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THE EFFECT OF BLADE CRACK ON MODE LOCALIZATION IN ROTATING BLADED DISKS
Journal of Sound and
THE EFFECT OF BLADE CRACK ON MODE LOCALIZATION IN ROTATING BLADED DISKS J. H. KUANG AND B. W. HUANG Department of Mechanical Engineering, National Sun >at-Sen ;niversity, Kaohsiung, ¹aiwan, R.O.C. (Received 2 January 1998, and in ,nal form 12 April 1999) The e!ects of blade crack on mode localization in rotating bladed disks are investigated in this study. A disk consisting of periodic shrouded blades is used to simulate the bladed disk of a turbo-rotor. Blades on the disk are approximated as Euler}Bernoulli beams. A crack in the blade is regarded as a local disorder of the system. This local disorder might introduce the so-called mode localization phenomenon in this mistuned system. The Galerkin method is employed to derive the equation of motion of the mistuned system, with the blade crack. The e!ects of position, depth of crack and rotating speed on the localization phenomenon in the rotating blade-disk system are studied. Numerical results indicate that the blade crack may be one of the reasons for the occurrence of the localization phenomenon in the rotating bladed disk. 1999 Academic Press
1. INTRODUCTION
The dynamic analysis of repetitious engineering structures is greatly simpli"ed by assuming perfect periodicity. Unfortunately, this mathematical idealization is invalid due to unavoidable manufacturing and material defects. Due to manufacturing #ows or cyclic fatigue during operation, cracks frequently appear in the rotating machinery [1, 2]. A blade crack may cause local change in the #exibility, i.e., structural irregularity which may change the dynamic behavior of the structure. As noted by several investigators [1, 9], the existence of a local defect in the coupled periodic system may introduce the so-called vibration localization phenomenon. However, these studies have not focused on the investigation of mode localization. Most types of crack are surface cracks in the rotating machinery, but it is di$cult to emphasize on fracture mechanics in this model. To make the analysis even easier, the investigation of a two-dimensional problem using a model of a crack extends over the entire chord, is necessary. Krawczuk et al. [3}9] have studied the vibrational behavior for the blade, beam and rotor with a crack by the two-dimensional crack model. In this study, the crack blade is still modelled as a two-dimensional model, i.e., a two-span beam model [9]. The localization phenomenon is known to occur in coupled periodic structures under certain circumstances by local structural or material irregularities. Such localization may in turn localize the vibrational modes and thereby con"ne the 0022-460X/99/410085#19 $30.00/0
1999 Academic Press
86
J. H. KUANG AND B. W. HUANG
vibrational energy. Recently, a number of studies have been conducted to investigate mode localization in periodic structures [10}14]. Studies on normal mode localization in nearly periodic system in one, two and three dimensions were published by Cai et al. [15, 16]. Di!erent models and parameter values used in the above analyses have, however yielded di!erent and even con#icting conclusions. The localization phenomenon may occur in a rotating bladed disk under certain circumstances, such as structural or material irregularity. The natural frequencies or mode shapes of the mistuned system can be seriously a!ected by such local irregularity. The amplitudes of individual defected blades may in turn be seriously excited. The localized vibrations further increase the amplitudes and stresses locally. The shrouded bladed disk of a turbo-rotor can be regarded as a periodic system if all the blades are assembled periodically. The dynamic behavior of such a shrouded blade-disk system has been studied by Cottney and Ewisn [17], whereas the fundamental aspects of mode localization in mistuned turbomachinery rotors have been studied by Bendisksen et al. [18}21]. More recent studies by Cha and Pierre et al. [22}25] have dealt with vibrational localization in the mistuned system with a multi-mode subsystem. Most of the studies on mode localization are limited to stationary structures, with assumed structural and material irregularity. Only a few studies on the e!ects of crack and rotation on localization in a shrouded bladed disk have been conducted. The individual blades of the shrouded blade-disk are approximated as Euler}Bernoulli beams, and a two-span beam with a torsional spring is used in modelling the cracked blade. The shrouded e!ect, which is regarded as a massless spring e!ect, is considered in order to introduce the constraint between the blades. The e!ects of crack size, crack position and rotation speed on mode localization in the shrouded blade disk have been investigated in this study.
2. EQUATIONS OF MOTION
The periodic shrouded blade structure is shown in Figure 1, consisting of a rigid hub and a cyclic assembly of N coupled blades. Each blade is coupled with the adjacent one through a shroud. A massless spring k is assumed to model the Q function of this shroud. The length of the cantilever beam is ¸"R !R , and M F every blade is coupled by a spring k to the adjacent one at position R . The Q A notation l (r, t) denotes the transverse #exible de#ection of the sth blade as Q a constant rotating speed X. The moment of inertia and the cross-sectional area of the sth blade are denoted as I and A respectively, while the Young's modulus of the blade is denoted as E. Q Q 2.1.
BLADES WITHOUT CRACK
According to Kuang and Huang [26], the e!ect of coriolis force on the mode localization phenomenon in a turbo disk system is not signi"cant. By neglecting the coriolis force and the vibration in the radial direction, the kinetic energy of the sth
87
MODE LOCALIZATION IN CRACKED BLADE DISKS
Figure 1. Geometry of the rotating bladed disk.
blade is given by
oA 0M *l (r, t) Q ¹" Q (1) #[Xl (r, t)] dr, Q Q 2 *t 0F where the subscript s denotes the blade number of the disk. The corresponding strain energy of the sth blade is
*l (r, t) 1 0M 1 0M Q dr# oA X EI ;" Q Q Q 2 *r 4 0F 0F 1 # k [l (R , t)!l (R , t)]. Q A 2 Q Q> A
(R #¸)!r F
*l (r, t) Q dr *r (2)
By applying Hamilton's principle, the equation of motion of the sth blade in the transverse direction can be obtained as
* *l (r, t) EI * *l (r, t) Q Q # Q *t *t *r oA *r Q *l (r, t) 1 * *l (r, t) Q #X r Q !l (r, t)! [(R #¸)!r] Q F *r 2 *r *r
#(k #k ) l d (r!R )!k l d (r!R )!k l d (r!R )"0 Q Q\ Q A Q Q> A Q\ Q\ A for s"1, 2,2, N.
