# The Vertical Dynamic Properties of Flexible Footbridges under Bipedal Crowd Induced Excitation

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## Abstract

**:**

## 1. Introduction

## 2. Dynamic Excitation Mechanism

_{B}is span length; ${m}^{\left(q\right)}$ is the lump mass of the bipedal model; the superscript “q” means the ordinal number of pedestrians; ${x}^{\left(q\right)}$ and ${z}^{\left(q\right)}$ are longitudinal and vertical displacements of the center of mass (CoM), respectively; ${x}_{t}^{\left(q\right)}$ and ${x}_{l}^{\left(q\right)}$ are the trialing and leading footholds positions, respectively; $\left({x}_{l}^{\left(q\right)},t\right)$ and $\left({x}_{t}^{\left(q\right)},t\right)$ are structural vertical displacements in the leading and trialing footholds, respectively.

## 3. Excitation Assumption

_{x}and N

_{y}, respectively. The total number of pedestrians can be obtained with $\chi ={N}_{x}{N}_{y}$.

## 4. Numerical Validation

_{p}/M

_{s}between crowd and structure increases from 0.1 to 0.5 under same step 1 Hz in Figure 4, the frequency approximately decreases from 1.985 Hz to 1.92 Hz and the damping ratio also shows an obvious change in Figure 4b. However, the change of step frequency under the same mass ratio M

_{p}/M

_{s}= 0.5 has a smaller effect on the model parameters. Figure 4c gives the detail of damping ratio under the M

_{p}/M

_{s}= 0.1 and f

_{s}= 1 Hz, which shows that the structure damping ratio is periodically changed along with walking gait. The frequency in Figure 4a also shows this effect and model property is disturbed by periodic walking gaits. The crowd size has a more significant impact on model properties than the walking excitation frequency.

_{s}= 1.0 Hz, the change of mass ratio between crowd and structure has a tiny effect on vibration amplitude. However, the variation of step frequency has an obvious impact on structural responses. Under the walking frequency f

_{s}= 2.0 Hz, which is the same as the structural natural frequency, the acceleration response is rapidly amplified, as shown in Figure 5b. This demonstrates that significantly varying the frequency of excitation from the structural natural frequency is a more efficient method for alleviating excessive vibration, than by limiting the crowd size.

_{s}= 2.0 Hz. Along with the increase of mass ratio between crowd and structure, crowd induced frequency approximately decreases from 2 Hz to 1.92 Hz and damping ratio approximately increases from 1% to 7%. It is noted that the both dynamic response displacement and acceleration amplitudes (Figure 7) increase along with the increase of mass ratio between crowd and structure under resonance excitation frequency. Thus it can be seen that the increase of crowd size under resonance excitation has the most prominent effect on structural dynamic properties.

_{p}/M

_{s}is less than 0.3. When the M

_{p}/M

_{s}is larger than 0.3, the acceleration approximately increases along with the increase of M

_{p}/M

_{s}. The influence of crowd size on model characteristics is consistent with measurement results [3].

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Effect of step frequency and crowd size on dynamic properties: (

**a**) the larger mass ratio M

_{p}/M

_{s}results the lower natural frequencies of structure, which is altered following the step rate; (

**b**) the larger mass M

_{p}/M

_{s}results the larger damping of structure; (

**c**) damping of occupied structure is altered following the step rate 1 Hz under the mass ratio M

_{p}/M

_{s}= 0.1.

**Figure 5.**Effect of step frequency and crowd size on dynamic responses: (

**a**) the larger mass ratio M

_{p}/M

_{s}= 0.5 induces the slightly larger vibrational displacement under the uniform non-resonant excitation rate f

_{s}= 1 Hz; however the resonant excitation rate f

_{s}= 2 Hz induces the noteworthy larger vibrational displacement amplitude under the same mass ratio M

_{p}/M

_{s}= 0.5; (

**b**) the larger mass ratio Mp/Ms = 0.5 induces the slightly larger vibrational acceleration under the uniform non-resonant excitation rate f

_{s}= 1 Hz; but the resonant excitation rate f

_{s}= 2 Hz induces the remarkable larger vibrational acceleration amplitude under the same mass ratio Mp/Ms = 0.5.

**Figure 6.**Effect of mass ratio between crowd and structure on dynamic properties: (

**a**) the larger mass ratio between crowd and structure induces the lower frequency of structure under resonant excitation; (

**b**) the larger mass ratio between crowd and structure results the larger damping of structure under resonant excitation.

**Figure 7.**Effect of mass ratio between crowd and structure on dynamic responses: the increase of mass ratio induces corresponding persistent growth of structural vibrational displacement (

**a**) and acceleration (

**b**) under resonant excitation.

**Figure 8.**Dynamic property peaks under different mass ratios between crowd and structure: (

**a**) structural frequency is approximately linearly decreased along with the growth of mass ratio; (

**b**) structural damping is approximately linearly increased along with the growth of mass ratio; (

**c**) the increase of mass ratio result in the persistent growth of peak acceleration under resonant excitation.

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**MDPI and ACS Style**

Gao, Y.-a.; Wang, J.; Liu, M. The Vertical Dynamic Properties of Flexible Footbridges under Bipedal Crowd Induced Excitation. *Appl. Sci.* **2017**, *7*, 677.
https://doi.org/10.3390/app7070677

**AMA Style**

Gao Y-a, Wang J, Liu M. The Vertical Dynamic Properties of Flexible Footbridges under Bipedal Crowd Induced Excitation. *Applied Sciences*. 2017; 7(7):677.
https://doi.org/10.3390/app7070677

**Chicago/Turabian Style**

Gao, Yan-an, Juan Wang, and Min Liu. 2017. "The Vertical Dynamic Properties of Flexible Footbridges under Bipedal Crowd Induced Excitation" *Applied Sciences* 7, no. 7: 677.
https://doi.org/10.3390/app7070677