Dynamical behavior of disordered rotationally periodic structures_A homogenization approach

SAVE OFFLINE

Journal of Sound and Vibration 330 (2011) 2608–2627

Contents lists available at ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Dynamical behavior of disordered rotationally periodic structures: A homogenization approach Paolo Bisegna a, Giovanni Caruso b, a b

Department of Civil Engineering, University of Rome ‘‘Tor Vergata’’, 00133 Rome, Italy ITC-CNR, 00016 Monterotondo stazione, Italy

a r t i c l e in fo

abstract

Article history: Received 20 July 2010 Received in revised form 7 December 2010 Accepted 7 December 2010 Handling Editor: H. Ouyang Available online 11 January 2011

This paper put forth a new approach, based on the mathematical theory of homogenization, to study the vibration localization phenomenon in disordered rotationally periodic structures. In order to illustrate the method, a case-study structure is considered, composed of pendula equipped with hinge angular springs and connected one to each other by linear springs. The structure is mistuned due to mass and/or stiffness imperfections. Simple continuous models describing the dynamical behavior of the structure are derived and validated by comparison with a well-known discrete model. The proposed models provide analytical closed-form expressions for the eigenfrequencies and the eigenmodes, as well as for the resonance peaks of the forced response. These expressions highlight how the features of the dynamics of the mistuned structure, e.g. frequency split and localization phenomenon, depend on the physical parameters involved. & 2010 Elsevier Ltd. All rights reserved.

1. Introduction A periodic system is spatially repetitive, consisting of an assembly of identical subsystems dynamically coupled in some identical manner. Systems with rotational periodicity, like turbine bladed rotors, cooling towers with legs, satellite antennae, stator–rotor assemblies of electrical machineries, are commonly employed in many fields of engineering. It is well known that, due to the periodicity they posses, their dynamical behavior is peculiar, being characterized by pairs of degenerate eigenmodes at coinciding eigenfrequencies [1–3]. Finite element analysis has been extensively used [2,4,5] in order to study the free and forced response of rotationally periodic structures. To this end, it is sufficient to model a single substructure only, enforcing suitable periodicity boundary constraints [2]: indeed, in a typical structural eigenmode, all the substructures exhibit the same modal shape with different phases. As a matter of fact, perfect periodicity is only an idealized assumption; in real-world applications discrepancies occur between the substructures, due to unavoidable manufacturing defects, material tolerances or damage arising during operative life (e.g. cracks), which destroy the periodicity of the structure. This break of periodicity, called disorder or mistuning, may significantly alter the system dynamics even for small disorder levels. In particular, the eigenmode degeneracy is removed, implying the frequency split phenomenon, and the vibration localization may appear, consisting of a vibrational-energy confinement in small regions of the structure, rapidly decaying far away. As a consequence, some components may vibrate with small amplitudes whereas some others with significantly larger amplitudes, causing a local stress increase, possibly leading to fatigue failure.

 Corresponding author.

E-mail addresses: [email protected] (P. Bisegna), [email protected] (G. Caruso). 0022-460X/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2010.12.009

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

2609

The first study concerning this phenomenon was in the field of solid state physics [6]. Studies of the vibration localization phenomenon occurring in mechanics appeared some years later [7]. Localization in simple systems, such as coupled pendula and strings, was analyzed by using the transfer-matrix method in [8,9]. Several other methods, such as the random matrix theory [10–12], asymptotic expansions [13,14], the probability theory [15], stochastic perturbation techniques [16–18], and genetic algorithms [19] have been adopted to study the localization phenomenon, usually considering simplified structural models. A modern approach to localization [20] recasts the issue of vibration confinement as a stability problem of differential equations with almost periodic coefficients. The difference between strong and weak localization in disordered periodic structures was pointed out in [13,21,18]. Recent works focus on the investigation of localization phenomena in real engineering structures like turbine bladed rotors: discrete models have been obtained from continuous formulations by adopting the Galerkin method in [22,23], whereas simplified lumped-parameter models have been studied in [24–27]. Both lumped-parameter models and a finite element analysis were employed in [28] to investigate the dependence of the localization phenomenon from the internal damping and interblade coupling in bladed disks. A detailed finite element model was employed in [29] for investigating the worst mistuning patterns, yielding the largest resonant amplitudes in the forced response of mistuned bladed disks. The problem of reducing the localization level was addressed in [30,31] by employing piezoelectric networking, and in [32] by introducing a specific intentional mistuning. In the cited literature, discrete models were considered and numerical simulations were performed in order to investigate the dynamical behavior of disordered periodic structures and the influence of parameters, like mistuning level or interblade coupling, on the localization phenomenon. Continuous models for the description of the localization phenomenon in periodic disordered structures are less developed. For example, a continuous model of a turbine composed of grouped blades mounted on a flexible disk was presented in [33]; however imperfections were not considered there. While finite element analysis or other numerical methods, applied to the study of disordered periodic structures with a large number of subcomponents, may require a significant amount of storage capacity and run time, a continuous model can yield accurate results with little computational effort. For this reason it is particularly useful for performing parametric analyses or designing innovative vibration control schemes. Finally, continuous models can easily yield analytical results providing a general understanding of the physics of the problem. In this paper the feasibility of employing the homogenization technique to derive spatially continuous models of disordered rotationally periodic structures is investigated. In order to illustrate the method and to assess its effectiveness, a case-study structure is considered, composed of pendula connected with springs, shown in Fig. 1(a). This structure has been extensively studied by many previous authors [24,34,30,27,19]: indeed, though simple, it is able to reproduce the essential features of the vibration localization phenomenon in rotationally periodic structures [24,25]. The paper is organized as follows. A homogenized model of the perfect structure is derived in Section 2. The method is straightforward: a family of structures is introduced, by increasing more and more the number of pendula and rescaling at the same time the involved physical parameters (mass, damping, stiffness, coupling compliance, force) in order to keep constant their angular densities. The homogenized model is obtained in the limit of number of pendula going to infinity. In the spirit of the homogenization theory, the continuous model here derived is expected to accurately estimate the structural eigenmodes of the original structure, whose wavelength is sufficiently larger than the angular spacing between adjacent pendula. In order to validate the model and to investigate its predictive capabilities, mistuning is first introduced as a variation of the stiffness or mass of a single pendulum [24,27,23] with respect to its nominal value (Section 3). The model accuracy is assessed by comparison with the numerical results supplied by the discrete model or found in the literature. The localization phenomenon, appearing as a concentration of the vibration amplitude around the imperfect pendulum and exponentially decaying far apart, is well predicted by the homogenized model for small-to-moderate imperfection level and moderate-tostrong inter-pendulum coupling. A modified version of the homogenized model, suitable for the description of the vibration localization phenomenon even in weakly coupled and/or heavily imperfect structures, is also derived (Section 4).

qj   j k, b

O

kr m

/r



m/r

k k /r, b /r

Fig. 1. (a) Case-study cyclic structure; (b) cut view of the rescaled structure with r = 3.

2610

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

Simple algebraic expressions of the eigenfrequencies and eigenmodes, as well as of the resonance peaks of the forced response, are obtained by computing asymptotic expansions of the solutions of the proposed homogenized models. These expressions demonstrate the influence of the various parameters entering the models on the eigenfrequency split phenomenon and on the localization level. The homogenization approach is then applied to study the free and forced response of the rotationally periodic structure depicted in Fig. 1(a) with distributed mistuning affecting all the pendula (Section 5). Numerical results show the effectiveness and the accuracy of the proposed approach also in the case of distributed imperfections. A perturbation expansion combined with a Fourier series solution is presented, and closed-form expressions of the eigenfrequencies, the eigenmodes and the forced response are obtained, highlighting the influence of the imperfection distribution pattern on the structural localized response. The proposed approach is quite general and may be applied to other structural schemes, able to realistically model the dynamical behavior of structures employed in the technology, like turbine bladed rotors [35,36]. For example, the homogenization technique was employed in [37] to develop a continuous model of a perfect bladed disk, taking into account the flexibility of both the support disk and the blades using the Euler–Bernoulli beam theory. That continuous model was then used for the design and optimization of a passive vibration control scheme. Extension of the proposed method to other structural models of rotationally periodic structures with imperfections is the subject of an ongoing work. 2. Perfect structure 2.1. Position of the problem In this section the rotationally periodic structure depicted in Fig. 1(a) is considered. It is a cyclic structure, composed of rigid pendula of length l and mass m, connected one to each other by means of linear springs of elastic constant k; moreover, a rotational spring of elastic constant ky and a rotational damper of viscous coefficient by are applied to the hinge of each pendulum. This structure has been widely studied in the literature (e.g. [24,34,25,30,27,19]). In particular, in [34,30,27] it was adopted as a simplified scheme of a bladed rotor, where the blades vibrate along a single local eigenmode; in particular, m and ky represented the blade mass and stiffness, whereas k was used to emulate the coupling effects between blades due to the elastic support disk and/or to the presence of shrouds. This structure, exhibiting the essential features of the vibration localization phenomenon in rotationally periodic structures [24,25,19], is here employed to illustrate the homogenization procedure. The equations governing the dynamical behavior of the structure are   b k mo2 þ 2y io þ 2y qj kðqj1 2qj þqj þ 1 Þ ¼ fj , j ¼ 1,2, . . . ,N: (1) l l Here N is the number of pendula, qj is the displacement of the j-th pendulum, which is placed at the angular location yj ¼ ðj1Þs, s ¼ 2p=N being the angular spacing between two adjacent pendula (Fig. 1(a)). The j-th pendulum is subjected to the load fj with spatially harmonic distribution fj ¼ f expðiFj Þ, where f is the load amplitude, Fj ¼ ne yj , and the integer number pffiffiffiffiffiffiffi ne, belonging to the range ½N=2,N=2, is the engine order. Moreover, i ¼ 1, and a harmonic time dependence (i.e. a proportionality to expðiotÞ of the involved quantities) has been assumed, where o is the circular frequency and t is the time. These equations were solved in [24,27] by using the U-transformation approach. 2.2. Homogenized model In this section, a homogenized version of Eq. (1) is derived, by increasing the number of pendula and rescaling at the same time m, by , ky , 1/k, and f, in order to keep constant their angular densities: M¼

m

s

,

By ¼

by

s

,

Ky ¼

ky

s

,

1=K ¼

1=k

s

,

F¼

f

s

:

(2)

More specifically, a family of periodic structures indexed by the integer number r is considered. The r-th structure has rN pendula, placed at angles ys ¼ ðs1Þs=r, s ¼ 1,2, . . . ,rN. Each pendulum has m/r tip mass, ky =r hinge stiffness, by =r hinge damper, (1/k)/r coupling compliance, and is subjected to a force ðf =rÞexpðiFs Þ, Fs ¼ ne ys . A drawing of the rescaled structure with r = 3 is depicted in Fig. 1(b). Its governing equation is obtained by substituting these rescaled quantities into Eq. (1). Multiplying the resulting equation by r=s, it turns out that   B K qs1 2qs þ qs þ 1 ¼ FexpðiFs Þ, s ¼ 1,2, . . . ,rN: (3) Mo2 þ 2y io þ 2y qs K l l ðs=rÞ2 By letting r-1, the quantity ys approaches the continuous angular variable y, and the following homogenized equation is obtained:   B K Mo2 þ 2y io þ 2y Q KQ 00 ¼ Fexpðine yÞ, y 2 ð0,2pÞ, (4) l l

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

2611

where Q is a scalar function of y, representing the displacement field of the homogenized pendula, and Q 00 denotes its second yderivative. Periodic solutions of Eq. (4) are sought for: accordingly, the boundary conditions Q ð0 þ Þ ¼ Q ð2p Þ and Q uð0 þ Þ ¼ Q uð2p Þ are enforced, where, e.g. Q ð0 þ Þ and Q ð2p Þ respectively denote the right limit of Q ðyÞ at y ¼ 0 and its left limit at y ¼ 2p. An estimate of the pendulum displacement qj can be obtained from the homogenized displacement Q by evaluating the latter at the pendulum angular location. In passing, it is noted that Eq. (4) may describe either the axial damped vibrations of a rod with periodic boundary conditions and uniformly distributed axial springs or the tangential damped vibrations of a radially constrained ring with uniformly distributed tangential springs. It is emphasized that the present approach yields a model (i.e. the homogenized model (4)), which is much easier to be solved than the original discrete model (1). This approach is different from the one adopted in [24,27], where the discrete model was solved using the U-transformation approach, and then a limit of the resulting solution was taken, for frequencies out of the structure pass band, by letting the number of pendula go to infinity. A similar homogenization approach could be extended to other models of rotationally periodic structures, provided that a suitable rescaling of the dynamical parameters of the representative substructure is performed during the limit process. As an example, the homogenization approach was applied to a bladed-rotor model with no imperfections in [37], accounting for distributed flexibility and mass of blades and supporting disk. The discrete model (1) and its homogenized version (4) can be respectively recast in the following dimensionless forms as ½O2 þ 2ziO þ 1qj gðqj1 2qj þ qj þ 1 Þ ¼ ½O2 þ2ziO þ 1Q GQ 00 ¼

F expðine yÞ, Ky =l2

f expðiFj Þ, ky =l2

Q ð0 þ Þ ¼ Q ð2p Þ,

j ¼ 1,2, . . . ,N;

Q uð0 þ Þ ¼ Q uð2p Þ,

(5)

(6)

where the following dimensionless parameters have been introduced:

O¼

o by k K , g¼ , z ¼ pffiffiffiffiffiffiffiffiffi , G¼ ¼ gs2 : oo ky =l2 Ky =l2 2 ky ml

(7)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here oo ¼ ðky =l2 Þ=m is a frequency scale (it is the eigenfrequency of uncoupled pendula), O is a dimensionless frequency, z is a damping coefficient, whereas g and its homogenized counterpart G measure the inter-pendulum coupling. In what follows the discrete model (5) and the homogenized model (6) are denoted by the acronyms DM and HM, respectively. 2.3. Free response analysis The free vibrations of the perfect structure are studied in this section using the homogenized model. The results are compared with the ones supplied by the discrete model. The eigenmodes Q ðyÞ of the homogenized perfect structure are obtained by solving the field equation (6)1 under the periodic boundary conditions (6)2,3, after having set to zero both the damping coefficient z and the external load amplitude F. Thus, the representation Q ðyÞ ¼ CexpðlyÞ is enforced, C being an arbitrary constant and l a candidate system pole, which must solve the equation: 2

Gl þ ð1O2 Þ ¼ 0:

(8)

The behavior of l as a function of O is reported in Fig. 2. Two branches appear: the solid-line branch, implying purely imaginary poles, holds for O 4 1, whereas the dashed-line branch, implying real poles, is relevant to O r 1. On the other hand, in order to accomplish the periodicity requirement, only imaginary poles can be accepted, satisfying

l ¼ 7 in, 6

ℜ(λ), ℑ(λ)

5 4

(9)

immaginary pole branch real pole branch poles, perfect structure poles, β=0.04 poles, β=?0.04

3 2 1 0 0.98

1

1.02 1.04 1.06 1.08 Ω

1.1

1.12

Fig. 2. Cyclic structure with N=20, g ¼ 0:1. System poles for perfect structure, or imperfect structure with b ¼ 7 0:04 and m ¼ 0, evaluated using the HM model.

2612

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

with n an integer number, referred to as mode number, 2p=n being the corresponding mode wavelength. Accordingly, poles of the perfect structure, denoted by the suffix ‘‘o’’, exist only on the solid-line branch, and the countable set of eigenfrequencies pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Oo,n ¼ 1 þ n2 G ¼ 1 þ ð2pn=NÞ2 g (10) is selected out from Eqs. (8) and (9). They are denoted by solid circles in Fig. 2. In passing, it is noted that the frequency spacing between consecutive eigenfrequencies reduces as the number N of pendula increases or the coupling g decreases. The relevant eigenmodes are given by Qo,n ðyÞ ¼ Ao,n cosðny þ jn Þ,

n ¼ 0,1,2, . . .

(11)

where Ao,n is an arbitrary amplitude and jn is an arbitrary phase shift, referred to as mode orientation. A couple of independent degenerate eigenmodes Qo,n are obtained for n 4 0 at the same eigenfrequency Oo,n , whereas a single eigenmode Qo,0 with constant modal shape is obtained for n = 0 at Oo,0 ¼ 1. A numerical validation of the homogenization limit is presented in Table 1. The structural eigenfrequencies relevant to n ¼ 0,1, . . . ,5 are evaluated using the rescaled discrete model (3): the row r= 1 is relevant to the original structure; the rows r = 2,3,4 correspond to structures with increased number of rescaled pendula. The eigenfrequencies are compared with the ones computed using the homogenized model (6), obtained in the limit r-1. It turns out that increasing r all the considered eigenfrequencies converge to their homogenized limit. The fundamental hypothesis underlying the homogenization approach is that the modal wavelength 2p=n be sufficiently greater than the angular spacing s ¼ 2p=N between adjacent pendula. Thus, the higher the mode number n, the higher the relative error on the relevant eigenfrequency. On the other hand, the relative error decreases with the increase in the number N of pendula. By comparing the first with the last row of Table 1, it follows that the first six structural eigenfrequencies are satisfactorily estimated by the homogenized model, even for a structure with only N = 20 pendula.

2.4. Forced response analysis The forced harmonic response under the load Fexpðine yÞ can be obtained by solving Eq. (6), yielding Qo ðy, OÞ ¼

Q expðine yÞ, Dne ðOÞ

(12)

Table 1 Cyclic structure with N =20 and g ¼ 0:1. Eigenfrequencies Oo,n computed using the DM and HM models. The number of pendula in the discrete model is increased by a factor r.

DM, DM, DM, DM, HM

r =1 r =2 r =3 r =4

0

1

2

3

4

5

1 1 1 1 1

1.0049 1.0049 1.0049 1.0049 1.0049

1.0189 1.0194 1.0195 1.0195 1.0195

1.0404 1.0427 1.0431 1.0433 1.0435

1.0669 1.0737 1.0750 1.0755 1.0761

1.0954 1.1110 1.1141 1.1152 1.1166

600 DM model HM model ne =0

500

ne =1

400 | |/

n

300 200 100 0 0.97

0.98

0.99

1

1.01

1.02

1.03

Ω Fig. 3. Cyclic structure with N = 20, g ¼ 0:1, z ¼ 0:001. Frequency response amplitude of any pendulum versus frequency.

