Fan Blade Vibrations in Synchronous Machines

Transcription

1 Fan Blade Vibrations in Synchronous Machines Oskar Jonsson Master of Science Thesis KTH School of Industrial Engineering and Management Energy Technology EGI Division of Heat and Power SE STOCKHOLM

2 Master of Science Thesis EGI 2016: EGI_ MCS EKV1162 Fan Blade Vibrations in Synchonous Machines Oskar Jonsson Approved Examiner Paul Petrie-Repar Commissioner Supervisor Nenad Glodic Contact person Abstract The design of a fan structure should be made in relation to both an aerodynamic aspect as well as a structural point of view to avoid both self-excited vibrations and forced response. If the blades start to vibrate it can cause high cycle fatigue until they break and cause severe damage. Research in the aerodynamic area is moving forward with an increase in computer power. It is however difficult and very time demanding to calculate the behavior of complex structures. Using an analytical approach on the other hand may lead to a very conservative design. This work has its focus on the structural point of view with a design of the blades towards best aerodynamic stability and to make the design procedure at ABB better towards an optimization. The theory describes the aerodynamic phenomena, structural dynamics as well as analyzing techniques to avoid high cycle fatigue. The analyzing techniques described in the theory have been applied to two different fan geometries used by ABB today. A new design procedure has been made to get a higher degree of freedom for more machine sizes as well as to save material and to get a safer structure. -2-

3 Abstrakt Designen på ett fläktblad måste vara utformat både i relation till en aerodynamisk aspekt lika mycket som till en strukturell aspekt för att undvika själv-exciterade vibrationer så väl som påtvingade vibrationer. Om bladen börjar vibrera och energi tillförs på bladen från luftflödet eller andra faktorer så att vibrationerna ökar för varje cykel kan detta leda till att bladen bryts av och orsakar stora skador. Forskning inom det aerodynamiska området går framåt i förhållande till ökad datorkraft. Ändå är det svårt och väldigt tidskrävande att räkna ut beteendet på komplicerade strukturer. Med hjälp av en analytisk tillämpning kan man dock hitta en konservativ design. Detta arbete har fokus på den strukturella aspekten med bästa möjliga aerodynamiska stabilitet och att kolla på ABB s design för bästa optimering. Teorin beskriver det aerodynamiska och strukturella fenomenet men också olika tekniker för hur man undviker utmattning av bladen. Analysteknikerna som är beskrivna i teorin är applicerade på två olika fläktgeometrier som används av ABB idag. En ny design har tagits fram som kan ge större möjligheter. -3-

4 Table of contents Abstract Introduction Thesis background Objective Aeroelasticity Forced response Flutter Damping Fluid mechanics Design of fan Calculation of mean line D Current design method Reduced frequency parameter Structural Dynamics Campbell diagram Effects on a bladed disk Mode shapes ZZENF diagram Approach Ansys Results Eigenfrequencies Campbell diagram Traveling wave pattern Design of new fan geometry Wing stress Final conclusion and future work Bibliography

5 Table of figures Figure 1: CAD model of an fan geometry Figure 2: CAD model of an fan geometry Figure 3: Rotor with fans Figure 4: Installed synchronous machine Figure 5: Velocity triangles over blade row Figure 6: Mean line according to velocity triangles Figure 7: Parameters to be included when calculating airfoil geometries Figure 8: Coordinates for mean line deviation Figure 9: Design parameters of a blade for equation Figure 10: A visualization of the reduced frequency Figure 11: Visualization of a Campbell diagram Figure 12: Effects due to centrifugal forces Figure 13: Gyroscopic effect on a disk ( 21 Figure 14: Bending modes for a beam Figure 15: Different mode families for every nodal diameter Figure 16: ZZENF diagram, Zig Zag-shaped excitation line in the nodal diameter versus frequency diagram Figure 17: Mesh size on the blisk Figure 19: First three eigenmodes of the fan geometry Figure 20: Different frequencies for different thickness of blade Figure 21: Modal analysis for 10 mm blade Figure 22: First two eigenmodes of fan geometry 1 with engine order line of interest Figure 23: First two eigenmodes for an fan geometry 2with engine orders of interest Figure 24: Campbell diagram for fan geometry 1 with engine order related to bindings Figure 25: Campbell diagram for fan geometry 2 with engine order related to bindings Figure 26: Travelling wave mode of fan geometry Figure 27: Travelling wave mode of fan geometry Figure 28: Families and nodal diameters for a 14 bladed blisk (fan geometry 1) Figure 29: Mode shapes and nodal diameters for an 18 bladed blisk (fan geometry 2) Figure 30: ZZENF diagram of an fan geometry Figure 31: ZZENF of fan geometry Figure 32: Flow chart for design process Figure 33: Static structural to modal analysis Figure 34: Setup in Mechanical

6 Nomenclature Blisk Tuned disk HCF Disk and blades in an assemble All the blades on the blisk is identical High Cycle Fatigue, structural failure due to a lot of cycles. Ω Rotational speed [Rad/s] ND N n δδ C W Nodal diameter Number of blades Number of nodal diameter Deviation angle Absolute velocity Relative velocity U Rotational speed [rpm] Q q Design flow [m 3 /s] A λλ vv bb Flow area Wing stress Approximation of the eigenfrequency -6-