(3)
For simplifying the notations, a number of non-dimensional parameters are de"ned as r!R F and rN "0P1, rN " (4) ¸
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J. H. KUANG AND B. W. HUANG
l (rN , t) lN (rN , t)" Q , Q ¸
(5)
(6)
XM "X
R !R F RM " A A ¸
EI , oA¸
and
R 1 F" , ¸ 2
(7)
k ¸ bM " Q . (8) EI Q By substituting these dimensionless variables into the equation of motion of the sth blade, it yields
EI * *lN *lN Q Q# oA ¸ *rN *rN *t Q *lN 1 #XM rN Q!lN ! Q 2 *rN
R * *lN R F#1 ! rN # F Q ¸ ¸ *rN *rN
#(bM #bM ) lN d (rN !RM )!b lN d (rN !RM )!b lN d (rN !RM )"0 Q Q\ Q A Q Q> A Q\ Q\ A for s"1, 2,2, N, with the boundary conditions lN (0, t)"0, Q *lN (0, t) Q "0, *rN
(9)
(10) (11)
*lN (1, t) Q "0, *rN
(12)
(13)
* *lN (1, t) Q "0. *rN *rN The solution of the above eigenvalue problem can be expressed as
K lN (rN , t)" qQ (t) tQ (rN ), (14) Q G G G where tQ (rN ) is the comparison function of equation (9) and qQ (t) the coe$cient, G G which is to be determined. For simplicity, three exact bending modes of the clamped cantilever beam tQ (rN ) are considered in this article. They are G cos j #cosh j G G (sinh j rN !sin j rN ) tQ (rN )"(cosh j rN!cos j rN )! G G G G G sin j #sinh j G G for i"1, 2,2, m, (15) The coe$cients j are the roots of G cos j rN cosh j rN #1"0. G G
(16)
89
MODE LOCALIZATION IN CRACKED BLADE DISKS
Using Galerkin method, the equation of motion of the sth blade can be derived in matrix form as EI +[[k] #[K] [m] +qK , # Q Q Q Q oA ¸ Q #b +t(RM ), +t(RM ),2#b +t(RM ), +t(RM ),2 ] +q, Q A Q A Q Q\ A Q A Q Q !b +t(RM ), +t(RM ),2 ] +q, !b +t(RM ), +t(RM ),2 +q, ,"0 Q\ A Q A Q\ Q\ Q A Q A Q> Q> for s"1, 2,2, N and +t(r), "[tQ (r), tQ (r),2, tQ (r)]2. Q K Due to the cyclic arrangement of blades +q, "+q, , ,> where the elements (m)Q , (k)Q and (K)Q are given as GH GH GH (m)Q "(m)Q " tQ (rN ) tQ (rN ) drN , GH HG G H dtQ (rN ) dtQ (rN ) G H drN , (k)Q "(k)Q " GH HG drN drN R dtQ (rN ) H !tQ (rN ), (K) "XM tQ (rN ) rN # F H GH G drN ¸ 1 R R * dtQ (rN ) F#1 ! rN # F H ! 2 ¸ ¸ *rN dr
(18)
2.2.
(17)
(19)
drN .
(20)
BLADE WITH A CRACK
Considering a crack on the mth blade of this system, it may be regarded as a mistuned system. The response of the mth cracked blade can be expressed as K lN (rN , t)" qK (t) tK (rN ), (21) K G G G where tK (rN ) are comparison functions of equation of motion of the cracked blade. G The qK (t) are the coe$cients to be determined. For simplicity, only three bending G modes are considered in this study, and they are calculated from the cracked beam model. The cracked blade, as shown in Figure 2, is modelled as a two-span beam coupled with a torsional spring. This cracked beam model has been studied by Rizos and Asprogathos [9]. The notation ¸ denotes the position of the crack, while A the depth of the crack is a. The #exibility G caused by the crack with depth a can be derived from Broke's approximation [27] as (1!k) K (Pb) dG '" , E 2b da
(22)
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J. H. KUANG AND B. W. HUANG
Figure 2. Geometry of the cracked blade.
where K is the stress intensity factor under mode I loading, k the Poisson Ratio, ' b the width of the blade, h the height of the blade, P the bending moment at the @ crack, and G the #exibility of the blade. The magnitude of stress intensity factor can be derived from Tada's formula [28] as 6P (23) K " @ (nch F (c), ' ' gh where
2 nc 0)923#0)199 [1!sin (nc/2)] F (c)" tan , ' nc 2 cos (nc/2)
(24)
a c" . h
(25)
By substituting the stress intensity factor K in equation (22), it leads to ' 6(1!k) h G" Q(c) EI and A Q(c)" nc F (c) dc. ' Since 1 k " 2 G
(26)
(27)
the bending sti!ness k of the cracked section of a blade can be expressed as 2 EI . (28) k " 2 6(1!k) hQ(c)
91
MODE LOCALIZATION IN CRACKED BLADE DISKS
The corresponding kinetic energy of the cracked blade is
0
*A *l (r, t) 0M *l (r, t) K K dr# dr . *t *t F *A The strain energy of the cracked blade can thus be found as oA ¹" K K 2
*l (r, t) 1 0M *l (r, t) 1 *A K K dr# dr EI EI ;" K K K 2 *r 2 *r 0F *A 1 *l *l K (¸ , t)! K (¸ , t) . # k A A 2 2 *r *r
(29)
(30)
In this case, the equation of motion of the cracked blade can be derived as EI *lN K K# oA ¸ *t K EI *lN K K# oA ¸ *t K
* *lN K *rN *rN
"0 for rN : 0 to ¸M , A
(31)
* *lN K *rN *rN
"0 for rN : ¸M to 1, A
(32)
where ¸ !R F. ¸M " A A ¸
(33)
The boundary conditions of this cracked blade are lN (0, t)"0, K *lN K (0, t)"0, *rN
(34)
lN (¸M , t)"l (¸M , t), K A K A *lN *lN K (¸M , t)" K (¸M , t), A A *rN *rN
(36)
*lN *lN K (¸M , t)" K (¸M , t), A A *rN *rN EI *lN *lN *lN K (¸M , t)" K (¸M , t), K (¸M , t)# K A A A k ¸ *rN *rN *rN 2 *lN K (l, t)"0, *rN *lN K (l, t)"0. *rN
(35)
(37) (38) (39) (40) (41)
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J. H. KUANG AND B. W. HUANG
The solutions of such a two-span beam with a torsional spring can be assumed to be as K lN (rN , t)" qK (t) K (rN ), (42) K G G G K lN (rN , t)" qK (t) K (rN ), (43) K G G G where
K (rN )"A sin (k rN )#A cos (k rN )#A sinh (k rN )#A cosh (k rN ), (44) G G G G G
K (rN )"B sin (k rN )#B cos (k rN )#B sinh (k rN )#B cosh (k rN ). (45) G G G G G Three modes, i.e., m"3 are assumed. The coe$cients of functions K (rN ) and G
K (rN ) can be obtained by substituting equations (44) and (45) in the above G boundary conditions. Table 1 lists coe$cients for the blade with a crack at the root. The coe$cients for the blade with a crack at di!erent positions, i.e., ¸M "0)2 and 0)4 A are presented in Table 2. The comparison functions of the whole cracked blade can thus be expressed as tK (rN )" K (rN ) u (rN )! K (rN ) u (rN !¸M )# K (rN ) u(rN !¸M ), A A G G G G where u(rN ) is a unit step function.
