# Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks
## 4 Conclusions  
The paper reports the validation of a numerical method based on a modal work approach to assess the forced response of bladed disks characterized by a cyclic symmetry. The method requires as inputs the forcing function distributions obtained by a time and space Fourier decomposition of an URANS CFD solution and the corresponding traveling wave mode shapes coming from a modal analysis of the cyclic sector. The method can be applied for a crossing on the ZZENF diagram (resonance condition) or for a generic out-of-resonance condition and computes the scaling factor to obtain actual displacements and stresses from modal results. The method is validated against numerical results obtained by dynamic analysis computed by the ANSYS solver with the same forcing function distributions that come from the decomposition of a URANS solution carried out with the CFD TRAF code. The FEM ANSYS code is also used for the prior modal analysis required by both numerical approaches, while the aerodynamic damping is again computed by the TRAF code with an uncoupled flutter method, which solves the unsteady pressure response of the flow field due to a vibrating blade row. For this exercise, the mechanical damping component is neglected but can be easily included by adding this value to the aerodynamic contribution. The validation of the method was firstly performed for a dummy test case representative of a bladed disk with cyclic symmetry for different arbitrary forcing distributions that are not subjected to any interpolation procedure. The comparison between Tremor and ANSYS shows an excellent agreement in terms of displacements. Finally, the validation has been extended to a more realistic test case representative of a low-pressure turbine bladed rotor excited by two different upstream sources: the IGV wakes with low engine order, and the stator wakes. The two spinning forcing functions were extracted from a multi-row unsteady simulation of the IGV and the downstream stage. This test case is based on a rotor geometry already studied during the EU project FUTRE dedicated to flutter and forced response, but the blade count and rotational speed were modified to obtain a resonance condition at the IGV passing frequency and one out of resonance for the stator passing frequency on the first mode family (bending mode). The results show a good agreement between numerical results and pinpoint the source of slight discrepancy in the approximations of the mapped forcing in the ANSYS environment due to a coarse mesh that smooths the pressure distributions and small errors in the overlapping of the forcing point cloud and FEM surface points before the interpolation. This underlines the higher accuracy of Tremor computations that are performed on a finer CFD mesh, where the interpolation of the mode shapes introduces less discrepancy than the interpolation of the forcing on the FEM superficial mesh. The validation confirms the capability of the presented approach to predict the forced response of a bladed disk. This approach is strictly connected to the CFD framework and only requires a prior modal analysis, making the tool attractive for its use of an integrated multi-objective optimization procedure that is able to include aeromechanical aspects. Future work concerns the extension of the method to intentionally mistuned bladed row configurations. All the authors decided on the methodology and selected the two test cases. [PERSON] formulated the method, implemented the Tremor tool and wrote the theoretical part of the manuscript. [PERSON] performed the numerical simulations and wrote the result section of the manuscript. [PERSON] and [PERSON] contributed to the final review and manuscript. [PERSON] supervised the work. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable.