AEROELASTICITY IN AXIAL FLOW TURBOMACHINES

von Karman Institute for Fluid Dynamics Lecture Series Programme 1998-99 AEROELASTICITY IN AXIAL FLOW TURBOMACHINES May 3-7, 1999 Rhode-Saint- Genèse Belgium STRUCTURAL DYNAMICS: BASICS OF DISK AND BLADE VIBRATION M. Imregun Mechanical Engineering Department Imperial College, London SW7 2BX, UK

STRUCTURAL DYNAMICS: BASICS OF DISK AND BLADE VIBRATION M. Imregun Mechanical Engineering Department Imperial College, London SW7 2BX, UK 1. Introduction. The first part of the lecture will deal with the basic vibration properties and characteristics of general structures. The second part of the lecture will provide a detailed account of blade, disk and bladed-disk vibration. The available analysis tools will also be discussed. More detailed information can be found in Chapters 13, 14 & 15 of the AGARD Monograph on Aeroelasticity. Some of the figures have been taken from this reference. As shown in Fig. BD-2, there are many types of stresses in a bladed assembly. The steady stresses are due to centrifugal loads, gas pressure and thermal effects. From a design point of view, these are relatively easy to deal with as well establishes methods exist for their prediction. However, generally speaking, the same is not true of alternating and unexpected stresses which may cause structural failure via some HCF mechanism. Finally, transient stresses may also become very large and need to be considered at the design stage. 2. Vibration Properties and Vibration Characteristics The consequences of vibration range from noise to total structural failure (Fig. BD-3). It is the mechanical designer s responsibility to insure the integrity of the structure, though the excitation sources may well be of aerodynamic origin. This multidisciplinary aspect makes aeroelasticity one of the more challenging subjects. The modal properties of a structure consist of natural frequencies, mode shapes and damping factors. The mass and stiffness properties determine the natural frequencies and mode shapes while the excitation levels and damping determine the actual amplitude if the vibration response (Fig. BD-4). Here one must distinguish between the vibration properties and the vibration characteristics of a structure. The structural features alone determine the vibration properties. The characteristics, such as resonance, response levels, response histories, etc, are determined by the vibration properties and external forcing (Fig. BD-5). There are two well established routes in structural dynamics: (i) Experimental route which consists of modal testing and analysis, and (ii) Theoretical route which includes many analysis techniques: tabulated data from closed-form solutions, lumped parameter models, FE method, boundary element method, etc. In structural dynamics, most techniques are based in the frequency domain. First, the structural behaviour is usually measured as a frequency response function (The data

acquisition may well be in the time domain but automated signal processing yields frequency-domain data). Therefore, it is convenient to predict frequency domain data for comparison purposes. A more important consideration is numerical feasibility. Frequency domain analyses are much cheaper than their time domain counterparts, especially when dealing with large systems (Fig. BD-7). In the frequency domain, there are three types of models: spatial model, modal model and response model (Fig. BD-8). For linear systems and simple harmonic motion, these models are interchangeable. Theoretical analyses start from a spatial model and produce a response model. Experimental analyses start from a response model and attempt to produce a spatial model. The comparison is usually made at the modal model level. 3. Basics of Blade and Disk Vibration 2.1 General Considerations The vibration of blades can be derived from those of much simpler structures, such as beams and plates. Indeed, it is easier to gain a better understanding from the study of simpler structures. The basic blade modes of vibration are flapwise, edgewise and torsional, though much more complicated patterns also exist. The first flap mode is abbreviated to 1F, the second torsion to 2T, etc. As shown in Fig. BD-0, the blade modes can be classified as: (i) Beam-like modes which can be flapwise, edgewise or torsional but not in chordwise bending, (ii) Plate-like modes where the presence of chordwise bending gives rise to complex patterns (iii) Fixture modes that arise from the boundary conditions at the root and shroud interfaces (iv) Assembly modes where disk effects may be dominant. This will be discussed later. Examples of blade and plate modes are given in Fig. BD-10. For more complicated (and realistic) blade geometries, more detailed analyses need to be undertaken. The FE techniques provide such a route. Examples are given in Fig. BD-11. 2.2 Factors Affecting Blade vibration In common with many other types of engineering analyses, the correct choice and modelling of the boundary conditions are crucial to the accuracy of the results. The blade vibration is influenced by both the root and shroud (if any) attachments. The additional flexibility provided by the root is very difficult to estimate. The inclusion of friction dampers makes the behaviour significantly non-linear. Similarly, the shroud contact mechanism is notoriously difficult to model. Again, the behaviour may be nonlinear. Other factors that affect the blade vibration are the centrifugal forces. These change both the datum position (known as untwist) of the blade and increase the stiffness. The gas bending loads also change the untwist position. Temperature effects change the

