Alejandro J. Rivas-Guerra

Optimization of intentional mistuning patterns for the reduction of the forced response effects of unintentional mistuning: Formulation and assessment

The focus of the present investigation is on the use of intentional mistuning of bladed disks to reduce their sensitivity to unintentional random mistuning. The class of intentionally mistuned disks considered here is limited, for cost reasons, to arrangements of two types of blades (A and B, say). A two-step procedure is then described to optimize the arrangement of these blades around the disk to reduce the effects of unintentional mistuning. First, a pure optimization effort is undertaken to obtain the pattern(s) of the A and B blades that yields small/the smallest value of the largest amplitude of response to a given excitation in the absence of unintentional mistuning. Then, in the second step, a pattern screening technique based on a recently introduced measure of localization is used to determine which of the patterns does have a large/small sensitivity to random unintentional mistuning. In this manner, expensive Monte Carlo simulations can be eliminated. Examples of application involving both simple bladed disk models and a 17-blade industrial rotor clearly demonstrate the significant benefits of using this class of intentionally mistuned disks.

Maximum Amplification of Blade Response due to Mistuning: Localization and Mode Shape Aspects of the Worst Disks

This paper focuses on the determination and study of the maximum amplification of the steady state forced response of bladed disks due to mistuning. First, an optimization strategy is proposed in which partially mistuned bladed disks are considered as physical approximations of the worst case disk and the mistuned properties are sought to maximize the response of a specific blade. This approach is exemplified on both a reduced order model of a blisk and a single-degree-of-freedom per blade disk model an extensive parametric study of which is conducted with respect to blade-to-blade coupling, damping, and engine order. A mode shape-based formulation of the amplification factor is then developed to clarify the findings of the parametric study in the strong coupling/small damping limit. In this process, the upper bound of Whitehead is recovered for all engine orders and number of blades and the conditions under which this limit is exactly achieved or closely approached are clarified. This process also uncovers a simple yet reliable approximation of the resonant mode shapes and natural frequencies of the worst disk.

Identification of Mistuning Characteristics of Bladed Disks From Free Response Data— Part II

The focus of the present two-part investigation is on the estimation of the dynamic properties, i.e., masses, stiffnesses, natural frequencies, mode shapes and their statistical distributions, of turbomachine blades to be used in the accurate prediction of the forced response of mistuned bladed disks. As input to this process, it is assumed that the lowest natural frequencies of the blades alone have been experimentally measured, for example in a broach block test. Since the number of measurements is always less than the number of unknowns, this problem is indeterminate in nature. In this second part of the investigation, the maximum likelihood method (ML) will first be revisited and a thorough assessment of its reliability in a wide variety of conditions, including the presence of measurement noise, different distributions of blade structural properties, etc., will be conducted. Then, a technique that provides a bridge between the two identification methods introduced in Part I, i.e., the random modal stiffnesses (RMS) and maximum likelihood (ML) approaches, will be presented. This technique, termed the improved random modal stiffnesses (IRMS) method is based on the maximum likelihood concepts but yields a mistuning model similar to that of the random modal stiffnesses technique. Finally, the accuracy of the RMS, ML, and IRMS methods in predicting the forced response statistics of mistuned bladed disks will be investigated in the presence of close blade alone natural frequencies.

Local/Global Effects of Mistuning on the Forced Response of Bladed Disks

The focus of the percent investigation is on the assessment and modeling of the local (spanning only a few blades) and global (encompassing the entire disk) effects of mistiming on the forced response of bladed disks. To this end, the concept of localization is first revisited and a new measure of this effect is introduced in terms of the number of blades the mistuning of which actually affects the forced response of a central blade. Using this new metric, it is demonstrated that high responding blades typically exhibit a high level of localization and that the reverse is not necessarily true. Thus, localization is not only disk dependent but also varies from blade-to-blade on the same disk. This observation is then used to validate a partial mistiming approach to the determination of the maximum amplitude of response over the entire population of disks. The results of this study indicate that the largest amplification due to the mistuning occurs at very strong blade-to-blade coupling levels, at the contrary of a general perception, but is associated with large mistuning levels. Finally, the above phenomenological observations are used to devise a modeling technique of both local and global components of mistuning. An example of application is presented that demonstrates the high accuracy of this approach through the entire blade-to-blade coupling domain.

Maximum amplification of blade response due to mistuning in multi-degree-of-freedom blade models