(46)
TABLE 1 ¹he magnitude of coe.cients for the blade with a crack at the root (a) ¸M "0)0, c"0)01 A B i"1 i"2 i"3
1)00010 1)00019 1)00024
B
B
!1)36214 !0)98167 !1)00037
!0)99987 !0)99981 !0)99973
B
B
k G
1)36214 0)981678 1)00051
1)8750 4)6939 7)8546
B
(b) ¸M "0)0, c"0)05 A B i"1 i"2 i"3
1)00245 1)00441 1)00745
!1)36042 !0)97737 !0)99328
!0)99755 !0)99559 !0)99254
B
B
k G
1)36042 0)977379 0)993322
1)8733 4)6896 7)8473
B
(c) ¸M "0)0, c"0)10 A B i"1 i"2 i"3
1)00929 1)01656 1)02795
!1)35542 !0)96499 !0)97302
!0)99070 !0)98343 !0)97211
Note: Only B , B , B , B , as the crack at the root of blade.
k G
1)35542 0)964995 0)972909
1)8683 4)6772 7)8269
B
93
MODE LOCALIZATION IN CRACKED BLADE DISKS
TABLE 2 ¹he magnitude of coe.cients for the blade with a crack at di+erent locations (a) ¸M "0)2, c"0)10 A A i"1 i"2 i"3
i"1 i"2 i"3
A
A
A
k G
1)0000 1)0000 1)0000
!1)3597 !0)98138 !1)01206
!1)0000 !1)0000 !1)0000
1)3397 0)98138 1)01206
1)8715 4)6940 7)8327
B
B
B
B
k G
1)00631 1)00071 1)02197
!1)36217 !0)98234 !0)97863
!0)99274 !1)99822 !1)24493
1)35710 0)98007 1)24573
1)8715 4)6940 7)8327
A
A
A
k G
(b) ¸M "0)4, c"0)10 A A i"1 i"2 i"3
i"1 i"2 i"3
1)0000 1)0000 1)0000
!1)36154 !0)98337 !1)00139
!1)0000 !1)0000 !1)0000
1)36154 0)98337 1)00139
1)8736 4)6882 7)8488
B
B
B
B
k G
1)00316 1)00280 1)01379
!1)36449 !0)97374 !1)00244
!0)99441 !1)03175 !1)14455
1)35800 0)01356 1)14534
1)8736 4)6882 7)8488
By substituting equations (42) and (43) into equations (31) and (32), the equation of motion of the cracked blade can be obtained in the matrix form as EI K +[[k] #[K] [m] +qK , # K K K K oA ¸ K #b +t(RM ), +t(RM ),2#b +t(RM ), +t(RM ),2 ] +q, K A K A K K\ A K A K K !b
K\
+t(RM ), +t(RM ),2 +q, !b +t(RM ), +t(RM ),2 +q, ,"0 A K A K\ K\ K A K A K> K>
(47)
with
(k)K "(k)K " HG GH
tKG (rN ) tKH (rN ) drN ,
(48)
dtK (rN ) dtK (rN ) G H drN , drN drN
(49)
(m)K "(m)K " HG GH
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J. H. KUANG AND B. W. HUANG
(K) "XM GH
1 ! 2
2.3.
R dtK (rN ) H !tK (rN ), rN # F H ¸ drN
tK (rN ) G
R R * dtK (rN ) F#1 ! rN # F H ¸ *rN drN ¸
drN .
(50)
EQUATION OF MOTION OF THE MISTUNED SYSTEM
For the sake of simplicity, the same comparison function is assumed for each individual blade, i.e., tQ (rN ),t (rN ). The equation of motion of the entire disk H H system can be expressed as [M ] +XG ,#[K] +X,"0,
(51)
where the system mass matrix [M] and the system sti!ness matrix [K ] are [m] 0 0
0
0
0 [m] 0 [m]
.
0
0
0
.
0
0
0
.
0
0
0
.
.
.
[M]" .
.
.
.
0
0
0
. [m]
0
0
0
.
0
0
0
.
EI [K]" oA¸
[aN ]
!b [U]
!b [U]
[aM ]
0
!b [U]
[aN ]
.
.
0
,
0 0 ,\ 0 [m] 0 ,\ 0 0 [m]
(52)
,
.
0
0
!b [U] ,
!b [U] .
0
0
0
.
0
0
0
.
.
.
.
.
0
0
.
[aN ]
0
0
0
. !b [U] ,\
!b [U] ,
0
0
.
0
,\
0
!b
,\
[aN ]
[U]
,\
!b
,\
[U]
,
0 !b [U] ,\ [aN ] ,
(53) +X,"[ +q,2 , +q,2 ,2, +q, 2 , +q,2 ]2 ,\ ,
(54)
[aN ] "[k ]#[K] #b [U]#b [U], Q Q Q Q Q\ [U]"+t (RM ) , +t (RM ),2, G A H A bM "bM . ,
(55)
and
(56) (57)
95
MODE LOCALIZATION IN CRACKED BLADE DISKS
The non-dimensional frequency u of equation (51), i.e., the natural frequency of the L mistuned system, is de"ned as uN "u L L
EI for n"1, 2,2 . oA¸
(58)
The lowest natural frequency of the tuned system without any rotational speed, i.e., uN *"3)516, is used for reference in the following numerical analysis. 2.4.
FORCE RESPONSE
Consider the blades to be excited by a uniformly distributed harmonic force h cos ut. The equation of motion of the sth blade can be rewritten in the matrix form as [M] +XG ,#[K] +X ,"+F , cos ut,
(59)
where +F,"[ + f ,2 , + f ,2 ,2, + f , 2 , + f ,2 ]2, ,\ , f " t (rN ) hM dr, Q Q h hM " . oA Q Let the dynamic response of the bladed-disk system be
(60)
(61) (62)
+X ,"[; ] +g(t),.
(63)
The modal matrix [; ] of the mistuned system is calculated from equation (51). With the initial condition as g (0)"0, gR (0)"0, the response of the system can be G G derived as
C 1!cos (uN #uN ) t 1!cos (uN !uN ) t G G # g (t)" G cos uN t G uN !uN 2uN uN #uN G G G sin (uN !uN ) t sin (uN #uN ) t G G #sin u t ! for uOu G uN #uN uN !uN G G
(64)
with +C ,"[;]2 +F,.
