# Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks
## 2 Numerical Methods
### Unsteady Forcing Decomposition  
The unsteady aerodynamic forcing generated by rotor-stator interactions can be extracted from the analyzed bladed disks by means of a proper decomposition of a multi-row URANS solution. A Discrete Fourier Transform (DFT), applied both in time and in space, allows the extraction of the pressure fluctuation components on the blade surfaces in the cyclic symmetry environment typical of turbomachinery bladed disks. To do this, the time-varying pressure fluctuations seen by an airfoil row are firstly decomposed in time by a TRAF code at run-time to extract the harmonic content at selected frequencies (Engine Order, EO) by the following formulation:  
\[P^{(h)}=\frac{1}{N_{div}}\sum_{n_{t}=0}^{N_{div}-1}p_{t}\ e^{-ih\frac{2\pi}{N_{div }}n_{t}} \tag{1}\]  
in which \(h\) is the time harmonic index (the engine order), \(p_{t}\) is the discrete equally spaced pressure signal in time and \(N_{div}\) is the total number of samples.  
The resulting complex of Fourier coefficients \(P^{(h)}\) extracted on blade surface grid points is further spatially decomposed by the implemented tool along the circumferential direction to determine the rotating perturbation that will excite the corresponding traveling wave mode shape. The time-space Fourier coefficients are extracted along corresponding points on the grid surface as follows:  
\[\hat{P}^{(h)}_{(\ell)}=\frac{1}{N}\sum_{k=1}^{N}P^{(h)}_{k}\ e^{-i\ell\frac{2 \pi}{N}k} \tag{2}\]  
in which \(\ell\) is the circumferential order, \(P^{(h)}_{k}\) is the discrete time Fourier coefficient of the tangential distribution on the blade's corresponding surface points and \(N\) is the blade count. The circumferential order \(\ell\) defines the number of rotating lobes, and its sign defines the spinning direction: a positive sign means a downward rotating wave and negative an upward rotating one when looking at the blade-to-blade view with the machine axis from left to right. The latter quantity can be easily converted to nodal diameter ND where the sign has a different definition related to the rotor rotational direction: if positive, it defines a forward rotating wave; if negative, a backward rotating wave.  
In the presented time-space decomposition, the circumferential DFT is performed on a coarse down-sampled set of only \(N\) samples taken on each blade at corresponding positions (grid points) on the surface. Hence, the extracted rotating pressure components have a number of lobes ranging from \(-N/2\) to \(+N/2\) or from \(-(N-1)/2\) to \(+(N-1)/2\) in the case of an odd blade count, as suggested by [PERSON]'s theorem. It is clear how this approach also takes into account the aliasing phenomenon experienced by the blade row when excited by a rotating perturbation with a lobe number higher than \(N/2\).  
Figure 1 provides a visual representation of a possible excitation scenario, where a rotating perturbation with 16 lobes, represented by the solid violet line, impinges on a row composed by of blades represented by the radial grey segments. The 12 blades experience a 4-lobe perturbation represented by the green dashed line due to the aliasing phenomenon. The orange dash-dot radial lines highlight the resulting four nodal diameters associated with the 4-lobe excitation. The implemented spatial decomposition on the \"blade-sampled\" corresponding N points along the circumferential direction is thus able to convert the 16-lobe incoming perturbation into the 4-lobe excitation experienced by the blade row, which will finally vibrate as a traveling wave with four nodal diameters.  
The implemented post-processor generates the blade surface pressure perturbation maps for each nodal diameter at the selected engine order in a format directly readable by ANSYS.