Bladed Disk Assembly Research Articles

Vibration Analysis of Bladed Disc Assemblies∗

Abstract The dynamic behavior of a bladed disc assembly is examined by developing a finite element model (FEM). The object of this work is to develop an understanding of the dynamic behavior of the system with respect to the various boundary conditions. The analysis includes variation of the boundary conditions at the blade-disc interface, between adjacent blades, and at the shroud. It is clearly shown that only certain mode patterns are affected with respect to a particular boundary condition, leaving the remaining modes unchanged. The quantitative effect of the various boundary conditions is calculated by finite element analysis of a cyclic symmetric model. A methodology is also presented where the associated modes of the higher harmonics resonant frequencies of the assembly are identified in a systematic and automated way without being plotted and viewed. This methodology, which uses the cross-orthogonality equations, is valuable in analyzing systems with a large number of nodal diameters, where the complete set of the associated mode shapes can be analyzed in an efficient and systematic way. The technique is illustrated through an example, where a complete analytical Campbell diagram is developed for the lower order modes.

The General Model of Free Vibrations of Mistuned Bladed Discs, Part I: Theory

This paper is divided into two parts. The author's purpose in Part I is to formulate the equations of motion describing the linear dynamics of bladed disc assemblies. The problem of calculating the natural frequencies of a rotating bladed disk is solved by the Ritz method. A consistent second order moderately thick plate theory is used for the disc. The blade is treated by a beam theory and includes shear deformation, rotary inertia effects, blade pretwist, centrifugal effects, and warping as a function of bending and torsion. The numerical results for tuned and mistuned bladed discs with the blades of complex geometry is the subject of Part II.

The General Model of Free Vibrations of Mistuned Bladed Discs, Part II: Numerical Results

This paper is divided into two parts. In Part I the equations of motion describing the free vibration of bladed disc assemblies were formulated. The numerical results for the tuned and mistuned bladed discs for blades of complex geometry are the subject of Part II. Answers to the question "Which of the blades in the mistuned system are most likely to experience the largest vibration amplitude?" are presented. The best and worst distributions of the detuned blades from the stress point of view are shown. The influence of the stagger angle, the cross-sectional area and the twist angle of the blades on the natural frequencies, stress distributions of the blades around the disc and the position of the maximal stress along the blade length of rotating and non-rotating tuned and mistuned bladed discs is presented. The accuracy of numerical calculations of the blades' stresses is dependent on the inclusion in the blade model of the bending centre. This problem is also considered.

Vibration analysis of mistuned bladed-disk assemblies - inverse approach

This paper deals with a method that aims at solving the inverse problem where some of the key structural parameters are determined from knowledge of the system's response. This method was applied to the specific case of a 12-bladed disk to determine the range of individual blade cantilever frequency scatter so that all blade responses lay below an allowable maximum level. This scatter or manufacturing tolerance was computed statistically, and the analysis was verified using a direct Monte Carlo simulation. The relationship between the allowable response increase due to mistuning and the corresponding manufacturing tolerances was also investigated, and the presence of a critical mistuning threshold was noted

Aeroelastic stability of cascades in turbomachinery

State-of-the-art prediction of the aeroelastic stability of cascades in axial-flow turbomachines is reviewed. The first main chapter of the article presents a comprehensive formulation of the two- and three-dimensional classical (unstalled) flutter problem of tuned and mistuned rotor blade rows and bladed disc assemblies. Within the framework of linearized analysis, a complete and generalized theory in modal form is outlined, comprising the various formulations of the cascade flutter problem distributed in fragments throughout the literature. Brief outlines are also made of recent advances in unsteady aero-dynamic methods for turbomachinery aeroelastic applications. The second main chapter contains a parametric study of the classical flutter stability characteristics of compressor and turbine cascades in subsonic and supersonic flow. Stability boundaries and dominant trends in flutter behaviour are outlined, and the significant effects of blade mistuning on the aeroelastic stability of turbomachine bladings are highlighted.

Use of networked workstations for parallel nonlinear structural dynamic simulations of rotating bladed-disk assemblies

The principal objective of this research is to investigate, develop and demonstrate coarse-grained, parallel-processing strategies for nonlinear dynamic simulations of rotating bladed-disk assemblies. The parallel-processing strategies addressed include numerical algorithms for parallel nonlinear solutions and techniques to effect load balancing among processors. The parallel environment employed is a distributed-memory, coarse-grained one consisting of networked workstations. A parallel explicit time integration method has been implemented for transient nonlinear solutions of rotating bladed-disk assemblies. Automatic domain partitioning techniques have been investigated for load balancing among processors. Advanced computing environments, data structures and interactive computer graphics all contribute to an integrated parallel finite element analysis system to facilitate more efficient and powerful dynamic simulations.

