Key Action Mechanisms of Intentional Mistuning

Intentional mistuning is a common procedure to decrease the uncontrolled vibration amplification effects of the (unavoidable) random mistuning, and to reduce the sensitivity to it. The idea is to introduce an intentional mistuning pattern that is small, but much larger than the existing random mistuning. The frequency of adjacent blades is moved apart by the intentional mistuning, reducing the blade-to-blade coupling and, thus, the effect of the random mistuning. In order to clearly show the action mechanisms of intentional mistuning, we focus in this work in a quite simple configuration: forced response of a blade dominated modal family in a mistuned rotor with linear material damping. The problem is analysed using the asymptotic mistuning model methodology. A more reduced order model is derived that allows us to understand the relevant parameters behind the effect of intentional mistuning, and gives a simple expression for the estimation of its beneficial effect. The results from the reduced model are checked against detailed FEM simulations of two mistuned rotors.

Vibration Analyses of an Axial Turbine Wheel with Intentional Mistuning

The last stage bladed disk of a steam turbine is analyzed with respect to both flutter susceptibility and limitation of forced response. Due to the lack of variable stator vanes unfavorable flow conditions may occur which increases the risk of flutter at part load conditions. For this reason, intentional mistuning is employed with the objective to prevent any self-excited vibrations. A first step in this direction is done by choosing alternate mistuning, which keeps the manufactural efforts in limits. In this sense, two different series of blades have been made. However, small deviations from the design intention are unavoidable due to the manufacturing procedure, which could be proved by bonk tests carried out earlier. The influence of these additional deviations is considered in numerical simulations. Moreover, the strong dependence of blade frequencies on the speed is taken into account since centrifugal stiffening effects significantly attenuate the blade-to-blade frequency difference. Focusing on the first flap mode it could be shown that a mitigation of flutter susceptibility is achieved by prescribing alternate mistuning, which indeed evokes an increase of originally small aerodynamic damping ratios. Nevertheless, the occurrence of negative damping ratios could not be completely precluded at part load conditions. That is why optimization studies are conducted based on genetic algorithms with the objective function of maximizing the lowest aerodynamic damping ratios. Finally, mistuning patterns could be identified featuring a tremendous increase of aerodynamic damping ratios. The robustness of the solutions could be proved by superimposing additional random mistuning.

Vibration Analyses of an Axial Turbine Wheel With Intentional Mistuning

The last stage bladed disk of a steam turbine is analyzed with respect to both flutter susceptibility and limitation of forced response. Due to the lack of variable stator vanes unfavorable flow conditions may occur which can lead to flow separation in some circumstances. Consequently, there is the risk of flutter in principle, particularly at nominal speed under part load conditions. For this reason, intentional mistuning is employed by the manufacturer with the objective to prevent any self-excited vibrations. A first step in this direction is done by choosing alternate mistuning, which keeps the manufactural efforts in limits since only two different blade designs are allowed. In this sense, two different series of blades have been made. However, it is well known that small deviations from the design intention are unavoidable due to the manufacturing procedure, which could be proved by bonk tests carried out earlier. The influence of these additional but unwanted deviations is considered in numerical simulations. Moreover, the strong dependence of blade frequencies on the speed is taken into account since centrifugal stiffening effects significantly attenuate the blade-to-blade frequency difference in this particular case. Focusing on the first flap mode it could be shown that a mitigation of flutter susceptibility is achieved by prescribing alternate mistuning, which indeed evokes an increase of originally small aerodynamic damping ratios. Nevertheless, the occurrence of negative damping ratios could not be completely precluded at part load conditions. That is why optimization studies are conducted based on genetic algorithms with the objective function of maximizing the lowest aerodynamic damping ratios. Again only two different blade designs are admitted. Finally, mistuning patterns could be identified causing a tremendous increase of aerodynamic damping ratios. The robustness of the solutions found could be proved by superimposing additional random mistuning.

Comparative Studies of Surrogate Models for Response Analysis of Mistuned Bladed Disks

The forced responses of bladed disks are highly sensitive to inevitable random mistuning. Considerable computational efforts are required for the sampling process to assess the statistical vibration properties of mistuned bladed disks. Therefore, efficient surrogate models are preferred to accelerate the process for probabilistic analysis. In this paper, four surrogate models are utilized to construct the relation between random mistuning and forced response amplitudes, which are polynomial chaos expansion (PCE), response surface method (RSM), artificial neural networks (ANN) and Kriging interpolation, respectively. A bladed disk with 2-degrees-of-freedom (2-DOF) each sector is used to validate the effectiveness of the surrogate models. The effects of number of training samples on the surrogate model accuracy are discussed. The responses results of one blade (single output) and maximum response of all blades (multi-output) indicate that PCE and Kriging interpolation could yield accurate and stable predictions of the statistical characteristics of the forced responses. PCE is recommended for the mistuned response predictions due to its accuracy and efficiency.

