EXPLANATION AND APPLICATION OF THE SAFE DIAGRAM

Transcription

1 11 International RASD Conference th 1-3 July 13 Pisa 3 EXPLAATIO AD APPLICATIO OF THE SAFE DIAGRAM Leonardo ertini 1, ernardo D. Monelli 1, Paolo eri 1*, Ciro Santus 1 and Alberto Guglielo 1 Civil and Engineering departent University of Pisa Largo Lucio Lazzarino, 56 Pisa, Italy E-ail * : paolo.neri@for.unipi.it uovo Pignone General Electric, Oil & Gas ia Felice Matteucci, 517 Florence, Italy Keywords: SAFE diagra, odal analysis, laded wheel resonance. ASTRACT ibration of industrial application bladed wheels is reason of noise and structural dynaic loads. Copressor and turbine bladed wheels interact with the fluid distributed by the stator vanes. The blades are excited by these fluctuating forces and their vibration can be reason of fatigue failures, especially when resonance conditions are excited. Obviously, avoid resonances is a strategic requireent in bladed wheels design. The Capbell diagra approach just excludes the atching between the natural ode frequency and the excitation frequency. However, bladed wheels show any natural frequencies that are very near each other, so it is difficult to avoid any resonance atching. The Singh s Advanced Frequency Evaluation (SAFE) diagra approach introduces also the atching instead of just the frequency atching. Many frequency atching can be identified as non-critical and then tolerated. The present paper explains the SAFE diagra and introduces an analytical expression to identify the critical interactions. The atching ap is introduced, showing the of the ode that fully interact with each excitation haronic order. Full resonances are distinguished fro partial resonances that are not identified on the SAFE diagra. Finally, an application is reported on a faily of bladed wheels, identifying the optial geoetry configuration to avoid any critical resonance. 1. ITRODUCTIO The basic idea behind the Singh's Advanced Frequency Evaluation (SAFE) diagra is that a load case is reason of dynaic resonance if there is a atch between working and natural frequency (the Capbell diagra criterion [1]), and between the revolution periodic of the excitation and of the natural ode related to that natural frequency [, 3, 4, 5, 6, 7]. This consideration is really useful for rotating ipeller or copressor wheel optiization, in order to avoid resonance and then high stress that could lead to fatigue failure. A scheatic exaple is reported in Fig.1, showing that a Capbell resonance can be not dangerous in ters of vibration aplification because of isatch.

2 Figure 1. SAFE no resonance scheatic exaple.. THE SAFE DIAGRAM Fig. shows the schee of a bladed wheel interacting with its stator. is the nuber of blades of the rotator wheel, while is the nuber of vanes of the stator, and is the rotating speed of the wheel. i-th lade i 1,,..., Figure. laded wheel and stator schee. The fluctuating force acting on each blade is a periodic function, whose ain haronic coponent frequency is given by the vane-fluid interaction, so its value is: 1= (1) The blades are equally spaced by the sector angle so the force of each consecutive blade is shifted by this phase with respect to the previous. The generic angular position is given by the angular speed plus the shift angle, Fig.3(a). The periodic blade force can be decoposed according to the Fourier series. The force is a periodic function of the angular position coordinate and can be decoposed in a steady coponent plus the ain haronic, period: π /, plus the second haronic, period: π / ( ), plus the third and so on, Fig.3(b). The i-th blade total force can then be written as follows: i n n n1 f () t F F cos( n ( t i ) ) () where F, F1,..., F n,... are the force haronics aplitude and n is a generic phase for each n-th haronic. So, the n-th haronic coponent angular frequency is: where n is a generic positive integer: n 1,,.... n n1 n (3)

