Intership Report

Multi-S aling and Averaging for HigherFidelity Lumped Model of Bladed Diskwith Centrifugal Pendulum VibrationAbsorbersStéphanie Polchi

Stage de Fin d’Études de l’École Centrale de Nantes

Structural Dynamics and Vibration Laboratory, McGill University

5/04/07 - 28/09/07

ContentsForeword v

Introduction vii

1 Background 1

1.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Kronecker Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Fourier Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Circulant Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.4 Diagonalization of Circulant Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Engine Order Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Traveling Wave Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Vibrations Characteristics of Cyclic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.1 Prototypical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Forced Response Under Engine Order Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2.1 Existence of a Traveling Wave Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.2.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.3 Eigenfrequency Characteristics and Normal Modes of Vibration . . . . . . . . . . . . . . . . . . 111.3.4 Resonance Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Centrifugal Pendulum Vibration Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Lumped-Parameter Bladed Disk Model with CPVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Mathematical Model 17

2.1 Geometry of an Arbitrary Absorber Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.1 Fundamental Relationship Between Path Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.2 Angle Between Radius Vector and Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Bladed Disk Fitted with General-Path Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Lumped-Parameter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.2 Kinetic and Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.3.1 Dimensional Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.3.2 Dimensionless Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.4 Generalized Two-Parameter Family of Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Bladed Disk Fitted with Circular-Path Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Linearized Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Absorber Tuning for the Linearized System 33

3.1 Sector Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Special Case: Locked Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Absorber Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Forced Response of the Nonlinear System 39

4.1 Analytical Approximation of the Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.1.3 Comparison between Linear and Scaled Solutions . . . . . . . . . . . . . . . . . . . . . 424.1.1.4 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.2 Traveling Wave Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Comparison: Analytical Solution vs. Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Isolated Nonlinear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.2 Coupled Nonlinear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Conclusion 57

ForewordI carried out my “Travail de Fin d’Études” at the Structural Dynamics and Vibration Laboratory at

McGill University in Montreal. The Laboratory is part of the Mechanical Engineering Department

at McGill University. It is concerned with theoretical knowledge as well as the development of new

tools and methods capable of bringing solutions to practical problems in the industrial sphere.

Related research and education are focused on Vibrations, Acoustics, Structural Dynamics, Non-

linear Dynamics, and Wave Propagation. During my internship, the lab was employing 8 people,

of which 4 were Ph.D. candidates, 2 were post-doctoral fellows and 2 were interns.

My project fall within the framework of Serif Gozen’s thesis entitled “Mistuned Bladed Disk Sys-

tems with Pendulum Absorbers”. Thus, Ph.D. student Serif Gozen was my supervisor, but I also

received much help from Mathias Legrand, who is a post-doctoral fellow at the Structural Dynam-

ics and Vibration Laboratory.

I wish to thank M. Christophe Pierre, dean of the Faculty of Engineering at McGill University, who

gave me the opportunity to work in the Structural Dynamics and Vibration Laboratory.

I am deeply grateful to my supervisors, Mathias Legrand and Serif Gozen, for their constructive

comments and advices, and for their support throughout this work.

I warmly thank Alain Batailly, Sébastien Roques, Vladislav Ganine, Galina Pilgun and Bruno Varin

who worked with me at the Structural Dynamics and Vibration Laboratory for their cordial wel-

come and their help.

The financial support of the McGill University is gratefully acknowledged.

In this document, full sentences, paragraphs, figures, and organization of technical content

were taken either verbatim or essentially from Brian Olson’s thesis [OLS 06] with his permis-

sion.

Introdu tionMany rotating flexible structures consist of an array of interconnected constituent parts whose

geometry and structural properties are rotationally periodic, and they are said to have cyclic sym-

metry. In a bladed disk, for example, the fundamental substructure is one blade plus the cor-

responding segment of the disk, which is referred to as a sector. During steady operation these

systems rotate at a constant speed and are subjected to traveling wave dynamic loading, which is

characterized by excitation frequencies that are proportional to the mean rotational speed of the

rotor. Such excitations result in component vibrations and can lead to high cycle fatigue failure,

noise, reduced performance, and other undesirable effects.

Order-tuned vibration absorbers exploit the centrifugal field from rotation of a primary system.

Since rotating flexible structures are dominated by forces that occur at orders of rotation, order-

tuned absorbers are suited for such applications. A class of order-tuned absorbers are centrifugal

pendulum vibration absorbers, or CPVAs. They essentially consist of mass particles that ride along

designer-specified paths relative to the primary system and their parameters are chosen such that

they counteract the fluctuating loads applied to that system. The dynamic performance, charac-

teristics, and features of CPVAs are well-understood in typical situations, and there are numerous

examples of their implementation.

In previous works [OLS 06] (see also [OLS b; OLS a]), lumped-parameter models of cyclically-coupled

bladed disk assembly fitted with order-tuned vibration absorbers was investigated. In such mod-

els, each sector is composed of two degrees of freedom (DOFs) which describe blade and ab-

sorber dynamics. The fundamental linearized dynamics of those systems was investigated in de-

tail. Multi-scaling and averaging methods were successfully applied to their nonlinear equations

of motions (EOMs). However, it is needed to investigate more realistic models for centrifugally-

driven vibration absorbers attached to bladed disk assembly which would consider disk as a flexi-

ble element. For this purpose, a higher-fidelity lumped-parameter model for bladed disk assembly

including disk DOFs and coupling between them is proposed.

This report is organized as follows. First chapter highlights pertinent theoretical background in-

cluding some mathematical preliminaries, engine order excitation, vibration characteristics of

cyclic system, order-tuned vibration absorbers. A higher-fidelity mathematical model for a bladed

disk assembly fitted with order-tuned vibration absorbers is then developed in Chapter 2. Equa-

tions of motions for this model are obtained by using Lagrange Method. Some fundamental fea-

tures of the linearized model are investigated in Chapter 3, which gives rise to a linear absorber

tuning strategy. In Chapter 4, nonlinear EOMs are reduced via scaling and averaging to a set of

nonlinear averaged sector models. Finally, results of scaling and averaging are verified by direct

simulation of original nonlinear equations of motion.

1Ba kground

This chapter highlights relevant background material that will be useful in the analysis of subse-

quent chapters. Some mathematical preliminaries are considered first in Section 1.1, including

the Kronecker product, the Fourier matrix, and the theory of circulant matrices. A model for en-

gine order (e.o.) excitation is subsequently developed in Section 1.2 and its traveling wave (TW)

nature discussed. In order to characterize the basic free and forced response of cyclically-coupled

systems under e.o. excitation, a cyclic chain of linear oscillators is investigated in Section 1.3.

The theory of Centrifugal Pendulum Vibration Absorber (CPVA) is given in Section 1.4. Finally, a

lumped-parameter model of a bladed disk assembly fitted with CPVA is presented in Section 1.5

and his linear and nonlinear dynamic characteristics are introduced.1.1 Mathemati al Preliminaries1.1.1 Krone ker Produ tThe Kronecker product of A and B, for A ∈Cm×n and B∈Cp×q , is the m p ×nq matrix :

A⊗B=

a 11B a 12B . . . a 1n B

a 21B a 22B . . . a 2n B...

..... .

...

a m 1B a m 2B . . . a m n B

(1.1)

Selected properties of the Kronecker product are given in what follows.

1. If A, B, C and D are square matrices such that AC and BD exist, then

(A⊗B)(C⊗D) = (AC)⊗ (BD)

2. If A and B are invertible matrices, then (A⊗B)−1 =A−1⊗B−1

3. If A and B are square matrices, then (A⊗B)H =AH ⊗BH

(·)H = (·)T is the Hermitian operator, or conjugate transpose.

2 Ba kground1.1.2 Fourier MatrixThe N ×N complex Fourier matrix is defined as

E= [ei k ]; ei k =1p

Nw (i−1)(k−1) =

1p

Nej (k−1)ϕi , i ,k ∈N (1.2)

whereN = 1,2, . . . ,N , w = e2jπ

N is the N th rooth of unity and

ϕi =2π(i −1)

N, i ∈N (1.3)

is the angle1 subtended from the positive real axis in the complex plane to the i th power of w . An

important property of the Fourier matrix is that it is unitary :

EH E= EEH = I (1.4)

where I is the identity matrix. It is noticeable that the matrices EH , E⊗ I and (E⊗ I)H =EH ⊗ I are

also unitary.1.1.3 Cir ulant Matri esAn N ×N circulant matrix is a matrix of the form:

C=

c1 c2 . . . cN

cN c1 . . . cN−1

......

. .....

c2 c3 . . . c1

(1.5)

Thus a circulant matrix is completely defined by an ordered set of generating elements c1,c2, . . . ,cN .

It is convenient to define the circulant operator circ(·) that takes as its argument these generating

elements and return the array given by Equation (1.5). Then,

C= circ(c1,c2, . . . ,cN ) (1.6)

Block circulant matrices are defined in a similar way as circulant matrices. In the representation

of Equations (1.5) and (1.6), each entry ck is replaced by the M ×M matrix Ck . The ordered set of

matrices C1,C2, . . . ,CN are called its generating matrices.

If a matrix is both circulant and symmetric it can be written as

C=

(

circ(c1,c2, . . . ,c N2

,c N+22

,c N2

, . . . ,c3,c2) N even

circ(c1,c2, . . . ,c N−12

,c N+12

,c N+12

,c N−12

, . . . ,c3,c2) N odd(1.7)

and necessarily has repeated generating elements.

1ϕi will be later used in a different context in the present report, without any obvious conflict requiring the use of another variablename

1.2 Engine Order Ex itation 31.1.4 Diagonalization of Cir ulant Matri esA circulant matrix C with generating elements c1,c2, . . . ,cN can be diagonalized via the unitary

transformation

EH CE=

λ1 0

λ2

. ..

0 λN

(1.8)

where

λi =

N∑

k=1

ck w (k−1)(i−1), i ∈N (1.9)

are the eigenvalues of C, which depends only on the generating elements ck . All circulant matrices

share the same eigenvectors, which are the N columns of the Fourier matrix E.

ei =1p

N

1,w (i−1),w 2(i−1), . . . ,w (N−1)(i−1)T

=1p

N

1,ejϕi ,e2jϕi , . . . ,ej (N−1)ϕi

T

, i ∈N (1.10)

Similarly, a M ×N block circulant matrix can be block diagonalized via the unitary transformation

(EHN ⊗ IM )C (EN ⊗ IM ) =

Λ1 0

Λ2

. . .

0 ΛN

(1.11)

where 0 and IM are the M ×M zeros and identity matrices and

Λi =

N∑

k=1

Ck w (k−1)(i−1), i ∈N (1.12)

Since (1.11) is a unitary transformation, it preserves the eigenvalues of C. Hence its eigenvalues

are the eigenvalues of the N matrices Λi . If vi is an eigenvector of Λi then the corresponding

eigenvector of C is ui = ei ⊗ vi .1.2 Engine Order Ex itation1.2.1 Mathemati al ModelFlow entering a jet engine must passes through static obstacles, such as struts or stator blades, and

also rotating constituents, such as fans, compressors, and turbines in its path to the exhaust. This

results in a flow slightly upstream and downstream of bladed disks that is spatially non-uniform

in pressure and temperature. Therefore, the rotor disks will encounter a force field that varies

circumferentially relative to the engine casing.

4 Ba kgroundFor example, n evenly-spaced struts slightly upstream or downstream of a bladed assembly will

produce a circumferential variation upon the mean axial gas pressure that is proportional to cosnθ,

where θ is an angular position. Thus a blade rotating through this static pressure field experiences

a force proportional to cosnΩt , whereΩ is the constant angular speed of the bladed disk assembly.

An adjacent blade experiences the same force, but at a constant fraction of time later. This type of

excitation is defined as engine order excitation and n is said to be the order of the excitation.

More precisely, the axial gas pressure of a steady flow through an engine may be described by a

function p (θ) = p (θ +2π), where θ is an angular coordinate measured relative to a fixed origin on

the structure. The pressure field is rotationally periodic and can therefore be expanded in a Fourier

series with spatial harmonics of the form p0 cosnθ. Assuming linear dynamics, the response of the

assembly to each of these harmonics may be analyzed separately.

Let us define the angular position of the i th blade relative to the same origin as

θi (t ) = Ωt +2π

N(i −1), i ∈N

where N is the total number of blades, andN = 1,2, . . . ,N is the set of blade, or sector numbers.

It follows that the total effective force exerted on blade i due to the n th harmonic of the pressure

field can be captured by

F cos

nΩt +2πn

N(i −1)

, i ∈N (1.13)

which gives rise to the following complex form

Fi (t ) = F ejφi ej nΩt , i ∈N (1.14)

It has period T = 2π/nΩ, strength F , and is said to have angular speed Ω. The inter-blade phase

angle is defined by

φi =φ(n)

i = 2πn

N(i −1) = nϕi , i ∈N (1.15)

where ϕi is the angle subtended from blade 1 to blade i and is given by :

ϕi =2π(i −1)

N, i ∈N (1.16)

Equation (1.14) is defined as n th engine order (e.o.) excitation and is used to model the dynamic

loading on the models of bladed disk assemblies presented in this work.1.2.2 Traveling Wave Chara teristi sEquation (1.14) is a function of continuous time t and it is discretized in space via the index i .

This leads up to two interpretations of e.o. excitation : the first one is discrete and the other one

is continuous and these can be visualized in Figure 1.1, which shows a dissection of the excitation

amplitudes along time and sector axes. In the first sense, Equation (1.14) is a discrete temporal

variation of the dynamic loading applied to individual blades. That is, as described just above,

under an e.o. n excitation each sector is forced with strength F and frequency nΩ , but with a fixed

phase difference relative to its nearest neighbors.

1.2 Engine Order Ex itation 5Sector 1 Sector 2 Sector N

−F−F−F

FFF

ttt

F1(t ) F2(t ) FN (t )

. . .

. . .

(a)

Am

plit

ud

e

Time

Sector Number

1 2 3 4 56 7 8 9 10 11

Discrete Temporal Variation

Continuous Spatial Variation (BTW)

Applied Loads (SW, BTW or FTW)

(b)

11 N N. . .. . .

Sector Number

i

−F

0

F

N/n

(c)

Figure 1.1 - Illustration of the discrete temporal and continuous spatial variations of the travelingwave excitation defined by Equation (1.14)

6 Ba kgroundIn the second sense, Equation (1.14) can be viewed as a traveling wave (TW) which means as

a continuous spatial variation of the excitation strength (along the sector axis) that evolves with

increasing time. In this sense, the instantaneous loading applied to individual blades is obtained

by sampling the continuous TW at each sector i ∈ N and, as time evolves, these sampled points

define N time-profiles of the force amplitudes. This interpretation is equivalent to the discrete

temporal one described above.

To explain the TW mathematically, it is convenient to define the following cosinusoidal waveform

with wavelength 2π/ϕk :

Φk (χ) = cos

2π(k −1)

Nχ

= cos(ϕkχ), (1.17)

Then Equation (1.14) can be rewritten as

Fi (t ) = F cos(ϕn+1(i −1)+nΩt ) = FΦn+1(i −1+C t ) (1.18)

which is a harmonic function with a wavelength of 2π/ϕn+1 = N /n (ϕn+1 is the wave number)

and angular frequency nΩ. Equation (1.18) demonstrates that the loading is a TW in the negative

i -direction with speed C = nΩ/ϕn+1 = NΩ/2π. An example plot of this continuous backward

traveling wave (BTW) is shown in Figure 1.1(c).

Then the e.o. excitation applied to the individual blades consists of a wave composed of these

N discrete points, examples of which are presented in Figure 1.2. Interestingly, this gives rise

to discrete SW or even forward traveling wave (FTW) applied dynamic loads even though Equa-

tion (1.18) is strictly a backward traveling waveform relative to the rotating hub. These additional

possibilities arise due to aliasing of the sampled points just as it occurs in elementary signal pro-

cessing theory. The TW nature of the discrete applied loads (SW, BTW, or FTW) depends only on

the value of n relative to N .1.3 Vibrations Chara teristi s of Cy li SystemsThe free vibration characteristics of cyclic systems are discussed next, in addition to forced vi-

bration under the e.o. excitation described in previous section. A prototypical linear model is

introduced in Section 1.3.1, which consists of a cyclic array of N identical, identically coupled

oscillators, each with a single degree of freedom (DOF). Its forced response is considered in Sec-

tion 1.3.2, including a decoupling strategy based on the cyclic symmetry of the system. The details

of its free response are given in Section 1.3.3, which discuss the eigenfrequency characteristics and

normal modes of vibration. Finally, conditions for resonance are given in Section 1.3.4.1.3.1 Prototypi al ModelThe undamped cyclic system represented in Figure 1.3 is considered. It is composed of N identi-

cal single-DOF oscillators, each of mass M , whose transverse displacements are captured by the

variable x i . They are uniformly attached around the circumference of a rigid hub via linear elas-

tic elements of stiffness kb and length L. Coupling between adjacent masses is modelized with

1.3 Vibrations Chara teristi s of Cy li Systems 7F

0

−F

1 2 3 4 5 6 7 8 9 10 i

BTW

BTW

(a) n = 1

F

0

−F

1 2 3 4 5 6 7 8 9 10 i

BTW

SW

(b) n = 5

F

0

−F

1 2 3 4 5 6 7 8 9 10 i

BTW

FTW

(c) n = 9

F

0

−F

1 2 3 4 5 6 7 8 9 10 i

BTW

SW

(d) n = 10

Figure 1.2 - Example plots of applied dynamic loading (represented by the dots) for a model with N =10 sectors and with (a) n = 1,(b) n = 5,(c) n = 9 and (d) n = 10. The BTW engine orderexcitation is represented by the solid lines.

