Vibration behavior of a two-crack shaft in a rotor disc-bearing system

Mechanism and Machine Theory 113 (2017) 67–84

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Research paper

Vibration behavior of a two-crack shaft in a rotor disc-bearing system Hamid Khorrami∗, Subhash Rakheja, Ramin Sedaghati Department of Mechanical and Industrial Engineering, Concordia University, Montréal, Canada

a r t i c l e

i n f o

Article history: Received 15 June 2016 Revised 12 March 2017 Accepted 13 March 2017

Keywords: Multiple cracks Rotor disc-bearing system Critical speeds Harmonic balance method Breathing function

a b s t r a c t The vibration response characteristics of a rotor disc-bearing system with one and two cracks are analytically investigated using a modified harmonic balance method. The analytical model is formulated considering rigid-short bearing supports to study the effects of cracks’ characteristics such as depth, location and relative angular position on selected vibrational properties, namely, critical speeds, harmonic and super-harmonic components of the unbalance lateral response and the shaft center orbit. Each crack is initially described by a breathing function proposed by Mayes and Davies, which is subsequently modified as a softly-clipped cosine function to accurately describe saturation in breathing phenomenon. The response characteristics of the cracked shaft are also compared with those of the system with an intact shaft in order to identify potential measures for detecting cracks. The results show that the presence of a second crack emphasizes changes in the critical speeds of the rotor disc-bearing system, while relative angular position of the cracks influences the shaft center orbit. The validity of the proposed analytical model and the solution method is also investigated through simulations of a finite element model of the system. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Emergence and propagation of fatigue cracks in rotating machines could adversely affect their operational efficiency and may lead to catastrophic failures. Considerable efforts have been made to identify various crack detection measures based on vibration responses of the rotor disc-bearing systems, which have been thoroughly reviewed in [1–3]. The changes in vibrational properties of the rotor system in the presence of a crack have been widely proposed for identification of crack parameters in majority of the studies. Changes in modal parameters, transient responses, shaft center orbit evolutions in sub-critical speeds, torsional-lateral vibrational coupling and equivalent fictitious loads have been applied to evaluate effects of crack parameters on vibrational properties of the rotor disc-bearing systems [4–8]. The effects of a single crack on vibrational properties of different structural systems have been widely reported, while the effects of double/ multiple cracks have been addressed in a relatively fewer studies. This may be due to the fact that the presence of multiple cracks on a shaft happens rarely and as the authors know, no case study has been reported. Different combinations of properties of single or multiple crack such as depth and location may yield similar effects on vibrational properties of the system. Sekhar [9] conducted a review of methods for identifying two or more cracks in components

∗

Corresponding author. E-mail addresses: [email protected] (H. Khorrami), [email protected] (S. Rakheja), [email protected] (R. Sedaghati).

http://dx.doi.org/10.1016/j.mechmachtheory.2017.03.006 0094-114X/© 2017 Elsevier Ltd. All rights reserved.

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such as beams, rotors and pipes and concluded that efficiency of identification of single or multiple cracks mostly depends on the applied signal processing techniques. Fourier transform is a suitable signal processing method for analyzing signals, which their frequency content does not change in time. However, in case of time-variable frequency contents of vibration responses of cracked mechanical systems, their features should be extracted using advanced signal processing techniques such as wavelet transform (WT) and Hilbert-Huang transform (HHT) [e.g., 10–13]. Some studies have presented changes in preselected vibrational properties due to known multiple crack parameters [e.g., 14,15], while others have presented inverse problem of determining multiple cracks’ parameters from known changes in vibrational properties [e.g., 16,17]. Sekhar [14] investigated the effects of two open cracks on the eigenfrequencies, modeshapes and threshold speed of a rotor system using Finite Element (FE) method. Darpe et al. [15] studied the effect of two breathing cracks on unbalance response of a simple Jeffcott rotor. Both studies suggested that relative angular position of the two cracks may exhibit significant changes in the shaft center orbit. Saridakis et al. [16] formulated an inverse problem of identifying cracks’ properties (depth, location and relative angular position) from known changes in vibrational properties of a flexible cantilever shaft with two breathing cracks. The inverse problem was solved using artificial neural networks, genetic algorithm and fuzzy logic methods. The study suggested that the method could provide real-time identification of crack parameters. Ramesh and Sekhar [17] proposed an alternate crack indicator, referred to as ‘amplitude deviation curve (ADC)’, based on the concept of operational deflection shape (ODS) and concluded that the method outperforms that based on the continuous wavelet transform (CWT) reported in [18] for detecting two breathing cracks. Sekhar [19] subsequently considered both forward and inverse problems to investigate the effects of two cracks on vibration responses of a shaft-rotor system. The method employed fictitious loads obtained from a FE model for identifying the depths and locations of the cracks. Han and Chu [20] investigated the stability of a rotor-bearing system in the presence of two transverse cracks on the shaft. They compared the effect of applying two different crack breathing models proposed by Mayes and Davies [21] and Al-Shudeifat and Butcher [22] on instability shaft speeds. Furthermore, they showed that the shaft speed instability regions are highly affected by the relative angular position of the cracks on the shaft. The reported analytical studies have invariably shown that the changes in vibrational properties of a cracked shaft-rotor system are strongly affected by the crack model. Modeling the crack has thus been an important challenge in many studies. The changes in shaft cross-section geometry in the vicinity of the crack, and thereby changes in the local stiffness of the shaft strongly depend on the crack depth, which are generally obtained using the concepts of strain energy release rate and stress intensity factor [23,24] or changes in area moment of inertia about the transverse coordinates of the cracked shaft [25,26]. In a rotating shaft, the crack-induced changes are also a function of the shaft rotation, when the shaft static deflection dominates the lateral vibration and the crack thickness in axial direction is negligible [22]. The crack has been considered as an open-crack in a number of studies, assuming negligible effect of shaft rotation on changes in the shaft local stiffness [e.g., 14,15,27]. Alternatively, crack is modeled as a breathing crack to incorporate the effect of shaft rotation on the localized stiffness [e.g., 21,28,29]. The majority of the breathing crack models employ explicit breathing functions such as step and cosine to describe local stiffness changes with shaft rotation [e.g., 27,30,31]. Such functions, however, describe the breathing behavior of the crack neglecting the effects of its properties. A few studies have employed linear fracture mechanics to evaluate local stiffness of the shaft cross-section near the crack location corresponding to different shaft angles of rotation and proposed breathing crack models considering the effects of crack depth and location [24,32,33]. Jun et al. [32] used the sign of the total stress intensity factor (SIF) at each point on the crack edge to determine whether the crack is open or closed. In compressive stress state, the sign of total SIF is negative and the crack is assumed to be closed, while in tensile stress state the sign of total SIF is positive and consequently the crack is considered to be open. Darpe et al. [24] introduced the concept of crack closure line (CCL), an imaginary line perpendicular to the crack edge, separating the open and closed parts of the crack, to study breathing behavior of the crack. The CCL position was obtained considering the sign of the total SIF at each point on the crack edge, as proposed by Jun et al. [32]. Owing to complexities associated with evaluations of the stress intensity factor for shaft angles corresponding to stress state transition between the vertical and horizontal moments, Chasalevris and Papadopoulos [33] employed B-spline curves to interpolate between the transient points. Al-Shudeifat and Butcher [22] used changes in area moment of inertia about the transverse coordinates of the cracked shaft cross-section area to obtain an accurate breathing function. The method considered exact locations of centroid and neutral axis of the cracked shaft cross-section at each shaft angle to estimate corresponding area moments of inertia and the local stiffness. Unlike the earlier studies [24,32,33], the breathing crack model employed an explicit breathing function. The reported studies have employed different analytical [6,15,33–38] and numerical models [5,22,39] of the rotor discbearing systems for evaluating reference vibration responses and the effects of cracks. The analytical models are generally based on the simplified Jeffcott rotor model [e.g., 6,15]. Chasalevris and Papadopoulos [33] used the Timoshenko beam theory to develop a continuous shaft-disc model to study the effects of a crack on the vibrational responses. A number of FE models of the rotating cracked rotor disc-bearing systems have also been reported considering either Euler–Bernoulli and Timoshenko beam theories [e.g., 5,9,22,26,39,40]. The studies have also employed different solution strategies, which satisfy the continuity and boundary conditions to obtain steady-state and transient responses, and changes in the selected vibrational properties. Some studies have employed a solution function in exponential form, which is only limited to the first harmonic component [e.g., 33,35,36,41], while

