Dynamic analysis of asymmetric bladed-rotors supported by anisotropic stator

Journal of Sound and Vibration 331 (2012) 5224–5246

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Dynamic analysis of asymmetric bladed-rotors supported by anisotropic stator Kyung-Taek Kim, Chong-Won Lee n Center for Noise and Vibration Control (NOVIC), Department of Mechanical Engineering, KAIST, Science Town, Daejeon 305–701, Republic of Korea

a r t i c l e in f o

abstract

Article history: Received 5 January 2012 Received in revised form 4 June 2012 Accepted 10 June 2012 Handling Editor: L.G. Tham Available online 28 July 2012

For the mathematical convenience of conventional rotor dynamic analysis, the components of the entire rotor system are often classified into two parts: the stationary parts and the rotating parts, depending upon whether or not the corresponding components rotate with respect to their axes of rotation. Even for bladed-rotors, the rotor blades have been treated along the same lines as the ordinary rotating components such as the rotor disk and the shaft. The distinct dynamic nature of the blades, therefore, has not been thoroughly taken into account in the conventional rotor dynamic analysis. In this paper, the rotating parts of a bladed-rotor system are further subdivided into the rotor blade-group and the other rotating components. The equation-of-motion for the bladed-rotor system is then developed, by modifying the conventional general rotor system equation to adopt the blade-group dynamics without loss of generality. Complex modal solutions to the bladed-rotor system are investigated based on a new modulated coordinate transformation approach, yielding newly defined directional frequency response functions that characterize the nature of asymmetry present in the rotating blade-array. Finally, the effects of the blade-group asymmetry on the rotor system dynamics are demonstrated with a pertinent numerical example. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction Traditionally, the dynamics of bladed-rotor systems has been investigated in two different ways, depending upon the simplicity in the modeling scheme. One way is to study first the dynamics of the bladed-rotor system, assuming that the blades are rigidly attached to the system, and then, separately, the local vibrations of the blades that are fixed to the ground. This approach essentially ignores the weak dynamic coupling between the blade assembly and the rest of the bladed-rotor system, mostly for mathematical convenience. Classical theory of rotor dynamics, which has been extensively developed in the past based on this simplistic bladed-rotor model, well applies to many industrial turbo-machines such as compressors, turbines and generators. For example, the complex modal analysis theory, developed for the rotating shaft systems with rigid disks and blades by Lee [1,2], characterizes the rotor system dynamics in the presence of either stationary or rotating asymmetry by means of directional frequency response functions (dFRFs). The dFRFs are a powerful tool for the rotating shaft system design, identification, damage and fault detection, and so on [3–5]. Another way to study the dynamics of the bladed-rotor system is to introduce a simplistic model for the rotor system excluding the blade assembly, when the rotor blades dominate the entire dynamics. Rotary wing vehicles, propeller engines and miscellaneous

n

Corresponding author. Tel.: þ 82 42 350 3016; fax: þ82 42 350 8220. E-mail addresses: [email protected] (K.-T. Kim), [email protected] (C.-W. Lee).

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.06.014

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

Nomenclature 0 A, B B BðOtÞ cB, cS C, C C

zero matrix or vector system matrices in generalized state-space form highest number of nodal-diameters in blade traveling waves basis multi-blade coordinate transformation matrix blade and stator damping generalized damping matrices in stationary and rotating coordinates basis complex coordinate transformation matrix

z}|{N diag½ U  block diagonal matrix augmented with N diagonal element matrices fðtÞ, f ðtÞ external force vectors in stationary and rotating coordinate systems f C bla ðtÞ, f C rot ðtÞ complex force vectors associated with rotor blade-array and rotor disk-shaft systems F(t) input vector in generalized state-space form g(t) external force vector for general bladedrotor system g;n(t) modulated external force vector (¼ g(t) ejnOt) G(jo) Fourier transform of g(t) H(jo) directional frequency response function HgC ;0pC ;0 ðjoÞ directional frequency response function for the system symmetry HgC ;0pC ;0 ðjoÞ directional frequency response function for the stator anisotropy HgC ;2pC ;0 ðjoÞ directional frequency response function for the system asymmetry HgC ;2pC ;0 ðjoÞ directional frequency response function for the coupled effect between the system anisotropy and asymmetry HgR ; þ 1pC ;0 ðjoÞ directional frequency response function for the blade-group asymmetry HgR ;1pC ;0 ðjoÞ directional frequency response function for the coupled effect between the bladegroup asymmetry and stator anisotropy HgR ;0pR ;0 ðjoÞ directional frequency response function for the blade-group symmetry H(jo) directional frequency response matrix HgC ;0pC ;0 ðjoÞ directional frequency response matrix for the system symmetry HgC ;0pC ;0 ðjoÞ directional frequency response matrix for the stator anisotropy HgC ;2pC ;0 ðjoÞ directional frequency response matrix for the system asymmetry HgC ;2pC ;0 ðjoÞ directional frequency response matrix for the coupled effect between the system anisotropy and asymmetry HgR ; þ 1pC ;0 ðjoÞ directional frequency response matrix for the blade-group asymmetry HgR ;1pC ;0 ðjoÞ directional frequency response matrix for the coupled effect between the bladegroup asymmetry and stator anisotropy HgR ;0pR ;0 ðjoÞ directional frequency response matrix for the blade-group symmetry I identity matrix pffiffiffiffiffiffiffi j imaginary number ( ¼ 1)

JD

5225

mean value of disk principal diametrical mass moments-of-inertia ( ¼(JDZ þJDx)/2) JDp disk polar mass moment-of-inertia JDZ, JDx disk principal diametrical mass moments-ofinertia kB, kS blade and stator stiffness K, K generalized stiffness matrices in stationary and rotating coordinates lB blade length l left eigenvector mB blade mass M number of rotor blades M, M generalized mass matrices in stationary and rotating coordinates N number of vibration modes NC , NR number of complex- and real-valued coordinates P fundamental period p(t) coordinate vector for general bladed-rotor system p;n(t) modulated coordinate vector ( ¼ p(t) ejnOt) P(jo) Fourier transform of p(t) qv ðtÞ,qw ðtÞ,qy ðtÞ coordinates for local blade motions qck ðtÞ,qsk ðtÞ multi-blade coordinates for the kth sine and cosine components in blade traveling waves qk ðtÞ complex multi-blade coordinate for the kth harmonic component in blade traveling waves ( ¼ qck ðtÞ þjqsk ðtÞ) qo ðtÞ,qd ðtÞ multi-blade coordinates for the collective and differential collective blade-group motions q(t), q(t) coordinate vectors in stationary and rotating coordinate systems qC bla ðtÞ,qC rot ðtÞ complex coordinate vectors associated with rotor blade-array and rotor disk-shaft systems t time variable TB ðOtÞ multi-blade coordinate transformation matrix TC complex coordinate transformation matrix rD rotor disk radius r right eigenvector u modal vector v(z,t) blade displacement in the in-plane (lead-lag) direction v adjoint modal vector w(z,t) blade displacement in the out-of-plane (flapwise) direction w(t) state vector in generalized state-space form z spatial variable y(t), z(t) real-valued displacement vectors in stationary coordinate system a shape factor for gyroscopic effect ( ¼JDp/JD) dB rotor blade-group asymmetry ( ¼(kB2  kB1)/ kB1 ¼  (kB3  kB1)/kB1) dD rotor disk asymmetry ( ¼(JDZ JDx)/2JD) k dirðmÞsðlÞ Kronecker delta ( ¼1 for i¼k, r ¼s and m¼l, otherwise 0) DS stator anisotropy ( ¼(kSy  kSz)/2kS) Z(t) principal coordinate y(z,t) blade displacement in the torsional (pitchwise) direction

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l eigenvalue fi azimuth angle of the ith blade (¼ 2p(i 1)/M) fv ðzÞ, fw ðzÞ, fy ðzÞ eigenfunctions for local blade

o O

motions circular frequency rotational speed

Superscripts ðUÞB ,ðUÞF (U)T ðUÞðlÞ ðUÞ

backward and forward modes transpose of a matrix or a vector vibration mode index complex conjugate

Subscripts ðUÞB ðUÞC ðUÞC (U)f, (U)b, ðUÞh

real multi-blade coordinate system complex-valued parameter complex multi-blade coordinate system (U)r system symmetry, anisotropy and asymmetry for conventional rotor system harmonic order index

blade index nodal diameter index for blade traveling waves (U)m, (U)q, (U)s matrix indices for the equation-of-motion associated with blade travelling wave excitation ðUÞ;n modulation index (U)o, (U)p matrix indices for the equation-of-motion associated with blade collective excitation (U)p(m) index for the eigensolutions in cluster m, involved in the coupled blade collective modes (U)r(m) index for the eigensolutions in cluster m, involved in the rotor whirling and blade traveling wave modes ðUÞR real-valued parameter (U)y, (U)z component directions in stationary coordinate system ðUÞ1 infinite dimension ðUÞi (U)k

Abbreviations dFRF dFRM

directional frequency response function directional frequency response matrix

fan-rotor systems for fluid flow control may belong to this category. For this sort of rotor system, one of the most distinctive dynamic behaviors is known as the self-excited mechanical instability, popularly referred to as ‘‘ground resonance’’ in helicopter fields, and the first theoretical examination of this phenomenon was given by Coleman [6]. In his work, the multiblade coordinate, which represents the blade-group motions from the viewpoint of the rotating disk-shaft system, was introduced and utilized to investigate the dynamic interactions between the blade-group and the other rotor components. Since then, the method of multi-blade coordinates has commonly been adopted for dynamic analysis of such bladed-rotor systems taking the dynamics of all the sub-components, including the rotor blades, into consideration [7–9]. Most applications of the multi-blade coordinate approach, however, have been confined to the bladed-rotor systems having identical rotor blades, because of difficulties not only in obtaining solutions of the governing equation for the rotor systems with dissimilar rotor blades but also in providing physical interpretation to the solutions. Even though many studies in the field of turbo-machinery have treated the mistuning (detuning) of the rotor blades, they essentially dealt with the local vibrations of the blade–disk assembly only and ignored the interaction with the rotor shaft and stationary parts [10–12]. The two conventional modeling schemes developed for dynamic analysis of bladed-rotors are too simplistic to accurately predict the dynamic interactions between the blade assembly and the rest of the bladed-rotor system because the modern bladed-rotary machines are designed to be more flexible and sophisticated than ever. Furthermore, intentionally or not, numerous dynamic effects leading to a non-symmetric nature in the rotating blade-array are often imposed through various routes. It is, therefore, necessary to consider the non-symmetric configurations of the rotating flexible blade-array, together with both the rotor asymmetry and the stator anisotropy, in order to secure better accuracy and reliability in analyzing the bladed-rotor system dynamics. The present paper derives a generalized model for bladed-rotor systems, in which the non-symmetric properties are taken into consideration for all the dynamic components, including the rotor blade-array, by subdividing its rotating components into the two parts, the rotor blade-array and the other rotating part, and by introducing the complex multi-blade coordinates. Since the dynamics associated with the system non-symmetry is characterized by periodic parameters in the equation-of-motion, the general bladed-rotor model takes the form of a typical periodically time-varying linear differential equation. The modulated coordinate transformation approach [13,14] is applied to treat the periodic terms in the equation-of-motion, which complicate the dynamic analysis. The complex modal solutions are then obtained by the complex modal analysis theory. Based on these modal solutions, three additional dFRFs are newly defined to characterize the distinct dynamic nature of the rotating bladearray, together with the four conventional dFRFs that characterize the dynamics of the bladed-rotor system as a whole [14]. Finally, a set of illustrative numerical examples, regarding various symmetry conditions on the blade-array and the other rotor and stator components, demonstrate the effects of the blade-group asymmetry on the rotor system dynamics and the features of the proposed approach in analyzing the bladed-rotor system dynamics. 2. Analysis model For bladed-rotor systems, a number of rotor blades are usually arranged in rows and integrated with the other rotating components such as the rotor disk and the shaft. Unlike the case of a single isolated blade, the blade-array may exert a

