Master thesis

Parameter study of bladed disk/casing

interactions through direct contact in

aero-engine assemblies

Anthony Guégan

École Centrale de Nantes

Structural Dynamics and Vibration Laboratory, McGill University

14/04/08 - 15/09/08

Contents

Foreword iii

Introduction v

1 Background 1

1.1 Contact mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Kinematic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Numerical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Component mode synthesis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Fixed interface methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Free interface methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 2D model 17

2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Mono-harmonic studies S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Intermediary subspace S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Damped motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Locked motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Two-diameter study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 Three-diameter study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 3D model 35

3.1 Model presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.2 Contact algorithm specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 CMS methods Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.1 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.2 Contact contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Friction contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.5 Sensitivity to rotational velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5.1 CB and CM methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.2 Angular velocities of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Approximation of an industrial mistuned bladed disk 53

4.1 Sequential CMS methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1.1 Guyan method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.2 CB method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.3 Tests on a 2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Applications to a 3D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.2 Industrial application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.3 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Conclusion 65

References 69

Foreword

The internship I performed in the Vibration and Structural Dynamics Laboratory at McGill

University was a great experience. First, working with passionate and competent scientists

was a good way to discover scientific research in the mechanical field; secondly, this work

was a mean to apply the knowledge I had gained during my studies, and to learn and en-

large my competencies. For those reasons I sincerely thank the Professor Pierre and Mathias

Legrand, who allowed me to perform my internship in their laboratory.

Then I am liable to Alain Batailly for his advices and his help during my internship. He

knew how to guide me on different subjects to achieve the best results I could.

The work presented in this thesis could not have been preformed without the cluster of

the GEM laboratory. So I would like to thank all the persons who allowed me to access this

computation server, especially Nicolas Chevaugeon who helped me in my first step using

the cluster.

Finally, I thank all the persons I met in the laboratory, researchers and interns. Their

attitude allowed to work in the best conditions possible.

Nomenclature

Φ Transformation matrix

∆t Time step

η Number of constraint modes in a reduction basis

λ Contact effort

a Physical DOF

CN Contact matrix

D Damping matrix

fext External load

g Gap function

I Identity matrix

K Stiffness matrix

M Mass matrix

R Residual flexibility matrix

u Solution vector

uFE Finite element solution

Ωc Critical speed

0 Subscript referring to time t=0

f Subscript referring to the boundary DOF

i Subscript referring to the internal DOF

η Subscript referring to constraint modes

n d Number of nodal diameters

Introduction

For the past decades, new technologies have efficiently served the turbomachinery manu-

facturers. In order to continuously improve jet engines, several issues have been investi-

gated in order to optimize machines, in terms of performance, cost, and weight.

In order to increase aerodynamic performance of an engine, designers tend to decrease

the gap between rotating components and their surroundings casings. The gap between

bladed disks and casings have a first-order impact on the thrust provided by the engine.

Even though the decrease of the clearance improves the aerodynamic efficiency, it also im-

plies new mechanical problems for the structure that must be taken into account during

the design. Among these mechanical problems, this study focuses on contacts occurring

between bladed disks and casings.

Based on the PhD thesis by Mathias Legrand [LEG 05], who developed contact algo-

rithms, and on the PhD research of Alain Batailly, who introduced different component

mode synthesis methods (CMS) to treat industrial problems, this internship aims at com-

paring different CMS methods on different situations, and highlighting better approaches

leading to the best accuracy.

The document will be organized as follows: first, the main useful works here will be pre-

sented, with theoretical knowledge, computational works and results obtained. The second

part deals with a two dimensional model study. Finally, the third chapter presents an inves-

tigation conducted on full industrial 3D model.

In order to respect confidentiality of industrial data, all the numerical values of this re-

port are normalized.

1.1 Contact mechanics 1

1.1.1 Kinematic equations 11.1.2 Solution method 41.1.3 Numerical issues 51.1.4 Conclusion 6

1.2 Component mode synthesis methods 6

1.2.1 Fixed interface methods 81.2.2 Free interface methods 101.2.3 Convergence 131.2.4 Conclusion 16

1Background

This chapter deals with theoretical statements required to perform this internship. In spite

of the regular increase of computational power, industrial non-linear calculations – such

as contact simulations – may require very long computation times. That is the reason why

component mode synthesis (CMS) methods are introduced in this chapter after presenting

a brief state of the art on contact mechanics.

1.1 Contact mechanicsThe PhD thesis [LEG 05] essentially deals with the problem of contact between a bladed disk

and its surrounding casing. This document has been used to help us present the unilateral

contact law, the friction law and kinematic equations. Figure 1.1 defines the different no-

Γ(m)u

Γ(s)u

Γ(m)σ

Γ(s)σ

Γ(s)c

Γ(m)c

Ω(m)

Ω(s)

Ω(s)

Ω(m)

x(m)

x(s)

gn

Figure 1.1 - Notations used and contact

tations used in this document to describe the contact equations based on a master/slave

approach.

1.1.1 Kinematic equations

Contact between bladed disks and casing involves large displacements because of the great

speed of the rotating gear. However, the aim is to detect interactions between the bladed

2 Background

disk and the casing leading to vibrations without considering potential critical deforma-

tions that may occur due to these interactions. That is the reason why the displacements

considered are small compared to the initial size of the structure. Moreover, the assumption

of small strain is taken into account in our work. These assumptions lead to two different

formulations of the problem: (1) the strong formulation, true formulation of the physical

problem, and (2) the weak formulation, which is considered when numerical methods are

applied such as the finite element method (FEM).

Strong formulation

This formulation of the problem is obtained using the general theory of continuum mechan-

ics. We consider the following equations:

• Dynamic equilibrium equation:

divσ+ρfd = ρ -u (1.1)

• Displacement boundary condition on Γu:

u= ud (1.2)

• Force boundary condition on Γc:

σn= td (1.3)

• Initial conditions in Ω:

u|t=0 = u0 , u|t=0 = u0 (1.4)

• Linear elasticity equation: this law links the Cauchy tensorσ and the Green-Lagrange

tensor with the assumption of small strains ǫ:

ǫij =1

2(u i,j+u j,i)⇐⇒ ǫ=

1

2(grad(u)+ grad

T(u))

σij =Dijklǫkl⇐⇒σ=D : ǫ

(1.5)

• Unilateral contact constraints:

tN ≥ 0

g ≤ 0

tN g = 0

(1.6)

with g the distance slave/master, and tN the normal effort due to contact,

• Friction law - Coulomb law:

‖ tT ‖≤µtN

‖ tT ‖<µtN⇒ vg = 0

‖ tT ‖=µtN⇒∃ α> 0 so that vg =αtT

‖ tT ‖

(1.7)

with tT the tangential effort due to contact,µ the friction coefficient, and vg the sliding

speed.

1.1 Contact mechanics 3

Weak formulation

The physical problem can be approached considering a physical system checking a varia-

tional inequality [CZE 01]. This variational problem takes into account a closed convex:

K(Ω) = u such as u|Γu = 0 and g (u)≤ 0 on Γc (1.8)

Let us note δu = v− u, the variational inequality consists in finding a displacements field

u ∈K(˙) such as ∀v∈K(˙),

∫

Ω

ρ -uδudV+

∫

Ω

σ : δǫdV+

∫

Γc

µtN(u)(‖ vT ‖ − ‖uT ‖)dS≥

∫

Ω

f dδudV+

∫

Γc

tdδudS (1.9)

This is a non-linear problem, particularly because of the friction conditions. Mathemati-

cally, there is no evidence that there is a unique solution for this inequality. In order to solve

such a problem, it is necessary to consider a few assumptions.

Virtual work principle

It is possible to define a space of admissible pressures [CHA 01]:

CN = tN : Γc→R so that tN ≥ 0 (1.10)

then the space of admissible tangential efforts:

CT(tN) = tT : Γc→R3 so that tT.n= 0 and ‖ tT ‖≤ µtN (1.11)

The variational inequality equivalent to the normal contact law is:

tT ∈CN,

∫

Γc

g N(rN− tN)dS≥ 0 ∀rN ∈ CN (1.12)

and the one equivalent to the friction law:

tT ∈ CT(tN),

∫

Γc

vg.(rT− tT)dS≥ 0 ∀rT ∈CT(tN) (1.13)

With these formulations, the spaces where the solution is found are different. It is also nec-

essary to define the two spaces:

U(Ω) = u so that u|`u = 0 and V(Ω) = u so that u|`u = ud (1.14)

The new formulation of the problem implies that we find a displacements field u so that

∀δu ∈U(˙),

∫

Ω

ρ -uδudV+

∫

Ω

σ : δǫdV=

∫

Γc

(tNδg + tTδuT)dS+

∫

Ω

f dδudV+

∫

Γc

tdδudS (1.15)

where the contact pressure tN and the friction forces tT have to satisfy conditions (1.12)

and (1.13).

4 Background

1.1.2 Solution method

There are several methods eligible to solve a system such as the contact problem presented

in (1.1). Most of these methods refer to optimization techniques. Indeed, the problem of

contact considered here can be seen as an optimization problem, for which we are looking

for a minimum solution, taking into account constraints. The most frequently used methods

in mechanics are the penalty method, the Lagrange multiplier method and the augmented

Lagrange multiplier method. These three methods, called regularization methods, allow to

write the non-linear friction and contact laws as functions depending only on the displace-

ments. In this section of the thesis, the problem considered is written explicitly, it consists

in minimizing the strain energy of the system:

Ed =1

2

∫

Ω

ǫ : D : ǫdV−

∫

Ω

fdudV−

∫

Γσ

tdudS (1.16)

minu∈K

Ed(u) (1.17)

Karush, Kuhn and Tucker have written the following optimality conditions:

Having u+ a local minimum of Ed(u), if Ed(u) and g (u) are differentiable at the point u+,

∃λ so that:

δEd(u+)+

∫

Γc

λδg (u+)dS= 0

with g (u+), λ≥ 0, λg (u+) = 0 on Γc

(1.18)

Lagrange multiplier method

This method is based on the theorem of Kuhn and Tucker. It allows to simplify the resolution

of the initial problem. First, it is necessary to define the indicator function

Φ(u) =

(

+∞ if u 6∈ K

0 otherwise(1.19)

The initial problem (1.17) is equivalent to:

minu ∈ K(Ed(u)+Φ(u)) (1.20)

The restriction on the displacements u has been replaced by the function Φ(u) non differen-

tiable. It does not really make the problem easier to solve, but it is now possible to make a

function Φ(u) satisfying the conditions (1.19):

Φ(u) =maxλ∈CN

∫

Γc

λg (u)dS (1.21)

With the definition of the lagrangian associated to the primal problem L(u,λ),∃(u+,λ+) so

that:

L(u,λ) = Ed(u)+

∫

Γc

λg (u)dS , L(u+,λ)≤ L(u+,λ+)≤ L(u,λ+) (1.22)

1.1 Contact mechanics 5

where u+ is the primal problem solution and λ+ the Kuhn and Tucker multiplier associated.