3 i-th traveling blade t ( i 1)-th traveling blade lade force t f i lade force f i /... (a) n F Anglecoordinate, F F1 Anglecoordinate, (b) Figure 3. lade Force: (a) angular shift, (b) Fourier decoposition. ibration of the blades can be written following the odal decoposition. The generic i-th blade displaceent, with respect to the undefored, is the superiposition of each ode: where X is the -th i t 1 x () t X X cos( t )cos( d i ) (4) ode coponent, is its angular frequency and, t are generic phases for the tie and the angle variables respectively. When the nuber of nodal diaeters ( d ) is zero the angular dependence vanishes: xi, Xcos( t t) (5) where is a zero nodal diaeters ode: d. There are soe odes that are not haronic d, so they do not show cyclic periodic dependence, Fig.4(a). These odes cannot be odelled by eans of a sector cyclic Finite Eleent (FE) analysis and they cannot be written according to the Eq.4 for. The Fig.4 also shows other exaples of zero (b), one (c) and two (d) nodal diaeters odes. There is no relation between the natural frequency and the nuber of nodal diaeters: odes with high natural frequency can have low nuber of nodal diaeters, and any different odes can share the sae nuber of nodal diaeters. The tie dependency of both the force haronic coponents and the displaceent odal coponents fro the bladed wheel point of view is haronic. Each force coponent is a traveling wave, since the bladed wheel is oving with respect to the stator. Each displaceent odal coponent is a stationary wave fro the bladed wheel point of view, Fig.5. Any stationary wave can be decoposed as the su of two traveling waves along the two opposite directions:

4 xi, Xcos( tt)cos( di ) X [cos( tdi1 ) cos( tdi)] (6) where the generic phases are: 1 t and t. (a) (b) (c) (d) Figure 4. Exaple of a ode with: (a) no cyclic period, (b) zero nodal diaeters, (c) one nodal diaeter, (d) two nodal diaeters. Traveling wave: cos( t ) angle coordinate, lades: Stationary wave: 1 1 cos( t)cos( ) cos( t) cos( t) angle coordinate, Figure 5. Relative tie dependency of force haronic coponent and displaceent odal coponent. As a result of the opposite direction traveling waves, the corresponding ode is always a twin couple of odes. o cyclic periodic ode, such as that reported in Fig.4(a), and any zero nodal diaeters ode are always single odes instead. Any bladed wheel has an infinite nuber of odes: 1,,... while the axiu nuber of nodal diaeters is liited: d d, where d ax is: ax

5 d d ax ax if iseven 1 if is odd (7) Apparently, the nuber of axiu nodal diaeters should be infinite since the bladed wheel is a continuu body. Indeed, this is true for a siple plate disc. However, in a bladed wheel only the displaceent at the blades is to be considered. The disc supporting the blades can have even larger nuber of nodal diaeters but the blades relative displaceent has a saller nuber of nodal diaeters, Fig.6: the black points represent the blade angular positions and the wavy line is the scheatic aplitude of the odal displaceent. The nuber of the displaceent sign reversals along the hoop coordinate of the odal defored gives the nuber of nodal diaeters. Though the disc can have several zeros, the bladed wheel reversals is liited by the nuber of the blades. d (bladed wheel) d (disc) 6 d (bladed wheel) d (disc) 6 Apparent bladed wheelodal Apparent bladed wheel odal (a) (b) Figure 6. Disc nuber of nodal diaeters and bladed wheel apparent nuber of nodal diaeters: odal displaceent at initial tie (a) and after half tie period (b). The speed of the i -th blade can be obtained as the partial derivative of the displaceent with respect to the tie. The odal superiposition is still valid because of the derivative operator linearity. y considering the traveling waves decoposition, the odal speed coponent can be written as the following: xi, vi, = Xcos( tt / )cos( di ) t X [cos( tdi1 / ) cos( tdi / )] (8) So the speed can be written as: X v ( t) [cos( t d i φ ) cos( t d i φ )] (9) i 1 1 where the phases are: φ 1 1 / and φ /. The work done over the blades is the integration of the haronic force coponents, Eq. and the vibration speed of the blades, Eq.9:

6 W i dw i xi dwi fidxi fi dt fv i idt (1) t fv FX cos( n ( Ωti) )[cos( td Δi φ ) cos( td Δi φ )] i i n n 1 n1 1 The high order ters of the two series can be neglected, liiting the series to n ax, ax instead of infinite. The steady ter F of the haronic decoposition of the force does not produce work, being the speed the su of fluctuating ters. Indeed, the ter F does not appear in Eq.1. The n-th haronic force and -th odal speed coponent cobination produces work, over an indefinite period of tie, only if the frequencies atch: or, ore siply: n = (11) n = (1) Eq.1 is the Capbell diagra criterion, and it is necessary only. The atching between the s of the ode and the haronic force is also required. Soe exaples are reported in Fig.7. Given that the frequency of the applied forces atches the excited natural ode frequency, the forces ust be in phase with the deforation displaceents in order to have the axiu work done (full resonance, Fig.7(a)). On the contrary, if the forces are copletely out of phase the ode is not excited, because one force produces positive work while the other force produces negative work. More precisely, the phase between force and odal displaceent is undefined (Fig.7(c)). There are also any possible other conditions (Fig.7(b)) where the forces excite the natural ode but the force phase cobination is not perfectly constructive with the excited natural ode. This latter condition still is a resonance, though in principle less dangerous than the full resonance case. Haronic forces Second natural ode, Haronic forces Second natural Haronic forces ode, Second natural ode, work: ax( ) work: ax( ) work: ( ) work: ( ) work: ax( ) work: ax( ) (a) (b) (c) Figure 7. Resonance condition: (a) full resonance, (b) partial resonance, (c) no resonance. This explains why the second condition of SAFE diagra identifies only those force haronics that fully atch the odal. Eq.1 proves that there is atching if one of the two following conditions is verified, for all blade indexes i 1,,...,, and for any integer k...,, 1,,1,,... : n = d k( ) n i = di k( ) The Eqs.13 can be suarized as: i i (13)

7 n i = d i k( )for alli 1,,..., and for anyinteger k...,, 1,,1,,... (14) So the resonance is considered dangerous for all the cobinations of indexes n, which satisfy Eq.1 and one of the two conditions in Eqs.13. When the Eq.14 is verified for i 1 it is verified also for any other index i. If i 1, k k1 solves the Eq.14, i, k k solves it too, and so on. The opposite iplication is obvious: if 1 Eq.14 is solved for all the blade indexes i 1,,..., then it is also solved for the single i 1. It can be concluded that: so the Eq.14 reduces to: Eq.14sovled for i 1 Eq.14sovled for all bladeindex i 1,,..., (15) n = d k( )for anyinteger k...,, 1,,1,,... () Finally, the SAFE resonance conditions are: n = n d integer (17) The zero nodal diaeters odes d are considered in the conditions of the Eq.17: n =, d n integer (18) The typical exaple is the situation with equal blades to vanes nuber:, where any frequency haronic n can excite a zero nodal diaeters ode, in case of frequencies atch. All not haronic odes, such as that reported in Fig.4(a), are not considered in Eq.17. In fact they are not included in the ode for of Eq.6 fro which the resonance conditions are derived. These odes can never produce full resonance, because they cannot be atching with the force which is haronic instead, siilarly to Fig.7(b). Soe exaples are reported in the following to better understand the second condition of Eq Exaples: no atching ode 7, 5, n, d n d n d 1.714, (19) The not atching is graphically reported in Fig.8. If the haronic force and the odal displaceent are in phase for one blade, they are not in phase for the other blades (Fig.8(b)). In this condition the resonance can never be full. It generally it is partial (such as Fig.7(b)) or, in soe special cases, copletely destructive (such as Fig.7(c)).

8 Figure 8. Exaple 1: (a) blades position, (b) tie dependency of force and displaceent. Force and displaceent blade isatch: (c) tie 1, (d) tie, (e) tie 3, (f) tie 4, (g) tie 5, (h) tie 6.. Exaples: zero nodal diaeters atching ode n d n 6, 3,, 1 () The atching is graphically reported in Fig.9, where it is clear that haronic forces and blade displaceents are all in phase once they are in phase for one blade. Since the odal is zero nodal diaeters, all the blade displaceents are in phase one with respect to each other. Figure 9. Exaple : (a) blades position, (b) tie dependency of force and displaceent. Force and displaceent blade isatch: (c) tie 1, (d) tie, (e) tie 3, (f) tie 4, (g) tie 5, (h) tie 6..3 Exaples: negative sign atching ode 1, 6, n, d n d n d 1.4, 1 (1) The atching is graphically reported in Fig.1, each blade has haronic force and displaceent in phase, though the different nuber of periods in the angular full circle.