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significant influence on the dynamics of the entire rotor system by interacting with the other components [15–17]. In this section, the rotating blade-array is modeled within the framework of the rotor disk-shaft system by employing the multiblade coordinates. The conventional rotor disk-shaft system model is then extended to account even for the blade-group dynamics, combining it with the blade-array model on the basis of the complex coordinate representation. 2.1. Conventional general rotor system The conventional general rotor system, whose rotating components are considered simply as the rotor disk-shaft assembly, can be represented, by the linear periodically time-varying system equation with the fundamental period of P ( ¼ p/O), as [14,18,19] Mf q€ C rot ðtÞ þ Cf q_ C rot ðtÞ þ Kf qC rot ðtÞ þ M q€ ðtÞ þ C q_ ðtÞ þ K q ðtÞ b C rot

b

(1)

b C rot

C rot

þ Mr ej2Ot q€ C rot ðtÞ þ Cr ej2Ot q_ C rot ðtÞ þ Kr ej2Ot qC rot ðtÞ ¼ f C rot ðtÞ and qC rot ðtÞ ¼ yðtÞ þ jzðtÞ,

f C rot ðtÞ ¼ f y ðtÞ þ jf z ðtÞ,

qC rot ðtÞ ¼ yðtÞjzðtÞ, f C rot ðtÞ ¼ f y ðtÞjf z ðtÞ, (2) pffiffiffiffiffiffiffi where the imaginary number is denoted by j ( ¼ 1); the symbol ‘–’ indicates the complex conjugate; O represents the rotational speed; the complex-valued generalized mass, damping and stiffness matrices are respectively denoted by M, C and K; the subscripts f, b and r classify the system matrices according to the system symmetric properties, i.e. f refers to the terms associated with the isotropic nature of the rotor system while b and r indicate the terms solely due to the presence of the stator anisotropy and the rotor asymmetry, respectively [14,18]; the orthogonal pairs of the real-valued displacement and external force vectors, y(t)  z(t) and fy(t) fz(t), respectively, take the real and imaginary parts of the complex displacement and force vectors, qC rot ðtÞ and f C rot ðtÞ, which describe the rotor whirling motion and excitation. Eq. (1) reveals that the description of the rotor asymmetry based on the stationary coordinate leads to the periodic coefficients in the equation-of-motion. Even though these become time-invariant with the observation on the rotating reference frame, new periodic terms will then arise due to the stator anisotropy, implying that the system inherently exhibits a periodically time-varying dynamic nature. 2.2. Rotating blade-array system We consider an array of rotating blades consisting of M flexible blades, rigidly attached to a rotor hub and evenly spaced in azimuth around the rotational axis as shown in Fig. 1. The blades undergo three types of elastic deformations: the out-ofplane (flap-wise) and in-plane (lead-lag) bending and the torsional (pitch-wise) vibrations, respectively, denoted by w(z,t), v(z,t) and y(z,t) in Fig. 2. The deformation of the ith blade, which is a function of both space and time, can be expressed, using the separation of variables, and approximated with only a finite, but sufficient, number of modes (N), as wi ðz,tÞ ffi

N X

ðlÞ fðlÞ vi ðz,tÞ ffi wi ðzÞqwi ðtÞ,

l¼1

N X

ðlÞ fvi ðzÞqðlÞ ðtÞ, yi ðz,tÞ ffi vi

l¼1

N X

ðlÞ fðlÞ yi ðzÞqyi ðtÞ,

(3)

l¼1

where z is the radial position along the length of the blade; the superscript l in parenthesis is the vibration mode index which ðlÞ ðlÞ ðlÞ ranges from 1 to N; hence, fwi ðzÞ, fvi ðzÞ and fyi ðzÞ indicate the lth eigenfunctions of the blade associated with the out-ofplane, in-plane and torsional vibrations, respectively, and qðlÞ ðtÞ, qðlÞ ðtÞ and qðlÞ wi vi yi ðtÞ represent the corresponding generalized coordinates. By means of various methods for deriving the equation-of-motion, such as those based on D’Alembert’s principle and on variational principles, the linear ordinary differential equation for the rotor blade-array model can then be derived as € þ C qðtÞ _ þ KqðtÞ ¼ f ðtÞ M qðtÞ

(4)

and 9 8 qð1Þ ðtÞ > > > > > > > = < qð2Þ ðtÞ > , qðtÞ ¼ > > ^ > > > > > ; : qðNÞ ðtÞ >

8 ðlÞ 9 > > > > > q1 ðtÞ > > > > = < qðlÞ ðtÞ > ðlÞ 2 , q ðtÞ ¼ > > ^ > > > > > > > > : qðlÞ ðtÞ ; M

8 ð1Þ 9 f ðtÞ > > > > > > > = < f ð2Þ ðtÞ > f ðtÞ ¼ , > > ^ > > > > > ; : ðNÞ > f ðtÞ

8 ðlÞ 9 > > > > > f 1 ðtÞ > > > > = < f ðlÞ ðtÞ > ðlÞ 2 , f ðtÞ ¼ > > ^ > > > > > > > > : f ðlÞ ðtÞ ;

(5)

M

where the total number of blades is denoted by M; the italic expression implies that they are defined in the rotating frame of reference; the constant matrices, M, C and K, are the generalized mass, damping and stiffness matrices, respectively, defined in the rotating local blade coordinates; the MN  1 coordinate vector q(t) consists of all the generalized blade coordinates, and f(t) is the corresponding generalized external force vector. Note here that, for simplicity, only an arbitrary

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Fig. 1. Rotating blade-array model consisting of M-blades evenly spaced at the outer radius of a support structure: O xyz, stationary reference frame; Ri  xiyizi, rotating reference frame aligning its zi-axis with the centerline of the ith blade; O, rotational speed; ji, azimuth angle of the ith blade from a reference position.

xi

i

vi (z , t) Pi

i (z , t) i wi (z , t)

Ω

yi Ri

Fig. 2. Arbitrary cross-section of the ith rotor blade undergoing the out-of-plane (flap-wise, wi(z,t)), in-plane (lead-lag, vi(z,t)) bending and the torsional (pitch-wise,yi(z,t)) vibrations: Ri  xiyizi, rotating reference frame; Pi  xiZizi, blade fixed frame.

single motion among the three in Eq. (3) is considered in deriving the equation-of-motion, yet the analysis can be extended to treat all the blade motions without difficulty, even together with their coupled dynamic behavior. Therefore, the subscripts w, v and y can be arbitrarily assigned to all the coordinate and force vectors. Since the equation-of-motion (4), frequently seen in the literature that deals with rotating beam vibration, describes the blades dynamics at the level of a single blade entity, its solution provides useful information on the local dynamics of the individual blades. For this reason, the dynamic analysis based on the local blade coordinate is often applied for the problems in which the local blade response itself is of paramount importance, such as rotor blade design for preventing the blade failure caused by excessive vibration and the resultant dynamic stress. Nevertheless, this approach is not suitable for investigating the dynamics of the bladed-rotor system as a whole, because the blades exert dynamic influences on the rotor disk and shaft at the blade-group level rather than the individual blade unit.

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2.2.1. Multi-blade coordinate transformation In order to examine the overall dynamic behavior of the rotor blade-array and the interactions between the bladegroup and the other rotor and stator components, the multi-blade coordinate transformation approach is employed. The multi-blade coordinates describe the blade motions, such as those mentioned in Eq. (3), in terms of the distinct vibratory patterns of the blade-group associated with the rotor disk–shaft system vibrations—namely, the torsional, axial (longitudinal), lateral (bending) and precessional (tilting) vibrations. Based on the multi-blade coordinate representation [6,20], the generalized local coordinate of the ith blade qðlÞ ðtÞ is i expressed, for an odd number of blades, as ðlÞ qðlÞ i ðtÞ ¼ qo ðtÞ þ

ðM1Þ=2 X n

o qðlÞ ðtÞcoskðOt þ ji Þ þ qðlÞ ðtÞsinkðOt þ ji Þ , ck sk

(6a)

o ðlÞ qck ðtÞcoskðOt þ ji Þ þ qðlÞ ðtÞsinkðOt þ ji Þ þ ð1Þi qðlÞ ðtÞ, sk d

(6b)

k¼1

and, for an even number of blades, as qiðlÞ ðtÞ ¼ qoðlÞ ðtÞ þ

ðM2Þ=2 X n k¼1

where ji ( ¼2p(i 1)/M) indicates the relative azimuth angle of the ith blade from a reference position, and the multi-blade coordinates are defined by qðlÞ o ðtÞ ¼

qðlÞ ðtÞ ¼ ck

M 1 X qðlÞ ðtÞ, Mi¼1 i

qðlÞ ðtÞ ¼ d

M 2 X qðlÞ ðtÞcoskðOt þ ji Þ, Mi¼1 i

M 1 X ð1Þi qiðlÞ ðtÞ, Mi¼1

qðlÞ ðtÞ ¼ sk

M 2 X qðlÞ ðtÞsinkðOt þ ji Þ: Mi¼1 i

(7)

Here, referring to Fig. 3(b), the collective multi-blade coordinate qðlÞ o ðtÞ represents the synchronous motion of all the blades; qðlÞ ðtÞ, referred to as the differential collective multi-blade coordinate that appears only for an even M, describes the d ðlÞ blade-group motion in which every blade exhibits out-of-phase movement with its adjacent blades; qðlÞ ðtÞ and qsk ðtÞ are, ck respectively, the kth order sine and cosine components of the cyclic multi-blade coordinates that characterize the bladegroup motions by the traveling waves of blade deformation, in which all the blades experience identical, cyclically varying, elastic deformation at every azimuth position. Note that the description of the M blade local motions by the M generalized multi-blade coordinates is rigorous, and it involves no approximations or arbitrary assumptions. The relation between Eqs. (6) and (7) can be conveniently rewritten as ðlÞ qðlÞ ðtÞ ¼ BðOtÞqB ðtÞ,

T ðlÞ qðlÞ B ðtÞ ¼ BðOtÞ q ðtÞ,

(8)

where for an odd M h ðlÞ qo ðtÞ qðlÞ B ðtÞ ¼

ðlÞ qc1 ðtÞ

qðlÞ s1 ðtÞ

...

qðlÞ cððM1Þ=2Þ ðtÞ

qðlÞ sððM1Þ=2Þ ðtÞ

iT

,

Fig. 3. Coordinates for in-plane (lead-lag) blade motion of an array of four rotating blades: (a) the local blade coordinates in the rotating frame, and (b) the multi-blade coordinates in the stationary frame.

