# Validation of a Modal Work Approach for Forced Response Analysis of Bladed Disks
## 1 Introduction  
Bladed-disk vibration has been studied by researchers and designers for more than half a century given the risks posed to turbomachineries, such as aeronautical engines and gas and steam turbines for energy production [1, 2]. Vibration issues can jeopardize the reliable design of these components, leading, in the worst cases, to catastrophic HCF failures. In this context, an accurate aeromechanical design able to prevent vibration risk is thus required to avoid or reduce the vibration amplitudes of components mainly due to forced response or flutter phenomena [3]. Despite all of the progress made in this field, the design of more robust and safer bladed disks with respect to HCF vibration is still a challenging task for the designers. Thanks to the increase of computing resources, more accurate computational methods for aeromechanical study have been applied to a wider part of turbomachinery modules. Such methods are implemented, for instance, to assess the flutter occurrence [4, 5] and the effect of mistuning and interlocking on the fluid-structure instability conditions [6, 7] or to compute the aerodynamic forcing that can excite the blade row in regards to the resonance condition in order to evaluate maximum displacements and stresses [8, 9, 10, 11]. There is a vast literature that addresses the latter phenomenon proposing numerical methodologies [12, 13] and experimental campaigns [14] to obtain a deeper insight into the physics and to evaluate the validity of design criteria to lower vibrations. Forced vibrations of turbomachinery bladed disks are caused by the reciprocal interaction of stationary and rotating components that generate rotating perturbations able to excite specific nodal diameters (ND) of a bladed disk. The most common perturbationsources are related to blade wakes that are convected downstream or pressure potential fields propagating upstream in a multi-row machine [15]. More recently, it has been proven that acoustic perturbations can also excite blade rows and deserve to be carefully considered in a multi-row environment [16; 17; 18]. Numerical tools for forced response analysis can be split into CFD solvers able to predict the flow unsteadiness [19; 20] and FEM codes used to perform modal and harmonic analysis of the excited component [21]. Such methods can be coupled together or used in a row depending on the type of approach and the complexity of the required simulations. The modern trend in bladed disk design is oriented toward a holistic approach that takes into account aeromechanical aspects together with aerodynamic, thermal and acoustic requirements. For this reason, integrated and fast approaches are preferable for gathering all these aspects in the design loop and are better in a multi-objective optimization procedure [22]. In this context, this work presents a fast approach to predict the forced response of a bladed disk based on the cyclic symmetry of the component. The approach takes advantage of a dedicated in-house tool able to extract the spinning perturbations acting on the blade row surface from the multi-row URANS solution by means of a temporal and spatial Fourier decomposition. After the forcing decomposition, a second in-house tool was implemented to compute the modal work done by the rotating force acting on the corresponding bladed disk mode shape in order to derive the scaling factor to compute actual oscillating displacements and stresses. The method was conceived for its integration in a CFD environment based on the URANS solver TRAF (developed at the University of Florence) capable of solving the force evaluation for the stator-rotor interaction [23; 24] and also performing flutter simulations to compute the aerodynamic damping to be added to forced response analyses together with the mechanical contribution [25].