# Modeling and Experimental Research of One Kind of New Planar Vortex Actuator Based on Shape Memory Alloy

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Mechanical Model of SMA Vortex Spring

#### 2.1. The Constitutive Model of SMA Material

_{L}is the maximum recoverable strain, η is the shape memory factor, t is the temperature, and the subscript 0 is the initial state. E(φ) means that for SMA, the E is related to the φ, and its calculation method can be expressed in Equation (5):

_{M}is the modulus of elasticity of martensite, and E

_{A}is the modulus of elasticity of austenite.

_{s}

^{cr}is the critical stress of martensite initial transformation, σ

_{f}

^{cr}is the critical stress of martensite termination transformation, σ

_{As}is the initial transformation stress of austenite, σ

_{Af}is the austenite final transformation stress, and A

_{s}is austenite The initial transformation temperature of austenite, A

_{f}is the termination temperature of austenite transformation, C

_{A}is the coefficient of austenite transformation temperature-critical stress.

_{As}and austenite final transformation stress σ

_{Af}is:

#### 2.2. The Mechanical Model of PVA

_{E}and Δψ

_{η}are the plastic deformation caused by the transformation of material under stress, and Δψ

_{t}is the thermal expansion deformation caused by temperature change.

- Based on the assumption of one-dimensional deformation, describe the deformation process of the SMA vortex spring, and set the material to have the same elastic modulus and maximum recoverable strain during tension and compression.
- All physical parameters of the material are the same in the deformation process.
- The material is pure bending strain and the cross-sectional shape remains unchanged.
- The strain caused by thermal expansion is much smaller than the elastic strain caused by torsion and the material phase change strain, so the third term in the equation is ignored.
- The length of the deformed section of the PVA remains unchanged.

## 3. Calculation of Driving Performance

- Each parameter in the initial state is represented by subscript 0. Temperature is lower than martensitic initial phase transition temperature, namely t
_{0}< M_{s}torque T_{0}= 0. The volume fraction of martensite is φ_{0}= 1, the memory factor is η_{0}= 0, the strain angle is ψ_{0}= 0. - Parameters in the pre-tightening stage are set as 1. Temperature t
_{1}= t_{0}< M_{s}. The deformation angle of PVA under the pretightening torque T_{1}is ψ_{1}:

- 3.
- The parameter in the in-use stage is set as subscript 2. Keep the temperature t
_{2}= t_{0}< M_{s}, T_{2}= 0 after the preload torque is removed. After PVA recovered the elastic deformation, the residual deformation angle was ψ_{2}:

- 4.
- The parameters of the excited output state are indicated by the subscript 3. When the temperature is increased to t
_{3}> A_{s}, the PVA generates shape recovery and outputs torque and angular displacement, which are T_{3}and ψ_{3}, respectively. Since the stress change is caused by the difference between the elastic modulus during the transformation from martensite to austenite, there are constraints:

_{3}, t

_{3}and T

_{1}.

## 4. Validations and Discussions

#### 4.1. Design and Manufacture of PVA Sample

_{1}is the polar diameter of the starting point of the spiral, r is the polar diameter of any point of the spiral, θ is the polar angle of any point, t

_{r}is the pitch of the spiral, and n is the winding number. The length of the working section can be calculated according to Equation (35):

#### 4.2. Test System Design and Construction

#### 4.3. Experimental Validations

^{−3}N·m, 50 × 10

^{−3}N·m and 75 × 10

^{−3}N·m, respectively. The test results are shown in Figure 5. Curve A indicates that the initial state of martensite is obtained after loading twin martensite, and curve B indicates that the initial state of martensite is obtained by lowering the temperature from austenite state.

_{en}, the pre-tightening torque is expressed in T

_{pre}, and the output torque is expressed in T

_{ex}. The process of the experiment is as follows:

- Heat the PVA to 80 °C (above A
_{f}) in a free state, and slowly decrease the temperature to 20 °C (below M_{f}). Repeat more than three times to make sure the material is in full twin martensite state. - Set the ambient temperature to 20 °C (lower than A
_{s}), and preload the PVA to 270° according to the constraints of Equation (20), and the T_{pre}is 0.069 N·m. After the restraint is released, the residual strain is 214°. Connect the test device to the test system. - The initial temperature of the water bath was set at 30 °C (lower than A
_{s}), and the temperature is raised at 2 °C/min. The values of t_{en}and T_{ex}of the water bath are recorded. The values recorded in the test are compared with those calculated by Equations (33) and (34), and the results are shown in Figure 6.