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pffiffiffi 1= 2 6 pffiffiffi rffiffiffiffiffi6 1= 2 26 6 ^ BðOtÞ ¼ 6 M 6 pffiffiffi 6 1= 2 4 pffiffiffi 1= 2 2

cosðOt þ j1 Þ

sinðOt þ j1 Þ



cosððM1Þ=2ÞðOt þ j1 Þ

cosðOt þ j2 Þ

sinðOt þ j2 Þ



cosððM1Þ=2ÞðOt þ j2 Þ

^

^

&

^

cosðOt þ jM1 Þ

sinðOt þ jM1 Þ



cosððM1Þ=2ÞðOt þ jM1 Þ

cosðOt þ jM Þ

sinðOt þ jM Þ



cosððM1Þ=2ÞðOt þ jM Þ

sinððM1Þ=2ÞðOt þ j1 Þ

3

7 sinððM1Þ=2ÞðOt þ j2 Þ 7 7 7 ^ 7, 7 sinððM1Þ=2ÞðOt þ jM1 Þ 7 5 sinððM1Þ=2ÞðOt þ jM Þ (9a)

and for an even M pffiffiffi 1= 2 p ffiffiffi 6 rffiffiffiffiffi6 1= 2 6 26 BðOtÞ ¼ 6 ^ M 6 pffiffiffi 6 1= 2 4 pffiffiffi 1= 2 2

h ðlÞ qo ðtÞ qðlÞ B ðtÞ ¼

qðlÞ c1 ðtÞ



qðlÞ cððM2Þ=2Þ ðtÞ

qðlÞ sððM2Þ=2Þ ðtÞ

qðlÞ ðtÞ d

iT

,

cosðOt þ j1 Þ

sinðOt þ j1 Þ



cosððM2Þ=2ÞðOt þ j1 Þ

sinððM2Þ=2ÞðOt þ j1 Þ

cosðOt þ j2 Þ ^ cosðOt þ jM1 Þ

sinðOt þ j2 Þ ^ sinðOt þ jM1 Þ

 & 

cosððM2Þ=2ÞðOt þ j2 Þ ^ cosððM2Þ=2ÞðOt þ jM1 Þ

sinððM2Þ=2ÞðOt þ j2 Þ ^ sinððM2Þ=2ÞðOt þ jM1 Þ

cosðOt þ jM Þ

sinðOt þ jM Þ



cosððM2Þ=2ÞðOt þ jM Þ

sinððM2Þ=2ÞðOt þ jM Þ

1

3

7 ð1Þ2 7 7 7 ^ 7: M1 7 7 ð1Þ 5 ð1ÞM

(9b) The global multi-blade coordinate and force vectors are then expressed respectively as 9 9 8 8 ð1Þ > > > qð1Þ f B ðtÞ > > > > B ðtÞ > > > > > > > > ð2Þ > > > > = < q ðtÞ = < f ð2Þ ðtÞ > B B T T , f B ðtÞ ¼ TB ðOtÞ f ðtÞ ¼ , qB ðtÞ ¼ TB ðOtÞ qðtÞ ¼ > > > > > > ^ > > ^ > > > > > > > > > > ; ; : qðNÞ ðtÞ > : f ðNÞ ðtÞ > B

(10)

B

where the symbol ‘T’ denotes the transpose of a matrix (or a vector); TB ðOtÞ, which represents the global multi-blade coordinate transformation matrix whose block diagonal elements consist of N basis multi-blade-coordinate transformation matrices BðOtÞ defined in Eq. (9), is given by N

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ TB ðOtÞ ¼ diag½BðOtÞ, BðOtÞ, . . ., BðOtÞ :

(11)

Substituting Eqs. (8) and (10) into Eq. (4) and pre-multiplying the transpose of the global multi-blade coordinate transformation matrix, i.e. TB ðOtÞT , we obtain the equation-of-motion expressed in the multi-blade coordinates as MB ðOtÞq€ B ðtÞ þ CB ðOtÞq_ B ðtÞ þKB ðOtÞqB ðtÞ ¼ f B ðtÞ,

(12)

where MB ðOtÞ ¼ TB ðOtÞT MTB ðOtÞ, n o CB ðOtÞ ¼ TB ðOtÞT CTB ðOtÞ þ 2M T_ B ðOtÞ , n o KB ðOtÞ ¼ TB ðOtÞT KTB ðOtÞ þC T_ B ðOtÞ þM T€ B ðOtÞ , f B ðtÞ ¼ TB ðOtÞT f ðtÞ: Note that the multi-blade coordinate transformed system matrices are given by the periodic functions of time, since the multi-blade coordinates describe the blade-group motions in the inertial frame of reference, whereas the local blade coordinates in Eq. (3) are defined in the rotating frame. By means of the multi-blade coordinate transformation, the blade-group motions are described by the real-valued multi-blade coordinates in the rotor disk-shaft frame, providing intuitive understanding of the dynamic coupling mechanism between the rotor blade-array and the other rotor components (e.g., for a rotating array of identical blades, only the collective and first-order cyclic multi-blade motions are involved in the coupling by inducing unbalanced resultant inertial force and moment to its mount structure). However, for the cyclic multi-blade motions, regarded as the blade traveling waves, the real-valued coordinate representation becomes too restrictive to thoroughly describe and characterize its distinct dynamic nature, such as the directivity of the traveling wave mode. 2.2.2. Complex coordinate transformation For rotor dynamic analysis, the complex-valued coordinate, already mentioned in Eq. (2), is often adopted to describe the whirling motion of rotor components, since it allows clear physical interpretation by incorporating the directivity information of the whirling modes, i.e. backward and forward modes [1,2]. Owing to the use of multi-blade coordinates, the blade-group motions associated with the traveling waves can be treated in the same way as the rotor whirling motions. The concept of complex coordinates, therefore, can also be employed to describe the traveling waves of blade deformation, entirely utilizing the benefits of using the complex notation. The complex-valued multi-blade coordinate,

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

5231

qðlÞ ðtÞ, is defined as k ðlÞ qðlÞ ðtÞ ¼ qck ðtÞ þjqðlÞ ðtÞ, k sk

qðlÞ ðtÞ ¼ qðlÞ ðtÞjqðlÞ ðtÞ: k ck sk

(13)

ðtÞ and qðlÞ ðtÞ, respectively take the real and imaginary parts, and Here, the kth order cyclic multi-blade coordinates, qðlÞ ck sk the harmonic order index k, which indicates the number of the nodal diameters of the traveling wave, ranges from 1 to B, where the highest number of the nodal diameter is determined by the total number of blades, M, as ( ðM1Þ=2 for odd M, B¼ (14) ðM2Þ=2 for even M: Based on Eq. (13), the relation between the real- and complex-valued multi-blade coordinates can be expressed as T

ðlÞ qðlÞ B ðtÞ ¼ CqC ðtÞ,

qCðlÞ ðtÞ

where the complex multi-blade coordinate vector for an odd M, as h ðlÞ qo ðtÞ qðlÞ qðlÞ 1 ðtÞ C ðtÞ ¼

ðlÞ qðlÞ C ðtÞ ¼ C qB ðtÞ,

(15)

and the complex coordinate transformation matrix C are defined,

ðlÞ ðlÞ qðlÞ 1 ðtÞ    qððM1Þ=2Þ ðtÞ qððM1Þ=2Þ ðtÞ 2 pffiffiffi 3 2 0 0  0 0 6 7 6 0 1 1  0 07 6 7 6 1 6 0 j j    0 0 7 7 C ¼ pffiffiffi 6 7, ^ ^ ^ & ^ ^7 26 6 7 6 0 0 0  1 17 4 5

0

0

0



j

iT

,

(16a)

j

and, for an even M, as h ðlÞ qo ðtÞ qðlÞ C ðtÞ ¼

q1ðlÞ ðtÞ

qðlÞ 1 ðtÞ 2 pffiffiffi 2 6 6 0 6 6 0 1 6 6 C ¼ pffiffiffi 6 ^ 26 6 0 6 6 4 0 0

qðlÞ ððM2Þ=2Þ ðtÞ

 0

0



0

1

1



0

j

j



0

^

^

&

^

0

0



1

0

0



j

0

0



0

ðlÞ qððM2Þ=2Þ ðtÞ 3 0 0 7 0 0 7 7 0 0 7 7 7 ^ ^ 7: 7 1 0 7 7 7 j 0 5 pffiffiffi 0 2

qðlÞ ðtÞ d

iT

,

(16b)

In a similar way as the derivation of Eqs. (10) and (11), the global complex multi-blade coordinate and force vectors, respectively, take the form of 9 9 8 8 ð1Þ ð1Þ > > > f C ðtÞ > > > > > > > > qC ðtÞ > > > > > > > = = < f ð2Þ ðtÞ > < qð2Þ ðtÞ > T T C C , (17) , f C ðtÞ ¼ TC f B ðtÞ ¼ qC ðtÞ ¼ TC qB ðtÞ ¼ > > > ^ > ^ > > > > > > > > > > > > > > > > ðNÞ ðNÞ : f ðtÞ ; : q ðtÞ ; C

C

where the global complex coordinate transformation matrix is given by N

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ TC ¼ diag½C, C, . . .,C :

(18)

Substitution of Eqs. (15) and (17) into Eq. (12) and subsequent pre-multiplication of the adjoint (the complex conjugate and T transpose) of the global complex coordinate transformation matrix, i.e. TC , lead to the equation-of-motion in the complex form of MC ðOtÞq€ C ðtÞ þCC ðOtÞq_ C ðtÞ þKC ðOtÞqC ðtÞ ¼ f C ðtÞ, where T

MC ðOtÞ ¼ TC MB ðOtÞTC , T

CC ðOtÞ ¼ TC CB ðOtÞTC , T

KC ðOtÞ ¼ TC KB ðOtÞTC , T

f C ðtÞ ¼ TC f B ðtÞ:

(19)

5232

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

Furthermore, we can rearrange the complex multi-blade coordinate vector qC ðtÞ as 9 8 q ðtÞ > = < C bla > qC ðtÞ ¼ qRbla ðtÞ , > ; : q ðtÞ > C bla

(20)

allowing one to classify the coordinate vector into the three groups: the first and the second groups, denoted by qC bla ðtÞ and qC bla ðtÞ, respectively, are the complex-valued multi-blade coordinates and its complex conjugate pair, obtained through the complex coordinate transformation (17); and the third group qRbla ðtÞ is the rest of the real-valued coordinates, which are not transformed into the complex coordinates. Then, a set of matrix equations that are equivalent to Eq. (19) can be obtained in a similar form to that of the conventional general rotor system (1) as ( ) ( ) ( ) B1 B 2B X X X jhOt jhOt jhOt € € M e M e M e q€ ðtÞ q ðtÞ þ q ðtÞ þ mh

(

þ ( þ

(

Cmh ejhOt q_ C bla ðtÞ þ

h ¼ B þ 1

)

(

Kmh ejhOt qC bla ðtÞ þ

h ¼ B þ 1

B X

h¼1 B X

)