material properties, causing natural frequency variations. For some modes, centrifugal and temperature effects may be self-cancelling. 2.3 Disc Vibration Disks vibrate in nodal diameter modes which are usually double modes. A double mode arises from circular symmetry and the two constituent modes have the same natural frequency but their mode shapes are oriented differently. As shown in Fig. BD-17, such modes can occur in axial, radial or tangential directions. As one goes up in frequency, the nodal diameter modes will start exhibiting nodal circles. Therefore, in general, disks vibrate in terms of nodal diameters and nodal circles. Examples can be seen in Fig. BD-18. As shown in Fig. BD-19, disk vibration is characterised by families of nodal diameters. Each family is associated with a nodal circle. All circular components, such as cylinders, rings and shafts, exhibit the same vibration properties. If there are non-uniformities (holes, manufacturing imperfections, etc), the double modes will split into close frequency pairs. The rotation will also have an effect and this will be discussed later. 4. Basics of Bladed-disk Vibration Bladed-disk assemblies exhibit the same vibration properties as simple disks, but the relative disk/blade flexibility determines the overall characteristics (Fig. BD-22). If the disk is rigid (e.g. fan assembly), the vibration modes will be dominated by the blade modes. If the disk is flexible, both disk and blade characteristics will co-exist: at low nodal diameters, the disk will dominate; at high nodal diameters the cantilevered blade modes will dominate (Fig. BD-21). For a discrete lumped-parameter system with N blades, the maximum nodal diameter value is N/2 (N even) or (N-1)/2 (N odd). For continuous systems, higher values are possible but these become indistinguishable from the corresponding lower nodal diameter values because of spatial aliasing. This feature will be discussed later. The addition of a shroud (part-span or tip) makes the vibration characteristics even more complicated. The general behaviour is similar to that of unshrouded disks, except for the asymptotic behaviour towards the blade s cantilever frequency. As shown in fig. BD-23, continuous or non-interlocking shrouds yield markedly different dynamic behaviour. The modes with 0 and 1 nodal diameters merit special discussion because they involve a net motion of the disk s centre. Motion in a 0 nodal diameter mode involves axial and torsional motion of the disk while motion in a 1 nodal diameter mode involves a rotation about that diameter. The other modes will not see such a coupling. Therefore, an accurate prediction of 0 and 1 nodal diameter modes may require the inclusion of shaft and bearing effects. Finally, when N is even, the N/2 mode is also a special case because it consists of two split modes. In the first one, the nodal diameters go through the blade attachment points. In the second one, they pass symmetrically between the blades.

5. Sources of Excitation In gas turbines, one may encounter the following types of excitation. (i) Self-excitation or flutter. This will discussed later. (ii) General unsteadiness and random turbulence This type of excitation is much more difficult to deal with as the controlling parameters and the exact excitation mechanism are poorly understood. However, the unsteady aerodynamic forcing function is known to be composed of low-order harmonics as it is responsible for exciting low-order nodal diameter assembly modes, hence the term LEO forced response. The main characteristics of the LEO forced response can be summarised as follows. It excites low nodal diameter fundamental blade modes which exhibit higher vibration levels. Hence, the likelihood of blade failure becomes high. Industrial experience suggests that any loss of symmetry might give rise to LEO forced response. The following parameters are thought to be the most significant ones: inherent non-uniform spacing of the stator blades (or throat width variation), flow exit angle variations, axial gap changes between the rotor and stator blades, general unsteadiness through the engine, density variation due to combustion effects, blade numbers through several stages, effects due to burner blockages. (iii) Non-uniformities in the working pressure (Fig. BD-26). Blade-passing forced response is due to the excitation forces generated by the rotation of the bladed system past a pressure field, the strength of which varies periodically with angular position around the turbine. Such flow variations are mainly caused by the stator blades which act as upstream obstructions, and the rotor blades experience their wakes as time-varying forces with a frequency or periodicity based on the rotation speed. The spatial distribution of the forcing function will primarily be determined by the number of upstream stator blades and by its aliases with respect to the rotor blades. A Fourier transform of this forcing function will reveal the harmonics that will excite the assembly modes. Typically, such harmonics will excite high nodal diameter modes as their actual order is related to the blade numbers in the rotor/stator row of interest. Although it is difficult to predict the corresponding rotor blade vibration levels accurately, turbomachinery designers rely on Campbell diagrams (Fig. BD-28)) which indicate the likelihood of encountering forced response resonances of the first type within the operating range. In principle, it is then possible to design the rotor wheels away from the primary resonances, subject to being able to predict the assembly s dynamic behaviour to a required degree of accuracy. As a general rule, two conditions need to be met to excite an assembly mode. First, the excitation frequency, a multiple of the engine speed, must coincide with an assembly natural frequency. Second, the excitation pattern must match the associated nodal diameter shape.

6. Critical Vibration Modes in Aeroelasticity For fan flutter, the critical nodal diameter family is the first one, though occasionally the second family may also need to be considered. As the disk is very stiff, the assembly modes are dominated by the blade characteristics. Flutter analyses usually focus on the 1-6 forward travelling nodal diameter modes, corresponding to the blade s 1F mode. For forced response, blade passing excitation affects high nodal diameter modes. In addition, the general unsteadiness creates low engine-order excitation which affects low nodal diameter modes. In both cases, the first 4-5 families may be affected. 7. Further Reading AGARD Manual 1987 Aeroelasticity in axial-flow turbomachines (ed. PLATZER, M. F. & CARTA, F. O.). AGARD-AG-298, Vol. 2 EWINS, D. J. 1973 Vibration characteristics of bladed disk assemblies. Journal of Mechanical Engineering Sciences 15, 165-186 LALANNE, M. 1985 Vibration Problems in Jet Engines, Shock and Vibration Digest 17, 19-24 RAO, J. S. 1987 Turbomachine blade vibration. The Shock and Vibration Digest 19, 3-10