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

2613

where Dne ðOÞ ¼ 1 þ n2e G þ2ziOO2 ,

Q¼

F : Ky =l2

(13)

It turns out that, for each engine order ne, there is only one resonance peak occurring at a frequency minimizing jDne ðOÞj and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ n2e G under the hypothesis that z 5 1. The

almost coinciding with the eigenfrequency of the perfect structure Oo,ne ¼ response at the resonance frequency Oo,ne reads as Qo ðy, Oo,ne Þ ¼

Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expðine yÞ, 2iz 1 þn2e G

(14)

whose amplitude jQ j is independent of y. Fig. 3 shows jQ j=Q for ne = 0 and 1, as a function of the frequency O, computed using the DM and HM models. The agreement is excellent. 3. Imperfect structure: lumped imperfection 3.1. Homogenized model In this section, an imperfection affecting a single pendulum [24] is considered in the modeling process. The imperfection consists of a variation Dky of the stiffness of the hinge spring connected to the pendulum (accounting, e.g. for a blade crack in a bladed disk [27]), or a variation Dm of the pendulum mass, with respect to its nominal value. Without loss of generality, let the imperfect pendulum be the first one, placed at y ¼ 0. The dynamical equilibrium equation of the imperfect pendulum is   b ðk þ Dk Þ (15) ðm þ DmÞo2 þ 2y io þ y 2 y q1 kðqN 2q1 þ q2 Þ ¼ f1 : l l It is recast as     b k Dk mo2 þ 2y io þ 2y q1 kðqN 2q1 þ q2 Þ ¼ f1  Dmo2 þ 2 y q1 : l l l

(16)

This equation suggests that the homogenization process can be carried out in the same way both for the perfect and the imperfect structure, up to taking into account the additional force acting in the latter case on the imperfect pendulum, appearing on the right-hand side of Eq. (16). Accordingly, the homogenized equation (4) is modified as follows:     B K Dk M o2 þ 2y io þ 2y Q KQ 00 ¼ Fexpðine yÞ Dmo2 þ 2 y Q ð0Þd0 , y 2 ð0,2pÞ, (17) l l l where d0 is the Dirac function supported at y ¼ 0. The Dirac source can be recast into a jump condition on the flux KQ u, by integrating Eq. (17) over a small interval centered at y ¼ 0 whose length is let go to zero. Hence, the following dimensionless problem is obtained: ½O2 þ 2ziO þ1Q GQ 00 ¼ Q expðine yÞ, Q ð0 þ ÞQ ð2p Þ ¼ 0,

y 2 ð0,2pÞ,

G½Q uð0 þ ÞQ uð2p- Þ ¼ sðbmO2 ÞQ ð0Þ,

(18) (19)

where

b¼

Dky ky

,

m¼

Dm m

(20)

are the relative variations of the stiffness and mass values of the imperfect pendulum, respectively. 3.2. Free response analysis The eigenfrequencies and eigenmodes of the imperfect structure can be obtained by solving Problem (18) and (19) with

z ¼ 0, Q ¼ 0, and b or m different from zero. The eigenmodes admit the same representation Q ðyÞ ¼ CexpðlyÞ previously considered in the case of perfect structure. In order to evaluate l, it is convenient to separately treat the cases O 41 or O r 1. 3.2.1. The case O 4 1 In this frequency range, the system poles are purely imaginary. Setting l ¼ ia, with a a real number to be determined, the following representations are obtained, symmetric or antisymmetric with respect to y ¼ p, respectively: Q ðyÞ ¼ Ccos½aðypÞ

or

Q ðyÞ ¼ Csin½aðypÞ,

(21)

with a and O related by the equation

O¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ a2 G:

(22)

2614

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

The antisymmetric representation (21)2, together with the boundary condition (19)1, yield Q ð0Þ ¼ 0, i.e. the imperfection location coincides with an eigenmodal node. As a consequence, the boundary condition (19)2 does not detect the stiffness or mass imperfection, and the corresponding eigenmodes and eigenfrequencies coincide with those relevant to the perfect structure, respectively Qo,n ðyÞ and Oo,n , for n 4 0. More interesting is the discussion about the symmetric representation (21)1. It satisfies the boundary condition (19)1; enforcing the boundary condition (19)2, it turns out that 2aGsinðapÞ ¼ sðbmO2 ÞcosðapÞ;

(23)

using Eq. (22), it follows that

aptanðapÞ ¼

ps 2G

½bmð1 þ a2 GÞ:

(24)

It is convenient, for the sake of clarity, to separately consider a stiffness or mass imperfection. Accordingly, it is assumed ba0, m ¼ 0 in Eq. (24). The case b ¼ 0, ma0 is reported in Appendix A. A graphical representation of the solutions of Eq. (24) is shown in Fig. 4(a) for a either positive or negative stiffness imperfection. It can be verified that Eq. (24) admits a countable set of solutions an , yielding the shifted eigenfrequencies Os,n by Eq. (22), and the relevant eigenmodes Qs,n by Eq. (21)1. Simple algebraic expressions for the eigenfrequencies Os,n can be obtained by performing a series expansion of the solution of Eq. (24) around b ¼ 0. For mode indexes n 4 0, this is accomplished by computing the MacLaurin series of the inverse of the function y ¼ ðnpÞ2 ðnp þ xÞtanðxÞ, with x ¼ pðan nÞ and y ¼ b=ðNGn2 Þ. The solutions

an n

¼ 1þ y þoðyÞ ¼ 1 þ

bN ð2pnÞ2 g

(25)

þoðbÞ,

Os,n n2 Gy b þoðbÞ ¼ 1þ þ oðyÞ ¼ 1þ Oo,n 1 þn2 G N þ ð2pnÞ2 g=N

(26)

are obtained. Forpn= 0, it is convenient to consider the MacLaurin series of the inverse of the function y ¼ p1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ a0 p and y ¼ b=ðNGÞ. Accordingly, the expansions sffiffiffiffiffiffiffi pffiffiffi 1 bN a0 ¼ yþ oðyÞ ¼ þ oð bÞ, 2p g

Os,0 ¼ 1þ

Gy2 2

þ oðy2 Þ ¼ 1 þ

b 2N

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xtanðxÞ, with

(27)

þ oðbÞ

(28)

prevail. Of course, they are acceptable for b 40 only. Exploiting these results, it turns out that (Fig. 4(a))

b-0 þ , a-0 þ ,1 þ ,2 þ ,3 þ    ;

b-0 , a-1 ,2 ,3    :

(29)

In particular, a positive [respectively, negative] stiffness imperfection leads to a positive [respectively, negative] shift of the eigenfrequencies Os,n with respect to Oo,n .

απ tan(απ) πσβ/(2Γ), β=−0.04 πσβ/(2Γ), β=0.04

5

0

−5

1

2

α

3

4

5

10

απ tanh(απ), −πσβ/(2Γ)

απ tan(απ), πσβ/(2Γ)

10

απ tanh(απ) −πσβ/(2Γ), β=−0.04 −πσβ/(2Γ), β=0.04

5

0

−5

−10 0

0.5

1

α

1.5

2

Fig. 4. Cyclic structure with N=20, g ¼ 0:1, b ¼ 7 0:04 and m ¼ 0. Graphical representation of the solution of: (a) the pole equation (24); (b) the pole equation (33).

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

2615

3.2.2. The case O r 1 In this frequency range, the system poles are real. Setting l ¼ a, with real a to be determined, the following representations are obtained: Q ðyÞ ¼ Ccosh½aðypÞ

or

Q ðyÞ ¼ Csinh½aðypÞ,

(30)

provided that a and O satisfy the equation

O¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1a2 G:

(31)

The hyperbolic trigonometric functions appearing in the modal shape expression implies that these eigenmodes, if existing, exhibit a vibration amplitude localized around y ¼ 0. The antisymmetric eigenmodes (30)2, together with the boundary condition (19)1, yield Q ð0Þ ¼ 0. As a consequence, the boundary condition (19)2 becomes homogeneous, implying that such antisymmetric eigenmodes does not exist. On the other hand, the symmetric eigenmodes (30)1 satisfy the boundary condition (19)1. Exploiting the boundary condition (19)2, it turns out that 2aGsinhðapÞ ¼ sðbmO2 ÞcoshðapÞ,

(32)

so that, using Eq. (31), it is obtained that

aptanhðapÞ ¼ 

ps 2G

½bmð1a2 GÞ:

(33)

As done before, the cases of stiffness or mass imperfection are separately discussed. Accordingly, it is assumed ba0, m ¼ 0 in Eq. (33). The case b ¼ 0, ma0 is reported in Appendix A. A graphical representation of the solutions of Eq. (33) is shown in Fig. 4(b). It can be verified that no solutions exist when b 4 0. On the other hand, one and only one solution a0 , denoted by aloc , exists provided that b o0, leading to the localized eigenmode Qs,0 , denoted by Qloc , through Eq. (30)1. The relevant eigenfrequency Os,0 , denoted by Oloc , is given by Eq. (31). A simple algebraic expression for the eigenfrequency Oloc can be obtained by performing a series expansion of the solution of Eq. (33) around b ¼ 0. This is accomplished by computing the MacLaurin series of the inverse of the function pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ¼ p1 xtanhðxÞ, with x ¼ aloc p and y ¼ b=ðNGÞ. Accordingly, the expansions sffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 bN (34) aloc ¼ yþ oðyÞ ¼ þ oð bÞ, 2p g