Point (u+,λ+) is called saddle point of the lagrangian problem. It is solution of the orig-

inal problem and is found solving the dual problem in which the constraints are easier to

obtain and applied on the dual variable. For the static approach it means writing the sta-

tionarity of the lagrangian depending on u and λ associated to the energy:

δL(u,λ) = 0 (1.23)

taking into account the conditions of Karush, Kuhn and Tucker. These conditions allow to

find the surface contact (unknown a priori).

The lagrangian multiplier method was chosen in [LEG 05] to treat the contact between

bladed disks and the casing. In fact, one of his previous studies has shown that the penalty

method was not indicated in this case of contact to obtain a good accuracy.

1.1.3 Numerical issues

After having considered the theoretical background on the contact problem, we now focus

on the selection of space and time methods to discretize the problem in order to compute

and analyze it numerically.

The space method chosen is probably the most popular one, the Finite Element Method

(FEM). The choice of the time method is more complicated. In fact there are many numeri-

cal time schemes, pros and cons of each one must be taken into account.

A previous study [LEG 05] on a simple case with has been led using both the penalty

method and the Lagrange multiplier method. It appeared that by definition, the penalty

method implied residual penetrations in the master/slave approach, and does not allow

to fulfill efficiently the contact conditions. As a contrary, the Lagrange multiplier method

allows to fulfill perfectly the contact conditions, with no residual penetrations, which has

encouraged to choose this method.

Concerning the time integration scheme, an explicit centered scheme has been chosen,

based on the algorithm of Carpenter [CAR 91]. This algorithm is user friendly and also very

efficient, ensuring the non-penetration and the compatibility of speed and acceleration. A

drawback of this algorithm that must be noticed is that there is a time increment between

the estimation of the acceleration and the consideration of the contact efforts.

Starting from the following system:

M -u+D_u+Ku= fext

u(t = t0) = u0 , _u(t = t0) = _u0

(1.24)

The algorithm used in the case of the interaction bladed disk/casing can be described as

follows:

1. prediction of the displacements

Using the explicit centered scheme, without taking into account any contact:

un+1,p =

M

∆t 2+

D

2∆t

−1

fextn +

2M

∆t 2−K

un+

D

2∆t−

M

∆t 2

un-1

(1.25)

6 Background

2. calculation of the penetrations

The contact matrix CN is initialized to 0. It is updated each time a penetration is de-

tected and is created using the distance function g introduced in (1.1).

3. correction

The annulation of the calculated penetrations leads to:

gn+1 =CNTun+1,c+gn+1,p = 0 (1.26)

Simultaneously, a new matrix CNT is created considering the corrections both in nor-

mal and tangential directions:

M

∆t 2+

D

2∆t

un+1,c+CNTλN,n+1 = 0

CNTun+1,c+gn+1,p = 0

(1.27)

This method enables an exact correction of the displacements, but remains compu-

tationally expensive because of the inversion, at each time step, of a non symmetric

matrix.

1.1.4 Conclusion

This part has introduced theoretical bases for studying contact problems between two flex-

ible entities. It has been underlined that small strain assumption is considered in our work.

However, contact problems may also be managed without this assumption. The work pre-

sented in [MAG 06], is one of the references where different contact algorithms are devel-

oped using the assumption of large strain. Actually, this formulation of the contact is gen-

eralized to every kind of strain with an elastic behavior, but it leads to more complicated

formulations and is more time consuming. Because of the small scale of the strains face to

the initial size of the components, the small strain formulation is a good approximation of

the behavior of the structures, it is so logical to consider it in our work.

In this report the only method taken into account to treat the contact is the Lagrangian

multiplier method, used in an explicit integration scheme developed in [CAR 91].

1.2 Component mode synthesis methodsComponent mode synthesis (CMS) methods are used in mechanics in the FEM field. The

principle of these methods lies in a change of variable to study a structure behavior. From

a mathematical point of view, FEM enables to study a structure in a vectorial space E, n-

dimensional – n being the number of degrees of freedom (DOF) of the model – and the CMS

methods consist in finding an appropriate subspace of E that keeps the dynamics of interest

of the original system. CMS methods are usually separated in four categories :

• loaded-interface methods.

• mixed-interface methods.

1.2 Component mode synthesis methods 7

• free interface methods such as Craig, Craig-Chang and Craig-Chang-Martinez meth-

ods.

• fixed interface methods Craing-Bampton method.

The methods considered in our study must match certain criteria such as the possibility to

access physical displacements of some DOF directly form the reduced order model (ROM).

This condition is necessary due to contact management in order to avoid backward and

forward between the ROM and the finite element model when displacements are corrected.

Among the different CMS methods available, two match this criteria: the Craig-Bampton

(CB) method and the Craig-Chang-Martinez (CCM) method. The fact that they belong to

different categories means that their reduction basis is composed of different kind of com-

ponent modes. This will be an asset in order to obtain non similar ROM. Figure 1.2 presents

∂ Ωu

∂ Ωf0

∂ Ωf1

Ω0∂ Ωf1 : T> 0∂ Ωf0 : T= 0

∂ Ωu : u = 0

T: external forcesu : displacements

Figure 1.2 - General presentation of a structure

the different notations used in the following to describe CMS methods applied on a struc-

ture. The structure is divided into two main parts:

• the boundary, divided into three parts:

1. ∂ Ωu: imposed displacements.

2. ∂ Ωf1: imposed forces.

• Ω0 and ∂ Ωf0: interior of the structure, considering the interior as the components

which are condensed using the CMS methods.

Due to the CMS methods chosen, only two types of methods are presented: fixed interface

methods, and free interface methods. Both consider a separation of the DOF between inter-

nal and boundary DOF. The dynamical behavior of the structure is then approximated with

a basis containing the eigenmodes of the system with a free boundary for the free interface

methods, and with cantilever boundary for fixed interface methods.

The description of CMS methods will be limited to the CB and CCM methods, and clearly

highlights the differences between free interface methods and fixed interface methods.

8 Background

1.2.1 Fixed interface methods - Craig-Bampton method

CB method is a very popular method in aeronautics. In fact, M. Bampton, who worked for

the Boeing Company, and R. Craig, who worked at the University of Texas in Austin, were

both searching new methods to treat in an easier way complex structures. Their results are

published in [CRA 68]. In this document, they present a method to treat a complex struc-

ture as an assembly of distinct regions or substructures. It is necessary to precise that dur-

ing this internship, the CB method will be used without any substructuration. Using basic

mass and stiffness matrices for the substructures, together with conditions of geometrical

compatibility along substructure boundaries, the method employs two forms of general-

ized coordinates. Boundary generalized coordinates give displacements and rotations of

points along the substructure boundaries and are related to the displacement modes of the

substructure known as “constraint modes”. Substructure normal-mode generalized coor-

dinates are related to the free vibration modes of the substructures relative to completely

restrained boundaries. Figure 1.3 illustrates the boundary conditions and the decomposi-

tion of the DOF of a structure into boundary and internal DOF. Figures 1.4 illustrates the two

first constraint modes of this plate, and 1.5 the six first fixed interface modes. In figure 1.3,

∂ Ωf1

∂ Ωf0

clamp

Ω0

∂ Ωu

Figure 1.3 - Boundary conditions and notations used on a plate

the decomposition of the structure is such that ∂ Ωu+∂ Ωf1 stand for the boundary DOF and

Ω0+ ∂ Ωf0, for the internal DOF.

Figure 1.4 - Two first constraint modes for a plate

DOF organization: Let us consider a structure divided into a boundary and an interior.

The use of the FEM leads to mass an stiffness matrices organized following the numbering of

1.2 Component mode synthesis methods 9

(a) mode 1 (b) mode 2

(c) mode 3 (d) mode 4

(e) mode 5 (f) mode 6

Figure 1.5 - First six fixed interface modes of a plate

the nodes composing the structure. It is possible to reorganize these matrices by operations

on rows and columns to obtain Kreorg and Mreorg organized by blocks, with an organization

of the DOF with first the internal DOF and secondly the boundary DOF. To simplify the no-

tations of the problems, we can note:

K=Kreorg and: M=Mreorg (1.28)

The mass and stiffness matrices are organized in such way that the DOF of the boundary

and the internal ones are separated:

Mii Mif

Mfi Mff

qi

qf

!

+

Kii Kif

Kfi Kff

qi

qf

!

=

Fext,i

Fext,f

!

(1.29)

The subscripts “ii” (respectively “ff”) match to the rows and columns linked to the internal

DOF (respectively boundary DOF).

Fixed interface modes: they are computed as matrix containing the fixed interface

modes of the structure when the boundary is clamped. The eigenvalue problem to solve is:

(Kii−ω2Mii)~x= 0 (1.30)

10 Background

Constrained modes: they are the static shapes of the structure obtained applying a unit

effort on a DOF of the boundary and building-in the other boundary DOF. On a numerical

point of view, considering that there is no effort applied on the internal DOF, the matrix of

these boundary generalized modes is:

(−Kii)−1Kif

I

(1.31)

The CB matrix: Φ allows to go from the physical basis to the CB basis

qi

qf

!

=

Φηη Φηf

Φfη Φff

η

qf

!

(1.32)

With

Φηη Φηf

Φfη Φff

=

xr (−Kii)−1Kif

0 I

(1.33)

• I is the identity matrix of dimension (n fnd d l )2 with n f number of nodes on the bound-

ary and nd d l the number of DOF per node,

• xr is the matrix containing the “η” first columns of the matrix x (η=number of modes

kept for the modal reduction).

Reduced stiffness and mass matrices: The stiffness and mass matrices are finally

obtained thanks to these equalities:

Kr =ΦTKΦ Mr =Φ

TMΦ (1.34)

It is then possible to solve the reduced problem:

Mηη Mηf

Mfη Mff

-η

-qf

!