9 Figure 1. Exaple : (a) blades position, (b) tie dependency of force and displaceent. Force and displaceent blade isatch: (c) tie 1, (d) tie, (e) tie 3, (f) tie 4, (g) tie 5, (h) tie 6..4 Exaples: positive sign atching ode 1, 9, n, d n d n d, 1.6 () The atching is graphically reported in Fig.11. Siilarly to before, each blade has haronic force and displaceent in phase. The difference between these two latter cases is that the positive sign atching has the odal counter rotating with respect to the haronic force, while in the negative sign atching case the odal and the haronic force are rotating in the sae direction. The reason is that in the odal decoposition of Eq.8 only one of the two counter rotating traveling waves is excited, depending on the sign plus or inus of the condition that produces the integer. Figure 11. Exaple : (a) blades position, (b) tie dependency of force and displaceent. Force and displaceent blade isatch: (c) tie 1, (d) tie, (e) tie 3, (f) tie 4, (g) tie 5, (h) tie 6..5 Shape atching ap To understand the SAFE diagra an iportant further step is the ap that shows the atching cobinations of haronic indexes and nodal diaeters. It is very iportant to point

10 out that for any, cobination, and any positive integer n, there is always one unique d lower than / (or ( 1)/ if is odd) that satisfies the second of the Eqs.17. Any rational nuber, such as ( n)/, can be separated as the integer part and the reainder: n d integer (3) where d is an integer either positive or negative whose value is: d when 1 1 d when is odd is even (4) Therefore, the atching nodal diaeter is: d d (5) The separation of Eq.3 is always possible and unique, so there is always a unique d given any cobination of,, n that satisfies the atching condition. The Eqs.5,3 can be expressed in a graph that is the atching ap. Given a cobination, for any haronic index n, the fraction decoposition (Eq.3) is considered and the value of d is reported in cobination with that haronic index (Fig.1). Exaple 1: 1, 7, n 1 n 1x (6) indeed, d 3 atches the first haronic index, as reported in Fig.1(a). Exaple : 1, 7, n n x (7) indeed, d 4 atches the first haronic index, as reported in Fig.1(a). Exaple 3: 9, 7, n 1 n 1x7 9 1 (8) indeed, d atches the first haronic index, as reported in Fig.1(b).

11 excitation haronic order nodal diaeter 7 6 excitation haronic order nodal diaeter (a) (b) Figure 1. Shape atching ap exaples: (a) even nuber of blades, (b) odd nuber of blades. Exaple 4: 9, 7, n 4 n 4x (9) indeed, d 1 atches the first haronic index, as reported in Fig.1(b). The atching ap can be also interpreted in a graphical way. Given a cobination of and an index n, it is possible to find the atching d by starting counting the value of d and by applying reflections at /, if ( 1)/, if This way is equivalent to the previous. In fact Eq. 3 can be written as:, n fro 1 to is even, Fig.1(a), or at is odd, Fig.1(b). The final position of the counting is the atching d. n integer d (3) Once a ultiple of is stepped ( integer ) the position reached is d, due to the reflections. Then adding or subtracting d nuber of steps, the position d d is reached. The counting technique can be considered ore practical because the search is straightforward: once d for n 1 is found, the counting can restart fro the sae position to find the d for n and so on..6 SAFE diagra explanation The two conditions of the Eq.17 can be suarized in a unique diagra: the Singh's Advanced Frequency Evaluation (SAFE) diagra. Along the horizontal axis there is the nuber (nodal diaeters) up to d, Eq.7. Along the vertical axis there is the frequency. The ax wheel angular frequency is ultiplied by the d index to draw the dotted line in Fig.13. Reflections are applied, according to the schees of the Figs.1(a),(b) and also at d. y counting along the d coordinate up to, the corresponding angular frequency is. Shape atching schee is followed, so the spotted point is at the position 1 ( d ( 1), n 1), where d( n 1) is the nuber of nodal diaeters atching the first haronic ( n 1). Counting and reflections are applied again for the next ties: the point ( d ( ), n ) is found, and so on. On the sae graph the natural odes are reported as

12 nuber of nodal diaeters and natural frequencies. Finally, the resonance conditions can be spotted as the points where the natural odes are coincident with the possible resonances points ( d( n), n). In Fig.13 the first ode d 1 is not a resonance because of isatch (such as in Fig.1). The second ode d 3 is not perfectly coincident in ters of frequency with the first possible resonance point in Fig.13(a). However, aplification is relevant even if the frequencies are not perfectly equal as long as the is atching (the angular speed of the wheel can change: the SAFE diagras are usually reported as a working frequency range). Finally, the third ode d3 4 is and frequency coincident with the second haronic possible resonance (Fig.13(a),(b)) d ( 1), n d1 d ( ), n 1 d 3 d d ( ), n d1 d3 4 d (a) (b) Figure 13. SAFE diagra exaples: (a) even nuber of blades, (b) odd nuber of blades. 3. SAFE DIAGRAM APPLICATIO TO A FAMILY OF IMPELLER WHEELS The SAFE diagra was applied to a faily of ipeller wheels that have sae diaeters but different body disc thickness and nuber of blades (and vanes). All the ten wheels investigated had the external diaeter of 4. The ain paraeters are reported in Tab.1. uber of the wheel Disc Thickness [] Table 1: Investigated wheel properties.

