# Natural Frequency Sensitivity Analysis of Fire-Fighting Jet System with Adaptive Gun Head

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## Abstract

**:**

## 1. Introduction

## 2. Establishment of the Dynamic Model and Equations of the Jet System

- (1)
- The jet system is modeled based on the lumped parameter method. It is considered that the density, stiffness, pressure, and other attribute parameters of each fluid unit in the jet system are evenly distributed in the control volume and equal everywhere;
- (2)
- Except for the fluid units and the spring, it is considered that the parts such as the spray core, the gun head enclosure, the barrel, and the outer wall of the pipe are rigid bodies, regardless of their deformation;
- (3)
- The spray core, the core rod-end cap structure, and each fluid unit are only affected by the axial force, and the force of the fluid units on the solid units is equivalent to the axial linear spring force;
- (4)
- The damping between the fluid units and the solid units is equivalent to axial linear damping;
- (5)
- Ignore the processing and installation errors of each component.

## 3. Derivation of the Sensitivity Formula of the Jet System

#### 3.1. Sensitivity of the Natural Frequency of the Jet System to the Mass of Fluid Unit 1

#### 3.2. Sensitivity of the Natural Frequency of the Jet System to the Stiffness of Fluid Unit 1

#### 3.3. Sensitivity of the Natural Frequency of the Jet System to the Mass of Fluid Unit 2

#### 3.4. Sensitivity of the Natural Frequency of the Jet System to the Stiffness of Fluid Unit 2

#### 3.5. Sensitivity of the Natural Frequency of the Jet System to the Mass of the Spray Core

#### 3.6. Sensitivity of the Natural Frequency of the Jet System to the Stiffness of the Spring in the Gun Head

## 4. Sensitivity Analysis of the Jet System

#### 4.1. Modal Analysis of the Jet System

#### 4.2. Sensitivity Analysis of the Natural Frequency of the Jet System to the Mass of Fluid Unit 1

#### 4.3. Sensitivity Analysis of the Natural Frequency of the Jet System to the Stiffness of Fluid Unit 1

#### 4.4. Sensitivity Analysis of the Natural Frequency of the Jet System to the Mass of Fluid Unit 2

#### 4.5. Sensitivity Analysis of the Natural Frequency of the Jet System to the Stiffness of Fluid Unit 2

#### 4.6. Sensitivity Analysis of the Natural Frequency of the Jet System to the Mass of the Spray Core

- (1)
- With the increase in the mass of the spray core, the first-order natural frequency sensitivity of the jet system increases slightly in the process, but it can be regarded as basically unchanged due to the small increase. The absolute value of the sensitivity is small and negative. In the variation of the natural frequency, the first-order natural frequency decreases slightly from 19.73 Hz to 19.54 Hz with the increase in the mass of the spray core.
- (2)
- With the increase in the mass of the spray core, the second-order natural frequency sensitivity of the jet system is negative, and its absolute value increases first and then decreases. Therefore, the second-order natural frequency of the jet system appears to decrease monotonically with the increase in the mass of the spray core, and the rate of decline increases slightly first and then decreases gradually.
- (3)
- With the increase in the mass of the spray core, the third-order natural frequency sensitivity of the jet system is negative, and its absolute value decreases gradually. Therefore, the third-order natural frequency of the jet system decreases with the increase in the mass of the spray core. The rate of decline decreases gradually and tends to zero eventually.

#### 4.7. Sensitivity Analysis of the Natural Frequency of the Jet System to the Stiffness of the Spring in the Gun Head

## 5. Experimental Verification

## 6. Conclusions

- (1)
- Considering the fluid-structure interaction and discrete-continuous coupling characteristics of the fire-fighting jet system with the adaptive gun head, the dynamic model and equations of the fire-fighting jet system with adaptive gun head were established based on the lumped parameter method. The sensitivity calculation formulas of the natural frequency of the jet system to typical design parameters were derived.
- (2)
- The modal analysis of the jet system was carried out, and the natural frequencies of the orders and the corresponding modal vectors under given conditions were obtained. The sensitivity analysis of the jet system was carried out, and the variation law of the first three natural frequencies and their sensitivities of the jet system to typical design parameters was revealed. Among the parameters involved, the first three natural frequencies of the jet system were the most sensitive to the change in the mass of fluid unit 2 in the range of a certain initial gas content.
- (3)
- The platform for the dynamic experiment of the fire-fighting jet system with the adaptive gun head was built, and the modal experiment of the jet system was carried out. There was only a 0.51% error between the value of the first-order natural frequency of the jet system determined by the modal experiment and the theoretical one determined by the analytical method, showing that good agreement with the first-order natural frequency of the jet system was found.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Structure of a traditional diversion gun head. 1. Joint; 2. Nut; 3. Regulator; 4. Gasket; 5. Enclosure; 6. Ring; 7. Outer nozzle; 8. Inner nozzle; 9. Spray core; 10. End cap; 11. Core rod.

**Figure 2.**Structure of the new adaptive gun head. 1. Joint; 2. Nut; 3. Regulator; 4. Gasket; 5. Enclosure; 6. Ring; 7. Outer nozzle; 8. Inner nozzle; 9. Spray core; 10. End cap; 11. Core rod; 12. Spring; 13. Core sleeve; 14. Seal ring.