+

Kηη Kηf

Kfη Kff

η

qf

!

=ΦTF (1.35)

1.2.2 Free interface methods - Craig-Chang-Martinez method

As in the CB method, the free interface methods take into account a structure divided into

internal DOF (Ω) and the boundary (∂ Ω) ones. The definitions of both the boundary and

the internal DOF are identical to the one given previously for CB method in 1.2.1. The CCM

method can be seen as an evolution of simpler methods, it will be introduced step by step

by introducing first the Craig and Craig-Martinez method with attachment modes. The fol-

lowing of this subsection is based on several articles, in particular in [CRA 77].

Craig method

Craig method uses a projection of the solution vector on a new basis thanks to:

x=Φ1x1+Φ2x2 (1.36)

1.2 Component mode synthesis methods 11

with Φ1 the matrix whose m columns are the m first normal-modes kept, and Φ2 the matrix

whose columns are the (n −m ) last modes not taken into account. The reduced matrices of

the dynamic problem can be calculated as follows:

KCraig =ΦT1 KΦ1

MCraig =ΦT1 MΦ1

(1.37)

With xCraig = x1 such as:

x=Φ1x1 (1.38)

Craig method is a simple truncation of a normal-mode basis. These modes are obtained

without any force applied on the structure. there are some limitations due to the simple

truncation that is made: in fact, this modal approximation can easily become inefficient to

describe the dynamical behavior of a structure. A classical way to improve the approxima-

tion of the solution is to take into account the influence of the non-kept modes through a

pseudo-static correction.

Pseudo-static correction: Considering equation: M -x+Kx= F. x projected on the trun-

cated modal basis: x = Φ1x1+Φ2x2 where Φi is the truncation of the normal-modes matrix

Φ.

M(Φ1x1+Φ2x2)+K(Φ1x1+Φ2x2) = F (1.39)

Equation (1.39) is then multiplied byΦT1 and then byΦT

2 . Thanks to the M and K-orthogonalities

of the matrices Φi, we obtain the following equations:

(Ω1−ω2I)x1 =Φ1

TF

(Ω2−ω2I)x2 =Φ2

TF(1.40)

Then we proceed to the pseudo-static assumption in equation (1.40), considering that any

pulsation of use ω is way smaller than the minimum eigenpulsation of the normal-modes

not kept in the modal reduction:

∀ ω2 ∈ diag(Ω2),ω<<ω2⇒Ω2x2 =Φ2TF (1.41)

Considering a static problem, and the previous assumption, we can write:

Φ2x2 = (K−1−Φ1Ω

−11 Φ1

T)F (1.42)

The term (K−1−Φ1Ω−11 Φ1

T) is called residual static flexibility and noted R. We finally obtain:

x=Φ1x1+Φ2x2 =Φ1x1+(K−1−Φ1Ω

−11 Φ1

T)F

x=Φ1x1+RF(1.43)

12 Background

Equations: Starting with Mx+Kx= F. The truncation and the pseudo-static correc-

tion introduced earlier involve two independent equations:(

Φ1TMΦ1x1+Φ1

TKΦ1x1 =Φ1TF

Φ2TMΦ2x2+Φ2

TKΦ2x2 =Φ2TF

(1.44)

In fact, to solve the system, we only have to solve the first equation, which corresponds to

the initial equation on the truncated normal-modes basis. We finally obtain x using the

equation (1.2.2).

Craig-Martinez method: Craig’s method has a main drawback compared to the CB

method: it is not possible to have a direct access to the degrees of freedom of ∂ Ω. The way

to access these DOF is complicated: first it is necessary to build the normal-modes basis to

the physical basis. The aim of the CCM method is to correct this, enlarging the unknown

vector of Craig’s method in the modal basis by adding physical degrees of freedom that can

be loaded.

The equation to consider is the equation (1.2.2):

x=

qf

qi

!

(1.45)

This order in the vector x implies a such order in Φ1 and R in the equation (1.2.2):

qf

qi

!

=

Φ1f

Φ1i

x1

+

Rf

Ri

F

(1.46)

Simplifying the notations to obtain a square matrix Rf:

Rf

Ri

F

⇔

Rf1 Rf2

Ri1 Ri2

F1

0

!

(1.47)

We note: Rf =Rf1 and Ri =Ri1

qf

qi

!

=

Φ1f Rf

Φ1i Ri

x1

F

!

(1.48)

Taking into account only the first equation:

F=Rf−1(qf−Φ1fx1) (1.49)

The equation (1.49) enables to define a transformation in order to find boundary degrees of

freedom in the unknown vector:

qf

qi

!

=

Iff 0

RiRf−1

Φ1i−RiRf−1Φ1f

qf

x1

!

=

PCM

qf

x1

!

(1.50)

We define the matrix PCM representative of CCM transformation which enables to go from

the physical degrees of freedom to a vector containing the “modal” displacements and bound-

ary degrees of freedom. The main equation remains

Mx+Kx= F (1.51)

1.2 Component mode synthesis methods 13

The matrices K and M are supposed already reduced to non built-in DOF. By organizing

differently the DOF we obtain the following system:

Mx+Kx= F→

Mff Mfi

Mif Mii

! ¨ qf

qi

!

+

Kff Kfi

Kif Kii

!

qf

qi

!

=

Ff

Fi

!

(1.52)

then we apply the Craig-Martinez transformation:

qf

qi

!

=

PCM

qf

x1

!

(1.53)

Mr

qf

qi

!

+

Kr

qf

qi

!

= (Fr)→

MCM

qf

x1

!

+

KCM

qf

x1

!

= (FCM) (1.54)

with: MCM = PCMTMrPCM, KCM = PCM

TKrPCM and FCM = PCMTFr. The fixed and free inter-

face CMS methods are two different point of view to describe the dynamic of a structure.

The user has to choose pertinently the type of method and the parameters retained. The

description of the two methods done previously is based on several articles, including the

article of Alain Batailly [BAT 07] in which he describes the methods he uses before highlight-

ing the computational convergence observed, the article of Michel A. Tournour, Noureddine

Atalla, Olivier Chiello and Franck Sgard on the convergence of CMS methods [TOU 01] and

the PhD thesis of Adrien Bobillot [BOB 02].

CCM method (with attachment modes): The attachment modes are the static

shapes of the structure when a unit force is applied on a DOF. Adding this modes to the

truncated basis enable to approximated more efficiently the different shapes linked to the

loaded. This enrichment of the modal basis constitutes the gap between Craig-Martinez and

CCM methods.

1.2.3 CMS convergence and comparison

As described in (1.2.1) and (1.2.2), component mode synthesis methods can be very useful to

describe the behavior of complex structures. In fact, what is earned in terms of computation

time using CMS methods must be balanced by the need of results with good accuracy. The

evaluation of this accuracy requires criteria to qualify the convergence of these methods.

Convergence criteria and error estimation

There are several ways to compare modes and eigenvalues. In fact, some criteria tend to be-

come natural when dynamic problems are considered. The criteria described in this para-

graph are used in several works, as in [TOU 01], [BOB 02], and [HUR 71].

Natural frequencies. This is one of the main criterion existing to establish the conver-

gence of component mode synthesis. It is defined usually as follows:

e fi ,CMS =‖ f i,CMS− f i ‖

f i

With f i,CMS the frequency calculated thanks to the CMS method, and f i the frequency calcu-

lated with the classical FE method.

14 Background

Displacements. The shape obtained with and without CMS is way to measure the ac-

curacy of a method. In fact it is useful to compare and estimate the difference between the

displacements obtained with or without CMS reduction.

Such as with the previous criteria, we can define for example an error on longitudinal

displacements for a beam submitted to traction forces:

euk,CMS =‖ uk,CMS−uk ‖

‖uk ‖

With uk,CMS the longitudinal displacement obtained with CMS and uk the longitudinal dis-

placement obtained with the finite element method.

MAC. The Modal Assurance Criterion (MAC) was introduced by D.J.Ewins and is based

on two notions, which are the collinearity and the orthogonality. This criterion is used by

Adrien Bobillot in his PhD thesis [BOB 02] to check the correspondence between the results

obtained by calculation and the behavior observed during experimental tests. It is a natural

correlation criterion which is usually defined as follows:

MACQ,i,j =(ZT

CMS,jQZi)2

(ZTCMS,jQZCMS,j)(Z

Ti QZi)

(1.55)

With Q a ponderation matrix, ZCMS,j the vector mode j calculated with CMS and Zi the vector

mode i calculated with a classical finite element method.

Several choices are possible for the matrix Q:

• Q= I, which corresponds to a classical MAC ;

• Q = M, with M the mass matrix of the initial problem. This corresponds to the M-

orthogonality of the modes calculated.

The second possibility on Q will be useful in the following. In fact it enables to have a sim-

ple criterion to measure convergence ratios: the closer the M-MAC is to 1, the better is the

method.

Comparison of dynamical behaviors

The criteria presented below allow to compare modes calculated with and without CMS. It

is so a way to access to much information at eigenvalue frequencies, but it is not enough to

assure a good description of the dynamical behavior of a structure. To show how accurate a

CMS method is to describe a dynamical behavior, it is necessary to introduce new criteria.

Sinus, K-norm and enriched criterion. Besides of the usual criteria that compare

ROM and FE models in terms of eigenvalues or eigenvectors, it might be of great interest

to evaluate the variations of the dynamic response for a frequency that does not match an

eigenfrequency. Moreover, the usual criteria may be inappropriate to assess ROM computed

with free-interface CMS methods: since in that case the first eigenvectors are included in

the reduction basis, the corresponding eigenfrequencies will be perfectly obtained with the

ROM. For these reasons, we propose the following criteria.

First, let us denote xCMS the displacements vector obtained with CMS, and xFE the dis-

placements vector obtained with a classical finite elements method.

1.2 Component mode synthesis methods 15

A way to analyze the accuracy of the vector xCMS is to calculate its sinus with Ve c t (xFE).

This sinus is defined as follows:

sin(xCMS,Vect(xFE)) =‖ (Id − PVe c t (xFE))xCMS ‖

‖ xCMS ‖

Basically, if this sinus is equal to 0, so ∃ a ǫR∗such that xCMS = a xFE.

However, this criterion can not be used alone to check the accuracy of a CMS method.