(

Kqh ejhOt qRbla ðtÞ þ

( þ

(

Mph ejhOt q€ C bla ðtÞ þ Mo q€ Rbla ðtÞ þ B X

)

B X

(

Cph ejhOt q_ C bla ðtÞ þ Co q_ Rbla ðtÞ þ )

Kph ejhOt qC bla ðtÞ þKo qRbla ðtÞ þ

h¼1

h¼2 2B X

Csh ejhOt q_ C bla ðtÞ )

Ksh ejhOt qC bla ðtÞ ¼ f C bla ðtÞ,

(21a)

) Mph ejhOt q€ C bla ðtÞ

h¼1

h¼1 B X

2B X

)

h¼2

h¼1

þ

(

Cqh ejhOt q_ Rbla ðtÞ þ

)

B X

C bla

h¼2

)

h¼1

(

(

sh

Rbla

h¼1

)

B1 X

B1 X

qh

C bla

h ¼ B þ 1

(

B X

h¼1 B X

) Cph ejhOt q_ C bla ðtÞ )

Kph ejhOt qC bla ðtÞ ¼ f Rbla ðtÞ,

(21b)

h¼1

where the system matrices are given by periodic functions with the fundamental frequency O, and the order of harmonic contents depends on the total number of rotor blades, whereas those for the conventional rotor systems only contain the single 2O-frequency component as seen in Eq. (1). Due to the presence of these periodically time-varying terms that result from the asymmetric blade-group configuration, Eqs. (21a) and (21b) are coupled implying that the blade traveling waves and the collective blade-group vibrations are mutually correlated. Note here that, in the particular case of the symmetric blade-array, i.e. the blade-array consisting of identical rotor blades, the periodic terms vanish during the multi-blade coordinate transformation (10), and consequently, the coupled periodically time-varying Eqs. (21a) and (21b) reduce to the two, completely decoupled, time-invariant system equations as Mm0 q€ C bla ðtÞ þ Cm0 q_ C bla ðtÞ þKm0 qC bla ðtÞ ¼ f C bla ðtÞ,

(22a)

Mo q€ Rbla ðtÞ þ Co q_ Rbla ðtÞ þKo qRbla ðtÞ ¼ f Rbla ðtÞ,

(22b)

where the subscripts m0 and o are equivalent to the subscript f in Eq. (1) in the sense that they denote the system symmetric property. Most studies on the bladed-rotor system, therefore, have adopted the simplified model with the underlying assumption of the symmetric blade-array, in order to conveniently utilize the ordinary eigenanalysis scheme for linear time-invariant system. 2.3. General bladed-rotor system To construct a dynamic analysis model for bladed-rotor systems in a general configuration, i.e. the asymmetric bladedrotors supported by an anisotropic stator, the two decoupled system Eqs. (1) and (21), which describe the rotating bladearray and the rest of the rotor components separately, are combined by introducing the generalized coordinates and force vectors given by 8 9 ( ) ( ) > < qC ðtÞ > = qC rot ðtÞ qC rot ðtÞ pðtÞ ¼ qR ðtÞ , qC ðtÞ ¼ , , qC ðtÞ ¼ qC bla ðtÞ qC bla ðtÞ > : q ðtÞ > ; C 8 9 (23) ( ) ( ) > < f C ðtÞ > = f C rot ðtÞ f C rot ðtÞ , gðtÞ ¼ f R ðtÞ , f C ðtÞ ¼ , f C ðtÞ ¼ f C bla ðtÞ > > f C bla ðtÞ : ; f C ðtÞ

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

5233

where the coordinates for the rotor whirling and the blade traveling wave motions are put together in a single complex coordinate vector. The periodically time-varying equation-of-motion for the general bladed-rotor system can then be obtained as € þ CðOtÞpðtÞ _ þKðOtÞpðtÞ ¼ gðtÞ, MðOtÞpðtÞ

(24)

where 2B X

MðOtÞ ¼

Mh ejhOt ,

2B X

CðOtÞ ¼

h ¼ 2B

Ch ejhOt ,

KðOtÞ ¼

h ¼ 2B

2B X

Kh ejhOt :

h ¼ 2B

Since the asymmetry of the rotor blade-array and that of the rotor disk-shaft are considered equivalent from a system perspective, the structure of the system matrices in Eq. (24) is not so different from that in Eq. (21). Yet, with the addition of the element matrices associated with the stator anisotropy, such as those denoted by the subscript b in Eq. (1), the complete equation-of-motion for the general bladed-rotor system is derived. 3. Complex modal analysis For linear systems possessing periodically time-varying parameters, the approach employing Floquet theory, whose analysis scheme is well established in the literature [21,22], can be applied to obtain the entire time response by the eigenvalues and the corresponding periodically time-varying eigenvectors. However, even though the eigensolutions in closed form can be obtained analytically by means of the direct Floquet analysis, the analytical approach becomes impractical, if not impossible, for most practical applications because of the mathematical complexity. On the other hand, the problem associated with numerical instability, which results from the accumulated error with an extensive numerical integration process, cannot be avoided, and this leads to difficulties in utilizing the numerical approach [23]. Therefore, the modulated coordinate transformation, an alternative approach distinct from the direct Floquet method, is utilized to analyze the, periodically time-varying, general bladed-rotor system (24). 3.1. Modulated coordinate transformation The equation-of-motion with periodically time-varying coefficients described in Eq. (24) can be transformed into an equivalent time-invariant matrix equation by introducing the modulated complex coordinate and force vectors defined by [14] p;n ðtÞ  pðtÞejnOt , g;n ðtÞ  gðtÞejnOt ,

p;n ðtÞ  pðtÞejnOt , g;n ðtÞ  gðtÞejnOt ,

(25)

where the modulation index n is an arbitrary integer. Substituting Eq. (25) into Eq. (24), we can obtain an equivalent linear differential equation in the form of M1 p€ 1 ðtÞ þ C1 p_ 1 ðtÞ þ K1 p1 ðtÞ ¼ g1 ðtÞ:

(26)

Here, the subscript symbol ‘N’ represents an infinite dimensional matrix (or vector); and the generalized coordinate and force vectors, and the system matrices, respectively, are defined as h iT T p;1 ðtÞT p;0 ðtÞT p; þ 1 ðtÞT p; þ 2 ðtÞT    , p1 ðtÞ ¼    p;2 ðtÞ h g1 ðtÞ ¼   

g;2 ðtÞT

g;1 ðtÞT

g;0 ðtÞT

g; þ 1 ðtÞT

g; þ 2 ðtÞT



iT

,

and 2

&

6  6 6 6  6 6  6 6 6  6 6 M1 ¼ 6    6 6  6 6  6 6 6  6 6  4 c

^

^

^

^

^

^

^

^

3

^

c

 7 7 7  7 7  7 7 7  7 7  7 7, 7  7 7  7 7 7  7 7  7 5 &

M0

Mþ1



M þ 2B

0

0



0

0

M1 ^

M0 ^



M þ 2B1 ^

M þ 2B ^

0 ^



0 ^

0 ^

M2B

M2B þ 1



M0

Mþ1

Mþ2



0

0

0

M2B



M1

M0

Mþ1



M þ 2B

0

0

0



M2

M1

M0



M þ 2B1

M þ 2B

^

^

^

^

^

0

0



0

M2B

M2B þ 1

0

0



0

0

^

^

^

^

^

^

^



M0

Mþ1

M2B



M1

M0

^

^

^

^

5234

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

2

& 6  6 6 6  6 6  6 6 6  6 6 C1 ¼ 6    6 6  6 6  6 6 6  6 6  4 c 2

&

6  6 6 6  6 6  6 6 6  6 6 K1 ¼ 6    6 6  6 6  6 6 6  6 6  4 c

^

^

^

C0;2B1

C þ 1;2B



C þ 2B;1

0

0



C1;2B1

C0;2B



C þ 2B1;1

C þ 2B;0

0



^

^

^

^

^



C0;1

C þ 1;0

C þ 2; þ 1



0

0

^

^

C2B;2B1

C2B-1;2B

^

^

^

^

^

3

^

c

0

0

0

0

 7 7 7  7 7  7 7 7  7 7  7 7, 7  7 7  7 7 7  7 7  7 5 &

0

C2B;2B



C1;1

C0;0

C þ 1; þ 1



C þ 2B; þ 2B

0

0

0



C2;1

C1;0

C0; þ 1



C þ 2B1; þ 2B

C þ 2B; þ 2B þ 1

^ 0

^ 0



^ 0

^ C2B;0

^ C2B þ 1; þ 1



^ C0; þ 2B

^ C þ 1; þ 2B þ 1

0

0



0

0

C2B ; þ 1



C1; þ 2B

C0; þ 2B þ 1

^

^

^

^

^

^

^

^

^

^

^

^

^

^

^

^

^

^

c

K0;2B1 K1;2B1

K þ 1;2B K0 ;2B

 

K þ 2B;1 K þ 2B1;1

0 K þ 2B;0

0 0

 

0 0

0 0

^

^

0

0

 7 7 7  7 7  7 7 7  7 7  7 7, 7  7 7  7 7 7  7 7  7 5 &

^

^

^

^

^

K2B;2B1

K2B-1;2B



K0;1

K þ 1;0

K þ 2; þ 1



0

K2B;2B



K1;1

K0;0

K þ 1; þ 1



K þ 2B; þ 2B

0

0

0



K2;1

K1;0

K0; þ 1



K þ 2B1; þ 2B

K þ 2B; þ 2B þ 1

^

^

^

^

^

^

^

0

0



0

K2B;0

K2B þ 1; þ 1



K0; þ 2B

K þ 1; þ 2B þ 1

0 ^

0 ^

 ^

0 ^

0 ^

K2B; þ 1 ^

 ^

K1; þ 2B ^

K0; þ 2B þ 1 ^

3

where Kh;n ¼ Kh jnOCh n2 O2 Mh ,

Ch;n ¼ Ch j2nOMh ,

h ¼ 0, 7 1, . . ., 7 2B,

n ¼ 0, 7 1, 72, . . .:

It is noted that the transformed equation (26) certainly describes a linear time-invariant system, whose rotor speed dependent system matrices take the form of Hill’s infinite dimensional matrix with (4B þ1)N bandwidth.