(65)
The corresponding resonant response can be written as C g (t)" Q [cos u t(1!cos 2uN t)#sin uN t(2uN t!sin 2uN t)] Q Q Q Q Q G 4uN Q where uN is one of the natural frequencies of system. Q
for uN "uN , (66) Q
96
J. H. KUANG AND B. W. HUANG
3. NUMERICAL RESULTS AND DISCUSSIONS
A rigid hub attached to 46 uniform blades is used to approximate the bladed disk. The existence of mode localization for a mistuned bladed disk system with a cracked blade is investigated. The e!ects of rotational speed of the disk, depth and location of the crack on mode localization have also been studied. As the crack propagates on a blade, it may not only alter the dynamic behavior of this blade, but may also introduce the so-called mode localization phenomenon in
Figure 3. Tip displacements of the uncracked and cracked systems with di!erent depths of crack (for the the 1st mode), RM "0)5, ¸M "0)0, b"0)2, XM /uN *"0)0, uN *"3)516: (a) system without crack A A c"0)0 (u "3)516); (b) system with a crack c"0)01 (u "3)516); (c) system with a crack c"0)1 (u "3)476).
MODE LOCALIZATION IN CRACKED BLADE DISKS
97
the whole mistuned system. In this numerical example, a crack is assumed on the 23rd blade. 3.1.
FREE VIBRATION ANALYSIS
Figure 3(a) shows the tip vibration pattern of the tuned system at the lowest resonant frequency uN *"3)516. The vibrational energy is uniformly distributed on the individual blades of the tuned system. The corresponding tip vibrational
Figure 4. Tip displacements of the system with a crack at di!erent locations (for the the 1st mode), RM "0)5, c"0)1, b"0)2, XM /uN *"0)0, uN *"3)516: (a) system without crack at ¸M "0)0 (u "3)476); A A (b) system with a crack at ¸M "0)2 (u "3)495); (c) system with a crack at ¸M "0)4 (u "3)511). A A
98
J. H. KUANG AND B. W. HUANG
patterns for the mistuned system, with di!erent depths of crack, i.e., c"0)01 and 0)1, are shown in Figures 3(b) and 3(c). These "gures indicate that the vibrational energy is con"ned on the blades near the cracked blade of these two mistuned systems. It can also be observed that the localization pattern varies with the depth of the crack. Generally, the model localization phenomenon is enhanced with increase in the crack depth. The localization pattern may also be a!ected by location of the crack. The tip vibration pattern for di!erent crack locations are shown in Figure 4(a)}4(c), which indicate that the
Figure 5. Tip displacements of the cracked system under di!erent rotating speeds (for the 1st mode), c"0)01, ¸M "0)0, RM "0)5, bM "0)2, uN *"3)516: (a) crack system without speed XM /u *"0)0 A A (uN "3)516); (b) crack system at the speed XM /u *"0)25 (uN "3)528); (c) crack system at the speed XM /u *"0)5 (uN "3)529).
MODE LOCALIZATION IN CRACKED BLADE DISKS
99
localization phenomenon is suppressed as the crack position is changed from ¸M "0)0 to 0)4. A Figures 5(a)}5(c) illustrate the e!ect of rotational speed X on mode localization in the mistuned system. These results appear to indicate that the con"nement of vibrations may be strongly localized as the rotational speed is increased. In other words, mode localization phenomenon may be enhanced as the rotational speed is increased. The weak or strong localization phenomenon depends upon the magnitude of disorder and the modal coupling e!ect. Now, the strong modal coupling e!ect is studied in this investigation. Figure 6(a) shows the weak localization at the lower order mode uN as the strong coupling sti!ness at the blade tip. However, as above, the strong localization may appear at the higher order mode as shown in Figure 6(b). 3.2.
FORCE RESPONSE
Consider that the blades of the system are excited by a uniformly distributed harmonic force with an excitation frequency u. A signi"cant resonant response of the cracked blade can be observed as the excitation frequency approaches the localization frequency, i.e., the lowest natural frequency of the mistuned system. Figure 7 shows the time response of the blades near the cracked blade, for excitation at the localization frequency uN "uN "3)529. Figure 8 shows the changes in frequency response of the 23rd blade with or without a crack located at the "xed end. There is only a single peak, uN "uN "3)516, in the frequency spectra of the 23rd blade in the tuned system, which shows that every individual blade possesses the same frequency and mode. However, a group of peaks is observed for the corresponding cracked blade, with a crack at the blade root. The lowest natural frequency at u "3)476 is the localization frequency. Figure 9 displays the e!ect of rotational speed on the frequency response of the cracked blade in this mistuned system, which reveals that the rotational speed is a signi"cant factor for mode localization. The mode localization phenomenon is enhanced signi"cantly with increase in the rotational speed.
4. CONCLUSIONS
The e!ects of crack on mode localization in the rotating blade-disk system have been studied. The major conclusions that can be drawn from this study are summarized as follows: (1) It has been found that the crack on a bladed is one of the parameters for localization in the shrouded blade-disk system. For a coupled blade system, the existence of a crack may change the vibrational mode drastically, i.e., from the extended to the localized. (2) The rotational speed has a signi"cant in#uence on mode localization in the mistuned system. As the rotational speed of the disk is increased, the mode localization phenomenon is enhanced.
100
J. H. KUANG AND B. W. HUANG
Figure 6. Tip displacements of the cracked system with strong modal coupling sti!ness (for the 1st and 2nd order modes), c"0)05, ¸M "0)0, RM "1)0, b"10)0, XM /uN *"0)5, uN *"3)516: (a) the "rst A A order mode (u "3)549); (b) the second order mode (uN "22)085).
Figure 7. Tip responses of the blades under the lowest localization frequency u "5)329; c"0)01, ¸M "0)0, RM "0)2, bM "0)2, XM /uN *"0)5. A A
MODE LOCALIZATION IN CRACKED BLADE DISKS
101
Figure 8. Di!erence between frequency responses of the 23rd blade with and without a crack, bM "0)2, RM "0)5, XM /uN *"0)0, ¸M "0)0: (a) c"0)0; (b) c"0)1. A A
Figure 9. Frequency response vibration of the cracked blade with or without considering the rotating speed, bM "0)2, RM "0)5, ¸M "0)0, c"0)01: (a) XM /uN *"0)0, (b) XM /uN *"0)5. A A
(3) The magnitude and location of the crack on the blade also a!ect mode localization in the shrouded blade-disk system. (4) Only the cracked blade of the mistuned system shows a signi"cant resonant response as the excitation frequency approaches the localization frequency. Obvious beat response on blades near the cracked blade is observed at localization frequency.
ACKNOWLEDGMENTS
The authors would like to acknowledge the support of the National Science Council, R.O.C., through grant no. NSC84-2212-E110}007.