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Fig. 1. (a) A continuous shaft-disc system with two cracks; and (b) Shaft cross-sections at the locations of the crack 1 and 2 in the stationary (x,y,z) and rotating (ζ , ξ , η) frames.

others have used numerical integration techniques to obtain super-harmonic components of the steady-state responses of the analytical models [e.g., 15,32,42] and the transient responses [e.g., 43–45]. The steady-state responses of FE models have also been obtained using different methods such as generalized harmonic balance and alternate frequency/time domain methods [e.g., 26,31]. In this study, an analytical model of a rotor disc-bearing system is formulated and analyzed to obtain its vibrational responses in the presence of two breathing cracks within the flexible shaft. A modified harmonic balance method is utilized to determine responses such as critical speeds, shaft center orbits and the unbalance lateral response. An explicit function is proposed to describe breathing behavior of the crack and facilitate implementation of the modified harmonic balance method. The proposed function also resulted in peaks with additional amplitudes in frequency spectrum of the lateral response. The effects of cracks’ parameters such as depths, locations and relative angular positions on the critical speeds and the shaft center orbit are investigated. The frequency spectra of lateral responses of the systems with single and two cracks are compared to that of an intact rotor in order to highlight the changes in responses caused by the cracks. The validation of the analytical model together with the proposed solution strategy is also demonstrated by comparing the analytical model results with those obtained from the FE model. 2. Analytical model of a continuous shaft with two breathing cracks The governing equation of lateral motion of a continuous rotating shaft with a disc located at the mid-span (Fig. 1) is obtained using the Timoshenko beam theory [36], considering the gyroscopic moments, rotary inertia and shear deformation, as:

EI

 ∂ 4u ρ 2 Ar02  ∂ 3 u ∂ 4 u  E Iρ ∂ 2 u ρ 2 Ar02 ∂ 4 u ∂2 u − + ρ Ar02 + 2iρ Ar02  2 + − 2i + ρA 2 = 0 4 2 2 4 3 kG kG ∂ t kG ∂x ∂ x ∂t ∂ x∂ t ∂t ∂t

(1)

where u(x, t ) = uy (x, t ) + iuz (x, t ) is lateral response of the system, in which uy and uz represent vertical and horizontal components of the response, respectively, and i denotes the imaginary unit. The parameters E, I, k, G, A, r0 , ρ and  are the modulus of elasticity, shaft area  moment of inertia about x-axis, shear factor, shear modulus, shaft cross-sectional area of the shaft, radius of gyration ( I/A ), mass density of the shaft and spin speed of the shaft, respectively. To obtain the lateral response u(x, t) the assumed solution function must satisfy boundary conditions of the rotating shaft in addition to compatibility relations at the crack and disc locations. The flexible shaft with a single disc is divided into two segments located on left- and right-sides of the disc. The coupled governing equations of lateral motion formulated for each segment are combined considering the appropriate continuity conditions. The governing equations describing lateral response of a flexible shaft with a single transverse crack can also be obtained in a similar manner [33,34]. The effect of disc

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is incorporated into the continuity conditions in shear force and bending moments, while the crack effect is represented by a small but abrupt change in the slope continuity condition of the shaft, as described in Section 2.3. The disc effects are related to its mass, and diametral and polar mass moment of inertia, while the crack effect depends on its depth and breathing behavior. In the presence of two transverse cracks, as shown in Fig. 1, the governing equation of motion is formulated considering four separate segments (s = 0, 1, 2, 3 ) of the shaft (0 ≤ x ≤ L0 ; L0 ≤ x ≤ L1 ; L1 ≤ x ≤ L2 ; and L2 ≤ x ≤ L3 ). In the model, x = L0 and x = L2 , define locations of the two cracks, while x = L1 describes the disc location on the shaft of length L3 . Satisfying the continuity conditions at the location of the disc and the cracks simultaneously would yield the lateral dynamic response of the entire cracked rotor disc-bearing system. It is to be noted that us (x, t) in Fig. 1 represents the lateral displacement of segment s of the flexible shaft, , while ω is the whirling speed of the shaft. 2.1. Crack modeling The presence of a transverse crack on a shaft causes a sudden change in the slope of the shaft in lateral direction, which depends on crack depth and exhibits a reduction in the shaft local stiffness at the crack location. The crack-induced changes in local stiffness can be obtained using linear fracture mechanics theory [23,24,43]. The compliance matrix, C, for a fullyopen crack in the y-z plane has been derived using the strain energy release rate and Castingliano’s theorem and generally described as [23]:



c C = yy czy

cyz czz



(2)