It has to be completed with another one, for example a K-norm. This norm is defined as

follows:

xTCMSKxFE

xTFEKxFE

(1.56)

with K the stiffness matrix of the structure. Obviously these two criteria are necessary and

sufficient to have xCMS = xFE, but separately they do not ensure the unicity of the solution.

Energy criterion. In order to assess the convergence of reduced order models, once may

also check that the eigenmodes of the computed ROM match the eigenmodes of the full

finite element model. Such a criterion has been introduced in [BOB 02]. Considering the

assumption that an accurate approximation of the solution searched can be obtained in a

subspace of the initial solving space, the method is organized as follows:

ri = (K−ω2CMS,iM)ΦzCMS,i (1.57)

first, a “residual” vector associated to the i th eigenmode ri is computed, K is the stiffness

matrix of the system, M the mass matrix, Φ the transformation matrix of the CMS method

considered, zCMS,i the i th eigenvector of the ROM andωCMS,i the associated eigenvalue. Vec-

tor ri is homogeneous to an effort, and the associated static displacement can be computed:

rs,i =K−1ri (1.58)

It is underlined in [BOB 02] that other operator than K−1 may be suitable depending on the

application considered. Finally, this residual displacement leads to the strain energy error

indicator:

εi =rT

s,iKrs,i

(ΦzCMS,i)TK(ΦzCMS,i)

(1.59)

εi may be seen as the ratio between the strain energy between the finite element eigenmode

i and the ROM computed eigenmode i and the strain energy of the ROM computed eigen-

mode i . If the eigenmodes of the FE model and the ROM match, εi = 0.

Other criteria used. Other criteria may be found in the literature such as the mean

square velocity, or mode shapes. This criterion is used in [TOU 01] to assess the accuracy

of different CMS methods on structures like plates, beams and boxes. An other criterion

noticed by Hurty in [HUR 71] is the local stresses. It is described as a slow convergence in all

free interface methods.

16 Background

1.2.4 Conclusion

The use of CMS methods allows to drastically reduce the size of heavy industrial finite ele-

ment models and consequently leads to minimize computation times. Both methods intro-

duced in this chapter have already been used and studied in the case of 2D and 3D models

of bladed disks and casing [BAT 07, LEG 05]. The aim of the work presented in this report

is to qualify and test these methods in 2D and 3D configurations among certain criteria: (1)

investigating in details the detection of modal interaction and its sensitivity toward the CMS

method used and (2) evaluating the most appropriate method for studying modal interac-

tion with 3D models.

The next chapter of this report focuses on the 2D study, the results of our study are

shown after a brief state of the art of detection of modal interaction with 2D models.

2.1 Modeling 17

2.2 Previous work 17

2.2.1 Mono-harmonic studies S1 182.2.2 Intermediary subspace S2 19

2.3 Convergence 20

2.3.1 Damped motion 202.3.2 Locked motion 20

2.4 Energy balance 24

2.5 Two-diameter study 28

2.6 Three-diameter study 31

2.7 Conclusion 32

22D model

2.1 ModelingThe model used in our study is composed of a casing made of curved beams and a bladed

disk with straight beams. Figure 2.1 illustrates the 2D model in the case where the radius of

the disk has is made of one single straight beam, and a blade is composed of four elements.

blade j-1

blade j

blade j+1

x j-1

x j

yj-1

yj

y0

Φ

x0

casing

Figure 2.1 - 2D model

2.2 Previous workOur study is the continuation of previous work with the 2D contact algorithm in order to de-

tect modal interaction cases between a bladed disk and a casing. More precisely, the contact

algorithm has been initially developed and introduced in [LEG 05], then a parametric study

has been performed in a mono-harmonic subspace to detect modal interaction in [IDO 05].

More recently, CMS methods have been combined with the contact algorithm in order to

avoid kinematic restrictions. Validation of this combination is shown in [BAT 07].

As mentioned above, the first studies made on 2D models of bladed disks and casing

used kinematic restrictions in order to improve computation times. These restrictions are

18 2D model

introduced by the projection of each structure on its two first n-nodal diameter free vibra-

tion modes meaning that each structure is represented by a 2-DOF system. While reducing

drastically computation times, these restrictions allowed to detect several interaction mo-

tions described in the following of this report.

Studies have been made to evaluate the impact of these kinematic restrictions on the

detection of modal interaction. In order to achieve this goal, three different spaces are con-

sidered to detect interactions, from the simplest – in which one both structures are projected

on their two first n-nodal diameter free vibration modes – to the most complex considering

a ROM of the bladed disk and the finite element model of the casing:

1. Subspace S1: this space implies kinematic restrictions on both the bladed disk and the

casing. The structures are projected on their two first n-nodal diameter free vibration

modes. This means that the dynamic of the system bladed disk/casing can only be

expressed with the two first n-diameter modes of each structure, which implies that

only the n t h harmonic is involved in the behavior of the system.

2. Subspace S2: intermediary space, no kinematic restriction is applied on the bladed

disk since a ROM is computed using CMS methods but the casing is still projected on

its two first n-diameter modes.

3. Subspace S3: a ROM is computed for modeling the bladed disk, the finite element

model of the casing is used. As a consequence, computations are very time-consuming.

2.2.1 Mono-harmonic studies S1

In this type of study, the casing and the bladed disk are both projected on their n-diameter

modes. Calculus are launched on two and three-diameter modes. These projections mean

an important assumption on the behavior of the two entities. Several results have been high-

lighted in [LEG 05], among them, an important condition is given for observing interaction

cases:

Ω≥ωc+ωra

n d(2.1)

with Ω the angular velocity of the bladed disk, ωc the eigenfrequency of the casing at the

number of diameters considered n d, ωra the eigenfrequency of the bladed disk at the num-

ber of diameters considered n d. This study showed the different behaviors of the system

bladed disk/casing:

1. locked: at the beginning of the simulation, several contacts occur. Then, one or sev-

eral blades stay in permanent contact with the casing, and both structures embrace,

due to an important coupling between the dynamical behaviors of the two struc-

tures. This behavior was first observed in [LEG 05] and then in [IDO 05] in the mono-

harmonic space;

2. damped: this behavior is observed as soon as the load is not strong enough, or as

soon as the friction coefficient or the angular velocity are too low. Despite the initial

2.2 Previous work 19

load and the first contacts between the casing and the bladed disk, the two structures

do not interact, and the vibrations of the system are damped until the initial gap is

achieved;

3. sustained: this kind of motion appears in the first studies when the number of blades

can not be divided by the number of diameters considered to load initially the casing.

It was observed in [IDO 05]. It is a very typical case, where the two structures do not

embrace but with many pseudo-periodic contacts;

4. divergent: can be observed if the angular velocity or the friction coefficient is too high.

In fact, the interaction between the two structures is so strong that the phenomena

is not only sustained, but amplified during the simulation. Practically, this kind of

behavior could be the worst possible for the engine ; but in our case, this can simply

be a divergence due to numerical problems.

L. Idoux pursued the study in the mono-harmonic space and obtained interesting results

emphasizing the link between the possible interactions and the geometry of the structure.

First, the large numbers of simulation he launched shew that locked motion can be ob-

served only if the number of blades is divisible by the number of diameters; if it is not, only

sustained behaviors can be seen. Secondly the coefficient of modal damped of the blades

have no influence on the results. Thirdly, he obtained that increasing the friction coefficient

shifts the areas of sustained behavior to low velocities. Finally a study in the curvature of the

beams showed there is a minimum curvature to observe sustained behaviors.

These results where confirmed by A. Batailly in his different first studies in the subspace

S1. Results have been obtained for a number of diameters between two and seven.

2.2.2 Intermediary subspace S2

The second step on the way to industrial application is the use of CMS methods to avoid time

consuming calculations. In the space of study named “S2”, the casing is projected on its n-

diameters modes (here n is 2) and the bladed disk is reduced thanks to CMS. As a result, it

appears that the interaction areas observed in this subspace are included in the interaction

zones observed in the mono-harmonic space. It is also noticed that the interaction appears

in this space for an angular velocity higher than in the mono-harmonic space. It appears

necessary to increase the efforts applied on the casing to detect an interaction.

Following the mono-harmonic study made by L. Idoux [IDO 05] on a 2D model, and the

work of A. Batailly to implement CMS methods into contact codes, this part deals with the

study of a 2D model using the CMS methods. The model used here is nearly the same than

in the previous studies 2.2.1. The bladed disk has 22 blades. The only difference with the

model used by M. Legrand is that the disk now modeled with curved beams instead of simple

beams. The number of curved beams in the radial direction is 1. Each blade has a fixed

number of beams elements with 6 DOF (u , v and θ on each node). The global curvature

of the blade is obtained considering an angle a i between the beams i and i + 1. The angle

between two bladesφ = 2π/22. The casing is modeled as a cylinder and composed of curved

beams with 2 nodes. The table 2.1 presents the mechanical properties of the models. The

20 2D model

Casing Bladed disk

Young Modulus Ec = 7 ·106 Eb = 2.1 ·1010

density (kg·m−3) ρc = 2800 ρb = 7800thickness hc = 0.005 hb = 0.005wide wc = 0.05 wb = 0.05radius Rc = 0.2505 Rb = 0.25modal damped ξc = 0.03 ξb = 0.005number of blades N= 22

Table 2.1 - Properties of the 2D models

aim of this part is to underline the effects of the use of CMS in this case, and to characterize

the convergence ratio and the different behaviors noticed compared to the mono-harmonic

study.

2.3 ConvergenceThe aim of this section is to demonstrate that the contact code is relevant when if CMS

methods are used. Applying successively CB and CCM method, several calculations have

been launched among an iterative process: for a unique interaction case, the modal reduc-

tion basis varies from 0 to 220 fixed/free interface modes (increment of 22).

2.3.1 Damped motion

The convergence of CB and CCM methods has been evaluated with different parameters.

First, it was necessary to check that below the critical angular velocity, we had a fast con-

vergence of the CMS methods with a damped behavior. Secondly, the convergence on two

points of the critical zone has been tested, to check if the CMS methods had a real influence

on the interaction bladed disk/casing observed above the critical angular velocity.

Figures 2.3(a) and 2.3(b) have been obtained with a angular velocity of 1 and a friction

coefficient: µ= 0.3. The load initially applied on the casing is a two-diameter one, and since

in this case Ωc = 1.075, we are below the critical velocity. Consequently, there is no reason

to observe any interaction. This is confirmed by the two figures, with only damped modes,

and a fast convergence of the CMS methods. The calculation made over 1 second shows the

stability of the system, with the bladed disk/casing distance approaching very quickly the

initial tip-clearance of 0.5.

