# Dynamics of Double-Beam System with Various Symmetric Boundary Conditions Traversed by a Moving Force: Analytical Analyses

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## Abstract

**:**

## 1. Introduction

## 2. Frequencies and Mode Shapes of the Double-Beam System

#### 2.1. Mathematical Formulation

#### 2.1.1. Fixed-Fixed Boundary Conditions

#### 2.1.2. Pinned-Pinned Boundary Conditions

#### 2.1.3. Fixed-Pinned Boundary Conditions

#### 2.1.4. Pined-Fixed Boundary Conditions

#### 2.1.5. Fixed-Free Boundary Conditions

#### 2.2. Verification of the Analytical Solutions in Section 2.1

#### 2.3. Parametric Studies

#### 2.3.1. The Order of Basic Mode

#### 2.3.2. Contact Stiffness Ratio

#### 2.3.3. Mass Ratio

#### 2.3.4. Beam Stiffness Ratio

## 3. Double-Beam System Traversed by a Moving Force

#### 3.1. Mathematical Formulation

#### 3.1.1. Fixed-Fixed and Fixed-Pinned Boundary Conditions

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#### 3.1.2. Pinned-Pinned Boundary Conditions

#### 3.1.3. Pinned-Fixed Boundary Conditions

#### 3.1.4. Fixed-Free Boundary Conditions

#### 3.2. Verification of the Analytical Solutions in Section 3.1

#### 3.2.1. Fixed-Fixed Boundary Conditions

#### 3.2.2. Pinned-Pinned Boundary Conditions

#### 3.3. Parametric Studies

#### 3.3.1. Speed Ratio

#### 3.3.2. Contact Stiffness Ratio

- ${\overline{w}}_{1,\mathrm{max}}$ generally decreases with ${s}_{\mathrm{e}}$ and the opposite trend is true for ${\overline{w}}_{2,\mathrm{max}}$;
- The varying rates of ${\overline{w}}_{i,\mathrm{max}}\text{}\left(i=1,\text{}2\right)$ are very large when ${s}_{\mathrm{e}}$ is below a turning point and become much smaller when ${s}_{\mathrm{e}}$ is beyond the turning point;
- With the increase of ${s}_{\mathrm{e}}$, ${\overline{w}}_{1,\mathrm{max}}$ and ${\overline{w}}_{2,\mathrm{max}}$ tend to be the same value which is half of that for a single beam with the same boundary conditions and at the same values of $\alpha $, ${s}_{\mathrm{m}}$ and ${s}_{\mathrm{k}}$;
- The difference between ${\overline{w}}_{1,\mathrm{max}}$ and ${\overline{w}}_{2,\mathrm{max}}$ is generally smaller for looser boundary conditions.

#### 3.3.3. Mass Ratio

#### 3.3.4. Beam Stiffness Ratio

## 4. Conclusions

- (1)
- Each wavenumber corresponds to two sub-modes of the system. The mode shapes of one beam of the system are the same as those for the single beam with the same boundary condition. The amplitudes of the mode shapes for one beam of the double-beam system are the multiple of those for the other beam of the system.
- (2)
- The two sub-modes corresponding to the first wavenumber both make significant contributions to the dynamics of the system under a moving load, which is different from the case for a single beam.
- (3)
- The maximum dynamic displacement of the primary beam generally decreases with the stiffness of the contact springs. The opposite trend is true for the maximum dynamic displacement of the secondary beam. The two beams vibrate together when the contact springs are very stiff. With the increase of the ratio between the mass of the primary beam and the secondary beam, the maximum dynamic displacement ratios of both beams increases first and then decreases. The maximum dynamic displacement ratios of both beams are smaller for a larger bending stiffness ratio of the primary beam to the secondary beam.
- (4)
- The primary beam tends to vibrate together with the secondary beam when the boundary condition of the system is looser.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Frequency ratios in two sub-modes versus the order of basic mode for different boundary conditions when ${s}_{\mathrm{e}}=100$, ${s}_{\mathrm{m}}=1$ and ${s}_{\mathrm{k}}=2$.

**Figure 4.**Mode amplitude ratios in two sub-modes versus the order of basic mode for different boundary conditions when ${s}_{\mathrm{e}}=100$, ${s}_{\mathrm{m}}=1$ and ${s}_{\mathrm{k}}=2$.

**Figure 5.**Frequency ratios in two sub-modes corresponding to the 1st basic mode versus contact stiffness ratio (${s}_{\mathrm{e}}$) for different boundary conditions when ${s}_{\mathrm{m}}=1$ and ${s}_{\mathrm{k}}=2$.

**Figure 6.**Mode amplitude ratios in two sub-modes corresponding to the 1st basic mode versus contact stiffness ratio (${s}_{\mathrm{e}}$) for different boundary conditions when ${s}_{\mathrm{m}}=1$ and ${s}_{\mathrm{k}}=2$.

**Figure 7.**Frequency ratios in two sub-modes corresponding to the 1st basic mode versus contact stiffness ratio for different boundary conditions when ${s}_{\mathrm{e}}=100$ and ${s}_{\mathrm{k}}=1$.