8 Ba kgroundlinear springs of stiffness kc . When x i = 0 for each i ∈ N , we assume that the elastic springs are

unstressed. An individual oscillator, together with the nearest-forward neighbor elastic coupling,

forms one fundamental sector and there are N such sectors in the overall system. Finally, the

system is subjected to e.o. excitation of order n and angular speed Ω, which can be modeled by

Equation (1.14). The linear equation of motion (EOM) for sector i is divided by the inertia term

MMM

L, kb

L, kbL, kb

i −1i

i +1

x i−1x i

x i+1

. . . . . .

kckc

Figure 1.3 - A prototypical linear cyclic system with cyclic boundary conditions x0 = xN and xN+1 = x1

M L and time is rescaled using the new parameter τ=ω0t , where ω0 =p

kb/M is the undamped

natural frequency of a single isolated sector. Then if qi = x i /L, the dynamics of the i th sector are

governed by

q ′′i +qi +ν2(−qi−1+2qi −qi+1) = f ejφi ej nστ, i ∈N (1.19)

where ( · )′ = d ( · )/dτ, ν =p

kc /kb is a nondimensional coupling strength, σ = Ω/ω0 and f =

F/Lkb are the dimensionless angular speed and strength of the e.o. excitation. As the N th oscillator

is coupled to the first, q0 = qN and qN+1 = q1 in Equation (1.19).

The EOM for the overall N -DOF system is obtained by ordering the N coordinates qi into the

vector q= (q1,q2, . . . ,qN )T :

q′′+Kq= fej nστ (1.20)

where f= (f e jφ1 , f e jφ2 , . . . , f e jφN )T is the system forcing vector.

K=

1+2ν2 −ν2 0 . . . 0 −ν2

−ν2 1+2ν2 −ν2 . . . 0 0

0 −ν2 1+2ν2 . . . 0 0...

......

. . ....

...

0 0 0 . . . 1+2ν2 −ν2

−ν2 0 0 . . . −ν2 1+2ν2

(1.21)

The N ×N matrix K is a circulant matrix and can be written as

K= circ(1+2ν2,−ν2,0, . . . ,0,−ν2) (1.22)

If there is no coupling, which means ν = 0, K is diagonal and Equation (1.20) represents a decou-

pled set of N harmonically forced, single-DOF oscillators.

1.3 Vibrations Chara teristi s of Cy li Systems 91.3.2 For ed Response Under Engine Order Ex itationThe steady-state response of Equation (1.20) can be obtained using standard techniques and is

given by

qs s (τ) = (K−n 2σ2I)−1fe j nστ (1.23)

where I is the N ×N identity matrix. However, this requires inversion of the impedance matrix

K−n 2σ2I, which can be computationally expensive for a large number of sectors. In what follows,

a transformation based on the cyclic symmetry of the system is exploited to fully decouple the

single, N -DOF system to a set of N , single-DOF oscillators from which the steady-state response

can easily be obtained. The procedure is similar to the usual modal analysis from vibration theory.1.3.2.1 Existen e of a Traveling Wave ResponseSince the e.o. excitation can be viewed as a TW, it is natural to expect steady-state solutions of the

same form, that is,

q s si (τ) =Aejφi ej nστ, i ∈N (1.24)

In Equation (1.24), it is assumed that each sector responds with the same amplitude A, but with a

constant phase difference relative to its nearest neighbors, and together all N such solutions form

a TW response among the sectors. By using this trial solution into Equation (1.19), the expression

of the amplitude A is found to be

A =f

1+2ν2(1− cosϕn+1)− (nσ)2(1.25)

where the identity

φi±1−φi =±ϕn+1 (1.26)

has been employed. It follows that one of the N natural frequencies of the coupled system is

ωn+1 =p

1+2ν2(1− cosϕn+1), corresponding to mode p = n+1. Equation (1.26) highlights a fun-

damental result when a linear cyclic system with nearest-neighbor elastic coupling is subjected to

e.o. excitation of order n : mode n +1 is excited.1.3.2.2 Modal AnalysisIt has been shown in Section 1.1 that circulant matrices, such as the stiffness matrix defined by

Equation (1.21), can be diagonalized via a unitary transformation involving the Fourier matrix

(see Equation (1.9)). In what follows this property is exploited to fully decouple the governing

equations of motion given by Equation (1.20). To this end, the change of coordinates

q(τ) = Eu(τ) =

N∑

k=1

ek u k (τ) or qp (τ) = eTp u(τ) (1.27)

is considered where E is the N×N complex Fourier matrix, ep is its p th column, and u= (u 1,u 2, . . . ,u N )T

is a vector of modal or cyclic coordinates. Substituting Equation (1.27) into Equation (1.20) and

10 Ba kgroundmultiplying from the left by EH yields

u′′+EH KEu= EH fej nστ (1.28)

which can be written as

u ′′1u ′′2

...

u ′′N

+

ω21 0

ω22

.. .

0 ω2N

u 1

u 2

...

u N

=

eH1 f

eH2 f...

eHN f

ej nστ (1.29)

For each p ∈N the system natural frequencies follow from Equation (1.9) :

ω2p =

ωp

ω0

2

= 1+2ν2−ν2(w (p−1)+w (N−1)(p−1))

= 1+2ν2(1− cosϕp )

(1.30)

The identity w (p−1) +w (N−1)(p−1) = 2 cosϕp has been employed. The overbar indicates that the

frequencies are in dimensionless form.

Equation (1.29) is a decoupled set of N , single-DOF harmonically forced oscillators and, in the

steady-state, the p th modal response is

u s sp (τ) =

eHp f

ω2p − (nσ)2

ej nστ, p ∈N (1.31)

In physical space, the steady state response of sector i can be obtained from the transformation of

Equation (1.27) and is given by q s si(τ) = eT

ius s (τ)

q s si (τ) =

N∑

k=1

1p

Nej (i−1)ϕk u s s

k(τ)

=1p

N

N∑

k=1

eHk

f

ω2k− (nσ)2

ej (i−1)ϕk ej nστ

(1.32)

which reflects that the total system response is simply a superposition of individual modal re-

sponses. Equation (1.32) shows that there are N possible resonances which appear if the excita-

tion frequency matches a system natural frequency. However, the p th modal forcing term reduces

to

eHp f=

N∑

k=1

1p

Nw (k−1)(p−1) f w n(k−1)

=fp

N

N∑

k=1

w (k−1)(n+1−p )

=

¨ pN f n +1−p =m N , m ∈Z

0 otherwise

(1.33)

from which it follows that only a single mode is excited. Indeed, the force vector f is orthogonal to

all but one of the modal vectors ep . Given an e.o. n ∈N, and since p ∈N , the excited mode is

p = n mod N +1 (1.34)

1.3 Vibrations Chara teristi s of Cy li Systems 11Finally, Equation (1.32) can be written as

q s si (τ) =

f

ω2n+1− (nσ)2

ejφi ej nστ i ∈N (1.35)

which is in agreement with the results of the previous section.1.3.3 Eigenfrequen y Chara teristi s and Normal Modes of VibrationThe dimensionless natural frequencies follow from Equation (1.30). It is captured by

ωp =ωp

ω0=p

1+2ν2(1− cosϕp ) (1.36)

which clearly exhibit the effect of the coupling. For ν = 0, we recover ωp = 1, or ωp = ω0 in

dimensional form. In this case the sectors are dynamically isolated and each has the same natural

frequency. For nonzero coupling, there will be repeated natural frequencies, a degeneracy that is

due to the circulant structure of K. This is manifested in the cyclic term cosϕp . For instance, the

natural frequency corresponding to mode p = 1 is distinct, but the remaining natural frequencies

appear in repeated pairs, except for the case of even N , in which case the p = (N +2)/2 frequency

is also distinct. It can be shown that there are (N − 1)/2 repeated natural frequencies if N is odd

and (N −2)/2 for even N .

In what follows mode shapes are characterized by investigating the free response of the system,

and it is shown that they are of the SW, BTW, or FTW variety. The free response of the system in

its p th mode of vibration can be described by q(p )(τ) = a p ep ej ωpτ , where a p is a modal amplitude.

Then, the free response of sector i is deduced :

q(p )

i(τ) = a p cos(ϕp (i −1)+ ωpτ) = a pΦp (i −1+Cpτ) (1.37)

where Cp = ωp/ϕp and the function Φk (χ) is defined by Equation (1.17). Equation (1.37) is a func-

tion of continuous time τ and it is discretized according to the sector number i . In this way, it

is endowed with the same discrete temporal and continuous spatial duality that was described in

Section 1.2.2. The propagating waveform is strictly a BTW in the negative i -direction with wave-

length 2π/ϕp = N /(p − 1) and speed Cp . However, depending on the value of p , this gives rise to

SW, BTW, or FTW mode shapes, a property that follows similarly from the features described in

Figure 1.2.

For the special case of p = 1 in Equation (1.37), each sector behaves identically with the same

amplitude and the same phase sinceϕ1 = 0. An additional special case occurs when p = (N+2)/2 if

N is even. Then ϕ(N+2)/2 =π and each sector has the same amplitude but adjacent sectors oscillate

out-of-phase. The remaining mode shapes correspond to repeated natural frequencies and are

either BTWs or FTWs.

Figure 1.4 illustrates the normal modes of free vibration for a model with N = 100 sectors. In

this figure, the radial lines represents sector displacements; those appearing outside (resp. inside)

the hub are to be interpreted as being positively (resp. negatively) displaced relative to their zero

positions. Modes 1 and 51 are SWs and modes 2-50 (resp. 52-100) consist of backward (resp.

12 Ba kground8, 94 9, 93 10, 92 11, 91 12, 90 13, 89

15, 87 16, 86 17, 85 18, 84 19, 83 20, 82 21, 81

22, 80 23, 79 24, 78 25, 77 26, 76 27, 75 28, 74

29, 73 30, 72 31, 71 32, 70 33, 69 34, 68

14, 88

35, 67

36, 66 37, 65 38, 64 39, 63 40, 62 41, 61 42, 60

43, 59 44, 58 45, 57 46, 56 47, 55 48, 54 49, 53

50, 52 51

1 2, 100 3, 99 4, 98 5, 97 6, 96 7, 95

Figure 1.4 - Normal modes of free vibration for a model with 100 sectors. Mode 1 and 51 consists ofSWs, mode 2-50 of BTWs and 52-100 of FTWs

forward) traveling waves. In this figure, it can be seen that the mode shapes are harmonic in the

circumferential direction. For bladed disk assemblies, this leads to nodal lines across the disk

called nodal diameters, and the system mode shapes are referred to as nodal diameter modes.

The number of nodal diameters (n.d.) can be clearly identified in Figure 1.4. For example, modes

4 and 98 feature 3 n.d.

1.4 Centrifugal Pendulum Vibration Absorber 131.3.4 Resonan e Stru tureIn light of Equation (1.35), a system resonance occurs when the excitation frequency of order n

matches the natural frequency corresponding to mode n + 1 that is, if nσ = ωn+1 or nΩ = ωn+1

in dimensional form. In order to identify these resonance, it is convenient to use a Campbell dia-

gram. An example of such diagram is presented in Figure 1.5(a) for a model with N = 10, ν = 0.5,

and for engine orders n ∈ N . In this figure, the natural frequencies are plotted in terms of the

dimensionless rotor speed and several e.o. lines nσ are superimposed. Resonance points are sit-

uated at the intersection of the e.o. line nσ and the eigenfrequency locus corresponding to mode

n + 1. They are indicated by black dots in Figure 1.5(a). The corresponding frequency response

curves are shown in Figure 1.5(b) for an excitation strength f = 0.01.

1.4 Centrifugal Pendulum Vibration AbsorberWhen an engineering structure experiences unwanted levels of vibration due to periodic excita-

tions acting on its constituent parts it may be impractical to change the makeup of the system to

improve its vibratory characteristics, or to eliminate the source of the excitation. In these cases

tuned vibration absorbers offer a possible solution. They consist of auxiliary components that are

attached to a primary system to eliminate, or otherwise reduce its steady-state motions. This is

done through a particular choice of absorber parameters.

In this work, we consider vibration absorbers belonging to the Centrifugal Pendulum Vibration

Absorber (CPVA) variety. Absorbers of this type are suitable for systems with rotating components,

such as bladed disk assemblies, which are characterized by varying speeds and forces that occur

at orders of rotation. Examples of CPVAs are shown in Figure 1.6. The system illustrated in Fig-

ure 1.6(a) captures the essential features of a typical CPVA. It is made of a rigid rotor (primary

system) with moment of inertia J and radius R , in rotation about a fixed axis at O. The primary

system is harmonically excited by a torque of the form T (t ) = T0+ T ej nΩt , where T is the strength

of the excitation, n is its order, Ω is the speed of the rotor, and T0 is the mean torque. A pendulum

absorber of length r and mass m is attached to the periphery of the rotor.

The absorber’s natural frequency is proportional to the rotor speed. That is,

ωn =

Ç

R

rΩ= nΩ (1.38)

where n =p

R/r is defined as the linear tuning order of the absorber. The steady-state torsional

oscillations of the rotor can be eliminated completely by setting the natural frequency of the ab-

sorber to that of the excitation. This gives rise to a tuning condition n = n which is independent

of the rotor speed. In this way, the absorber tuning is effective over the full range of possible ro-

tor speeds. Similar statements can be made for the more general-path absorber system shown in

Figure 1.6(b).

14 Ba kground

σ

ωp

1 e.o.

2 e.o.3 e.o.4 e.o.. . .N e.o.

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

(a)

σ

log |X |

0 0.2 0.4 0.6 0.8 1 1.2-3

-2

-1

0

1

2

(b)

Figure 1.5 - (a) Campbell diagram and (b) corresponding frequency response curves for N = 10, ν =0.5, f = 0.01 and n = 1, . . . ,N1.5 Lumped-Parameter Bladed Disk Model with CPVA

This intern work follow logically the research work led by B. J. Olson in his thesis [OLS 06]. In his pa-

per, he investigates the use of CPVA to suppress the steady-state vibrations of a rotating bladed disk

assembly under e.o. excitation. For this purpose he has employed a lumped-parameter model.

It is shown schematically in Figure 1.5. It consists of a nominally-cyclic array of N blades, and

each is modeled by a simple pendulum of length L i and mass M i . These are uniformly attached

around the periphery of a rigid disk of radius H , which rotates at a constant speed about a fixed

axis through O. The single-mode flexural stiffness of blade i is modeled with a linear torsional

spring of stiffness k bi , and the elastic inter-blade coupling is captured by linear springs of stiff-

1.5 Lumped-Parameter Bladed Disk Model with CPVA 15T (t )

RO

r m

J

φ

(a) Circular Path CPVA

T (t )

O

m

J

R0

R(S)

S

(b) General Path CPVA

Figure 1.6 - Order-tuned Centrifugal Pendulum Vibration Absorbers

ness k ci . The blades are fitted with identical vibration absorbers, which consist of particle masses

m i riding on user-specified paths. The system is divided in N fundamental sectors, each of them

composed of a blade fitted with a general-path absorber and a portion of the rigid disk. Thus, the

O

Ωt

H

k bi−1 k b

i

k bi+1

k ci

k ci−1

m i−1

m i

m i+1

M i−1

M i

M i+1

Figure 1.7 - Lumped-parameter model of a rotating bladed disk assembly investigated in B. J. Olson’sthesis [OLS 06]

dynamics of each sector model is described with two DOF: one for the blade and one for the ab-

sorber. In this intern work, the same type of lumped-parameter model of bladed disk assembly

fitted with CPVA is investigated. But flexibility is added to the system as we let one part of the rigid

disk be free to oscillate. It means that one more DOF is necessary to model the dynamics of one

sector. This 3-DOF sector model is presented and investigated in detail in the following chapters.