Figures 2.4(b) and 2.4(a) have been obtained for a angular velocity of 1.35 and a friction

coefficient: µ = 0.2. The casing is initially loaded with a three-diameter effort. Since in the

three-diameter case, Ωc = 1.225, we are in the critical zone. In this case it is clear that a fast

convergence of the CMS methods is achieved. In both CB and CCM methods, we observe a

damped behavior despite the fact that the angular velocity is over the critical one Ωc.

2.3.2 Locked motion

In order to ensure the quality of the model, convergence must be checked in the case of an

interaction motion. Because of its high numerical sensitivity, it has been decided to focus

2.3 Convergence 21

0 2 4 6 8 100

0.35

0.7

1.05

1.4

1.75d

ista

nce

s

time ×103

Figure 2.2 - Blade tip 1/casing distance for CB method (Ω = 1,µ = 0.3); η = 0 ( ); η = 44( ); η= 88 ( ); η= 220 ( )

0 10

0.35

0.7

1.05

1.4

1.75

dis

tan

ces

time ×103

(a) Blade tip 1/casing distance for CBmethod

0 10

0.35

0.7

1.05

1.4

1.75

dis

tan

ces

time×103

(b) Blade tip 1/casing distance for CCMmethod

Figure 2.3 - Blade tip 1/casing distance for CB and CCM method (Ω= 1,µ= 0.3)

on the convergence of the reduced order models in the case of a locked motion.

Using CB method, the simulations are launched with Ω= 1.9 and µ= 0.2. Figures 2.5(a),

2.5(b), 2.6(a) and 2.6(b) enable to observe locked modes, whatever the value of η, reduction

parameter. The convergence is obtained as we increase the number of modes in the reduc-

tion basis. Figure 2.5(b) highlights that the blades which are in permanent contact with the

casing depend on the number of modes in the synthesis. For example, from 88 to 220 con-

straint modes in CB method, after 5 ms, the blade number 5 is constantly in contact with

the casing, whereas it vibrates when only 44 modes are kept for the CMS.

As a conclusion, increasing of the number of constraint modes kept in the modal reduc-

tion basis enables to get a better approximation of the displacements on blade tips. An in-

sufficient modal basis can lead to a gap in the initiation of the locked modes. In the present

example, the initiation happens on blades 8 and 19 or on blades 9 and 20.

The convergence of the results is more difficult to obtain with CCM method. In fact, the

method is more sensitive to numerical issues. Figures 2.7(a), 2.7(b) et 2.7(c) highlight those

numerical problems: even if the motion of the system is locked, the blades in permanent

contact with the casing vary depending on the modal reduction basis.

22 2D model

0 10

0.35

0.7

1.05

1.4

1.75

dis

tan

ces

time ×103

(a) Blade tip 7/casing distance for CB method

0 10

0.35

0.7

1.05

1.4

1.75

dis

tan

ces

time×103

(b) Blade tip 7/casing distance for CCM method

Figure 2.4 - Blade tip 7/casing distance for CB and CCM method (Ω = 1.35,µ = 0.2); ; η = 0( ); η= 44 ( ); η= 88 ( ); η= 220 ( )

0

0 1

2 4 6 8 100

0

0.35

0.35

0.7

0.7

1.05

1.05

1.4

1.4

1.75

1.75

dis

tan

ces

dis

tan

ces

time ×103

time ×103

(a) Blade tip 1/casing distance for CB method

0

0 1

2 4 6 8 100

0

0.35

0.35

0.7

0.7

1.05

1.05

1.4

1.4

1.75

1.75

dis

tan

ces

dis

tan

ces

time×103

time×103

(b) Blade tip 5/casing distance for CB method

Figure 2.5 - Blade tip 1 and 5/casing distance for CB method; (Ω= 1.9,µ= 0.2); η= 44 ( );

η= 88 ( ); η= 220 ( ); u ef(t ) ( )

2.3 Convergence 23

0

0

1

2 4 6 8 10

0.35

0.35

1.05

1.05

1.75

1.75

dis

tan

ces

dis

tan

ces

time ×103

time ×103

(a) Displacement u on the tip of blade 1 for CBmethod

0

0 1

2 4 6 8 10

-0.5

-0.5

0

0

0.5

0.5

dis

pla

cem

ent

dis

pla

cem

ent

time×103

time×103

(b) Displacement v on the tip of blade 1 for CBmethod

Figure 2.6 - Evolution of the displacement u (top) and v (bottom) on the tip of blade 1 and

focus between t = 0 and t = 1000 for CB method; (Ω= 1.9,µ= 0.2); η= 44 ( );

η= 88 ( ); η= 220 ( ); u FE(t ) ( )

24 2D model

0 200 400 600 800 1000

time

0

0.5

1

1.5

2

2.5d

ista

nce

s

(a) Blade tip 2/casing distance

0 200 400 600 800 1000

time

0

0.5

1

1.5

dis

tan

ces

(b) Blade tip 6/casing distance

0 200 400 600 800 1000

time

0

0.5

1

dis

tan

ces

(c) Blade tip 8/casing distance

Figure 2.7 - Blade tip 2,6 and 8/casing distance for CCM method (Ω = 2.5,µ = 0.1); φ = 44( ); φ = 88 ( ); φ = 220 ( ); u FE(t ) ( )

These studies on the convergence of the use of CMS methods highlight several points:

outside the critical zone, the convergence is fast. Also we can observe a fast convergence

to a case of damped motion with both CMS methods when the angular velocity is lower

than Ωc and the friction coefficient is low. The convergence is also obtained clearly for CB

method on locked modes whenΩ≥Ωc. The convergence is only obtained in terms of general

behavior with CCM method, with locked modes observed in every simulations, but initiated

on different blades as we enrich the CMS.

2.4 Energy balanceThe energy balance has not yet been taken into account in our study. Insuring the con-

servation of energy among time is necessary to confirm the quality of the algorithm used,

showing that it is non dissipative. The calculations made and the method introduced in the

following are not depending on the model considered and will be applied for the 2D model

as well as for the 3D model:

• kinetic energy: to define the kinetic energy, we need to calculate the velocity of each

DOF during the computation. Taking into account that the contact algorithm uses a

centered scheme, the velocity at time n is the mean of the displacement at time n −1

2.4 Energy balance 25

and n +1,

_xn =xn+1−xn-1

2∆t(2.2)

with ∆t the time step of the simulation. As a consequence we define the classical

kinetic energy:

Ec = _xT·M · _x (2.3)

with M the mass matrix of the system, and _x the velocity at the moment considered

• strain energy: it is define more easily with the stiffness matrix and the displacement:

Ed = xT ·K · x (2.4)

• damped energy: the damped energy is an energy absorbed by the system, and so it is

necessary to consider a sum of energies dissipated between each time. To compute

this easily, we introduce a damped power, which is integrated over the time:

Ed =

∫

_xT ·D · _xdt ≃∆t

N∑

i=0

_xiT ·D · _xi (2.5)

with h the time step, D the damped matrix and N the total number of time step in the

simulation.

In order to confirm the good behavior of the contact algorithm on the 2D model, several

computations have been launched with the parameters given in table 2.2. Some assump-

initial conditions for the casing displacement conditions over a 2-diameter modeangular velocity Ω= 1.2loading time between t = 0 and t = 20bladed disk reduction CB with 66 constraint modescasing model FEtime step ∆t = 5 ·10−4

gap 0.5damping ξRA = 0.005 and ξC = 0.03friction µ= 0 for the first simulationsinitial conditions for the casing displacement conditions over a 2-diameter mode

Table 2.2 - Simulation parameters

tions are made for the energy study: the centrifugal efforts are not taken into account, and so

is the kinetic energy due to the global rotation of the bladed disk. In fact, this is in agreement

with the algorithm which considers neither the centrifugal effects nor the kinetic energy due

to the global rotation.

Figure 2.8 presents the distances blades/casing during the first simulation launched

without friction. This case is a good way to check the characteristic non dissipative of the

algorithm, because theoretically the total energy has to be constant despite the contacts.

26 2D model

0 20 40 60 80 100time

-0.25

0

0.25

0.5

0.75

1

1.25

dis

tan

ces

contact line

Figure 2.8 - Blades/casing distances - Energy balance: distances ( ), areas ( ) corre-

spond to the moment when one or several blades are in contact with the casing

0 20 40 60 80 100

time

0

10

20

30

40

ener

gy

(a) Energy balance of the casing

0 20 40 60 80 100

time

0

2

4

6

8

ener

gy

(b) Energy balance of the bladed disk

0 20 40 60 80 100time

0

10

20

30

40

ener

gy

ε=−3.7%

(c) energy balance of the whole system

Figure 2.9 - Energy balances or the 2D model: for each balance the total energy ET ( ) is

computed as the sum of damped energy Ea ( ), kinetic energy Ec ( ) and

strain energy Ed ( ). Areas ( ) correspond to the moment when one or sev-

eral blades are in contact with the casing

2.4 Energy balance 27

Figure 2.9(a) presents a logical evolution of the energies on the casing. Launched initially

on a two-diameter mode, the casing tends to vibrate and both the kinetic energy Ec and the

strain energy Ed oscillate with a phasing of π. We can see that because of the contacts (four

zones), the total energy of the carter decreases. This is due to a transfer between the casing

and the bladed disk, which can be seen in the different increases of the total energy of the

bladed disk in figure 2.9(b). The damped energy on the two structures can also be seen on

figures 2.9(a) and 2.9(b), in agreement with the damped behavior of the system.

Figure 2.9(c) shows the energies on the whole system. We observe a low decrease of

the total energy at each contact, due to numerical approximations in the detection of the

contact. This highlights the sensitivity of the contact algorithm, which remains depending

on many numerical parameters as the time step or the length of the elements, etc.