13 Wheel nuber 1 initial odes distribution is reported in Fig.14, and soe of the initial odes are shown in the Fig.4. The haronic odes with d always appear in twins: they are the two counter rotating odes so they are to be reported as a single ode in the SAFE diagra. Soe other odes are single, being either not haronic odes or zero nodal diaeters odes (Fig.4(a) and (b) respectively). The different disc thickness is not affecting the natural frequency significantly, for this geoetry, Fig.15. Figure 14. Wheel 1 initial odes distribution. (a) (b) Figure 15. Disc thickness effect on the natural frequency: low (a) and high (b) thickness. Also the nuber of blades does not affect the natural frequency notably, especially for the initial odes, which ore involve the entire body instead of the blades. Indeed, the distribution of the initial haronic odes are alost coincident for the entire wheel set (Fig.). 6 Frequency, Hz odal diaeters Figure. Coparison of the initial haronic odes of the entire wheel set.

14 It is evident that the distribution of the initial siple odes is along a line where the nodal diaeters increase with the natural frequency. Above the frequency of 4 Hz, odes are alost uniforly distributed both in ters of nuber of nodal diaeters and in ters of frequencies. Though siilar distribution of odes, each wheel has different cobination of blades and vane nubers, so the SAFE diagras are different. The SAFE diagras for each wheel are reported in Fig.17. The range of investigated wheel angular speed is: 9-1 rp ( 15 - Hz) (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Figure 17. SAFE diagras for the investigated wheels: (a) wheel 1, (b) wheel, (c) wheel 3, (d) wheel 4, (e) wheel 5, (f) wheel 6, (g) wheel 7, (h) wheel 8, (i) wheel 9, (j) wheel 1.

15 The third wheel is the best cobination of nuber of blades and vanes ( 15, 4, Fig.17(c)) because it does not have any SAFE diagra resonance, at least up to the tenth haronic. The other wheels have high nuber of vanes and it is alost ipossible to avoid resonances. The reported odes are liited to the natural frequency value of 6 Hz. The third wheel tenth haronic is alost equal to this value. The tenth haronic frequency of the other wheels is higher, so ore odes should be considered. Anyway resonances were already found in this liited analyses, proving the wrong design. It is evident that a low nuber of vanes would be recoended. Indeed, if the nuber of vanes is high the initial haronics are going to atch high frequency odes whose distribution is alost unifor in ters of frequencies and nuber on nodal diaeters, so avoiding atches is ipossible. 4. COCLUSIOS The Singh's Advanced Frequency Evaluation (SAFE) diagra was copletely explained and then applied to a faily of wheels to find the best cobination of geoetry and blade / vane nubers. The set of investigated wheels showed very siilar distribution of odes, in ters of frequencies and s. The way to find a not resonance design was just anipulating the nuber of blades, or ost effectively the nuber of vanes. As a general rule it is better to prescribe a low nuber of vanes (when it is possible) in order to avoid initial haronic atch with high frequencies odes, which are really close each other. REFERECES [1] R.E.D. ishop and D.C. Johnson, The Mechanics of ibration. Cabridge University Press, [] H.P. loch and M. Singh, Stea Turbines: design, applications and re-rating. McGraw- Hill, 8. [3] M.P. Singh. SAFE Diagra. Technical report, Dresser Rand Copany, ST. [4] M.P. Singh. SAFE Diagra a design and reliability tool for turbine blading. Technical report, Dresser Rand Copany,. TP4. [5] M.P. Singh. SAFE Diagra A Dresser Rand Evaluation Tool For Packeted laded Disc Assebly. Technical report, Dresser Rand Copany,. TP4. [6] J.D. Dello. Frequency evaluation of a stea turbine bladed disk. Technical report, Dresser Rand Copany,. TP3. [7] G. ordwall and M. Leduc, A. Deeulenaere, Unsteady blade and disk resonant stress analysis due to supersonic inlet guide vane wakes. In Proceedings of GT8 ASME Turbo Expo 8: Power for Land, Sea and Air, erlin, Geran, 8, GT