An overview of relevant theoretical background has been given. Key mathematical concepts were

presented first, including the Kronecker product, the Fourier matrix, circulant matrices and their

diagonalization. A mathematical model for e.o. excitation was then developed. It was described

as a discrete temporal variation of dynamic loading applied to individual blades as well as a TW,

16 Ba kgroundwhich is a continuous spatial variation of the excitation strength. The vibration characteristics

of a prototypical cyclic system under e.o. excitation of order n was given in Section 1.3. It was

shown by using modal analysis to decouple the equations of motion that for such a system only a

single mode is excited, which is mode n + 1. The basic theory of order-tuned dynamic vibration

absorbers was subsequently highlighted. A key feature is that the absorbers are tuned to a given

order of rotation, rather than to a fixed frequency. Finally, a lumped-parameter model of bladed

disk assembly fitted with CPVA was introduced. Mathematical models are developed next for CPVA

fitted to higher-fidelity lumped-parameter bladed disk model under e.o. excitation.

2Mathemati al Model

A general mathematical model is developed for the bladed disk assemblies of interest fitted with

arbitrary-path vibration absorbers, from which a number of specific models to be considered in

subsequent chapters are derived. The chapter begins with the description of the geometry of an ar-

bitrary absorber path in Section 2.1, which forms a kinematic model for a general-path absorber,

and the desired model for a nominally-cyclic bladed disk assembly fitted with such absorbers is

formulated in Section 2.2. This general model forms the basis for all of the analysis to follow. The

EOMs for the case of circular-path absorbers are then deduced in Section 2.3, and these are sub-

sequently linearized for small blade and absorber responses and cast into a form that is employed

in Chapter 3.2.1 Geometry of an Arbitrary Absorber PathThis section describes in general terms the geometry of the absorber paths, which prescribe their

positions relative to the blades. Key results are relationships between the path variables, which are

described in Section 2.1.1 and an expression for the tangent angle of each path, which is given in

Section 2.1.2. These are employed in Section 2.2, where the EOMs for a bladed disk assembly fitted

with general-path absorbers are derived.

The geometry of a general absorber path is illustrated in Figure 2.1. A point P of the path is located

relative to a basepoint O by the radius vector Ri (Si ) = Ri (Si )eRi

, where Si is the curvilinear abscissa

along the path relative to an origin at its vertex V. The angle subtended by Si is denoted by ϑi (Si )

and the distance from the basepoint to the path vertex P is given by Ri (0) = R0i = αi d i + γi d i .

An alternative to this parametrization is the use of variables (ρi ,ψi ) instead of (Ri ,ϑi ). A circular

path is obtained by restricting ρi = γi d i = constant, which is indicated by the dashed lines in

Figure 2.1. Finally, the paths are generally assumed to be symmetric about their vertices at Si = 0,

which means that Ri (Si ) = Ri (−Si ), or that Ri is an even function in Si . Relationships between the

path variables Ri , Si , and ϑi are derived next and the expression of the angle ςi between the radius

and tangent unit vectors is subsequently deduced.

18 Mathemati al Model

γi L i

αi L i

R0i

ψi

ϑiδϑi

Ri Ri +δRi

łi

βi

ςi

π/2− ςi

O

N

V

P

M

Q

eθi

eLi

eϑi

eRi

General path

Circular path

eSi

eϑi

V P =S i

PQ = δS i

PQ = δc i

OV = R0i

OP =Ri

OQ = Ri +δRi

ON =αi d i

N V = γi d i

N P = ρi

Figure 2.1 - Geometry of a general absorber path.2.1.1 Fundamental Relationship Between Path VariablesLet P be an arbitrary position along the general path corresponding to Ri , Si , ϑi > 0 and let Q

correspond to Ri + δRi , Si + δSi , and ϑi + δϑi , where δRi , δSi , δϑi are taken to be positive. Then

relationships between the path variables can be obtained by considering the triangle PQM in Fig-

ure 2.1. If δc i = PQ, it follows from (PQ)2 = (PM)2+(MQ)2 that

δc i

δϑi

2

=

Ri

sinδϑi

δϑi

2

+

δRi

δϑi

+Ri sin(δϑi /2)sinδϑi /2

δϑi /2

2

.

Then,

δSi

δϑi

2

=

δSi

δc i

δc i

δϑi

2

=

δSi

δc i

2

Ri

sinδϑi

δϑi

2

+

δRi

δϑi

+Ri sin(δϑi /2)sinδϑi /2

δϑi /2

2

.

In the limit as Q→ P or δϑi → 0 it follows that

d Si

dϑi

2

= (1)2

(Ri (1))2+

d Ri

dϑi

+Ri (0)(1)

2

,

or

(d Si )2 = (Ri dϑi )

2+(d Ri )2. (2.1)

2.1 Geometry of an Arbitrary Absorber Path 19This is the fundamental relationship between path variables Si , Ri and ϑi . By dividing through by

(dϑi )2 and (d Si )

2, we obtain

d Si

dϑi

=

È

R2i+

d Ri

dϑi

2

, (2.2a)

Γi = Ri

dϑi

d Si

=

È

1−

d Ri

d Si

2

. (2.2b)

Then the angle subtended by Si is given by the integral

ϑi (Si ) =

∫ Si

0

Γi (χ)

Ri (χ)dχ. (2.3)2.1.2 Angle Between Radius Ve tor and Tangent

In Figure 2.1 consider the ratio

tanβi =MP

MQ=

Ri sinδϑi

δRi +2Ri sin2(δϑi /2)=

Risinδϑi

δϑi

δRi

δϑi+2Ri sin(δϑi /2)

sinδϑi /2δϑi /2

, (2.4)

where the relationships MP= Ri sinδϑi and

MQ=OQ−OM

= Ri +δRi −Ri cosδϑi = δRi +Ri (1− cosδϑi ) = δRi +2Ri sin2(δϑi /2)

have been employed. In the limit as Q→ P or δϑi → 0, βi → ςi and it follows from Equation (2.4)

that

tanςi = limδϑi→0

tanβi =Ri (1)

d Ri

dϑi+2Ri (0)(1)

= Ri

dϑi

d Ri

. (2.5)

Expressions for sinςi and cosςi , which will be needed in next sections, are obtained as follows :

d Si

d Ri

=d Si

dϑi

dϑi

d Ri

=

È

R2i+

d Ri

dϑi

2 dϑi

d Ri

=

È

1+

Ri

dϑi

d Ri

2

=p

1+ tan2 ςi

= 1/cosςi .

From this result, together with Equation (2.5), it follows that

sinςi = tanςi cosςi = Ri

dϑi

d Ri

d Ri

d Si

= Ri

dϑi

d Si

.

20 Mathemati al ModelTo summarize and in light of Equation (2.2b),

Γi = sinςi = Ri

dϑi

d Si

=

È

1−

d Ri

d Si

2

, (2.6a)

cosςi =d Ri

d Si

, (2.6b)

tanςi = Ri

dϑi

d Ri

. (2.6c)

We now concentrate ourselves more particularly on the governing EOMs for a bladed disk assem-

bly fitted with general-path absorbers, from which a number of specific models are deduced.2.2 Bladed Disk Fitted with General-Path AbsorbersIn this section an idealized mathematical model of a bladed disk assembly under engine order

excitation is developed. Each blade on the rotating assembly is fitted with a CPVA and the gov-

erning nonlinear EOMs for the overall coupled system are derived. The model is described in Sec-

tion 2.2.1, followed by the development of the system kinetic and potential energy in Section 2.2.2.

The general nonlinear EOMs are subsequently derived in Section 2.2.3 by employing the method

of Lagrange and a particular two-parameter family of paths, which are used in Chapter 4, is de-

scribed in Section 2.2.42.2.1 Lumped-Parameter ModelA lumped-parameter model of a rotating bladed disk assembly is shown schematically in Fig-

ure 2.2. It consists of a cyclic array of N sectors, each of them being one blade and one absorber

plus the corresponding segment of the disk. The disk part of one such substructure is composed

of a rigid part of radius H which rotates at a constant speedΩ about a fixed axis through O (in what

follows, we refer to it as the rotor). A second part, more flexible, vibrates about a fixed point D of

the periphery of the rigid disk (we refer to it as the disk). This vibrating component is modeled by

a pendulum of length d i and mass M di . A second pendulum of length L i and mass M b

i , attached

to the first at vertex B, captures blade dynamics. There are N such double-pendula systems, which

are uniformly attached around the periphery of the rotor. Thus, in this model, the disk and blade

DOFs are defined as pendulums. We might also consider other models where the disk DOF can be

designed to oscillate in translational direction.

As indicated in Figure 2.2(b), the flexural stiffness of blade i is modeled with a linear torsional

spring of stiffness K bi

, and the elastic inter-blade coupling due to shrouds and aerodynamic ef-

fects is captured by linear springs of stiffness k bi . Similarly, a torsional spring of stiffness K d

i and

linear springs of stiffness k di

are defined for the disk DOF. The translational coupling springs con-

nect adjacent blades (resp. disks) at a distance b (resp. h) relative to their attachment points to the

disk (resp. rotor). It is assumed that the springs are unstressed when the disks and blades are in a

purely radial configuration, that is, when ϕi = 0 and θi = 0 for each i ∈ N . The blades are fitted

with identical vibration absorbers, which consist of particle masses M ai riding on user-specified

2.2 Bladed Disk Fitted with General-Path Absorbers 21

O

Ωt

H

eHi

eΩi

d i

ϕi

L i

θi

eϕ

i

edi M b

i

eθi eLi

M di

B

D

M ai

Fi (t )

(a)

O

i +1

i

i −1

K di+1 K d

i

K di−1

K bi+1

K bi

K bi−1k d

i

k di−1

k bi

k bi−1

h

b

(b)

Figure 2.2 - Lumped parameter model of a rotating bladed disk assembly : (a) Parametrization for asingle sector model; (b) Modelization of inter-disk and inter-blade couplings and flexuralstiffnesses.

paths. Figure 2.3 shows a schematic of the i th blade fitted with a general-path absorber, which

together with a portion of the disk composes the i th fundamental sector. If we require that point O

in Figure 2.1 coincides with the attachment point of blade i to the disk, and that the unit vectors eLi

and eθi are aligned and rotate with the blade as shown in Figure 2.3, then the mathematical details

for the i th absorber path are given in Section 2.1. Linear viscous damping models are employed

S i

Ri

Ri

M a

ρi

ψi

d i

N

V

ϑi

Mb

d i

a i

O

Figure 2.3 - Sector model of a bladed disk assembly fitted with a general-path absorber.

22 Mathemati al Modelto model the system damping. The disk and inter-disk (resp. blade and inter-blade) damping is

captured by linear torsional and translational dampers, which are not shown in Figure 2.2 and Fig-

ure 2.3, with constants C di and c d

i (resp. C bi and c b

i ). The effective translational absorber damping

constant is denoted by C ai

, where a bar has been added to the superscript a to distinguish it from

the torsional damping constant C ai to be employed in the linearized model in Section 2.3.

Finally, the blades are harmonically excited in the transverse sense by e.o. excitation of order n

and the model described in Section 1.2 is considered for this purpose. The e.o. is restricted such

that n ∈N .2.2.2 Kineti and Potential EnergyThe total kinetic energy of the system is that of the N disks, blades and their attendant absorbers

and is given by

T =1

2

N∑

i=1

M di ‖vd

i ‖2+1

2

N∑

i=1

M bi ‖vb

i ‖2+1

2

N∑

i=1

M ai ‖va

i ‖2, (2.7)

where

vdi =HΩeΩi +d i (Ω+ ϕi )e

ϕ

i

vbi =HΩeΩi +d i (Ω+ ϕi )e

ϕ

i + L i (Ω+ ϕi + θi )eθi

vai =HΩeΩi +d i (Ω+ ϕi )e

ϕ

i +Ri (Ω+ ϕi + θi )eϑi + Si eS

i

, i ∈N

are the absolute velocities of the i th disk, blade and absorber masses, respectively. The correspond-

ing speeds are then derived

‖vdi ‖2 =H 2Ω2+d 2

i (Ω+ ϕi )2+2HΩd i (Ω+ ϕi )cosϕi ,

‖vbi ‖2 =H 2Ω2+d 2

i (Ω+ ϕi )2+ L2

i (Ω+ ϕi + θi )2

+2HΩd i (Ω+ ϕi )cosϕi

+2HΩL i (Ω+ ϕi + θi )cos(ϕi +θi )

+2d i L i (Ω+ ϕi )(Ω+ ϕi + θi )cosθi ,

‖vai ‖2 =H 2Ω2+d 2

i (Ω+ ϕi )2+R2

i (Ω+ ϕi + θi )2+ S2

i

+2HΩd i (Ω+ ϕi )cosϕi

+2HΩRi (Ω+ ϕi + θi )cos(ϕi +θi +ϑi )

+2HΩSi

Γi cos(ϕi +θi +ϑi )+d Ri

d Si

sin(ϕi +θi +ϑi )

+2d i Ri (Ω+ ϕi )(Ω+ ϕi + θi )cos(θi +ϑi )

+2d i Si (Ω+ ϕi )

Γi cos(θi +ϑi )+d Ri

d Si

sin(θi +ϑi )

+2Ri Si (Ω+ ϕi + θi )Γi ,

2.2 Bladed Disk Fitted with General-Path Absorbers 23where the expressions given by Equation (2.6) as well as the following products have been em-

ployed

eΩi · eϕ

i = cosϕi

eΩi · eθi = cos(ϕi +θi )

eΩi · eϑi = cos(ϕi +θi +ϑi )

eΩi · eSi = Γi cos(ϕi +θi +ϑi )+

d Ri

dSisin(ϕi +θi +ϑi )

eϕ

i · eθi = cosθi

eϕ

i · eϑi = cos(θi +ϑi )

eϕ

i · eSi = Γi cos(θi +ϑi )+

d Ri

dSisin(θi +ϑi )

eϑi · eSi = Γi .

Ignoring gravitational effects, the system potential energy arises only from the flexural stiffness of

the blades and elastic coupling among the sectors. It is given by

V =1

2

N∑

i=1

K di ϕ

2i +

1

2

N∑

i=1

K bi θ

2i +

1

2

N∑

i=1

k di h2(ϕi+1−ϕi )

2

+1

2

N∑

i=1

k bi

d (ϕi+1−ϕi )+b (θi+1−θi )2

(2.9)

where θN+1 = θ1. The coupling elements are meant to capture only the basic flexibility between

adjacent blades and hence their nonlinear kinematic contributions have been neglected in Equa-

tion (2.9). This approximation is done independently of any assumptions on the blade amplitudes

and does not imply small blade motions.2.2.3 Equations of Motion2.2.3.1 Dimensional FormThe EOMs are deduced from the kinetic and potential energy terms T and V by employing La-

grange’s method with the generalized coordinates q(a )

i =ϕi , q(b )

i = θi and q(c )

i =Si :

d

d t

∂ T

∂ q(k )

i

!

+∂ V

∂ q(k )

i

− ∂ T

∂ q(k )

i

=Q(k )i , k = a,b,c. (2.10)

The i th set of generalized forces arises from the e.o. excitation and the linear viscous damping.

They are

Q(a )

i=−C d

i ϕi +C bi θi +C a

i

RiΓi +d i

Γi sin(θi +ϑi )+d Ri

d Si

cos(θi +ϑi )

Si

− c bi−1bd i (θi − θi−1)− c b

i bd i (θi − θi+1)− (c di−1h2+ c b

i−1d 2i )(ϕi − ϕi−1)

− (c di h2+ c b

i d 2)(ϕi − ϕi+1)+ F0(L i +d i )ejφi ej nΩt ,

Q(b )i =−C b

i θi +C ai RiΓi Si − c b

i−1b 2(θi − θi−1)− c bi b 2(θi − θi+1)

− c bi−1bd i (ϕi − ϕi−1)− c b

i bd i (ϕi − ϕi+1)+ F0L i ejφi ej nΩt ,

Q(c )i =−C a

i Si .

