# High-Fidelity Fin–Actuator System Modeling and Aeroelastic Analysis Considering Friction Effect

^{*}

## Abstract

**:**

## Featured Application

**A high fidelity aeroelastic model of a typical fin–actuator system is established, in which the friction model is more accurate. The influence of freeplay and friction on the system stability is analyzed using the time-domain method and frequency-domain method, and some suggestions for flutter suppression design are put forward.**

## Abstract

## 1. Introduction

## 2. Study Object

^{2}. A branch mode analysis of the fin was performed, which was adopted for aeroelastic analysis, and the first two elastic natural frequencies and the corresponding mode shapes of branch 1 are listed in Table 2.

## 3. Modeling of the System

#### 3.1. Modeling of the Fin Structure

#### 3.2. Modeling of the Unsteady Aerodynamics

#### 3.3. State-Space Form of the Fin Model

#### 3.4. Modeling of the Electromechanical Actuator

#### 3.4.1. Model of DC Motor

#### 3.4.2. Model of the Gear Pair

#### 3.4.3. Model of the Screw–Nut Pair and Fork

#### 3.4.4. Model of the Controller and Sensor

#### 3.4.5. Model of the System

## 4. Flutter Analysis Method

#### 4.1. Frequency Domain Method

#### 4.1.1. Dynamic Stiffness

#### 4.1.2. Flutter Analysis with V–g Method

#### 4.2. Time Domain Method

## 5. Aeroelastic Analysis Results and Discussion

#### 5.1. Identification of Actuator Parameters

#### 5.2. Preliminary Flutter Results for No-Freeplay Gap ($e\text{}=\text{}0$) and No-Friction

#### 5.3. Aeroelastic Characteristics of the Fin–Actuator System with Freeplay

#### 5.3.1. Influence of Different Initial Deflection Angle with $e={0.2}^{\circ}$

#### 5.3.2. Influence of Different Freeplay with ${\delta}_{0}={1}^{\circ}$

#### 5.4. Aeroelastic Characteristics of the System with Freeplay and Friction

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Matrices and Vectors

## References

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**Figure 1.**Model comparison for flutter analysis, where (

**a**) is the model of the conventional fin–actuator system and (

**b**) is the fin–actuator system considering the characteristics of an actuator.

**Figure 3.**Schematic diagram of the fin–actuator system, where (

**a**) is the whole system structure and (

**b**) is the schematic diagram of the friction sheet position.

**Figure 4.**The modal synthesis method was used to build the structure model of the fin, where (

**a**) is the elastic branch and (

**b**) is the rotational rigid branch.

**Figure 12.**The time domain flutter results of the system without freeplay and friction, where (

**a**) is the response of the fin shaft angle at 760 m/s, and (

**b**) is at 765 m/s.

**Figure 14.**The influence of ${\delta}_{0}$ with $e={0.2}^{\circ}$, where (

**a**) was the influence of ${\delta}_{0}$ on the actuator dynamic stiffness and (

**b**) was the influence of ${\delta}_{0}$ on the response of the fin deflection angle at different flight speeds.

**Figure 15.**Comparison of the results of the flutter analysis in the time domain and frequency domain with ${\delta}_{0}={1}^{\circ}$ and $e={0.2}^{\circ}$, where (

**a**) is the flight speed–$A/e$ curve and (

**b**) is the flight speed–oscillation frequency curve.

**Figure 16.**The influence of $e$ with ${\delta}_{0}={1}^{\circ}$, where (

**a**) is the influence of $e$ on the dynamic stiffness of the actuator and (

**b**) is the influence of $e$ on the response of the fin deflection angle at different flight speeds.

**Figure 17.**Influence of the parameters of the LuGre model on the dynamic stiffness of the actuator, where (

**a**–

**f**), respectively, show the effects of ${\sigma}_{0}$, ${\sigma}_{1}$, ${\sigma}_{2}$, ${F}_{\mathrm{c}}$, ${F}_{\mathrm{s}}$, and ${V}_{\mathrm{s}}$.

**Figure 18.**Comparison of the results of flutter analysis in time domain and frequency domain with friction, when ${\delta}_{0}={1}^{\circ}$ and $e={0.2}^{\circ}$, where (

**a**) is the flight speed–$A/e$ curve and (

**b**) is the flight speed–oscillation frequency curve.

**Figure 20.**Influence of the parameters of the LuGre model on the response of the fin deflection angle at different flight speeds with ${\delta}_{0}={1}^{\circ}$ and $e={0.2}^{\circ}$, where (

**a**–

**f**), respectively, show the effects of ${\sigma}_{0}$, ${\sigma}_{1}$, ${\sigma}_{2}$, ${F}_{\mathrm{c}}$, ${F}_{\mathrm{s}}$, and ${V}_{\mathrm{s}}$.

**Table 1.**Friction characteristics reflected by different models. Generalized Maxwell-Slip, GMS; two-state friction model, 2SEP.