A crack can be modeled as an open crack or a breathing crack. The crack maybe modeled as an open crack, when dependence of the local stiffness change on the shaft angle is considered negligible. It has been shown that the stiffness reduction is strongly related to shaft angle of rotation. The crack is thus more accurately modeled as a breathing crack, where the crack opens and closes during each revolution of the shaft. A periodic breathing function, proposed by Mayes and Davies [29], has been widely employed, given by:

1 − cos(t + χr ) 2 = | r − 1 |, r = 1, 2

f (t ) =

χr

(3)

where r (r = 1, 2 ) define the angles between the crack edge normal line and axis ξ , as shown in Fig. 1(b). The compliance matrix for the breathing crack corresponding to a shaft angle is subsequently obtained from the product of Eqs. (2) and (3). The breathing function in Eq. (3) can also be expressed in the exponential form, which can facilitate solution of equations of motion of the rotor disc-bearing system using the modified harmonic balance method, as described in Section 4, such that:

f (t ) =

1 1 − (ei(t+χr ) + e−i(t+χr ) ) 2 4

(4)

2.2. An alternate breathing function A crack in a shaft may exhibit saturation in opening and closing during rotation. This saturation may alter the amplitude of peaks in frequency spectrum of lateral response of the rotor disc-bearing system, which is believed to serve as a better indicator of presence of the second crack. An alternate explicit breathing function is thus proposed to account for saturation of crack breathing behavior using a softly-clipped cosine function, such that:

f (t ) =

5 1 1 − cos(t + χr ) − cos(3(t + χr ) − π ) 2 9 18

(5)

It is evident that the periodic function in Eq. (3) leads to fully-open or closed crack at an instant as shaft angle approaches (θ = ± n2π , n = 0, 1, 2, . . . ). The modified explicit breathing function in Eq. (5), on the other hand yields saturation in opening or closing over a range of shaft angle. In these saturation intervals, the crack locates completely in either tensile or compressive stress zones of the shaft and remain either fully-open or closed. Fig. 2 compares the normalized crack opening obtained from the proposed periodic breathing function and that of Mayes and Davies function. The comparison clearly illustrates that the proposed function can describe crack opening and closing saturation. The proposed breathing function also contains the term cos(3t + r ) which leads to distribution of the energy dissipation originated from crack saturation among more number of harmonic components of the lateral response using the modified harmonic balance method, and may reduce amplitudes of harmonic and super-harmonic components. Briefly, trying to model the crack breathing behavior considering the saturation phenomenon may give lateral response peaks in frequency spectrum with lower amplitudes. Mayes and Davies breathing function [29], Eq. (3), does not account for saturation in crack breathing and subsequently the energy dissipation due to crack saturation, which may lead to lateral response peaks with relatively higher amplitudes. Moreover, a few studies have reported that the compliance matrix for a breathing crack varies periodically versus the shaft angle [22,24,32,33]. An explicit function in the form of Eq. (5) has not yet been provided for application in the modified

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71

Fig. 2. Comparison of the proposed breathing function with that reported by Mayes and Davies [29].

harmonic balance method, described in Section 4. The proposed explicit function can also be expressed in the exponential form to facilitate its application to the modified harmonic balance method, such that:

f (t ) =

 1  i((3t+χ )−π ) −i((3t+χ )−π )  5  i(t+χr ) 1 r r − e + e−i(t+χr ) − e +e 2 18 36

(6)

2.3. Boundary and continuity conditions The boundary conditions of rotor disc-bearing systems with rigid-short bearing supports [3] have been described in a number of studies. Assuming negligible resistance to bending moments, the boundary conditions can be expressed as [33]:

∂ 2 u0 ∂ 2 u3 ( 0, t ) = (L , t ) = 0 2 ∂x ∂ x2 3

(7)

where L3 is shaft length. Rigidity of supports also leads to following displacement boundary conditions:

u0 ( 0, t ) = u3 ( L3 , t ) = 0

(8)

The continuity conditions for the shaft near the crack location have also been described by Chasalevris and Papadopoulos [33], when the disc and the crack are located at the same position on the shaft. The continuity conditions for the shaft considering multiple cracks and different locations of the disc and cracks may also be formulated in a similar manner. Assuming negligible contributions of disc mass and inertia to the slopes and displacements on the left- and right-sides of the disc, the displacement and slope continuity equations at the disc location can be expressed as:

u1 ( L1 , t ) = u2 ( L1 , t )

(9)

∂ u1 ∂ u2 (L , t ) = (L , t ) ∂x 1 ∂x 1

(10)

where L1 denotes the disc location on the shaft, as shown in Fig. 1. The disc mass, however, yields abrupt change in shear force developed in adjacent segments of the shaft on both sides of the disc, which is given by:



3 2

∂ 3 u1 ( L1 , t ) ∂ 3 u1 ( L1 , t ) ∂ 3 u2 ( L1 , t ) 2 ∂ u ( L1 , t ) EI − + ρ Ar0 − ··· ∂ x3 ∂ x3 ∂ x∂ t 2 ∂ x∂ t 2 2 1

∂ 2 u1 ( L1 , t ) ∂ u ( L1 , t ) ∂ 2 u2 ( L1 , t ) + 2iρ Ar02  − = Md + mu ru 2 ei(t+ u ) ∂ x∂ t ∂ x∂ t ∂t2

(11)

where Md , mu , ru and u represent the disc mass, unbalance mass, eccentricity and angular position of the unbalance. u is defined as the angle between the normal line of the first crack and the unbalance mass, as shown in Fig. 1(b). The above force continuity equation is formulated considering an unbalance mass located on the disc. Furthermore, the disc mass moments of inertia affects the shaft bending moment, such that:



∂ 2 u2 ( L1 , t ) ∂ 2 u1 ( L1 , t ) EI − ∂ x2 ∂ x2



= ID,d

∂ 3 u1 ( L1 , t ) ∂ 2 u1 ( L1 , t ) − iI p,d  ∂ x∂ t ∂ x∂ t 2

(12)

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Fig. 3. FE model of a rotor disc bearing system with two cracks.