7 1 Introduction A synchronous machine can be used both as generator and machine. ABB produces large synchronous AC machines in sizes from 3-60 MW/MVA with a design that is customized for each costumers needs. 1.1 Thesis background A synchronous machine needs cooling to maintain a temperature that corresponds to a good efficiency at operating speed. Cooling is done by an air flow with two axial fans which are located at the ends of the rotor. Air flow passes through the fans into the rotor and cools down the poles. The air is then pushed out from the rotor up to a heat exchanger that cools down the air flow and before it goes to the fans again. A potential problem with dimensioning of fans is that the fan blades may break due to high cycle fatigue caused by vibration and cause severe damage when a blade breaks at the wedges and is tossed at the stator of the machine. High cycle fatigue is mainly due to one of two different aeroelastic phenomena, forced response or flutter. Forced response is caused by physical objects that influence the airstream in a way that will make the airstream affect the fan in an interaction with the structures eigenfrequency. Flutter is caused by self-excited vibrations due to an interaction between the fluid and the structure. The main reason to high cycle fatigue when flutter occurs is when the structure receives energy from the flow and the amplitude of the oscillation grows bigger until the structure break. The design criteria today at ABB for the fan blades is the reduced frequency parameter (sometimes also called wing stress), which was the most important parameter for identifying flutter when computers were not commonly used for fluid dynamics simulations. The calculations are based on the peripheral speed, wing bending frequency and the wing chord at the inner chord length at the hub. An order specific dimensioning of the fan geometry is made for each machine. Two different sizes of fans are investigated with different geometries on the blades and diameter on the hub. Figure 1: CAD model of smaller fan. The smallest size of the fan is presented in figure 1 above and has an operational speed of 1500 RPM. -7-

8 Figure 2: CAD model of larger fan. The largest fan is presented in figure 2 above and has the operational speed of 1800 rpm. Figure 3: Rotor with fans. The fan blades are installed on a rotor to be able to cool down the generating part in the machine, see figure

9 Figure 4: Installed synchronous machine. The rotor is installed in the casing, see figure 4. The cooling air in the machine is in a closed system where the top part in figure 4 is a heat exchanger where the air is cooled down and circulated back down to the rotor again. 1.2 Objective ABB wants to get better understanding of their design procedure for fans. The design criteria that is used have its focus on getting a high enough eigenfrequency of the structure. There is also a need to get better knowledge about the aerodynamic point of view for the fan blade. How it is designed and how to check if it gets effected by the aerodynamic influences. 2 Aeroelasticity An interaction between three forces acting on a structure or structures that are exposed to an airstream is called aeroelasticity. The different forces included are the inertial, elastic and the aerodynamic forces. Design of a blade is limited by these forces and how they cause vibration. Some of the different phenomena of aeroelasticity can be listed as static aeroelasticity, flutter and forced response. The static aeroelastic phenomena handles the calculation of the blade running shape and takes into account the design geometry and flow conditions. The blades tend to deform elastically under for example centrifugal loads which can make the blade twist and large blades usually have a quite large bending displacement. Another effect in static aeroelasticity is the divergence which will occur if the structure is not stiff enough and will then deflect and can in that case break if it is bad enough (Marshall et al, 1996). A lot of parameters influence the aeroelastic phenomena and a small change in some of them can make a huge difference. (Srinivasan. 1997) did set up a long list with all the parameters, see table 1, that -9-

10 influences the vibrations by aeroelasticity and it can be seen that it is almost impossible to account for all of them. There are although a few of them that is considered to be the more important. Table 1 Aeroelastic parameters Blade characteristics: Stiffness Mass ratio Mode shape (frequency, position of torsion axis, phase angle between bending and torsion, bending direction,) Shroud location Shroud angle Blade attachment Mechanical damping (in blade, at snubbers, under platform dampers) Elastic coupling between blades Structural mistuning (in frequency, damping) Centrifugal softening/stiffening Untwisting (because of centrifugal effects) Steady stress (rotational tip speed) Blade row geometry: Blade form (thickness/chord-ratio) Blade twist Hub/tip-ratio Aspect ratio Stagger angle Flow conditions: Inlet and outlet flow velocities Incidence angle Blade loading Blade surface pressure distribution Separation Transition Shock position Shock motion Incoming velocity defects Incoming pressure defects Aerodynamic mistuning (in blade shape, pitch, stagger) Absolute value of inlet and exit conditions (pressures, temperatures) General: Reduced frequency Interblade phase angle The mass ratio blade/air is considered very large if the blade is consistently steel and therefore is the time dependent aerodynamic forces too small to influence the natural mode shapes which means that it is not that common that self-excited vibrations occur at an engine order line in the Campbell diagram. The first parameter to consider is the blade vibration natural frequency, the higher natural frequency the less likely it is that the problem is due to flutter. The torsional axis of the blade is the thicker part of the blade and is considered one of the most important parameters. Each center of the torsional axis has a different aerodynamic damping. If an area has a positive aerodynamic damping and a torsional axis is located here it gives a damped vibration and the other way around for a negative aerodynamic work. If the mode shape is between a bending or torsion is it important to consider the phase angle between the bending and torsion vibration. If the phase angle is 0 it gives an excitation and if it is