Oloc ¼ 1

Gy2 2

þ oðy2 Þ ¼ 1þ

b 2N

þ oðbÞ

(35)

prevail. Of course, they are acceptable for b o0 only. Exploiting these results, it turns out that (Fig. 4(b))

b-0 , a-0 þ , O-1 ,

(36)

implying that a negative stiffness imperfection shifts the first eigenfrequency Oo,0 ¼ 1 of the perfect structure to the eigenfrequency Oloc o1. 3.2.3. Discussion The previous results can be summarized as follows. When a lumped stiffness or mass imperfection, positive or negative, affects one pendulum, the couples of degenerate structural eigenmodes lose the degeneracy they possess in the case of perfect structure, and split into non-degenerate eigenmodes at different eigenfrequencies (Fig. 2). The eigenfrequencies relevant to odd eigenmodes with respect to the imperfection location coincide with the eigenfrequencies Oo,n of the perfect structure, whereas the eigenfrequencies Os,n relevant to even eigenmodes are shifted from Oo,n , in agreement with the findings in [24,27]. In particular, increasing [respectively, decreasing] the stiffness of a hinge spring, the eigenfrequencies Os,n are increased [respectively, decreased] with respect to the corresponding eigenfrequencies Oo,n . An opposite behavior is obtained in the case of a mass imperfection (see Appendix A). This behavior can be observed in Table 2, also reporting a comparison between the results supplied by the DM and HM models: the first six eigenfrequencies of an imperfect structure with N = 20 and g ¼ 0:1 are evaluated, for several imperfection levels b, showing a good agreement between the two model predictions. An analytical account of those results is given by the asymptotic expansions (26), (28) and (35) above, up to oðbÞ terms [or (A.2), (A.4), and (A.6), up to oðmÞ terms]. In particular, the relative shift ðOs,n Oo,n Þ=Oo,n turns out to be proportional to b [or m]. The eigenmodes relevant to the shifted eigenfrequencies, given by Eq. (21)1, are nonlocalized when n 40 (Fig. 5(a)). The case n =0, corresponding to a single eigenmode in the perfect structure, exhibits a different behavior. Indeed, the eigenfrequency Oo,0 ¼ 1 in the perfect structure is shifted to Oloc o1 when a negative stiffness imperfection or a positive mass imperfection is present: the relative shift ðOloc Oo,0 Þ=Oo,0 is approximated by b=ð2NÞ [or m=ð2NÞ], and the relevant eigenmode, having a constant amplitude in the perfect structure, has an exponential shape implying the localization phenomenon, i.e. a large modal amplitude concentrated around the imperfect pendulum and exponentially decaying far apart (Figs. 5(b–d)). Fig. 5

2616

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

Table 2 Cyclic structure with N = 20, g ¼ 0:1, m ¼ 0. Comparison between shifted eigenfrequencies Os,n predicted by the DM and HM models. Eigenfrequencies Oloc corresponding to localized eigenmodes are reported in boldface. n

0

1

2

3

4

5

b = 0.08

DM HM

1.0008 1.0008

1.0077 1.0077

1.0223 1.0230

1.0439 1.0470

1.0704 1.0796

1.0989 1.1201

b = 0.04

DM HM

1.0006 1.0006

1.0066 1.0066

1.0208 1.0214

1.0423 1.0453

1.0687 1.0779

1.0972 1.1183

b= 0

DM HM

1.0 1.0

1.0049 1.0049

1.0189 1.0195

1.0404 1.0435

1.0669 1.0761

1.0954 1.1166

b =  0.04

DM HM

0.9978 0.9978

1.0030 1.0030

1.0170 1.0176

1.0385 1.0416

1.0650 1.0742

1.0936 1.1148

b =  0.08

DM HM

0.9922 0.9919

1.0021 1.0021

1.0154 1.0159

1.0368 1.0398

1.0633 1.0724

1.0936 1.1130

1

1

DM model HM model

0.8 modal amplitude

0.5 modal amplitude

DM model HM model

0

0.6

0.4

-0.5 0.2

-1

01

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

pendulum #

pendulum #

1

1

DM model HM model

DM model HM model

0.8 modal amplitude

modal amplitude

0.8

0.6

0.4

0.2

0

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

0.6

0.4

0.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

pendulum #

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

pendulum #

Fig. 5. Cyclic structure with N= 20, g ¼ 0:1, m ¼ 0. Comparison between eigenmodes evaluated using the DM and HM models. (a) Nonlocalized eigenmode (n= 2) with b ¼ 0:04; (b) localized eigenmode (n= 0) with b ¼ 0:04; (c) localized eigenmode (n= 0) with b ¼ 0:08; (d) localized eigenmode (n =0) with b ¼ 0:16.

also shows the excellent agreement between the (localized or non-localized) eigenmodes supplied by the DM and HM models. The closed-form expression (30)1 of the localized eigenmode Qloc allows one to easily compute the angular amplitude L of the localization region, defined as the subtangent to the eigenmodal shape at y ¼ 0 (Figs. 5(b–d)). Recalling Eq. (33) and

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

2617

setting m ¼ 0, it turns that L ¼ Qloc ð0Þ=Q uloc ð0Þ ¼

1

aloc tanhðaloc pÞ

¼

2gs 4pg : ¼ b bN

(37)

Accordingly, the higher the ratio bN=g is, the more pronounced the localization phenomenon. In the case m 4 0, b ¼ 0, it results: L ¼ 2gs=ðmO2loc Þ ¼ 4pg=ðmO2loc NÞ, with Oloc given by Eq. (A.6). In the spirit of the homogenization theory, the homogenized model is expected to accurately estimate the localized eigenfrequency and eigenmode as long as L is larger than the angular spacing between adjacent pendula, i.e. L 4 s. This yields the condition: b=ð2gÞ o1 [or mO2loc =ð2gÞ o 1]. For heavily imperfect or weakly coupled structures this condition may fail, and the HM model may not be able to accurately describe the structural dynamics in the localization frequency region O o1, still being effective in the region O 4 1. A more complex homogenized model is derived in Section 4, suitable for the analysis of the localization phenomenon of imperfect structures with b=ð2gÞ Z1 [or mO2loc =ð2gÞ Z 1]. 3.3. Forced response analysis The forced response of the cyclic structure is given by the solution of Problem (18) and (19), allowing the presence of an inherent structural damping and an external force. It can be written as Q ðy, OÞ ¼

Q ½expðine yÞ þ C1 expðlyÞ þ C2 expðlyÞ, Dne ðOÞ

(38)

where C1 and C2 are constants to be determined, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1O2 þ 2izO l¼ :

(39)

G

By a priori enforcing the boundary condition (19)1, it follows that C1 ¼ CexpðlpÞ, C2 ¼ CexpðlpÞ, with C to be determined, and Eq. (38) becomes Q ðy, OÞ ¼

Q fexpðine yÞ þ Ccosh½lðypÞg: Dne ðOÞ

(40)

The constant C is evaluated by enforcing the boundary condition (19)2 and reads as C¼

sðbmO2 Þ : sðbmO2 ÞcoshðlpÞ þ 2GlsinhðlpÞ

(41)

The forced response Q ðy, OÞ of the imperfect structure is thus obtained by substituting C from Eq. (41) into Eq. (40). It exhibits several resonance peaks. One peak, appearing also in the frequency response of the perfect structure, occurs at a frequency minimizing jDne ðOÞj, almost coinciding with Oo,ne for z 51. Then, a countable set of resonance peaks, not appearing in the case of perfect structure, arises at frequencies minimizing the denominator of Eq. (41). These frequencies almost coincide with the shifted eigenfrequencies Os,n of the imperfect structure for z 5 1, as follows by comparing the denominator of Eq. (41) to Eqs. (23) or (32), and Eq. (39) to Eqs. (22) or (31). The forced-response amplitude, evaluated at the imperfect pendulum and at another pendulum, is reported in Fig. 6 as a function of O, for two different values of ne . The computations are performed using the DM and HM models. The main resonance peak at O ¼ Oo,ne appears, together with other resonance peaks at

900

900

DM model HM model blade #1 blade #10

800

700

600

600

500

500 | |/

| |/

700

400

400

300

300

200

200

100

100

0 0.97

0.98

0.99

1 Ω

1.01

1.02

1.03

DM model HM model blade #1 blade #10

800

0 0.97

0.98

0.99

1

1.01

1.02

1.03

Ω

Fig. 6. Cyclic structure with N = 20, g ¼ 0:1, z ¼ 0:001, b ¼ 0:04, m ¼ 0. Frequency response amplitude of a single pendulum versus frequency. Engine order: (a) ne ¼ 0; (b) ne ¼ 1.

