# Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks
## 3 Results and Comparisons
### LPT Bladed Rotor Test Case
#### 3.2.7. Tremor-ANSYS Comparison  
The Tremor tool was finally used to predict the forced response of the rotor blade due to the IGV and the stator wakes. The modal displacements for ND \(\,=\,-7\) and ND \(\,=\,+21\) (first family) are provided from the previous modal analysis on the blade surfaces and for the pair of control nodes. The pressure harmonic decomposed in space and time coming from the unsteady multi-row analysis is extracted at EO \(\,=\,7\) and EO \(\,=\,126\). A rotation matrix to transfer the mode displacements from the FEM frame of reference  
Figure 14. Check on forcing distribution mapping (EO \(\,=\,7\)). Figure 15. Check on forcing distribution mapping (EO \(\,=\,126\)). to the CFD one is obtained by means of the open-source CloudCompare program able to find out the rigid transfomation to overlap the FEM surface to the CFD one by means of an optimization procedure. The aerodynamic damping resulting from the flutter analyses is specified for each nodal diameter in the input file. The complex modal forcing distribution is evaluated on cell center points of the CFD blade surface elements to compute the scaling factors to be applied to modal displacements and stresses. The module of the modal force surface density, which is computed as the local modal force contribution divided by the corresponding cell area for each surface cell, is shown in Figure 16.  
Note that the module of this quantity is only useful to qualitatively observe the regions of the blade surface that contribute the most to the overall forced response amplitude; the quantitative comparison must be done separately on the real and imaginary parts of this quantity, which leads to the scaling factor when integrated over the surface and composed together, as shown in Equation (4). In the case of IGV excitation, the greater values are observed near the tip of the PS (pressure side), where the first bending mode presents higher modal displacements, and on the LE, where the pressure perturbation has a higher amplitude. Considering the stator excitation, this quantity is more than one order of magnitude higher than the other case, and the greater values are observed near the tip on the SS, where the first bending mode presents higher modal displacements and the pressure perturbation is stronger. The actual displacements were finally obtained by scaling the modal displacements with the complex scaling factor resulting from the modal work computation. The module of these displacements, i.e., the amplitude of total deformation, which is calculated by the square root of the sum of the squares of the real and the imaginary parts in all directions, is depicted in Figure 17.  
The contours represent the scaling of the first bending mode, and the displacements are almost the same between PS and SS: this further confirms that the mode shape is well transferred from the FEM grid to the CFD one where the scaling factor is computed. The comparison between the harmonic cyclic symmetry analysis in ANSYS and Tremor computations for the first bending mode is listed in Table 4. For the case of IGV excitation, the validation is performed at the resonance condition for ND \(=\)\(-\)7 backward corresponding to 361.67 Hz, while in the case of stator excitation, the results are compared at the 6510 Hz  
Figure 16: Modal force surface density module. out-of-resonance condition for ND \(=\) +21 forward that exhibits a natural frequency of 362.48 Hz. The agreement is good for the stator excitation and satisfactory for the IGV excitation. However, the ANSYS results strongly depend on the mapping of the pressure perturbation from the CFD grid to the FEM one. The Tremor tool computes the modal work on the blade surface directly on the CFD discretization without interpolating the forcing function and transferring the blade mode shape, which usually shows a more gradual distribution with respect to the forcing function. It is more likely that the transfer of the forcing functions from CFD to the FEM model can lead to higher transfer errors.