Maximum Forced Response Research Articles

АНАЛИЗ ДИНАМИЧЕСКИХ ХАРАКТЕРИСТИК ОБЛОПАЧЕННЫХ ДИСКОВ ТУРБОМАШИН С ПРЕДНАМЕРЕННОЙ РАССТРОЙКОЙ

To increase technical level of energy turbomachine in modern turbomachinery, high reliability and durability of structures are required in the design, manufacture and operation of turbomachine. Any change geometry, mass, material properties of the bladed disk of turbomachine in the design is called mistuning parameters. With a small value of mistuning blades can significantly increase amplitude, displacement or stresses of the blades structures. So, analysis influence of the effect mistuning parameters on the dynamic characteristics in the field of turbomachine is an important and urgent task. This article analyzes the effect intentional mistuning of the axial bladed disk turbomachine in order to reduce forced response due to low-order engine excitation. The maximum value forced response of rotor blades turbomachine with mistuning parameters is usually much more than that of the tuned rotors. An increase level mistuning of this critical value actually leads to a decrease magnifications of the forced response. Thus, the actual work has been introducing some degree of intentional mistuning in the design to achieve these purposes. In this paper, we study the effectiveness of intentional mistuning at the design stage bladed disk turbomachine, which is introduced into the rotor design by changing the nominal mass of the blades in harmonic Формаls.

Mistuning sensitivity and optimization for bladed disks using high-fidelity models

Abstract An effective method is developed for efficient calculations of the sensitivity of the maximum forced response levels for mistuned bladed disks with respect to blade frequency mistuning. The expressions for 1st and 2nd order sensitivity coefficients are derived in an analytical form which provides high accuracy and computational efficiency. Then, the optimization methods are used for searching the best and worst mistuning patterns of bladed disks. Two major types of the mistuning optimization problems are considered: (i) a continuous optimization problem when the blade mistuning can take any values from a prescribed range and (ii) a combinatorial optimization problem, when the set of mistuned blades is given and the optimization can be achieved by blade re-arrangement in a disk. For the first type of the optimization problem a set of sensitivity-based optimization algorithms is applied and for the second type a variant of a genetic algorithm is developed. The analysis of mistuning sensitivity coefficients and results of optimization searching are shown on an example of a realistic turbine bladed disk.

Modal Analyses of an Axial Turbine Blisk With Intentional Mistuning

The potential of intentional mistuning to reduce the maximum forced response is analyzed within the development of an axial turbine blisk for ship diesel engine turbocharger applications. The basic idea of the approach is to provide an increased aerodynamic damping level for particular engine order (EO) excitations and mode shapes without any significant distortions of the aerodynamic performance. The mistuning pattern intended to yield a mitigation of the forced response is derived from an optimization study applying genetic algorithms. Two blisk prototypes have been manufactured a first one with and another one without employing intentional mistuning. Hence, the differences regarding the real mistuning and other modal properties can be experimentally determined and evaluated as well. In addition, the experimental data basis allows for updating structural models which are well suited to compute the forced response under operational conditions. In this way, the real benefit achieved with the application of intentional mistuning is demonstrated.

Effect of coriolis force on forced response magnification of intentionally mistuned bladed disk

Blade manufacturing tolerance and wear in operation may induce mistuning, and mistuning will lead to vibration localization which will result in destruction of bladed disk. Generally, intentional mistuning has been widely investigated to control the maximum forced response. On the other hand, it should be noted that the bladed disk with high rotational speed is obviously subjected to the Coriolis force. However, the Coriolis force is not included in intentionally mistuned bladed disk in previous studies. Therefore, this paper is to study the effect of the Coriolis force on forced response magnification of intentionally mistuned bladed disk. Finite element method is used to calculate the harmonic response of the intentionally mistuned bladed disk with and without the Coriolis force. The effects of intentional mistuning strength and different integer harmonic order on the response magnification factor with the Coriolis force are discussed. It should be pointed out that, when the integer harmonic order is 1, 3 and 5, the response magnification factor with the effect of the Coriolis force increase by 3.9%, 3.53% and 3.76% respectively compared to the system of non-Coriolis force. In addition, forced response magnification factor of intentionally mistuned bladed disk with and without the Coriolis force under different rotational speed is researched in contrast. It shows that, when the rotational speed is 3000rpm, the response magnification factor with the Coriolis force increases by 0.65% compared to the system of non-Coriolis force, while the response magnification factor with the Coriolis force decreases by 6.28% compared to the system of non-Coriolis force when the rotational speed is 12000rpm.