Viscous | Stribeck Effect | Pre-Sliding | Hysteresis | |
---|---|---|---|---|

Coulomb | No | No | No | No |

Viscous | Yes | No | No | No |

Stribeck | Yes | Yes | No | No |

Dahl | No | No | Yes | Yes |

LuGre | Yes | Yes | Yes | Yes |

Leuven | Yes | Yes | Yes | Yes |

GMS | Yes | Yes | Yes | Yes |

2SEP | Yes | Yes | Yes | Yes |

Elastic Branch | Frequency | Modal Shape |
---|---|---|

1st mode | 50.3 Hz | |

2nd mode | 593.0 Hz |

Description | Symbol | Value | Unit |
---|---|---|---|

Inductance | $L$ | $6.53\times {10}^{-4}$ | mH |

Resistance | $R$ | 1.1 | $\mathsf{\Omega}$ |

Rotor inertia | ${J}_{\mathrm{m}}$ | $1.094\times {10}^{-6}$ | $\mathrm{kg}\cdot {\mathrm{m}}^{2}$ |

Torque coefficient | ${K}_{\mathrm{m}}$ | 0.0238 | Nm/A |

Back EMF coefficient | ${C}_{\mathrm{e}}$ | 0.034 | V/(rad/s) |

Connection stiffness | ${k}_{\mathrm{m}}$ | 1000 | Nm/rad |

Moment of inertia of gear 1 | ${J}_{1}$ | $2.4\times {10}^{-8}$ | $\mathrm{kg}\cdot {\mathrm{m}}^{2}$ |

Radius of gear 1 | ${r}_{1}$ | 0.005 | m |

Moment of inertia of gear 2 | ${J}_{2}$ | $4\times {10}^{-6}$ | $\mathrm{kg}\cdot {\mathrm{m}}^{2}$ |

Radius of gear 2 | ${r}_{2}$ | 0.0225 | m |

Meshing stiffness | ${k}_{\mathrm{g}}$ | ${10}^{8}$ | $\mathrm{kg}\cdot {\mathrm{s}}^{2}/\mathrm{rad}$ |

Connection stiffness | ${k}_{\mathrm{z}}$ | 1000 | Nm/rad |

Screw inertia | ${J}_{\mathrm{sg}}$ | $3.98\times {10}^{-5}$ | $\mathrm{kg}\cdot {\mathrm{m}}^{2}$ |

Radius of screw | ${r}_{sg}$ | 0.006 | m |

Screw efficiency | $\eta $ | 0.85 | |

Fork length | ${L}_{\mathrm{bc}}$ | 0.0285 | m |

Comprehensive stiffness | ${k}_{\mathrm{c}}$ | $2.27\times {10}^{6}$ | ${\mathrm{kg}/\mathrm{s}}^{2}$ |

Bristle stiffness | ${\sigma}_{0}$ | 300 | Nm/rad |

Bristle damping | ${\sigma}_{1}$ | 2.5 | Nm/(rad/s) |

Viscous damping | ${\sigma}_{2}$ | 0.02 | Nm/(rad/s) |

Coulomb friction | ${F}_{\mathrm{c}}$ | 1.2565 | Nm |

Maximum static friction force | ${F}_{\mathrm{s}}$ | 0.774 | Nm |

Stribeck velocity | ${V}_{\mathrm{s}}$ | 1 | rad/s |

Description | Symbol | PARAMETER RANGE | Unit | Parameter Depends Principally upon [26,59,63] |
---|---|---|---|---|

Bristle stiffness | ${\sigma}_{0}$ | $10.23\text{~}6.5\times {10}^{4}$ | Nm/rad | Material properties |

Bristle damping | ${\sigma}_{1}$ | $2.719\times {10}^{-3}$~45.2 | Nm/(rad/s) | Contact geometry and lubricant |

Viscous damping | ${\sigma}_{2}$ | $9.216\times {10}^{-6}$~1.819 | Nm/(rad/s) | Lubricant |

Coulomb friction | ${F}_{\mathrm{c}}$ | $1.895\times {10}^{-3}$~2646.856 | Nm | Lubricant, contact geometry, and pressure |

Maximum static friction force | ${F}_{\mathrm{s}}$ | $3.124\times {10}^{-3}$~8.558 | Nm | Boundary lubrication and pressure |

Stribeck velocity | ${V}_{\mathrm{s}}$ | $6.109\times {10}^{-2}$~88.1 | rad/s | Lubricant and pressure |

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**MDPI and ACS Style**

Lu, J.; Wu, Z.; Yang, C. High-Fidelity Fin–Actuator System Modeling and Aeroelastic Analysis Considering Friction Effect. *Appl. Sci.* **2021**, *11*, 3057.
https://doi.org/10.3390/app11073057

**AMA Style**

Lu J, Wu Z, Yang C. High-Fidelity Fin–Actuator System Modeling and Aeroelastic Analysis Considering Friction Effect. *Applied Sciences*. 2021; 11(7):3057.
https://doi.org/10.3390/app11073057

**Chicago/Turabian Style**

Lu, Jin, Zhigang Wu, and Chao Yang. 2021. "High-Fidelity Fin–Actuator System Modeling and Aeroelastic Analysis Considering Friction Effect" *Applied Sciences* 11, no. 7: 3057.
https://doi.org/10.3390/app11073057