Theoretically, the friction of the blades on the casing acts against the rotation of the

bladed disk, implying a lose of energy by dissipation. This analysis can not be observed in

our case, because the angular velocity of the bladed disk is forced to be constant, meaning

that the kinetic energy of the bladed disk may increase to be match the condition of con-

stant angular velocity. Another phenomenon may occur due to the contact algorithm used:

we can obtain an increase of the displacement on blades’ tips during the contacts and an

increase of the energy transferred to the casing (EcCA and EdCA ), not taking into account any

decrease of the angular velocity of the bladed disk (EcRA ) due to friction effects. This leads

to an increase of the total energy while µ increases. This assumption is confirmed by figure

2.10. This energy study shows the good behavior of the contact algorithm. Despite a low

0 20 40 60 80 100time

0

10

20

30

40

50

ener

gy

µ

µ

µ

Figure 2.10 - Evolution of the total energy ET ( ) of the system bladed disk/casing depend-

ing on the friction coefficient µ. The evolutions of the total energy of both the

casing ETCA( ) and the bladed disk ETRA ( ) highlight that an increase of

the friction coefficient implies an increase of the total energy of the bladed disk.

Areas ( ) correspond to the moment when one or several blades are in contact

with the casing

decrease of the total energy at each contact without friction, it remains very efficient to de-

scribe the interaction between the casing and the bladed disk. Another aspect highlighted

by this study is the effect of the coefficient of friction, which increases the total energy of the

system. This last phenomenon had been anticipated, and is clearly due to strong assump-

tions on the dynamics of the system.

28 2D model

2.5 Two-diameter study

In this section, the casing is projected on its two first two-diameter modes, and the bladed

disk model is reduced thanks to CMS methods. The problem is so considered in the inter-

mediary subspace S2. The casing is load initially during 2 · 10−1 with a two-diameter effort,

and is then released to let it interact with the bladed disk.

Figure 2.11 - Shape n d=2

A representation of the shape of the casing is

presented in figure 2.11: the casing initially cir-

cular is loaded with a two-diameter effort and

adopt an elliptic shape. The simulations pre-

sented in this section last over a time of 500.

Figures 2.12(a), 2.12(b), 2.12(c) and 2.12(d) high-

light the different behaviors adopted by the sys-

tem bladed disk/casing. On each figure, the

curve ( ) is the blade tip 13/casing distance and

the curve ( ) is the blade tip 20/casing distance.

In fact, as it was already mentioned in 2.2.2, there

are four main kinds of motions:

1. Damped

2. Blocked

3. Sustained without locking

4. Divergent

Figure 2.12(b) shows a sustained mode has been obtained when the casing is loaded with a

two-diameters effort, for a angular velocity equal to 1.25 and a friction coefficient of 0.2. Ob-

serving a sustained motion with a bladed disk composed of 22 blades and with a 2-diameter

load may not appear compatible with M. Legrand’s previous conclusions. It is remarkable

that only one sustained motion is observed, only with the CCM method. As a conclusion

we assume that this sustained motion confirms the high numerical sensitivity of the CCM

method. This kind of mode, rarely observed and only when CCM method is concerned,

proves the influence of the CMS method chosen to reduce the size of the system, and im-

plies also questions about the relevance and the accuracy of the methods.

As we can see in figure 2.12(c), a damped behavior implies that the distance between

each blade and the casing tends to come back to the initial tip-clearance of 1 mm. Figure

2.12(d) illustrates a divergent behavior.

The figure 2.13(a) has been obtained using the CMS method of CCM, with 66 attachment

modes and 44 free interface modes. The figure 2.13(b) has been obtained using the CMS

method of CB, with 66 constraint modes and 44 fixed interface modes.The figure 2.14(b)

has been obtained using the CMS method of CCM, with 66 attachment modes and 88 free

interface modes.The figure 2.14(a) has been obtained using the CMS method of CB, with 66

constraint modes and 88 free interface modes.

2.5 Two-diameter study 29

dis

tan

ce

1.5

1.25

1.0

0.75

0.5

0.25

0.0

500 1000 1500 2000time

(a) locked mode

1.0

0.75

0.5

0.25

0.0

dis

tan

ce

500 1000 1500 2000

time(b) sustained mode

dis

tan

ce

1.0

0.75

0.5

0.25

0.0

500 1000 1500 2000time

(c) damped mode

35

30

25

20

15

10

5

0

dis

tan

ce

200 400 600 800 1000

time

(d) divergent mode

Figure 2.12 - Examples of locked, sustained, damped and divergent motions

The maps presented use different colors to underline the fact that several behaviors can

be observed. Areas [ ], [ ], [ ] and [ ] respectively correspond to damped, sus-

tained, locked and divergent motions. Thanks to a modal calculation on the structure, we

can define the critical angular velocity for the system, which is equal for a two-diameters

excitation, to

Ωc =ωc+ωra

n d= 1.075 (2.6)

in the present case. Below this critical velocity, the system always reacts following a damped

mode. It was already one of the conclusion presented by Mathias Legrand in the mono-

harmonic study, and it is here one of the first significative results of the use of CMS methods.

Another aspect highlighted by 2.13(a) and 2.13(b) is the role of the CMS method used. In fact,

even if the different behaviors of the structures are localized in similar zones on both maps,

there are some differences between the results obtained with the two methods. CB method

seems to be more favorable to obtain an interaction between the casing and the bladed disk.

In fact, there are more locked conditions when this method is used than with CB method,

and the zone containing these locked modes on the figure are more homogeneous than with

CCM method.

Taking into account the cyclic symmetry of the structures, it is important to use effi-

ciently the CMS methods, and to chose the number of modes kept for the synthesis as ef-

ficiently as possible. The bladed disk is composed of twenty-two blades, and the decom-

30 2D model

Ω4

3.5

3

2.5

2

1.5

1

0.5

0.1 0.3 0.5 0.7 0.9

µ

(a) 2-diameter load, CCM, 44 fixed interfacemodes, 66 attachment modes

0.1 0.3 0.5 0.7 0.9

µ

Ωc

(b) 2-diameter load, CB, 44 constraint modes

Figure 2.13 - n d = 2: Maps of behaviors for CB method with 44 constraint modes and Craig-

Chang-Marinez with 44 free-interface modes; Ωc = 1.075

position of the DOF and the relationship between boundary and internal DOF imposed us

to chose and add modes by groups of twenty-two modes to ensure a certain homogeneity

in the calculation. The figures 2.14(b) and 2.14(a) are realized using now 88 fixed or free

interface modes.

Comparing 2.13(b) and 2.14(a) enables to see the influence of the number of constraints

modes considered for the synthesis with CB method. The maps look really similar, and the

increase of the number of constraint modes kept do not influence the results obtained. This

highlights the efficiency of CB method, which enables to approach the dynamical behavior

of the structure with a low number of constraint modes in the synthesis.

The comparison between 2.13(a) and 2.14(b) highlights differences between the results

obtained enriching the modal basis. Some differences can qualify the influence of the num-

ber of constraint modes. In fact, the comparison between the two maps shows that the

increase of the number of modes in the synthesis gives more homogeneous results. CCM

method is highly sensitive numerically speaking, and on 2.13(a) it appears that the modal

reduction basis is not rich enough to describe the behavior of the system: we observe sus-

tained modes, rare and dispersed on the map, which are in contradiction with the conclu-

sion of Legrand and Idoux for a two-diameter load. This sensitivity of the method is due to

the attachment modes: these modes are obtained by putting one the DOF of the boundary

to 1, and so the two attachment modes concerning two close nodes can be, by definition,

quasi-equal numerically, and lead to singular mass and stiffness matrices in the algorithm.

The two-diameter study in the intermediary subspace S2 has given significant results

on the use of CMS methods. As it was already observed in the mono-harmonic studies (S1

subspace), there are clearly four kind of motions may be observed for the system depending

2.6 Three-diameter study 31

Ω4

3.5

3

2.5

2

1.5

1

0.5

0.1 0.3 0.5 0.7 0.9

µ

(a) 2-diameter, CB, 88 constraint modes

0.1 0.3 0.5 0.7 0.9

µ

Ωc

(b) 2-diameter load, CCM, 88 free inter-face modes, 66 attachment modes

Figure 2.14 - n d = 2: Maps of behaviors for CB method with 88 constraint modes and Craig-

Chang-Marinez with 88 free-interface modes; Ωc = 1.075

on the simulation parameters. This study also emphasizes the stability and the efficiency of

CB method, and the sensitivity of CCM method.

2.6 Three-diameter study

In this section, the casing is projected on its two first three-diameter modes, and the bladed

disk is reduced using CMS methods. The initial load applied on the casing during 2 · 10−1

is a three-diameter effort. Figure 2.15 shows an example of shape of the casing under a

three-diameter effort. The aim here is to do a similar study as in the two-diameter case,

and to highlight different behaviors. In fact, according to 2.16(a) and 2.16(b), there are no

real interaction between the casing and the bladed disk and we only observe damped or

divergent behaviors, probably due to a too heavy load. Following the results obtained, other

calculations have been launched to see if the number of modes taken into account in the

CMS method had an influence on the global aspect of the map obtained. The figure 2.17(a)

and 2.17(b) have been obtained considering 88 free or fixed interface modes, depending on

the CMS method used.

The first conclusion is that even if a certain influence can be observed on the behav-

ior of the structure, the increase of the number of modes considered does not improve the

quality of the interaction between the casing and the bladed disk. It seems that in the three-

diameter case, the bladed disk has not the good specifications to interact strongly with the

casing.

32 2D model

Figure 2.15 - Shape n d=3

Ω

4

3.5

3

2.5

2

1.5

1

0.5

0.1 0.3 0.5 0.7 0.9

µ

(a) 3-diameter load, CCM, 44 fixed inter-face modes fixes, 66 attachment modes

0.1 0.3 0.5 0.7 0.9

µ

Ωc

(b) 3-diameter load, CB, 44 constraintmodes

Figure 2.16 - n d = 3: Maps of behaviors for CB method with 44 constraint modes and Craig-

Chang-Marinez with 44 free-interface modes; Ωc = 1.225

2.7 Conclusion

to validate and check the convergence of the contact algorithm used with linear CMS meth-

ods in the case of contact non-linear simulations. To sum up, we have obtained a conver-

gence on displacements for both CB and CCM methods, with damped conditions, and more

complex, with locked conditions. This highlights the efficiency of both the CMS methods

and the contact algorithm used in the study.