102
J. H. KUANG AND B. W. HUANG
REFERENCES 1. H. L. BERNSTEIN and J. M. ALLEN 1992 ¹ransactions of the American Society of Mechanical Engineers, Journal of Engineering for Gas ¹urbines and Power 114, 293}301. Analysis of cracked gas turbine blades. 2. J. R. GALVELE 1990 Corrsion Science 30, 955}958. Surface mobility}stress corrsion cracking mechanism of steels for steam turbine rotor. 3. M. KRAWCZUK and W. M. OSTACHOWICZ 1993 ¹ransactions of the American Society of Mechanical Engineers, Journal of
MODE LOCALIZATION IN CRACKED BLADE DISKS
103
19. D. AFOLABI 1985
THE EFFECT OF BLADE CRACK ON MODE LOCALIZATION IN ROTATING BLADED DISKS J. H. KUANG AND B. W. HUANG Department of Mechanical Engineering, National Sun >at-Sen ;niversity, Kaohsiung, ¹aiwan, R.O.C. (Received 2 January 1998, and in ,nal form 12 April 1999) The e!ects of blade crack on mode localization in rotating bladed disks are investigated in this study. A disk consisting of periodic shrouded blades is used to simulate the bladed disk of a turbo-rotor. Blades on the disk are approximated as Euler}Bernoulli beams. A crack in the blade is regarded as a local disorder of the system. This local disorder might introduce the so-called mode localization phenomenon in this mistuned system. The Galerkin method is employed to derive the equation of motion of the mistuned system, with the blade crack. The e!ects of position, depth of crack and rotating speed on the localization phenomenon in the rotating blade-disk system are studied. Numerical results indicate that the blade crack may be one of the reasons for the occurrence of the localization phenomenon in the rotating bladed disk. 1999 Academic Press
1. INTRODUCTION
The dynamic analysis of repetitious engineering structures is greatly simpli"ed by assuming perfect periodicity. Unfortunately, this mathematical idealization is invalid due to unavoidable manufacturing and material defects. Due to manufacturing #ows or cyclic fatigue during operation, cracks frequently appear in the rotating machinery [1, 2]. A blade crack may cause local change in the #exibility, i.e., structural irregularity which may change the dynamic behavior of the structure. As noted by several investigators [1, 9], the existence of a local defect in the coupled periodic system may introduce the so-called vibration localization phenomenon. However, these studies have not focused on the investigation of mode localization. Most types of crack are surface cracks in the rotating machinery, but it is di$cult to emphasize on fracture mechanics in this model. To make the analysis even easier, the investigation of a two-dimensional problem using a model of a crack extends over the entire chord, is necessary. Krawczuk et al. [3}9] have studied the vibrational behavior for the blade, beam and rotor with a crack by the two-dimensional crack model. In this study, the crack blade is still modelled as a two-dimensional model, i.e., a two-span beam model [9]. The localization phenomenon is known to occur in coupled periodic structures under certain circumstances by local structural or material irregularities. Such localization may in turn localize the vibrational modes and thereby con"ne the 0022-460X/99/410085#19 $30.00/0
1999 Academic Press
86
J. H. KUANG AND B. W. HUANG
vibrational energy. Recently, a number of studies have been conducted to investigate mode localization in periodic structures [10}14]. Studies on normal mode localization in nearly periodic system in one, two and three dimensions were published by Cai et al. [15, 16]. Di!erent models and parameter values used in the above analyses have, however yielded di!erent and even con#icting conclusions. The localization phenomenon may occur in a rotating bladed disk under certain circumstances, such as structural or material irregularity. The natural frequencies or mode shapes of the mistuned system can be seriously a!ected by such local irregularity. The amplitudes of individual defected blades may in turn be seriously excited. The localized vibrations further increase the amplitudes and stresses locally. The shrouded bladed disk of a turbo-rotor can be regarded as a periodic system if all the blades are assembled periodically. The dynamic behavior of such a shrouded blade-disk system has been studied by Cottney and Ewisn [17], whereas the fundamental aspects of mode localization in mistuned turbomachinery rotors have been studied by Bendisksen et al. [18}21]. More recent studies by Cha and Pierre et al. [22}25] have dealt with vibrational localization in the mistuned system with a multi-mode subsystem. Most of the studies on mode localization are limited to stationary structures, with assumed structural and material irregularity. Only a few studies on the e!ects of crack and rotation on localization in a shrouded bladed disk have been conducted. The individual blades of the shrouded blade-disk are approximated as Euler}Bernoulli beams, and a two-span beam with a torsional spring is used in modelling the cracked blade. The shrouded e!ect, which is regarded as a massless spring e!ect, is considered in order to introduce the constraint between the blades. The e!ects of crack size, crack position and rotation speed on mode localization in the shrouded blade disk have been investigated in this study.
2. EQUATIONS OF MOTION
The periodic shrouded blade structure is shown in Figure 1, consisting of a rigid hub and a cyclic assembly of N coupled blades. Each blade is coupled with the adjacent one through a shroud. A massless spring k is assumed to model the Q function of this shroud. The length of the cantilever beam is ¸"R !R , and M F every blade is coupled by a spring k to the adjacent one at position R . The Q A notation l (r, t) denotes the transverse #exible de#ection of the sth blade as Q a constant rotating speed X. The moment of inertia and the cross-sectional area of the sth blade are denoted as I and A respectively, while the Young's modulus of the blade is denoted as E. Q Q 2.1.
BLADES WITHOUT CRACK
According to Kuang and Huang [26], the e!ect of coriolis force on the mode localization phenomenon in a turbo disk system is not signi"cant. By neglecting the coriolis force and the vibration in the radial direction, the kinetic energy of the sth
87
MODE LOCALIZATION IN CRACKED BLADE DISKS
Figure 1. Geometry of the rotating bladed disk.
blade is given by
oA 0M *l (r, t) Q ¹" Q (1) #[Xl (r, t)] dr, Q Q 2 *t 0F where the subscript s denotes the blade number of the disk. The corresponding strain energy of the sth blade is
*l (r, t) 1 0M 1 0M Q dr# oA X EI ;" Q Q Q 2 *r 4 0F 0F 1 # k [l (R , t)!l (R , t)]. Q A 2 Q Q> A
(R #¸)!r F
*l (r, t) Q dr *r (2)
By applying Hamilton's principle, the equation of motion of the sth blade in the transverse direction can be obtained as
* *l (r, t) EI * *l (r, t) Q Q # Q *t *t *r oA *r Q *l (r, t) 1 * *l (r, t) Q #X r Q !l (r, t)! [(R #¸)!r] Q F *r 2 *r *r
#(k #k ) l d (r!R )!k l d (r!R )!k l d (r!R )"0 Q Q\ Q A Q Q> A Q\ Q\ A for s"1, 2,2, N.