Fatigue Life Prediction Using Frequency Response Functions

This paper presents a method for fatigue life prediction of engineering components subjected to dynamic loads. It is based on the determination of the nominal stress at the crack position using frequency response functions and this in turn enables the prediction of dynamic fatigue life under forced vibration. The main advantage of this approach lies in the fact that stresses used for fatigue life prediction are determined via a vibration analysis and hence not only elastic but also inertia and damping forces are included in the model. The implementation of the technique is discussed in the case of a bladed-disc assembly where single-blade mistuning is caused by a fatigue crack. It is believed that the proposed method has promising implications for safer designs and also for the prediction of inspection intervals, especially in rotating machinery applications where such considerations are of paramount importance.

Catastrophe Theory, Curve Veering and the Vibration of Bladed Discs

The problem of vibration in bladed disc assemblies is discussed in terms of catastrophe theory and the ‘curve veering’ phenomenon. A region in the bifurcation diagram of a bladed disc where unstable vibration amplitudes could occur as a result of coupling and mistumng is identified.

Random vibrations of bladed-disk assemblies under cyclostationary excitation

Random vibration of bladed-disk assemblies is studied. The blade excitation is modeled as a nonstationary random process whose statistics vary periodically in time. To the best Of our knowledge, this is the first stochastic model that accounts for the unsteadiness of the turbulent forces applied to the blades. It is shown that a conventional approach, which treats the blade excitation as a stationary process, is likely to underestimate the response of the blades. Moreover, the amplitude of the nonstationary random part of the response is found to be comparable to the deterministic part. Consequently, it is important to model the flow around the blades as a nonstationary stochastic process.

Shaft flexibility effects on the forced response of a bladed-disk assembly

A modal analysis approach is used to study the forced response of an actual flexible bladed-disk-shaft system. Both in-plane and out-of-plane flexible deformations of the bladed-disk assembly are considered, in addition to its rigid-body translations and rotations, resulting from the bending of the supporting flexible shaft in two orthogonal planes. The effects of Coriolis forces and structural coupling between flexible and rigid disk motions on the system's response are investigated. Aerodynamic loads acting on the rotating and vibrating bladed-disk assembly are accounted for through a simple quasi-steady representation, to evaluate their influence, combined with shaft flexibility and Coriolis effects.

Probabilistic Method to Compute the Optimal Slip Load for a Mistuned Bladed Disk Assembly With Friction Dampers

In this paper, an analytical technique has been developed to compute the statistics of forced response of a mistuned bladed disk assembly with friction dampers (blade-to-blade or blade-to-ground). The method is based on the statistical linearization approach and predicts the probability distribution function of a blade’s amplitude. The validity of this technique has been corroborated by comparison with the results from numerical simulations. Using this technique, the optimal value of the slip load at the friction joint has been computed to minimize the probability that a blade’s amplitude will exceed a critical value.

Shaft flexibility effects on aeroelastic stability of a rotating bladed disk

A comprehensive study of Coriolis forces and shaft flexibility effects on the structural dynamics and aeroelastic stability of a rotating bladed-disk assembly attached to a cantilever, massless, flexible shaft is presented. Analyses were performed for an actual bladed-disk assembly, used as the first stage in the fan of the 'E3' engine. In the structural model, both in-plane and out-of-plane elastic deformation of the bladed-disk assembly were considered relative to their hub, in addition to rigid disk translations and rotations introduced by shaft flexibility. Besides structural coupling between blades (through the flexible disk), additional coupling is introduced through quasisteady aerodynamic loads. Rotational effects are accounted for throughout the work, and some mode shapes for the whole structure are presented at a selected rpm.

A higher order technique to compute the statistics of forced response of a mistuned bladed disk assembly

A higher order technique is presented in this paper to compute the probability distribution function of the forced response of a mistuned bladed disk assembly. The modal stiffness of each blade is assumed to be a random variable with a Gaussian distribution. The validity of the analytical technique is established by comparison with results from numerical simulations.