11 it gives a damped motion. The most important parameter is said to be the interblade phase angle when investigating self-excited vibrations and is the difference in vibration of a blade and the blade neighbor blades. The difference is the time-delay angle because of a wave around the disc for all the blades (Torsten H. Fransson). 2.1 Forced response Forced response comes from when the rotating blades pass through flow defects from an interaction by upstream and downstream blade-rows that will cause large unsteady aerodynamic forces and can cause vibrations. Forced response comes from an interaction from something else than just the airstream and the blade (Marshall et al, 1996). Every time a blade passes this disturbance it creates an excitation whose frequency is a multiple of the fan speed. If this frequency is close to one of the blades eigenmodes the blade will oscillate and the amplitudes can grow significantly (Victor Guillard). If there is a crossing between the operational speed and the engine order line in the Campbell diagram explained in chapter 4.3 it is a chance that a forced response will excite the vibrations and cause high cycle fatigue (Mayorca. M. A). 2.2 Flutter Flutter is a self-excited vibration due to and interaction between the structure and the fluid where the fluid gives energy to the structure and can be expressed as an uninterrupted oscillation occurring by an interaction between aerodynamic forces, elastic response and inertial forces. One main reason for flutter is when the airstream velocity is higher than the critical air stream velocity. If the structure to fluid mass ratio is low as if the structure is a wing for example, then flutter tends to occur as a result from coupling between torsional and bending modes. If that ratio is high as in turbo machinery, then the flutter tends to occur from a single-mode phenomenon as the aerodynamic forces cannot cause modal coupling. The aerodynamic forces are much smaller than the inertial and the stiffness forces in this case (Marshall et al, 1996). The time delay between the vibrating structure and the unsteady aerodynamical forces is named the force face angle. Depending on the force face angle, the structure will either absorb energy from the flow or give energy to the flow. It can also continue neutral in relation to the vibration cycle and if the structure gives energy to the flow the vibrations will be damped for every cycle (Vogt, D.M., 2013) To prevent flutter on blades, shrouds were introduced to give an elastic support at part-span. Shrouds are installed on the blade and protuberances on every individual blade to make the blades stiffer. To get a maximum stability should the shrouds be installed at two-thirds of the distance from the blade root to the tip. In time will the shrouds wear and this can lead to a change in dynamic characteristics due to less tightness of contact at interface (Marshall et al, 1996) Damping There are several kinds of damping in a system. The viscous/ visco-elastic damping is associated with bodies moving in fluid. Friction damping is usually related to when a body is sliding on a dry surface and structural damping comes from frictions within the material. The aerodynamic damping is almost the same as the viscous damping with the difference that this can become negative. If the aerodynamic damping becomes negative, it will lead to self-excited vibration in form if flutter. (Vogt, D.M., 2013) -11-

12 3 Fluid mechanics The fundamental laws of fluid dynamics are used in a turbo machinery frame of reference to calculate the behavior of the flow. These equations are the core for computational fluid dynamics (CFD) which is a method for calculating how a fluid will act over different geometries. CFD is based on the governing equations of fluid dynamics which represents the conservation laws of physics. (Teodora, P, M, 2009) derives these equations where the following physic laws are adopted. Mass is conserved for the fluid. Newton s second law, the rate of change of momentum equals the sum of all forces acting on the fluid. First law of thermodynamics, the rate of change of energy equals the sum rate of heat addition to the fluid and the rate of work done on the fluid. The result is equation 1 and is the Navier-Stokes system of equations for incompressible flow and is based on four equations with four unknown parameters in a closed system where c is the velocity, t is time, P is pressure, ρρ the density, µ the viscosity and g the gravitation. cc jj ρρ t + cc jj = P + μμ 2 cc ii xx ii xx 2 jj xx + ρρg jj ii c ii xx ii = 0 The Navier-Stokes equation must however be solved in a rotation system to get the accurate flow behavior. When writing the Navier-Stoke system in a rotating frame of reference will two more forces appear to conserve the momentum balance and are the Coriolis force and the centrifugal force. (Teodora, P, M, 2009) derives the equations with the assumptions that the flow is stationary, neglecting molecular moment transport and considering constant angular velocity. This will lead to equation 2 and is related to the energy equation where c is the absolute velocity, u is the circumferential velocity and w is the relative velocity. WW = 1 2 [(cc 1 2 cc 2 2 ) + (uu 1 2 uu 2 2 ) + (ww 2 2 ww 1 2 )] This is the Euler equation and will give a negative sign for pumps and positive for turbine. The first term, 1 (cc cc 2 2 ) relates to a change in absolute kinetic energy of fluid going from entrance to exit section in for example a fan passage and is related to the pressure rise over the passage. The kinetic energy is then transferred from the blades to the flow according to a fan or pump. 1 [(uu uu 2 2 ) + (ww 2 2 ww 2 1 )] is related to a pressure rise due to the rotor itself where the first part is from the centrifugal effect and the last is due to a change in relative kinetic energy. If in for example a fan u 1 is equal to u 2 because the flow particles will enter and leave the rotor at same radius. Bernoulli s equation describes the static pressure difference for a particle when it has a change in radial direction. If a particle is moving towards for example the hub and changes direction against axial direction, the particle will accelerate and the static pressure will be reduced. The particle will rise in static pressure and decelerate if the particle is deflected away from the hub. This is the reason for the lift force on an airfoil. Bernoulli s equation is not valid near a surface where the velocity of the air flow will decrease the closer to the surface the air particle gets and is zero at the surface. This is due to neglecting viscosity in Bernoulli s equation. The layer near the surface is called a boundary layer where the thickness is defined to start where the velocity of the flow is 99 % of the main flow. The boundary layer thickness grows with distance from the moment the airflow touches the surface due to the acceleration or deceleration as Bernoulli s (1) (2) -12-

13 equation explains. The larger boundary layer the more losses there will be due to a pressure gradient. An acceleration of the flow over a blade will keep the boundary layer thinner and reduce losses. One of the major problems in fluid mechanics is the so called flow separation where the flow decreases in velocity and forces the air particles close to the surface to go towards the flow and will therefore lift from the surface which will cause turbulence and large losses (Piolenc M., Wright, G. E,1996). The flow in turbo machines is mostly turbulent and requires a mesh in the scale of grid to be able to solve the time-dependent Navier-Stokes equations of the fully turbulent flows at high Reynolds numbers which requires high computational power. One method to be used is done by closing the system and includes two more equations, the turbulent kinetic energy and the rate of dissipation of the turbulent energy, the k-ε model. 3.1 Design of fan There are several different methods developed when designing the flow path to get the best aerodynamic performance in turbo machinery. There are although three general basic features to be followed. First is the one-dimensional design use to get the mean line design. Then the two and threedimensional methods to include the blade and vane definition based on the potential flow analysis. Last is the advanced viscous 3D calculations based on Reynolds equations that includes complex structural calculations. The fan to be designed should be based on max efficiency corresponding to the design flow Calculation of mean line The design procedure when the chord has been chosen is done by deciding the velocity triangles. The power given from the blade to the fluid is the mass flow multiplied with the rotational speed and the change in absolute velocity. The velocity components can be explained by two vectors, the relative velocity vector and the absolute velocity vector. When the fluid enters the rotor blades it has the absolute velocity C 1 and the angle αα 1 away from axial direction. The relative velocity W 1 has the angle ββ 1 and can be explained as the velocity that an observer sees while sitting on the rotor. When the air flow leaves the rotor it will have the absolute velocity C 2 with angle αα 2 and the relative velocity W 2 and angle ββ 2 and U is the rotational speed of the rotor, figure 5 (Cumpsty, N.A. 2004). Figure 5: Velocity triangles over blade row. -13-