Vibration Characteristics of a Mistuned Bladed Disk considering the Effect of Coriolis Forces

To investigate the influence of Coriolis force on vibration characteristics of mistuned bladed disk, a bladed disk with 22 blades is employed and the effects of different rotational speeds and excitation engine orders on the maximum forced response are discussed considering the effects of Coriolis forces. The results show that if there are frequency veering regions, the largest split of double natural frequencies of each modal family considering the effects of Coriolis forces appears at frequency veering region. In addition, the amplitude magnification factor considering the Coriolis effects is increased by 1.02% compared to the system without considering the Coriolis effects as the rotating speed is 3000 rpm, while the amplitude magnification factor is increased by 2.76% as the rotating speed is 10000 rpm. The results indicate that the amplitude magnification factor may be moderately enhanced with the increasing of rotating speed. Moreover, the position of the maximum forced response of bladed disk may shift from one blade to another with the increasing of the rotational speed, when the effects of Coriolis forces are considered.

Probabilistic dynamics of mistuned bladed disc systems using subset simulation

The work describes an assessment of subset simulation (SubSim) techniques to increase the computational efficiency for the predictions of probabilistic dynamic behaviour in mistuned bladed disc systems. SubSim is an adaptive stochastic procedure to efficiently compute small failure probabilities, which are expressed as a product of large conditional failures probabilities by introducing intermediate failure events. The original version of SubSim with a classical modified Markov chain Monte Carlo (MCMC) method is used in this work to generate samples related to intermediate failure events. A 2-DOFs model with lumped parameters identified from a high-fidelity finite element model is used to represent a bladed disc. The statistics associated to the maximum forced frequency response amplitudes are evaluated from different levels of the blade mistuning using stiffness perturbations of the blades. Direct Monte Carlo simulations (MCS) are used to benchmark the results from the SubSim. The proposed methodology is shown to capture efficiently the statistical properties of the mistuned blades with less than 5% samples compared to the direct MCS method. Trade-off parametric studies of the SubSim method indicate that 2000 samples at each level yield an overall good computational efficiency and accuracy for the bladed disk system considered in this work. The study confirms that SubSim techniques can be effectively used in stochastic analysis of bladed disc systems with uncertainty related to the blade configurations.

Reduction of Forced Response Levels for Bladed Disks by Mistuning: Overview of the Phenomenon

The newly revealed phenomenon of reduction of forced response levels in a mistuned bladed disk to levels significantly (e.g., by a factor of 2 and more) lower than that of its tuned counterpart is studied in detail on an example of a realistic bladed disk. Statistical properties of the amplification factor of the mistuned forced response calculated with aero-effects included have been studied for cases of random blade mistuning and for mistuned blade rearrangements. The optimization search for the best mistuning patterns providing maximum forced response reduction effect have been performed and the robustness of the optimum mistuning patterns has been demonstrated. The combined effect of the aerodynamic and structural damping on the response reduction is assessed. It is shown that the new phenomenon is of major practical significance and has to be taken into account in analysis of the forced response and design decisions.

Mistuning Forced Response Characteristics Analysis of Mistuned Bladed Disks

The problem of determining the worst-case mistuning pattern and robust maximum mistuning forced response of a mistuned bladed rotor is formulated and solved as an optimization problem. This approach is exemplified on a two-degrees-of-freedom per blade disk model, two three-degrees-of-freedom per blade disk models, and a mistuned two-stage bladed rotor. The results of the optimum search of the worst-case mistuning patterns for the lumped parameter models are analyzed, which reveals that the maximum blade forced response in a mistuned bladed disk is associated with mistuning jump, which causes strong localization of the vibration response in a particular blade. The mistuning jump-localization phenomenon has been observed for all of the numerical examples, and it is also demonstrated that the highest response was always experienced by a blade of mistuning value jump. The two- and three-degrees-of-freedom per blade disk models are also for determination of its sensitivity coefficients with respect to mistuning variation. Studies show that there is not a threshold of mistuning beyond which the maximum forced response levels off, or even drops, as the degree of mistuning is increased further. The maximum magnification factor is found to increase as the mistuning level is increased and reaches a maximum value at the upper limit of the mistuning level. The influence of the multistage coupling is revealed by comparing the results of single-stage analysis with that of the multistage case. The computed results have been compared with the Monte Carlo simulation produced, and it is demonstrated that the accuracy and efficiency of the maximum amplitude magnification factor computed by the presented method can be better than that of Monte Carlo simulations.

