# The Stiffness and Damping Characteristics of a Dual-Chamber Air Spring Device Applied to Motion Suppression of Marine Structures

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}, causes a compression of the air in the upper chamber. The pressure variation of the air spring chamber is of the same phase and amplitude as the external load. For the gas-liquid coupling air spring device applied to motion suppression of marine structures, a liquid column can also move in an independent manner. As shown in Figure 1b, the amount of compressed air in the air chamber is the relative displacement between the liquid column and the top of the air spring, i.e. x

_{1}−x

_{0}. The movement of the liquid column (similar to the piston) and the external excitation can differ in phases and amplitude. The stiffness and damping characteristics of this kind of dual-chamber air spring have not been investigated.

## 2. Theoretical Analysis

#### 2.1. Parallel Model

**,**substituted into Equation (1) and using Euler′s Formula, ${e}^{i\mathsf{\phi}}=\mathrm{cos}\mathsf{\phi}+i\mathrm{sin}\mathsf{\phi}$, the stiffness and damping of the vibration system can be written as:

_{s}refers to the sampling frequency of the experiment, f

_{s}= 1024 Hz. Due to the equal-interval sampling of the experiment data, ${t}_{i}=\frac{1}{1024}$.

#### 2.2. Series Model

#### 2.3. Work Done by Excitation Force

## 3. Experimental Method

#### 3.1. Design of Experimental Device

#### 3.2. Setting of Test Parameters

_{2}, Volume of lower air chamber V

_{1}, opening diameter d, amplitude of outer excitation A, and frequency of outer excitation ω.

_{1}, and the frequency of outer excitation, ω, were used as the basic variables with the following three dimensionless numbers:

_{1}= 0.2 m

#### 3.3. Orthogonal Experiment and Orthogonal Array

^{4}) experiments need to be carried out. Hence, a method with the minimum numbers of experiments is needed to find out the effect of each parameter on the vibrating characteristics of the air spring.

## 4. Experimental Results and Analysis of a Dual-Chamber Air Spring

#### 4.1. Influence of Various Parameters on Air Spring Stiffness

#### 4.2. Influence of Various Parameters on Air Spring Damping

## 5. Rapid Calculation Model of Dual-Chamber Air Spring Stiffness and Damping

#### 5.1. Normalization Process of Each Factor

#### 5.2. Determination of Rapid Calculation Model of Stiffness and Damping

#### 5.3. Test of Calculation Model

^{2}is the ratio between the regression sum of squares and sum of squares for total. R

^{2}is between 0 and 1. The closer it is to 1, the better the fit of the regression. A goodness of fit of more than 0.8 is typically considered acceptable.

^{2}expression:

^{2}is related to the number of independent variables. In order to accurately test the accuracy of the model, it is necessary to take into account the degrees of freedom and make adjustments of R

^{2}expression according to the size of the formula.

_{a}

^{2}is denoted as the adjusted coefficient of determination; n is the total number of experiments, and p is the number of variables. The adjusted coefficient of determination more accurately reflects the degree of fit between the model and experimental data.

- For the stiffness model R
^{2}= 0.99154, R_{a}^{2}= 0.96378, the goodness of fit is 96.378%, indicating that the model can well simulate and predict the experimental results. This indicates the effectiveness of the cubic polynomial regression equation fit. - For the damping model, R
^{2}= 0.98970, R_{a}^{2}= 0.95470, the goodness of fit is 95.470%, again indicating that the model can well simulate and predict the experimental results. This again supports the effectiveness of the cubic polynomial regression equation fit.

#### 5.4. Prediction of Experimental Results by the Rapid Calculation Model

## 6. Conclusions

- Based on energy consumption results, the goodness of fit of the parallel model was 89.43%, and the goodness of fit of the series model was 99.88%. The parallel model is more consistent with the real physical model.
- The effects of volume ratio and orifice ratio on dual-chamber vibration absorber stiffness were not monotonic, but the loading amplitude ratio and frequency tended toward monotonic increasing.
- The effects of the volume ratio, orifice ratio, and loading amplitude ratio on the dual-chamber vibration absorber stiffness did not behave in a monotonic manner, but the loading frequency on damping tended toward monotonic increasing.
- A polynomial rapid calculation model for stiffness and damping was constructed. The accuracy of the rapid calculation model results was verified by the experimental results, and the predicted values were in good agreement with the experimental values.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