Secondly, the maps obtained to compare the effects of CMS methods on the global be-

havior of the system bladed disk/casing are sufficient to have clear observations: CB is fast

convergent, even with locked conditions, whereas the results obtained with CCM method

are numerically sensitive, and the convergence is more difficult to obtain. We assume that

the origin of this numerical sensitivity is linked with attachment modes. Work is in progress

2.7 Conclusion 33

Ω4

3.5

3

2.5

2

1.5

1

0.5

0.1 0.3 0.5 0.7 0.9

µ

(a) 3-diameter load, CCM, 88 fixed interfacemodes, 66 attachment modes

0.1 0.3 0.5 0.7 0.9

µ

Ωc

(b) 3-diameter load, CB, 88 constraintmodes

Figure 2.17 - n d = 3: Maps of behaviors for CB method with 44 constraint modes and Craig-

Chang-Marinez with 44 free-interface modes; Ωc = 1.225

to evaluate more precisely the impact of mathematically similar modes (such as attachment

modes) on the quality of the modal reduction.

Finally, an energy study allows us to confirm the good behavior of the algorithm, with

a low dissipative aspect (without friction) and a low increase of total energy due to the as-

sumption of a constant angular velocity of the bladed disk when friction is considered.

The results obtained will be taken into account for the following of this study: the study

of modal interaction on 3D industrial models.

34 2D model

3.1 Model presentation 35

3.1.1 Boundary conditions 353.1.2 Contact algorithm specifications 36

3.2 CMS methods Convergence 36

3.2.1 Modal analysis 363.2.2 Contact contribution 373.2.3 Consequences 39

3.3 Energy balance 39

3.4 Friction contribution 41

3.5 Sensitivity to rotational velocity 43

3.5.1 CB and CM methods 443.5.2 Angular velocities of interest 46

3.6 Conclusions 49

33D model

Numerical methods and convergence criteria have been introduced in previous chap-

ters in the case of a 2D model of bladed disk and casing. Considering the interesting results

we obtained, a logical continuation of this study is the application of these methods to in-

dustrial 3D models that require higher complexity and more precised parameters. These

industrial models have been obtained from Snecma as finite element mesh files. The point

of this chapter is first to present the obtention of the reduced order models computed from

these finite element models and to evaluate the convergence of these reduced models to-

ward the finite element ones. Subsequently, we will process to the evaluation of the influ-

ence of several parameters such as angular velocity of the bladed disk and the friction coef-

ficient on the dynamic behavior of the structures. One of the point of this study is to assess

the use of a free-interface CMS method (CM method) comparing it to a fixed-interface CMS

method (CB method). The pro and cons of each method are considered to justify which

method is more appropriate for studying modal interaction on 3D industrial structures.

3.1 Model presentationThe size of the finite element models justifies by itself the use of CMS methods: the finite ele-

ment model of the bladed disk is composed of NBD DOF spread over the 56 sectors (one blade

per sector) and the casing is composed of NCA DOF spread over 112 sectors. Consequently,

the use of FEM method for studying contact interactions between the two structures would

lead to excessive time consuming computations. The assumption is made that linear CMS

methods such as the Craig-Bampton and Craig-Martinez ones can be used for non-linear

contact simulations. This assumption is strengthened by the convergence results obtained

in the case of 2D models contact interactions 2.3. The industrial models are pictured in

figure 3.1.

3.1.1 Boundary conditions

Specific boundary conditions are applied on both the casing and the bladed disk. These

boundary conditions are slightly simplified from the typical industrial ones usually applied

on this kind of structures since we only consider built-in nodes. Aware that the FE model

of the casing used only represents a little part of the whole structure, we choose to clamp

the casing FE model on both side. The bladed disk is practically linked to a rigid gear, link

modeled here by clamps.

36 3D model

Figure 3.1 - Representation of the 3D model of the system bladed disk/casing

3.1.2 Contact algorithm specifications

The contact algorithm has already been explained, but several things have to be precised

concerning the industrial application. Three nodes are chosen on the top of each blade for

managing contact forces. In the present study, on node belongs to the leading edge, one

is at the middle of the chord and the last one belongs on the trailing edge. Computation

of contact forces implies to locate at any time on the casing the projection of the nodes

on the tip of the blades. The shape of the bladed disk is modeled in the 3D code with a

spline, parametric surface which takes into account the position of the nodes selected on

the casing. This group of nodes contains 4 nodes by sector1 of the casing. As pictured in

figure 3.2: in green, we can see the nodes selected on the casing, 4 per sector, and in red the

nodes selected on the blades tips. The arrows from the blade tip to the spline illustrate the

projection of the red nodes on the spline used to identify the carter element crashed and

compute efficiently the blade tip/casing distance.

3.2 CMS methods Convergence

3.2.1 Modal analysis

Reduced order models of the 3D industrial models have been computed by Alain Batailly

during his PhD thesis. He implemented CMS methods on the 3D model and checked the

convergence of the methods in terms of eigenfrequencies for the bladed disk and the casing,

and in terms of energy criteria (presented in 1.2.3, and in [BOB 02]) on the casing2.

1The representation of the casing on figure 3.2 only explains visually the positions of the differentboundary nodes. With 10 lines of 4 nodes, we can see 9 sectors of the casing.

2Computing energy criteria on the bladed disk model leads to very time consuming computationsdue to the number of DOF of the FE model.

3.2 CMS methods Convergence 37

casing

bladed disk

spline

Figure 3.2 - Entities of interest for the contact calculation

We focus on the convergence of the eigenfrequencies relative to the first bending and

torsion modes for the bladed disk. This represents 112 modes (2 per blade). He obtained

a very fast convergence on frequencies, and the gap between FE eigenfrequencies and CB

ones is lower than 0.5 for every modes as soon as 6 groups of 56 modes are kept for the CMS

method. About the error indicator considered on the casing, it tends to 0 for each of the 112

first modes as we enrich the modal basis, and the eigenfrequencies gap also tends to 0.

The gap between FE eigenfrequencies and CM3 ones is lower than 0.5% for every modes

as soon as 448 free interface modes are kept for the CMS method. About the error indicator

considered on the casing, it converges to 0 for each of the 112 first modes as we enrich the

modal basis. So do the frequencies gap.

3.2.2 Contact contribution

Once convergence of CB and CM reduced models is observed in terms of eigenfrequencies

and eigenmodes, it is necessary to check the convergence of the results for a contact simula-

tion. The goal is to evaluate the size of the modal basis required for our contact simulations

for each CMS method. The simulations are launched with a casing loaded such as the pen-

etration of the casing into the bladed disk would be 0.1 mm4.

CB method

Figure 3.3 presents the blade tip 1 leading edge node/casing distance depending on the

number of constraint modes families kept for CB method. As we can see, a good approx-

imation of the behavior of the system is achieved when the number modes exceeds 336.

3Although in the 2D study CCM was used, in the 3D study it is only CM method. In fact, thenumerical problems noticed in 2.3.2 due to attachment modes lead to singular mass and stiffnessmatrices, and the choice has been made to delete the attachment modes and recover a CM methodin order to avoid numerical problems.

4The load on the casing is applied among a 2-diameter vibration mode. The amplitude of this loadis computed so that the static deformation of the casing under that load (without considering thebladed disk) would lead to a penetration of 0.1 mm between the two structures.

38 3D model

More information can be obtained considering the Fourier transform of the radial displace-

0

0.5

1

1.5

2

10−4

50 100 150 200 250

dis

tan

ce

time

Figure 3.3 - Blade tip 1 leading edge node/casing distance depending on the size of the modal

reduction basis for CB method; η = 44 ( ); η = 88 ( ); η = 220 ( );

u e f (t ) ( )

ment of the leading edge node on blade tip 1. This gives us more details on the frequency

response of the system, particularly for the bladed disk. Figure 3.4 illustrates the evolution

of this response depending on the number of free interface modes kept, and shows clearly

the convergence of the frequency response as we increase the number of constraint modes.

For over 336 constraint modes in CB method, we have only low amplitudes variations as

we increase the number of constraint modes in the modal reduction basis, and we obtain

consequently a good approximation of the behavior of the system.

0 5 10 15 20 25 30 35 40 45 50

0

0.5

1

1.5

2

2.5

3

3.510−5

frequency

amp

litu

de

Figure 3.4 - Fourier transform of the radial displacement of the leading edge node of blade 1

depending on the size of the CMS for CB method; η = 56 ( ); η = 112 ( );

η= 336 ( ); η= 1120 ( )

3.3 Energy balance 39

CM method

In the same way as what has been done for CB reduced order models, we now focus on the

convergence of the results in the case of a contact simulation for the CM reduced order mod-

els. Figure 3.5 pictures the evolution of the distance between the leading edge node blade

tip 1 and the casing while the number of free vibration modes kept in the modal reduction

basis is increased. Only few differences are observed between each simulation. The main

drawback in CM method is that if the number of mode kept is not high enough, the signal

of the radial displacement is noisy and high frequencies are to take into account. This noise

is widely reduced as we increase the modal basis to 448 modes. From a frequency content

0

0.5

1

1.5

210−4

50

100

100

150

150 200 250

dis

tan

ce

time

Figure 3.5 - Blade tip 1/casing distance depending on the size of the CMS for CM method;

η= 56 ( ); η= 112 ( ); η= 224 ( ); η= 448 ( )

point of view, CM responses converge has we increase the size of the reduced model. This

is illustrated by figures 3.6, with maximum of amplitudes observed at fixed frequencies for

every number of modes kept.

3.2.3 Consequences

The convergence study enables to check the efficiency of the CB and CM methods to approx-

imate the dynamical behavior of the system. Considering the results observed, the following

studies will take into account fixed-sized systems: the simulations with CB method will be

done with 336 constraint modes, and the computations with CM method 448 free interface

modes. This equivalence of the two models on a dynamical point of view is illustrated by

figure 3.7, on which we can observe similar maxima on the Fourier transform of the leading

edge blade tip 1 node radial displacement.