Forced Response of Mistuned Bladed Disks in Gas Flow: A Comparative Study of Predictions and Full-Scale Experimental Results

An integrated experimental-numerical study of forced response for a mistuned bladed disk has been performed. A full chain for the predictive forced response analysis has been developed including data exchange between the computational fluid dynamics code and a code for the prediction of the nonlinear forced response for a bladed disk. The experimental measurements are performed at a full-scale single stage test rig with excitation by aerodynamic forces from gas flow. The numerical modeling approaches and the test rig setup are discussed. Comparison of experimentally measured and predicted values of blade resonance frequencies and response levels for a mistuned bladed disk without dampers is performed. A good correspondence between frequencies at which individual blades have maximum response levels is achieved. The effects of structural damping and underplatform damper parameters on amplitudes and resonance frequencies of the bladed disk are explored. It is shown that the underplatform damper significantly reduces scatters in values of the individual blade frequencies and maximum forced response levels.

Asymptotic Description of Maximum Mistuning Amplification of Bladed Disk Forced Response

The problem of determining the maximum forced response vibration amplification that can be produced just by the addition of a small mistuning to a perfectly cyclical bladed disk still remains not completely clear. In this paper we apply a recently introduced perturbation methodology, the asymptotic mistuning model (AMM), to determine which are the key ingredients of this amplification process and to evaluate the maximum mistuning amplification factor that a given modal family with a particular distribution of tuned frequencies can exhibit. A more accurate upper bound for the maximum forced response amplification of a mistuned bladed disk is obtained from this description, and the results of the AMM are validated numerically using a simple mass-spring model.

Robust Strategies for Forced Response Reduction of Bladed Disks Based on Large Mistuning Concept

In this paper, robust maximum forced response reduction strategies based on a “large mistuning” concept are introduced, including both (i) random and (ii) deterministic approaches. An industrial bladed fan disk serves as an application example for a reliability assessment of the aforementioned strategies using two well-established tools for uncertainty analysis: (i) statistics and (ii) sensitivity and robustness. The feasibility and other practical aspects of implementing large mistuning as a means of preventing excessive forced response levels caused by random mistuning and ensuring the predictability of the response are discussed.

Minimizing Maximum Modal Force in Mistuned Bladed Disk Forced Response

Amplitude magnification is defined as the maximum forced response amplitude of any blade on a mistuned bladed disk divided by the maximum response amplitude of any blade on a tuned bladed disk over a range of engine order excitation frequencies. This paper shows that amplitude magnification can be approximated as the maximum ratio of modal force divided by modal vector magnitude in an isolated family of turbine engine bladed disk modes. An infinite linear mistuning pattern, defined by a constant interblade stiffness increment between an infinite number of blades, is found to minimize the maximum modal force when subjected to engine order N/4 excitation. Linear mistuning, an approximation of the infinite linear mistuning pattern, approximately minimizes the maximum modal force for bladed disks with a finite number of blades when subjected to engine order N/4 excitation. From this theory, 2/N is proposed to be a lower boundary for amplitude magnification. The linear mistuning method is demonstrated to produce very low amplitude magnifications numerically and experimentally. The numerical examples suggest that linear mistuning may produce amplitude magnifications near the absolute minimum possible.

Parametric study of the aeroelastic response of mistuned bladed disks

Mistuning, imperfections in the cyclical symmetry of bladed disks, is unavoidable due to many factors, including manufacturing tolerances and in-service wear and tear. Even small mistuning levels can cause large changes in the dynamics of bladed disks because mistuning destroys cyclic symmetry. In particular, mistuning can cause mode localization and a drastic increase of the maximum forced response level. To study mistuning, classical finite element analyses (FEA) approaches become computationally expensive because mistuning requires the analysis of the full disk model. Recently, compact and accurate reduced order models (ROMs) of mistuned bladed disks considering only structural coupling between sectors have been developed. However, the incorporation of unsteady aerodynamic effects into these ROMs is limited despite the fact that accurately accounting for the aerodynamic blade-to-blade coupling is essential for predicting mistuning effects in turbomachinery. Previous studies by the authors have shown that aerodynamic coupling can have important effects on the dynamics of bladed disks for subsonic flows. In particular, differences in the aerodynamic damping affect significantly the forced response amplification factor, and aeroelastic mistuned mode shapes differ significantly from the structural ones. These phenomena are accentuated in the transonic regime, where the presence of the shock creates a stronger blade-to-blade coupling than in the subsonic regime, as demonstrated herein. Parametric studies are conducted for various inflow Mach numbers and incidence angles. It is shown that the effect of aerodynamic coupling diminishes when the inflow Mach number is smaller, while the differences in the aerodynamic damping values still affect significantly the dynamics of mistuned bladed disks. Also, severe mistuned mode localizations are observed for modes having separated frequencies due to frequency veering.