(3)
For simplifying the notations, a number of non-dimensional parameters are de"ned as r!R F and rN "0P1, rN " (4) ¸
88
J. H. KUANG AND B. W. HUANG
l (rN , t) lN (rN , t)" Q , Q ¸
(5)
(6)
XM "X
R !R F RM " A A ¸
EI , oA¸
and
R 1 F" , ¸ 2
(7)
k ¸ bM " Q . (8) EI Q By substituting these dimensionless variables into the equation of motion of the sth blade, it yields
EI * *lN *lN Q Q# oA ¸ *rN *rN *t Q *lN 1 #XM rN Q!lN ! Q 2 *rN
R * *lN R F#1 ! rN # F Q ¸ ¸ *rN *rN
#(bM #bM ) lN d (rN !RM )!b lN d (rN !RM )!b lN d (rN !RM )"0 Q Q\ Q A Q Q> A Q\ Q\ A for s"1, 2,2, N, with the boundary conditions lN (0, t)"0, Q *lN (0, t) Q "0, *rN
(9)
(10) (11)
*lN (1, t) Q "0, *rN
(12)
(13)
* *lN (1, t) Q "0. *rN *rN The solution of the above eigenvalue problem can be expressed as
K lN (rN , t)" qQ (t) tQ (rN ), (14) Q G G G where tQ (rN ) is the comparison function of equation (9) and qQ (t) the coe$cient, G G which is to be determined. For simplicity, three exact bending modes of the clamped cantilever beam tQ (rN ) are considered in this article. They are G cos j #cosh j G G (sinh j rN !sin j rN ) tQ (rN )"(cosh j rN!cos j rN )! G G G G G sin j #sinh j G G for i"1, 2,2, m, (15) The coe$cients j are the roots of G cos j rN cosh j rN #1"0. G G
(16)
89
MODE LOCALIZATION IN CRACKED BLADE DISKS
Using Galerkin method, the equation of motion of the sth blade can be derived in matrix form as EI +[[k] #[K] [m] +qK , # Q Q Q Q oA ¸ Q #b +t(RM ), +t(RM ),2#b +t(RM ), +t(RM ),2 ] +q, Q A Q A Q Q\ A Q A Q Q !b +t(RM ), +t(RM ),2 ] +q, !b +t(RM ), +t(RM ),2 +q, ,"0 Q\ A Q A Q\ Q\ Q A Q A Q> Q> for s"1, 2,2, N and +t(r), "[tQ (r), tQ (r),2, tQ (r)]2. Q K Due to the cyclic arrangement of blades +q, "+q, , ,> where the elements (m)Q , (k)Q and (K)Q are given as GH GH GH (m)Q "(m)Q " tQ (rN ) tQ (rN ) drN , GH HG G H dtQ (rN ) dtQ (rN ) G H drN , (k)Q "(k)Q " GH HG drN drN R dtQ (rN ) H !tQ (rN ), (K) "XM tQ (rN ) rN # F H GH G drN ¸ 1 R R * dtQ (rN ) F#1 ! rN # F H ! 2 ¸ ¸ *rN dr
(18)
2.2.
(17)
(19)
drN .
(20)
BLADE WITH A CRACK
Considering a crack on the mth blade of this system, it may be regarded as a mistuned system. The response of the mth cracked blade can be expressed as K lN (rN , t)" qK (t) tK (rN ), (21) K G G G where tK (rN ) are comparison functions of equation of motion of the cracked blade. G The qK (t) are the coe$cients to be determined. For simplicity, only three bending G modes are considered in this study, and they are calculated from the cracked beam model. The cracked blade, as shown in Figure 2, is modelled as a two-span beam coupled with a torsional spring. This cracked beam model has been studied by Rizos and Asprogathos [9]. The notation ¸ denotes the position of the crack, while A the depth of the crack is a. The #exibility G caused by the crack with depth a can be derived from Broke's approximation [27] as (1!k) K (Pb) dG '" , E 2b da
(22)
90
J. H. KUANG AND B. W. HUANG
Figure 2. Geometry of the cracked blade.
where K is the stress intensity factor under mode I loading, k the Poisson Ratio, ' b the width of the blade, h the height of the blade, P the bending moment at the @ crack, and G the #exibility of the blade. The magnitude of stress intensity factor can be derived from Tada's formula [28] as 6P (23) K " @ (nch F (c), ' ' gh where
2 nc 0)923#0)199 [1!sin (nc/2)] F (c)" tan , ' nc 2 cos (nc/2)
(24)
a c" . h
(25)
By substituting the stress intensity factor K in equation (22), it leads to ' 6(1!k) h G" Q(c) EI and A Q(c)" nc F (c) dc. ' Since 1 k " 2 G
(26)
(27)
the bending sti!ness k of the cracked section of a blade can be expressed as 2 EI . (28) k " 2 6(1!k) hQ(c)
91
MODE LOCALIZATION IN CRACKED BLADE DISKS
The corresponding kinetic energy of the cracked blade is
0
*A *l (r, t) 0M *l (r, t) K K dr# dr . *t *t F *A The strain energy of the cracked blade can thus be found as oA ¹" K K 2
*l (r, t) 1 0M *l (r, t) 1 *A K K dr# dr EI EI ;" K K K 2 *r 2 *r 0F *A 1 *l *l K (¸ , t)! K (¸ , t) . # k A A 2 2 *r *r
(29)
(30)
In this case, the equation of motion of the cracked blade can be derived as EI *lN K K# oA ¸ *t K EI *lN K K# oA ¸ *t K
* *lN K *rN *rN
"0 for rN : 0 to ¸M , A
(31)
* *lN K *rN *rN
"0 for rN : ¸M to 1, A
(32)
where ¸ !R F. ¸M " A A ¸
(33)
The boundary conditions of this cracked blade are lN (0, t)"0, K *lN K (0, t)"0, *rN
(34)
lN (¸M , t)"l (¸M , t), K A K A *lN *lN K (¸M , t)" K (¸M , t), A A *rN *rN
(36)
*lN *lN K (¸M , t)" K (¸M , t), A A *rN *rN EI *lN *lN *lN K (¸M , t)" K (¸M , t), K (¸M , t)# K A A A k ¸ *rN *rN *rN 2 *lN K (l, t)"0, *rN *lN K (l, t)"0. *rN
(35)
(37) (38) (39) (40) (41)
92
J. H. KUANG AND B. W. HUANG
The solutions of such a two-span beam with a torsional spring can be assumed to be as K lN (rN , t)" qK (t) K (rN ), (42) K G G G K lN (rN , t)" qK (t) K (rN ), (43) K G G G where
K (rN )"A sin (k rN )#A cos (k rN )#A sinh (k rN )#A cosh (k rN ), (44) G G G G G
K (rN )"B sin (k rN )#B cos (k rN )#B sinh (k rN )#B cosh (k rN ). (45) G G G G G Three modes, i.e., m"3 are assumed. The coe$cients of functions K (rN ) and G
K (rN ) can be obtained by substituting equations (44) and (45) in the above G boundary conditions. Table 1 lists coe$cients for the blade with a crack at the root. The coe$cients for the blade with a crack at di!erent positions, i.e., ¸M "0)2 and 0)4 A are presented in Table 2. The comparison functions of the whole cracked blade can thus be expressed as tK (rN )" K (rN ) u (rN )! K (rN ) u (rN !¸M )# K (rN ) u(rN !¸M ), A A G G G G where u(rN ) is a unit step function.