(2.11)

24 Mathemati al ModelThen the governing EOMs for the i th sector follow from Equation (2.10) and take the form

• ϕi -dynamics

M di d 2

i ϕi +C di ϕi −C b

i θi −C ai

RiΓi +d i

Γi sin(θi +ϑi )+d Ri

d Si

cos(θi +ϑi )

Si

+K di ϕi +M d

i d i HΩ2 sinϕi

+M bi

d 2i ϕi +L2

i (ϕi + θi )+d i L i cosθi (2ϕi + θi )−d i L i sinθi (2Ω+2ϕi + θi )θi

+d i HΩ2 sinϕi + L i HΩ2 sin(ϕi +θi )

+M ai

d 2i ϕi +R2

i (ϕi + θi )+d i Ri cos(θi +ϑi )(2ϕi + θi )

+d i

Γi cos(θi +ϑi )+d Ri

dSis i n(θi +ϑi )

Si +RiΓi Si

−d i Ri sin(θi +ϑi )(2Ω+2ϕi + θi )θi +2Rid Ri

dSi(Ω+ ϕi + θi )Si

+2d i

d Ri

dSicos(θi +ϑi )−Γi sin(θi +ϑi )

(Ω+ ϕi + θi )Si

+d id

dSi

Γcos(θi +ϑi )+d Ri

dSisin(θi +ϑi )

Si2+

d (RiΓi )

dSiSi

2

+d i HΩ2 sinϕi +Ri HΩ2 sin(ϕi +θi +ϑi )

+(k di−1h2+k b

i−1d 2i )(ϕi −ϕi−1)+ (k

di h2+k b

i d 2i )(ϕi −ϕi+1)

+k bi−1bd i (θi −θi−1)+k b

i bd i (θi −θi+1)

+ (c di−1h2+ c b

i−1d 2i )(ϕi − ϕi−1)+ (c

di h2+ c b

i d 2i )(ϕi − ϕi+1)

+ c bi−1bd i (θi − θi−1)+ c b

i bd i (θi − θi+1) = F0(L i +d i )ejφi ej nΩt ,

(2.12a)

• θi -dynamics

M bi L2

i θi +M bi L2

i ϕi +M bi d i L i cosθi ϕi +C b

i θi −C ai RiΓi Si

+M bi d i L i sinθi (Ω+ ϕi )

2+K bi θi +M b

i L i HΩ2 sin(ϕi +θi )

+M ai

R2i (ϕi + θi ) +d i Ri cos(θi +ϑi )ϕi +RiΓSi

+d i Ri sin(θi +ϑi )(Ω+ ϕi )2+2Ri

d Ri

dSi(Ω+ ϕi + θi )Si

+d (RiΓ)

dSiSi

2+Ri HΩ2 sin(ϕi +θi +ϑi )

+k bi−1bd i (ϕi −ϕi−1)+k b

i bd i (ϕi −ϕi+1)

+k bi−1b 2(θi −θi−1)+k b

i b 2(θi −θi+1)

+ c bi−1bd i (ϕi − ϕi−1)+ c b

i bd i (ϕi − ϕi+1)

+ c bi−1b 2(θi − θi−1)+ c b

i b 2(θi − θi+1) = F0L i ejφi ej nΩt ,

(2.12b)

• Si -dynamics

M ai Si +M a

i d i

Γi cos(θi +ϑi )+d Ri

d Si

s i n(θi +ϑi )

ϕi +M ai RiΓi (ϕi + θi )+C a

i Si

−M ai Ri

d Ri

d Si

(Ω+ ϕi + θi )2+M a

i HΩ2

Γi sin(ϕi +θi +ϑi )−d Ri

d Si

cos(ϕi +θi +ϑi )

+M ai d i (Ω+ ϕi )

2

Γi sin(θi +ϑi )−d Ri

d Si

cos(θi +ϑi )

= 0.

(2.12c)

2.2 Bladed Disk Fitted with General-Path Absorbers 252.2.3.2 Dimensionless FormIt is advisable to work with a dimensionless form of EOMs. This is done by restricting d i = d ,

M di=M d , and K d

i= K d for all i ∈N . The undamped natural frequency of a single isolated disk is

defined as

ω0 =

r

K d /d 2

M d. (2.13)

Time is rescaled according to τ = ω0t . Then if s i = Si /d and ri = Ri /d denote respectively the

i th nondimensional arc and radial length, a dimensionless form of the EOMs follows by divid-

ing Equations (2.12a) and (2.12b) through by the term M d d 2ω20 and Equation (2.12c) by the term

M d dω20. They are given by

• ϕi -dynamics

ϕ′′i +ξdi ϕ′i −ξb

i θ′i −ξa

i

riΓi +Γi sin(θi +ϑi )+d ri

d s i

cos(θi +ϑi )

s ′i +ϕi +δσ2 sinϕi

+µbi

ϕ′′i +ρ2i (ϕ′′i +θ

′′i )+ρi cosθi (2ϕ′′i +θ

′′i )−ρi sinθi (2σ+2ϕ′i +θ

′i )θ′i

+δσ2 sinϕi +ρiδσ2 sin(ϕi +θi )

+µai

ϕ′′i +r 2i (ϕ′′i +θ

′′i )+ ri cos(θi +ϑi )(2ϕ′′i +θ

′′i )

+

Γi cos(θi +ϑi )+d ri

d s isin(θi +ϑi )

s ′′i + riΓi s ′′i−ri sin(θi +ϑi )(2σ+2ϕ′i +θ

′i )θ′i +2ri

d ri

d s i(σ+ϕ′i +θ

′i )s′i

+2

d ri

d s icos(θi +ϑi )−Γi sin(θi +ϑi )

(σ+ϕ′i +θ′i )s′i

+ d

d s i

Γi cos(θi +ϑi )+d ri

d s isin(θi +ϑi )

s ′2i +d (riΓi )

d s is ′2i

+δσ2 sinϕi + riδσ2 sin(ϕi +θi +ϑi ))

+ξa ci−1(ϕ

′i −ϕ′i−1)+ξ

a ci (ϕ

′i −ϕ′i+1)+ξ

bci−1(θ

′i −θ′i−1)+ξ

bci (θ

′i −θ′i+1)

+νa 2i−1(ϕi −ϕi−1)+ν

a 2i (ϕi −ϕi+1)+ν

b 2i−1(θi −θi−1)+ν

b 2i (θi −θi+1) = fϕejφi ej nστ,

(2.14a)

• θi -dynamics

µbi ρ

2i θ′′i +µ

bi ρ

2i ϕ′′i +µ

bi ρiϕ

′′i cosθi +ξ

bi θ′i −ξa

i riΓi s ′i +µbi ρi sinθi (σ+ϕ

′i )

2

+λ2i θi +µ

bi ρiδσ

2 sin(ϕi +θi )

+µai

r 2i (ϕ′′i +θ

′′i ) +ri cos(θi +ϑi )ϕ

′′i + riΓi s ′′i

+ri sin(θi +ϑi )(σ+ϕ′i )

2+2rid ri

d s i(σ+ϕ′i +θ

′i )s′i

+d (ri Γi )

d s is ′2i + riδσ2 sin(ϕi +θi +ϑi )

+ξbci−1(ϕ

′i −ϕ′i−1)+ξ

bci (ϕ

′i −ϕ′i+1)+ξ

c ci−1(θ

′i −θ′i−1)+ξ

c ci (θ

′i −θ′i+1)

+νb 2i−1(ϕi −ϕi−1)+ν

b 2i (ϕi −ϕi+1)+ν

c 2i−1(θi −θi−1)+ν

c 2i (θi −θi+1) = f θejφi ej nστ,

(2.14b)

26 Mathemati al Model• s i -dynamics

µai s ′′i +µ

ai

Γi cos(θi +ϑi )+d ri

d s i

sin(θi +ϑi )

ϕ′′i +µai riΓi (ϕ

′′i +θ

′′i )+ξ

ai s ′i

−µai ri

d ri

d s i

(σ+ϕ′i +θ′i )

2+µai δσ

2

Γi sin(ϕi +θi +ϑi )−d ri

d s i

cos(ϕi +θi +ϑi )

+µai (σ+ϕ

′i )

2

Γi sin(θi +ϑi )−d ri

d s i

cos(θi +ϑi )

= 0,

(2.14c)

where (·)′ = d (·)/dτ and it follows from Equation (2.15) and Equation (2.6a) that

ϑi (s i ) =

∫ s i

0

Γi (χ)

ri (χ)dχ, (2.15)

Γi (s i ) =

È

1−

d ri

d s i

2

. (2.16)

The dimensionless parameters appearing in Equation (2.14) are defined in Table 2.1 and the nondi-

mensional distance from the blade basepoint O to the path vertex V is denoted by r0i = ri (0) =

αi +γi .2.2.4 Generalized Two-Parameter Family of PathsIn what follows, a two-parameter family of paths is derived in terms of linear and nonlinear tun-

ing parameters, which will be used as the absorber design variables in the subsequent chapters.

The idea is to assume an expanded form of the radial lengths ri (s i ). Since the paths are symmetric

about their vertices, only even terms are included in the expansions. The expansions are intro-

duced to the full nonlinear EOMs and, by restricting zero disk and blade motions relative to the

rotor and truncating nonlinear terms, a set of nonlinear systems results. These reduced systems

depend only on the absorber dynamics and they motivate the selection of two tuning parameters.

The first parameter sets the linear absorber tuning order by setting the path curvature at its vertex.

The second parameter prescribes the nonlinear tuning by varying the curvature along the path,

and can be thought of as the strength of the path nonlinearity. In what follows, each absorber path

is assumed to be identical and identically fitted to the blades by imposing αi =α and γi = γ for all

i ∈N . Then

r0i = r0 =α+γ, ∀i ∈N . (2.17)

By assuming no damping and restricting ϕi = ϕ′i = ϕ

′′i = 0 and θi = θ

′i = θ

′′i = 0, Equation (2.14c)

reduces to

s ′′i −σ2ri

d ri

d s i

+(δ+1)σ2

Γi sinϑi −d ri

d s i

cosϑi

= 0. (2.18)

Next the dimensionless radial length ri (s i ) is expanded according to

r 2i (s i ) = b0+b2s 2

i +b4s 4i +O (s 6

i ). (2.19)

2.2 Bladed Disk Fitted with General-Path Absorbers 27Parameter Description

ri = Ri /d Radial length from blade base point O to i th absorber at P

s i =S i /d Arc length from path vertex V to i th absorber at P

r0i =αi +γi Radial length from blade base point O to path vertex V

δ= H

dRadius of the rotor rigid disk

ρi =L i

dLength of i th blade pendulum

αi Distance from blade base point O to absorber base point N

γi Length of i th absorber pendulum

µai =

M ai

Mdi th absorber mass

µbi =

M bi

Mdi th blade mass

fϕ =F0(L+d )

K d Strength of the e.o. excitation in ϕ-dynamics

f θ =F0 L

K d Strength of the e.o. excitation in θ-dynamics

λi =

q

K bi

K d Ratio between blade torsional stiffness and disk torsional stiffness

νai =

q

(k di h2+k b

i d 2)

K d Strength of stiffness coupling between sectors i and i +1

νbi =

q

k bi

d b

K d Strength of stiffness coupling between sectors i and i +1

νci =

q

k bi b 2

K d Strength of stiffness coupling between sectors i and i +1

ξa ci =

1d 2

c di

h2+c bi

d 2

pK d M d /d 2

Strength of damping coupling between sectors i and i +1

ξb ci =

1d 2

c bi

d bpK d M d /d 2

Strength of damping coupling between sectors i and i +1

ξc ci =

1d 2

c bi b 2

pK d M d /d 2

Strength of damping coupling between sectors i and i +1

ξai =

1d 2

C aip

K d M d /d 2i th absorber torsional damping constant

ξai =

1d 2

C aip

K d M d /d 2i th absorber translational damping constant

ξbi =

1d 2

C bip

K d M d /d 2i th blade torsional damping constant

ξdi =

1d 2

C dip

K d M d /d 2i th disk torsional damping constant

σ=Ω/ω0 Angular speed of the rotor

Table 2.1 - List of dimensionless variables and parameters

Each has the same constant coefficients, implying that all of the paths are identical. Since r0 =

ri (0) =p

b0, the first parameter in Equation (2.19) is automatically defined and is given by b0 =

r 20 = (α + γ)

2, and the remaining parameters b2 and b4 are to be specified. Substituting Equa-

tion (2.19) into Equation (2.18) and expanding in s i yields

s ′′i + n 2σ2s i +ησ2s 3

i +O (s 5i ) = 0, (2.20)

where

n =

s

−b2−b2−1p

b0

(δ+1) and η=−2b4−

(b2−1)2+12b0b4

6b0

p

b0

!

(δ+1) (2.21)

28 Mathemati al Modelare defined to be the linear absorber tuning order and the nonlinear absorber tuning parameter.

Equation (2.20) is recognized to be a standard undamped and unforced Duffing oscillator. For

small amplitudes the nonlinear term can be neglected and the oscillator exhibits free harmonic

responses with frequency nσ. Linear tuning of the centrifugally-driven absorbers under consid-

eration can be done by setting the absorber tuning order n relative to the order of the excitation

n . When these match identically, and in the absence of damping, a complete elimination of vi-

brations of the primary system is possible. For larger motions the nonlinearity in Equation (2.20)

becomes important and the oscillations become amplitude-dependent. In the context of absorber

path design, therefore, the nonlinear tuning parameter η is used to modify the absorber behavior

without compromising the small-motion linear tuning n ≈ n . When η> 0 the response is harden-

ing, and it is softening for η< 0.

The remaining expansion coefficients b2 and b4 can be obtained from Equation (2.21) in terms of

the system geometry and the linear and nonlinear tuning parameters n and η, that is

b0 = r 20 ,

b2 =δ+1− n 2r0

δ+1+ r0,

b4 =−(δ+1)(n 2+1)2

12(δ+1+ r0)3− r0

2(δ+1+ r0)η.

(2.22)

Next we consider a special case of the EOMs in which circular-path absorbers are employed.

2.3 Bladed Disk Fitted with Cir ular-Path AbsorbersIn what follows, the nonlinear EOMs for a cyclic bladed disk assembly fitted with circular-path

absorbers are deduced from the general system given by Equation (2.12). These are linearized for

small blade/absorber responses in Section 2.3.2. The resulting mathematical model is employed

in the linear analysis of Chapter 3.2.3.1 Equations of MotionConsider the sector model of Section 2.2. In this case the blades are fitted with circular-path ab-

sorber pendulums of mass M ai

and radius γi d i , the motions of which are described by the angles

ψi . The governing EOMs for the overall system could be derived from this model in the usual

manner via the method of Lagrange. However, it will be more convenient to deduce them from

the general results of Section 2.2 by restricting the arbitrary path to be circular.

By restricting each ρi = γi d i to be constant the general path reduces to a circular path and it can

2.3 Bladed Disk Fitted with Cir ular-Path Absorbers 29be shown that

Si = γi d iψi

R2i =α

2i d 2

i +γ2i d 2

i +2αiγi d 2i cosψi

Ri sinϑi = γi d i sinψi

Ri cosϑi =αi d i +γi d i cosψi

RiΓi =αi d i cosψi +γi d i

d (RiΓi )

d Si

=−αi d i sinψi

γi d i

Ri

d Ri

d Si

=−αi d i sinψi

Γi sin(θi +ϑi )−d Ri

d Si

cos(θi +ϑi ) = sin(θi +ψi )

Γi cos(θi +ϑi )+d Ri

d Si

sin(θi +ϑi ) = cos(θi +ψi )

Γi sin(ϕi +θi +ϑi )−d Ri

d Si

cos(ϕi +θi +ϑi ) = sin(ϕi +θi +ψi )

, (2.23)

which relate the general-path variables Ri , Si , and ϑi to the circular-path angle ψi . The expres-

sions given in Equation (2.23) can be employed to deduce the circular-path EOMs term-by-term

from Equation (2.12). For the proposed circular-path model it is natural to express the absorber

damping in terms of a torsional viscous damping constant instead of the effective translational

representation of Section 2.2. The generalized forces are re-formulated to account for this and are

given by

Q(a )i =−C d

i ϕi +C bi θi +C a

i ψi − c bi−1bd i (θi − θi−1)− c b

i bd i (θi − θi+1)

− (c di−1h2+ c b

i−1d 2i )(ϕi − ϕi−1)− (c d

i h2+ c bi d 2)(ϕi − ϕi+1)+ F0(L i +d i )e

jφi ej nΩt ,

Q(b )i =−C b

i θi +C ai ψi − c b

i−1b 2(θi − θi−1)− c bi b 2(θi − θi+1)

− c bi−1bd i (ϕi − ϕi−1)− c b

i bd i (ϕi − ϕi+1)+ F0L i ejφi ej nΩt ,

Q(c )i =−C a

i ψi ,

(2.24)

where C ai is the torsional damping constant for the i th absorber and is not to be confused with C a

i .