where ID, d and Ip, d denote the disc diametral and polar mass moment of inertias. The displacement continuity at the cracks’ location can be expressed as:

u p (L p , t ) = u p+1 (L p , t )

(13)

where Lp ( p = 0, 2 ) describes locations of cracks on the shaft. The presence of a crack between two consecutive shaft segments imposes a sudden change in slopes of displacements of shaft segments in the lateral direction. The slope continuity equation of the two adjacent segments at the crack location is thus expressed as:

2 p

∂ u p+1 ∂ up ∂ 2 up ∂ u r r L , t +  − i δ f ( t ) con j L , t (L p , t ) − (L p , t ) = · · · EI (β r − iν r ) f (t ) ( ) ( ) ( ) ∂x ∂x ∂ x2 p ∂ x2 p

(14)

where the parameters β r , ν r ,  r and δ r represent the effect of a breathing crack and depend on the crack depth. The superscript r = 1, 2 denotes the number of the crack, as shown in Fig. 1. These parameters are defined using the compliance matrix, described in Eq. (2), such that:



 = r

r r czz − cyy

2





;β = r

r r czz + cyy

2





;δ = r

r r cyz + czy



2



;ν = r

r r cyz − czy



2

(15)

The shear force and bending moment continuity between the adjacent shaft segments in the vicinity of the crack are independent of the crack properties, and are formulated, as:



EI

3 p+1

∂ 3 u p (L p , t ) ∂ 3 u p+1 (L p , t ) (L p , t ) ∂ 3 u p (L p , t ) 2 ∂ u − + ρ Ar − ··· 0 ∂ x3 ∂ x3 ∂ x∂ t 2 ∂ x∂ t 2 2 p

∂ u (L p , t ) ∂ 2 u p+1 (L p , t ) 2 + 2iρ Ar0  − =0 ∂ x∂ t ∂ x∂ t

∂ 2 u p+1 (L p , t ) ∂ 2 u p (L p , t ) = ∂ x2 ∂ x2

(16)

(17)

The boundary and continuity conditions, described in Eqs. (7)–(17), are used in conjunction with the governing equation of motion to obtain lateral vibration response of the rotor disc-bearing system using the modified harmonic balance method. 3. FE model of the cracked rotor disc-bearing system The validity of the proposed analytical model of the rotor disc-bearing system is evaluated through comparisons of lateral responses with those obtained from a FE model [26,31,46]. In the FE model, the shaft is modeled using Timoshenko beam elements considering different number of elements in the shaft segments with a rigid disc mounted on the mid-span [26,31]. The governing equations of motion for the FE model of the shaft-disc system with two cracks and supported on rigid-short bearings, as shown in Fig. 3, can be described in the finite element form, in a manner described in [31] for a single crack system, such that:



MU¨ + DU˙ +

K−

2

fr (t )KC r U = F

(18)

r=1

where M is mass matrix integrating shaft and disc mass matrices, D includes shaft and rigid disc gyroscopic effects, K describes stiffness of the shaft, and F represents unbalance and gravitational forces imposed on the rotor disc-bearing system. KC r represents the stiffness matrix of the shaft element containing rth (r = 1, 2. ) crack and fr (t) is corresponding crack breathing function. 4. Modified harmonic balance method of solutions The harmonic balance method has been widely used for dynamic response analysis of nonlinear mechanical structures. This approximation technique is based on development of a truncated Fourier series to obtain periodic vibration responses of

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non-linear systems [46]. Sinou and Lee [31] and AL-Shudeifat et al. [26] employed harmonic balance method to obtain nonlinear lateral vibration response of a FE model of a rotating shaft-disc system with a breathing crack. The super-harmonic components of lateral displacement response of the model comprising a breathing crack were obtained assuming the periodic lateral response u(x, t) of the form:

u(x, t ) =

m

n (x )(An cos(nωt ) + Bn sin(nωt ))

(19)

n=1

where m and n denote number of harmonic components considered in the response and the order of the super-harmonics, respectively.  n (x) represents the space-dependent part of the nth super-harmonic component of the lateral response. The majority of the studies on analytical models of a rotating shaft-disc system with a breathing crack, however, have considered the assumed solution function leading only to the first harmonic [e.g., 33,34], such that:

u(x, t ) = (x )eiωt

(20)

In this study, a modified harmonic balance method is used to determine super-harmonic components of lateral response of the rotor disc-bearing system subject to synchronized unbalance force, ω = . The assumed solution function is subsequently formulated as:

u(x, t ) =

m

n (x )einωt

(21)

n=1

Unbalance force may excite backward whirling of an asymmetric shaft or a symmetric shaft supported on anisotropic bearing supports [47,48]. In this study, a symmetric shaft mounted on rigid bearings is considered, thus, the assumed solution is limited only to forward whirling term, einωt , and the backward whirling term, e−inωt , is neglected. The space-dependent portion of the proposed solution,  n (x), may be expressed as:

n (x ) = An eαn x

(22)

where An is coefficient of the nth super-harmonic component and α n denotes roots of the characteristic equation, which are obtained upon substitution of assumed solution into coupled governing equation of lateral motion, described in Eq. (1), such that:

E Iα 4jn +

 E I ρ kG



+ ρ Ar02 n2 − 2ρ Ar02 n

  ρ 2 Ar02  4 α 2jn 2 + n − 2n3 4 − ρ An2 2 = 0 kG

(23)

The above characteristic equation is a polynomial of order four with characteristic roots α jn , ( j = 1, 2, 3, 4 ). Considering ω = , the assumed solution in Eq. (21) can be written as:



u(x, t ) =

m 4 n=1

Anj eα jn x eint

(24)

j=1

4.1. Methods of solution The coefficients of the assumed solution functions are determined while ensuring that the continuity conditions between adjacent segments of the shaft and support boundary conditions are satisfied. The analyses are performed considering two different types of crack breathing functions. These include the cosine function proposed by Mayes and Davies [29] and the proposed softly-clipped cosine function, described in Eq. (6). Substituting the assumed solution function in Eq. (24) into the boundary and continuity conditions formulated in Section 2.3 yields a system of algebraic equations, which can be expressed in the matrix form as:

[ ] { A } = { B }

(25)

where [] is matrix of coefficients related to the selected breathing function, vector {A} contains the coefficients of harmonic and super-harmonic components and vector {B} describes the harmonic external forces acting on each segment of the shaft. The above system of linear equations is solved to obtain lateral displacement response of the rotor disc-bearing system in the presence of either one or two cracks. The system of linear equations are presented in details in the Appendix. The lateral responses of the rotor disc-bearing system are also evaluated from the FE model considering eigen solutions of Eq. (18) together with the assumed solution in Eq. (19). The resultant eigenvalues are coefficients of the harmonic and superharmonic components of the lateral response, An and Bn , as described in Eq. (19). Owing to important effect of length of the cracked element on the shaft local stiffness and consequently the lateral response of the FE model, the cracked element length for each crack depth is chosen so as to yield the first critical speed identical to that obtained from the analytical model. The length of the cracked elements corresponding to the each normalized crack depth is given in Table 2. It is noted that the summarized length of the cracked element in Table 2 are corresponding to the Normalized crack locations of 0.45 and 0.55 on the shaft.