(46)
TABLE 1 ¹he magnitude of coe.cients for the blade with a crack at the root (a) ¸M "0)0, c"0)01 A B i"1 i"2 i"3
1)00010 1)00019 1)00024
B
B
!1)36214 !0)98167 !1)00037
!0)99987 !0)99981 !0)99973
B
B
k G
1)36214 0)981678 1)00051
1)8750 4)6939 7)8546
B
(b) ¸M "0)0, c"0)05 A B i"1 i"2 i"3
1)00245 1)00441 1)00745
!1)36042 !0)97737 !0)99328
!0)99755 !0)99559 !0)99254
B
B
k G
1)36042 0)977379 0)993322
1)8733 4)6896 7)8473
B
(c) ¸M "0)0, c"0)10 A B i"1 i"2 i"3
1)00929 1)01656 1)02795
!1)35542 !0)96499 !0)97302
!0)99070 !0)98343 !0)97211
Note: Only B , B , B , B , as the crack at the root of blade.
k G
1)35542 0)964995 0)972909
1)8683 4)6772 7)8269
B
93
MODE LOCALIZATION IN CRACKED BLADE DISKS
TABLE 2 ¹he magnitude of coe.cients for the blade with a crack at di+erent locations (a) ¸M "0)2, c"0)10 A A i"1 i"2 i"3
i"1 i"2 i"3
A
A
A
k G
1)0000 1)0000 1)0000
!1)3597 !0)98138 !1)01206
!1)0000 !1)0000 !1)0000
1)3397 0)98138 1)01206
1)8715 4)6940 7)8327
B
B
B
B
k G
1)00631 1)00071 1)02197
!1)36217 !0)98234 !0)97863
!0)99274 !1)99822 !1)24493
1)35710 0)98007 1)24573
1)8715 4)6940 7)8327
A
A
A
k G
(b) ¸M "0)4, c"0)10 A A i"1 i"2 i"3
i"1 i"2 i"3
1)0000 1)0000 1)0000
!1)36154 !0)98337 !1)00139
!1)0000 !1)0000 !1)0000
1)36154 0)98337 1)00139
1)8736 4)6882 7)8488
B
B
B
B
k G
1)00316 1)00280 1)01379
!1)36449 !0)97374 !1)00244
!0)99441 !1)03175 !1)14455
1)35800 0)01356 1)14534
1)8736 4)6882 7)8488
By substituting equations (42) and (43) into equations (31) and (32), the equation of motion of the cracked blade can be obtained in the matrix form as EI K +[[k] #[K] [m] +qK , # K K K K oA ¸ K #b +t(RM ), +t(RM ),2#b +t(RM ), +t(RM ),2 ] +q, K A K A K K\ A K A K K !b
K\
+t(RM ), +t(RM ),2 +q, !b +t(RM ), +t(RM ),2 +q, ,"0 A K A K\ K\ K A K A K> K>
(47)
with
(k)K "(k)K " HG GH
tKG (rN ) tKH (rN ) drN ,
(48)
dtK (rN ) dtK (rN ) G H drN , drN drN
(49)
(m)K "(m)K " HG GH
94
J. H. KUANG AND B. W. HUANG
(K) "XM GH
1 ! 2
2.3.
R dtK (rN ) H !tK (rN ), rN # F H ¸ drN
tK (rN ) G
R R * dtK (rN ) F#1 ! rN # F H ¸ *rN drN ¸
drN .
(50)
EQUATION OF MOTION OF THE MISTUNED SYSTEM
For the sake of simplicity, the same comparison function is assumed for each individual blade, i.e., tQ (rN ),t (rN ). The equation of motion of the entire disk H H system can be expressed as [M ] +XG ,#[K] +X,"0,
(51)
where the system mass matrix [M] and the system sti!ness matrix [K ] are [m] 0 0
0
0
0 [m] 0 [m]
.
0
0
0
.
0
0
0
.
0
0
0
.
.
.
[M]" .
.
.
.
0
0
0
. [m]
0
0
0
.
0
0
0
.
EI [K]" oA¸
[aN ]
!b [U]
!b [U]
[aM ]
0
!b [U]
[aN ]
.
.
0
,
0 0 ,\ 0 [m] 0 ,\ 0 0 [m]
(52)
,
.
0
0
!b [U] ,
!b [U] .
0
0
0
.
0
0
0
.
.
.
.
.
0
0
.
[aN ]
0
0
0
. !b [U] ,\
!b [U] ,
0
0
.
0
,\
0
!b
,\
[aN ]
[U]
,\
!b
,\
[U]
,
0 !b [U] ,\ [aN ] ,
(53) +X,"[ +q,2 , +q,2 ,2, +q, 2 , +q,2 ]2 ,\ ,
(54)
[aN ] "[k ]#[K] #b [U]#b [U], Q Q Q Q Q\ [U]"+t (RM ) , +t (RM ),2, G A H A bM "bM . ,
(55)
and
(56) (57)
95
MODE LOCALIZATION IN CRACKED BLADE DISKS
The non-dimensional frequency u of equation (51), i.e., the natural frequency of the L mistuned system, is de"ned as uN "u L L
EI for n"1, 2,2 . oA¸
(58)
The lowest natural frequency of the tuned system without any rotational speed, i.e., uN *"3)516, is used for reference in the following numerical analysis. 2.4.
FORCE RESPONSE
Consider the blades to be excited by a uniformly distributed harmonic force h cos ut. The equation of motion of the sth blade can be rewritten in the matrix form as [M] +XG ,#[K] +X ,"+F , cos ut,
(59)
where +F,"[ + f ,2 , + f ,2 ,2, + f , 2 , + f ,2 ]2, ,\ , f " t (rN ) hM dr, Q Q h hM " . oA Q Let the dynamic response of the bladed-disk system be
(60)
(61) (62)
+X ,"[; ] +g(t),.
(63)
The modal matrix [; ] of the mistuned system is calculated from equation (51). With the initial condition as g (0)"0, gR (0)"0, the response of the system can be G G derived as
C 1!cos (uN #uN ) t 1!cos (uN !uN ) t G G # g (t)" G cos uN t G uN !uN 2uN uN #uN G G G sin (uN !uN ) t sin (uN #uN ) t G G #sin u t ! for uOu G uN #uN uN !uN G G
(64)
with +C ,"[;]2 +F,.
