# Analytical Solutions for Stochastic Vibration of Orthotropic Membrane under Random Impact Load

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Study

#### 2.1. Model Description

- (1)
- Both membrane and impact load are symmetric. Furthermore, the center of impact load coincides with the center of membrane, namely (0,0);
- (2)
- The impact load is the mass of homogeneous intensity ${\rho}^{*}$ and area with the length of $2{a}^{\ast}$ and the width of $2{b}^{\ast}$;
- (3)
- The space distribution of impact load is symmetric and non-uniform;
- (4)
- According to the central limit theorem, the amplitude of impact load follows a Gaussian distribution;
- (5)
- The applied area of impact load can be defined as ${\mathsf{\Omega}}_{0}=\left\{(x,y)|-{a}^{*}\le x\le {a}^{*},-{b}^{*}\le y\le {b}^{*}\right\}$.

#### 2.2. Statistical Characteristics of Impact Load

#### 2.3. Establishing the FPK Equation

#### 2.4. Solving the FPK Equation

## 3. Experimental Study

#### 3.1. Experimental System

#### 3.1.1. Experimental Pretension Device

#### 3.1.2. Load Device

#### 3.1.3. Data Collection Device

#### 3.2. Experimental Schedule

#### 3.2.1. Experimental Samples

#### 3.2.2. Load Program

#### 3.2.3. Measurement Parameters

## 4. Validation of Theoretical Model

#### 4.1. Impact Load

#### 4.2. Stochastic Vibration Characteristic

## 5. Results and Discussions

#### 5.1. Effect of Pretension Force

#### 5.1.1. Effect of Pretension Force on Mean Value of Displacement

#### 5.1.2. Effect of Pretension Force on Variance Value of Displacement

#### 5.2. Effect of Velocity of Impact Load

#### 5.2.1. Effect of Velocity of Impact Load on Mean Value of Displacement

#### 5.2.2. Effect of Velocity of Impact Load on Variance Value of Displacement

#### 5.3. Effect of Membrane Material

#### 5.3.1. Effect of Membrane Material on Mean Value of Displacement

#### 5.3.2. Effect of Membrane Material on Variance Value of Displacement

## 6. Conclusions

- The theoretical model proposed can predict the stochastic dynamic characteristics of the membrane accurately subjected to random impact load;
- When the strong stochastic vibration with obvious nonlinearity occurs, membrane will function as the nonlinear filter system, with the PDF results of dynamic response prone to approximate the Rayleigh Distribution. However, when the stochastic vibration proves weak, membrane will function as the linear filter system, with the corresponding PDF results more likely to follow the Gaussian distribution.
- The developed experimental system and program paves a way to study the stochastic vibration problem of membrane subjected to random impact load;
- The mean and variance value declines nonlinearly with pretension force and elastic modulus increasing. However, it will rise with velocity of impact load increasing. Furthermore, pretension force affects stochastic vibration results most dominantly among different variables.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The statistical function of random impact load. (

**a**) Auto-correlation function; (

**b**) power spectral density function.

**Figure 10.**Some experimental data samples at Point A2. (

**a**) Impact velocity with respect to time; (

**b**) displacement with respect to time.

**Figure 15.**PDF and CDF results of displacement at specific positions: (

**a**) Point O, (

**b**) Point A1, (

**c**) Point A2, (

**d**) Point B1, (

**e**) Point B2, and (

**f**) Point C.

Blower Type | Power (W) | Flow (m^{3}/h) | Pressure (Pa) | Rotational Speed (r/s) |
---|---|---|---|---|

CZR-CZT | 1100 | 1140 | 1080 | 2800/60 |

Y90L-2 | 2200 | 1191 | 2562 | 2840/60 |

Y100L-2 | 3000 | 1704 | 3253 | 2880/60 |

Type | Density (kg/m²) | Thickness (mm) | Poisson Ratio (Warp/Weft) | Elastic Modulus (Warp/Weft) (MPa) | Tensile Strength (Warp/Weft) (N/cm) |
---|---|---|---|---|---|

Hextex | 0.95 | 0.8 | 0.3/0.4 | 1520/1290 | 4000/3800/5 |

ZZF | 0.95 | 0.8 | 0.3/0.4 | 1590/1360 | 4300/4000/5 |

XYD | 0.95 | 0.8 | 0.3/0.4 | 1720/1490 | 4400/4200/5 |

Wind Type | Mean Value | Variance Value | ||||
---|---|---|---|---|---|---|

Theory (m/s) | Experiment (m/s) | Error (%) | Theory (m^{2}/s^{2}) | Experiment (m^{2}/s^{2}) | Error (%) | |

Minimum wind | 2.716 | 2.652 | 2.41 | 1.164 | 1.142 | 1.93 |

Medium wind | 13.997 | 14.187 | −1.34 | 2.749 | 2.628 | 4.60 |

Maximal wind | 21.912 | 22.017 | −0.48 | 3.829 | 3.874 | −1.16 |

Position | Theory (mm) | Experiment (mm) | Error (%) |
---|---|---|---|

Point O | 3.035 | 2.983 | 1.743 |

Point A1 | 1.994 | 1.982 | 0.605 |

Point A2 | 2.457 | 2.408 | 2.035 |

Point B1 | 1.248 | 1.224 | 1.961 |

Point B2 | 1.368 | 1.342 | 1.937 |

Point C | 1.952 | 1.974 | 1.114 |

Position | Theory (mm^{2}) | Experiment (mm^{2}) | Error (%) |
---|---|---|---|

Point O | 0.033 | 0.031 | 6.452 |

Point A1 | 0.031 | 0.029 | 5.969 |

Point A2 | 0.032 | 0.030 | 5.864 |

Point B1 | 0.018 | 0.017 | 3.053 |

Point B2 | 0.024 | 0.023 | 4.013 |

Point C | 0.030 | 0.028 | 6.078 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Li, D.; Zheng, Z.-L.; Yang, R.; Zhang, P.
Analytical Solutions for Stochastic Vibration of Orthotropic Membrane under Random Impact Load. *Materials* **2018**, *11*, 1231.
https://doi.org/10.3390/ma11071231

**AMA Style**

Li D, Zheng Z-L, Yang R, Zhang P.
Analytical Solutions for Stochastic Vibration of Orthotropic Membrane under Random Impact Load. *Materials*. 2018; 11(7):1231.
https://doi.org/10.3390/ma11071231

**Chicago/Turabian Style**

Li, Dong, Zhou-Lian Zheng, Rui Yang, and Peng Zhang.
2018. "Analytical Solutions for Stochastic Vibration of Orthotropic Membrane under Random Impact Load" *Materials* 11, no. 7: 1231.
https://doi.org/10.3390/ma11071231