# FSI Application in the Parametric Analysis of Blade-Disk Connection of Turbojet Engine Compressor
## 1 Introduction
The way the blade is connected to the disk is critical for the operation of the entire compressor rotor assembly. Such an attachment must ensure the transfer of loads from centrifugal forces, aerodynamic loads, vibration loads, etc. The loads are transferred from the blade to the root and then from the root to the disk. Hence, the dimensions and shape of the attachment used have a significant impact on the stress distribution throughout the rotor stage. And, this area is the main interest of the author in the present work. The most common type of connection used in fans and rotor stages of axial compressors is a trapezoidal root with a cross-section like an isosceles trapezoid. The method of attaching the blade depends largely on the type of supporting structure. In the case of disk structures, the blades are fixed in specially made channels parallel or diagonal to the axis of rotation of the engine, and in the case of drum or drum-disk structures, in channels turned circumferentially [8; 9]. The main dimensional loads in the root are centrifugal forces acting on the weight of the part of the blade located above the root (they cause tension in the cross-section and  
Figure 1: Schematic diagram of a turbofan engine (W—fan, S—high-pressure compressor, KS—combustion chamber, T\({}_{\text{NC}}\)—low-pressure turbine, T\({}_{\text{WC}}\)—high-pressure turbine, D—exhaust nozzle, n—rotation speed) [8]. pressure of the mating surfaces at the root and the disk rim) and aerodynamic forces acting on the blade, causing bending and twisting. In the case of perimeter-fixed trapezoidal roots, the greatest tensile force acts in the 0-0 cross-section (Figure 2). The maximum tensile stresses in this section are functions of the loading force and blade dimensions:  
\[\sigma_{r}=f(F_{r},\ b,c) \tag{1}\]  
where \(F_{r}\) is the resultant loading force of the part of the blade above the section with dimensions b and c [10]. The Huber-von Mises-Hencky hypothesis is used to determine the maximum equivalent stresses in the rotor stage. Accordingly, the stress \(\sigma_{e}\) is calculated according to the following equations [8]:  
\[\sigma_{e}=\left(\frac{1}{2}\Big{[}(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}- \sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\Big{]}\right)^{\frac{1}{2}} \tag{2}\]  
or  
\[\sigma_{e}=\left(\frac{1}{2}\Big{[}(\sigma_{x}-\sigma_{y})^{2}+\big{(}\sigma_ {y}-\sigma_{z}\big{)}^{2}+(\sigma_{z}-\sigma_{x})^{2}+6\Big{(}\sigma_{xy}^{2} +\sigma_{yz}^{2}+\sigma_{xz}^{2}\Big{)}\Big{]}\right) \tag{3}\]  
An important aspect of the methods of designing blade connections with disks is the use of numerical methods and parametric models to optimize and assess the impact of specific variables on the strength and operation of such a structure [11; 12; 13; 14; 15; 16; 17; 18; 19]. Parameters, in this case, may be characteristic geometric dimensions and operating factors, e.g., rotational speeds or pressures affecting the distribution of aerodynamic loads acting on the blade. Several types of loads require the use of advanced tools, such as coupled analyses examining the fluid-structure interaction (FSI). An example of the use of FSI coupling in the analysis of rotor assemblies is a publication by [PERSON] et al. [20], where the authors presented the FSI methodology in the analysis of the centrifugal compressor rotor stage and the impact of the applied coupling on the stress distribution in the rotor blade. In the present work, the strength properties of the rotor stage with a trapezoidal blade-disk connection were analyzed using parametric analysis considering FSI. The analysis was based on the relationship between the selected geometrical parameters and the maximum stresses in the structure. The analysis algorithm is shown in Figure 3.  
Figure 2: Scheme of a perimeter-fixed trapezoidal root.