**Figure 6.**The variation law of the first three natural frequencies and their sensitivities of the jet system under a different initial gas content of the fluid with the mass of fluid unit 1. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 7.**The variation law of the first three natural frequencies and their sensitivities of the jet system under a different pressure of the fluid with the mass of fluid unit 1. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 8.**The variation law of the first three natural frequencies and their sensitivities of the jet system under a different initial gas content of the fluid with the stiffness of fluid unit 1. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 9.**The variation law of the first three natural frequencies and their sensitivities of the jet system under a different pressure of the fluid with the stiffness of fluid unit 1. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 10.**The variation law of the first three natural frequencies and their sensitivities of the jet system under a different initial gas content of fluid with the mass of fluid unit 2. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 11.**The variation law of the first three natural frequencies and their sensitivities of the jet system under a different pressure of the fluid with the mass of fluid unit 2. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 12.**The variation law of the first three natural frequencies and their sensitivities of the jet system under a different initial gas content of fluid with the stiffness of fluid unit 2. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 13.**The variation law of the first three natural frequencies and their sensitivities of the jet system under a different pressure of the fluid with the stiffness of fluid unit 2. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 14.**The variation law of the first three natural frequencies and their sensitivities of the jet system with the mass of the spray core. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 15.**The variation law of the first three natural frequencies and their sensitivities of the jet system with the stiffness of the spring in the gun head. (

**a**) Variation law of natural frequency sensitivity. (

**b**) Variation law of natural frequency.

**Figure 16.**Platform for dynamic experiment of the fire-fighting jet system with the adaptive gun head. (

**a**) Prototype of gun head. (

**b**) Experimental equipment.

**Figure 17.**Amplitude-frequency characteristic curve of the fire-fighting jet system with the adaptive gun head.

Fluid Unit | Segmentation | Equivalent Stiffness of Fluid Unit Subsection | Average Area of Flow Cross Section S_{ai}/(mm^{2}) | Axial Length of the Fluid Domain L_{i}/(mm) |
---|---|---|---|---|

Fluid unit 1 | l_{1} | k_{f11} | 6221 | 1534.6 |

l_{2} | k_{f12} | 6200 | 38 | |

l_{3} | k_{f13} | 3200 | 39 | |

l_{4} | k_{f14} | 3800 | 35.5 | |

l_{5} | k_{f15} | 4400 | 38 | |

l_{6} | k_{f16} | 3700 | 64 | |

Fluid unit 2 | l_{1’} | k_{f21’} | 300 | 46.1 |

l_{2’} | k_{f22’} | 800 | 11 | |

l_{3’} | k_{f23’} | 2800 | 40.5 |

Parameter Name | Parameter Symbol | Unit | Value |
---|---|---|---|

Mass of core rod | m | kg | 0.0354 |

Mass of end cap | m_{d} | kg | 0.1177 |

Mass of spray core | m_{6} | kg | 0.3163 |

Mass of Fluid unit 1 | m_{7} | kg | 16.4593 |

Mass of Fluid unit 2 | m_{5} | kg | 0.1362 |

Stiffness of discrete unit of the core rod | k_{1} | kN/m | 333,460 |

Equivalent stiffness 1 of fluid | k_{f1} | kN/m | 354.05 |

Equivalent stiffness 2 of fluid | k_{f2} | kN/m | 424.07 |

Equivalent stiffness 3 of fluid | k_{f3} | kN/m | 424.07 |

Stiffness of the spring in the gun head | k_{5} | kN/m | 18 |

Fluid pressure | p | MPa | 0.6 |

Initial gas content of fluid | x | % | 2 |

Bulk elastic modulus of fluid | B_{f} | MPa | 161.1432 |

Temperature | T | K | 293 |

Order | First-Order | Second-Order | Third-Order | Fourth-Order | Fifth-Order | Sixth-Order | Seventh-Order |
---|---|---|---|---|---|---|---|

Natural frequency f_{ni}/Hz | f_{n1} | f_{n2} | f_{n3} | f_{n4} | f_{n5} | f_{n6} | f_{n7} |

19.6 | 230.8 | 427.4 | 4040 | 24,276.9 | 43,906 | 57,146.6 | |

Principal mode of each mode Φ_{ni} | 0.0004 | 0.001 | 0.0013 | 0.261 | 0.7231 | 1 | −0.7044 |

0.0009 | 0.002 | 0.0025 | 0.5176 | 1 | −0.0186 | 1 | |

0.0015 | 0.003 | 0.0038 | 0.7653 | 0.6597 | −0.9997 | −0.7153 | |

0.0019 | 0.0041 | 0.005 | 1 | −0.0877 | 0.0372 | 0.0155 | |

0.3549 | 0.7578 | 1 | −0.0049 | 0 | 0 | 0 | |

0.7062 | 1 | −0.3205 | 0.0001 | 0 | 0 | 0 | |

1 | −0.0252 | 0.0023 | 0 | 0 | 0 | 0 |

Natural Frequency | Experimental Data | Theoretical Value | Error |
---|---|---|---|

The first-order natural frequency | 19.5 Hz | 19.6 Hz | 0.51% |

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## Share and Cite

**MDPI and ACS Style**

Yuan, X.; Zhu, X.; Wang, C.; Zhang, L.; Zhu, Y.
Natural Frequency Sensitivity Analysis of Fire-Fighting Jet System with Adaptive Gun Head. *Processes* **2019**, *7*, 808.
https://doi.org/10.3390/pr7110808

**AMA Style**

Yuan X, Zhu X, Wang C, Zhang L, Zhu Y.
Natural Frequency Sensitivity Analysis of Fire-Fighting Jet System with Adaptive Gun Head. *Processes*. 2019; 7(11):808.
https://doi.org/10.3390/pr7110808

**Chicago/Turabian Style**

Yuan, Xiaoming, Xuan Zhu, Chu Wang, Lijie Zhang, and Yong Zhu.
2019. "Natural Frequency Sensitivity Analysis of Fire-Fighting Jet System with Adaptive Gun Head" *Processes* 7, no. 11: 808.
https://doi.org/10.3390/pr7110808