14 In the case of an axial fan as mentioned in section 1.2 the flow particles are leaving and entering the blades at the same radial section, therefore u 1 =u 2 =u. Now Euler s equation 2 states that the total pressure difference is made by a static and a dynamic pressure. Figure 6: Mean line according to velocity triangles. As the inlet has an axial component w u1 that is equal to u and u is equal to the rotational speed, the pressure difference from Euler s equation can be written as equation 3 where C u2 is equal to CC θθ2. pp = ρuucc uu2 (3) The inlet and outlet angles can be expressed as: tanγγ 1 = ww mm = cc 1 ww uu1 uu tanγγ 2 = ww mm ww uu2 The fan blades in this thesis are to be design for the design flow to cool engines and needs therefore a specific design flow, Q q. As the relative velocity is the same throughout the passage and is equal to the ratio of the flow rate to the flow area can the inlet angle be expressed as equation 6 because the flow area is, AA = ππ(rr tt 2 rr h 2 ) where t is the tip radius, h the hub radius and n the rotational speed. γγ 1 = QQ dd ππ(rr 2 tt rr 2 h ) 1 2ππππππ For the outlet angle can Euler s equation be derived in a pressure frame of reference and the maximum flow can then be expressed as equation 7. QQ mmmmmm = 2ππππππππππππππγγ 2 (7) The maximum flow is then obtained when the outlet angle is 90. This value is radius dependent and should only be on hub radius because 90 will lead to the highest possible load on the blade. If the pressure difference over the blade is written in terms of loading and u=2πrnρ will equation 7 above instead be equation 8. (4) (5) (6) -14-

15 pp = ρuucc uu2 = 2ππππππππcc uu2 (8) 2πnρ is constant in radial direction and can be named as K, therefore: cc uu2 = KK rr C u2 is the swirl velocity and is an expression of losses and for a good design this component should be minimized (Teodora, P, M, 2009). (9) Figure 7: Parameters to be included when calculating airfoil geometries. Table 2 Cascade parameters. All parameters for the geometry of the airfoil are shown in figure 7 and table 2 (Falck F., 2008). The incidence angle for a given airfoil is the difference between the inlet blade angle and the flow angle achieved at the inlet section, ii = γγ 1 ββ 1. The deviation angle is the difference in flow angle and the blade angle at the outlet section, δδ = γγ 2 ββ 2. For a chock free inlet angle the incidence angle should be, i=0. (Howell, A. R.1980) developed a method for how to calculate the deviation according to a shock free inlet and the channels proportions according to equation 10. δδ = mmmm σσ 1 2 σσ is as shown in table 2 the cascade solidity and is the ratio between the length of the blade chord and the spacing between the blades (10)

16 σσ = ll tt (11) θθ = γγ 1 γγ 2 is the camber angle and m is calculated by equation 12. mm = γγ (McKenzie, A. B. 1980) presents a suggestion on an empirical solution for the calculation of the deviation angle by equation 13. δδ = ( θθ)σσ 1 3 The usual design of an air foil has the shape of an ellipse at the trailing edge and going towards max thickness and after a curvature going towards a wedge like shape. It is possible to maintain large accelerations of the flow with a small radius of the leading edge but there is then a risk for separations from the leading edge nose. A useful airfoil section for fan blade has in general a quite large leading edge radius but a cambered plate is an exception. There are a lot of airfoil families that has been developed with a specific aim. The incidence angle has a large influence of the lifting force where an increase in the incidence angle will accelerate the velocity on the top side of the airfoil and the other way around for the lower side of the airfoil. This will according to Bernoulli s equation give a rise in pressure on the lower side and a decrease on the top side and the pressure difference will lead to the lift force. LL (14) CC LL = 1 2 ρρuu 0 2 AA The overall lift force in equation 14 over an airfoil is a function of the incidence angle and Reynolds number (Piolenc M., Wright, G. E,1996). However, (Teodora, P, M, 2009) show that at low pressures raises, around a couple of thousands Pa is a variable thickness along the blade span is unnecessary since measures of two blades with the same overall parameters other than that one had an airfoil profile and the other a constant thickness, shows exactly the same pressure distribution. To make a fan as efficient as possible there are some major parameters to consider. One of them is the ratio between the swirl velocity and the axial velocity component at a given radius. The swirl is explained as a measure of the torque of the rotor. The flow coefficient is the ratio between the axial component and the rotational rotor speed at a given radius and is another important parameter when designing the blade. The design is almost only a function of these two coefficients. The flow and swirl usually varies along the blade span and these two coefficients are often obtained by assuming the axial velocity component to be constant everywhere. Another assumption is to design the swirl velocity inversely proportional to the radius. This gives the free vortex flow. The vortex flow is described by the circulatory flow about an axis and there are at least two different vortices in a fan. One of them is when the fan rotates and the other when the flow that is passing through a blade stage and the third is due to for example boundary layer effects. If a particle is exposed to a centrifugal force in radial direction the particle will move in that one direction. In fans there are more forces acting to counter the centrifugal force to have particles remain in the axial direction. (R.a. Wallis) derives the vortex flow as equation 15 and is a key for establish an arbitrary vortex flow. (12) (13) dddd dddd = ρρωω2 rr + 1 dduu2 ρρ 2 dddd + 1 dd(ωωωω)2 ρρ 2 dddd (15) -16-