This paper focuses on the determination and study of the maximum amplification of the steady state forced response of bladed disks due to mistuning. A general multi-degree-of-freedom dynamic model is adopted for each blade/disk sector and optimization techniques are used to maximize a weighted quadratic norm of the response of the degrees-of-freedom of blade 1 (overall response of blade 1). First, a mathematical optimization effort is conducted in which the resonant mistuned mode shape(s) (1 for engine orders 0 and N/2 where N is the number of blades, 2 otherwise) is selected to maximize the overall response of blade 1. The form of these optimum mode shapes is derived for all weighting matrices. The specific mode shapes are also derived for two particular weights the first one of which depends on the tuned bladed disk mass matrix and for which the overall response is akin to the kinetic energy. A closed form solution is also derived when the analysis focuses solely on the response of a specific degree-of-freedom or a specific stress component. In these cases, the ratio of the corresponding overall response to its tuned counterpart, i.e. the amplification factor, is found to be the product of two terms. The first one is an amplification obtained by tuned variations of the blade properties/mode shapes and thus is referred to as the modal amplification factor. The second term is an amplification obtained by proper mistuning. Interestingly, the modal amplification factor may take on very large values while a representative value of the largest mistuned factor is often the Whitehead limit of (1 + N)/2 as in the single-degree-of-freedom per blade model. The above formulation and results are readily extended to the optimization of the blade alone response (as opposed to blade and disk sector). Numerical optimization efforts were also undertaken on both a two-degree-of-freedom per blade disk model and a 24-blade blisk reduced order model. The results of these computational efforts not only confirm the assumptions and findings of the theoretical developments but also demonstrate that substantially larger amplification factors can be obtained with a general natural frequency mistuning as opposed to Young’s modulus mistuning. Finally, an amplification due to mistuning (no tuned amplification) slightly larger than the Whitehead limit was obtained with relative variations in blade alone frequencies less than 0.5%.Copyright

Maximum Amplification of Blade Response Due to Mistuning: Localization and Mode Shapes Aspects of the Worst Disks

This paper focuses on the determination and study of the maximum amplification of the steady state forced response of bladed disks due to mistuning. First, an optimization strategy is proposed in which partially mistuned bladed disks are considered as physical approximations of the worst case disk and the mistuned properties are sought to maximize the response of a specific blade. This approach is exemplified on both a reduced order model of a blisk and a single-degree-of-freedom per blade disk model an extensive parametric study of which is conducted with respect to blade-to-blade coupling, damping, and engine order. A mode shape-based formulation of the amplification factor is then developed to clarify the findings of the parametric study in the strong coupling/small damping limit. In this process, the upper bound of Whitehead is recovered for all engine orders and number of blades and the conditions under which this limit is exactly achieved or closely approached are clarified. This process also uncovers a simple yet reliable approximation of the resonant mode shapes and natural frequencies of the worst disk.Copyright

Optimization of Intentional Mistuning Patterns for the Reduction of the Forced Response Effects of Unintentional Mistuning: Formulation and Assessment

The focus of the present investigation is on the use of intentional mistuning of bladed disks to reduce their sensitivity to unintentional random mistuning. The class of intentionally mistuned disks considered here is limited, for cost reasons, to arrangements of two types of blades (A and B, say). A two-step procedure is then describe to optimize the arrangement of these blades around the disk to reduce the effects of unintentional mistuning. First, a pure optimization effort is undertaken to obtain the pattern(s) of the A and B blades that yields small/the smallest value of the largest amplitude of response to a given excitation in the absence of unintentional mistuning. Then, in the second step, a pattern screening technique based on a recently introduced measure of localization is used to determine which of the patterns does have a large/small sensitivity to random unintentional mistuning. In this manner, expensive Monte Carlo simulations can be eliminated. Examples of application involving both simple bladed disk models and a 17-blade industrial rotor clearly demonstrate the significant benefits of using this class of intentionally mistuned disks.Copyright

Towards a comprehensive direct prediction strategy of the effects of mistuning on the forced response of turbomachinery blades

The objective of the present article is to provide a progress report on, and highlight, some ongoing efforts regarding the available techniques for the direct (i.e. not based on Monte Carlo simulations) prediction of the distribution of the forced response of turbomachinery bladed disks that exhibit small blade‐to‐blade variations in their structural properties (random mistuning). The focus of this effort is on the statistical distributions of the amplitudes of response of a typical blade at a given frequency (level 1), of the maximum responding blade on the disk at a given frequency (level 2), and finally of the maximum responding blade on the disk over a frequency sweep (level 3). When appropriate, emphasis will be placed on the reliability of these techniques as a function of the blade‐to‐blade coupling strength.

Identification of Mistuning Characteristics of Bladed Disks From Free Response Data — Part II

The focus of the present investigation is on the estimation of the dynamic properties, i.e. masses, stiffnesses, natural frequencies, mode shapes and their statistical distributions, of turbomachine blades to be used in the accurate prediction of the forced response of mistuned bladed disks. As input to this process, it is assumed that the lowest natural frequencies of the blades alone have been experimentally measured, for example in a broach block test.Since the number of measurements is always less than the number of unknowns, this problem is indeterminate in nature. Three distinct approaches will be investigated to resolve the shortfall of data. The first one relies on the imposition of as many constraints as needed to insure a unique solution to this identification problem. Specifically, the mode shapes and modal masses of the blades are set to their design/tuned counterparts while the modal stiffnesses are varied from blade-to-blade to match the measured natural frequencies. The second approach, based on the maximum likelihood principle, yields estimates of all the structural parameters of the blades through the minimization of a specified “cost function”. Finally, the third approach provides a bridge between the first two methods being based on the second but yielding a mistuning model similar to that of the first approach. The accuracy of these three techniques in predicting the forced response of mistuned bladed disks will be assessed on simple dynamic models of the blades.Copyright