# Research on the Vibration Reduction Mechanism of a New Tensioning Platform with an Embedded Superstructure

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## Abstract

**:**

## 1. Introduction

## 2. Design and Optimization of the Main Scheme of the Actuation Platform

#### 2.1. Principal Scheme

#### 2.2. Optimization of Main Structure Parameters

_{1}, the radius of the arc profile of the flexible hinge is x

_{2}, and the width of the input end beam is x

_{3}. In Figure 2b, d

_{in}is the input displacement, d

_{out}

_{1}is the displacement after primary amplification, and d

_{out}

_{2}is the displacement after secondary amplification. The improved Scott–Russell mechanism is equivalent to a series of lever and Scott–Russell mechanisms. In engineering applications, the magnification ratio of the entire compliant tensioning platform is mainly determined by the motion characteristics of Scott–Russell mechanisms via the velocity projection theorem. The total platform amplification ratio can be simply estimated as the product of the Scott–Russell mechanism and the lever mechanism amplification ratio: $\left({l}_{3}/{l}_{4}\right)\times \left({l}_{2}/{l}_{1}\right)={l}_{3}{l}_{2}/{l}_{4}{l}_{1}$.

_{1,}x

_{2,}x

_{3}have little impact on the platform’s basic structure; since they are not dependent on one another, a sensitivity analysis can be carried out. The starting value of the selected optimal size is defined by the size that, at the time of preliminary design, can achieve the performance of the entire flexible tensioning platform. As a result, the selected size range of the platform is mainly determined by the actual operation of the flexible tensioning platform. As can be observed from previous studies, the first-order natural frequency Y

_{p}and amplification ratio Y

_{t}of the flexible tension platform are chosen as the results of sensitivity analysis so that the larger motion output range and higher working frequency band are fulfilled. ANSYS software was used to generate 16 sets of orthogonal experimental parameters, followed by sensitivity analyses to obtain analysis results under different combinations; the specific values are shown in Table 1. Figure 3 shows the histogram of the effects of these three structural parameters on both the amplification ratio and the intrinsic frequency of the platform. It is evident that x

_{3}has a smaller effect on the overall platform performance compared to x

_{1}and x

_{2}. In addition, x

_{1}shows a positive correlation with the deformation of the platform and a negative correlation with the frequency of the platform, while x

_{2}has the opposite effect.

_{t}and first-order natural frequency Y

_{p}with the three parameters is obtained [23]:

_{1}and x

_{2}parameters have a significant influence on the platform performance. The larger the x

_{1}and the smaller the x

_{2}, the larger the amplification ratio of the tensioning platform. The smaller the x

_{1}and the larger the x

_{2}, the greater the natural frequency of the tensioning platform. It is evident that the theoretical model fits well and better reflects the mapping relationship between relevant parameters and platform performance indicators.

_{1}= 75, x

_{2}= 0.8, and x

_{3}= 4.5, respectively. The amplification ratio of the scheme is close to 3.4. The first-order solid frequency reached 365 Hz, compared with the finite element simulation results (as shown in Figure 6 and Figure 7), and basically remained the same. This demonstrates the effectiveness of the optimization method; therefore, the presented scheme provides a basis for avoiding external disturbances in critical low- and medium-frequency bands. Subsequently, as shown in Figure 8, the prototype is manufactured and assembled according to the above optimization results.

## 3. Vibration Suppression Mechanism of the Embedded Superstructure

#### 3.1. Equivalent Dynamic Modeling

_{0}, where m

_{0}, k

_{0}, and c

_{0}represent the mass, stiffness, and damping of the main structure, respectively. x

_{i}(I = 1, …, 4), m

_{i}(I = 1, …, 4), k

_{i}(I = 1, …, 4), and c

_{i}(I = 1, …, 4) represent the displacement, mass, stiffness, and damping of each phase of the superstructure, respectively.

#### 3.2. Vibration Suppression Mechanism

_{s}is the superstructure’s overall mass. With other parameters unchanged, different mass ratios are selected to assess the influence of different mass ratios on vibration characteristics of the main system. The results are shown in Figure 11.

## 4. Numerical Research and Result Analysis

#### 4.1. Static Performance

#### 4.2. Dynamic Tensioning Performance

#### 4.3. Harmonic Response Tests of the Overall Platform

#### 4.4. Vibration Suppression Tests under Fixed Frequencies Disturbance

_{i}(i = 1, …, 4) is the corresponding resonant frequency of each phase of the superstructure, which can be determined via the previously mentioned modal simulation results. A

_{i}(i = 1, …, 4) is the equivalent vibration amplitude, respectively, and can be set to 0.1 mm, 0.05 mm, 0.01 mm and 0.005 mm. After transient dynamic analysis [29], the simulation results in the time and frequency domains are processed using MATLAB, as plotted in Figure 19.

