Transient Response of an Airfoil

The project objective is the modeling of an airfoil using Nastran to familiarize with the graphical environment and to analyze/validate the solutions offered by the program through a secondary analytical method when a sine force is applied at the edge of the wing.

Preliminary Design

The model used for the airfoil design is the one proposed in the figure. This model will make the assumption of zero wing sweep and will be designed with rigid panels and strings attached to the degrees of freedom . This strings will simulate the flexion and torsion of the wing and their strength will cedrease gradually from the end to the wing's center and will be kept constant after this point. Point masses will be localized in the front and bottom of the wing (leading edge).

Meshing

The first step into modelin the airfoil is the mesh. This part of the design is one of the most important to have a good understanding of the problem. Simple meshings lead to wrong approaches to the problem and large meshings to high computational timing and redundant efforts.

Since this is a very simple panel model with few relevant points it can be meshed with the 3 main elements:

Corners of the panels
Connection points
Point masses

Since the the final problem will involve a force application in the middle point of the panel, the meshing needs to be refined and adapted to the final problem proposition. Both meshings are shown in the figures.

First meshing from the panel model

Meshing refined adapted to the problem

The model implementation in Nastran is shown below with codes implemented for both the normal and refined meshing

Modeling Elements

Next thing once the meshing is done is to specify the elements used as materials for the airfoil. This can be:

Rigid rods (RBAR)
Beams (CBEAM)
Rigid elements (RBEx)

Both RBAR and RBEx are very similar, although the main difference between them is that RBAR stablishes relationships between individual nodes and RBEx considers relationships between one node and all of the different nodes proposed.

Since the structure will be formed of 10 rigid panels, the degrees of freedom of the nodes of one panel will move consequently along the airfoil. Therefore each panel, whether connected to several nodes, it will only account fo one independent node and that's why RBEx elements are more efficient in this case.

For the connection among the panels, the elements used will be RJOINTs. MPCs were also considered but for this case, even though these last ones are more versatile, RJOINTs work perfectly in our model and are defined easire than MPCs.

Structural Analysis

For a structural analysis of the system the first step is consider the natural frequencies of the system and the eigenmodes with zero damping. This way, the basic structural behavior of the system is obtained for further dynamic force application.

The reasons to analyze the structure with eigenvalues are based on the interaction between all the components of the system. Eigenmode formulations allows the calculus of the structure displacements as a composition of the individual displacements of each mode. All of them contribute to the frequency response and will increase as closer to the excitation frequency of the mode.

Eigenmodes Analysis

Damping Analysis

Damping is considered a mathematical approach to account of energy dissipation in a real system. There are two main dampings used for analysis in lineal-elastic materials:

Viscous damping: Based on velocity
Structural damping: Based on displacements

Depending on the physics behind the dissipation mechanism, the user can choose to apply viscous or structural damping. A graph showing both damping types depending on the damping force and force frequency is shown.

Damping in MSC.Nastran

There are 6 different possible dampings used in Nastran:

Structural damping
Viscous damping
Modal damping
Rayleigh damping
Hybrid damping
No-lineal damping

The one used for the model is the hybrid damping HYBDAMP adding the damping table TABDAMP where the damping coefficients will be considered.

Analytical Problem Resolution

The goal of this section is to solve the system response when a sine force of the form 1-cos is applied, that is a forced vibrational problem. The point of study is (x = -0.5, y = 3-75) and it is responsible for considering a more complex meshing at the begining. The function of the applied force will be:

Next step is obtain the rigid and mass matrices of the system. This is an easy step in Nastran and thank to the MATLAB code provided from the Aerodynamics department at UPM the results can be easily obtained in an analytical approach. The eigenvalue problem proposition will be based on the equationbelow where M is the mass matrix and K the rigid matrix.

By solving the equation, 18 eigenvalues along with their frequencies can be obtained, they are shown in the figure on the left.

When solving the analytical model the process involves a first step to obtain modal coordinates using the equation:

After applying the Duhamel equation along with the force applied in matrix form the results of the system response can be obtained. In the figure are shown the results for the panels 4 and 10 in a small time range as showing the whole matrix can be ecexively long for this brief presentation.

Flexion-Torsion Analysis

To validate the results, a secondary approach has been proposed uncoupling the flexion and torsion displacements and then combined them for the final resolution. Both graphs from the Nastran matrices model as well as the MATLAB model are shown in the figure and, as expected, they are practically the same.

Nastran-based results

MATLAB results

NASTRAN CODE

Nastran Solution

The section above analyzed the MATLAB model with the matrix results from Nastran applied to this MATLAB approach. In this case the solution will be obtained directly from the software Nastran.

The solution applied will be the 112 Nastran which requires the introduction of specific cards in the main BDF to define the structure and analyze the eigenvalues from the SOL 103. The modifications in the BDF imply:

Force definition
Damping mechanism
Time stepping
Calculation method
Results desired

The results from the panel 4 and 10 are shown in the figure below:

Graphic Resulst

The results from Nastran are shown below for force application in different nodes and closer detail to the damping region.

3D Simulation of the structure

The figures below show the simulation of the whole airfoil when the force is applied in the specific point mentioned before. As it can be seen, the oscillations for the structure follow the damping way of the graphs above and decreases with the time step.

Piano Structure

Result Comparison and Conclusion

Looking at the tables shown above it can be seen that the results for different models are practically the same with a total error of 1% and being kept under 0.1% on the first time steps.

Although the results for the first steps are very solid, as the time increases the error does it too reaching a 5% for 5 secons of time step.

After the project termination it can be concluded that it is a great introduction to the Nastran software not only for the result obtaining but to understand the physics behind it and how a simple model can recreate the results (obviously not as precise as Nastran). The analytical study allowed us to understand the eigenvalue problem proposition and introducing to the Duhamel equation and matrix solving. Additionally, during the project, sever hypotthesis were made and considered having the freedom to analyze and decide which was the optimal solution.

Overall this was one of my hardest projects I've been involved to not say the most. It implied several concepts I wasn't familiar with and that I had to understand. It helped me work along my teammates and contribute in the best way possible as well as learning everything I could.

Analytical model

Duhamel

Page updated

Google Sites

Report abuse