Prediction of mistuning effect of bladed disks using eigensensitivity analysis

https://doi.org/10.1016/j.engstruct.2020.110416Get rights and content

Highlights

  • Development of global design variable from grouped multivariate design modification.

  • Formulate eigensensitivities for repeated eigenvalues from vibration characteristic.

  • Numerical vibration response for realistic discrete parameter model of bladed disk.

Abstract

Vibration modes with repeated eigenvalues often occur in engineering design and analysis practices due to geometrical symmetries of structural systems such as the cyclic symmetry of bladed disk assemblies of turbomachinery. However, the degeneration of eigenvector space and the resulting discontinuities have prevented useful applications of eigensensitivities to vibration analysis of a wide class of problems with repeated modes. To overcome such difficulties, a new concept of a global design variable is developed in which all intended multivariate design modifications are grouped into a single global design variable. Eigensensitivities have then been formulated for repeated eigenvalues from which accurate predictions of vibration characteristics can be made. Further, second order eigenvalue derivatives are also employed to further improve the accuracy of predicted vibration properties. Such newly formulated eigensensitivity analysis has been effectively applied for the first time to the prediction of mistuning effect of bladed disk assemblies. Numerical results from a realistic discrete parameter model of a bladed disk have demonstrated that not only natural frequencies and mode shapes can be predicted very accurately, but also blade vibration responses under engine order excitations. In addition, the proposed eigensensitivty analysis can predict the statistical variations of blade vibrations under random mistuning. Finite element modeling and vibration testing of a practical bladed disk structure have been carried out to demonstrate the practical potential of the proposed method to be possibly integrated into finite element analysis for structural modification predictions, especially for structures with repeated eigenvalues such as bladed disks.

Introduction

Accurate and effective vibration modeling and characterization of bladed disk assemblies have been and remain to be one of the central issues of gas turbine engine design and development. Small variations of vibration characteristics of individual blades arise in practice due to manufacturing tolerances, in-service wears, as well as lack of material homogeneity, leading to inevitable mistuning of practical bladed disk assemblies. It has been well known that the dynamical response of a mistuned bladed disk can become significantly different from that of its tuned counterpart so that each individual blade may experience large stress variations within the same assembly [1]. As a result, our ability to predict vibration responses of blades given known mistuning patterns within a disk assembly becomes paramount in assessing accurately and reliably the integrity and fatigue life of engines as well as possibly controlling mistuning effect to enhance engine performance. Significant research has been carried out and considerable progress has been made to date in this important area over recent decades as witnessed by recent reviews [2], [3].

Early studies on vibrations of mistuned bladed disks have been based on simplified discrete mass spring damper models in which each blade disk segment is modeled as one, two or few lumped masses connected by springs and dampers. Such models are computationally very efficient and have been found to be able to provide decently accurate results to aid our general understanding on the effects of blade mistuning on vibration of bladed disks. These analyses allow us to identify the critical blades with the largest responses and hence stresses as well as increases in forced vibrations caused by blade mistuning [4], [5], [6], [7], [8], [9]. However, discrete lumped parameter models have limited applications due to the many approximations involved in deriving the model parameters, as well as the inadequate DOFs used to describe the dynamic behavior of practical bladed disk assemblies.

More accurate vibration analysis of bladed disks can be realized by modeling disks as plates and blades as beams whose vibration charateristics are generally known. These plates and beams are then structurally coupled based on receptance coupling [10], [11], [12], [13], [14], [15], [16], [17] to predict the overall vibration responses. These investigations often led very accurate results for a wider class of bladed disk systems but their range of applicability is still limited due to the ideal beam and plate theoretical models used for blades and disks while actual blades may have more complex geometries.

The current trend of design approach in gas turbine industry seeks to avoid any unrealistic assumptions in order to establish vibration characteristics closer to the actual behavior of bladed disks in operation, especially in relation to mistuning studies. As a results, full scale finite element modeling are routinely employed for design calculations and design modifications. However, a complete finite element (FE) model of a bladed disk may typically involve millions of degrees of freedom, making parametric studies of blade mistuning too expensive, in spite of the rapid recent advances in FE modeling and high performance computing. Accurate model reduction techniques, as well as more efficient numerical methods become essential for vibration analysis of mistuned bladed disks. Coupled vibration modes of bladed disks were computed using FE analysis and were found to be in close agreement with those from experimental modal tests [18]. Shrouded bladed disk vibration analysis was carried out and accurate natural frequencies were obtained [19]. By taking blade shear centre effects into account, vibration properties of bladed disks were established from which it was observed that the shear centre effects could be substantial under certain circumstances [20]. The problem of determining the worst mistuning patterns was formulated and solved as an mathematical optimization problem based large scale FE models of a mistuned bladed disk [21]. An effective method for direct parametric analysis of nonlinear forced vibration response of bladed disks with friction contact interfaces acting as mechanical dampers was developed with numerical results from realistic FE analysis [22]. An efficient method for analysis of nonlinear vibrations of mistuned bladed disks with frictional damping interfaces was developed through exact condensation of nonlinear FE models [23]. An integrated multi-stage bladed disk FE model was established to predict forced vibration response under synchronous engine order excitations [24]. Based on large scale FE models, a method for computing sensitivities of resonance peak frequency and forced response to nonlinear friction contact interfaces and excitations was investigated [25]. Nevertheless full scale FE analysis has been proven to be computationally exhaustive, especially in parametric studies of mistuning of bladed disks, as a result, accurate model reduction methods [26], [27] and more efficient computational methods [28], [29] have been developed to significantly reduce computational costs while maintaining required analysis accuracy.