- 4.
- When the torque output value T
_{ex}is stable, keep the excitation temperature t_{en}unchanged (70 °C). Through the limit device, the angular displacement of the SMA vortex spring is increased by 3 (°)/s, and the value of the angular displacement λ and the torque output T_{ex}are recorded. Compare the value of the test record with the value calculated according to Equations (33) and (34). The result is shown in Figure 7.

#### 4.4. Error Discussions

_{A}, the angular displacement and torque output should be linear. In Figure 7a, the slope of the calculated value of the model is −9.9 × 10

^{−4}N·m/(°). The slope of the fitting straight line of the test data points is −7.7 × 10

^{−4}N·m/(°).

- More experiments were conducted to increase the correction coefficient of the effect of spring vortex number on the mechanical model.
- The deformation model of large deformation and non-pure bending is established.
- It is no longer assumed that the stress and strain of each part of the strip are the same, but the distribution law of stress and strain is emphatically discussed.
- The influence of thermal expansion caused by temperature on the mechanical model is considered, especially when the number of vortex springs is large and the material length is long.
- The one-dimensional constitutive equation in the mechanical model is replaced by three-dimensional constitutive equation.
- The finite element method is used as the numerical calculation method to provide a verification means for the establishment of the theoretical model.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Relationship between temperature and angular deformation under different torques. (

**a**) 25 × 10

^{−3}N·m; (

**b**) 50 × 10

^{−3}N·m; (

**c**) 75 × 10

^{−3}N·m; (Curve A represents martensite after loading twin martensite, and curve B represents martensite after austenite state is lowered temperature.)

**Figure 6.**Relationship between torque and excitation temperature. (

**a**) Test value and calculated, (

**b**) error between test value and calculated.

**Figure 7.**Relationship between torsion force and angular displacement. (

**a**) Test value and calculated; (

**b**) error between test value and calculated.

r_{1} (mm) | N | tr (mm) | l (mm) | h/(mm) | b (mm) |
---|---|---|---|---|---|

6 | 4.5 | 3.5 | 390 | 0.8 | 4.5 |

M_{s}/°C | M_{f}/°C | A_{s}/°C | A_{f}/°C | E_{M}/GPa | E_{A}/GPa |
---|---|---|---|---|---|

27 | 22.5 | 35 | 55 | 22 | 50 |

C_{M}/MPa·(°C)^{−1} | C_{A}/MPa·(°C)^{−1} | ${\sigma}_{s}^{cr}$/MPa | ${\sigma}_{\mathrm{f}}^{cr}$/MPa | ${\epsilon}_{L}$/% | |

8 | 17 | 35 | 150 | 5.5 |

b/mm | h/mm | R_{s}/mm | p/mm | n_{0} |
---|---|---|---|---|

6.5 | 0.83 | 3 | 5 | 3.5 |

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**MDPI and ACS Style**

Kong, X.; Gu, Y.; Wu, J.; Yang, Y.; Shen, X.
Modeling and Experimental Research of One Kind of New Planar Vortex Actuator Based on Shape Memory Alloy. *Actuators* **2022**, *11*, 8.
https://doi.org/10.3390/act11010008

**AMA Style**

Kong X, Gu Y, Wu J, Yang Y, Shen X.
Modeling and Experimental Research of One Kind of New Planar Vortex Actuator Based on Shape Memory Alloy. *Actuators*. 2022; 11(1):8.
https://doi.org/10.3390/act11010008

**Chicago/Turabian Style**

Kong, Xiangsen, Yilei Gu, Jiajun Wu, Yang Yang, and Xing Shen.
2022. "Modeling and Experimental Research of One Kind of New Planar Vortex Actuator Based on Shape Memory Alloy" *Actuators* 11, no. 1: 8.
https://doi.org/10.3390/act11010008