Rotor-pala sistemlerinde mil burulma-pala eğilme bağlaşık titreşimleri

Özet

Uçak ve helikopter pervaneleri, kompresörler, turbomakinalar gibi önemli uygulamalara sahip olmaları nedeniyle rotor–pala sistemlerinin dinamiği, özellikle de titreşimleri önemli bir araştırma konusudur. Anılan uygulama alanlarında prototip geliştirme ve test maliyetlerinin yüksek, hizmete alınma sonrasında ortaya çıkan hasarların sonucunun ise genellikle gerçek bir felaket olması nedeniyle bu sistemlerin çalışma koşullarında sahip olacakları titreşim karakteristiklerinin daha tasarım aşamasındayken gerçeğe uygun olarak öngörülebilmesi büyük önem taşır. Bu çalışmada, her kademesinde birden çok özdeş pala taşıyan, tek ve çok kademeli rotor–pala (mil–disk–pala) sistemleri ele alınmış ve mil burulma titreşimleriyle, palaların dönme düzlemi içerisindeki eğilme titreşimlerinin bağlaşıklığına ilişkin bir inceleme gerçekleştirilmiştir. Bu yapılırken, Euler–Bernouilli kirişi olarak göz önüne alınan palaların Galerkin yöntemi ile, burulma elastikliğine sahip milin ise sonlu elemanlar yöntemi ile modellendiği karma bir modelleme yöntemi önerilmiş ve uygulanmıştır. Yöntem, ele alınan sistemin nispeten düşük serbestlik dereceli bir modelle başarılı bir şekilde modellenmesini sağlamış ve modele ilişkin özdeğer problemi analitik olarak geliştirilerek, birbirinden bağımsız, çok düşük boyutlu, “mil burulma–pala eğilme bağlaşık modları” ve “rijid mil modları” adı verilen ve iki farklı mod şekli sınıfına karşılık gelen alt problemlere ayrılmıştır. Bu alt problemlerin incelenmesiyle bağlaşıklığın, ele alınan sistemlerin doğal frekansları ve titreşim biçimleri üzerindeki ciddi etkileri gösterilmiş ve bazı sistem parametrelerinin titreşim davranışı üzerindeki etkileri ortaya konmuştur.

Abstract

Vibrations of rotor–blade systems is an important research topic due to very important applications such as aeroplane and helicopter propellers, compressors, fans, turbo–machines, etc. In those application areas, accurate prediction of vibration characteristics is crucial in the design stage because prototyping and testing costs are exceptionally high and failure is generally disastrous. As the vibratory failures generally occur in the blades, most of the researches are concentrated on the blade vibrations. Bending and torsional vibrations of the shaft are also treated in the literature as other important sources of vibration failure. However, vibrations of different elements are generally considered separately, overlooking thus possible coupling effects, though the few works considering the coupling, pointed to serious interaction between shaft and blade vibrations. The aim of this study is to study the coupling effects between shaft torsional and blade in–plane bending vibrations in single and multi stage rotor–blade systems through an analytical approach. For this purpose, an idealized model, that consists in a torsionally elastic shaft carrying a number of rigid disks, which in turn, carry a number of identical blades modelled as cantilevered, uniform Euler–Bernouilli beams is considered. The equations of this multi body system is derived through a synthetical (elements of the system are considered first), multiframe (the motion of each element is referred to the most appropriate frame) and mixed (each element is modelled by a different method) approach. Thus, the torsionally elastic shaft is modelled by finite element method to obtain a discrete model and the blades are modelled by Galerkin’s method to obtain a similar model. The resulting equations of motion are linear in shaft’s torsional coordinates, nonlinear in blades’ bending coordinates, and include both linear and nonlinear coupling between them. A consistent small vibrations assumption leads to a fully linear model. Then, the resulting linearized equations are brought together into a single hyper–matrix–vector equation whose matrices have repetitive structures owing to the identity of the blades on each stage. The related eigenvalue analysis problem is developed analytically and splitted into two independent sub-problems corresponding to two kinds of possible normal mode motions of the system. These are referred to as coupled shaft torsion–blade bending modes and rigid shaft modes by the authors. The dimensions of the sub–problems are independent of the number of blades on each disk. This corresponds to a considerable model reduction and enables systems with high number of blades to be examined without difficulty. In the coupled shaft torsion–blade bending modes that exist in both single and multi blade systems, all the blades of the same disk have the same mode shape. The dimension of the related sub–problem equals the model degree of freedom of the shaft plus that of one blade per stage. In these modes, eigenvalue loci veering phenomena occur and this makes the coupling to have a strong effect on the vibration characteristics of the system at certain combinations of the design parameters. In the rigid shaft modes that are peculiar to multi– blade systems, the shaft behaves like a rigid body and the blades of all the disks except one are at rest. The blades of the remaining disk have similar mode shapes but their total effect on the disk vanishes. The number of these sub-problems equals the number of stages of the system and the dimension of each sub– problem equals the model degree of freedom of one blade of the related disk. In this study, a single stage rotor–blade system, which consists of a torsionally elastic shaft carrying a bladed rigid disk attached at its right end, is first considered. The variation of the natural frequencies with certain dimensionless system parameters, such as shaft rotation speed, disk radius, is examined. Then, a rotor–blade system with three identical and equally spaced stages is considered and the variation of the natural frequencies with the dimensionless rotation speed is studied. In these studies, uncoupled analysis results are also given along with the coupled ones and it is shown that eigenvalue loci veering phenomena occur between shaft torsion and blade in–plane bending modes. This makes it necessary the coupling to be considered in the eigenanalysis whenever accuracy is required in the calculations.