(65)
The corresponding resonant response can be written as C g (t)" Q [cos u t(1!cos 2uN t)#sin uN t(2uN t!sin 2uN t)] Q Q Q Q Q G 4uN Q where uN is one of the natural frequencies of system. Q
for uN "uN , (66) Q
96
J. H. KUANG AND B. W. HUANG
3. NUMERICAL RESULTS AND DISCUSSIONS
A rigid hub attached to 46 uniform blades is used to approximate the bladed disk. The existence of mode localization for a mistuned bladed disk system with a cracked blade is investigated. The e!ects of rotational speed of the disk, depth and location of the crack on mode localization have also been studied. As the crack propagates on a blade, it may not only alter the dynamic behavior of this blade, but may also introduce the so-called mode localization phenomenon in
Figure 3. Tip displacements of the uncracked and cracked systems with di!erent depths of crack (for the the 1st mode), RM "0)5, ¸M "0)0, b"0)2, XM /uN *"0)0, uN *"3)516: (a) system without crack A A c"0)0 (u "3)516); (b) system with a crack c"0)01 (u "3)516); (c) system with a crack c"0)1 (u "3)476).
MODE LOCALIZATION IN CRACKED BLADE DISKS
97
the whole mistuned system. In this numerical example, a crack is assumed on the 23rd blade. 3.1.
FREE VIBRATION ANALYSIS
Figure 3(a) shows the tip vibration pattern of the tuned system at the lowest resonant frequency uN *"3)516. The vibrational energy is uniformly distributed on the individual blades of the tuned system. The corresponding tip vibrational
Figure 4. Tip displacements of the system with a crack at di!erent locations (for the the 1st mode), RM "0)5, c"0)1, b"0)2, XM /uN *"0)0, uN *"3)516: (a) system without crack at ¸M "0)0 (u "3)476); A A (b) system with a crack at ¸M "0)2 (u "3)495); (c) system with a crack at ¸M "0)4 (u "3)511). A A
98
J. H. KUANG AND B. W. HUANG
patterns for the mistuned system, with di!erent depths of crack, i.e., c"0)01 and 0)1, are shown in Figures 3(b) and 3(c). These "gures indicate that the vibrational energy is con"ned on the blades near the cracked blade of these two mistuned systems. It can also be observed that the localization pattern varies with the depth of the crack. Generally, the model localization phenomenon is enhanced with increase in the crack depth. The localization pattern may also be a!ected by location of the crack. The tip vibration pattern for di!erent crack locations are shown in Figure 4(a)}4(c), which indicate that the
Figure 5. Tip displacements of the cracked system under di!erent rotating speeds (for the 1st mode), c"0)01, ¸M "0)0, RM "0)5, bM "0)2, uN *"3)516: (a) crack system without speed XM /u *"0)0 A A (uN "3)516); (b) crack system at the speed XM /u *"0)25 (uN "3)528); (c) crack system at the speed XM /u *"0)5 (uN "3)529).
MODE LOCALIZATION IN CRACKED BLADE DISKS
99
localization phenomenon is suppressed as the crack position is changed from ¸M "0)0 to 0)4. A Figures 5(a)}5(c) illustrate the e!ect of rotational speed X on mode localization in the mistuned system. These results appear to indicate that the con"nement of vibrations may be strongly localized as the rotational speed is increased. In other words, mode localization phenomenon may be enhanced as the rotational speed is increased. The weak or strong localization phenomenon depends upon the magnitude of disorder and the modal coupling e!ect. Now, the strong modal coupling e!ect is studied in this investigation. Figure 6(a) shows the weak localization at the lower order mode uN as the strong coupling sti!ness at the blade tip. However, as above, the strong localization may appear at the higher order mode as shown in Figure 6(b). 3.2.
FORCE RESPONSE
Consider that the blades of the system are excited by a uniformly distributed harmonic force with an excitation frequency u. A signi"cant resonant response of the cracked blade can be observed as the excitation frequency approaches the localization frequency, i.e., the lowest natural frequency of the mistuned system. Figure 7 shows the time response of the blades near the cracked blade, for excitation at the localization frequency uN "uN "3)529. Figure 8 shows the changes in frequency response of the 23rd blade with or without a crack located at the "xed end. There is only a single peak, uN "uN "3)516, in the frequency spectra of the 23rd blade in the tuned system, which shows that every individual blade possesses the same frequency and mode. However, a group of peaks is observed for the corresponding cracked blade, with a crack at the blade root. The lowest natural frequency at u "3)476 is the localization frequency. Figure 9 displays the e!ect of rotational speed on the frequency response of the cracked blade in this mistuned system, which reveals that the rotational speed is a signi"cant factor for mode localization. The mode localization phenomenon is enhanced signi"cantly with increase in the rotational speed.
4. CONCLUSIONS
The e!ects of crack on mode localization in the rotating blade-disk system have been studied. The major conclusions that can be drawn from this study are summarized as follows: (1) It has been found that the crack on a bladed is one of the parameters for localization in the shrouded blade-disk system. For a coupled blade system, the existence of a crack may change the vibrational mode drastically, i.e., from the extended to the localized. (2) The rotational speed has a signi"cant in#uence on mode localization in the mistuned system. As the rotational speed of the disk is increased, the mode localization phenomenon is enhanced.
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Figure 6. Tip displacements of the cracked system with strong modal coupling sti!ness (for the 1st and 2nd order modes), c"0)05, ¸M "0)0, RM "1)0, b"10)0, XM /uN *"0)5, uN *"3)516: (a) the "rst A A order mode (u "3)549); (b) the second order mode (uN "22)085).
Figure 7. Tip responses of the blades under the lowest localization frequency u "5)329; c"0)01, ¸M "0)0, RM "0)2, bM "0)2, XM /uN *"0)5. A A
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Figure 8. Di!erence between frequency responses of the 23rd blade with and without a crack, bM "0)2, RM "0)5, XM /uN *"0)0, ¸M "0)0: (a) c"0)0; (b) c"0)1. A A
Figure 9. Frequency response vibration of the cracked blade with or without considering the rotating speed, bM "0)2, RM "0)5, ¸M "0)0, c"0)01: (a) XM /uN *"0)0, (b) XM /uN *"0)5. A A
(3) The magnitude and location of the crack on the blade also a!ect mode localization in the shrouded blade-disk system. (4) Only the cracked blade of the mistuned system shows a signi"cant resonant response as the excitation frequency approaches the localization frequency. Obvious beat response on blades near the cracked blade is observed at localization frequency.
ACKNOWLEDGMENTS
The authors would like to acknowledge the support of the National Science Council, R.O.C., through grant no. NSC84-2212-E110}007.
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REFERENCES 1. H. L. BERNSTEIN and J. M. ALLEN 1992 ¹ransactions of the American Society of Mechanical Engineers, Journal of Engineering for Gas ¹urbines and Power 114, 293}301. Analysis of cracked gas turbine blades. 2. J. R. GALVELE 1990 Corrsion Science 30, 955}958. Surface mobility}stress corrsion cracking mechanism of steels for steam turbine rotor. 3. M. KRAWCZUK and W. M. OSTACHOWICZ 1993 ¹ransactions of the American Society of Mechanical Engineers, Journal of
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19. D. AFOLABI 1985