Case | Volume Ratio $\left({\mathsf{\xi}}_{1}\right)$ | Orifice Ratio $\left({\mathsf{\xi}}_{2}\right)$ | Loading Amplitude Ratio $\left({\mathsf{\xi}}_{3}\right)$ | Loading Frequency $\mathrm{ln}\mathsf{\omega}$ | Real Loading Frequency (Hz) | Parallel Model | Series Model | Work Done by Excitation Force | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

K | C | Q | k | c | q | W | ||||||

1 | 0.5 | 0 | 0.01 | −0.7 | 0.497 | 109.192 | 14.772 | 0.0004292 | 128.627 | 97.766 | 0.003081 | 0.0004297 |

2 | 0.5 | 0.1 | 0.03 | 0.5 | 1.649 | 1249.035 | 19.870 | 0.02411 | 1282.923 | 752.237 | 0.02483 | 0.02437 |

3 | 0.5 | 0.2 | 0.05 | 1.7 | 5.474 | 24512.047 | 58.106 | 0.1492 | 24674.823 | 8808.250 | 0.1752 | 0.1516 |

4 | 0.5 | 0.3 | 0.07 | 0.1 | 1.105 | 503.285 | 15.400 | 0.06352 | 525.984 | 356.854 | 0.07156 | 0.06384 |

5 | 0.5 | 0.4 | 0.02 | 1.3 | 3.669 | 7208.513 | 47.043 | 0.04246 | 7371.526 | 2127.292 | 0.04434 | 0.04304 |

6 | 0.5 | 0.5 | 0.04 | −0.3 | 0.741 | 158.223 | 7.249 | 0.005029 | 165.412 | 166.801 | 0.005102 | 0.005048 |

7 | 0.5 | 0.6 | 0.06 | 0.9 | 2.460 | 3121.949 | 36.450 | 0.2058 | 3223.487 | 1157.179 | 0.1996 | 0.1887 |

8 | 0.75 | 0 | 0.07 | 1.3 | 3.669 | 10804.935 | 10.771 | 0.03249 | 10810.636 | 20423.049 | 0.1299 | 0.03209 |

9 | 0.75 | 0.1 | 0.02 | −0.3 | 0.741 | 152.925 | 5.989 | 0.001248 | 158.002 | 186.403 | 0.001331 | 0.001287 |

10 | 0.75 | 0.2 | 0.04 | 0.9 | 2.460 | 3298.782 | 35.056 | 0.08677 | 3387.664 | 1336.123 | 0.08447 | 0.08329 |

11 | 0.75 | 0.3 | 0.06 | −0.7 | 0.497 | 98.908 | 8.265 | 0.008236 | 105.624 | 129.970 | 0.09743 | 0.008478 |

12 | 0.75 | 0.4 | 0.01 | 0.5 | 1.649 | 1277.190 | 20.470 | 0.002832 | 1312.362 | 763.784 | 0.002948 | 0.002881 |

13 | 0.75 | 0.5 | 0.03 | 1.7 | 5.474 | 17619.902 | 102.952 | 0.2962 | 18330.771 | 2654.775 | 0.2812 | 0.2846 |

14 | 0.75 | 0.6 | 0.05 | 0.1 | 1.105 | 519.751 | 15.930 | 0.03474 | 543.270 | 367.970 | 0.03593 | 0.03468 |

15 | 1 | 0 | 0.06 | 0.5 | 1.649 | 1412.352 | 24.368 | 0.09738 | 1457.423 | 787.955 | 0.09871 | 0.09771 |

16 | 1 | 0.1 | 0.01 | 1.7 | 5.474 | 17009.600 | 98.264 | 0.03743 | 17680.432 | 2589.838 | 0.04081 | 0.0374 |

17 | 1 | 0.2 | 0.03 | 0.1 | 1.105 | 561.650 | 16.714 | 0.01086 | 585.609 | 408.524 | 0.01111 | 0.01092 |

18 | 1 | 0.3 | 0.05 | 1.3 | 3.669 | 7642.988 | 36.205 | 0.17 | 7734.053 | 3074.820 | 0.1937 | 0.178 |

