In turbomachinery one major problem is still the calculation and the optimization of the spatial vibrations of mistuned bladed disk assemblies with friction contacts. Friction contacts are widely used to reduce dynamic stresses in turbine blades. Due to dry friction and the relative motion of the contact planes energy is dissipated. This effect results in a reduction of blade vibration amplitudes. In the case of a tuned bladed disk cyclic boundary conditions can be used for the calculation of the dynamic response. For a mistuned bladed disk the complete system has to be modeled and simulated. To reduce the computation time the so-called substructure method is applied. This method is based on the modal description of each substructure, especially disk and blades, combined with a reduction of the degrees of freedom, to describe the dynamics of each component. The spatial dynamical behavior of each component is considered and described by the mode shapes, natural frequencies and modal damping ratios. Using the Harmonic Balance Method the nonlinear friction forces can be linearized. From here it is possible to calculate the frequency response functions of a mistuned bladed disk assembly with friction contacts. In many cases Monte-Carlo simulations are used to find regions, where the system response is sensitive to parameter uncertainties like the natural frequencies of the blades. These simulations require a large computation time. Therefore, an approximate method is developed to calculate the envelopes of the frequency response functions for statistically varying natural frequencies of the blades. This method is based on a sensitivity analysis and the Weibull-distribution of the vibration amplitudes. From here, a measure for the strength of localization for mistuned cyclic systems is derived. Regions, where localization can occur with a high probability, can be calculated by this method. The mean value and the standard deviation of the vibration amplitudes are calculated by simulation and by the approximate method. The comparisons between the approximate method and the Monte-Carlo simulations show a good agreement. Therefore, applying this method leads to remarkable reduction of computation time and gives a quick insight into the system behavior. The approximate method can also be applied to systems, that include the elasticity of the disk and/or the coupling by shrouds or other friction devices.
On Solving One-Dimensional Partial Differential Equations With Spatially Dependent Variables Using the Wavelet-Galerkin Method