Intentional Mistuning With Predominant Aerodynamic Effects

Intentional mistuning is a well known procedure to decrease the uncontrolled vibration amplification effects of the inherent random mistuning and to reduce the sensitivity to it. The idea is to introduce an intentional mistuning pattern that is small but much larger that the existing random mistuning. The frequency of adjacent blades is moved apart by the intentional mistuning, reducing the effect of the blade-to-blade coupling and thus the effect of the random mistuning. The situation considered in this work is more complicated because the main source for the blade damping is the effect of the aerodynamic forces (as it happens in a blisk for a family of blade dominated modes with very similar frequencies). In this case the damping is clearly defined for the tuned traveling waves but not for each blade. The problem is analyzed using the Asymptotic Mistuning Model methodology. A reduced order model is derived that allows us to understand the action mechanism of the intentional mistuning, and gives a simple expression for the estimation of its beneficial effect. The results from the reduced model are compared with those from a finite element model of a more realistic rotor under different forcing conditions.

Practical Optimization of Mistuned Bladed Disk of Steam Turbine With Free-Standing Blade Structure for Forced and Self-Excited Vibration

Although bladed disks are nominally designed to be cyclically symmetric (tuned system), the vibration characteristics of all the blades on a disk are slightly different due to the manufacturing tolerance, deviations in the material properties, and wear during operation. These small variations break the cyclic symmetry and split the eigenvalue pairs. Bladed disks with small variations are referred to as a mistuned system. Many researchers suggest that while mistuning has an undesirable effect on the forced response, it has a beneficial (stabilizing) effect on blade flutter (the self-excited vibration). Therefore, it is necessary to optimize a bladed disk for forced vibration and blade flutter. In this study, firstly, the stability analysis of a mistuned bladed disk of a steam turbine that experienced the blade flutter in the field is carried out by use of the reduced order model, the Fundamental Mistuning Model. It is reported that the bladed disk analyzed failed due to unstalled flutter of the 1st mode, and the problem was solved by alternating mistuning. By comparing the analysis results with these field experiences, the analysis method is validated. Secondly, a parametric study on the mistuning effect is carried out for typical mistuning patterns, such as periodic and random mistuning, for both forced and self-excited vibrations. Finally, based on the above-mentioned results, a practical optimization method considering both forced vibration and self-excited vibration with respect to the bladed disk of a steam turbine with a free-standing blade structure is proposed.

Theoretical Study of the Vibration Suppression on a Mistuned Bladed Disk Using a Bi-periodic Piezoelectric Network

The paper deals with the vibration suppression of a bladed disk with a piezoelectric network. The piezoelectric network has a different period (so called bi-period) from that of the bladed disk and there is no inductor in it. The system is simulated by an electromechanical lumped parameter model with two DOFs per sector. The research focuses on suppressing the amplitude magnification or reducing the vibration localization of the mistuned bladed disk. The dynamic equations of the system are derived. Both mechanical mistuning and electrical mistuning have been taken into account. The Modified Modal Assurance Criterion (MMAC) is used to evaluate the vibration suppression ability of the bi-periodic piezoelectric network. The Monte Carlo simulation is used to calculate the MMAC of the system with the random mistuning. As a reference, the forced responses of the bladed disk with and without the piezoelectric network are given. The results show that the piezoelectric network would effectively suppress amplitude magnification induced by mistuning. The vibration amplitude is even smaller than that of the tuned system. The robustness analysis shows that the bi-periodic piezoelectric network can provide a reliable assurance for avoiding the forced response amplification of the mistuned bladed disk. The amplified response induced by the mechanical mistuning with standard deviation 0.2 can be effectively suppressed through the bi-periodic piezoelectric network.

Analysis of the Effect of Multirow and Multipassage Aerodynamic Interaction on the Forced Response Variation in a Compressor Configuration—Part I: Aerodynamic Excitation

The aeroelastic prediction of blade forcing is still a very important topic in turbomachinery design. Usually, the wake from an upstream airfoil and the potential field from a downstream airfoil are considered as the main disturbances. In recent years, it became evident that in addition to those two mechanisms, Tyler–Sofrin modes, also called scattered or spinning modes, may have a significant impact on blade forcing. It was recently shown in literature that in multirow configurations, not only the next but also the next but one blade row is very important as it may create a large circumferential forcing variation, which is fixed in the rotating frame of reference. In the present paper, a study of these effects is performed on the basis of a quasi three-dimensional (3D) multirow and multipassage compressor configuration. For the analysis, a harmonic balancing code, which was developed by DLR Cologne, is used for various setups and the results are compared to full-annulus unsteady calculations. It is shown that the effect of the circumferentially different blade excitation is mainly contributed by the Tyler–Sofrin modes and not to blade-to-blade variation in the steady flow field. The influence of various clocking positions, coupling schemes and number of harmonics onto the forcing is investigated. It is also shown that along a speed-line in the compressor map, the blade-to-blade forcing variation may change significantly. In addition, multirow flutter calculations are performed, showing the influence of the upstream and downstream blade row onto aerodynamic damping. The effect of these forcing variations onto random mistuning effects is investigated in the second part of the paper.