PSI - Issue 8

N. Di Domenico et al. / Procedia Structural Integrity 8 (2018) 422–432

424

N. Di Domenico et al. / Structural Integrity Procedia 00 (2017) 000–000

3

The FP7 EU Project RBF4AERO (RBF4AERO (2017)), running since 2013, aims to extend the RBF mesh morphing approach to relevant aeronautical applications providing, among other tools, 2-way and modal superposition FSI routines. Proposed approach is demonstrated on a practical application (NACA0009 Hydrofoil) for which experimental data are available and compared to literature. The transient coupled solver is employed for the computation of eigenvalues in water by post processing the free vibration response in calm fluid.

2. Theoretical background

In this section a brief recall of the basic theory behind RBF and modal theory is given.

2.1. Modal analysis

Structural modes are a set of inherent properties that each body possess and is defined by its geometry, material composition and working boundary conditions. Each mode is defined by a modal frequency and is characterized by a modal damping and shape. A variation in one of the parameters defining the geometry, the material or the constraints reflects in a mode change in terms of shapes and frequencies. A number of modes equal to the number of degrees of freedom exist; infinite in the case of a continuum structure. Modes define how a structure reacts to a static loading and its dynamic behavior, making them of essential importance during the design phase. Modal analysis is the study that aims to calculate the undamped structural modes of a structure (Meirovitch (2001)). For an n-degree-of-freedom undamped system the di ff erential equation of motion can be written as:

[ M ] ¨ q + [ K ] q = Q

(1)

Where M is the mass matrix, K the sti ff ness matrix and Q the external forces vector. Modes can be obtained by solving the eigenvalue problem:

[ K ] u = ω 2 [ M ] u

(2)

In which the eigenvectors u are the modal shapes and the eigenvalues ω the circular frequencies. Modal frequencies can be directly calculated ( ω = 2 π f ). By normalizing the equation of motion with respect to masses, meaning that each mode has a unit modal mass, can be written: [ u ] T [ M ] [ u ] = 1 [ u ] T [ K ] [ u ] = ω 2 (3) In which mass and sti ff ness matrices are diagonal. Exploiting the spectral decomposition modes are orthogonal and form a basis in the modal coordinate’s space (Cook (2001)): each mode acts as a single DOF dynamic system. System response can be then calculated as a linear superimposition of each mode response, allowing to take into account only a subset of structural modes. In a static case the structure behaves as a low-pass filter, being modes at lower frequency more important and influent in the structural response. By employing a numerical solution method such as Lanczos modes can be extracted in ascending order, stopping the calculation when a required number of modes have been found. When dealing with dynamic analyses the number of modes retained is instead defined on the basis of excited frequencies.