17 The free vortex condition ensures the radial displacement of the flow for the design condition, due to the balance of the centrifugal forces. It is possible to increase the capacity if the free vortex is increased towards the tip (Piolenc M., Wright, G. E, 1996). It is proven by means of stream line analysis that a parabolic distribution of the mean line is appropriate for axial fans in a way of equation 16 in relation to figure 8. Figure 8: Coordinates for mean line deviation. γγ(yy) = AAyy 2 + BBBB + CC (16) The geometry is iterated until the geometry is correct with the axial chords and in relation to flow parameters. Both the inlet and outlet angles for the mean live are calculated since before for the given design flow D calculations To make the energy exchange between the blade and the fluid equal at different radius the product of Euler s turbine equation, U (C 2 -C 1 ) should be constant with increase in radius which will lead to a twist of the blade. U is the speed of the blade and will change in radial direction and gives the idea to the change of (C 2 -C 1 ) until the product is equal over the blade height (Alvarez, H. 2010). The pressure produced by the fan is proportional to the number of blades times blade width, hence the same pressure can be produced by doubling the number of blades and making the width half the size and the other way around. One consideration is that fewer and wider blades will result in a better fan efficiency and lower noise level but also a bulkier hub that will become harder to balance. The blades should not overlap each other to choke the airflow. This can usually be calculated if the width of the blade is according to equation 17. LL < dd/nn bb (17) d is the hub diameter and n b is the number of blades (BASF Corporation, 2003). 4 Current design method With the first design of the jet engine, one of the main parameter established were the reduced velocity and is in fact the inverse to reduced frequency. If there is a disturbance at a point on a body and this -17-

18 disturbance oscillates together with the body, the air that is influenced by the disturbance will move downstream with a mean velocity of the air relative to the blade as U. One can also say that the reduced frequency is characterized by the way a disturbance is felt on other points of a body. Every point on a body disturbs the flow if the body is oscillating, the reduced frequency can also be explained as the mutual influence between the motions at various points of the body (C. Fung 2002). To see if the ABB fan blades can withstand the forces and not break in the wedges the so called wing stress is calculated. The wing stress is in fact the reduced velocity and is in turn the inverse to the commercial used reduced frequency parameter. The wing stress parameter is defined in equation 18 below where U is the peripheral speed at the blade inner chord, C vi is the chord length at the hub and v b is the eigenfrequency of the structure. UU λλ = 2222CC vvvv vv bb (18) There are a number of hub diameters to be selected according the specific machine of interest. When the hub diameter is chosen the height of the blade is an easy choice in relation to the height to the windings around the fan. Number of blades on the rotor is specified according the size of the fan. If the blades that are modeled have a low eigenfrequency the chord of the blade at the inner diameter is enlarged to raise the eigenfrequency. The eigenfrequency of the structure is calculated by equation 19. C vi is the chord diameter at the shroud, R m is the bending radius of the blade, h v is the height of the blade, p h is the width accounting for the bending from leading edge to trailing edge and the width is measured in the middle of the chord and x,p and q are constants, figure 9. Figure 9: Design parameters of a blade for equation 19. vv bb = xx vv ii RR mm hh pp + qq pp 22 hh vv CCvv ii (19) -18-

19 4.1 Reduced frequency parameter The reduced frequency parameter is a parameter that establishes if the air stream is stable and can be seen as a first indicator of flutter margin, often used as a stability key parameter. (Srinivasan A. V., 1997) explains the reduced frequency parameter as the ratio of time taken for a fluid particle to travel past the length of a half a chord to the time taken for the blade to execute one cycle of vibration. Consider a blade chord and let a fluid particle travel across the chord. The fluid particle will then continue to travel in a wave shaped path after the chord, related to the structure and the geometry of the chord. If the length of a chord is set to be 2b and it is oscillating with a frequency of ω=2π/t. The chord is exposed to a stream with a velocity of V and the wavelength after the chord will be λ=vt=2πv/ω, ω is the frequency for which the blade is oscillating. When dividing the chord length with the wavelength is k/π obtained. At low values of k the wavelength will be large in relation to the chord length while at high k-values the wavelength will be short in relation to the chord length. The meaning of this is that the smaller the k-value is the steadier the effects is, figure 10. Figure 10: A visualization of the reduced frequency. 4.2 Structural Dynamics Every structure has an eigenfrequency and the eigenfrequency of a blade is associated with the mode shape of the blade, or eigenmode. These eigenmodes are determined by a free response analysis and numerically is done by solve the dynamic equation of motion but only look at the structural mass and stiffness constant. The vibration of a body or a structure can be described by the equation of motion 20. mmxx + ccxx + kkkk = FF(tt) (20) In equation 20 above m is the mass, c is the damping and k is the stiffness. When there are forced vibrations the time varying force on the right hand side will be set to be finite while otherwise it is set to zero and is thereby representing the free vibration case (natural vibration). When going through a modal analysis to find the natural frequency it is done by the following steps where mmxx IIIIIIIIIIIIIIII ffffffffff, ccxx DDDDDDDDDDDDDD ffffffffff and kkkk SSSSSSSSSSSSSSSSSS ffffffffff. Assuming a complex solution where; xx(tt) = xx ee λλλλ ee λλλλ mmmmmmmmmm haaaaaaaaaaaaaa rrrrrrrrrrrrrrrr, xx (tt) = λλxx ee λλλλ = iiiixx ee iiiiii ; λλ = iiii and xx (tt) = λλ 2 xx ee λλλλ = ωω 2 xx ee iiiiii Rewrite the equation of motion from the time domain to the frequency domain gives ( ωω 2 mm + iiiiii + kk)xx ee iiiiii = 0 ; xx is the eigenvector which decides the mode shape and how the structure will oscillate. (21), (Vogt, D.M., 2013). xx ee iiiiii Cannot be zero and therefore; ( ωω 2 mm + iiiiii + kk) = 0. When calculating the natural frequency the damping is zero and therefore will the natural frequency be; ( ωω 2 mm + kk) = 0 and therefore. -19-

