# Dynamic Response Analysis of a Simply Supported Double-Beam System under Successive Moving Loads

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Vertical Dynamic Response of a Simply Supported Double-Beam System

#### 2.1. Mathematical Model Building and Parameter Solving

^{th}moving load. It is assumed that ${F}_{1}$ acts on $x=0$ at the initial moment ($t=0$); ${t}_{i}={d}_{i}/v$, ${d}_{\mathrm{i}}$ represents the distance from ${F}_{i}$ to ${F}_{1}$.

^{th}column of matrix ${\mathsf{\Gamma}}_{k}$.

#### 2.2. Expression of Fourier Series of Successive Moving Loads

#### 2.3. Dynamic Response of Double-Beam Model Under Load Series

## 3. Analysis of Calculation Examples

#### 3.1. Effect of Speed of Loads on Dynamic Response of Double-Beam System

_{p}is the ratio of peak dynamic deflection response of midspan of the primary beam to that of the secondary beam. As shown in Figure 4 and Table 1, under a load series of multiple different speeds, the analytical calculation results of time-history curves and peaks of dynamic deflection response of the midspan for the simply supported double-beam system were consistent with the calculation results obtained from the ANSYS finite-element numerical method, thus demonstrating the rationality of the analytical calculation method proposed in this paper. Compared with the secondary beam, the primary beam had a significantly increased peak dynamic deflection response and a high-frequency component in the time-history curve of the dynamic deflection response of the midspan. Under the four speeds, λ

_{p}values were 1.950, 2.093, 1.706, and 2.467; the peak dynamic deflection responses of the midspan of the primary and secondary beams did not increase with the increase in the speed of the loads, indicating that the simply supported double-beam system under successive moving loads had critical speeds.

#### 3.2. Effect of Flexural Rigidity on Dynamic Response of Double-Beam System

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**The 3D dynamic graphs of vertical deflection of a simply supported double-beam system for: (

**a**) primary beam; (

**b**) secondary beam.

**Figure 4.**The response of the beams obtained by different method: (

**a**,

**b**) ${v}_{1}=40\text{}\mathrm{m}/\mathrm{s}$; (

**c**,

**d**) ${v}_{2}=100\text{}\mathrm{m}/\mathrm{s}$; (

**e**,

**f**) ${v}_{3}=122\text{}\mathrm{m}/\mathrm{s}$; (

**g**,

**h**) ${v}_{4}=180\text{}\mathrm{m}/\mathrm{s}$.

**Figure 6.**The max response versus the speed: (

**a**) ${E}_{1,1}{I}_{1,1}=0.0001{E}_{2}{I}_{2}$; (

**b**) ${E}_{1,2}{I}_{1,2}=0.001{E}_{2}{I}_{2}$; (

**c**) ${E}_{1,3}{I}_{1,3}=0.01{E}_{2}{I}_{2}$; (

**d**) ${E}_{1,4}{I}_{1,4}=0.1{E}_{2}{I}_{2}$.

**Figure 7.**The magnification factor under different flexural rigidity for: (

**a**) primary beam; (

**b**) secondary beam.

$\mathit{v}\text{}\left(\mathbf{m}/\mathbf{s}\right)$ | Layer | ${\mathit{p}}_{\mathit{a}\mathit{n}}$ | ${\mathit{p}}_{\mathit{f}\mathit{e}}$ | ${\mathit{e}}_{\mathit{p}}$ |
---|---|---|---|---|

40 | Primary beam | −3.067 | −3.112 | −1.46% |

Secondary beam | −1.572 | −1.571 | 0.06% | |

λ_{p} | 1.950 | 1.981 | ||

100 | Primary beam | −2.922 | −2.933 | −0.40% |

Secondary beam | −1.396 | −1.399 | −0.23% | |

λ_{p} | 2.093 | 2.096 | ||

122 | Primary beam | −3.754 | −3.815 | −1.60% |

Secondary beam | −2.201 | −2.200 | 0.04% | |

λ_{p} | 1.706 | 1.734 | ||

180 | Primary beam | −2.769 | −2.754 | 0.53% |

Secondary beam | −1.122 | −1.121 | 0.07% | |

λ_{p} | 2.467 | 2.456 |

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**MDPI and ACS Style**

Jiang, L.; Zhang, Y.; Feng, Y.; Zhou, W.; Tan, Z.
Dynamic Response Analysis of a Simply Supported Double-Beam System under Successive Moving Loads. *Appl. Sci.* **2019**, *9*, 2162.
https://doi.org/10.3390/app9102162

**AMA Style**

Jiang L, Zhang Y, Feng Y, Zhou W, Tan Z.
Dynamic Response Analysis of a Simply Supported Double-Beam System under Successive Moving Loads. *Applied Sciences*. 2019; 9(10):2162.
https://doi.org/10.3390/app9102162

**Chicago/Turabian Style**

Jiang, Lizhong, Yuntai Zhang, Yulin Feng, Wangbao Zhou, and Zhihua Tan.
2019. "Dynamic Response Analysis of a Simply Supported Double-Beam System under Successive Moving Loads" *Applied Sciences* 9, no. 10: 2162.
https://doi.org/10.3390/app9102162