Using a Fully Fractional Generalised Maxwell Model for Describing the Time Dependent Sinusoidal Creep of a Dielectric Elastomer Actuator
Abstract
1. Introduction
2. Materials and Methods
2.1. Principles of the DEA
2.2. Derivation of the Fully Fractional Generalised Maxwell Model
2.3. Experiments
2.4. Optimisation
3. Results
4. Discussion
- The number of fully fractional Maxwell elements slightly affected the effectiveness of the model.
- Adding more than two branches did not increase the effectiveness of the model.
- The fully fractional Maxwell model was reduced to the model seen in Figure 8.
- The middle frequency of 1/7 Hz had the best agreement of 0.88 between data.
- Optimising each frequency individually drastically improved the overall agreement between data to 0.745.
- Optimising each frequency individually has a drawback since each frequency requires its own material parameters.
- Topology optimisation cannot be included into the Pattern Search algorithm.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Symbol | Unit | Meaning |
---|---|---|
Area. | ||
/ | Order or fractional derivation of the springpots. | |
Time limits of the fractional derivation. | ||
Material properties of the springpots. | ||
Modul of elasticity. | ||
Absolute permittivity. | ||
/ | Relative permittivity. | |
1 | Strain. | |
Maxwell force. | ||
Electrical force. | ||
Force in individual branch. | ||
Calculated data. | ||
/ | Current number of fractional Maxwell element. | |
Spring constant. | ||
Dimensions of the DEA. | ||
Initial length. | ||
Displacement. | ||
/ | Integer order of derivation by the definition. | |
Mass of weight. | ||
/ | Number of fractional Maxwell elements. | |
Viscosity. | ||
/ | Fractional order of derivation by the definition. | |
/ | Coefficient of determination. | |
/ | Mean value of coefficient of determination. | |
/ | Laplace operator. | |
/ | Total sum of squares. | |
/ | Residual sum of squares. | |
Stress. | ||
Voltage. | ||
Displacement of individual branch. | ||
Measured data. | ||
Averaged measured data. |
Fully Fractional Generalised Maxwell Model Number of Branches | |
---|---|
n= 1 | 0.5456 |
n= 2 | 0.5489 |
n = 3 | 0.5456 |
n = 4 | 0.5456 |
n = 5 | 0.5456 |
Param. | n= 1 | |||||||||||
Initial | 0.2 | 1 | 1 | |||||||||
Optimised | 0.2 | 0.002 | 1 | |||||||||
Parameters | ||||||||||||
Initial | 500 | 500 | 500 | |||||||||
Optimised | 62.952 | 0.036 | 0.142 | |||||||||
Param. | n= 2 | |||||||||||
Initial | 0.2 | 1 | 1 | 1 | 1 | |||||||
Optimised | 0.2 | 1 | 1 | 0.523 | 0.046 | |||||||
Param. | ||||||||||||
Initial | 500 | 500 | 500 | 500 | 500 | |||||||
Optimised | 62.740 | 47.958 | 173.21 | 430.381 | 0.215 | |||||||
Param. | n= 3 | |||||||||||
Initial | 0.2 | 1 | 1 | 1 | 1 | 1 | 1 | |||||
Optimised | 0.2 | 1 | 1 | 1 | 0.002 | 1 | 1 | |||||
Param. | ||||||||||||
Initial | 500 | 500 | 500 | 500 | 500 | 500 | 500 | |||||
Optimised | 62.950 | 0.267 | 0.464 | 0.140 | 0.088 | 0.036 | 0.237 | |||||
Param. | n= 4 | |||||||||||
Initial | 0.2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||
Optimised | 0.2 | 0.002 | 1 | 0.002 | 1 | 1 | 0.002 | 1 | 0.002 | |||
Param. | ||||||||||||
Initial | 500 | 500 | 500 | 500 | 500 | 500 | 500 | 500 | 500 | |||
Optimised | 62.950 | 0.103 | 0.036 | 0.321 | 0.036 | 0.094 | 0.157 | 0.036 | 0.225 | |||
Param. | n= 5 | |||||||||||
Initial | 0.2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
Optimised | 0.2 | 0.002 | 1 | 1 | 1 | 1 | 0.002 | 1 | 0.002 | 1 | 1 | |
Param. | ||||||||||||
Initial | 500 | 500 | 500 | 500 | 500 | 500 | 500 | 500 | 500 | 500 | 500 | |
Optimised | 62.950 | 0.097 | 0.315 | 0.036 | 0.356 | 0.036 | 0.097 | 0.036 | 0.095 | 0.036 | 0.285 |
Parameters | F = 1/13 Hz n = 3 | ||||||||
Initial | 0.2 | 1 | 1 | 1 | 1 | ||||
Optimised | 0.1795 | 1 | 1 | 1 | 1 | ||||
Parameters | |||||||||
Initial | 500 | 500 | 500 | 500 | 500 | 0.658 | |||
Optimised | 51.773 | 0.002 | 0.002 | 0.002 | 0.002 | ||||
Parameters | F = 1/7 Hz n = 3 | ||||||||
Initial | 0.2 | 1 | 1 | 1 | 1 | ||||
Optimised | 0.188 | 0.255 | 0.046 | 0.225 | 0.880 | ||||
Parameters | |||||||||
Initial | 500 | 500 | 500 | 500 | 500 | 0.907 | |||
Optimised | 57.696 | 569.346 | 0.003 | 406.744 | 0.479 | ||||
Parameters | F = 1/5 Hz n = 3 | ||||||||
Initial | 0.2 | 1 | 1 | 1 | 1 | ||||
Optimised | 0.2 | 1 | 1 | 1 | 0.002 | ||||
Parameters | |||||||||
Initial | 500 | 500 | 500 | 500 | 500 | 0.665 | |||
Optimised | 73.133 | 19.199 | 188.074 | 305.285 | 191.367 | ||||
0.743 |
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Karner, T.; Belšak, R.; Gotlih, J. Using a Fully Fractional Generalised Maxwell Model for Describing the Time Dependent Sinusoidal Creep of a Dielectric Elastomer Actuator. Fractal Fract. 2022, 6, 720. https://doi.org/10.3390/fractalfract6120720
Karner T, Belšak R, Gotlih J. Using a Fully Fractional Generalised Maxwell Model for Describing the Time Dependent Sinusoidal Creep of a Dielectric Elastomer Actuator. Fractal and Fractional. 2022; 6(12):720. https://doi.org/10.3390/fractalfract6120720
Chicago/Turabian StyleKarner, Timi, Rok Belšak, and Janez Gotlih. 2022. "Using a Fully Fractional Generalised Maxwell Model for Describing the Time Dependent Sinusoidal Creep of a Dielectric Elastomer Actuator" Fractal and Fractional 6, no. 12: 720. https://doi.org/10.3390/fractalfract6120720
APA StyleKarner, T., Belšak, R., & Gotlih, J. (2022). Using a Fully Fractional Generalised Maxwell Model for Describing the Time Dependent Sinusoidal Creep of a Dielectric Elastomer Actuator. Fractal and Fractional, 6(12), 720. https://doi.org/10.3390/fractalfract6120720