The EOMs for blades fitted with circular-path absorbers are then

30 Mathemati al Model• ϕi -dynamics

M di d 2

i ϕi +C di ϕi −C b

i θi −C ai ψi +K d

i ϕi +M di d i HΩ2 sinϕi

+M bi

d 2i ϕi +L2

i (ϕi + θi )+d i L i cosθi (2ϕi + θi )−d i L i sinθi (2Ω+2ϕi + θi )θi

+d i HΩ2 sinϕi + L i HΩ2 sin(ϕi +θi )

+M ai

d 2i ϕi +α2

i d 2i (ϕi + θi )+γ

2i d 2

i (ϕi + θi + ψi )+αi d 2i (2ϕi + θi )cosθi

+γi d 2i(2ϕi + θi + ψi )cos(θi +ψi )+αiγi d 2

i(2ϕi +2θi + ψi )cosψi

−αi d 2i sinθi (2Ω+2ϕi + θi )θi −αiγi d 2

i sinψi (2Ω+2ϕi +2θi + ψi )ψi

−γi d 2i

sin(θi +ψi )(2Ω+2ϕi + θi + ψi )(θi + ψi )

+d i HΩ2 sinϕi +αi d i HΩ2 sin(ϕi +θi )+γi d i HΩ2 sin(ϕi +θi +ψi )

+(k di−1h2+k b

i−1d 2i )(ϕi −ϕi−1)+ (k

di h2+k b

i d 2i )(ϕi −ϕi+1)

+k bi−1bd i (θi −θi−1)+k b

i bd i (θi −θi+1)

+ (c di−1h2+ c b

i−1d 2i )(ϕi − ϕi−1)+ (c

di h2+ c b

i d 2i )(ϕi − ϕi+1)

+ c bi−1bd i (θi − θi−1)+ c b

i bd i (θi − θi+1) =

F0(L i +d i )ejφi ej nΩt ,

(2.25a)

• θi -dynamics

M bi L2

i θi +M bi L2

i ϕi +M bi d i L i cosθi ϕi +C b

i θi −C ai ψi +M b

i d i L i sinθi (Ω+ ϕi )2

+K bi θi +M b

i L i HΩ2 sin(ϕi +θi )

+M ai

α2i d 2

i θi +α2i d 2

i ϕi +γ2i d 2

i (ϕi + θi + ψi )+αi d 2i cosθi ϕi

+γi d 2i

cos(θi +ψi )ϕi +αiγi d 2i

cosψi (2ϕi +2θi + ψi )

+αi d 2i sinθi (Ω+ ϕi )

2+γi d 2i sin(θi +ψi )(Ω+ ϕi )

2

−αiγi d 2i sinψi (2Ω+2ϕi +2θi + ψi )ψi

+αi d i HΩ2 sin(ϕi +θi )+γi d i HΩ2 sin(ϕi +θi +ψi )

+k bi−1bd i (ϕi −ϕi−1)+k b

i bd i (ϕi −ϕi+1)+k bi−1b 2(θi −θi−1)+k b

i b 2(θi −θi+1)

+ c bi−1bd i (ϕi − ϕi−1)+ c b

i bd i (ϕi − ϕi+1)+ c bi−1b 2(θi − θi−1)+ c b

i b 2(θi − θi+1) =

F0L i ejφi ej nΩt ,

(2.25b)

• ψi -dynamics

M ai γ

2i d 2

i ψi +M ai γ

2i d 2

i (ϕi + θi )+M ai γi d 2

i cos(θi +ψi )ϕi +M ai αiγi d 2

i cosψi (ϕi + θi )

+C ai ψi +M a

i γi d 2i sin(θi +ψi )(Ω+ ϕi )

2+M ai αiγi d 2

i sinψi (Ω+ ϕi + θi )2

+M ai γi d i HΩ2 sin(ϕi +θi +ψi ) = 0.

(2.25c)

Equation (2.25) is made dimensionless in the same way as it was done in Section 2.2.3. The di-

mentionless nonlinear EOMs for sector i are then

2.3 Bladed Disk Fitted with Cir ular-Path Absorbers 31• ϕi -dynamics

µdi ϕ′′i +ξ

di ϕ′i −ξb

i θ′i −ξa

i ψ′i +ϕi +µ

di δσ

2 sinϕi

+µbi

ϕ′′i +ρ2i (ϕ′′i +θ

′′i )+ρi cosθi (2ϕ′′i +θ

′′i )−ρi sinθi (2σ+2ϕ′i +θ

′i )θ′i

+δσ2 sinϕi +ρiδσ2 sin(ϕi +θi )

+µai

ϕ′′i +α2i (ϕ′′i +θ

′′i )+γ

2i (ϕ′′i +θ

′′i +ψ

′′i )+αi (2ϕ′′i +θ

′′i )cosθi

+γi (2ϕ′′i +θ′′i+ψ′′

i)cos(θi +ψi )+αiγi (2ϕ′′i +2θ′′

i+ψ′′

i)cosψi

−αi sinθi (2σ+2ϕ′i +θ′i )θ′i −αiγi sinψi (2σ+2ϕ′i +2θ′i +ψ

′i )ψ′i

−γi sin(θi +ψi )(2σ+2ϕ′i+θ′

i+ψ′

i)(θ′

i+ψ′

i)

+δσ2 sinϕi +αiδσ2 sin(ϕi +θi )+γiδσ2 sin(ϕi +θi +ψi )

+νai−1

2(ϕi −ϕi−1)+ν

ai

2(ϕi −ϕi+1)+ν

bi−1

2(θi −θi−1)+ν

bi

2(θi −θi+1)

+ξa ci−1(ϕ

′i −ϕ′i−1)+ξ

a ci (ϕ

′i −ϕ′i+1)+ξ

bci−1(θ

′i −θ′i−1)+ξ

bci (θ

′i −θ′i+1)

= fϕejφi ej nσt ,

(2.26a)

• θi -dynamics

µbi ρ

2i θ′′i +µ

bi ρ

2i ϕ′′i +µ

bi ρi cosθiϕ

′′i +ξ

bi θ′i −ξa

i ψ′i +µ

bi ρi sinθi (σ+ϕ

′i )

2

+λ2i θi +µ

bi ρiδσ

2 sin(ϕi +θi )

+µai

α2i θ′′i +α2

iϕ′′i +γ

2i (ϕ′′i +θ

′′i +ψ

′′i )+αi cosθiϕ

′′i

+γi cos(θi +ψi )ϕ′′i +αiγi cosψi (2ϕ′′i +2θ′′i +ψ

′′i )

+αi sinθi (σ+ϕ′i )

2+γi sin(θi +ψi )(σ+ϕ′i )

2

−αiγi sinψi (2σ+2ϕ′i +2θ′i +ψ′i )ψ′i

+αiδσ2 sin(ϕi +θi )+γiδσ2 sin(ϕi +θi +ψi )

+νbi−1

2(ϕi −ϕi−1)+ν

bi

2(ϕi −ϕi+1)+ν

ci−1

2(θi −θi−1)+ν

ci

2(θi −θi+1)

+ξbci−1(ϕ

′i −ϕ′i−1)+ξ

bci (ϕ

′i −ϕ′i+1)+ξ

c ci−1(θ

′i −θ′i−1)+ξ

c ci (θ′i −θ′i+1)

= f θejφi ej nσt ,

(2.26b)

• ψi -dynamics

µai γ

2iψ′′i +µ

ai γ

2i (ϕ′′i +θ

′′i )+µ

ai γi cos(θi +ψi )ϕ

′′i +µ

ai αiγi cosψi (ϕ

′′i +θ

′′i )

+ξai ψ′i +µ

ai γi sin(θi +ψi )(σ+ϕ

′i )

2+µai αiγi sinψi (σ+ϕ

′i +θ

′i )

2

+µai γiδσ

2 sin(ϕi +θi +ψi ) = 0.

(2.26c)

2.3.2 Linearized ModelEquation (2.25) is linearized for small disk, blade and absorber motions. The dynamics of the i th

sector are governed by

32 Mathemati al Model• ϕi -dynamics

µbi (ρ

2i +1)+µa

i (αi +γi +1)2+1

ϕ′′i +

µbi ρi (ρi +1)+µa

i (αi +γi )(αi +γi +1)

θ′′i

+µai γi (αi +γi +1)ψ′′i +ξ

di ϕ′i −ξb

i θ′i −ξa

i ψ′i

+

δσ2

µbi (ρi +1)+µa

i (αi +γi +1)+1

+1

ϕi

+δσ2

µbi ρi +µ

ai (αi +γi )

θi +µai δσ

2γiψi

+νai−1

2(ϕi −ϕi−1)+ν

ai

2(ϕi −ϕi+1)+ν

bi−1

2(θi −θi−1)+ν

bi

2(θi −θi+1)

+ξa ci−1(ϕ

′i −ϕ′i−1)+ξ

a ci (ϕ

′i −ϕ′i+1)+ξ

bci−1(θ

′i −θ′i−1)+ξ

bci (θ

′i −θ′i+1)

= fϕejφi ej nσt ,

(2.27a)

• θi -dynamics

µbi ρi (ρi +1)+µa

i (αi +γi )(αi +γi +1)

ϕ′′i +

µbi ρ

2i +µ

ai (αi +γi )

2

θ′′i

+µai γi (αi +γi )ψ

′′i +ξ

bi θ′i −ξa

i ψ′i +δσ

2

µbi ρi +µ

ai (αi +γi )

ϕi

+

(δ+1)σ2

µbi ρi +µ

ai (αi +γi )

+λ2i

θi +µai (δ+1)σ2γiψi

+νbi−1

2(ϕi −ϕi−1)+ν

bi

2(ϕi −ϕi+1)+ν

ci−1

2(θi −θi−1)+ν

ci

2(θi −θi+1)

+ξbci−1(ϕ

′i −ϕ′i−1)+ξ

bci (ϕ

′i −ϕ′i+1)+ξ

c ci−1(θ

′i −θ′i−1)+ξ

c ci (θ

′i −θ′i+1)

= f θejφi ej nσt ,

(2.27b)

• ψi -dynamics

µai γi (αi +γi +1)ϕ′′i +µ

ai γi (αi +γi )θ

′′i +µ

ai γ

2iψ′′i

+ξai ψ′i +µ

ai δσ

2γiϕi +µai (δ+1)σ2γiθi +µ

ai (αi +δ+1)σ2γiψi = 0.

(2.27c)

A lumped-parameter mathematical model of a bladed disk assembly fitted with centrifugally-

driven, general-path vibration absorbers has been developed, which serves as the basis for all

of the analysis to follow. We shall be interested in three specific cases of this general nonlinear

system:

1. The linearized system which is given by Equation (2.27).

2. The fully nonlinear system given by Equation (2.14) with zero inter-sector coupling (νa =

0, νb = 0 and νc = 0), together with the two-parameter family of paths defined by Equa-

tion (2.19).

3. The fully-coupled nonlinear system (νa 6= 0, νb 6= 0 and νc 6= 0) given by Equation (2.14),

together with the two-parameter family of paths defined by Equation (2.19).

These three systems are analyzed in the next two chapters. We begin in the next chapter with an

analysis based on the coupled linearized system.

3Absorber Tuning for theLinearized System

This chapter is a transition chapter towards Chapter 4. In next chapter, the basic effects of non-

linearity are investigated for the fully nonlinear system. In order to understand how nonlinearity

affects system dynamics, we need first to highlight some fundamental features of the linearized

system.

This chapter is organized as follows. The linearized system under consideration is described in

Section 3.1. A special case of these governing equations is presented in Section 3.2. It motivates the

absorber tuning order, which is employed to tune the absorbers to a given order of the excitation.

Finally, absorber tuning or design is discussed in Section 3.3. The aim is to eliminate or otherwise

reduce blade motions relative to the rotor. A detuning parameter is defined in order to consider

operational and manufacturing uncertainties. It also assigns the absorber tuning order relative to

the order of the excitation.3.1 Se tor ModelThe linearized bladed disk model to be considered was presented in Section 2.3. The sectors are

assumed to be identical, and identically-coupled which implies that the parameter subscripts can

be dropped in the EOMs of the linearized system. The governing EOM for the ith 3-DOF sector

follow from Equation (2.27). They are

• ϕi -dynamics

µb (ρ2+1)+µa (α+γ+1)2+1

ϕ′′i +

µbρ(ρ+1)+µa (α+γ)(α+γ+1)

θ′′i

+µaγ(α+γ+1)ψ′′i +ξdϕ′i −ξbθ′i −ξaψ′i

+

δσ2

µb (ρ+1)+µa (α+γ+1)+1

+1

ϕi +δσ2

µbρ+µa (α+γ)

θi

+µaδσ2γψi +νa 2(−ϕi−1+2ϕi −ϕi+1)+ν

b 2(−θi−1+2θi −θi+1)

+ξa c (−ϕ′i−1+2ϕ′i −ϕ′i+1)+ξbc (−θ′i−1+2θ′i −θ′i+1) = fϕejφej nσt ,

(3.1)

34 Absorber Tuning for the Linearized System• θ-dynamics

µbρ(ρ+1)+µa (α+γ)(α+γ+1)

ϕ′′i +

µbρ2+µa (α+γ)2

θ′′i

+µaγ(α+γ)ψ′′i +ξbθ′i −ξaψ′i +δσ

2

µbρ+µa (α+γ)

ϕi

+

(δ+1)σ2

µbρ+µa (α+γ)

+λ2

θi +µa (δ+1)σ2γψi

+νb 2(−ϕi−1+2ϕi −ϕi+1)+ν

c 2(−θi−1+2θi −θi+1)

+ξbc (−ϕ′i−1+2ϕ′i −ϕ′i+1)+ξc c (−θ′i−1+2θ′i −θ′i+1) = f θejφej nσt ,

(3.2)

• ψ-dynamics

µaγ(α+γ+1)ϕ′′i +µaγ(α+γ)θ′′i +µ

aγ2ψ′′i

+ξaψ′i +µaδσ2γϕi +µ

a (δ+1)σ2γθi +µa (α+δ+1)σ2γψi = 0.

(3.3)

In matrix-vector form, Equations (3.1), (3.2) and (3.3) become

Mz′′i +Czi +Kzi +Cc (−z′i−1+2z′i − z′i+1)+Kc (−zi−1+2zi − zi+1) = fejφi ej nστ (3.4)

where the vector zi = (ϕi ,θi ,ψi )T captures the sector dynamics and the elements of the sector

mass, damping, and stiffness matrices are given in Table 3.1. The matrices

M 11 =µb (ρ2+1)+µa (α+γ+1)2+1 C11 = ξd K11 = δσ2(µb (ρ+1)+µa (α+γ+1)+1)+1

M 12 =µbρ(ρ+1)+µa (α+γ)(α+γ+1) C12 =−ξb K12 = δσ2(µbρ+µa (α+γ))

M 13 =µaγ(α+γ+1) C13 =−ξa K13 =µaδσ2γ

M 21 =M 12 C21 = 0 K21 = K12

M 22 =µbρ2+µa (α+γ)2 C22 = ξb K22 = (δ+1)σ2(µbρ+µa (α+γ))+λ2

M 23 =µaγ(α+γ) C23 =−ξa K23 =µa (δ+1)σ2γ

M 31 =M 13 C31 = 0 K31 = K13

M 32 =M 23 C32 = 0 K32 = K23

M 33 =µaγ2 C33 = ξa K33 =µa (α+δ+1)σ2γ

Table 3.1 - Elements of the sector mass, damping and stiffness matrices M, C, K

Cc =

ξa c ξbc 0

ξbc ξc c 0

0 0 0

, Kc =

νa 2 νb 2 0

νb 2νc 2 0

0 0 0

, (3.5)

capture the inter-sector coupling. The sector forcing vector is given by f= (fϕ , f θ ,0)T .3.2 Spe ial Case: Lo ked AbsorbersWe consider the special case when the disks and blades are locked in their zero positions relative

to the rotor. This leads to a system of dynamically isolated absorbers that oscillate freely under

3.3 Absorber Tuning 35the influence of centrifugal effects. The governing equations follow from Equation (??) by setting

ϕi =ϕ′i =ϕ

′′i = 0 and θi = θ

′i = θ

′′i = 0, and are given by

ψ′′i µaγ2+ξaψ′i +ψiµ

a (α+δ+1)σ2γ= 0. (3.6)

Equation (3.6) is a set of N uncoupled and unforced single-DOF harmonic oscillators. Their di-

mensionless undamped natural frequencies are given by

ω33 =ω33

ω0=

r

α+δ+1

γσ= nσ. (3.7)

where

n =

r

α+δ+1

γ(3.8)

is defined to be the absorber tuning order.3.3 Absorber TuningAbsorber tuning refers to a particular choice of absorber parameters to attenuate as much as pos-

sible the response of the blades over a range of operating speeds, and in particular near resonance.