_{1}and f

_{3}, and the amplitude is significantly reduced. Similarly, from Figure 19c,d, it is evident that the platform has a significant vibration suppression effect on the harmonic disturbances with frequencies of f

_{1}and f

_{4}, and the amplitude is significantly reduced. Finally, from Figure 19g,h, it is evident that the platform also shows a significant vibration suppression effect for harmonic disturbances with frequencies of f

_{1}, f

_{2}, f

_{3}and f

_{4}. The frequency domain results in Figure 19h show that four resonance peaks corresponding to the four excitation frequencies appear, and the amplitudes of these resonance peaks are obviously weakened, which demonstrates that the designed four-phase superstructure suppresses the vibration with desired frequency through local resonance function. The vibration suppression effects are further collated and listed in Table 2. It is evident that the vibration corresponding to the inherent frequency of each phase is suppressed to a certain extent, that the maximum vibration suppression effect appears at the first phase, and that the vibration suppression effect reaches −1.705 dB. Overall, although the designed superstructure achieves a certain vibration suppression capability, the suppression effect is limited. Because each branch of the planned superstructure does not form an array, only one single cell is used to perform vibration suppression, which has a restricted impact. Combined with the vibration suppression mechanism, future research should focus on the design of multiple embedded arrays in each branch, so as to improve the vibration suppression performance of the overall platform.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 12.**The frequency response with different natural frequency ratios of superstructure to main structure.

**Figure 13.**The frequency response with different damping ratios of superstructure to main structure.

**Figure 16.**Output response of the actuation platform. (

**a**) Output response at an input displacement amplitude of 10 μm. (

**b**) Output response at an input displacement amplitude of 30 μm. (

**c**) Output response at an input displacement amplitude of 50 μm. (

**d**) Output response at an input displacement amplitude of 100 μm.

Serial Number | x_{1}/mm | x_{2}/mm | x_{3}/mm | Y_{t}/mm | Y_{p}/Hz |
---|---|---|---|---|---|

1 | 75.78 | 0.700931 | 4.080975 | 0.033334 | 294.9327 |

2 | 70 | 0.75 | 4 | 0.029667 | 301.51 |

3 | 76.14604 | 0.776976 | 4.333236 | 0.031466 | 371.4842 |

4 | 71.55851 | 0.796421 | 4.175428 | 0.029454 | 303.9678 |

5 | 66.78863 | 0.711865 | 4.358513 | 0.029252 | 299.9477 |

6 | 72.04422 | 0.76038 | 4.321502 | 0.030315 | 361.7926 |

7 | 75.42766 | 0.712418 | 4.225901 | 0.032516 | 297.1097 |

8 | 73.88042 | 0.728615 | 4.121734 | 0.03181 | 298.5136 |

9 | 66.0245 | 0.789504 | 4.294115 | 0.027685 | 374.7236 |

10 | 76.78263 | 0.811695 | 4.146414 | 0.030889 | 302.6956 |

11 | 69.05379 | 0.791876 | 4.203257 | 0.02856 | 303.5848 |

12 | 70.41992 | 0.777609 | 4.398086 | 0.029325 | 363.1416 |

13 | 66.96664 | 0.700444 | 4.055081 | 0.029682 | 299.0751 |

14 | 65.94436 | 0.791094 | 4.278291 | 0.027621 | 374.4772 |

15 | 73.36935 | 0.716163 | 4.399461 | 0.031607 | 298.5916 |

16 | 70.58632 | 0.756065 | 4.155923 | 0.030033 | 380.6868 |

Frequency/Hz | Vibration Amplitude without Superstructure/μm | Vibration Amplitude with Superstructure/μm | Attenuation/dB |
---|---|---|---|

34.6 | 331 | 272 | −1.705 |

44.4 | 173 | 145 | −1.533 |

57.1 | 40 | 35 | −1.15 |

65.7 | 20 | 17 | −1.41 |

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**MDPI and ACS Style**

Sun, X.; Yang, Z.; Wang, J.; Hou, X.; Yang, Y.
Research on the Vibration Reduction Mechanism of a New Tensioning Platform with an Embedded Superstructure. *Actuators* **2023**, *12*, 279.
https://doi.org/10.3390/act12070279

**AMA Style**

Sun X, Yang Z, Wang J, Hou X, Yang Y.
Research on the Vibration Reduction Mechanism of a New Tensioning Platform with an Embedded Superstructure. *Actuators*. 2023; 12(7):279.
https://doi.org/10.3390/act12070279

**Chicago/Turabian Style**

Sun, Xiaoqing, Zhengyin Yang, Ju Wang, Xiusong Hou, and Yikun Yang.
2023. "Research on the Vibration Reduction Mechanism of a New Tensioning Platform with an Embedded Superstructure" *Actuators* 12, no. 7: 279.
https://doi.org/10.3390/act12070279