Numerical Approach for Detecting the Resonance Effects of Drilling during Assembly of Aircraft Structures †
Abstract
:1. Introduction
2. Numerical Approach
2.1. Guyan Reduction
2.2. Time Discretization Algorithm
2.3. Basic Assumptions of the Numerical Algorithm
3. Verification with Abaqus
4. Drilling Process Simulation
5. Search for Resonance Effects
5.1. Determination of Natural Frequencies
- The upper panel is affected by a sudden impulse to the drilling node (indicated by the red circle in Figure 5). The impulse causes free oscillations of both panels in the assembly. The damping is set to zero, so the oscillations do not decay. These oscillations are simulated by solving Equation (13) with initial conditions corresponding to the installed fasteners, as it was described in the previous section.
- The frequency response function is obtained as a Fourier transform of the computed residual gap in the drilling node.
- The natural frequencies of the assembly correspond to the local maxima of .
5.2. Building the Dependence of the Vibration Amplitude on the Drill Rotational Speed by Massive Computations
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
MIC | Method of influence coefficients |
FEM | Finite element model |
DOFs | Degrees of freedom |
The vector of displacements and rotations in all nodes of the FEM | |
The number of degrees of freedom | |
The time | |
T | The maximal time |
The mass matrix | |
The damping matrix | |
The stiffness matrix | |
The vector of the applied load | |
The vector of the initial displacement | |
The vector of the initial velocity | |
The set of indices of nodes with restricted degrees of freedom | |
The vector of the initial gap in the junction area | |
The number of nodes in the junction area | |
The vector of constrained degrees of freedom | |
. | The vector of unconstrained degrees of freedom |
The diagonal blocks of the mass matrix | |
The off-diagonal blocks of the mass matrix | |
The diagonal blocks of the damping matrix | |
The off-diagonal blocks of the damping matrix | |
The diagonal blocks of the stiffness matrix | |
The off-diagonal blocks of the stiffness matrix | |
The vector of the load applied to constrained degrees of freedom | |
The vector of the load applied to unconstrained degrees of freedom | |
The reduced mass matrix | |
The reduced damping matrix | |
The reduced stiffness matrix | |
The time step | |
The number of time steps | |
The discrete time | |
The discrete displacement | |
The discrete velocity | |
The discrete acceleration | |
α, , κ, φ | The parameters of the generalized α method |
The equivalent matrix of stiffness | |
The equivalent load vector | |
The initial gaps | |
The residual gap | |
Rayleigh damping mass matrix scale factor | |
Rayleigh damping stiffness matrix scale factor | |
The damping factors |
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Initial Gap | Mean Residual Gap (μm) | Mean Amplitude (μm) |
---|---|---|
137.0 | 6.4 | |
102.7 | 7.6 | |
112.5 | 8.6 | |
138.0 | 7.7 |
Initial Gap | Natural Frequencies (Hz) | ||||
---|---|---|---|---|---|
193 | 220 | 387 | 580 | 615 | |
197 | 223 | 443 | 588 | 605 | |
192 | 217 | 385 | 577 | 602 | |
195 | 228 | 392 | 587 | 618 |
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Vasiliev, A.; Lupuleac, S.; Shinder, J. Numerical Approach for Detecting the Resonance Effects of Drilling during Assembly of Aircraft Structures. Mathematics 2021, 9, 2926. https://doi.org/10.3390/math9222926
Vasiliev A, Lupuleac S, Shinder J. Numerical Approach for Detecting the Resonance Effects of Drilling during Assembly of Aircraft Structures. Mathematics. 2021; 9(22):2926. https://doi.org/10.3390/math9222926
Chicago/Turabian StyleVasiliev, Alexey, Sergey Lupuleac, and Julia Shinder. 2021. "Numerical Approach for Detecting the Resonance Effects of Drilling during Assembly of Aircraft Structures" Mathematics 9, no. 22: 2926. https://doi.org/10.3390/math9222926
APA StyleVasiliev, A., Lupuleac, S., & Shinder, J. (2021). Numerical Approach for Detecting the Resonance Effects of Drilling during Assembly of Aircraft Structures. Mathematics, 9(22), 2926. https://doi.org/10.3390/math9222926