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H. Khorrami et al. / Mechanism and Machine Theory 113 (2017) 67–84 Table 1 Material properties and dimensions of the shaft-disc system. Description

Value

Modulus of elasticity, E (Pa) Modulus of rigidity, G (Pa) Shaft diameter (m) Shaft length (m) Shaft density, ρ (kg/m3 ) Disc density, ρ d (kg/m3 ) Disc diameter (m) Disc thickness (m) Normalized disc location Unbalance mass, mu (kg) Unbalance eccentricity, ru (m) Unbalance angular position, u (deg)

69 × 109 34 × 109 0.01905 1.27 2700 2700 0.1524 0.0254 0.5 0.01 0.0508 0

Table 2 Comparisons of first cr,1 and third cr,3 critical speeds of the rotor disc-bearing system obtained from the modified harmonic balance method and the FE model, (γ1 = 0.45 and γ2 = 0.55). Normalized

Element length

μ2 = 0

crack depth μ1

of crack 1 (mm)

Proposed method

FE analysis

Proposed method

μ2 = 0.2 FE analysis

Proposed method

μ2 = 0.6 FE analysis

First critical speed, cr,1 (rpm) 0 0 0.2 4.9 0.4 10.5 0.6 17.8 0.8 29.4 1 52.6

747 746.7 745.3 742.7 738.1 730

747 746.7 745.3 742.7 738.1 730

746.7 746.3 745 742.4 737.8 729.7

746.7 746.4 745 742.4 737.8 729.7

742.7 742.4 741.1 738.5 734 726.1

742.7 742.4 741.1 738.6 734.2 726.4

Third critical speed, cr,3 (rpm) 0 0 0.2 4.9 0.4 10.5 0.6 17.8 0.8 29.4 1 52.6

9525.4 9522.6 9511 9488.4 9449.4 9382.7

9532 9530.6 9525.5 9515.6 9490.4 9466

9522.6 9519.7 9508.2 9485.6 9446.8 9380.2

9530.6 9529.5 9524.9 9515.3 9490.2 9464.3

9488.4 9485.6 9474.6 9452.9 9415.5 9351.3

9515.6 9515.3 9512.5 9503.9 9479.5 9428.3

5. Results and discussions 5.1. Model verification The modified harmonic balance method described in Section 4 is used to determine critical speeds of the analytical model of the rotor disc-bearing system with two breathing cracks mounted on rigid-short bearing supports. Table 1 summarizes the material properties and dimensions used in simulations. The cracks’ breathing behavior is initially modeled L L using the Mayes and Davies [29] function, while their locations are taken as γ1 = L0 = 0.45 and γ2 = L2 = 0.55 normalized 3 3 with respect to the shaft length. The resulting first and third critical speeds of the analytical model are compared with those obtained from the FE model, provided in Table 2 considering different cracks’ depth values. The cracks’ depths are normalized with respect to the shaft radius and denoted by μ1 and μ2 in the table. It should be noted that the first critical speed obtained from the analytical model with a single crack is used to calibrate crack elements’ lengths in the FE model, as described in Section 4.1. Since the crack element length also depends on the crack location on the shaft, the identified length could also be used for the second crack, provided they occur on the same location or at mirrored locations of the shaft considering the plane of symmetry of the rotor disc-bearing system. For the purpose of the model verification, mirrored locations of the two cracks are considered, while perpendicular to the shaft at the disc location is taken as the plane of symmetry. The comparisons in Table 2 suggest very good agreements in the first and third critical speeds obtained from the two methods for the ranges of crack depth (μ1 : 0 to 1; μ2 : 0 to 0.6 ) considered. The average deviations in the first and third critical speeds obtained from the two models are in the order of 0.016% and 0.250%, respectively, for the entire range of crack depths. It should be noted that the second deflection mode of the system is not excited since the node coincides with the position of the disc with unbalance. Consequently, the resonant peak corresponding to the second critical speed does not emerge in the lateral response.

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Table 3 First and third critical speeds of the rotor disc-bearing system with two cracks (μ1 = μ2 = 1 and γ1 = 0.5).

γ2

|γ2 − γ1 |

cr,1 (rpm)

cr,3 (rpm)

Breathing function

1 0.9 0.8 0.7 0.6

0.5 0.4 0.3 0.2 0.1

cosine [29]

clipped cosine

727.1 726.1 723.4 719.3 714.4

727.2 726.2 723.5 719.5 714.6

Deviation (%)

0.007 0.007 0.007 0.014 0.014

Breathing function cosine [29]

clipped cosine

9185.4 9102.4 9043.2 9131.4 9171.5

9187.8 9105.9 9047.2 9133.7 9174.4

Deviation (%)

0.013 0.019 0.022 0.012 0.016

Fig. 4. Changes in critical speeds of the system with one crack (μ1 = 0.5; 0  γ1  1 ) and two cracks (μ1 = μ2 = 0.5; γ1 = 0.5; 0  γ2  1) versus crack normalized location (a) first critical speed (cr,1 ); (b) third critical speed (cr,3 ).