20 (21) ωω = kk mm 4.3 Campbell diagram The Campbell diagram is a good tool to investigate vibration phenomena s. On the y-axis in figure 11 below the blades eigenfrequencies is found and on the x-axis the rotational speed is plotted. If the excitation frequencies coincide with the eigenfrequency there is a possibility for a resonance condition. As can be seen in figure 11 the engine order is plotted which is the rotational speed multiplied with an integer. The engine order of a blade is the perturbation the blade gets when it rotates 360. Figure 11: Visualization of a Campbell diagram. When looking at the Campbell diagram is the lowest modest often the most important to look at because they get excited by low engine orders and experience high excitation forces (Mayorca. M. A). 4.4 Effects on a bladed disk When a bladed disk is rotating with high speed are there two different effects counteracting each other. The stiffening of the structure caused by centrifugal forces. Figure 12: Effects due to centrifugal forces. -20-

21 and spin softening which is due to a change in distance from the center of rotation to different points of the structure when the structure vibrates. If the stiffening effects are larger than the softening the eigenfrequency will be smaller with an increase in rotational speed and the other way around if the softening forces are larger as can be seen in figure 12 above (Myhre, M,. 2003). Figure 13: Gyroscopic effect on a disk ( The gyroscopic effects are neglected if the assembly doesn t include a shaft along x-axis. The shaft can then make a displacement for the blisk in x-axis in relation to rotational centrum, figure Mode shapes Mode shapes or eigenmodes can be considered to be blade-alone or disk-alone mode shapes, or a combination of both. In a simulation for a given blisk the eigenfrequencies should be the same amount as the number of nodal diameters at the mode that is being excited. Eigenmodes for the total bladed disk are called mode families and refer to a group of bladed disk modes that respond with the same blade mode but with different nodal diameters. If the frequency of the bladed disk does not change with nodal diameter the mode is called blade dominant where the disk is very stiff. If the bladed disk frequency is highly affected by the nodal diameters it is called a disk dominant mode. How the frequency is varying with the nodal diameter becomes very important when determining the resonance crossings in the Campbell diagram. The names of the mode shapes are referred to a beam and the number of inflection lines on the beam. Bending or flexing (F), Torsion (T) and edge wise bending (E) as can be seen in figure 14 below. -21-

22 Figure 14: Bending modes for a beam. The bending mode of the whole disk is called nodal diameter as the inflection lines lay across the whole disk. The bladed disk modes are specified, first by the disk mode and then the blade modes and are named like 1F-2ND which refers to first flexion - second nodal diameter. Max number of nodal diameter is calculated by the number of blades divided by two if there are an even number of blades or number of blades minus one divided by two if the number of blades are un even. When the nodal diameter is rotating along the disk there will be blades that aren t moving for the moment, this phenomenon is called travelling wave modes or inter blade phase angle as will be mentioned in section 3.0. Here there will be a difference between the blade to blade bending that is called the inter blade phase angle where the angle is referred to the difference in deviation from one blade to the other. The travelling wave mode is calculated by using equation 22 below (Mayorca. M. A). σσ = 22 ππ NNNN NN In figure 15 below a schematic picture of a plot for different modes with different nodal diameter is shown. These are the different families for a bladed disk. (22) -22-

23 Figure 15: Different mode families for every nodal diameter. A mode family that has a close eigenfrequency to another mode family as demonstrated in figure 15 is called frequency veering. This can lead to a combination of high coupling strength and high modal density and can be critical for the mistuned response behavior. The higher the nodal diameter, the closer to the blade alone eigenfrequency it gets. 4.5 ZZENF diagram ZZENF stand for Zig Zag excitation line in the nodal diameters versus frequency and is named by the look of the excitation lines in the diagram, see figure 16. The blades on the disk are exerted by a concentrated force that is rotating with the rotational speed and if the force is concentrated opposite one blade the strength of the force will be fully transferred to the blade (Wildheim, S, J, 1980). -23-

24 Figure 16: ZZENF diagram, Zig Zag-shaped excitation line in the nodal diameter versus frequency diagram. (Wildheim, S, J, 1980) also derives how the displacement of the blades vary over the blade span and the resulting equation to be plotted in a frequency versus nodal diameter is as. ωω nn = (kkkk ± nn)ω kk = 0,1,2, (23) When k=0 the plot will result in a forward traveling wave where the wave follows the force. The plot in figure 16 comes from when k=1 and ωω nn = (NN nn)ω which leads to a backward travelling wave mode. The third line is k=1 as well but ωω nn = (NN + nn)ω and results in the next forward traveling wave mode. It can be a good idea to study the ZZENF diagram for the rotational speed from 0 to operational speed if problem occurs in the startup or the shutdown of the engine. The ZZENF diagram shows all the nodal diameter of interest with respect to travelling wave modes. 5 Approach The different CAD models that were analyzed are available in figure 1 and figure 2. The different geometries were then put into ANSYS to be able to analyze how the different eigenfrequencies of the geometries look both with a rotational speed and a regular modal analysis in a stationary reference of frame. An analysis of the eigenfrequencies of the assembly is done and then a Campbell diagram is created to check if the eigenfrequencies of the vibrations is close to any of the engine order lines. -24-