3.2. Complex modal solution The state-space representation of Eq. (26) is _ 1 ðtÞ ¼ B1 w1 ðtÞ þ F1 ðtÞ A1 w

(27)

and " A1 ¼

01

M1

M1

C1

#

" ,

B1 ¼

M1

01

01

K1

(

# ,

w1 ðtÞ ¼

p_ 1 ðtÞ

(

) ,

p1 ðtÞ

F1 ðtÞ ¼

01 g1 ðtÞ

) ,

(28)

where 0N represents the zero matrix or vector of infinite dimension. The eigenvalue and its adjoint problems associated with Eq. (27) are then given by

lirðmÞ A1 ri 1rðmÞ ¼ B1 ri1rðmÞ , lipðmÞ A1 ri 1pðmÞ ¼ B1 ri1pðmÞ , i

T

i

T

i

i

T

i

T

i

lrðmÞ A1 l1rðmÞ ¼ B1 l1rðmÞ , lpðmÞ A1 l1pðmÞ ¼ B1 l1pðmÞ ,

(29)

where the eigenvectors and the adjoint eigenvectors, rN and l1 , corresponding to the eigenvalues l, are given, respectively, as ( )i ( )i lu1 lu1 i i , r1pðmÞ ¼ , r1rðmÞ ¼ u1 u1 rðmÞ

i

l1rðmÞ ¼

(

lv1 v1

pðmÞ

)i , rðmÞ

i

l1pðmÞ ¼

(

lv1 v1

)i , pðmÞ

(30)

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

5235

with h ui1rðmÞ ¼    h ¼ 

^ TC; þ 1 u

uTR;1

T u;1 u;0T

^ TC;0 u

uTC;1

h ui1pðmÞ ¼    h

^ TC; þ 1 u

¼ 

uTR;1

^ TC;0 u

uTC;1

h vi1rðmÞ ¼    h ¼ 

T v^ C; þ 1

vTR;1

T v^ C; þ 1

vTR;1

uTR;0

vTR;0

uTC;0

vTC;0

vTR;0

vTC;0

T

rðmÞ

^ TC;1 u

uTR; þ 1 ii

uTC; þ 1



uTC; þ 1



ii

T

rðmÞ

,

T

pðmÞ

^ TC;1 u

uTR; þ 1

ii

ii

T

pðmÞ

,

T

rðmÞ

T v^ C;1

T T T v;1 v;0 v; þ 1 

T v^ C;0

vTC;1

ii

T u;1 uT;0 u;Tþ 1   

T v^ C;0

vTC;1

uTC;0

T T T v;1 v;0 v; þ 1 

h vi1pðmÞ ¼    h ¼ 

uTR;0

u;Tþ 1   

T v^ C;1

vTR; þ 1 ii

vTC; þ 1



vTC; þ 1



ii

T

rðmÞ

,

T

pðmÞ

vTR; þ 1

ii

T

pðmÞ

,

and r ¼ 7 1, 7 2,. . ., 7 NC ,

p ¼ NC þ 1,NC þ 2,. . .,NC þ NR ,

m ¼ 0, 7 1, 7 2,. . .,

i ¼ B,F:

Here, uN and vN are respectively the right and left latent vectors for the latent value problem associated with Eq. (26); the subscripts, r(m) and p(m), refer to the rth and pth eigensolutions in cluster m, respectively; the superscripts, B and F, indicate the directivity of the whirling and traveling wave modes referred to as ‘‘backward’’ and ‘‘forward’’ modes, respectively [18]; the number of the complex- and real-valued coordinates are, respectively, denoted by NC and NR; and the vibration modes are subdivided into the modes dominantly involved in the rotor whirling and blade traveling wave motions and in the other blade collective motions, being indicated by the subscripts r and p, respectively. Note that the modes with the subscript p only appear in the case of an asymmetric blade-group configuration, leading to the mode coupling between the directional whirling and traveling wave modes and the collective multi-blade mode that has no directivity. In addition, each pair of the eigenvalues, equal in value but different in the sign of the subscript, form a complex conjugate pair, and cluster m consists of only the set of eigensolutions associated with the modulation index m, or equivalently, with the shifted eigenvalues by jmO [14,24]. Substituting the complex state vector 2 3 w1 ðtÞ ¼

1 X m ¼ 1

NC N CX þ NR X6 i  i 7 6 X  7 r1 ZðtÞ rðmÞ þ r1 ZðtÞ pðmÞ 7 6 4 5 p ¼ N þ1 i ¼ B,F r ¼ N C

(31)

C

ra0

into the homogeneous part of Eq. (27), and applying the bi-orthonormality condition given by k

T

k

T

i

k

l1sðlÞ A1 ri1rðmÞ ¼ drðmÞsðlÞ , i

i

k

T

k

T

i

k

l1qðlÞ A1 ri1pðmÞ ¼ dpðmÞqðlÞ , k

l1sðlÞ B1 ri 1rðmÞ ¼ lrðmÞ drðmÞsðlÞ ,

i

i

k

l1qðlÞ B1 ri 1pðmÞ ¼ lpðmÞ dpðmÞqðlÞ ,

(32)

where the Kronecker delta is defined as ( k dirðmÞsðlÞ

¼

1

for i ¼ k, r ¼ s, m ¼ l,

0

otherwise,

(33)

we obtain the modal equations

Z_ ðtÞirðmÞ ¼ lirðmÞ ZðtÞirðmÞ þvi1rðmÞT g1 ðtÞ, Z_ ðtÞipðmÞ ¼ lipðmÞ ZðtÞipðmÞ þvi1pðmÞT g1 ðtÞ, and consequently the complex principal coordinates Z(t) are determined.

(34)

5236

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

The forced response of the system then becomes 2

3

NC N CX þ NR X6 i  i 7 6 X  7 u1 ZðtÞ rðmÞ þ u1 ZðtÞ pðmÞ 7, p1 ðtÞ ¼ 6 4 5 m ¼ 1 i ¼ B,F r ¼ N p ¼ N þ1 1 X

C

(35)

C

ra0

and 2 pðtÞ ¼

3

NC N CX þ NR X6 i  i 7 6 X  7 u;0 ZðtÞ rðmÞ þ u;0 ZðtÞ pðmÞ 7 6 4 5 m ¼ 1 i ¼ B,F r ¼ N p ¼ N þ1 1 X

C

C

ra0

¼

*Z

1 X m ¼ 1

0

t

2 3 + NC n NCX þ NR n oi oi 7 X6 X 6 7 elðttÞ u;0 v;n T þ u;0 v;n T 6 7 g;n ðtÞdt : 4 rðmÞ pðmÞ 5 n ¼ 1 i ¼ B,F r ¼ N p ¼ N þ1 1 X

C

(36)

C

ra0

The above Eq. (36), representing the contribution of the modulated excitation force vectors g;n(t) to the output response, implies that this approach is essentially based on the frequency response of the system, which is contrary to the Floquet approach based on the periodic time response of the system. This eventually results in a significant discrepancy between the two approaches in numerically calculating the eigensolutions, even though the Floquet approach is theoretically equivalent to the modulated coordinate transformation and hence may provide the complex response solution that corresponds to Eq. (36). For numerical calculation of the eigensolutions, the modulated coordinate method approximates the infinite order system matrices in Eq. (26) to the reduced, finite, order Hill’s matrix, whereas the Floquet approach simply utilizes a partial set of Fourier coefficient vectors of the periodically time-varying eigenvectors [14]. The coordinate transformation approach is, therefore, not only numerically robust in the sense that it involves no numerical integration process, but it is also computationally efficient for the frequency domain analysis on the basis of the definite set of modal parameters.

3.3. Directional frequency response matrix (dFRM) Fourier transformation of Eq. (35) leads to P1 ðjoÞ ¼ H1 ðjoÞG1 ðjoÞ,

(37)

where PN(jo) and GN(jo) are the Fourier transforms of pN(t) and gN(t), respectively, and the directional frequency response matrix (dFRM) of infinite order takes the form of 2 NC X6 6 X H1 ðjoÞ ¼ 6 4 m ¼ 1 i ¼ B,F r ¼ N 1 X

C

( )i u1 v1 T jol

þ

rðmÞ

( )i u1 v1 T jol þ1

3

NCX þ NR p ¼ NC

pðmÞ

7 7 7 5

ra0

2

& 6  6 6 6  6 6  6 6 ¼6 6  6  6 6 6  6 6  4 c

^

^

^

^

^

^

^

HgR ;1pR ;1

HgC ;1pR ;1

HgC ;0pR ;1

HgR ;0pR ;1

HgC ;0pR ;1

HgC ;1pR ;1

HgR ; þ 1pR ;1

HgR ;1pC ;1

HgC ;1pC ;1

HgC ;0pC ;1

HgR ;0pC ;1

HgC ;0pC ;1

HgC ;1pC ;1

HgR ; þ 1pC ;1

HgR ;1pC ;0

HgC ;1pC ;0

HgC ;0pC ;0

HgR ;0pC ;0

HgC ;0pC ;0

HgC ;1pC ;0

HgR ; þ 1pC ;0

HgR ;1pR ;0

HgC ;1pR ;0

HgC ;0pR ;0

HgR ;0pR ;0

HgC ;0pR ;0

HgC ;1pR ;0

HgR ; þ 1pR ;0

HgR ;1pC ;0

HgC ;1pC ;0

HgC ;0pC ;0

HgR ;0pC ;0

HgC ;0pC ;0

HgC ;1pC ;0

HgR ; þ 1pC ;0

HgR ;1pC ;1

HgC ;1pC ;1

HgC ;0pC ;1

HgR ;0pC ;1

HgC ;0pC ;1

HgC ;1pC ;1

HgR ; þ 1pC ;1

HgR ;1pR ; þ 1

HgC ;1pR ; þ 1

HgC ;0pR ; þ 1

HgR ;0pR ; þ 1

HgC ;0pR ; þ 1

HgC ;1pR ; þ 1

HgR ; þ 1pR ; þ 1

^

^

^

^

^

^

^

3 c  7 7 7  7 7  7 7 7  7 7,  7 7 7  7 7  7 5 &

(38)

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

5237

where 2

u1 v1 T

&

6 6  6 6 6 6  6 6 6  6 6 6 ¼6  6 6 6  6 6 6 6  6 6 6  4 c

^

^

^

^

^

^

uR;1 vTR;1

uR;1 vTC;1

uR;1 v^ C;0

uC;1 vTR;1

uC;1 vTC;1

uC;1 v^ C;0

^ C;0 vTR;1 u

^ C;0 vTC;1 u

^ C;0 v^ C;0 u

uR;0 vTR;1

uR;0 vTC;1

uR;0 v^ C;0

uC;0 vTR;1

uC;0 vTC;1

uC;0 v^ C;0

^ C;1 vTR;1 u

^ C;1 vTC;1 u

^ C;1 v^ C;0 u

uR; þ 1 vTR;1

uR; þ 1 vTC;1

^

^

^

T

uR;1 vTR;0

uR;1 vTC;0

uR;1 v^ C;1

T

uC;1 vTR;0

uC;1 vTC;0

uC;1 v^ C;1

T

^ C;0 vTR;0 u

^ C;0 vTC;0 u

^ C;0 v^ C;1 u

T

uR;0 vTR;0

uR;0 vTC;0

uR;0 v^ C;1

T

uC;0 vTR;0

uC;0 vTC;0

uC;0 v^ C;1

T

^ C;1 vTR;0 u

^ C;1 vTC;0 u

^ C;1 v^ C;1 u

uR; þ 1 v^ C;0

T

uR; þ 1 vTR;0

uR; þ 1 vTC;0

^

^

^

T

uR;1 vTR; þ 1

T

uC;1 vTR; þ 1

T

^ C;0 vTR; þ 1 u

T

uR;0 vTR; þ 1

T

uC;0 vTR; þ 1

T

^ C;1 vTR; þ 1 u

uR; þ 1 v^ C;1

T

uR; þ 1 vTR; þ 1

^

^

c

3

7  7 7 7 7  7 7 7  7 7 7 7    7: 7 7  7 7 7 7  7 7 7  7 5 &

Here, the infinite number of block dFRMs in Eq. (38) are not independently determined as is the case of the general rotor system [14], but two rows (or columns) of the infinite order dFRM – one is for the complex-valued responses and the other one is for the real-valued – constitute a set of independent dFRMs. However, the independent dFRM set still consists of an infinite number of dFRMs, and therefore the most representative dFRMs are introduced to characterize the bladedrotor system dynamics. The whirling behavior of the bladed-rotor system, described by the complex generalized coordinates, can be reflected by the following six block dFRMs: The first four dFRMs, which are associated with the complex-valued responses and excitations, pC(t) and gC(t), and play roles similar to those of the conventional general rotor system [14], are given by 2