This is done by prescribing the dimensionless parameters µa , γ, and α. It can be shown that there

exists an absorber tuning which enables the suppression of one of the two blade resonances, al-

though this may require large-amplitude vibrations of the absorbers. This tuning is realized by

matching the natural frequency of the isolated absorbers to that of the excitation. It is called exact

absorber tuning and is given by

n = n , or ω33 = nσ= nσ. (3.9)

The exact tuning is valid at all rotation speeds, a feature that is made possible by the structure of

ω33(σ) = nσ. However, any perturbation of the model or absorber parameters, due to in-service

wear or environmental effects, will destroy the exact tuning. To account for such effects, and to

allow for intentionally detuned designs, we let

n = n(1+β), (3.10)

where β is a detuning parameter. Perfect linear tuning corresponds to β = 0, while undertuning

(resp. overtuning) corresponds to β< 0 (resp. β> 0).

For the linear system under consideration, there are 3 natural frequencies ω(p ) corresponding to

each mode p ∈N . The 3N dimensionless natural frequencies ω(p )1,2,3 are plotted in Campbell dia-

grams in Figures 3.2 and 3.1 for N = 10, n = 3 and for two detuning values (β= 0.1 in Figure 3.1 and

β = 0 in Figure 3.2). The disk/blade/absorber frequency response amplitude curves are also rep-

resented in these figures. Possible resonances can be identified by the intersections of the eigen-

frequency loci ω(p )1,2,3(σ) with the order line nσ. However, it was shown that only mode n + 1 is

excited in the steady-state, and hence there is a system resonance only when nσ = ω(n+1)1,2,3 is sat-

isfied, which corresponds to rotor speeds σ = σr e s 1,2. These resonances are indicated by dots in

Figure 3.2 and 3.1.

36 Absorber Tuning for the Linearized System

0

0

0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1

1σ

σ-6

-4

-2

0

2

0

1

2

3

ω(p )

1

ω(p )

2

ω(p )

3

log |X |

log |Y |

log |Z |

nσ

σr e s 1 σr e s 2

Low Rotor Speeds Eigenvalue Veering Region High Rotor Speeds Eigenvalue Veering Region

Figure 3.1 - Campbell diagram and disk, blade and absorber frequency response curves for N = 10,

n = 3, α = 1.4, δ = 1.12, ρ = 1.67, µa = 1.5 · 10−3, µb = 0.1, λ = 0.32, νa = 0.5, νb = 0.1,νc = 0.1, fϕ = 1.6 · 10−4, f θ = 10−4, β= 0.1 and zero damping

3.3 Absorber Tuning 37

0 0.2 0.4 0.6 0.8 1

σ

σ

σr e s 2

0

1

2

3

ω(p )

1

ω(p )

2

ω(p )

3

nσ

Low Rotor Speeds Eigenvalue Veering Region High Rotor Speeds Eigenvalue Veering Region

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

log |X |

log |Y |

log |Z |

Figure 3.2 - Campbell diagram and disk, blade and absorber frequency response curves for N = 10,

n = 3, α = 1.4, δ = 1.12, ρ = 1.67, µa = 1.5 · 10−3, µb = 0.1, λ = 0.32, νa = 0.5, νb = 0.1,νc = 0.1, fϕ = 1.6 · 10−4, f θ = 10−4, β= 0 (absorber perfectly tuned) and zero damping

38 Absorber Tuning for the Linearized SystemIt is known that the frequencies ω(p )1,2,3 are nearly coincident when the inter-sector coupling is weak

and that they spread out for increasing coupling. In order to show clearly which modes are ex-

cited, and also the effects of absorber detuning, a rather large (possibly unrealistic) value of the

coupling is employed in Figures 3.2 and 3.1; this does not qualitatively affect the approach nor the

conclusions.

The frequency loci in Figures 3.2 and 3.1 exhibit the eigenvalue veering phenomenon, or mutual

repulsion of the eigenfrequencies, which arises due to small dynamic coupling between the disks

and blades and between the blades and absorbers. In general, the eigenfrequency loci nσ= ω(p )

1,2,3

crosses the order line nσ twice, firstly in the Low Rotor Speeds Eigenvalue Veering Region (LR-

SEVR) and secondly in the High Rotor Speeds Eigenvalue Veering Region (HRSEVR). This results

in two system resonances. It is clear by looking at Figures 3.2 that if absorber is perfectly tuned

(β = 0), the LRSEVR resonance disappears although the HRSEVR resonance still occurs. In this

case, there is only one system resonance over the full range of possible rotor speeds.

One of the main findings of Shaw, Olson, and Pierre [SHA 06; OLS 06; OLS 05] is the existence of

a no-resonance zone in the isolated sector and coupled sector dynamics.. It consists of a finite

range of absorber undertuning values for which there are no system resonances over the full range

of possible rotor speeds. The upper bound of this range consists of exact tuning and it is bounded

from the bottom by a critical linear detuning. Proper absorber design involves intentional under-

tuning within this generally small gap. For the case of 3-DOF sector model, the no-resonance zone

is replaced by a region of absorber undertuning where one of the two resonances (LRSEVR reso-

nance) is avoided. This range is still bounded on one side by the exact tuning and on the other side

by a critical linear tuning.

The path parameters and dimensions set the tuning order of the absorber. They also determine

the nonlinear behavior of the system at larger amplitudes. Since added mass penalties are often

very stiff, especially in aerospace applications, it is crucial that designs minimize the amount of

mass used. This leads to complications, however, since small absorber mass necessarily implies

large absorber responses, where nonlinear effects become very important. The potentially devas-

tating effects due to nonlinear responses are well known. As torque levels increase, the system can

experience a jump, due to nonlinear effects, which results in absorber motion that actually amplify

vibration amplitudes. Not only nonlinearity but also type of absorber path should be investigated

for optimum performance.

In next chapter, nonlinear behavior of the system is investigated. Analytical treatment of the non-

linear EOMs leads to a simplified nonlinear model, where nonlinearity is concentrated on ab-

sorber motion.

4For ed Response of theNonlinear System

The aim of this chapter is to apply scaling and averaging methods to solve analytically the non-

linear vibration EOMs and to validate the methodology by comparison with numerical data. It is

convenient to employ the generalized, two-parameter family of paths that were developed in Sec-

tion 2.2.4 to introduce the nonlinearity. These allow the final path design to be specified directly by

choosing a linear tuning order n and a nonlinear tuning parameter η. The nonlinear sector mod-

els from Section 2.2 are employed. In Section 4.1, these are reduced via scaling and averaging to a

set of nonlinear sector models from which traveling wave responses are determined, where each

sector behaves identically, except for a fixed phase difference among its nearest neighbors. In Sec-

tion 4.2, the accuracy of the averaged sector models is validated by comparing analytical solutions

to numerical results. The analysis is carried out first for the isolated nonlinear system, consist-

ing of a single linear disk, linear blade and nonlinear absorber and second for the fully coupled

nonlinear system.

4.1 Analyti al Approximation of the Solution4.1.1 Formulation4.1.1.1 Equations of MotionThe cyclically-coupled model to be considered was introduced in Section 2.2. The EOMs in dimen-

tionless form were given in Equation (2.14). The sectors are assumed to be identical and identically

coupled which means that the parameter subscripts can be removed in Equation (2.14), and the

remaining subscripts i are taken modN . Then the governing EOMs for the i th sector take the form

40 For ed Response of the Nonlinear System• ϕi -dynamics

ϕ′′i +ξdϕ′i −ξbθ′i −ξa

riΓi +Γi sin(θi +ϑi )+d ri

d s i

cos(θi +ϑi )

s ′i +ϕi +δσ2 sinϕi

+µb

ϕ′′i +ρ2(ϕ′′i +θ′′i )+ρ cosθi (2ϕ′′i +θ

′′i )−ρ sinθi (2σ+2ϕ′i +θ

′i )θ′i

+δσ2 sinϕi +ρδσ2 sin(ϕi +θi )

+µa

ϕ′′i+r 2

i(ϕ′′

i+θ′′

i)+ ri cos(θi +ϑi )(2ϕ′′i +θ

′′i)

+

Γi cos(θi +ϑi )+d ri

d s isin(θi +ϑi )

s ′′i + riΓi s ′′i−ri sin(θi +ϑi )(2σ+2ϕ′

i+θ′

i)θ′

i+2ri

d ri

d s i(σ+ϕ′

i+θ′

i)s ′

i

+2

d ri

d s icos(θi +ϑi )−Γi sin(θi +ϑi )

(σ+ϕ′i +θ′i )s′i

+ d

d s i

Γi cos(θi +ϑi )+d ri

d s isin(θi +ϑi )

s ′2i+

d (ri Γi )

d s is ′2

i

+δσ2 sinϕi + riδσ2 sin(ϕi +θi +ϑi ))

+ξa c (−ϕ′i−1+2ϕ′i −ϕ′i+1)+ξbc (−θ′i−1+2θ′i −θ′i+1)

+νa 2(−ϕi−1+2ϕi −ϕi+1)+ν

b 2(−θi−1+2θi −θi+1) = fϕejφi ej nστ,

(4.1a)

• θi -dynamics

µbρ2θ′′i +µbρ2ϕ′′i +µ

bρϕ′′i cosθi +ξbθ′i −ξa riΓi s ′i +µ

bρ sinθi (σ+ϕ′i )

2

+λ2θi +µbρδσ2 sin(ϕi +θi )

+µa

r 2iθ′′

i+r 2

iϕ′′

i+ ri cos(θi +ϑi )ϕ

′′i+ riΓi s ′′

i

+ri sin(θi +ϑi )(σ+ϕ′i )

2+2rid ri

d s i(σ+ϕ′i +θ

′i )s′i

+d (riΓi )

d s is ′2i + riδσ2 sin(ϕi +θi +ϑi )

+ξbc (−ϕ′i−1+2ϕ′i −ϕ′i+1)+ξc c (−θ′i−1+2θ′i −θ′i+1)

+νb 2(−ϕi−1+2ϕi −ϕi+1)+ν

c 2(−θi−1+2θi −θi+1) = f θejφi ej nστ,

(4.1b)

• s i -dynamics

µa s ′′i +µa

Γi cos(θi +ϑi )+d ri

d s i

sin(θi +ϑi )

ϕ′′i +µa riΓi (ϕ

′′i +θ

′′i )+ξ

a s ′i

−µa ri

d ri

d s i

(σ+ϕ′i +θ′i )

2+µaδσ2

Γi sin(ϕi +θi +ϑi )−d ri

d s i

cos(ϕi +θi +ϑi )

+µa (σ+ϕ′i )2

Γi sin(θi +ϑi )−d ri

d s i

cos(θi +ϑi )

= 0.

(4.1c)

The absorber paths are assumed to be of the form given by Equation (2.19), where

b0 =α(n 2+1)+δ+1

n 2

2

b2 =−αn 2

α+δ+1

b4 =−(δ+1)n 6

12(α+δ+1)3(n 2+1)− α(n 2+1)+δ+1

2(α+δ+1)(n 2+1)η

(4.2)

are the path coefficients if γ is written in terms of the absorber tuning order according to Equa-

tion (2.22). In this representation, each path depends only on the linear and nonlinear tuning

4.1 Analyti al Approximation of the Solution 41parameters n and η. Once these are set by design, and given the disk radius δ, then α and γ are

automatically prescribed.

Next the full nonlinear system given by Equation (4.2) is reduced to a set of weakly nonlinear oscil-

lators, and perturbation techniques [MUR 91; NAY 73; NAY 79; GUC 83] are subsequently carried

out on these reduced equations.4.1.1.2 S alingIn one nonlinear sector model, the blade mass is low compared to that of the disk. Similarly, the

absorber mass is much smaller than that of the blade due to restrictions on its dimensions and

mass. This motivates the following relationships as the basis for the scaling

µa = ǫm and µb = ǫn ,

where 0 < ǫ≪ 1 is a small dimensionless parameter and the constants m and n are to be deter-

mined. The disk, blade and absorber dynamics are assumed to scale with ǫ according to

ϕi = ǫj ϕi , θi = ǫ

k θi and s i = ǫl s i

for each i ∈N in a manner that is to be determined. The dimensionless disk, blade and absorber

damping constants are scaled according to

ξa = ǫp ξa , ξb = ǫq ξb and ξd = ǫr ξd .

It is additionally assumed that the inter-disk and the inter-blade elastic couplings and coupling

dampings are weak, that is,

νa 2= ǫ f νa 2

, νb 2= ǫg νb

2and νc 2

= ǫh νc 2;

ξa c = ǫu ξa c , ξbc = ǫv ξbc and ξc c = ǫw ξc c .

The dimensionless blade stiffness parameter is scaled according to

λ2 = ǫe λ2.

Finally, weak forcing is assumed, that is,

fϕ = ǫs fϕ and f θ = ǫ

t f θ .

The scaling parameters m , n , j , k , l , p , q , r , f , g , h, u , v , w , e , s and t are chosen such that,

to leading order, a simplified and solvable system is obtained, one that is used as the basis for

the method of averaging. In order to investigate the potentially rich dynamics when the system is

weakly coupled and lightly damped, the scaling is chosen so that the nonlinearity, damping, and

coupling all appear at O (ǫ). To this end, a suitable choice for the scaling parameters is found to be

m = 4, n = 2, j =5

2, k =

3

2, l =

1

2, p = 5, q = 3, r = 1, f = 1,

g = 2, h = 3, u = 1, v = 2, w = 3, e = 2, s =7

2, t =

9

2. (4.3)

Upon substitution and simplification the i th scaled sector model becomes

42 For ed Response of the Nonlinear System• ϕi -dynamics

ϕ′′i + ω211ϕi + ǫ

ξa c (−ϕ′i+1+2ϕ′i − ϕ′i−1)+ ξbc (−θ′i+1+2θ′i − θ′i−1)

+ νa 2(−ϕi+1+2ϕi − ϕi−1)+ νb

2(−θi+1+2θi − θi−1)

= ǫ

−ρ(ρ+1)θ′′i −ρδσ2θi − ξd ϕ′i + fϕejφi ej nστ

+O (ǫ3/2),

(4.4a)

• θi -dynamics

ρ2θ′′i +ρ2ω2

22θi + ǫ

ξbc (−ϕ′i+1+2ϕ′i − ϕ′i−1)+ ξc c (−θ′i+1+2θ′i − θ′i−1)

+ νb2(−ϕi+1+2ϕi − ϕi−1)+ νc 2

(−θi+1+2θi − θi−1)

= ǫ

−ρ(ρ+1)ϕ′′i −ρδσ2ϕi −p

b0s ′′i − (δ+1)σ2s i − ξb θ′i + f θejφi ej nστ

+O (ǫ3/2),

(4.4b)

• s i -dynamics

s ′′i + ω233s i = ǫ

−p

b0θ′′i − (δ+1)σ2θi − ξa s ′i −ησ2s 3

i

+O (ǫ3/2), (4.4c)

where

ω11 =p

δσ2+1, ω22 =

p

ρ(δ+1)σ2+ λ2

ρ, ω33 = nσ. (4.5)

When ǫ = 0, Equation (4.4) reduces to a pair of uncoupled, undamped, and unforced linear os-

cillators. The general case of small ǫ 6= 0 is simply a perturbation of these uncoupled systems.

Equations (4.4a) and (4.4b) are linear, weakly forced oscillators that approximates the motions of

disk and blade i while Equation (4.4c), which captures the i th absorber dynamics, is unforced and

weakly nonlinear due to the cubic absorber path term. Thus, the reason we do scaling is obtaining

linear dynamics of disk and blade DOFs and nonlinear dynamics of absorber DOF. These oscilla-

tors are weakly coupled due to the assumptions of small µa and µb and small inter-blade stiffness

coupling.

Absorber design is carried out by choosing the linear absorber tuning order n and the nonlinear

tuning parameter η, both of which appear only in the i th absorber equation. This in turn pre-

scribes the absorber paths by setting the constants b0, b2, and b4, and it fixes the effects of the

absorbers on the disk and blade dynamics.

Before proceeding with a further reduction of Equation (4.4) via averaging, it should be verified

that the scaled sector models capture the linear resonance structure that was described in Chap-

ter 3. This is achieved in next section.4.1.1.3 Comparison between Linear and S aled SolutionsIf the scaling is applied to the i th linearized sector model defined by Equations (3.1), (3.2) and (3.3),

then we get back the i th scaled sector model given by Equation (4.2) when the nonlinear tuning pa-

rameter equals to zero. In this way, the scaling is seen to essentially linearize the system dynamics

4.1 Analyti al Approximation of the Solution 43while at the same time capturing the basic first-order effects of the absorber path nonlinearity with

a cubic term.