2618

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

frequencies Os,n , due to the imperfection. For ne = 0 the main resonance peak has the largest amplitude, larger than the resonance peak of the forced response of the perfect structure (Fig. 3). These findings, derived by the homogenized model, are in agreement with the results obtained in [27] using a discrete model and the U-transformation: as a matter of fact, the HM and DM curves in Fig. 6 are virtually coincident. The forced response Q ð0, OÞ of the imperfect pendulum, obtained setting y ¼ 0 into Eq. (40), turns out to be Q ð0, OÞ ¼

Q 2GlsinhðlpÞ : Dne ðOÞ sðbmO2 ÞcoshðlpÞ þ 2GlsinhðlpÞ

(42)

Simple algebraic expressions of Q ð0, OÞ, evaluated at the structural eigenfrequencies Oo,n or Os,n , can be obtained by assuming that the damping coefficient z and the imperfection level b [or m] are small quantities, by replacing the numerator and the denominator of Eq. (42) respectively by their McLaurin expansions with respect to z and b [or m], truncated to the first order. The case of a stiffness imperfection ba0, m ¼ 0 is considered here; the case of a mass imperfection is reported in Appendix A. After some algebra, it turns out that, for O ¼ Oo,n :

 if n =ne, Q ð0, Oo,n Þ

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2iz 1 þ n2e G þ cn b=N

(43)

z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Gðn2e n2 Þ½zicn b=ð2N 1þ n2 GÞ

(44)

Q

¼

 if nane , Q ð0, Oo,n Þ Q

¼

whereas, for O ¼ Os,n :

 if n =ne, Q ð0, Os,n Þ Q

 if nane , Q ð0, Os,n Þ Q

¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2iz 1 þn2e G

(45)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

¼

z þ icn b=ð2N 1þ n2 GÞ , zGðn2e n2 Þ

(46)

where cn = 1 if n =0, and cn = 2 if n 4 0. These expressions highlight how a small imperfection in the periodic structure affects its forced response. At frequencies O ¼ Oo,n with nane , the response amplitude is proportional to z (Eq. (44)): hence, these frequencies are antiresonances for the forced response. At O ¼ Oo,n with n= ne, the response amplitude remains finite even if the damping vanishes, with amplitude proportional to N=b (Eq. (43)), and so no resonance occurs. On the other hand, the frequencies Os,n are all resonances, since the response amplitude is proportional to 1=z. If nane , the peak amplitude, assuming z 5 jbj=N, is proportional to Nb=g and inversely proportional to jn2e n2 j (Eq. (46)), whereas if n= ne the corresponding peak amplitude, which is maximum for ne = 0, is independent on the imperfection level b and coupling g and coincides with the resonance amplitude of the perfect structure (compare Eqs. (46) and (14)). Finally, keeping z 5 1 fixed and letting b-0, the eigenfrequencies Os,n tend to the corresponding eigenfrequencies Oo,n , and the resonance peaks occurring at Os,n tend to the values given by Eq. (12) describing the forced response of the perfect structure. The above discussion holds independently of the sign of b. To assess the occurrence of localization in the forced response, the ratio Q ðy, OÞ=Q ð0, OÞ between the response at a generic angle y and the response of the imperfect pendulum is computed. From Eqs. (40) to (42) it is obtained that   Q ðy, OÞ cosh½lðypÞ 1 Q cosh½lðypÞ ¼ þ expðine yÞ : (47) Q ð0, OÞ coshðlpÞ Dne ðOÞ Q ð0, OÞ coshðlpÞ In order to have localization, the second term at the right-hand side, which is not localized, must be negligible with respect to the first one, which, in turn, is localized provided that the real part of l is large enough. Those conditions are fulfilled at the frequency O ¼ Oloc , provided that

b o 0, z 5 1, bN=g 4 1, z o b=N,

(48)

since in that case: (a) l  aloc by Eq. (39), with aloc large enough by Eq. (34); (b) jDne ðOÞQ ð0, OÞ=Q j b1 by Eqs. (13) and (45) or (46). The other shifted eigenfrequencies (Os,n with n 40, or Os,0 in the case b 4 0), or the non-shifted frequencies Oo,n do not imply localization, since l has small real part in those cases by Eq. (39). These considerations are confirmed by the results in Figs. 7 and 8. In particular, in Fig. 7 the forced-response amplitude of each pendulum is reported, evaluated at the resonance

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

900

DM model, Ω=Ω loc

HM model, Ω=Ω loc

800

700

DM model, Ω=Ωs,1

700

DM model, Ω=Ωs,1

600

HM model, Ω=Ωs,1

600

HM model, Ω=Ωs,1

500

|Q|/ Q

| |/

900

DM model, Ω=Ω loc

800

2619

400

500 400

300

300

200

200

100

100

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

HM model, Ω=Ω loc

01

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

pendulum #

pendulum #

Fig. 7. Cyclic structure with N= 20, g ¼ 0:1, z ¼ 0:001, b ¼ 0:04, m ¼ 0. Frequency response amplitude at a fixed frequency versus pendulum index. Engine order: (a) ne ¼ 0; (b) ne ¼ 1.

1 0.8

ζ| |/

0.6

DM model, ζ=0.001 DM model, ζ=0.005 DM model, ζ=0.01 HM model, ζ=0.001 HM model, ζ=0.005 HM model, ζ=0.01

0.4 0.2 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

pendulum # Fig. 8. Cyclic structure with N = 20, g ¼ 0:1, b ¼ 0:04, m ¼ 0, ne = 0. Frequency response amplitude at O ¼ Oloc versus pendulum index.

frequencies Oloc and Os,1 , for the two engine orders ne = 0 and 1. The vibration localization around the imperfect pendulum occurs in the response relevant to O ¼ Oloc . The largest amount of localization arises for ne =0, since in that case the external load is mainly exciting the lowest frequency eigenmode, which is the localized one. The significance of condition (48)4 is shown in Fig. 8, reporting the forced-response amplitude of each pendulum, evaluated at O ¼ Oloc , for ne =0 and different values of z. It turns out that the localization amount increases with the decrease of z. In the considered case, the imperfection level b=N ¼ 0:002 was chosen, thus violating the condition (48)4 for the curves relevant to z ¼ 0:005 and 0.01: indeed, a sinusoidal contribution superimposed to an exponential one appears on those curves. The comparison between the DM and HM models, used in the computations, is very satisfactory. 4. Heavily imperfect or weakly coupled structure: lumped imperfection 4.1. Homogenized model As remarked at the end of Section 3.2.3, the HM model may not be accurate in the localization frequency region O o1 for a heavily imperfect or weakly coupled structure, i.e. with large b=ð2gÞ [or mO2loc =ð2gÞ] ratio. A modified version of that model, denoted by MHM, is developed here. Though not being as simple as the HM model, it is effective in that case. For large b [or m], or small g, the dynamical behavior of the imperfect pendulum is respectively so different from, or so weakly coupled to the other perfect pendula, that it would be reasonable to exclude it from the homogenization process. Accordingly, the homogenization process is now applied only to the perfect pendula, whereas the imperfect one is modeled according to Eq. (5) with j= 1, suitably modified for the present purpose. The modified model reads as ðO2 þ 2ziO þ1ÞQ GQ 00 ¼ Q expðine yÞ, ½O2 ð1 þ mÞ þ 2ziO þð1 þ bÞq1 G

y 2 ð0,2pÞ,

Q ðsÞ2q1 þ Q ð2psÞ

s2

¼Q,

(49) (50)

2620

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

where q1 is the displacement of the imperfect pendulum and Q ðsÞ and Q ð2psÞ recover the displacement of the pendula adjacent to the imperfect one from the homogenized displacement Q. The two boundary conditions (19) are replaced by KQ uðs=2Þ ¼ k½Q ðsÞq1 ,

KQ uð2ps=2Þ ¼ k½q1 Q ð2psÞ,

(51)

enforcing the equilibrium between the stress KQ u emerging from the homogenized region and the stress in the two springs attached to the imperfect pendulum not involved in the homogenization process. Eqs. (51) can be rewritten as follows: Q uðs=2Þ þ Q uð2ps=2Þ Q ðsÞQ ð2psÞ ¼ , 2 2s Q uðs=2ÞQ uð2ps=2Þ

s

¼

Q ðsÞ2q1 þQ ð2psÞ

s2

(52)

(53)

:

4.2. Free response analysis In order to use the MHM model for evaluating the localized eigenfrequency and eigenmode, Problem (49) and (50) is solved, in the hypothesis of vanishing damping and external load, together with the boundary conditions (52) and (53). To this end, the same procedure described in Section 3.2.2 is adopted, and a solution of Eq. (49) is assumed of the form (30), where a is related to O by Eq. (31). By substituting the antisymmetric eigenmode (30)2 into Eq. (53), it is obtained that q1 =0 and sinhaðpsÞ þ as ¼ 0, coshaðps=2Þ

(54)

which admits the only solution a ¼ 0. Accordingly, no such eigenmode exists. The symmetric eigenmode (30)1 satisfies Eq. (52); by substituting Eq. (30)1 into Eq. (53) it is obtained that

asinh½aðps=2Þ þ

cosh½aðpsÞq1

s

¼ 0:

(55)

By substituting q1 from Eq. (50) into Eq. (55), the following equation is derived:      s bmð1a2 GÞ þ a2 G cosh½aðpsÞ þ a 2G þ s2 bmð1a2 GÞ þ a2 G sinh½aðps=2Þ ¼ 0,

(56)

yielding a solution aloc which leads to the localized eigenfrequency Oloc through Eq. (31). Asymptotic formulas for aloc and Oloc for weakly coupled structures (i.e. very small G) can be derived from Eq. (56). When G-0, it turns out that aloc - þ 1. Dividing Eq. (56) by acosh½aðpsÞ, it follows that the term in the curly brackets tends to zero. The case of mass imperfection is reported in Appendix A; in the case of a stiffness imperfection, the following expansions are obtained: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N b aloc ¼ 2 þ oð1Þ, Oloc ¼ 1ðbÞ þ 2g þ oðgÞ: (57) 2p g Fig. 9 reports Oloc , computed using the DM, HM and MHM models, for a cyclic structure with a stiffness imperfection b ¼ 0:04 or 0.08, as a function of the inter-pendulum coupling g. As discussed in Section 3.2.3, the fully homogenized model HM fails in computing the localized eigenfrequency in the range b=ð2gÞ Z1: the relevant curves have been truncated accordingly. On the other hand, the modified homogenized model MHM turns out to accurately estimate the localized eigenfrequency everywhere. The behavior of Oloc in Fig. 9 exhibits an asymptotic trend for small and for large values of g,

1.02 1.01

Ωloc

1 0.99 0.98 DM model HM model MHM model β=-0.04 β=-0.08

0.97 0.96 0.95 -3 10

-2

10

-1

10 γ

0

10

1

10

Fig. 9. Cyclic structure with N =20, m ¼ 0. Localized eigenfrequency Oloc versus inter-pendulum coupling g, for different values of stiffness imperfection b. Results computed according to the DM, HM, MHM models.