( )i T NC 1 X6 X 6 X uC;0 vC;0 HgC ;0pC ;0 ðjoÞ ¼ 6 4 jol m ¼ 1 i ¼ B,F r ¼ N C

rðmÞ

3

( )i NCX þ NR uC;0 vTC;0 þ jol p ¼ N þ1

pðmÞ

C

7 7 7, 5

ra0

2

8 9i T NC < 1 X6 X uC;0 v^ C;0 = 6 X HgC ;0pC ;0 ðjoÞ ¼ 6 : jol ; 4 m ¼ 1i ¼ B,F r ¼ N C

rðmÞ

ra0

C

pðmÞ

2

8 9i T NC < 1 X6 X uC;0 v^ C;2 = 6 X HgC ;2pC ;0 ðjoÞ ¼ 6 : jol ; 4 m ¼ 1i ¼ B,F r ¼ N C

ra0

2 NC X6 6 X HgC ;2pC ;0 ðjoÞ ¼ 6 4 m ¼ 1i ¼ B,F r ¼ N 1 X

C

3

8 9i T NCX þ NR < uC;0 v^ C;0 = þ : jol ; p ¼ N þ1

þ

p ¼ NC

rðmÞ

( )i uC;0 vTC;2 jol

N CX þ NR

þ

3

9

i T ^ C;2 = C;0 v

: jol ; þ1

pðmÞ

( )i uC;0 vTC;2 jol þ1

N CX þ NR p ¼ NC

rðmÞ

8
7 7 7, 5

pðmÞ

7 7 7, 5

3 7 7 7, 5

(39)

ra0

where the dFRM, HgC ;0pC ;0 ðjoÞ, is insensitive to the system anisotropy and asymmetry, and hence it characterizes the effect of system symmetry; whereas, the system anisotropy is represented by HgC ;0pC ;0 ðjoÞ; and the effect of system asymmetry and the coupled effect between the system anisotropy and asymmetry are characterized by HgC ;2pC ;0 ðjoÞ and HgC ;2pC ;0 ðjoÞ, respectively. Note here that not only the asymmetry of the rotor disk and shaft but also the blade-group asymmetry is represented by the preceding two dFRMs, since these two sort of asymmetric properties are taken into consideration as a whole. On the other hand, the relations between the complex-valued responses and the real-valued excitations, pC(t) and gR(t), are described by the two dFRMs given by 2 3 ( )i ( )i T T N N þ N 1 6 7 C C R X X 6 X uC;0 vR; þ 1 X uC;0 vR; þ 1 7 þ HgR ; þ 1pC ;0 ðjoÞ ¼ 6 7, 4 5 j o  l j o  l m ¼ 1i ¼ B,F r ¼ N p ¼ N þ1 rðmÞ

C

pðmÞ

C

ra0

2 NC X6 6 X HgR ;1pC ;0 ðjoÞ ¼ 6 4 m ¼ 1i ¼ B,F r ¼ N 1 X

ra0

C

( )i uC;0 vTR;1 jol

rðmÞ

þ

( )i uC;0 vTR;1 jol þ1

NCX þ NR p ¼ NC

pðmÞ

3 7 7 7, 5

(40)

5238

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

where HgR ; þ 1pC ;0 ðjoÞ is affected solely by the effect of asymmetric blade-group configuration, while HgR ;1pC ;0 ðjoÞ is sensitive to the coupled effect between the blade-group asymmetry and the stator anisotropy. Since these two dFRMs fall under the influence of the blade-group asymmetry, they vanish for the case of the symmetric blade-array, regardless of the presence of the rotor disk–shaft asymmetry. Unlike the previous dFRMs that represent the rotor whirling behavior, the dFRM associated with the real-valued responses and excitations, pR(t) and gR(t), which takes the form of 2

( )i T NC 1 X6 X 6 X uR;0 vR;0 HgR ;0pR ;0 ðjoÞ ¼ 6 4 jol m ¼ 1 i ¼ B,F r ¼ N

rðmÞ

C

( )i N CX þ NR uR;0 vTR;0 þ jol p ¼ N þ1

pðmÞ

C

3 7 7 7, 5

(41)

ra0

describes the collective blade-group dynamics, which is insensitive to all types of system non-symmetry, i.e. the bladegroup asymmetry, the asymmetry in the other rotating components and the stator anisotropy, and consequently, it characterizes the group symmetry of the blade-array. Since the dFRF is a complex-valued scalar function that constitutes the dFRM [18], the dFRFs associated with the dFRMs mentioned in Eqs. (39)–(41) are respectively denoted by the normal letters, namely, HgC ;0pC ;0 ðjoÞ,

HgC ;0pC ;0 ðjoÞ, HgR ; þ 1pC ;0 ðjoÞ,

HgC ;2pC ;0 ðjoÞ,

HgC ;2pC ;0 ðjoÞ,

HgR ;1pC ;0 ðjoÞ,

(42)

(43)

and HgR ;0pR ;0 ðjoÞ:

(44)

4. Numerical example As an illustrative numerical example to demonstrate the proposed dynamic analysis procedure, a simple bladed-disk rotor supported by an elastic torsional spring, as shown in Figs. 4 and 5, is treated here. The torsional spring is a simplistic representation of the stator property. The three blades mounted on the rotating rigid-disk are assumed to be elastic pendulums, only taking a single degree-of-freedom out-of-plane bending deflection into account. Based on the complex multi-blade coordinates, the equation-of-motion for the simple bladed-rotor system can be derived from Eq. (24) with the coordinate and external force vectors, and the system matrices, respectively, given by h pðtÞ ¼ qB1 ðtÞ

qD ðtÞ

h gðtÞ ¼ f B1 ðtÞ

f D ðtÞ

qB0 ðtÞ

f B0 ðtÞ

qB1 ðtÞ

qD ðtÞ

f B1 ðtÞ

f D ðtÞ

iT

,

iT

,

(45)

qi (t)

Ω

ξ

η Fig. 4. An array of 3-blades mounted on a rotating rigid-disk: (a) front view and (b) side view; qi(t), local coordinate of the ith blade defined in the rotating frame; x  Z, principal axes of the bladed-disk; O, rotational speed.

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

5239

Ω

Fig. 5. A simple bladed-disk supported by torsional springs: qD(t), coordinate for the disk tilting motion defined in the stationary frame; O, rotational speed.

and 2

2

mB lB 6 6 qffiffi 6 3 6 2mB lB ðlB þ r D Þ 6 6 0 MðOtÞ ¼ 6 6 6 6 0 6 6 4 0 2

2

cB þ j2mB lB O 6 pffiffiffi 6 j 6m l ðl þ r ÞO 6 B B B D 6 6 0 CðOtÞ ¼ 6 6 6 0 6 4 0 2 6 6 6 6 KðOtÞ ¼ 6 6 6 6 4

qffiffi 3 2mB lB ðlB þ r D Þ

0

0

J D þ 32mB ðlB þ r D Þ2

0

0

2 mB lB

0

0

0

mB lB

dD J D ej2Ot

0

pffiffiffi j 6mB lB ðlB þ r D ÞO n o cS þ j J Dp þ3mB ðlB þ r D Þ2 O

3

0

7 7 7 dD J D e 7 7 7 7, 0 7 qffiffi 7 3 7 l ðl þr Þ m D 7 2 B B B 7 5 JD þ 32mB ðlB þ r D Þ2 j2Ot

0

2

qffiffi 3 2mB lB ðlB þr D Þ

0

0

0

0

7 7 7 7 7 0 7, pffiffiffi 7 7 j 6mB lB ðlB þ r D ÞO n o 7 5 2 cS j JDp þ 3mB ðlB þ r D Þ O jdD JD O ej2Ot

0

cB

0

0

0

jdD JD O ej2Ot

0

cB j2mB lB O pffiffiffi j 6mB lB ðlB þ r D ÞO

2

kB þ jcB O þmB lB r D O2

0

p1ffiffi3jdB kB ejOt

0

kS

0

0

0

kB þ mB lB O2 þmB lB r D O2

p1ffiffi3jdB kB ejOt

p1ffiffijdB kB 3

p1ffiffi3jdB kB

ejOt 2jOt

e

0

2

p1ffiffijdB kB 3

0

DS kS

jO t

e

0

3

0

p1ffiffijdB kB 3

e2jOt

kB jcB O þ mB lB r D O

0

3 7

DS kS 7 7 2

0

7 0 7, 7 7 0 7 5 kS

(46)

where the system parameters are mB ¼ mB1 ¼ mB2 ¼ mB3 , J D ¼ ðJ DZ þ JDx Þ=2,

cB ¼ cB1 ¼ cB2 ¼ cB3 ,

kB ¼ ðkB1 þ kB2 þ kB3 Þ=3,

a ¼ JDp =JD , cS ¼ cSy ¼ cSz , kS ¼ ðkSy þkSz Þ=2,

DS ¼ ðkSy kSz Þ=ð2kS Þ, dD ¼ ðJ DZ JDx Þ=ð2J D Þ, dB ¼ ðkB2 kB1 Þ=kB1 ¼ ðkB3 kB1 Þ=kB1 : Here, lB and rD are the blade length and the disk radius, respectively; qB0 ðtÞ (f B0 ðtÞ) is the real-valued, collective multi-blade coordinate (force); qB1 ðtÞ and qD(t) are the complex coordinates for the blade traveling and the disk precessional motions, respectively, and f B1 ðtÞ and fD(t) represent the corresponding external moments; mBi, cBi and kBi (i ¼1, 2, 3) denote the lumped mass, damping and stiffness of the ith blade, respectively; JDZ and JDx are the diametrical mass moment-of-inertia of the disk with respect to the principal, Z and x axes, respectively; JDp is the disk polar mass moment-of-inertia, and a represents the ratio between the polar and the mean diametrical moment-of-inertia of the disk; cSy and cSz (kSy and kSz) denote the stator damping (stiffness) in the y–z directions, respectively; the stator anisotropy DS is defined as the deviatoric value of the stiffness in the two orthogonal directions normalized by the mean value, and, in a similar way, the rotor disk asymmetry dD is given by the normalized diametrical inertia deviation; and the blade-group asymmetry dB is defined as the normalized stiffness deviation of two of the three blades from the remaining reference blade. Note that the terms associated with the gyroscopic and circulatory effects due to the rotor rotation are certainly found in the system matrices, even though these are not in similar form to those of conventional gyroscopic non-conservative systems, owing to the multi-blade and complex coordinate transformations. For the analysis model, the modal frequencies and the dFRFs, which are an element of the dFRMs, are calculated based on the complex modal analysis theory, employing the complex multi-blade and the modulated coordinates. The analysis results are then compared for various asymmetry conditions imposed on the blade-array and the rest of the rotor and stator components. Table 1 summarizes the parameter values used in the simulation.