More recently, by employing the cyclically symmetric nature of tuned bladed disk assemblies and by analyzing just a single disk-blade sector together with required boundary conditions, eigensensitivities of the overall bladed disk assembly have been developed [30]. The method is believed to be very efficient for practical vibration and eigensensitivity analyses of tuned bladed structures. However, mistunings of blades tend to destroy the cyclic symmetry, leading to practical difficulties when applied to vibration prediction of mistuned bladed disks. The eigensensitivities computed from the ideally cyclic symmetric structures are not useful in general vibration analysis of mistuded bladed disks where general variations to several blades are made.

The present work seeks to develop an effective and efficient method for vibration analysis of bladed disks using eigensensitivity analysis. Eigenvalue and eigenvector sensitivities and their applications to structural vibration analysis are not new and much work has been carried out in structural analytical model updating and structural modification prediction [31]. However, useful applications of eigensensitivity analysis to vibration analysis of bladed disks have not been conducted to date due to the simple fact that these systems are somehow degenerate in a sense that they possess many repeated eigenvalues due to their cyclic symmetry and the presence of such repeated eigenvalues renders the direct application of eigensensivity to such problems impossible. As a result, some additional development is required before eigensensitivity becomes readily applicable. Eigenvalue and eigenvector sensitvities of repeated eigenvalues have attracted great research interests in recent decades due to their important applications in vibration and control engineering. Dailey [32] presented an effective numerical method for computing eigenvector derivatives with repeated eigenvalues, together with rigorous mathematical derivations and discussions. Since Dailey’s early work, a number of further developments have been made to date, by extending the original method to more general dynamic systems such as non-self-adjoint and damped vibration systems [33], [34], [35], [36], [37], [38], [39], [40], [41]. Though theoretically elegant, these methods in their very original form are not readily applicable to structural modification predictions of general multivariate design parameter changes due to the degeneration of eigenvector space in the case of repeated eigenvalues in which different design variable leads to different base reference eigenvectors, thereby making consistent predictions an impossible task to achieve.

To overcome existing difficulties in the application of eigensensitivity analysis to vibration systems with repeated eigenvalues, a new concept of a global design variable is developed in this paper in which all intended multivariate design modifications are grouped into a single global design variable. Eigensensitivities with respect to such gobal design variable have then been formulated for repeated eigenvalues from which accurate predictions of vibration characteristics can be made. Second order eigenvalue derivatives, which become readily available in the derivation of eigenvector derivatives, are further employed to improve the accuracy of predicted vibration properties. Such newly formulated eigensensitivity analysis has been effectively applied for the first time to the prediction of mistuning effects of bladed disk assemblies. Numerical results from a realistic discrete parameter model of a bladed disk have demonstrated that not only natural frequencies and mode shapes can be predicted very accurately, but also blade vibration responses under engine order excitations. Furthermore, the proposed eigensensitivity analysis can predict the statistical variations of blade vibration characteristics in the case of random mistuning. Finite element modeling and vibration testing of a practical bladed disk structure have been carried out to demonstrate the practical potential of the proposed method as well as its possibly integration into finite element analysis for general structural modification predictions, especially for structures with repeated eigenvalues such as bladed disks.

Section snippets

Derivation of eigenvalue and eigenvector sensitivities

Consider a structural vibration system with a mass matrix M, stiffness matrix K and structural damping matrix H, which are assumed to be functions of system design variable p, the eigenvalue problem, together with the orthonormal constraints, can be written as(K+jH)X=MXΛXTMX=I

where j=-1 is the complex notation, I is a unit matrix, Λ and X are the eigenvalue and eigenvector matrices of the repeated eigenvalue λ of multiplicity m with Λ=λI. Since the eigenspace is degenerated in this case due to

Applications of eigensensitivities to prediction of structural modifications

The ability to predict accurately the effect of given known modifications to a structure becomes very important to structural operations and designs. These modifications may be imposed by external factors such as design alterations for operational reasons and in this case, it will generally be necessary to determine what changes in vibration properties will ensue from introducing the modifications as these might be detrimental, for example by moving closer to a resonance condition than applied

Eigensensitivity analysis of bladed disk assemblies

A simplified turbine bladed disk vibration model shown diagrammatically in Fig. 2 is used as an application example of the proposed eigensensitivity analysis. Flap motion of a blade is represented by the vibration of mass m on a cantilever spring of stiffness k. The mass-spring subsystem is then mounted on a frictionless trolley of mass Md, which simulates the effective mass of the blade root and section of rotor disk. The trolleys, arranged in a ring, are restrained by springs kg to earth, to

Finite element modeling

To further validate the proposed eigensensitivity analysis, finite element modeling of a bladed disk with 24 blades has been carried out. The geometrical and material properties of the bladed disk are shown in Fig. 9. Commercial finite element software ANSYS 19.2 was used and typical vibration mode shapes for the tuned case are shown in Fig. 10. The same structure was carefully fabricated and tested using Laser Dopler Vibraometer (LDV) based modal testing and analysis system and some typical

Conclusions

Eigensensitivities have found important practical applications in system identifications, control system designs, gradient-based mathematical programming and optimization, assessment of design changes on system performance, as well predictions of system responses to changes in system design parameters. However, when vibration systems with repeated eigenvalues are concerned, the eigenvector space becomes mathematically degenerate, resulting in different base eigenvectors to be defined for

CRediT authorship contribution statement

R.M. Lin: Conceptualization, Methodology, Validation. T.Y. Ng: Resources.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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