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

2621

given by the asymptotic expansions (57)2 and (35), respectively. As a matter of fact, in the former case the coupling between pffiffiffiffiffiffiffiffiffiffiffi pendula is vanishing and thus the imperfect pendulum vibrates according to its eigenfrequency Oloc ¼ 1 þ b independent from the other pendula, in agreement with a finding in [27], whereas in the latter case the coupling is so strong that all the pendula experience almost the same displacement; accordingly the effect of the weakening imperfection on a pendulum is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi partitioned among all the pendula, yielding Oloc ¼ 1 þ b=N C 1 þ b=ð2NÞ. The frequency equation (56) is more involved than the corresponding one (33) relevant to the HM model, so it is not easy to compute L from Eq. (56) in closed form. However, an estimation of L in the range b=ð2gÞ b 1 can be obtained from the asymptotic expansion (57). It turns out that rffiffiffiffiffiffiffi 1 1 g C L¼ Cs , (58) aloc tanhðaloc pÞ aloc b showing that strong localization occurs for very weakly coupled structures, as discussed in [21].

4.3. Forced response analysis In order to compute the forced response by using the MHM model, the field equations (49) equipped with the boundary conditions (52) and (53) are solved. A general solution of Eq. (49)1 is given by Eq. (38). Then, exploiting Eqs. (50), (52) and (53), the frequency response q1 ðOÞ of the imperfect pendulum is computed and reads as q1 ðOÞ Q

¼

1 l½s2 Dne ðOÞ þ 2Gcosðne sÞS þ½sDne ðOÞ2Gne sinðne s=2ÞC , ~ OÞ þ 2GS þ sDð ~ OÞC Dne ðOÞ l½s2 Dð

(59)

~ OÞ ¼ 1 þ b þ 2ziOð1 þ mÞO2 , C ¼ cosh½lðpsÞ, S ¼ sinh½lðps=2Þ. where Dð Asymptotic formulas for q1 ðOÞ=Q for weakly coupled structures can be derived from Eq. (59). To this end, by Eq. (39) it is noted that when O o1 the real part of l tends to þ 1 as G-0. Hence, Eq. (59) can be approximated as follows: q1 ðOÞ Q



1 s2 Dne ðOÞ þ 2Gcosðne sÞ 1 þ 2gcosðne sÞðDne ðOÞÞ1 ¼ : ~ ~ OÞ þ2g 2 Dne ðOÞ s DðOÞ þ 2G Dð

(60)

At the localized frequency O ¼ Oloc given by Eq. (57), the denominator of the last expression reduces to 2ziOloc þ oðgÞ, thus exhibiting a resonance peak. Fig. 10 shows the frequency response amplitude of the imperfect pendulum with b ¼ 0:04, evaluated at the localized eigenfrequency Oloc , versus the inter-pendulum coupling g. The curves are computed in correspondence of ne = 0 or 1, using the DM, HM and MHM models. It can be seen that the HM model is accurate for sufficiently large values of g, whereas the MHM model accurately reproduces the frequency response amplitude of the imperfect pendulum for every g. The appearance of an optimal value of inter-pendulum coupling g maximizing q1 ðOloc Þ can be readily noticed, as pointed out in [27]. The asymptotic behavior for g-0 can be computed from Eq. (60): the frequency response of an isolated imperfect pendulum, i.e. ~ OÞ, is recovered. Finally, it is interesting to observe that, for large g, the response amplitude is very different depending on 1=Dð the engine order. In particular, as stated by the asymptotic expansion (45), if ne 40 the amplitude tends to 0, whereas for ne = 0 it tends to 1=ð2zÞ, equal to the amplitude in a perfect structure with ne = 0 (Eq. (14)). As previously pointed out, in heavily coupled structures all the pendula experience almost the same displacement and, accordingly, loads with ne 4 0, whose resultant is zero, do not elicit any appreciable response.

1200

DM model HM model MHM model ne=0

1000

ne=1

800 600 400 200 0 -3 10

-2

10

-1

10 γ

0

10

1

10

Fig. 10. Cyclic structure with N = 20, z ¼ 0:001, b ¼ 0:04, m ¼ 0, ne ¼ 0. Frequency response amplitude at O ¼ Oloc versus inter-pendulum coupling g.

2622

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

5. Imperfect structure: distributed imperfections In this section a homogenized continuous model of the rotationally periodic structure depicted in Fig. 1(a) is developed, taking into account distributed stiffness and mass imperfections affecting the pendula. The dynamical equilibrium equations are   ðky þ Dkyj Þ b ðm þ Dmj Þo2 þ 2y io þ (61) qj kðqj þ 1 2qj þ qj1 Þ ¼ fj , j ¼ 1,2, . . . ,N, l2 l where Dmj and Dkyj are the mass and rotational-stiffness imperfections on the j-th pendulum, respectively. Following a limiting procedure analogous to the one described in Section 2.2, the following homogenized equation is obtained:   B K þ DK ðyÞ ðM þ DMðyÞÞO2 þ 2y iO þ y 2 y (62) Q KQ 00 ¼ Fexpðine yÞ, y 2 ð0,2pÞ; Q ð0 þ Þ ¼ Q ð2p Þ, Q uð0 þ Þ ¼ Q uð2p Þ, l l where DMðyÞ and DKy ðyÞ are piece-wise constant functions accounting for the imperfections, reading as

DMðyÞ ¼

Dmj

s

,

DKy ðyÞ ¼

Dkyj

s

,

y 2 ððj1Þss=2,jss=2Þ, j ¼ 1,2, . . . ,N:

(63)

Eq. (62) can be rewritten in the following dimensionless form: ½O2 ð1 þ md ðyÞÞ þ2ziO þð1 þ bd ðyÞÞQ GQ 00 ¼ Q eine y ,

y 2 ð0,2pÞ;

Q ð0 þ Þ ¼ Q ð2p Þ,

Q uð0 þ Þ ¼ Q uð2p Þ,

(64)

where the dimensionless parameters have been defined in Eq. (7) and, moreover,

md ðyÞ ¼

DMðyÞ M

,

bd ðyÞ ¼

DKy ðyÞ Ky

(65)

:

In passing, it can be easily verified that the homogenized equation (17), describing the case of periodic structure with lumped imperfections b and m, can be recovered from Eq. (64) by choosing bd and md as Dirac distributions, i.e. bd ðyÞ ¼ sbd0 ðyÞ and md ðyÞ ¼ smd0 ðyÞ. 5.1. Free response analysis The eigenfrequencies and eigenmodes of the imperfect structure can be obtained by solving Problem (64) with z ¼ 0, Q ¼ 0. Due to the distributed imperfections, the eigenmodes of the degenerate eigenmode pair at frequency Oo,n of the perfect structure are split into two non-degenerate eigenmodes, respectively at frequencies OIs,n and OIIs,n , with OIs,n o OIIs,n . Moreover, the lowest eigenfrequency Oo,0 of the perfect structure is shifted to Os,0 . In order to show the accuracy of the homogenized model in the case of distributed imperfections, a stiffness mistuning, randomly distributed over the pendula, has been considered. It was assumed to have a Gaussian distribution with zero mean value and standard deviation sb . The numerically computed eigenfrequencies are reported in Table 3 and compared with the corresponding eigenfrequencies supplied by the discrete model. Moreover, in Fig. 11 the first three eigenmodes are reported, computed using the HM and DM models: the agreement is quite satisfactory, even in the case of a large mistuning level. Approximate analytical expressions are obtained under the hypothesis of small imperfections, employing a perturbation approach. To this end, an asymptotic expansion of the solution of Eq. (64) is performed, by perturbing the n-th eigenmode Qo,n of the perfect structure as Q ¼ Qo,n þ dQn , and the relevant eigenfrequency Oo,n as O ¼ Oo,n þ dOn , where dOn and dQn are unknown frequency and eigenmode perturbations to be computed. Using Eqs. (10) and (11) and retaining only the leadingorder terms, the following equation is obtained:

dQn00 þ n2 dQn ¼ gðyÞAo,n cosðny þ jn Þ, dQn ð0 þ Þ ¼ dQn ð2p Þ, dQnuð0 þ Þ ¼ dQnuð2p Þ,

(66)

where gðyÞ ¼

1

G

½2dOn

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þn2 Gð1 þ n2 GÞmd ðyÞ þ bd ðyÞ:

(67)

Table 3 Cyclic structure with N= 20, g ¼ 0:1, md ¼ 0. Randomly generated stiffness imperfection bd , normally distributed with zero mean value and standard deviation sb . Comparison between the eigenfrequencies predicted by the DM and HM models.