The finite dynamic annular element for the vibration analysis of variable thickness discs

According to the gyro-periodicity of dynamic displacements, the two-dimensional problems of circular plates with variable thickness are simplified into one-dimensional ones in this paper. Taking the expanded form of frequency power series of the dynamic displacement functions as the dynamic shape functions of the finite annular element, the mass and stiffness matrices as well as their one-order revised matrices are given succinctly. The dynamic method is used to analyse the vibration characteristics of a bladed disc assembly and is compared with conventional finite element method and experiment, and is proved to be superior to other numerical methods.

On Predicting the Resonant Response of Bladed Disk Assemblies

The vibratory responses of blades in gas turbine engines vary because of mistuning. An approach is developed for calculating the statistical distribution of peak resonant stresses in engine blading. It is used to predict the vibratory response of an un-shrouded fan stage. The results of the calculation compare favorably with test data for the lower frequency modes. The procedure can be used to predict fleet durability and offers a practical engineering approach for dealing with stage mistuning.

Natural frequencies of cantilever blades with resilient roots

It is difficult to enforce infinite rigidity at the root of a cantilever during vibration tests. The “cantilever” frequencies measured from a single response curve are the resonant frequencies of the entire system, including the effects of indeterminate root conditions. In some structural systems, such as bladed disk assemblies, system resonances may occur in a cluster near a blade cantilever frequency, so that it is difficult to distinguish between assembly modes and the blade cantilever frequency. It is shown that in such cases, a more accurate determination of actual cantilever frequencies may be made by computing the ratio of tip and root displacements as a function of excitation frequency.

Calculating the statistics of forced response of a mistuned bladed disk assembly

An analytical technique has been developed to calculate the statistics of the forced response of a mistuned bladed disk assembly. The modal mass and stiffness of each blade have been assumed to be random variables with Gaussian distribution. The comparison with results from numerical simulations indicate that the statistics of distributions of the blades' amplitudes predicted by the analytical technique are acceptable.

Vibration and flutter of mistuned bladed-disk assemblies

An analytical model for investigating vibration and flutter of mistuned bladed disk assemblies is presented. This model accounts for elastic, inertial and aerodynamic coupling between bending and torsional motions of each individual blade, elastic and inertial couplings between the blades and the disk, and aerodynamic coupling among the blades. The disk was modeled as a circular plate with constant thickness and each blade was represented by a twisted, slender, straight, nonuniform, elastic beam with a symmetric cross section. The elastic axis, inertia axis, and the tension axis were taken to be noncoincident and the structural warping of the section was explicitly considered. The blade aerodynamic loading in the subsonic and supersonic flow regimes was obtained from two-dimensional unsteady, cascade theories. All the possible standing wave modes of the disk and traveling wave modes of the blades were included. The equations of motion were derived by using the energy method in conjunction with the assumed mode shapes for the disk and the blades. Continuities of displacement and slope at the blade-disk junction were maintained. The equations were solved to investigate the effects of blade-disk coupling and blade frequency mistuning on vibration and flutter. Results showed that the flexibility of practical disks such as those used for current generation turbofans did not have a significant influence on either the tuned or mistuned flutter characteristics. However, the disk flexibility may have a strong influence on some of the system frequencies and on forced response.

The Interaction Between Mistuning and Friction in the Forced Response of Bladed Disk Assemblies

This paper summarizes the results of an investigation to establish the impact of mistuning on the performance and design of blade-to-blade friction dampers of the type used to control the resonant response of turbine blades in gas turbine engines. In addition, it discusses the importance of friction slip force variations on the dynamic response of shrouded fan blades.

Structural Dynamics of Rotating Bladed-Disk Assemblies Coupled with Flexible Shaft Motions

In order to analyze the dynamic behavior of the first stage compressor/fan of the 'E3' turbofan engine, a classical structural dynamics approach is employed to couple the motions of a flexible bladed disk to a rotating flexible shaft. The analysis accounts for flexible disk displacements which are transverse to the plane of rotation, and radial as well as tangential, and also accounts for rigid disk translations along, and rotations about, axes normal to the undeformed shaft axes. In the case of a wide range of E3 engine shaft flexibilities and speeds, some of the one-diametral node frequencies are shown to be affected by shaft degrees of freedom whose stiffness values are in general range of design practice. Coriolis forces are also found to significantly affect natural frequencies where strong coupling between certain modes is present.