5.2. Vibration response characteristics The vibration response characteristics of the rotor disc-bearing system are evaluated using the modified harmonic balance method considering either one or two cracks within the shaft. The effects of crack parameters such as depth, location and relative angular position, χ2 = | 2 − 1 | in case of two cracks, are investigated in view of different responses, namely the critical speeds, shaft center orbit and lateral vibration of the rotor disc-bearing system. Table 3 illustrates the effect of relative axial positions of the two cracks (γ 1 and γ 2 ) with unity normalized depths (μ1 = μ2 = 1 ) on the first and third critical speeds. The critical speeds of the system have been obtained using the Campbell diagram. The table also compares the critical speeds of the analytical model incorporating the proposed softly-clipped cosine breathing function and cosine function of Mayes and Davies [29]. The results are obtained considering fixed location of the first crack (γ1 = L0 /L3 = 0.5 ), while varying position of the second crack from the right bearing support (γ2 = L2 /L3 = 1 ) to a location close to mid-span of the shaft (γ2 = L2 /L3 = 0.6 ). The relative position, |γ2 − γ1 | = 0.5 in Table 3 implies that the second crack is located on right bearing and has no effect on the critical speeds. The results show that both the breathing function models yield comparable critical speeds of the system with cracks. Using the proposed breathing function, however, leads to only slightly higher critical speeds compared to those obtained with Mayes and Davies breathing function. The average deviation between first and third critical speeds obtained from both breathing functions are about 0.009% and 0.016%, respectively. This is likely due to saturation in closing and opening of the crack in the proposed breathing function. The first critical speed of the system with two cracks tends to be lower compared with that with one crack, as the spacing between the two cracks, |γ2 − γ1 |, decreases. This trend, however, is not consistently observed for changes in the third critical speed. The third critical speed tends to be lower when second crack is located close to the extrema points of the corresponding mode at normalized locations of 0.16, 0.5 and 0.83 on the shaft. The change in third critical speed is greatest when the cracks relative position is close to 0.3 (γ2 = 0.8 ). This is also evident from Fig. 4, which shows the changes in critical speeds with reference to those of the intact shaft (shaft without crack) as a function of normalized crack locations. The results in the figure are obtained using cosine breathing function [29] with normalized crack depth (μ1 = μ2 = 0.5) and the 0 relative angular position χ 2 (χ2 = | 2 − 1 | = 0 ). The Fig. 4(a) and (b) also illustrate changes in first and third critical speeds of the system with one crack, respectively, due to the changes in normalized crack location. Results show greatest change in critical speeds when the single crack is located at the mid-span of the shaft. Fig. 5 shows the effect of the cracks’ depths on the first and third critical speeds for crack normalized locations of (γ1 = 0.4 and γ2 = 0.5 ) and 0 relative angular position χ 2 . The critical speeds trend to decrease with increasing depth of the first crack μ1 , which is especially evident for the first critical speed. The results further show notable reduction in shaft

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Fig. 5. Effects of cracks’ depth on critical speeds of the rotor disc-bearing system (γ1 = 0.4 and γ2 = 0.5); (a) first critical speed (cr,1 ); (b) third critical speed (cr,3 ).

Fig. 6. Shaft orbit centers of the two-crack system close to half of the first critical speed ( 12 cr,1 ) corresponding to χ 2 , (μ1 = μ2 = 1; γ1 = 0.4, γ2 = 0.5).

critical speeds with increasing depth of second crack, irrespective of the first crack depth. The maximum changes in critical speeds occur when normalized depths of both cracks are taken as unity. Furthermore, higher changes observed on critical speeds of the system with one small crack compared with one deep crack, as the second crack depth increases. For example, as shown in Fig. 5(b), third critical speeds for the system with two cracks in the presence of first crack normalized depth of 0.2 and 1 are changing from 9518.5 to 9491.3 rpm and from 9185.4 to 9171.5 rpm, respectively, as the second crack normalized depth increases from 0 to unity. The total reductions in

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77

Fig. 7. Shaft orbit center close to half of the third critical speed ( 13 cr,1 ) corresponding to χ 2 , (μ1 = μ2 = 1; γ1 = 0.4, γ2 = 0.5).

Fig. 8. Changes in first and third critical speeds (cr,1 and cr,3 ) of the two-crack shaft due to changes in relative angular position (μ1 = μ2 = 1, γ1 = 0.5 and γ2 = 0.4).

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Fig. 9. Frequency spectra of the cracked system vertical responses at the shaft speed close to 13 cr,1 using the Mayes and Davies breathing function, (a) single crack shaft, γ1 = 0.5, μ1 = 1; (b,c and d) two-crack shaft (γ1 = 0.5 and γ2 = 0.4, μ1 = μ2 = 1 ).

the third critical speeds in these two cases are 27.2 and 13.9 rpm, respectively. In other words, the effect of the first crack on critical speeds is less sensitive to the presence of the second crack, as its depth increases. Breathing crack excites the super-harmonic components of the lateral vibration, which lead to shaft center orbits with inner loops at fractional critical speeds. Figs. 6 and 7 show the effect of relative angular positions (χ2 = | 2 − 1 | ) of the breathing cracks, modeled with cosine function [29]. The crack normalized depth are taken as (μ1 = μ2 = 1 ) at normalized locations of (γ1 = 0.4 and γ2 = 0.5). As illustrated in these Figures, shaft center orbits consist of 2 and 3 inner loops at shaft spin speeds close to 21 and 13 of the first critical speed corresponding to χ 2 , respectively. Positions of crossing points of the

H. Khorrami et al. / Mechanism and Machine Theory 113 (2017) 67–84

Fig. 10. Frequency spectra of the cracked system vertical responses at the shaft speed close to crack shaft, γ1 = 0.5, μ1 = 1; (b,c and d) two-crack shaft (γ1 = 0.5 and γ2 = 0.4, μ1 = μ2 = 1 ).

1 3

79

cr,1 using the proposed breathing function, (a) single

inner loops, point O in Fig. 6 and points O1 and O2 in Fig. 7, show that shaft center orbits rotate clockwise, as the relative angular position increases clockwise. Furthermore, Fig. 8 shows the effect of the relative angular position χ 2 on the critical speeds of the two-crack shaft using the cosine breathing function. As χ 2 changes from 0 to π , first and third critical speeds decrease, which can be attributed to the fact that the shaft stiffness reduction due to the cracks is maximized. Unbalance force Fu , which depends on the shaft speed  (Fu = mu ru 2 ), also decrease. Unbalance forces decline in higher rates than the shaft stiffness, which lead to decreases in lateral displacement amplitudes. It is clearly observed in Figs. 6 and 7 that the lateral displacement amplitude decreases, as the relative angular position χ 2 increases from 0 to π .