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Table 1 Parameter values of the simple bladed-rotor system model. Rotor system type

Isotropic bladed-rotor Anisotropic bladed-rotor Asymmetric bladed-rotor General bladed-rotor

Parameters lB

rD

mB, kB, JD, kS

cB, cS

a

DS

dD

dB

1 1 1 1 1 1

0.5 0.5 0.5 0.5 0.5 0.5

1 1 1 1 1 1

0.04 0.04 0.04 0.04 0.04 0.04

1.5 1.5 1.5 1.5 1.5 1.5

– 0.1 – – 0.1 0.1

– – 0.1 – 0.1 –

– – – 0.1 – 0.1

5 4 3

Whirl speed [rad/s]

2 1 0 -1 -2 -3 -4 -5 0.0

0.5

1.0 1.5 Rotational speed [rad/s]

2.0

2.5

Fig. 6. Whirl speed chart for the simple isotropic bladed-rotor system (DS ¼0, dD ¼ 0, dB ¼0): —m—, rotor precessional mode; —K—, blade traveling wave mode; ————, collective multi-blade mode. n As extreme system characteristics of the simple isotropic bladed-rotor system, the precessional mode of the rigid-disk supported by a torsional spring ( ) and the traveling wave mode of the rotating blade-array supported by a rigid stator ( ) are given for comparison.

4.1. Isotropic bladed-rotor system In Fig. 6, the whirl speeds for the simple bladed-rotor system with isotropic conditions are compared with those for both the bladed rotor supported by rigid stator and the rigid rotor-disk supported by a torsional spring (marked by gray lines), which represent the extreme characteristics of the original system. The rotor precessional modes and the blade traveling wave modes, respectively marked by triangles and circles, are significantly altered due to the blade–disk coupling effect in which the traveling modes with one nodal diameter generate the coupling moment to the disk. On the other hand, the broken lines that correspond to the collective multi-blade mode, which does not generate the coupling moment to the disk, remain unaffected by the blade–disk coupling. Figure 7 is the magnitude plots of the dFRFs for the isotropic bladedrotor system with O ¼1 rad/s. Since the system does not involve any of the non-symmetric properties, the dFRFs associated with non-symmetry vanish, and hence, only the remaining two dFRFs are given in the figure. The dFRF for the system isotropy, HgC ;0pC ;0 ðjoÞ, demonstrates that the well separated backward and forward modes of the blade traveling wave are heavily damped compared with those of the rotor precessional motion. The dFRF for the blade-group symmetry,

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

5241

dFRF for system isotropy / blade-group symmetry

Magnitude

102

100

10-2

10-4 -5

-4

-3

-2

-1 0 1 2 Frequency [rad/s]

3

4

5

Fig. 7. Magnitude plots of dFRFs for the simple isotropic bladed-rotor system (DS ¼ 0, dD ¼ 0, dB ¼0, O ¼ 1 rad/s): —, 9HgC ;0pC ;0 9 for the system symmetry; ————, 9HgR ;0pR ;0 9 for the blade-group symmetry.

5 4 3

Whirl speed [rad/s]

2 1 0 -1 -2 -3 -4 -5 0.0

0.5

1.0 1.5 Rotational speed [rad/s]

2.0

Fig. 8. Whirl speed chart for the simple anisotropic bladed-rotor system (DS ¼ 0.1, dD ¼0, dB ¼0): precessional mode; —K—, blade traveling wave mode; and ————, collective multi-blade mode.

2.5 , strong mode; —, weak mode; —m—, rotor

HgR ;0pR ;0 ðjoÞ, identifies the collective multi-blade mode and shows the conjugate even property with respect to frequency, similarly to that of the zero nodal diameter (umbrella) mode of a flexible disk. 4.2. Anisotropic bladed-rotor system Figures 8 and 9 show the whirl speeds and the dFRFs, respectively, for the simple bladed-rotor system with the anisotropic stator property, DS. The whirl speed chart reveals that the parasitic modes, indicated by thin lines, which are not present in the isotropic system (refer to Fig. 6), appear because of the presence of the stator anisotropy. Unlike the original modes, marked by thick lines and considered as strong modes likely to significantly contribute to system response from all possible excitation sources, the parasitic modes are classified as weak modes that are less significant in response contribution than the strong ones [24]. It is noted that we can identify the relative severity of modes by means of

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dFRF for system anisotropy

dFRF for system isotropy / blade-group symmetry 102 Magnitude

Magnitude

102 100 10-2 10-4

100 10-2 10-4

-5

-4

-3

-2

-1 0 1 2 Frequency [rad/s]

3

4

5

-5

-4

-3

-2

-1 0 1 2 Frequency [rad/s]

3

4

5

5

5

4

4

3

3

2

2 Whirl speed [rad/s]

Whirl speed [rad/s]

Fig. 9. Magnitude plots of dFRFs for the simple anisotropic bladed-rotor system (DS ¼ 0.1, dD ¼ 0, dB ¼ 0, O ¼1 rad/s): (a) —, 9HgC ;0pC ;0 9 for the system symmetry; ————, 9HgR ;0pR ;0 9 for the blade-group symmetry; and (b) 9HgC ;0pC ;0 9 for the system anisotropy.

1 0 -1 -2

1 0 -1 -2

-3

-3

-4

-4

-5 0.0

-5 0.5 1.0 1.5 2.0 Rotational speed [rad/s]

2.5

0.0

0.5 1.0 1.5 2.0 Rotational speed [rad/s]

2.5

Fig. 10. Whirl speed charts for the simple asymmetric bladed-rotor system considering (a) the asymmetric disk inertia (DS ¼ 0, dD ¼ 0.1, dB ¼0), and (b) the dissimilar blade stiffness (DS ¼0, dD ¼ 0, dB ¼0.1): , strong mode; —, weak mode; —m—, rotor precessional mode; —K—, blade traveling wave mode; —J—, coupled collective multi-blade mode; and ————, collective multi-blade mode.

dFRF for system isotropy / blade-group symmetry 10 Magnitude

Magnitude

10

dFRF for system asymmetry

2

100 10-2 10-4 -5 -4 -3 -2 -1 0 1 2 3 Frequency [rad/s]

2

100 10-2 10-4

4

5

-5 -4 -3 -2 -1 0 1 2 3 Frequency [rad/s]

4

5

Fig. 11. Magnitude plots of dFRFs for the simple bladed-rotor system with the asymmetric disk inertia (DS ¼ 0, dD ¼0.1, dB ¼0, O ¼1 rad/s): (a) —, 9HgC ;0pC ;0 9 for the system symmetry; ————, 9HgR ;0pR ;0 9 for the blade-group symmetry; and (b) 9HgC ;2pC ;0 9 for the system asymmetry.

the dFRFs, rather than the whirl speed chart, since the dFRFs utilize not only the modal frequency and damping, but also the modal and adjoint vectors, which may carry the information on the significance of modes. As shown in Fig. 9, the strong modes are identified from HgC ;0pC ;0 ðjoÞ, and the weak modes are observed in the dFRF for the system anisotropy, HgC ;0pC ;0 ðjoÞ.

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

5243

4.3. Asymmetric bladed-rotor system Two different types of rotating asymmetry are imposed on the bladed-rotor model: one is the asymmetric inertia of the rotor disk, dD, and the other one is the stiffness asymmetry of the blade-group, dB. Applying these two distinct asymmetry conditions, the whirl speeds and the dFRFs are respectively compared in Fig. 10 and in Figs. 11 and 12. The whirl speed charts for the two asymmetric bladed-rotor systems are analogous in the sense that the weak modes arise from the rotating asymmetry as is the case of that for the anisotropic bladed-rotor system (Fig. 8). It can also be considered in the

dFRF for system isotropy / blade-group symmetry

dFRF for system asymmetry 102 Magnitude

Magnitude

102 100 10-2 10-4

100 10-2 10-4

-5 -4 -3 -2 -1 0 1 2 Frequency [rad/s]

3

4

5

-5 -4 -3 -2 -1 0 1 2 Frequency [rad/s]

3

4

5

dFRF for blade-group asymmetry

Magnitude

10

2

100 10-2 10-4 -5 -4 -3 -2 -1 0 1 2 Frequency [rad/s]

3

4

5

Fig. 12. Magnitude plots of dFRFs for the simple bladed-rotor system with the dissimilar blade stiffness (DS ¼ 0, dD ¼ 0, dB ¼0.1, O ¼1 rad/s): (a) —, 9HgC ;0pC ;0 9 for the system symmetry; ————, 9HgR ;0pR ;0 9 for the blade-group symmetry; (b) 9HgC ;2pC ;0 9 for the system asymmetry; and (c) 9HgR ; þ 1pC ;0 9 for the blade-group asymmetry. Table 2 Modal solutions for the simple bladed-rotor system with the asymmetric disk inertia: dS ¼0, dD ¼ 0.1, dB ¼0, O ¼ 1 rad/s. Mode

lB2

l2

F

lB1

l1

F

lF1

l1

B

lF2

l2

B

Whirl speed Decay rate Modal vector uBo uBc uDc uBs uDs

 3.58  0.18

 2.26  0.13

 0.07  0.00

 0.15  0.01

1.85  0.01

2.07  0.00

4.26  0.13

5.58  0.18

 0.6047 0.2703  0.6088 0.2724

 0.1379 0.0562  0.1199 0.0482

0.0612 0.2068 0.0639 0.2049

 8E 6 0.0004  8E 6  0.0004

0.0051 0.2300 0.0048  0.2349

 0.0013  0.0043 0.0011 0.0036

0.6017  0.2453  0.6059 0.2441

0.1644  0.0735  0.1463 0.0655

Table 3 Modal solutions for the simple bladed-rotor system with the asymmetric blade configuration: dS ¼0, dD ¼ 0, dB ¼0.1, O ¼ 1 rad/s. Mode

lB2

l2

F

lB3

lB1

l1

F

lF1

l1

B

lF3

lF2

l2

B

Whirl speed Decay rate Modal vector uBo uBc uDc uBs uDs

 3.55  0.18

 2.27  0.13

 0.73  0.02

 0.07  0.00

0.15  0.01

1.85  0.01

2.07  0.00

2.73  0.02

4.27  0.13

5.55  0.18

0.0075  0.0225 0.0103 0.8687  0.3864

0.0121 0.1087  0.0491 0.0022  0.0011

0.5587  0.0003 0.0001  0.0225 0.0118

0.0040  0.0065  0.0287  0.0880  0.2880

0.0003  0.0003  8E 8  2E 5 8E  6

0.0003  0.0002  0.3212  0.0068 0.0013

0.0043  0.0039 0.0015  0.0003 0.0003

0.5674  0.0029 0.0010  0.0294 0.0118

0.0129 0.0215  0.0063 0.9824  0.4001

0.0075  0.0898 0.0367 0.0042  0.0015

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5

5

4

4

3

3

2

2

Whirl speed [rad/s]

Whirl speed [rad/s]

same context that the dFRFs that characterize the rotor asymmetry at the system level, HgC ;2pC ;0 ðjoÞ, given in Figs. 11(b) and 12(b), do not show a significant difference between each other. For the model with the blade-group asymmetry, however, additional whirling modes are observed by the modal lines with hollow circles in Fig. 10(b). The corresponding

