at the Last Node

- Rayleigh damping:$${\overline{C}}^{e}=\alpha {\overline{K}}^{e}+\beta {\overline{M}}^{e},$$
- Hysteretic damping:The dynamic stiffness matrix is demonstrated as:$${D}^{e}=\left(1+0.01\ast i\right){K}^{e}-{\omega}^{2}{M}^{e}.$$

#### 3.1.2. Truss under Seismic Load

#### 3.2. Frame Application under Seismic Load

## 4. Results

#### 4.1. Truss Application under Forced Vibration

#### 4.2. Truss Application under Seismic Load

#### 4.3. Frame Application under Seismic Load

#### 4.4. Analysis

#### 4.4.1. Periodic Structures Having Steel Material

#### 4.4.2. Periodic Structures Having Plastic Material

#### 4.4.3. The Effect of the Number of Forces Applied

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Mode shapes for the natural frequencies. (

**a**) The first mode shape with a modal frequency of $f{n}_{1}=1.0235\text{}\mathrm{Hz}$; (

**b**) the second mode shape with a modal frequency of $f{n}_{2}=5.7271\mathrm{Hz}$ .

**Figure 10.**Displacement of the excited node under Rayleigh damping for the truss application that is applied to forced vibration. The graph is logarithmically scaled. : low frequencies; : high frequencies. RM, recursive method.

**Figure 12.**Displacement of the excited node under hysteretic damping for the truss application that is applied to forced vibration. The graph is logarithmically scaled.

**Figure 13.**Displacement of the last top node of the truss structure under seismic load, applied to hysteretic damping. The graph is logarithmically scaled.

**Figure 14.**Displacement of the last top node for the frame application under seismic loading, applied to hysteretic damping. The graph is logarithmically scaled.

**Figure 15.**Mode shapes for the natural frequencies. (

**a**) The mode shapes relative to the modal frequency for the truss application of $f{n}_{T}=133.63\mathrm{Hz}$; (

**b**) the mode shape for the frame application of $f{n}_{F}=33.135\mathrm{Hz}$ .

**Figure 16.**The repeated cells are presented where: (

**a**) the new repeated cell is double the previous one and it is repeated eight times; (

**b**) the new repeated cell consists of four-times the old cell, and it is repeated four times; (

**c**) the cell will be repeated two times.

Young’s Modulus of Elasticity | $E=200\mathrm{GPa}$ |

Density | $\rho =7800\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

Damping Ratio | $\zeta =0.004$ [22] |

Larger Cross-Sectional Area | $A1=0.001175{\mathrm{m}}^{2}$ |

Smaller Cross-Sectional Area | $A2=2.91\times {10}^{-4}{\mathrm{m}}^{2}$ |

Length for the Vertical and Horizontal Bars, Respectively | $l=1.5-2\mathrm{m}$ |

**Table 2.**Time comparison between the two methods, for the truss application under forced vibrations.

Method | Elapsed Time (s) | Time Ratio | |
---|---|---|---|

Rayleigh Damping | Hysteretic Damping | ||

RM | 0.618 | 0.638 | $\frac{{t}_{RM}}{{t}_{FEM}}\tilde{=}\frac{1}{18}$ |

FEM | 11.26 | 11.43 |

Method | Elapsed Time (s) | Time Ratio |
---|---|---|

RM | 0.505 | $\frac{{t}_{RM}}{{t}_{FEM}}\tilde{=}\frac{1}{22}$ |

FEM | 11.08 |

Method | Elapsed Time (s) | Time Ratio |
---|---|---|

RM | 2.078 | $\frac{{t}_{RM}}{{t}_{FEM}}\tilde{=}\frac{1}{31.6}$ |

FEM | 65.80 |

Type of Application | Type of Loading | Type of Damping | Range of Frequency Studied | Percentage Difference |
---|---|---|---|---|

Crane | Forced vibrations | Rayleigh damping | 1:1:700 | 17.519% |

Hysteretic damping | 1:1:700 | 11.783% | ||

Harmonic displacement | Hysteretic damping | 1:1:700 | 12.544% | |

Building | Harmonic displacement | Hysteretic damping | 1:1:2000 | 18.920% |

**Table 6.**The modulus of displacement in the x-direction for the excited node under three different natural frequencies, for a truss application under forced vibrations.