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Fig. 11. Amplitudes of the harmonic and super-harmonic components of the vertical response of the two-crack rotor disc-bearing system at the shaft speed close to 13 cr,1 using the cosine breathing function [29] and softly-clipped cosine function, (γ1 = 0.5 and γ2 = 0.4, μ1 = μ2 = 1 ); (a) first harmonic components; (b) second super-harmonic components; (c) third super-harmonic components and(d) fourth super-harmonic components.

It is to be noted that, modeling the cracks’ breathing behavior using the proposed softly-clipped cosine function leads to similar shaft center orbits close to sub-critical speeds of the system corresponding to χ 2 with smaller lateral displacement amplitude, which is likely attributed to the fact that the crack is considered to be saturated in fully-open and fully-closed cracks in each shaft rotation. Figs. 9 and 10 compare frequency spectra of vertical responses of the single crack with two-crack rotor disc-bearing systems utilizing cosine [29] and softly-clipped cosine breathing functions, respectively. It is noted that vertical responses of the two-crack system are obtained at shaft spin speeds close to 31 of first critical speed corresponding to χ 2 . As evident from the Figs. 9(a and b) and 10(a and b), the presence of the two aligned cracks can effectively increase the third super-harmonic amplitude compared to that of the single crack system. The amplitudes of the second and fourth super-harmonics are also increased but not to the extent of the third super-harmonic component. This is likely due to the shaft spin speed value, which is close to 13 of first critical speed, in which the third super-harmonic component of vertical response is highly excited and much more sensitive to changes in shaft stiffness in comparison with other super-harmonic components. However, as the relative angular position χ 2 increases to (χ2 = π2 and χ2 = π ), the amplitudes of the harmonic and super-harmonic components change. As illustrated in the Figs. 9(e and f) and 10(e and f), at (χ2 = 34π and χ2 = π ) the amplitudes of second and fourth super-harmonic components are much lower in comparison with the first harmonic and third super-harmonic component’s amplitudes. These changes in the frequency spectrum are also observed in the Fig. 10(d), in which the relative angular position is (χ2 = π2 ) and the cracks are modeled using the proposed softly-clipped cosine function. While in the Fig. 9(d), which shows the frequency spectrum of the vertical response of the two-crack system using the cosine breathing function, the amplitudes of the second and fourth super-harmonic components are nearly identical to the first harmonic component amplitude. Furthermore, as described in Section 2.2, consideration of the breathing crack saturation phenomenon in the proposed breathing function may lead to reductions in amplitudes of harmonic and super-harmonic components compared to those obtained from the cosine breathing function. Fig. 11 compares the harmonic and super-harmonic amplitudes of the vertical responses of the two-crack system versus relative angular position χ 2 using cosine and softly-clipped cosine functions. In Fig. 11(a), the results show that the first harmonic components amplitudes are almost unaffected by the relative angular position χ 2 and corresponding obtained values from both breathing crack models are also almost identical. The maximum deviations in first harmonic components amplitudes as χ 2 changes from 0 to π are 6.87% and 6.91% based on cosine and softly-clipped cosine breathing functions. These deviations are much less than the average deviation (95%) in higher superharmonic components amplitudes obtained from both breathing functions, shown in Fig. 11(b,c and d). It can also be con-

H. Khorrami et al. / Mechanism and Machine Theory 113 (2017) 67–84

81

cluded that the amplitude of the first harmonic component is independent of the crack parameters and may only depend to the unbalance force. It is observed in Fig. 11(b,c and d) that the expected effect of saturation phenomenon on decreasing the amplitudes of super-harmonic components is verified. The presence of second and fourth super-harmonic components with lower amplitudes compared with first harmonic and third super-harmonic components in frequency spectrum can be considered as an indicator of the presence of two cracks. Based on this indicator, the two cracks are detectable if the relative angular position is in the ranges of ( 34π  χ2  π ) and ( π2  χ2  π ) using the cosine and softly-clipped cosine breathing functions, respectively. 6. Conclusion A modified harmonic balance method proposed to compute the vibrational properties of a cracked rotor disc-bearing system. The analytical model of the cracked rotor disc-bearing system is formulated using the Timoshenko beam theory. The crack modeled as a breathing crack using Mayes and Davies model and an alternate breathing function is also proposed to model the breathing behavior of the cracks. The effect of the crack parameters such as depth, location and relative angular position were investigated on the critical speeds, shaft center orbit and lateral response of the analytical model of the rotor disc-bearing system. The results are summarized as below: • The presence of the second crack intensifies the effect of the first crack on the critical speeds considering its depth and location on the shaft; • The small depth cracks are more sensitive to the propagation of the second crack compared with relatively deep cracks; • The detection of the two cracks on a rotating shaft is more feasible considering the frequency spectrum of the lateral vibration while the crack has been modeled using the softly-clipped cosine breathing function. • Using the cosine breathing function yields good estimations of the critical speeds and shaft center orbits of the cracked rotor disc-bearing system. Acknowledgment Support from Natural Science and Engineering Research Council of Canada is gratefully acknowledged. Appendix A. System of linear equations description and solution using modified harmonic balance method The vectors {A} and {B} in the system of linear equations, as described in Eq. (25), are given as:

 T {A}(4×m×(l+1))×1 = {A }1×(4×m×(l+1)) [C](4×m×(l+1))×(4×m×(l+1))    0 1 0 2 0 3 0 4 s 1 s 2 s 3 s 4 l 1 l 2 l A

=

A1 , A1 , A1 , A1 , . . . , An , An , An , An , . . . , Am , Am , A3m ,l A4m



[C ] = . . .



diag e

it

,e

it

,e

it

,e



it

,...,e

int

,e

int

,e

int

,e

int

,...,e

2 i(t+ u )

{B}1×(4×m×(l+1)) = 0, 0, 0, 0, 0, 0, mu ru  e

imt

,e

imt

,e

imt

,e

imt

, 0, . . . , 0, 0, 0, 0, 0, 0, 0, 0

(26)





The vector {A} is considered as a multiplication of vector {A } and matrix [C]. The vector {A } represents the coefficients of proposed solution in modified harmonic method, Eq. (24); and matrix [C] is a diagonal matrix with elements of eint . The parameter n (n = 1, 2, . . . , m ) represents the order of the super-harmonic component. The elements of vector {A} are j considered as: s An eint , in which the superscripts s (s = 0, 1, 2, . . . , l ) and j ( j = 1, 2, 3, 4 ) denote the number of the shaft segment and the jth root α jn of the characteristic equation, Eq. (23), respectively. Considering that the unbalance mass and the disc coincide on the shaft, thus vector {B} describes the external harmonic forces on shaft segments. Matrix [] is the coefficient matrix, which the value of its elements depend on type of the breathing function and location of the cracks and the disc on the shaft. For the cracked rotor disc-bearing system, as shown in Fig. 1, and considering the exponential form of the Mayes and Davis breathing function, the matrix [] is expressed as:

⎡

[](4×m×(l+1) )×(4×m×(l+1) )

1



⎢0 1 ⎢s1 ⎢ ⎢ ⎢ 0 ⎢ =⎢ ⎢ .. ⎢ . ⎢ ⎢ ⎢ ⎣ 0

0 s2

0

2

...