25 5.1 Ansys When analyzing a problem using FE it is good to know what the objectives of the analysis is and how large the analysis should be, if it s just a part or the whole system, how detailed it s should be and how refined the mesh should be. There are several disciplines in Ansys and the structural analysis is done with a modal analysis to find the natural frequency and mode shapes of the fan. There are circumstances that require a nonlinear solution and can arise from nonlinear geometry or material nonlinearities. The two main geometrical nonlinearities are the large deflection and rotation and the stress stiffening. The large deflection and rotation analyze becomes necessary if the structure undergoes large displacements compared to its smallest dimensions and position. The stress stiffening analysis is necessary if the stress in one direction affects the stiffness in another direction (Madenci, E. Guven). The following steps were used to analyze the fan geometry 2 geometry and same approach was applied for the AMS 800 with 14 blades instead of 18. The geometry has a total of 19 bodies, the hub and all the blades. The element size of mesh is 10 mm for the blades and 20 mm for the hub and in figure 17 picture of the mesh is shown. Figure 17: Mesh size on the blisk. To get the correct values for a Campbell diagram a static structural analysis was made with the corresponding rotational speed for the engine with large deflections activated in analysis. The solution from the static structural analysis was put into a modal analysis. Ansys is doing these calculations according to equation 24 below. [MM]{uu } + ([CC] + CC gggggggg ){uu } + [KK]{uu} = {FF} (24) M = mass matrix C = damping matrix K = stiffness matrix F = external forces and the centrifugal force C gyro = Gyroscopic matrix -25-

26 If the assembly that is being analyzed not containing a shaft the gyroscopic effect is neglected see 4.4 (Samuelsson, J, 2009). By changing the rotational speed with a few different numbers in the range from 0 to the operating speed the Campbell diagram can be plotted. The maximum nodal diameters of a bladed disk are half the number of blades see chapter , if there are 18 blades there are 9 nodal diameters where the first and last are single and the rest are double modes. The following procedure turned out to give the best results in ANSYS for the fan geometry 2 where the 18 first modes where found first and then the 18 modes after them, that is the second mode family and so on until the 6 first mode families where found and plotted in a frequency versus nodal diameter plot using MATLAB. -26-

27 6 Results The results are presented where an analysis is made of two fan geometries that are used at ABB in the synchronous machines. 6.1 Modal analysis A modal analysis was made for the largest geometry to get a picture of how the three different eigenmodes look. In table 3 a list of these frequencies is presented. Figure 18 shows mode shapes of the first bending mode, first torsion and the second bending mode. Table 3: First three eigenfrequencies of fan geometry 1 1:st Torsion 1:st Bending 2:nd Torsion Hz Hz 1521 Hz Figure 18: First three modes of fan geometry 1 Table 4 First three eigenfrequencies of fan geometry 2. 1:st Bending Hz 1:st Torsion Hz 2:nd Bending Hz Table 4 shows the different eigenfrequencies for the three first bending modes for fan geometry 1-27-

28 Figure 19: First three eigenmodes of the fan geometry 2. To get higher eigenfrequencies the thickness of the geometry can be raised. This has been analyzed for the thickness of 4-10 mm, see figure 19. Figure 20: Different frequencies for different thickness of blade. It can be seen in figure 20 that the eigenfrequencies changes significantly when thickness is increased from 4 to 6 mm. Figure 21: Modal analysis for 10 mm blade. -28-

29 Figure 21 shows a picture of the thickest blades modal analysis with the first bending mode. 6.2 Campbell diagram The first analysis that was made was with the two first eigenmodes for the structure with corresponding engine order lines of interest. These eigenmodes are of most interest because these are the easiest to excite and the ones one should consider first. Figure 22: First two eigenmodes of 800 with engine order line of interest. It can be seen in figure 22 that engine orders and are of interest. These numbers are related to the possibility of creating an excitation of the blades. If there is anything around the geometry that can excite the blades times or times it can cause a HCF until the blades break. Engine orders 19 and 18 are of most interest. -29-

30 Figure 23: First two eigenmodes for fan geometry 2 with engine orders of interest. In figure 23 the engine order is plotted. Engine order 15 and 17 are of most interest. None of the engine order lines has a crossing with the mode lines. Two Campbell diagram has been plotted for the fan geometry 1 and fan geometry 2 for the first 6 eigenmodes with an engine order line that is related to how many windings the machine has. If the engine order line is close to or has a crossing between one of the eigenmodes and the operational speed there are possibilities for a force that can excite the eigenmodes of the blades. -30-

31 3500 Engine order Eigenmodes Rotational speed Figure 24: Campbell diagram for fan geometry 1 with engine order related to bindings. In figure 24 above is the engine order line 72 is plotted. This is because the fan geometry 1 engine has 72 bindings which could possibly cause a force field 72 times annually. The engine order line is close to a crossing at eigenmode 4 and the operational speed line. It can be estimated that a change in blade design would be necessary to make a change in eigenfrequency so this engine order will be further away from the crossing point. -31-

32 Figure 25: Campbell diagram for fan geometry 2 with engine order related to bindings. In figure 25 above is the 6 first eigenmodes plotted for the blades on fan geometry 2 fan. The engine order line plotted is related to the 84 bindings in the machine and it can be seen that the fan has a safe blade design according the Campbell diagram and the first 6 mode lines. 6.3 Traveling wave pattern As explained in section the structure can be considered to be disk-alone shapes, blade-alone shapes or a combination as is often the case at higher frequencies. In the case of a blade alone structure the disk is very stiff in relation to the blades and the other way around. -32-

33 Figure 26: Travelling wave mode of AMS 800. For the fan geometry 1 as it can be seen in figure 26 that the structure has a blade alone shape for the first eigenmode. Figure 26 is showing a traveling wave mode, 1T-7ND (First torsional- seventh nodal diameter). Figure 27: Travelling wave mode of fan geometry