Experimental Demonstration of Maximum Mistuned Bladed Disk Forced Response

A theory was previously developed for predicting robust maximum forced response in mistuned bladed disks from distortion of a structural mode. This paper describes an experiment to demonstrate the theory. A bladed disk is designed to be sufficiently sensitive to mistuning to obtain maximum response. The maximum amplitude magnification from mistuning is predicted using the theory, 1.918. The bladed disk is intentionally mistuned to obtain the maximum response, and the response to an engine order traveling wave excitation is measured. The measured amplitude magnification is in close agreement with the theory. The robustness of the maximum response is demonstrated.

Maximum Bladed Disk Forced Response From Distortion of a Structural Mode

A method is presented for obtaining maximum bladed disk forced response from distortion of a structural mode. It is shown that maximum response from mode distortion in a bladed disk occurs when the harmonic components of a distorted mode superimpose in a certain manner, causing localization of the mode and strong response in a particular blade. In addition, it is shown that the response of an intentionally mistuned system with maximum response does not change significantly when small random mistuning is added to the system. A method is described for calculating the structural mistuning necessary to obtain the distorted mode that gives maximum response. The theory is validated numerically.

Vibration localization of simplified mistuned cyclic structures undertaking external harmonic force

Critical fatigue problems often occur in mistuned cyclic structures since their forced vibration responses are often much larger than those of perfectly tuned structures. Therefore, it is of great importance to predict the forced vibration responses of mistuned cyclic structures for their safe and reliable designs. In this paper, a simplified model for mistuned cyclic structures is chosen to investigate vibration localization phenomena. The effects of mistuning, stiffness coupling, and damping on the variations of maximum forced vibration responses of the model are examined through numerical study. It is found that strong vibration localization occurs under certain relations among mistuning, stiffness coupling, and damping. It is also found that the maximum forced vibration response asymptotically increases as the number of repeated subcomponents of the cyclic structure increases.

Forced Response of Turbine Engine Bladed Disks and Sensitivity to Harmonic Mistuning

The mistuned forced response of turbine engine bladed disks is treated using harmonic perturbations in the properties of a continuous ring. A continuous shear spring is attached to the ring in which the stiffness is allowed to vary along the ring annulus. The modes of such a structure with a single harmonic mistuning pattern are shown to obey the Mathieu equation, which is solved to obtain the natural frequencies and modes of the mistuned system. The forced response of the system is then examined to determine the sensitivity of the system to small mistuning. The model is extended to include multiple harmonics, allowing for the possibility of general mistuning. An expression for the maximum amplitude magnification due to small mistuning is developed by showing that high response is caused by distortion of the structural modes. A method to intentionally mistune systems for maximum forced response is demonstrated, and numerical results demonstrate the accuracy of the analytical prediction. The intentionally mistuned system response is shown to be robust with respect to small random mistuning. Such a result might be useful for designing a test rotor for screening new bladed disk designs or for establishing the root cause of fatigue problems.

The dynamic behaviour of a finite railway under the high-speed multiple moving forces by using finite element method

Based on the equivalence between the resonant frequency, the critical moving speed, the frequency spectrum and the maximum forced vibration response of the 'finite' railway and those of the 'infinite' one, an infinite railway is replaced by a finite one so that the finite element method is available for the dynamic analysis of the 'finite' railway subjected to the high-speed multiple moving forces. To agree with the practical situations, the dynamic behaviour of the 'discontinuous' railway is also investigated in addition to the traditional 'continuous' one. The influence of the moving-load speed, the total number of moving loads and the spacing between any two adjacent moving loads on the dynamic behaviours of the continuous railway and the discontinuous railway is studied.

Maximum Resonant Response of Mistuned Bladed Disks

The turbomachinery bladed disks used in today’s advanced turbine engines must meet strict standards with regard to aeroelastic stability and forced resonant response. One structural characteristic of bladed disks that can significantly impact both of these areas is that of bladed disk mistuning. Mistuning occurs when some circumferential asymmetry exists in the bladed disk. This asymmetry can be due to such things as mass or stiffness eccentricity or slight variations in the individual blade properties and occurs in all bladed disks to a greater or lesser extent. One important structural phenomenon resulting from mistuning is the splitting of the bladed disk’s diametral modes of vibration into “twin” or “dual” modes. The presence of dual mode characteristics in a bladed disk can significantly affect either or both of its aeroelastic stability and resonant response characteristics. The present paper, expanding upon the earlier works of Tobias and Arnold [1] and of Ewins [2] addresses the prediction of the maximum resonant response of a mistuned bladed disk having closely spaced dual modes as a function of mode mistuning and modal damping. A closed form expression is derived for the maximum forced resonant response. A discussion of mistune and damping characteristics of typical turbomachinery bladed disks is also presented.