1 0 -1 -2

1 0 -1 -2

-3

-3

-4

-4 -5 0.0

-5 0.0

0.5 1.0 1.5 2.0 Rotational speed [rad/s]

2.5

0.5 1.0 1.5 2.0 Rotational speed [rad/s]

2.5

Fig. 13. Whirl speed charts for the simple general bladed-rotor system based on reduced Hill’s matrices of order 3N considering the stator anisotropy (DS ¼ 0.1) along with (a) the asymmetric disk inertia (dD ¼0.1, dB ¼ 0), and (b) the dissimilar blade stiffness (dD ¼ 0, dB ¼0.1): , strong mode; —, weak mode; —m—, rotor precessional mode; —K—, blade traveling wave mode;—J—, coupled collective multi-blade mode; ————, collective multi-blade mode; hatched speed range indicates the instability region.

dFRF for system isotropy / blade-group symmetry

dFRF for system anisotropy 102 Magnitude

Magnitude

102 100 10-2 10-4

100 10-2 10-4

-5

-4

-3

-2 -1 0 1 2 Frequency [rad/s]

3

4

5

-5

dFRF for system asymmetry

-2 -1 0 1 2 Frequency [rad/s]

3

4

5

102 Magnitude

Magnitude

-3

dFRF for system asymmetry & anisotropy

102 100 10-2 10-4

-4

-5

-4

-3

-2 -1 0 2 1 Frequency [rad/s]

3

4

5

100 10-2 10-4

-5

-4

-3

-2 -1 0 2 1 Frequency [rad/s]

3

4

5

Fig. 14. Magnitude plots of dFRFs for the simple bladed-rotor system with both the stator anisotropy and asymmetric disk inertia (DS ¼0.1, dD ¼0.1, dB ¼ 0, O ¼ 1 rad/s): (a) —, 9HgC ;0pC ;0 9 for the system symmetry; ————, 9HgR ;0pR ;0 9 for the blade-group symmetry; (b) 9HgC ;0pC ;0 9 for the system anisotropy; (c) 9HgC ;2pC ;0 9 for the system asymmetry; and (d) 9HgC ;2pC ;0 9 for the coupled effect between the system asymmetry and anisotropy.

K.-T. Kim, C.-W. Lee / Journal of Sound and Vibration 331 (2012) 5224–5246

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bi-orthonormalized modal vectors, given in Tables 2 and 3, explain that these modes are associated with the collective multi-blade motion partially coupled with the rotor precessional and blade traveling wave motions. Furthermore, the dFRF for the blade-group asymmetry, HgR ; þ 1pC ;0 ðjoÞ, shown in Fig. 12(c), demonstrates that the collective multi-blade modes may also contribute to the bladed-rotor whirling responses through the dynamic coupling caused by the blade-group asymmetry.

4.4. General bladed-rotor system Lastly, the asymmetric bladed-rotor supported by the anisotropic stator is considered taking the values given in Table 1 for the rotor asymmetry and the stator anisotropy. Since this system is characterized by periodically time-varying parameters due to the non-symmetry in both the rotating and the stationary parts, we use the modulated coordinate transformation approach to obtain an equivalent infinite-order time-invariant system equation. Then, the transformed system is approximated to the reduced system of order 3N, but it is sufficient to yield the modal solutions of practical interest [14,19]. As for the case of the asymmetric blade-rotor systems, the whirl speed charts in Fig. 13(a) and (b) certainly reflect the dynamic influence of the two different types of rotating asymmetry on the system modal characteristics, despite the presence of many modulated modes generated from the periodically time-varying nature of the system. The dFRFs, however, are capable of providing a more clear and intuitive understanding of the system dynamics regarding the system non-symmetry as follows: Since, from a global system dynamics perspective, the two models are considered to be similar, no distinct differences are found between the conventional dFRFs in Figs. 14(b)–(d) and 15(b)–(d), which represent the dFRF for system isotropy / blade-group symmetry

dFRF for system anisotropy 102 Magnitude

Magnitude

102 100 10-2 10-4 -5 -4 -3 -2 -1 0 1 2 Frequency [rad/s]

100 10-2 10-4

3

4

-5 -4 -3 -2 -1 0 1 2 Frequency [rad/s]

5

102

5

4

5

102 Magnitude

Magnitude

4

dFRF for system asymmetry & anisotropy

dFRF for system asymmetry

100 10-2 10-4 -5 -4 -3 -2 -1 0 1 2 Frequency [rad/s]

100 10-2 10-4

3

4

-5 -4 -3 -2 -1 0 1 2 Frequency [rad/s]

5

102 Magnitude

102

100 10-2 10-4 -5 -4 -3 -2 -1 0 1 2 Frequency [rad/s]

3

dFRF for blade-group asymmetry & system anisotropy

dFRF for blade-group asymmetry

Magnitude

3

100 10-2 10-4

3

4

5

-5 -4 -3 -2 -1 0 1 2 Frequency [rad/s]

3

4

5

Fig. 15. Magnitude plots of dFRFs for the simple bladed-rotor system with both the stator anisotropy and dissimilar blade stiffness (DS ¼0.1, dD ¼0, dB ¼0.1, O ¼ 1 rad/s): (a) —, 9HgC ;0pC ;0 9 for the system symmetry; ————, 9HgR ;0pR ;0 9 for the blade-group symmetry; (b) 9HgC ;0pC ;0 9 for the system anisotropy; (c) 9HgC ;2pC ;0 9 for the system asymmetry; (d) 9HgC ;2pC ;0 9 for the coupled effect between the system anisotropy and asymmetry; (e) 9HgR ; þ 1pC ;0 9 for the blade-group asymmetry; and (f) 9HgR ;1pC ;0 9 for the coupled effect between the blade-group asymmetry and the system anisotropy.

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global rotor asymmetry, stator anisotropy, and coupled effect between them; whereas, the blade-group asymmetry can be identified from HgR ; þ 1pC ;0 ðjoÞ in Fig. 15(e); and in addition, another type of the dFRF, HgR ;1pC ;0 ðjoÞ, is found to characterize the coupled effect between the blade-group asymmetry and the stator anisotropy, as seen in Fig. 15(f). 5. Conclusions A general bladed-rotor system model, in which the non-symmetry in all the dynamic components is taken into account, is developed by correlating the blade-group deformation with rotor whirling motion. Based on the complex modal analysis theory and the modulated coordinate transformation approach, three additional dFRFs are newly defined to characterize the blade-group dynamics, allowing us to separately investigate the dynamics of the rotating blade-array and the other components, and their coupled effect, in detail. We found that the blade-group asymmetry leads to the dynamic coupling between the bladed-rotor whirling and the collective multi-blade modes, which are completely decoupled in the case of bladed-rotors with a symmetric blade-configuration, and it introduces additional directional whirling modes that increase the possibility of resonance. References [1] C.W. Lee, A complex modal testing theory for rotating machinery, Mechanical Systems and Signal Processing 5 (2) (1991) 119–137. [2] C.W. Lee, C.Y. Joh, Development of the use of the directional frequency response functions for the diagnosis of anisotropy and asymmetry in rotating machinery: theory, Mechanical Systems and Signal Processing 8 (6) (1994) 665–678. [3] C.Y. Joh, C.W. Lee, Use of dFRFs for diagnosis of asymmetric/anisotropic properties in rotor-bearing system, ASME Journal of Vibration and Acoustics 118 (1996) 64–69. [4] Y.H. Seo, C.W. Lee, K.C. Park, Crack identification in a rotating shaft via the reverse directional frequency response functions, ASME Journal of Vibration and Acoustics 131 (2009) 011012. [5] C.W. Lee, Y.H. Seo, Enhanced Campbell diagram with the concept of HN in rotating machinery: Lee diagram, ASME Journal of Applied Mechanics 77 (2010) 021012. [6] R.P. Coleman, A.M. Feingold, Theory of Self-excited Mechanical Oscillations of Helicopter Rotors With Singed Blades, Technical Report NACA-TR1351, 1958, available from /http://ntrs.nasa.govS. [7] S.H. Crandall, J. Dugundji, Resonant whirling of aircraft propeller-engine systems, ASME Journal of Applied Mechanics 48 (1981) 929–935. [8] K.H. Hohenemser, S.K. Yin, Some applications of the method of multiblade coordinates, Journal of the American Helicopter Society 17 (3) (1972) 3–12. [9] P.F. Skjoldan, M.H. Hansen, On the similarity of the Coleman and Lyapunov–Floquet transformations for modal analysis of bladed rotor structures, Journal of Sound and Vibration 327 (3-5) (2009) 424–439. [10] D.J. Ewins, Vibration characteristics of bladed disc assemblies, Journal of Mechanical Engineering Science 15 (3) (1973) 165–186. [11] E.P. Petrov, K.Y. Sanliturk, D.J. Ewins, A new method for dynamic analysis of mistuned bladed disks based on the exact relationship between tuned and mistuned systems, ASME Journal of Engineering for Gas Turbines and Power 124 (3) (2002) 586–597. [12] M.P. Castanier, C. Pierre, Modeling and analysis of mistuned bladed disk vibration: status and emerging directions, Journal of Propulsion and Power 22 (2) (2006) 384–396. [13] J.H. Suh, S.W. Hong, C.W. Lee, Modal analysis of asymmetric rotor system with isotropic stator using modulated coordinates, Journal of Sound and Vibration 284 (2005) 651–671. [14] C.W. Lee, D.J. Han, J.H. Suh, S.W. Hong, Modal analysis of periodically time-varying linear rotor systems, Journal of Sound and Vibration 303 (2007) 553–574. [15] S.B. Chun, C.W. Lee, Vibration analysis of the shaft-bladed disk system using the substructure synthesis and assumed modes method, Journal of Sound and Vibration 189 (5) (1996) 587–608. [16] S.K. Sinha, Dynamic characteristics of a flexible bladed-rotor with Coulomb damping due to tip-rub, Journal of Sound and Vibration 273 (2004) 875–919. [17] M. Gruin, F. Thouverez, L. Blanc, P. Jean, Nonlinear dynamics of a bladed dual-shaft, European Journal of Computational Mechanics 20 (1–4) (2011) 207–225. [18] C.W. Lee, Vibration Analysis of Rotors, Kluwer Academic Publishers, Dordrecht, 1993. [19] G. Genta, Whirling of unsymmetrical rotors: a finite element approach based on complex co-ordinates, Journal of Sound and Vibration 124 (1) (1988) 27–53. [20] W. Jhonson, Helicopter Theory, Princeton University Press, New Jersey, 1980. [21] R.A. Calico, W.E. Wiesel, Control of time-periodic systems, Journal of Guidance 7 (6) (1984) 671–676. [22] J.A. Richards, Analysis of Periodically Time-varying Systems, Springer, Berlin, 1983. [23] A.R. Gourlay, G.A. Watson, Computational Methods for Matrix Eigenproblems, Wiley, New York, 1973. [24] C.W. Lee, D.J. Han, Strength of modes in rotating machinery, Journal of Sound and Vibration 313 (2008) 268–289.