34 For the fan geometry 2 is has the blisk a blade alone shape for the first eigenmode as can be seen in figure 27 and shows the 1B-9ND as the blisk has 18 blades. All blades are excited simultaneously in a traveling wave mode. Figure 28: Families and nodal diameters for a 14 bladed blisk on fan geometry 1. Figure 28 shows the first 6 mode families in which the first and second modes are close to each other which have a possibility for so called frequency veering where a high coupling strength and high modal density is critical for a mistuned response behavior. With a change in geometry the eigen frequency will be changed and therefore a change in the first and second mode family s frequency by nodal diameter. -34-

35 Figure 29: Mode shapes and nodal diameters for an 18 bladed blisk fan geometry 2 In figure 29 the mode family 5 and 6 are close to each other at nodal diameter 7 which means that this point is interesting. There are also two points on mode family 4 and 5 that are of interest because of the frequency veering phenomena as can be seen in figure 29. Figure 30: ZZENF diagram of fan geometry 1. Figure 30 shows the ZZENF diagram for fan geometry 1. This diagram shows all the critical crossings and if these are related to number of bindings or number of blades should it be plotted in the Campbell -35-

36 diagram for a better investigation. The engine order lines of interest in figure 30 above is, EO21 and EO70. The zig zag line should cross a nodal diameter and the mode family line to be of interest. These engine order has nothing to do with the bindings on the engine since they are 72. Figure 31: ZZENF of fan geometry 2. The ZZENF diagram for the fan geometry 2 has interest crossings at engine order line EO18 and is not of interest in this case because of the 84 bindings on the machine, see figure Design of new fan geometry With the help of an inbuilt program in Workbench called Axial Fan Designer (AFD) (in ANSYS the geometry of a blade can be calculated with inserted parameters as shown in table 5. The number of blades are proportional to the blade width and should not be wider than so that they overlap each other, see section Table 5: Parameters for an AMS 800 blade for ANSYS workbench. Rotational speed 1500 RPM Inlet total pressure Pa Total head rise 1000 Pa Mass flow rate 22.6 kg/s Inlet total temperature 288 K Efficiency estimate 0.85 ωω 157 rad/s C x 50 m/s u 62 m/s Flow coefficient 0.8 r_hub m r_tip m -36-

37 r_mean 0.4 m Area-hub 0332 m 2 Area-shroud 0.71 m 2 Area-flow 0.4 m 2 Density-air 1.2 kg/m 3 Mass-flow kg/s hub/tip 0.68 Another parameter that has to be considered is the aspect ratio. The aspect ratio is the relation between the length of the blade to the width of the chord. The smaller aspect ratio the wider blade if the height is decided. The amount of blades on the fan is chosen according to how many blades there are on the blades on the old model and the longer the blades are the wider they have to be to get as high eigenfrequency as possible. Figure 32: New design of fan geometry 1. Figure 32 above shows a model that has a smaller diameter on the hub in relation to the old model. The new model has although a thicker hub but it has an increased aerodynamic efficiency. The new geometry contains 16 blades instead of 14 because with fewer blades should the blade chord be increased corresponding to an aerodynamic efficiency and with this will the hub be even thicker. Blade length 190 mm Blade chord 189 mm Eigenfrequency 362 Hz Wing stress parameter 0,146 Hub thickness 172 mm Aerodynamic efficiency

38 6.5 Wing stress The wing stress is calculated for the two different geometries with eigenfrequencies calculated by both Ansys and the equation that ABB is using. In table 7 the results are shown. Table 6: Wing stress for the different eigenfrequencies. Fan geometry 1: Eigenfrequency Wing stress parameter Analytic calculation Ansys Fan geometry 2: Eigenfrequency Wing stress parameter analytic calculation Ansys The wing stress parameter at ABB today has a limit and with eigenfrequency from the analysis by ANSYS made on the fan geometry 1 the value of the wing stress parameter become With the equation used by the eigenfrequency becomes and the wing stress parameter The higher the eigenfrequency the safer is the structure and when the eigenfrequency is calculated by Ansys is it a bit lower than the equation used by ABB. -38-

39 7 Final conclusion and future work The method for how to design the fan blades in the synchronous machine at ABB can be better. The formula that has been generated for calculating the eigenfrequency is not as accurate as it should be to get as safe structure as possible, to decrease the possibility for HCF. The eigenfrequency is then used to calculate the reduced frequency or reduced velocity, called wing stress at ABB where the limit is set to be very low. A suggestion for a new design procedure is as follows. - Get a new inclination angle related to synchronous machine and twist the blades for an even distributed force over the blade. Or calculate a new design using AFD in ANSYS. - Iterate best amount of blades. - Calculate the eigenfrequency using Ansys for the new design of the blade. - Use the eigenfrequency to calculate the wing stress parameter and see if it is under the limit. If the eigenfrequency is over the limit, make a new design and calculate the reduced frequency again. - Calculate the eigenfrequencies related to rotational speed in Ansys and generate a Campbell diagram. Draw engine order lines and see if these have any crossing with the eigenfrequency and the operational speed line. Check if the crossings are related to the amount of bindings or anything that can make a perturbation every cycle. Figure 33: Flow chart for design process. A flow chart of the suggested design procedure is presented in figure 33. To make sure what kind of vibrations there are on the blades experiments should be made on the blades to see if it is flutter or forced response that are acting on the blades and to see exactly at what frequency the vibrations occur. Changes in the geometry around the fan can be done to get a more stable aerodynamic flow. The bindings around will make an unstable area that is very hard to analyze. With some kind of plane surface that covers these binding is it easier to make a CFD analysis and thereby see what is going on around the fan blades. -39-

40 Appendix Setup for calculating the eigenfrequency using Ansys Figure 34: Static structural to modal analysis. To make Ansys receive the correct number it was necessary to first make a static structural analysis and then import the solution into a modal analysis, figure

41 Figure 35: Setup in Mechanical. The settings where made by a fixed support for the blade where it is connected to the hub (point A in figure 35 and a rotational speed with the center of the rotational speed of the hub (point B in figure 35) -41-