Since ǫ will generally be small, it is expected that the linear resonance structure qualitatively per-

sists under the scaling of the previous section. First, this is verified in Figure 4.1 which shows

sample plots of the rotor speeds σr e s 1,2 corresponding to resonance for the linearized and scaled

systems described above versus the linear absorber detuning parameter β for zero damping and

for various values of ǫ. To simplify matters, the curves are shown for the special case of a single iso-

lated sector. The linear resonance structures of the two systems appear to be in good agreement for

small ǫ. On the contrary, as ǫ increases, so too does the error in σr e s 1,2 between the linearized and

scaled systems. Logarithmic displacements of each DOF of the system are plotted in Figure 4.2

β

σr e s 1,2

Scaled Equations with η= 0

Linearized Equations

ǫ= 0.08

ǫ= 0.1

ǫ= 0.12ǫ= 0.14ǫ= 0.16

-0.2 -0.1 0 0.1 0.20

0.5

1

1.5

Figure 4.1 - The rotor speeds σr e s 1,2 corresponding to resonance for the linearized system of Chapter 3and the scaled system formed by Equation (4.2) with η = 0 versus the linear absorberdetuning parameter β for a model with n = 3, α = 1.40, δ = 1.12, ρ = 1.67, λ = 0.32(λ= 5.27), and for 0.08< ǫ< 0.16

versus the dimentionless rotor speed σ for various detuning values β and for the linearized and

scaled systems introduced previously. A sufficiently small value of ǫ (ǫ= 0.06) is chosen in order to

obtain good agreement between the linearized and scaled results. As for Figure 4.1, we consider a

single undamped isolated sector. For the special case β= 0, there is a shift between the two curves

in disk dynamics, blade scaled response disappears (in fact, blade response is zero which implies

its logarithmic response is minus infinity). Only absorber dynamics exhibits good match between

the linearized and scaled system. Those features arises due to the sensitivity of the system to pa-

44 For ed Response of the Nonlinear Systemrameters when β = 0. Indeed, when absorber is perfectly tuned, e.o. line is almost parallel to the

second mode shape frequency loci in Figure 3.2 of Chapter 3 which can be the source of numerical

problems when solving the linearized and scaled (with η= 0) EOMs.

σ

σ

σ

σ

σ

log |ϕ| log |θ| log |ψ|

(a) β=−0.1

(b) β=−0.01

(c) β= 0

(d) β=+0.01

(e) β=+0.1

Linearized Equations

Scaled Equations with η= 0

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

-6

-4

-2

0

2

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-6

-4

-2

0

Figure 4.2 - Disk/blade/absorber frequency response curves for the undamped isolated nonlinear sys-tem for the linearized system of Chapter 3 and the scaled system formed by Equation (4.2)with η = 0, for various detuning values β, and for n = 3, α = 1.40, δ = 1.12, ρ = 1.67,

ǫ= 0.06, λ= 0.141 (λ= 2.36), fϕ = 1.6 · 10−4 ( fϕ = 3.02), f θ = 10−4 ( f θ = 31.5)

In what follows averaging is carried out on the scaled sector models of Section 4.1.1.2. This is done

both in polar and cartesian form.

4.1 Analyti al Approximation of the Solution 454.1.1.4 AveragingPolar FormThe weakly nonlinear set of oscillators defined by Equation (4.4) can be put into a form that is

suitable for averaging via the transformation

ϕi (τ) = u i (τ)cos(nστ+φi +i (τ))

ϕ′i (τ) =−nσu i (τ)sin(nστ+φi +i (τ))

θi (τ) = vi (τ)cos(nστ+φi + ςi (τ))

θ′i (τ) =−nσvi (τ)sin(nστ+φi + ςi (τ))

s i (τ) =w i (τ)cos(nστ+φi +i (τ))

s ′i (τ) =−nσw i (τ)sin(nστ+φi +i (τ))

(4.6)

Equation (4.6) represents a standard variation of parameters to transform the dependent variables

from ϕi to u i and i , from θi to vi and ςi and from s i to w i andi . It allows to look for solutions

with slowly-varying amplitudes and phases in individual sector responses, and it also serves to

capture the desired traveling wave response among the sectors by including the inter-blade phase

angle. Upon substitution of Equation (4.6) into Equation (4.4), using constraint equations

0= u ′i (τ)cos(nστ+φi +i (τ))−u i (τ)′i (τ)sin(nστ+φi +i (τ)),

0= v ′i (τ)cos(nστ+φi + ςi (τ))− vi (τ)ς′i (τ)sin(nστ+φi + ςi (τ)),

0=w ′i (τ)cos(nστ+φi +i (τ))−w i (τ)′i (τ)cos(nστ+φi +i (τ)),

(4.7)

and elimination of terms at O (ǫ3/2) and higher we obtain

u ′i =1

nσ

(ω211−n 2σ2)u i cos(nστ+φi +i )+ ǫ f

(i )11

sin(nστ+φi +i )

u i′i =

1

nσ

(ω211−n 2σ2)u i cos(nστ+φi +i )+ ǫ f

(i )11

cos(nστ+φi +i )

v ′i =1

nσ

(ω222−n 2σ2)vi cos(nστ+φi + ςi )+ ǫ f

(i )22

sin(nστ+φi + ςi )

vi ς′i =

1

nσ

(ω222−n 2σ2)vi cos(nστ+φi + ςi )+ ǫ f

(i )22

cos(nστ+φi + ςi )

w ′i =1

nσ

(ω233−n 2σ2)w i cos(nστ+φi +i )+ ǫ f

(i )33

sin(nστ+φi +i )

w i′i =

1

nσ

(ω233−n 2σ2)w i cos(nστ+φi +i )+ ǫ f

(i )33

cos(nστ+φi +i )

, (4.8)

where the functions

f(i )11 = ν

a 2

−u i−1 cos(nστ+φi−1+i−1)+2u i cos(nστ+φi +i )−u i+1 cos(nστ+φi+1+i+1)

+ νb2− vi−1 cos(nστ+φi−1+ ςi−1)+2vi cos(nστ+φi + ςi )− vi+1 cos(nστ+φi+1+ ςi+1)

−nσξa c

−u i−1 sin(nστ+φi−1+i−1)+2u i sin(nστ+φi +i )−u i+1 sin(nστ+φi+1+i+1)

−nσξbc

− vi−1 sin(nστ+φi−1+ ςi−1)+2vi sin(nστ+φi + ςi )− vi+1 sin(nστ+φi+1+ ςi+1)

−nσξd u i sin(nστ+φi +i )− fϕ cos(nστ+φi )−ρ((ρ+1)ω222−δσ2)vi cos(nστ+φi + ςi )

(4.9a)

46 For ed Response of the Nonlinear Systemf(i )22 =

1

ρ2

νb2−u i−1 cos(nστ+φi−1+i−1)+2u i cos(nστ+φi +i )−u i+1 cos(nστ+φi+1+i+1)

+ νc 2

− vi−1 cos(nστ+φi−1+ ςi−1)+2vi cos(nστ+φi + ςi )− vi+1 cos(nστ+φi+1+ ςi+1)

−nσξbc

−u i−1 sin(nστ+φi−1+i−1)+2u i sin(nστ+φi +i )−u i+1 sin(nστ+φi+1+i+1)

−nσξc c

− vi−1 sin(nστ+φi−1+ ςi−1)+2vi sin(nστ+φi + ςi )− vi+1 sin(nστ+φi+1+ ςi+1)

−nσξb vi sin(nστ+φi + ςi )− f θ cos(nστ+φi )−ρ((ρ+1)ω211−δσ2)u i cos(nστ+φi +i )

− (r0ω233− (δ+1)σ2)w i cos(nστ+φi +i )

(4.9b)

f(i )33 =−nσξa w i sin(nστ+φi +i )− (r0ω

222− (δ+1)σ2)vi cos(nστ+φi + ςi )

+ησ2w 3i cos(nστ+φi +i )

3(4.9c)

capture all of the O (ǫ) terms in Equation (4.4). The differences

ω211−n 2σ2 =−

σ2−σ2r1

σ2r1

=−ǫ∆1 (4.10a)

ω222−n 2σ2 =

λ2

ρ2

−σ2−σ2

r2

σ2r2

=−ǫ∆2 (4.10b)

ω233−n 2σ2 = (n 2−n 2)σ2 = ǫλσ2 (4.10c)

give a measure of proximity of the rotor speed relative to the disk and blade resonances and to the

absorber design relative to exact linear tuning, respectively. Equation (4.10) motivates the speed

and order detunings

σ2 =σ2r1(1+ ǫ∆1), (4.11a)

σ2 =σ2r2(1+ ǫ

ρ2

λ2∆2), (4.11b)

n 2−n 2 = ǫλ. (4.11c)

The speed detunings ∆1 and ∆2 play the role of the rotor speeds and the order detuning λ is the

counterpart to the linear order detuning β presented in Section 3. The two order detuning param-

eters λ and β are related by

λ=βn 2(β+2)

ǫ(4.12)

which follows from Equation (4.11c) and Equation (3.10). After the appropriate substitutions are

made, Equation (4.8) is averaged over one period T = 2π/nσ. The result is divided into two vector

4.1 Analyti al Approximation of the Solution 47functions, one that defines the stationary points of the i th averaged sector model and the other

inherits any remaining terms. To O (ǫ3/2) and for each i ∈N , the desired form is

(u ′i , u i ¯ ′i , v ′i , vi ς′i ,w ′i ,w i ¯ ′i )

T =ǫ

2nσG(vi−1,vi ,vi+1)+

ǫ

2nσg(vi−1,vi ,vi+1) (4.13)

where vi = (u i , ¯i , vi , ςi ,w i , ¯ i )T . The elements of the 6×1 vector G are given by

G1 = νb2

vi−1 sin(ςi−1− ¯ i −ϕn+1)−2vi sin(ςi − ¯ i )+ vi+1 sin(ςi+1− ¯ i +ϕn+1)

+nσξbc

vi−1 cos(ςi−1− ¯ i −ϕn+1)−2vi cos(ςi − ¯ i )+ vi+1 cos(ςi+1− ¯ i +ϕn+1)

−nσξd u i − fϕ sin ¯ i − α2vi sin( ¯ i − ςi )

(4.14a)

G2 = νb2

− vi−1 cos(ςi−1− ¯ i −ϕn+1)+2vi cos(ςi − ¯ i )− vi+1 cos(ςi+1− ¯ i +ϕn+1)

+nσξbc

vi−1 sin(ςi−1− ¯ i −ϕn+1)+2vi sin(ςi − ¯ i )+ vi+1 sin(ςi+1− ¯ i +ϕn+1)

− fϕ cos ¯ i − α2vi cos( ¯ i − ςi )−∆1u i

(4.14b)

G3 =1

ρ2

νb2

u i−1 sin( ¯ i−1− ςi −ϕn+1)−2u i sin( ¯ i − ςi )+ u i+1 sin( ¯ i+1− ςi +ϕn+1)

+nσξbc

u i−1 cos( ¯ i−1− ςi −ϕn+1)−2u i cos( ¯ i − ςi )+ u i+1 cos( ¯ i+1− ςi +ϕn+1)

−nσξb vi − f θ sin ςi + α1u i sin( ¯ i − ςi )− β3w i sin(ςi − ¯ i )

(4.14c)

G4 =1

ρ2

νb2− u i−1 cos( ¯ i−1− ςi −ϕn+1)+2u i cos( ¯ i − ςi )− u i+1 cos( ¯ i+1− ςi +ϕn+1)

+nσξbc

u i−1 sin( ¯ i−1− ςi −ϕn+1)−2u i sin( ¯ i − ςi )+ u i+1 sin( ¯ i+1− ςi +ϕn+1)

− f θ cos ςi − α1u i cos( ¯ i − ςi )− β3w i cos(ςi − ¯ i )−ρ2∆2vi

(4.14d)

G5 =−nσξa w i + β2vi sin(ςi − ¯ i ) (4.14e)

G6 =−β2vi cos(ςi − ¯ i )+λσ2w i +

3

4ησ2w 3

i(4.14f)

and the elements of g are

g 1= νa 2

u i−1 sin( ¯ i−1− ¯ i −ϕn+1)+ u i+1 sin( ¯ i+1− ¯ i +ϕn+1)

+nσξa c

u i−1 cos( ¯ i−1− ¯ i −ϕn+1)−2u i + u i+1 cos( ¯ i+1− ¯ i +ϕn+1) (4.15a)

g 2= νa 2

− u i−1 cos( ¯ i−1− ¯ i −ϕn+1)+2u i − u i+1 cos( ¯ i+1− ¯ i +ϕn+1)

+nσξa c

u i−1 sin( ¯ i−1− ¯ i −ϕn+1)+ u i+1 sin( ¯ i+1− ¯ i +ϕn+1) (4.15b)

g 3=1

ρ2

νc 2

vi−1 sin(ςi−1− ςi −ϕn+1)+ vi+1 sin(ςi+1− ςi +ϕn+1)

+nσξc c

vi−1 cos(ςi−1− ςi −ϕn+1)−2vi + vi+1 cos(ςi+1− ςi +ϕn+1)

(4.15c)

48 For ed Response of the Nonlinear Systemg 4=

1

ρ2

νc 2

− vi−1 cos(ςi−1− ςi −ϕn+1)+2vi − vi+1 cos(ςi+1− ςi +ϕn+1)

+nσξc c

vi−1 sin(ςi−1− ςi −ϕn+1)+ vi+1 sin(ςi+1− ςi +ϕn+1)

(4.15d)

g 5= g 6 = 0 (4.15e)

where the following parameters have been employed

α1(σ) = ρ((ρ+1)ω211(σ)−δσ2)

α2(σ) = ρ((ρ+1)ω222(σ)−δσ2)

β2(σ) = r0ω222(σ)− (δ+1)σ2

β3(σ) = r0ω233(σ)− (δ+1)σ2

. (4.16)

The averaged sector models defined by Equation (4.13) serve as the basis for the analysis in the

rest of this chapter. The corresponding cartesian form is also useful, which is given next.Cartesian FormIn cartesian form, the transformation which is carried out is

ϕi (τ) = Ai (τ)cos(nστ+φi )+ Bi (τ)sin(nστ+φi )

ϕ′i (τ) = nσ

Bi (τ)cos(nστ+φi )−Ai (τ)sin(nστ+φi )

θi (τ) =C i (τ)cos(nστ+φi )+Di (τ)sin(nστ+φi )

θ′i (τ) = nσ

Di (τ)cos(nστ+φi )−C i (τ)sin(nστ+φi )

s i (τ) = E i (τ)cos(nστ+φi )+ Fi (τ)sin(nστ+φi )

s ′i (τ) = nσ

Fi (τ)cos(nστ+φi )− E i (τ)sin(nστ+φi )

. (4.17)

After substitution of Equation (4.17) into Equation (4.4) and averaging over one period T = 2π/nσ,

to O (ǫ3/2) and for each i ∈N , it can be shown that

w′i =ǫ

2nσP(wi−1,wi ,wi+1)+

ǫ

2nσp(wi−1,wi ,wi+1), (4.18)

if wi = (Ai , Bi ,C i ,Di , E i , Fi )T . The elements of P and p are given by

P1= νa 2

2Bi − (Bi−1+ Bi+1)cosϕn+1

+ νb2

2Di − (Di−1+ Di+1)cosϕn+1

+nσξa c

−2Ai +(Ai−1+ Ai+1)cosϕn+1

+nσξbc

−2C i +(C i−1+ C i+1)cosϕn+1

− α2Di −∆1Bi −nσξd Ai

(4.19a)

P2= νa 2

−2Ai +(Ai−1+ Ai+1)cosϕn+1

+ νb2−2C i +(C i−1+ C i+1)cosϕn+1

+nσξa c

−2Bi +(Bi−1+ Bi+1)cosϕn+1

+nσξbc

−2Di +(Di−1+ Di+1)cosϕn+1

+ α2C i +∆1Ai −nσξd Bi + fϕ

(4.19b)

4.1 Analyti al Approximation of the Solution 49P3= νb

2

2Bi − (Bi−1+ Bi+1)cosϕn+1

+ νc 2

2Di − (Di−1+ Di+1)cosϕn+1

+nσξbc

−2Ai +(Ai−1+ Ai+1)cosϕn+1

+nσξc c

−2C i +(C i−1+ C i+1)cosϕn+1

− α1Bi − β3Fi −ρ2∆2Di −nσξb C i

(4.19c)

P4= νb2−2Ai +(Ai−1+ Ai+1)cosϕn+1

+ νc 2

−2C i +(C i−1+ C i+1)cosϕn+1

+nσξbc

−2Bi +(Bi−1+ Bi+1)cosϕn+1

+nσξc c

−2Di +(Di−1+ Di+1)cosϕn+1

+ α1Ai + β3E i +ρ2∆2C i −nσξb Di + f θ

(4.19d)

P5=−β2Di +λσ2Fi −nσξa E i +

3

4ησ2(F 3

i + E 2i Fi ) (4.19e)

P6= β2C i −λσ2E i −nσξa Fi −3

4ησ2(E i F 2

i + E 3i ) (4.19f)

and

p1 = νa 2(Ai+1− Ai−1)sinϕn+1+ νb

2(C i+1− C i−1)sinϕn+1

+nσξa c (Bi+1− Bi−1)sinϕn+1+nσξbc (Di+1− Di−1)sinϕn+1

(4.20a)

p2 = νa 2(Bi+1− Bi−1)sinϕn+1+ νb

2(Di+1− Di−1)sinϕn+1

−nσξa c (Ai+1− Ai−1)sinϕn+1−nσξbc (C i+1− C i−1)sinϕn+1

(4.20b)

p3 =1

ρ2

νb2(Ai+1− Ai−1)sinϕn+1+ νc 2

(C i+1− C i−1)sinϕn+1

+nσξbc (Bi+1− Bi−1)sinϕn+1+nσξc c (Di+1− Di−1)sinϕn+1

(4.20c)

p4 =1

ρ2

νb2(Bi+1− Bi−1)sinϕn+1+ νc 2

(Di+1− Di−1)sinϕn+1

−nσξbc (Ai+1− Ai−1)sinϕn+1−nσξc c (C i+1− C i−1)sinϕn+1

(4.20d)

p5 = p6 = 0 (4.20e)

Existence and determination of the desired traveling wave response of the coupled system are

discussed next.4.1.2 Traveling Wave ResponseA traveling wave response (TW) is characterized by identical dynamics of individual sectors to-

gether with a fixed phase difference in these dynamics among neighboring sectors. If v= (u , ¯ , v , ς,w , ¯ )T

and w= (A, B ,C ,D , E , F )T , then such a response corresponds to

(vi−1,vi ,vi+1) = (v,v,v), ∀ i ∈N (Polar Form)

(wi−1,wi ,wi+1) = (w,w,w), ∀ i ∈N (Cartesian Form)(4.21)

where the phase difference among adjacent sectors is built into the transformation defined by

Equation (4.6) via the inter-blade phase angles φi .