0

0

 ..

0 sn

.

0 s(n−1 )

1

0

n

 ..

...

0

.

0 s(m−1 )

1

⎤

⎥ ⎥ ⎥ .. ⎥ . ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0  ⎦ 0 sm m 

(27)

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H. Khorrami et al. / Mechanism and Machine Theory 113 (2017) 67–84

where

0

sn

g

 ((l+1 )×4 )×((l+1 )×4 )

0

sn

0

sn

=

−1 0 1 0 2 0 3 0 4  0 2  0 4  sn g |sn g |sn g |sn g , sn g = sn g = [0], g = 0, 1. ⎡2 ⎤ [0]3×4

 1g = ⎣ gc11n1  3

g

c12n1

g

c13n1

g

c14n1 ⎦

[0]12×4

⎡

⎤

[0]11×4

g = ⎣ g c31n2

g

c32n2

g

c33n2

g

c34n2 ⎦

(28)

[0]4×4

[n ](4×(l+1))×(4×(l+1)) = [n 1 |n 2 |n 3 |n 4 ]

⎡

1

⎢ 11n ⎢ ⎢ ⎢ 211n ⎢ 3 c [ n 1 ] = ⎢ ⎢ 11n + 11n1 ⎢ 411n ⎢ ⎢ ⎣ 511n

⎤

1

1

1

12n

13n

14n

212n

213n

214n

312n + c12n1

313n + c13n1

314n + c14n1

412n

413n

414n

512n

513n

514n

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

[0]10×4

⎡

0

⎢ 0 ⎢ ⎢ −211n ⎢ ⎢ ⎢ −311n ⎢ ⎢ −411n ⎢ ⎢ [ n 2 ] = ⎢ −511n ⎢ ⎢ 221n ⎢ ⎢ ⎢ 321n ⎢ ⎢ 4 ⎢ 21n + d21n ⎢ ⎣ 521n + i21n

0

0

0

0

0

0

−212n

−213n

−214n

−312n

−313n

−314n

−

−

4 13n

−414n

−512n

−513n

−514n

222n

223n

224n

322n

323n

324n

4 12n

422n + d22n

423n + d23n

424n + d24n





524n + i24n

5 22n

+

i 22n

5 23n

+

i 23n

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

[0]6×4

⎡ ⎢ −221n ⎢ ⎢ −321n ⎢ −421n ⎢ ⎢ −521n [ n 3 ] = ⎢ 2 ⎢ ⎢ 3 31n c ⎢ 31n + 31n2 ⎢ 431n ⎢ ⎣ 531n

⎤

[0]6×4 −222n −322n −422n −522n

232n  + c32n2 432n 532n 3 32n

−223n −323n −423n −523n

233n  + c33n2 433n 533n 3 33n

[0]2×4

−224n −324n −424n −524n

234n  + c33n2 434n 534n 3 34n

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

H. Khorrami et al. / Mechanism and Machine Theory 113 (2017) 67–84

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ [ n 4 ] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

[0]10×4 −231n

−232n

−233n

−234n

−331n

−332n

−333n

−334n

−

−

−

4 33n

−434n

4 31n

4 32n

83

−531n

−532n

−533n

−434n

241n

242n

243n

244n

641n

642n

643n

644n

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

1jn = EIλ2jn , 2s jn = eLs λ jn , 3s jn = λ jn 2s jn     4s jn = 2s jn EIλ3jn + n2 − 2n ρ Ar02 ω2 λ jn

  5s jn = 1jn 2s jn , 6s jn = λ2jn 2s jn , ds jn = md n2 ω2 2s jn , is jn = 3s jn Ip nω2 − It n2 ω2   1 r 1 g g c (β − iγ r )5s jn ei(−1) χr + ( r − iδ r )con j 5s jn ei(−1) χr , r = 1, 2. g s jnr = 2 2   7jn = −EIλ3jn + ρ Ar02 ω2 λ jn 2n − n2    8s jn = 2s jn −EIλ3jn + ρ Ar02 ω2 λ jn 2n − n2 0  n

The matrices sn g and [ ] in Eq. (25) have been determined considering that the disc is located between the cracks on the shaft as shown in Fig. 1. These matrices have to be updated for other arrangements of the cracks and disc on the shaft. Modeling the breathing cracks using the proposed breathing function described in Eq. (6), the matrix [] is updated as:

[](4×m×l )×(4×m×l )

⎡

1



 10 0 9

s2

0

0

⎢  10 0 ⎢  10 0 2 ⎢ 9 s 1 1   9 s3 0 ⎢ ⎢  10 0 ⎢ 3 0   ⎢ 9 s2 1 ⎢ ⎢   10 0 ⎢ eiπ 0  0  ⎢ 9 s1 1 9 s3 1 ⎢ ⎢ .. .. ⎢ . . 0 =⎢ ⎢ ⎢ ⎢ ⎢ ..  eiπ 0 ⎢ ⎢ .  9 s(n−3 ) 1 ⎢ ⎢ .. ⎢ . ⎢ ⎢ ⎢ ⎢ ⎣ 0

 e−iπ 0

s4

9

0

0

 10 0

s4

9 4

0



..

...

0 ..

.

..

.

..

.

..

.

 e−iπ 0

sn

9

.

 10 0 9

s(n−1 )

.. ...

.

0

0

 10 0

sn

9

0

0

1

n

0



..

.

0

 10 0 9

..

.

..

.

s(m−1 )

1

⎤

⎥ ⎥ ⎥ ⎥ ⎥ .. ⎥ . ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (29) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥  10 0  ⎥ 9 sm 0 ⎦ m 

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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