# of Repeating Times | Mode Shape #3 f = 13.994 Hz | Mode Shape #7 f = 44.4651 Hz | Mode Shape #14 f = 105.294 Hz | |||
---|---|---|---|---|---|---|

Displacement (m) | Elapsed Time (s) | Displacement (m) | Elapsed Time (s) | Displacement (m) | Elapsed Time (s) | |

16 | 3.29 ×${10}^{-5}$ | 0.008694 | 2.63 ×${10}^{-6}$ | 0.008967 | 4.50 ×${10}^{-7}$ | 0.008403 |

8 | 3.29 ×${10}^{-5}$ | 0.008946 | 2.63 ×${10}^{-6}$ | 0.008710 | 4.50 ×${10}^{-7}$ | 0.008837 |

4 | 3.29 ×${10}^{-5}$ | 0.008666 | 2.63 ×${10}^{-6}$ | 0.009064 | 4.50 ×$\text{}{10}^{-7}$ | 0.008530 |

2 | 3.29 ×${10}^{-5}$ | 0.008368 | 2.63 ×${10}^{-6}$ | 0.008437 | 4.50 × ${10}^{-7}$ | 0.008868 |

**Table 7.**The modulus of displacement in the x-direction for the excited node under three different natural frequencies, for a truss application under forced vibrations, upon assignment of PVC.

# of Repeating Times | Mode Shape #3 f = 4.4185 Hz | Mode Shape #7 f = 14.0399 Hz | Mode Shape #14 f = 33.2466 Hz | |||
---|---|---|---|---|---|---|

Displacement (m) | Elapsed Time (s) | Displacement (m) | Elapsed Time (s) | Displacement (m) | Elapsed Time (s) | |

16 | 1.94 ×${10}^{-3}$ | 0.008859 | 1.55 ×${10}^{-4}$ | 0.009104 | 2.65 ×${10}^{-5}$ | 0.009558 |

8 | 1.94 ×${10}^{-3}$ | 0.008405 | 1.55 ×${10}^{-4}$ | 0.009659 | 2.65 ×${10}^{-5}$ | 0.010223 |

4 | 1.94 ×${10}^{-3}$ | 0.008891 | 1.55 ×${10}^{-4}$ | 0.008590 | 2.65 ×${10}^{-5}$ | 0.013492 |

2 | 1.94 ×${10}^{-3}$ | 0.008651 | 1.55 ×${10}^{-4}$ | 0.008518 | 2.65 ×${10}^{-5}$ | 0.008253 |

Type of Application | Type of Loading | Type of Damping | Time Elapsed (s) | Time Ratio ($\frac{{\mathit{t}}_{\mathit{R}\mathit{M}}}{{\mathit{t}}_{\mathit{F}\mathit{E}\mathit{M}}})$ | Range of Frequency Studied (Hz) | Percentage Difference | |
---|---|---|---|---|---|---|---|

FEM | RM | ||||||

Crane | Forced vibrations | Hysteretic damping | 12.64 | 0.572 | $\tilde{=}\frac{1}{22}$ | 1:700 | 3.3584% |

Harmonic displacement | Hysteretic damping | 11.03 | 0.509 | $\tilde{=}\frac{1}{22}$ | 1:700 | 17.4930% | |

Building | Harmonic displacement | Hysteretic damping | 86.34 | 0.575 | $\tilde{=}\frac{1}{151}$ | 1:800 | 14.3027% |

**Table 9.**Results upon adding different numbers of forces distributed equally over the periodic structure.

# of Forces Added | Elapsed Time (s) | Time Ratio $\frac{{\mathit{t}}_{\mathit{R}\mathit{M}}}{{\mathit{t}}_{\mathit{F}\mathit{E}\mathit{M}}}$ | Range of Frequency Studied (Hz) | Range of Percentage Difference | |
---|---|---|---|---|---|

FEM | RM | ||||

2 | 70.70 | 0.6680 | $\tilde{=}\frac{1}{106}$ | 1:1:700 | $9\le \text{\%}\mathrm{E}\le 12$ |

4 | 84.46 | 0.4909 | $\tilde{=}\frac{1}{172}$ | 1:1:700 | $8\le \text{\%}\mathrm{E}\le 10$ |

8 | 73.85 | 0.6727 | $\tilde{=}\frac{1}{110}$ | 1:1:700 | $9\le \text{\%}\mathrm{E}\le 11$ |

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**MDPI and ACS Style**

Yassine, R.; Salman, F.; Al Shaer, A.; Hammoud, M.; Duhamel, D.
Application of the Recursive Finite Element Approach on 2D Periodic Structures under Harmonic Vibrations. *Computation* **2017**, *5*, 1.
https://doi.org/10.3390/computation5010001

**AMA Style**

Yassine R, Salman F, Al Shaer A, Hammoud M, Duhamel D.
Application of the Recursive Finite Element Approach on 2D Periodic Structures under Harmonic Vibrations. *Computation*. 2017; 5(1):1.
https://doi.org/10.3390/computation5010001

**Chicago/Turabian Style**

Yassine, Reem, Faten Salman, Ali Al Shaer, Mohammad Hammoud, and Denis Duhamel.
2017. "Application of the Recursive Finite Element Approach on 2D Periodic Structures under Harmonic Vibrations" *Computation* 5, no. 1: 1.
https://doi.org/10.3390/computation5010001