Os,0

OIs,1

OIIs,1

OIs,2

OIIs,2

OIs,3

OIIs,3

sb = 0.01

DM HM

0.9992 0.9993

1.0051 1.0053

1.0079 1.0080

1.0191 1.0199

1.0215 1.0221

1.0403 1.0437

1.0432 1.0461

sb = 0.05

DM HM

0.9753 0.9791

1.0045 1.0051

1.0159 1.0167

1.0246 1.0274

1.0274 1.0296

1.0400 1.0450

1.0543 1.0584

0.6

0.6

0.4

0.4 modal amplitude

modal amplitude

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

0.2 0 DM model, n=0 HM model, n=0

-0.2 -0.4

DM model, n=1

I

HM model, n=1

I

DM model, n=1

II

HM model, n=1

II

2623

0.2 0 DM model, n=0 HM model, n=0

-0.2

I

DM model, n=1

I

HM model, n=1

-0.4

II

DM model, n=1

II

HM model, n=1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

pendulum #

pendulum #

Fig. 11. Cyclic structure with N =20, g ¼ 0:1, z ¼ 0:001, md ¼ 0. Comparison between eigenmodes evaluated using the DM and HM models. Randomly generated stiffness imperfection bd , normally distributed with zero mean value and standard deviation: (a) sb ¼ 0:01; (b) sb ¼ 0:05.

A solution of Eq. (66) is obtained using Fourier series. To this end, the following representations are introduced:

md ðyÞ ¼

þ1 X

fmd gl eily ,

l ¼ 1

bd ðyÞ ¼

þ1 X

fbd gl eily ,

þ1 X

dQn ðyÞ ¼

l ¼ 1

fdQn gl eily ,

(68)

l ¼ 1

where fcg is a vector whose l-th component fcgl is the l-th Fourier coefficient of the function cðyÞ. It is remarked that the representation (68)3 of the solution automatically satisfy the periodicity requirement. Without loss of generality, we may assume that fdQn gn ¼ 0. Substituting Eq. (68) into Eq. (66), multiplying by eily , with integer l, and integrating over ð0,2pÞ, the following expression is obtained, assuming jljan: fdQn gl 1 ½fb g ¼ eijn þ fbd gn þ l eijn ð1 þn2 GÞðfmd gn þ l eijn þfmd gn þ l eijn Þ: Ao,n 2Gðn2 l2 Þ d n þ l

(69)

On the other hand, choosing l = n, dOn is computed and reads as 1

ffi ½fbd g0 ð1þ n2 GÞfmd g0 þe2ijn ðfbd g2n ð1 þ n2 GÞfmd g2n Þ: dOn ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1þ n2 G

(70)

The mode orientation jn , which is undetermined in the perfect structure, can be computed in the presence of imperfections. It can be obtained from Eq. (70) enforcing that dOn is a real number: the following expression is thus obtained: 1 p jn ¼ arg½fbd g2n ð1 þ n2 GÞfmd g2n  þk , k ¼ 0,1: 2

2

(71)

It is pointed out that two different values of jn , corresponding to k= 0 and 1, are obtained. They yield two values for the frequency shift dOn , supplying in turn the two shifted frequencies OIs,n and OIIs,n . In the particular case n= 0, the eigenmode of the perfect structure is not degenerate and has a constant amplitude; accordingly the following expressions are obtained, for la0: fdQ0 gl fb g fm g ¼ d l 2 d l, Ao,0 Gl

dO0 ¼

fbd g0 fmd g0 : 2

(72)

The closed form expressions (69)–(72) are interesting, since they explicitly relate the frequency shifts and the harmonic content of the eigenmodes to the harmonic content of the imperfections. These formulas could be used to design intentional mistunings aimed at reducing the frequency shift and/or the localization phenomenon [38]. 5.2. Forced response analysis The forced response of the homogenized structure is obtained by solving Eq. (64) for any fixed positive value of the frequency O. The maximum value of the forced-response amplitude over all the pendula, for a randomly generated distribution of stiffness imperfections, is reported in Fig. 12 as a function of O. The computations have been performed using the DM and HM models, which turned out to supply nearly coincident results. The largest value is achieved for ne ¼ 0, attained at the lowest structural eigenfrequency Os,0 , whose corresponding eigenmode is localized (Fig. 11). Indeed, for ne ¼ 0 the external load is mainly exciting the first eigenmode, whose eigenfrequency, being lower than 1 according to Table 3, is out of the structural pass band [1]. The forced-response amplitude of each pendulum, evaluated at the resonance frequencies Os,0 , OIs,1 and OIIs,1 , is reported in Fig. 13. The vibration localization appears at the lowest structural eigenfrequency Os,0 . The comparison between the DM and HM models is very satisfactory.

2624

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

900 800 700

700

600

600

500

500

400

400

300

300

200

200

100

100

0 0.97

0.98

0.99

1 Ω

1.01

1.02

DM model HM model

800

| |/

| |/

900

DM model HM model

0 0.97

1.03

0.98

0.99

1

1.01

1.02

1.03

Ω

Fig. 12. Cyclic structure with N = 20, g ¼ 0:1, z ¼ 0:001, md ¼ 0. Randomly generated stiffness imperfection bd , normally distributed with zero mean value and standard deviation sb ¼ 0:01. Maximum frequency response amplitude over all the pendula versus frequency. Engine order: (a) ne ¼ 0; (b) ne ¼ 1.

900 800

DM model, Ω=Ω s,0

800

HM model, Ω=Ω s,0

HM model, Ω=Ω s,0

700

DM model, Ω=Ωs,1

700

DM model, Ω=Ωs,1

600

HM model,

I Ω=Ωs,1

600

HM model, Ω=Ωs,1

DM model,

II Ω=Ωs,1

500

DM model, Ω=Ωs,1

HM model,

II Ω=Ωs,1

400

HM model, Ω=Ωs,1

400

I

| |/

500 | |/

900

DM model, Ω=Ω s,0

300

300

200

200

100

100

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

0

I I

II II

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

pendulum #

pendulum #

Fig. 13. Cyclic structure with N = 20, g ¼ 0:1, z ¼ 0:001, md ¼ 0. Randomly generated stiffness imperfection bd , normally distributed with zero mean value and standard deviation sb ¼ 0:01. Frequency response amplitude at a fixed frequency versus pendulum index. Engine order: (a) ne ¼ 0; (b) ne ¼ 1.

Similarly to what has been done for the free-response analysis, an approximate analytical expression of the forced response can be achieved by performing an asymptotic expansion of the solution. To this end, perturbing the forced response Qo ðy, OÞ of the perfect structure (Eq. (12)) as Q ðy, OÞ ¼ Qo ðy, OÞ þ dQ ðy, OÞ, substituting into Eq. (64), and retaining only the leading-order terms, the following equation is obtained: ðO2 md þ bd ÞQo ðy, OÞ þ ðO2 þ 2ziO þ1ÞdQ ðy, OÞGdQ 00 ¼ 0:

(73)

The distributed imperfections md ðyÞ and bd ðyÞ appear in the non-sinusoidal forcing term of the second-order differential equation (73). This term implies that all the structural resonances contribute to the forced-response perturbation dQ ðy, OÞ, whereas only a single resonance peak related to the engine order ne appears in the forced response Qo ðy, OÞ of the perfect structure. As in the case of free-response analysis, a Fourier series approach is adopted to compute dQ ðy, OÞ. The following expression is obtained: fdQ gl Q

¼

fbd gne þ l þ O2 fmd gne þ l , Dne ðOÞDl ðOÞ

(74)

highlighting how the Fourier coefficients of the imperfections, modulated by the quantity Dl ðOÞ ¼ 1 þ l2 G þ 2ziOO2 , determine the Fourier coefficients of the perturbation dQ ðy, OÞ. At the frequency O ¼ Oo,n (i.e. at the n-th eigenfrequency of the perfect structure), the magnitude of the Dl term appearing in the denominator of (74) is minimized for l = 7n, and becomes very small for small damping z. Thus, at that frequency, the response is mainly dominated by the imperfection harmonics ne þ n and ne þ n. In particular, choosing n ¼ ne , the frequency response at O ¼ Oo,ne , which is the largest one, is dominated by the imperfection harmonics 0 and 2ne . This finding is in agreement with the analogous result obtained in [25] using a discrete model of a similar periodic structure. Finally, it is pointed out that Eq. (74) could be used to design and optimize intentional mistunings aimed at reducing the forced-response amplitude.