50 For ed Response of the Nonlinear SystemWhen they exist, the stationary points can be obtained from either Equation (4.13) or Equation (4.18).

The former is employed throughout the remainder of this section. By setting u i−1 = u i+1 = u ,

vi−1 = vi+1 = v , w i−1 = w i+1 = w , ¯ i−1 = ¯ i+1 = ¯ , ςi−1 = ςi+1 = ς, ¯ i−1 = ¯ i+1 = ¯ in Equa-

tion (4.13), it thus follows that

0=−2νb2(cosϕn+1−1)v sin( ¯ − ς)+2nσξbc (cosϕn+1−1)v cos( ¯ − ς)

+2nσξa c (cosϕn+1−1)u −nσξd u − fϕ sin ¯ − α2v sin( ¯ − ς)(4.22a)

0=−2νb2(cosϕn+1−1)v cos( ¯ − ς)−2nσξbc (cosϕn+1−1)v sin( ¯ − ς)

−2νa 2(cosϕn+1−1)u − fϕ cos ¯ −∆1u − α2v cos( ¯ − ς)

(4.22b)

0=+2νb2(cosϕn+1−1)u sin( ¯ − ς)+2nσξbc (cosϕn+1−1)u cos( ¯ − ς)

+2nσξc c (cosϕn+1−1)v −nσξb v − f θ sin ς

+ α1u sin( ¯ − ς)− β3w sin(ς− ¯ )

(4.22c)

0=−2νb2(cosϕn+1−1)u cos( ¯ − ς)+2nσξbc (cosϕn+1−1)u sin( ¯ − ς)

−2νc 2(cosϕn+1−1)v − f θ cos ς−ρ2∆2v

− α1u cos( ¯ − ς)− β3w cos(ς− ¯ )

(4.22d)

0=−nσξa w + β2v sin(ς− ¯ ) (4.22e)

0=−β2v cos(ς− ¯ )+λσ2w +3

4ησ2w 3 (4.22f)

That is, if a stationary point v can be found that satisfies Equation (4.22), then there exists a corre-

sponding TW response. Existence, therefore, follows from the equilibria of an individual averaged

sector model, a simplification that follows from the assumption of identical, identically-coupled

sectors.

The analytical part of the analysis ends there because the determination of the equilibrium points

cannot be carried out analytically. Moreover, numerical results are needed to validate the accuracy

of the scaling and averaging technique.4.2 Comparison: Analyti al Solution vs. Simulation4.2.1 Isolated Nonlinear SystemAnalysis should be first carried out for an isolated sector of the nonlinear system, consisting linear

blade and disk DOFs and nonlinear absorber DOF.

Plots for the disk, blade and absorber TW amplitudes |u |, |v | and |w | are shown in Figure 4.3 ver-

sus the rotor speed σ for a hardening absorber path (η = 1), zero damping, and for various levels

of the order detuning β, and a corresponding set of plots is shown in Figure 4.4 for a softening

path (η = −1). For comparison, the linearized frequency response curves of Chapter 3 are also

included. Investigation of linearized system is an easy way to understand general behavior of the

system such as resonance frequencies. Comparison of this linearized and nonlinear forms give

4.2 Comparison: Analyti al Solution vs. Simulation 51us informations about whether linearization is able to capture general characteristics of the sys-

tem or not. Figure 4.6 features the same frequency response loci shown in Figure 4.4 for softening

absorber paths, but with nonzero dampings ξa = 1× 10−4, ξb = 1× 10−5 and ξd = 1× 10−6. Fi-

nally, disk, blade and absorber frequency response curves are plotted in Figure 4.5 for a hardening

absorber path (η= 1), small linear undertuning β=−0.005 and for various values of λ.

Those curves corresponds to analytical solutions obtained from scaling and averaging. They were

determined numerically according to Equation (4.22). A Gauss-Newton algorithm is used to solve

the system of nonlinear algebraic equations. Simulation data corresponding to the full nonlinear

system are also indicated in Figure 4.6. They are derived from Equation (4.1), where the absorber

paths are assumed to be of the form given by Equation (2.19).(Expressions of Γi (s i ) and ϑi (s i ) are

deduced from Equation (2.2b) and Equation (2.15).) The matlab solver ode45 is employed to solve

the nonlinear differential EOMs. This solver is based on an explicit fourth- and fifth-order Runge-

Kutta method called the Dormand-Prince method.

In all of the figures presented below, nonlinear frequency response amplitudes are globally lower

for disk DOF than for blade DOF and the same tendency appears between blade and absorber

DOFs, as expected. This is also logical verification of the results. In Figures 4.3 and 4.4, there

are similarities on resonance frequencies between the linearized and nonlinear responses of the

system. Indeed, for β values far from zero (Figures (a) and (e)), their two resonance frequencies

match. For β value equal to zero (Figures (c)) or very close to zero (Figures (b) and (d)), there is a

shift between linearized and nonlinear curves and resonance frequencies do not match anymore,

which implies that linearized model do not capture the global level of the nonlinear response nor

the nonlinear resonance structure.

Straight-up/-down lines from green to red (or red to green) curves are representation of jump phe-

nomenon in nonlinear oscillations. They are indicated by circles in Figures 4.3 and 4.5. Jump

phenomenon mostly happens in absorber motion, which was expected since we forced the sys-

tem to have linear behavior for disk and blade DOFs and nonlinear behavior for absorber DOF.

Some of the jumps cannot be seen easily in the graphs so that they have been highlighted.

In Figure 4.5, the resonance frequencies that occurs in the lower rotor speeds are not changing so

much when λ increases whereas second resonance frequencies which occurs in the higher rotor

speeds shift to the right in both linear and nonlinear cases. This property could be used to design

substructures and select material properties for disk and blade.

Simulation data are obtained by time solving directly the full differential EOMs. To get steady state

motion in time response of the system, damping is thus necessary. For this reason, there is no

simulation data when the system is undamped. In Figure 4.6, addition of damping results in the

disappearance of nonlinear jumps. Simulation data dots in blade motion superimpose perfectly

to the scaled and averaged curve. They also captures disk dynamics of scaled and averaged model,

even if there is a discrepancy at high rotor speeds. For the absorber case, results are in agreement

for resonance frequencies and moderate rotor speeds.

52 For ed Response of the Nonlinear System

σ

σ

σ

σ

σ

log |u | log |v | log |w |

(a) β=−0.1

(b) β=−0.001

(c) β= 0

(d) β=+0.001

(e) β=+0.1

LinearizedNonlinear (1rst Branch)Nonlinear (2nd Branch)Jump phenomenon

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

Figure 4.3 - Disk/blade/absorber frequency response curves for the undamped isolated nonlinear sys-tem with a hardening absorber path (η= 1), for various detuning values β, and for n = 3,

α = 1.40, δ = 1.12, ρ = 1.67, ǫ = 0.06, λ2 = 0.02 (λ = 2.36), fϕ = 1.6 · 10−4 ( fϕ = 3.02),f θ = 10−4 ( f θ = 31.5)4.2.2 Coupled Nonlinear System

A new formulation of the nonlinear EOMs has been obtained with scaling and averaging methods.

This analytical solution has been compared to simulation data obtained by solving numerically

4.2 Comparison: Analyti al Solution vs. Simulation 53

σ

σ

σ

σ

σ

log |u | log |v | log |w |

(a) β=−0.1

(b) β=−0.001

(c) β= 0

(d) β=+0.001

(e) β=+0.1

LinearizedNonlinear (1rst Branch)Nonlinear (2nd Branch)

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

Figure 4.4 - Disk/blade/absorber frequency response curves for the undamped isolated nonlinear sys-tem with a softening absorber path (η=−1), for various detuning values β, and for n = 3,

α = 1.40, δ = 1.12, ρ = 1.67, ǫ = 0.06, λ2 = 0.02 (λ = 2.36), fϕ = 1.6 · 10−4 ( fϕ = 3.02),f θ = 10−4 ( f θ = 31.5)

the full nonlinear EOM. It was demonstrated that analytical and numerical solutions matches very

well for blade DOF which was the primary concern of the investigation.

54 For ed Response of the Nonlinear System

σ

σ

σ

σ

σ

log |u | log |v | log |w |

(a)λ2 = 0.001

λ= 0.527

(b)λ2 = 0.01

λ= 1.67

(c)λ2 = 0.02

λ= 2.36

(d)λ2 = 0.03

λ= 2.89

(e)λ2 = 0.04

λ= 3.33

LinearizedNonlinear (1rst Branch)Nonlinear (2nd Branch)Jump phenomenon

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1

-2

0

2

4

-4

-2

0

2

-8

-6

-4

-2

0

-2

0

2

4

-4

-2

0

2

-8

-6

-4

-2

0

-2

0

2

4

-4

-2

0

2

-8

-6

-4

-2

0

-2

0

2

4

-4

-2

0

2

-6

-4

-2

0

-2

0

2

4

-4

-2

0

2

-8

-6

-4

-2

0

Figure 4.5 - Disk/blade/absorber frequency response curves for the undamped isolated nonlinear sys-tem with a hardening absorber path (η = 1), linear undertuning β = −0.005, for various

values of λ, and for n = 3, α= 1.40, δ= 1.12, ρ = 1.67, ǫ= 0.06, fϕ = 1.6 · 10−4 ( fϕ = 3.02),f θ = 10−4 ( f θ = 31.5)

4.2 Comparison: Analyti al Solution vs. Simulation 55

σ

σ

σ

σ

σ

log |u | log |v | log |w |

(a) β=−0.1

(b) β=−0.005

(c) β= 0

(d) β=+0.005

(e) β=+0.1

AnalyticalSimulation

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1-6

-4

-2

0

-4

-2

0

-8

-6

-4

-2

0

-6

-4

-2

0

-4

-2

0

-8

-6

-4

-2

0

-6

-4

-2

0

-4

-2

0

-8

-6

-4

-2

0

-6

-4

-2

0

-4

-2

0

-8

-6

-4

-2

0

-6

-4

-2

0

-4

-2

0

-8

-6

-4

-2

0

Figure 4.6 - Disk/blade/absorber frequency response curves for the damped isolated nonlinear systemwith a softening absorber path (η = −1) for various detuning values β, n = 3, α = 1.40,

δ = 1.12, ρ = 1.67, ǫ = 0.06, λ2 = 0.02 (λ = 2.36), fϕ = 1.6 · 10−4 ( fϕ = 3.02), f θ = 10−4

( f θ = 31.5), ξa = 10−4, ξb = 10−5, ξd = 10−6

56 For ed Response of the Nonlinear System

σ

σ

σ

σ

σ

log |u | log |v | log |w |

(a)νb 2=νc 2

= 0

νb = νc = 0

(b)νb 2=νc 2

= 0.0001

νb = 0.167

νc = 0.680

(c)νb 2=νc 2

= 0.001

νb = 0.527

νc = 2.15

(d)νb 2=νc 2

= 0.01

νb = 1.67

νc = 6.80

(e)νb 2=νc 2

= 0.1

νb = 5.27

νc = 21.5

LinearizedNonlinear (1rst Branch)Nonlinear (2nd Branch)

0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

-4

-2

0

2

4

-6

-4

-2

0

2

-8

-6

-4

-2

0

Figure 4.7 - Disk/blade/absorber frequency response curves for the undamped coupled nonlinear sys-tem with N = 5 sectors, softening absorber paths (η=−1), linear undertuning β=−0.005,various coupling levels νb = νc , and for νa 2 = 0.5 (νa = 2.89), n = 3, α = 1.40, δ = 1.12,

ρ = 1.67, ǫ= 0.06, λ2 = 0.02 (λ= 2.36), fϕ = 0.00016 ( fϕ = 3.02), f θ = 0.0001 ( f θ = 31.5)

Con lusionA general model for higher-fidelity lumped-parameter of bladed disk assembly fitted with CPVA

excited by e.o. excitation has been introduced. It consists of a cyclic array of N coupled sec-

tors, each with 3 DOFs representing disk, blade and absorber vibrations. Linearized and nonlinear

models has been deduced from this general model.

Some features of the system linear behavior have been highlighted. It has been shown that there

exists a region of absorber undertuning where one of the two resonances is avoided. This range is

bounded on one side by the exact tuning and on the other side by a critical linear tuning.

Nonlinear EOMs has been solved in two different ways. The first method was semi-analytical. A

first step was to apply scaling and averaging techniques to the original nonlinear EOMs. This led

to a set of nonlinear algebraic EOMs. Those equations were subsequently solved numerically. The

second method was fully numerical. A Runge-Kutta algorithm was employed to solve the fully

nonlinear system. Finally, we compared the results of the two methodologies. It has been shown

that they are in good agreement. This demonstrates the accuracy of scaling and averaging methods

to solve the nonlinear EOMs of present system.

When we consider linear behavior, for the perfect linear absorber tuning, we avoid resonance in

the first eigenvalue veering region. But, for the second eigenvalue veering region a new definition

for absorber tuning should be described to eliminate that resonance condition.

Absorber tuning strategies for the sampled linear and nonlinear tuning parameters are not giving

better for the investigation of the nonlinear responses. Next step would be to consider an opti-

mization process among system parameters especially linear and nonlinear tuning parameters to

suppress all of the nonlinear resonances.

58 Con lusion

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of Vector Fields. Springer-Verlag, 1983. 41

[MUR 91] Murdock J. A., Perturbations : Theory and Methods. Wiley, 1991. 41

[NAY 73] Nayfeh A. H., Perturbations Methods. Wiley, 1973. 41

[NAY 79] Nayfeh A. H. and Mook D. T., Nonlinear Oscillations. Wiley, 1979. 41

[OLS a] Olson B. J. and Shaw S. W., “Vibration Absorbers for a Rotating Flexible Structure with Cyclic

Symmetry: Nonlinear Path Design”, Nonlinear Dynamics (in preparation). vii

[OLS b] Olson B. J., Shaw S. W., and Pierre C., “Vibration Absorbers for a Rotating Flexible Structure

with Cyclic Symmetry: Linear Order Tuning”, Journal of Sound and Vibration (in review). vii

[OLS 05] Olson B. J., Shaw S. W., and Pierre C., “Order-Tuned Vibration Absorbers for Cyclic Rotating

Flexible Structures”, in ASME 2005 International Design Engineering Technical Conferences

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September 2005. 38

[OLS 06] Olson B. J., Order-Tuned Vibration Absorbers for System with Cyclic Symmetry with Applica-

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[SHA 06] Shaw S. W. and Pierre C., “The Dynamic Response of Tuned Impact Absorbers for Rotating

Flexible Structures”, Journal of Computational and Nonlinear Dynamics, Vol. 1, No. 1, pp. 13–

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