# Mass and Force Lumping: An Essential Enhancement to the Intrinsic Beam Finite Element Discretization

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## Abstract

**:**

## 1. Introduction

## 2. Aeroelastic System

#### 2.1. Fully Intrinsic Formulation

#### 2.2. External Forces and Moments

#### 2.2.1. Gravity

#### 2.2.2. Aerodynamics

#### 2.2.3. Thrust

#### 2.3. Attitude and Rotation Matrix

## 3. Spatial Time Discretization Scheme

#### 3.1. Spatial Finite Element Discretization

#### 3.2. Mass and Force Lumping

#### 3.3. Final Differential-Algebraic Equations

#### 3.4. Time Domain Simulation

## 4. Linearization and Index Reduction

#### 4.1. Trimming

#### 4.2. Index Reduction

## 5. Numerical Results

#### 5.1. Trim Results

#### 5.2. Nonlinear Time Domain Simulation

#### 5.3. Linearization and Index Reduction

#### 5.4. Model Order Reduction

#### 5.5. Analysis of Flight Dynamics

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Trimming results of the flexible flying wing with payload varying from 0 to 227 kg. (

**a**) Angle of attack, thrust, and flap deflection. (

**b**) Root locus. The format of the comparison figure is similar to that of Wang [14].

**Figure 5.**(

**a**) The percentage of input curve; (

**b**) nonlinear time domain simulation results with ${\delta}_{e}$ input with a peak value of 5 degrees.

**Figure 6.**Comparison of eigenvalues before and after mass and force lumping, where the part indicated by the arrow represents the distribution of the eigenvalues near the origin.

**Figure 7.**Comparison of linear and nonlinear time domain simulation results in (

**a**) longitudinal direction and (

**b**) lateral direction with peak values of ${\delta}_{e}$ and ${\delta}_{a}$ at 2 degrees and 10 degrees, respectively.

**Figure 8.**Eigenvalues selected from Figure 6 to execute model reduction, where the part indicated by the arrow represents the distribution of the eigenvalues near the origin.

**Figure 9.**Comparison of Bode plots from $ds$ to (

**a**) pitch angle, (

**b**) forward speed, (

**c**) pitch rate, and (

**d**) upward velocity before and after model reduction. The red lines represent the full-order system, while the blue lines represent the reduced-order system.

**Figure 10.**Comparison of Bode plots from $da$ to (

**a**) roll angle, (

**b**) roll rate, (

**c**) yaw angle, and (

**d**) yaw rate before and after model reduction. The red lines represent the full-order system, while the blue lines represent the reduced-order system.

**Figure 11.**Comparison of reduced-order and full-order linear time domain simulation in (

**a**) longitudinal direction and (

**b**) lateral direction.

**Figure 12.**From left to right are the first four modes of longitudinal reduced-order model. The blue lines represent the shape of the aircraft, while the red lines represent the shape of the eigenvector.

**Figure 13.**From left to right are the first four modes of the lateral reduced-order model. The blue lines represent the shape of the aircraft, while the red lines represent the shape of the eigenvectors.

**Figure 14.**The dashed lines represent the time-domain simulation results for the reduced-order system, while the solid lines depict modal contributions of several main modes in (

**a**) longitudinal direction and (

**b**) lateral direction.

**Table 1.**Relevant properties of the flying wing [13].

Parameter | Value |
---|---|

Elastic/reference axis | 25% chord |

Aerodynamic center | 25% chord |

Center of gravity | 25% chord |

$GJ$ | $1.65\times {10}^{5}$ N$\xb7{\mathrm{m}}^{2}$ |

$E{I}_{2}$ | $1.03\times {10}^{6}$ N$\xb7{\mathrm{m}}^{2}$ |

$E{I}_{3}$ | $1.24\times {10}^{7}$ N$\xb7{\mathrm{m}}^{2}$ |

m | 8.93 kg/m |

${I}_{11}$ | 4.15 kg·m |

${I}_{22}$ | 0.69 kg·m |

${I}_{33}$ | 3.46 kg·m |

Wing ${C}_{{l}_{\alpha}}$ | 2$\pi $ |

Wing ${C}_{{l}_{\delta}}$ | 1 |

Wing ${C}_{{d}_{0}}$ | 0.01 |

Wing ${C}_{{m}_{0}}$ | 0.025 |

Wing ${C}_{{m}_{\delta}}$ | −0.25 |

Pod ${C}_{{l}_{\alpha}}$ | 5 |

Pod ${C}_{{d}_{0}}$ | 0.02 |

Pod ${C}_{{m}_{0}}$ | 0 |

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**MDPI and ACS Style**

Wang, J.; Zhou, Z.
Mass and Force Lumping: An Essential Enhancement to the Intrinsic Beam Finite Element Discretization. *Aerospace* **2023**, *10*, 957.
https://doi.org/10.3390/aerospace10110957

**AMA Style**

Wang J, Zhou Z.
Mass and Force Lumping: An Essential Enhancement to the Intrinsic Beam Finite Element Discretization. *Aerospace*. 2023; 10(11):957.
https://doi.org/10.3390/aerospace10110957

**Chicago/Turabian Style**

Wang, Jiachen, and Zhou Zhou.
2023. "Mass and Force Lumping: An Essential Enhancement to the Intrinsic Beam Finite Element Discretization" *Aerospace* 10, no. 11: 957.
https://doi.org/10.3390/aerospace10110957