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

2625

6. Conclusions The feasibility of applying the mathematical theory of homogenization to derive continuous models of disordered rotationally periodic structures was explored. The effectiveness of the method was demonstrated by taking into account a case-study structure, which is both simple, thus allowing closed-form solutions, and widely adopted in the literature to illustrate the vibration localization phenomenon. The free and forced responses of the structure mistuned due to mass and/or stiffness imperfections were investigated. The proposed method was able to reproduce the features of the structural response, like the frequency split and the localization phenomenon, yielding results in excellent agreement with the ones supplied by the discrete model. Simple closed-form expressions of the eigenfrequencies and of the forced-response resonance peaks were obtained by performing asymptotic expansions of the homogenized-model solution, highlighting the effect of the structural parameters and the mistuning level on the dynamical behavior of the structure. Appendix A. Lumped imperfection: the case of mass imperfection, i.e. la0 and b ¼ 0 A.1. Free vibration In the case O Z 1, for mode indexes n 40, one computes the MacLaurin series of the inverse of the function y ¼ ðnp þxÞtanðxÞ½n2 p2 þ ðnp þxÞ2 n2 G1 , with x ¼ pðan nÞ and y ¼ m=ðNGn2 Þ. It turns out that " # an ð2pnÞ2 g mN 2 ¼ 1þ ð1 þn GÞy þoðyÞ ¼ 1 1 þ þoðmÞ, (A.1) n N2 ð2pnÞ2 g

Os,n m ¼ 1 þ n2 Gy þoðyÞ ¼ 1 þoðmÞ: Oo,n N

(A.2) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi For n = 0,p it ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is convenient to consider the MacLaurin series of the inverse of the function y ¼ xtanðxÞðp2 þ x2 GÞ1 , with x ¼ a0 p and y ¼ m=ðN GÞ. Accordingly, the solution sffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 mN þ oð mÞ, a0 ¼ y þ oðyÞ ¼ (A.3) 2p g

Os,0 ¼ 1 þ

Gy2 2

þoðy2 Þ ¼ 1

m 2N

þ oðmÞ:

(A.4)

prevails, which is acceptable for m o0 only. In the case O r1, a solution can be obtained only for n = 0. One computes the MacLaurin series of the inverse of the function qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ¼ xtanhðxÞðp2 x2 GÞ1 , with x ¼ aloc p and y ¼ m=ðNGÞ. This yields sffiffiffiffiffiffiffi pffiffiffiffi 1 mN þoð mÞ, aloc ¼ yþ oðyÞ ¼ (A.5) 2p g

Oloc ¼ 1

Gy2 2

þ oðy2 Þ ¼ 1

m 2N

þ oðmÞ:

(A.6)

On the other hand, the model MHM yields

aloc ¼

N 2p

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2  þ oð1Þ, gð1 þ mÞ 1 þ m

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2g þ oðgÞ Oloc ¼ 1 þ 1þ m 1 þ m

(A.7)

(A.8)

obtained by setting ma0, b ¼ 0 in Eq. (56) and reasoning as explained in Section 4.2. Expansions (A.5)– (A.8) are acceptable for m 40 only. A.2. Forced vibration For frequencies O ¼ Oo,n , it is obtained that

 if n ¼ ne , Q ð0, Oo,n Þ Q

¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ½2izcn m 1 þn2e G=N 1þ n2e G

(A.9)

2626

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

 if nane , Q ð0, Oo,n Þ Q

¼

z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Gðn2e n2 Þ½z þ icn m 1 þn2 G=ð2NÞ

(A.10)

whereas, for frequencies O ¼ Os,n , it is obtained that:

 if n ¼ ne , Q ð0, Os,n Þ Q

¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2iz 1 þ n2e G

(A.11)

 if nane , Q ð0, Os,n Þ Q

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

¼

zicn m 1 þ n2 G=ð2NÞ : zGðn2e n2 Þ

(A.12)

References [1] D.J. Mead, Wave propagation and natural modes in periodic systems: I. Mono-coupled systems, Journal of Sound and Vibration 40 (1) (1975) 1–18. [2] D.L. Thomas, Dynamics of rotationally periodic structures, International Journal for Numerical Methods in Engineering 14 (1979) 81–102. [3] I.Y. Shen, Vibration of rotationally periodic structures, Journal of Sound and Vibration 172 (1994) 459–470. [4] A.J. Fricker, S. Potter, Transient forced vibration of rotationally periodic structure, International Journal for Numerical Methods in Engineering 17 (1981) 957–974. [5] C.W. Cai, Y.K. Cheung, H.C. Chan, Uncoupling of dynamic equations for periodic structures, Journal of Sound and Vibration 139 (1990) 253–273. [6] P.W. Anderson, Absence of diffusion in certain random lattices, Physical Review 109 (1958) 1492–1505. [7] D.J. Ewins, Effects of detuning upon forced vibrations of bladed disks, Journal of Sound and Vibration 9 (1) (1969) 65–79. [8] C.H. Hodges, Confinement of vibration by structural irregularity, Journal of Sound and Vibration 82 (3) (1982) 411–424. [9] C.H. Hodges, J. Woodhouse, Vibration isolation from irregularity in a nearly periodic structure: theory and measurement, Journal of the Acoustical Society of America 74 (3) (1983) 894–905. [10] R.A. Ibrahim, Structural dynamics with parameter uncertainties, Applied Mechanics Reviews 40 (3) (1987) 309–328. [11] G.J. Kissel, Localization in Disordered Periodic Structures, PhD Dissertation, Massachusetts Institute of Technology, 1988. [12] W.-C. Xie, S.T. Ariaratnam, Vibration mode localization in disordered cyclic structures. II: multiple substructure mode, Journal of Sound and Vibration 189 (5) (1996) 647–660. [13] C. Pierre, E.H. Dowell, Localization of vibration by structural irregularity, Journal of Sound and Vibration 114 (3) (1987) 549–564. [14] G.S. Happawana, O.D.I. Nwokah, A.K. Bajaj, M. Azene, Free and forced response of mistuned linear cyclic systems: a singular perturbation approach, Journal of Sound and Vibration 211 (5) (1998) 761–789. [15] G.J. Kissel, Localization factor for multichannel disordered systems, Physics Review A 44 (2) (1991) 1008–1014. [16] C.W. Cai, Y.K. Lin, Localization of wave propagation in disordered periodic structures, AIAA Journal 29 (1991) 450–456. [17] O.O. Bendiksen, Mode localization phenomena in large space structures, AIAA Journal 25 (9) (1987) 1241–1248. [18] C. Pierre, Weak and strong vibration localization in disordered structures: a statistical investigation, Journal of Sound and Vibration 139 (1) (1990) 111–132. [19] S.H. Shin, M.K. Kang, H.H. Yoo, Mistuned coupling stiffness effect on the vibration localization of cyclic systems, Journal of Mechanical Science and Technology 22 (2008) 269–275. [20] O.O. Bendiksen, Localization phenomena in structural dynamics, Chaos, Solitons and Fractals 11 (2000) 1621–1660. [21] C. Pierre, Mode localization and eigenvalue loci veering phenomena in disordered structures, Journal of Sound and Vibration 126 (3) (1988) 485–502. [22] B.W. Huang, Effect of numbers of blades and distribution of cracks on vibration localization in a cracked pre-twisted blade system, International Journal of Mechanical Sciences 48 (2006) 1–10. [23] Y.-J. Chiu, S.-C. Huang, The influence on coupling vibration of a rotor system due to a mistuned blade length, International Journal of Mechanical Sciences 49 (4) (2007) 522–532. [24] G.Q. Cai, Y.K. Cheung, H.C. Chan, Mode localization phenomena in nearly periodic systems, Journal of Applied Mechanics 62 (1) (1995) 141–149. [25] M.P. Mignolet, W. Hu, I. Jadic, On the forced response of harmonically and partially mistuned bladed disks. Part i: harmonic mistuning. Part ii: partial mistuning and applications, International Journal of Rotating Machinery 6 (1) (2000) 29–56. [26] H.H. Yoo, J.Y. Kim, D.J. Inman, Vibration localization of simplified mistuned cyclic structures undertaking external harmonic force, Journal of Sound and Vibration 261 (2003) 859–870. [27] X. Fang, J. Tang, E. Jordan, K.D. Murphy, Crack induced vibration localization in simplified bladed-disk structures, Journal of Sound and Vibration 291 (2006) 395–418. [28] Y.J. Chan, D.J. Ewins, Management of the variability of vibration response levels in mistuned bladed discs using robust design concepts. Part 1: parameter design, Mechanical Systems and Signal Processing 24 (2010) 2777–2791. [29] E.P. Petrov, D.J. Ewins, Analysis of the worst mistuning patterns in bladed disk assemblies, Transactions of the ASME. Journal of Turbomachinery 125 (4) (2003) 623–631. [30] J. Tang, K.W. Wang, Vibration delocalization of nearly periodic structures using coupled piezoelectric networks, Journal of Vibration and Acoustics 125 (2003) 95–108. [31] H. Yu, K.W. Wang, Vibration suppression of mistuned coupled-blade-disk systems using piezoelectric circuitry network, Journal of Vibration and Acoustics 131 (2009) 021008. [32] Y.J. Chan, D.J. Ewins, Management of the variability of vibration response levels in mistuned bladed discs using robust design concepts. Part 2: tolerance design, Mechanical Systems and Signal Processing 24 (2010) 2792–2806. [33] L.F. Wagner, J.H. Griffin, A continuous analog model for grouped-blade vibration, Journal of Sound and Vibration 165 (1993) 421–438. [34] J. Tang, K.W. Wang, Vibration control of rotationally periodic structures using passive shunt networks and active compensation, Journal of Vibration and Acoustics 121 (1999) 379–390.

P. Bisegna, G. Caruso / Journal of Sound and Vibration 330 (2011) 2608–2627

2627

[35] S.K. Sinha, Dynamic characteristics of flexible-bladed rotor with Coulomb damping due to tip-rub, Journal of Sound and Vibration 273 (4–5) (2004) 875–919. [36] N. Lesaffre, J.-J. Sinou, F. Thouverez, Contact analysis of flexible bladed-rotor, European Journal of Mechanics—A/Solids 26 (2007) 541–557. [37] P. Bisegna, G. Caruso, Optimization of a passive vibration control scheme acting on a bladed rotor using an homogenized model, Structural and Multidisciplinary Optimization 39 (6) (2009) 625–636. [38] P. Bisegna, G. Caruso, Frequency split and vibration localization in imperfect rings, Journal of sound and Vibration 306 (3–5) (2007) 691–711.