19 | 1 | 0.4 | 0.07 | −0.3 | 0.741 | 179.227 | 8.004 | 0.02 | 186.963 | 193.429 | 0.02007 | 0.02 |

20 | 1 | 0.5 | 0.02 | 0.9 | 2.460 | 3121.603 | 28.248 | 0.02371 | 3182.590 | 1474.095 | 0.02413 | 0.02384 |

21 | 1 | 0.6 | 0.04 | −0.7 | 0.497 | 99.429 | 8.566 | 0.004021 | 106.606 | 127.237 | 0.04211 | 0.004046 |

22 | 1.25 | 0 | 0.05 | −0.3 | 0.741 | 175.538 | 7.100 | 0.00899 | 181.754 | 207.608 | 0.009773 | 0.009058 |

23 | 1.25 | 0.1 | 0.07 | 0.9 | 2.460 | 3294.971 | 45.801 | 0.4436 | 3446.867 | 1039.328 | 0.452 | 0.446 |

24 | 1.25 | 0.2 | 0.02 | −0.7 | 0.497 | 93.417 | 19.146 | 0.001581 | 131.580 | 66.013 | 0.004269 | 0.001549 |

25 | 1.25 | 0.3 | 0.04 | 0.5 | 1.649 | 1337.610 | 35.092 | 0.07292 | 1436.307 | 510.685 | 0.07293 | 0.07287 |

26 | 1.25 | 0.4 | 0.06 | 1.7 | 5.474 | 39581.430 | 324.946 | 0.365 | 42733.898 | 4404.865 | 0.3508 | 0.3427 |

27 | 1.25 | 0.5 | 0.01 | 0.1 | 1.105 | 507.804 | 26.241 | 0.002135 | 573.125 | 230.241 | 0.002108 | 0.002117 |

28 | 1.25 | 0.6 | 0.03 | 1.3 | 3.669 | 7085.175 | 47.500 | 0.1258 | 7254.263 | 2037.839 | 0.1285 | 0.1262 |

29 | 1.5 | 0 | 0.04 | 1.7 | 5.474 | 25749.741 | 114.565 | 0.184 | 26352.096 | 5012.048 | 0.1885 | 0.1795 |

30 | 1.5 | 0.1 | 0.06 | 0.1 | 1.105 | 578.651 | 17.741 | 0.05048 | 604.853 | 409.543 | 0.05246 | 0.05053 |

31 | 1.5 | 0.2 | 0.01 | 1.3 | 3.669 | 7492.063 | 34.781 | 0.01139 | 7577.801 | 3074.069 | 0.01196 | 0.0116 |

32 | 1.5 | 0.3 | 0.03 | −0.3 | 0.741 | 191.848 | 8.807 | 0.003483 | 200.600 | 201.881 | 0.003591 | 0.003496 |

33 | 1.5 | 0.4 | 0.05 | 0.9 | 2.460 | 3380.530 | 47.534 | 0.2329 | 3540.001 | 1055.189 | 0.2364 | 0.2345 |

34 | 1.5 | 0.5 | 0.07 | −0.7 | 0.497 | 83.222 | 14.337 | 0.01418 | 107.242 | 64.009 | 0.03212 | 0.01412 |

35 | 1.5 | 0.6 | 0.02 | 0.5 | 1.649 | 1376.694 | 34.954 | 0.01857 | 1471.837 | 540.732 | 0.01881 | 0.01865 |

36 | 1.75 | 0 | 0.03 | 0.9 | 2.460 | 3187.457 | 27.938 | 0.05243 | 3245.880 | 1552.170 | 0.05043 | 0.05119 |

37 | 1.75 | 0.1 | 0.05 | −0.7 | 0.497 | 85.259 | 12.130 | 0.006464 | 102.043 | 73.747 | 0.01677 | 0.006484 |

38 | 1.75 | 0.2 | 0.07 | 0.5 | 1.649 | 1529.732 | 46.205 | 0.2168 | 1679.347 | 518.626 | 0.2152 | 0.2133 |

39 | 1.75 | 0.3 | 0.02 | 1.7 | 5.474 | 18778.734 | 133.334 | 0.2094 | 19897.493 | 2371.391 | 0.2137 | 0.2089 |