**Figure 8.**Mode amplitude ratios in two sub-modes corresponding to the 1st basic mode versus contact stiffness ratio for different boundary conditions when ${s}_{\mathrm{e}}=100$ and ${s}_{\mathrm{k}}=1$.

**Figure 9.**Frequency ratios in two sub-modes corresponding to the 1st basic mode versus beam stiffness ratio for different boundary conditions when ${s}_{\mathrm{e}}=100$ and ${s}_{\mathrm{m}}=1$.

**Figure 10.**Mode amplitude ratios in two sub-modes corresponding to the 1st basic mode versus beam stiffness ratio for different boundary conditions when ${s}_{\mathrm{e}}=100$ and ${s}_{\mathrm{m}}=1$.

**Figure 12.**Comparison between analytical results and numerical results for $\mathsf{\alpha}=0.5$, (

**a**,

**b**) ${s}_{\mathrm{e}}=10$, (

**c**,

**d**) ${s}_{\mathrm{e}}=100$.

**Figure 13.**Comparison between the results from the Modal Superposition (MS) method in this study and the results from the Equation Decoupled (ED) method by Hilal [19] for $\mathsf{\alpha}=0.5$, (

**a**,

**b**) ${s}_{\mathrm{e}}=10$, (

**c**,

**d**) ${s}_{\mathrm{e}}=100$.

**Figure 14.**Maximum displacement ratios ${\overline{w}}_{1,\mathrm{max}}$ and ${\overline{w}}_{2,\mathrm{max}}$ versus speed ratio $\alpha $ for different boundary conditions when ${s}_{\mathrm{e}}=100$, ${s}_{\mathrm{m}}=1$ and ${s}_{\mathrm{k}}=1$.

**Figure 15.**Maximum displacement ratios ${\overline{w}}_{1,\mathrm{max}}$ and ${\overline{w}}_{2,\mathrm{max}}$ versus contact stiffness ratio ${s}_{\mathrm{e}}$ for different boundary conditions when $\alpha =0.5$, ${s}_{\mathrm{m}}=1$ and ${s}_{\mathrm{k}}=1$.

**Figure 16.**Maximum displacement ratios ${\overline{w}}_{1,\mathrm{max}}$ and ${\overline{w}}_{2,\mathrm{max}}$ versus mass ratio ${s}_{\mathrm{m}}$ for different boundary conditions when $\alpha =0.5$, ${s}_{\mathrm{e}}=100$ and ${s}_{\mathrm{k}}=1$.

**Figure 17.**Maximum displacement ratios ${\overline{w}}_{1,\mathrm{max}}$ and ${\overline{w}}_{2,\mathrm{max}}$ versus mass ratio ${s}_{\mathrm{k}}$ for different boundary conditions when $\alpha =0.5$, ${s}_{\mathrm{e}}=100$ and ${s}_{\mathrm{m}}=1$.

Mode | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

${r}_{1}l$ | 4.73 | 7.8532 | 10.9956 | 14.1372 | 17.2788 | 20.4204 |

Mode | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

${r}_{1}l$ | 3.9266 | 7.0686 | 10.2102 | 13.3518 | 16.4934 | 19.6350 |

Mode | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

${r}_{1}l$ | 1.8751 | 4.6941 | 7.8548 | 10.9955 | 14.1372 | 17.2788 |

Mode Number | Frequency (Hz) | Difference | |
---|---|---|---|

FE | Analytical | (%) | |

1 | 16.12 | 16.12 | 0 |

2 | 44.44 | 44.44 | 0 |

3 | 87.08 | 87.08 | 0 |

4 | 143.66 | 143.68 | 0 |

5 | 179.90 | 180.72 | 0.5 |

6 | 180.03 | 180.85 | 0.5 |

7 | 180.49 | 181.31 | 0.5 |

8 | 181.77 | 182.59 | 0.5 |

9 | 182.20 | 183.01 | 0.4 |

10 | 185.72 | 186.57 | 0.5 |

11 | 190.52 | 191.38 | 0.5 |

12 | 197.36 | 198.23 | 0.4 |

13 | 206.69 | 207.61 | 0.4 |

14 | 216.22 | 216.24 | 0 |

15 | 218.82 | 219.77 | 0.4 |

16 | 233.99 | 234.96 | 0.4 |

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**MDPI and ACS Style**

Yang, J.; He, X.; Jing, H.; Wang, H.; Tinmitonde, S.
Dynamics of Double-Beam System with Various Symmetric Boundary Conditions Traversed by a Moving Force: Analytical Analyses. *Appl. Sci.* **2019**, *9*, 1218.
https://doi.org/10.3390/app9061218

**AMA Style**

Yang J, He X, Jing H, Wang H, Tinmitonde S.
Dynamics of Double-Beam System with Various Symmetric Boundary Conditions Traversed by a Moving Force: Analytical Analyses. *Applied Sciences*. 2019; 9(6):1218.
https://doi.org/10.3390/app9061218

**Chicago/Turabian Style**

Yang, Jing, Xuhui He, Haiquan Jing, Hanfeng Wang, and Sévérin Tinmitonde.
2019. "Dynamics of Double-Beam System with Various Symmetric Boundary Conditions Traversed by a Moving Force: Analytical Analyses" *Applied Sciences* 9, no. 6: 1218.
https://doi.org/10.3390/app9061218