Effects of the Geometric Uncertainties of Bladed Disks Through the Analytical Derivatives of the Finite Element Matrices

Manufacturing processes and wear are known to create imperfections in mechanical components that may result in a significant deviation of the modal properties from their predicted values. This is especially true for periodic structures such as bladed disks, where even small variations in the blade geometry can cause mistuning. The effect of parameter variations on the model is commonly assessed through sensitivity analysis within the framework of uncertainty propagation or structural optimization. Different formulations are available when dealing with the variation of a single scalar parameter. The problem becomes more complicated when the source of the variation is the whole component geometry. To deal with this case, it is necessary to define a representation able to describe the shift from the nominal geometry through a finite set of parameters and compute the derivatives of the system matrices with respect to those parameters. This paper proposes to compute analytically the derivatives of the system matrices with respect to the geometric variations. The derivatives are calculated at the finite element (FE) level and are then assembled into global derivative matrices. Using directional derivatives, the sensitivities of the system matrices are computed with standard FE numerical schemes based on the nominal geometry.