40 | 1.75 | 0.4 | 0.04 | 0.1 | 1.105 | 573.199 | 29.053 | 0.03713 | 644.131 | 263.825 | 0.03814 | 0.03742 |

41 | 1.75 | 0.5 | 0.06 | 1.3 | 3.669 | 10461.160 | 45.520 | 0.1921 | 10566.332 | 4573.206 | 0.2358 | 0.2106 |

42 | 1.75 | 0.6 | 0.01 | −0.3 | 0.741 | 174.026 | 10.879 | 0.0004944 | 188.745 | 139.501 | 0.0004885 | 0.0004824 |

43 | 2 | 0 | 0.02 | 0.1 | 1.105 | 564.645 | 14.696 | 0.005067 | 583.071 | 465.062 | 0.005049 | 0.005031 |

44 | 2 | 0.1 | 0.04 | 1.3 | 3.669 | 9798.212 | 48.073 | 0.179 | 9923.452 | 3809.107 | 0.1693 | 0.1718 |

45 | 2 | 0.2 | 0.06 | −0.3 | 0.741 | 198.357 | 16.858 | 0.02693 | 229.366 | 124.692 | 0.02689 | 0.02692 |

46 | 2 | 0.3 | 0.01 | 0.9 | 2.460 | 3509.969 | 47.140 | 0.009806 | 3661.021 | 1142.528 | 0.009924 | 0.009847 |

47 | 2 | 0.4 | 0.03 | −0.7 | 0.497 | 83.851 | 13.458 | 0.002517 | 104.857 | 67.178 | 0.005397 | 0.002638 |

48 | 2 | 0.5 | 0.05 | 0.5 | 1.649 | 1490.134 | 39.147 | 0.1286 | 1600.387 | 568.245 | 0.1297 | 0.129 |

49 | 2 | 0.6 | 0.07 | 1.7 | 5.474 | 65716.150 | 495.144 | 0.2703 | 70124.859 | 7875.753 | 0.368 | 0.3109 |

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**Figure 1.**Schematic diagram of two kinds of dual-chamber air spring: (

**a**) the traditional device; (

**b**) a gas-liquid coupling device.

**Figure 18.**Comparison of experimental and predicted values on the dual-chamber air spring stiffness.

Level | Volume Ratio $({\mathsf{\xi}}_{1})$ | Orifice Ratio $({\mathsf{\xi}}_{2})$ | Loading Amplitude Ratio $({\mathsf{\xi}}_{3})$ | Loading Frequency lnω | Real Loading Frequency (Hz) | |
---|---|---|---|---|---|---|

Factor | ||||||

1 | 0.5 | 0.0 | 0.01 | −0.7 | 0.497 | |

2 | 0.75 | 0.1 | 0.02 | −0.3 | 0.741 | |

3 | 1.0 | 0.2 | 0.03 | 0.1 | 1.105 | |

4 | 1.25 | 0.3 | 0.04 | 0.5 | 1.649 | |

5 | 1.5 | 0.4 | 0.05 | 0.9 | 2.460 | |

6 | 1.75 | 0.5 | 0.06 | 1.3 | 3.669 | |

7 | 2.0 | 0.6 | 0.07 | 1.7 | 5.474 |

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**MDPI and ACS Style**

Zeng, X.; Zhang, L.; Yu, Y.; Shi, M.; Zhou, J. The Stiffness and Damping Characteristics of a Dual-Chamber Air Spring Device Applied to Motion Suppression of Marine Structures. *Appl. Sci.* **2016**, *6*, 74.
https://doi.org/10.3390/app6030074

**AMA Style**

Zeng X, Zhang L, Yu Y, Shi M, Zhou J. The Stiffness and Damping Characteristics of a Dual-Chamber Air Spring Device Applied to Motion Suppression of Marine Structures. *Applied Sciences*. 2016; 6(3):74.
https://doi.org/10.3390/app6030074

**Chicago/Turabian Style**

Zeng, Xiaohui, Liang Zhang, Yang Yu, Min Shi, and Jifu Zhou. 2016. "The Stiffness and Damping Characteristics of a Dual-Chamber Air Spring Device Applied to Motion Suppression of Marine Structures" *Applied Sciences* 6, no. 3: 74.
https://doi.org/10.3390/app6030074