# Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks
## 2 Numerical Methods
### Modal Work Approach  
The implemented tool, named Tremor (Tool for Response Evaluation by Modal wORk), is capable of computing a forced response analysis of a bladed disk in a cyclic symmetry environment. The tool computes the forced response of a single traveling wave mode shape (defined by the nodal diameter and resulting from a modal cyclic symmetry analysis), excited by the corresponding rotating forcing function, obtained from an unsteady analysis and spatially decomposed to match the nodal diameter of the mode shape.  
The forced response evaluation is based on the modal work computation that is integrated on the blade surface CFD mesh, where a nearer point interpolation strategy is used to transfer the blade mode shape from the FEM to the CFD blade surface. The computation of the modal work on the CFD discretization ensures a more accurate result due to the higher density of the CFD mesh.  
The tool requires the blade mode shape, the decomposed forcing functions and the total damping as inputs in order to compute the scaling factor to be applied to modal displacements and stresses to obtain actual values. The scaling factor can be computed both in and out of resonance conditions. The modal work approach determines the maximum energy transfer of a rotating pressure perturbation (decomposed on blade corresponding points, as described in the previous section) applied to the corresponding traveling wave mode shape defined by the nodal diameter and is defined as follows:  
\[max[W_{\text{mod}}(\alpha)]=\pi\sqrt{\left[\iint_{\Sigma}\mathfrak{Im}\left[U_{ \text{mod}}^{*}\,F_{\text{aero}}\right]\,d\Sigma\right]^{2}+\left[\iint_{ \Sigma}\mathfrak{Re}\left[U_{\text{mod}}^{*}\,F_{\text{aero}}\right]d\Sigma \right]^{2}} \tag{3}\]  
where \(W_{\text{mod}}\) is the modal work, \(U_{\text{mod}}^{*}\) is the complex conjugate of the modal displacement phasor and \(F_{\text{aero}}\) the unsteady aerodynamic pressure phasor.  
The modal work approach leads to the computation of the modal force \(F_{\text{mod}}\) by the integration of the complex dot product between the complex conjugate of the modal displacements and the forcing function distribution, both evaluated at the cell center for each point of the CFD grid.  
\[F_{\text{mod}}=\sqrt{\left[\iint_{\Sigma}\mathfrak{Im}\left[U_{\text{mod}}^{* }\,F_{\text{aero}}\right]d\Sigma\right]^{2}+\left[\iint_{\Sigma}\mathfrak{Re} \left[U_{\text{mod}}^{*}\,F_{\text{aero}}\right]d\Sigma\right]^{2}} \tag{4}\]  
Figure 1: Example of a spinning forcing function and aliasing phenomenon.  
The scaling factors can be computed under resonance conditions or not by using the following equation:  
\[s(\omega_{e})=\frac{F_{\text{mod}}}{\sqrt{\left({\omega_{\text{mod}}}^{2}-{\omega_ {\text{e}}}^{2}\right)^{2}+\left(2\;\omega_{\text{mod}}\;\omega_{e}\;\xi\right) ^{2}}} \tag{5}\]  
where \(\omega_{\text{mod}}\) is the natural angular frequency, \(\omega_{\text{e}}\) is the angular frequency of interest and \(\xi\) is the damping, which can include the material, mechanical and aerodynamic damping. The scaling factors can be used to scale the modal displacements or stresses in order to obtain the actual values:  
\[\begin{split} U&=U_{\text{mod}}\left|s(\omega_{ \text{e}})\right|\\ \sigma&=\sigma_{\text{mod}}\left|s(\omega_{\text{e}} )\right|\end{split} \tag{6}\]  
where \(d_{\text{mod}}\) and \(\sigma_{\text{mod}}\) are the modal displacements and stresses from the modal cyclic symmetry analysis, and \(\left|s(\omega_{\text{mod}})\right|\) is the module of the scaling factor.