# Electrostatic and aerodynamic forced vibrations of a thin flexible electrode: Quasi-periodic vs. chaotic oscillations

## Abstract

In this paper, transverse vibrations of an electrostatically actuated thin flexible cantilever perturbed by low-speed air flow are studied using both experiments and numerical modeling. In the experiments, the dynamic characteristics of the cantilever are studied by supplying a DC voltage with an AC component for electrostatic forcing and a constant uniform air flow around the cantilever system for aerodynamic forcing. A range of control parameters leading to stable vibrations are established using a dimensionless operating parameter that is the ratio of the induced and the free stream velocities. Numerical results are validated with experimental data. Assuming the amplitude of vibrations are small, then a non-linear dynamic Euler-Bernoulli beam equation with viscous damping and gravitational effects is used to model the equation of motion. Aerodynamic forcing is modelled as a temporally sinusoidal and uniform force acting perpendicular to the beam length. The forcing amplitude is found to be proportional to the square of the air flow velocity. Numerical results strongly agree with the experiments predicting accurate vibration amplitude, displacement frequency, and quasi-periodic displacement of the cantilever tip.

- Authors:

- Department of Aerospace Engineering, Iowa State University, Ames, Iowa 50010 (United States)

- Publication Date:

- OSTI Identifier:
- 22596696

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 26; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; AIR FLOW; CHAOS THEORY; ELECTRODES; EQUATIONS OF MOTION; NONLINEAR PROBLEMS; OSCILLATIONS; PERIODICITY; SIMULATION

### Citation Formats

```
Madanu, Sushma B., Barbel, Stanley I., and Ward, Thomas.
```*Electrostatic and aerodynamic forced vibrations of a thin flexible electrode: Quasi-periodic vs. chaotic oscillations*. United States: N. p., 2016.
Web. doi:10.1063/1.4954025.

```
Madanu, Sushma B., Barbel, Stanley I., & Ward, Thomas.
```*Electrostatic and aerodynamic forced vibrations of a thin flexible electrode: Quasi-periodic vs. chaotic oscillations*. United States. doi:10.1063/1.4954025.

```
Madanu, Sushma B., Barbel, Stanley I., and Ward, Thomas. Wed .
"Electrostatic and aerodynamic forced vibrations of a thin flexible electrode: Quasi-periodic vs. chaotic oscillations". United States.
doi:10.1063/1.4954025.
```

```
@article{osti_22596696,
```

title = {Electrostatic and aerodynamic forced vibrations of a thin flexible electrode: Quasi-periodic vs. chaotic oscillations},

author = {Madanu, Sushma B. and Barbel, Stanley I. and Ward, Thomas},

abstractNote = {In this paper, transverse vibrations of an electrostatically actuated thin flexible cantilever perturbed by low-speed air flow are studied using both experiments and numerical modeling. In the experiments, the dynamic characteristics of the cantilever are studied by supplying a DC voltage with an AC component for electrostatic forcing and a constant uniform air flow around the cantilever system for aerodynamic forcing. A range of control parameters leading to stable vibrations are established using a dimensionless operating parameter that is the ratio of the induced and the free stream velocities. Numerical results are validated with experimental data. Assuming the amplitude of vibrations are small, then a non-linear dynamic Euler-Bernoulli beam equation with viscous damping and gravitational effects is used to model the equation of motion. Aerodynamic forcing is modelled as a temporally sinusoidal and uniform force acting perpendicular to the beam length. The forcing amplitude is found to be proportional to the square of the air flow velocity. Numerical results strongly agree with the experiments predicting accurate vibration amplitude, displacement frequency, and quasi-periodic displacement of the cantilever tip.},

doi = {10.1063/1.4954025},

journal = {Chaos (Woodbury, N. Y.)},

number = 6,

volume = 26,

place = {United States},

year = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}