3.3 Energy balanceThe energy balance on the 3D model follows the same assumptions as the one on the 2D

model: the centrifugal effects are neglected and we do not take into account the kinetic en-

ergy implied by the global rotation of the bladed disk. The computation follows the same

40 3D model

PSfr

0 5 10 15 20 25 30 35 40 45 50

frequency

0

0.5

1

1.5

2

2.5

3

3.510−5

amp

litu

de

Figure 3.6 - Fourier transform of the radial displacement of the leading edge node of blade 1

depending on the size of the CMS for CM method; η= 112 ( ); η= 224 ( );

η= 448 ( )

0 5 10 15 20 25 30 35 40 45 50

0

0.5

1

1.5

2

2.5

3

3.510−5

frequency

amp

litu

de

Figure 3.7 - Fourier transform of the radial displacement of the leading edge node of blade 1

for both CB ( ) and CM methods ( )

parameters than the previous, excepted the angular velocity which is now 5. We use CB re-

duced order models for both the bladed disk and the casing. In fact, in order to facilitate

the energy balance of the system, we can delete the load on the casing once the contact has

been initiated. From this moment to the end of the simulation, physical laws imply a con-

servation of the total energy. Figure 3.8 presents evolution of the blade tips/casing distances

for every boundary nodes. Blue areas correspond to the contact periods, and highlight that

the contact frequency decreases during the simulation, expressing the damping of the vibra-

tions of the bladed disk and the casing. Figures 3.9(a) and 3.9(b) illustrate the evolution of

the energies of the casing and the bladed disk during the simulation. The total energy of the

casing increases until 5 ms because of the load applied on the casing. As soon as the load is

stopped, the total energy is stable. Whereas in the 2D case the energy transfers between the

3.4 Friction contribution 41

0 50 100 150 200 250

time

contact line0

0.1

0.2

0.3

0.4

dis

tan

ces

Figure 3.8 - 3D case: evolution of blade tips/casing distances ( ) during the computation.

Areas ( ) correspond to the period when one or several blades are in contact

with the casing

casing and the bladed disk were clear, it is now difficult to identify them, because of the high

number of contact and because of the high stiffness of the casing which limit its vibrations

and its energy variations.

As for the previous energy balance, the evolution of the total energy of the system can

be divided into two parts: first it increase because of the load on the casing, and then it is

constant until the end of the simulation. A little increase appears just after the stop of the

load. As a global conclusion, these studies on the energy balance of the system allows to

consider the contact algorithm as non-dissipative.

3.4 Friction contribution

For first study, and following 3.3, the energy balance has been studied increasing the friction

coefficient. The parameters are exactly the same as in the other energy balance computa-

tions.

Mechanically, friction between the bladed disk and the casing tends to slow the rotation

of the bladed disk, implying a decrease of the kinetic energy, and so a decrease of the total

energy. this analysis is not valid here: in fact we consider the angular velocity of the bladed

disk as constant, assumption which can be considered as an addition of kinetic energy to

the bladed disk to balance friction effects. So, has we increase the friction coefficient, we

obtain an increasing total energy, phenomenon illustrated by figure 3.10 which highlight

that the increase of the total energy of the whole system is implied by the increase of the

total energy of the bladed disk. Considering the previous results on the convergence of the

CMS methods, the different studies following have been launched with 336 fixed interface

modes for CB and 448 free interface modes for CM method. The casing is here again loaded

such as the penetration would be 0.1 mm. The angular velocity considered is Ω= 12.5.

Figures 3.11(a) and 3.11(b) illustrate the evolution of the radial displacement of the lead-

42 3D model

0 50 100 150 200 250

time

1

2

3

4

5

ener

gy(×

10+

2)

(a) 3D case: energy balance of the casing

0 50 100 150 200 250

time

2

6

10

14

18

ener

gy

(b) 3D case: energy balance of the bladed disk

0 50 100 150 200 250

time

1

2

3

4

5

ener

gy(×

10+

2)

(c) 3D case: energy balance for the whole system

Figure 3.9 - Energy balances for the 3D case: for each balance, the total energy ET ( ) is

computed as the sum of the damped energy Ea ( ), the kinetic energy Ec ( )and the strain energy Ed ( ). Areas ( ) correspond to the periods when one

or several blades are in contact with the casing

ing edge node blade tip 1 when we increase the friction coefficient for CM and CB methods.

The radial displacements are similar while µ increases. It seems that the friction coefficient

has a limited impact on the radial vibration of the blades at this angular velocity. Figures

3.12(a) and 3.12(b) present the evolution of the maximum blade tips/casing distance when

we increase the friction coefficient for CM and CB methods. The behaviors of the two CMS

methods are not equivalent face to the increase of µ. The increase of the friction coefficient

seems to have no consequences using CM method, whereas it implies more noise on the

response of the system using CB. This trend is confirmed by figures 3.13(a) and 3.13(b) with

no evolution in the frequential response using CM, and an increase and the appearance of

some pics in the frequential response using CB as we increase µ.

3.5 Sensitivity to rotational velocity 43energy

time

0 50 100 150 200 250

2

4

6

8×10+2

1

3

5

7×10+2

1

2

3

4

×10+1

ener

gyµ

µ

µ

Figure 3.10 - 3D case: evolution of the total energy ET ( ) of the whole system depending

on the friction coefficient µ. the evolution of the total energy of the casing ETCA

( ) and the bladed disk ETRA ( ) highlight that the increase of the friction

coefficient implies an increase of the total energy of the bladed disk

10−1

1

0.5

0

−0.5

−150 100 150 200 250

time

dis

pla

cem

ent

(a) CM

10−1

1

0.5

0

−0.5

−150 100 150 200 250

time

dis

pla

cem

ent

(b) CB

Figure 3.11 - Friction effects on the radial displacement of the blade 1 leading edge node; µ=

0 ( ); µ= 0.3 ( ); µ= 0.6 ( ); µ= 0.8 ( )

3.5 Sensitivity to rotational velocity

The next step in the detection of the interaction between the bladed disk and the casing is to

observe the influence of the increase of the angular velocity on the vibrations of the assem-

bly. In this section, the friction coefficient is fixed to µ= 0.15, and the casing is loaded dur-

ing the whole simulation such as the penetration of the bladed disk into the casing would be

0.1mm. Iterations on the angular velocity with a step of 0.5 from 5 to 30 have been launched

using both CB and CM method. The study is organized as follows: first a global study on the

44 3D model

10−1

2.5

2

1.5

time

dis

tan

ce

50 100 150 200 250

(a) CM

10−1

2.5

2

1.5

time

dis

tan

ce

50 100 150 200 250

(b) CB

Figure 3.12 - Friction effects maximum of the blade tips/casing distance; µ= 0 ( ); µ= 0.3( ); µ= 0.6 ( ); µ= 0.8 ( )

10−4

0.5

0.4

0.3

0.2

0.1

02.5 5 7.5 10 12.5 15 17.5

frequency

amp

litu

de

(a) CM

10−4

0.5

0.4

0.3

0.2

0.1

02.5 5 7.5 10 12.5 15 17.5

frequency

amp

litu

de

(b) CB

Figure 3.13 - Friction effects on the frequency response of the radial displacement of the blade

1 leading edge node; µ= 0 ( ); µ= 0.3 ( ); µ= 0.6 ( ); µ= 0.8 ( )

results obtained is made to compare the two CMS methods; secondly, and according to the

global study conclusion, studies of the behavior of the system are made.

3.5.1 CB and CM methods

Results have been obtained for both CB and CM method for a large panel of angular velocity.

Despite the convergence studies made for an angular velocity Ω = 12.5, the validity of the

reduced order models used can not be predicted a priori for higher angular velocity. A way to

check this validity is to compare the behaviors of CB and CM methods on the whole angular

velocity panel.

Figure 3.5.1 illustrates the differences between blade 1 leading edge radial displacement

Fourier transform with CB and CM method. It appears that the differences increase as soon

as Ω is over 20. This is also visible in figures 3.15(a) and 3.15(b). It may also be seen in

figures 3.5.1 and 3.15(b) that these differences between CB and CM method are increased

for frequencies which are on motor harmonics. This illustrates that the contact frequency

increases as we increase the angular velocity, and that the two methods can not be compared

anymore as we go over Ω= 20.

Separated presentations of the blade 1 leading edge node radial displacement Fourier

transform can be seen in figures 3.16(a) and 3.16(b) for CM method and on figures 3.17(a)

3.5 Sensitivity to rotational velocity 45

510

1520

2530

0

5

10

15

20

25

30

frequency

amp

litu

de

Ω

0

0.75

1.5

×10−4

Figure 3.14 - Gap between the Fourier transform on the radial displacement of blade 1 lead-

ing edge node using CB method and the one using CM method, depending on

the angular velocity

5 10 150

1

2

3

4

10×10−5

0

5

10

15

20

25

30

Ω

freq

uen

cy

(a) Ω from 0 to 15

20 25 30150

0.5

1

3×10−4

5

10

15

20

25

30

0

Ω

freq

uen

cy

(b) Ω from 15 to 30

Figure 3.15 - Gap between the Fourier transform on the radial displacement of blade 1 lead-

ing edge node using CB method and the one using CM method, depending on

the angular velocity (from 0 to 30)

and 3.17(b) for CB method. In red are presented the engine harmonics, meaning frequencies

f such that:

f =k

πΩ (3.1)

with k an integer. We have chosen to use two figures to present the evolution of the Fourier

transform for each CMS method because of the scale: as soon as we exceed the critical ve-

locity Ωc = 155, we observe an increase of the vibrations which would mask the vibration

under Ωc on a unique graphic. As we already saw CB and CM methods have comparable

behaviors until 20, with an increase of the vibrations as we approach and exceed the criti-

cal velocity. In figures 3.16(a) and 3.17(a), we can notice that excepted the low response of

several frequencies, the main behavior can be describe thanks to frequencies on motor har-

monics. This can also be seen in figures 3.16(b) and 3.17(b). As we use a two-diameter load

5The definition of the critical velocity has been introduced in 2.2.1; if we consider the eigenfre-quency corresponding to 2 nodal diameters for both the casing and the bladed disk, we obtain acritical velocity Ωc = 15.

46 3D model

0 5 10 15 20 25 30

57.5

1012.5

15

frequency

amp

litu

de

Ω

1

2×10−4

(a) Ω from 0 to 15

0 5 10 15 20 25 30

15

20

25

30

frequency

amp

litu

de

Ω

2.5×10−3

(b) Ω from 15 to 30

Figure 3.16 - Fourier transform of the radial displacement of blade 1 leading edge node using

CM method, depending on the angular velocity (from 0 to 30)

during the simulation, the casing takes an ovale shape, an the frequencies of the bladed disk

which are supposed to be excited are on motor harmonics. Although CB and CM method

0 5 10 15 20 25 30

5

7.5

10

12.5

15

frequency

amp

litu

de

Ω

0.5

1×10−4

(a) Ω from 0 to 15

0 5 10 15 20 25 30

15

20

25

0

frequency

amp

litu

de

Ω

2.5×10−3

(b) Ω from 15 to 30

Figure 3.17 - Fourier transform of the radial displacement of blade 1 leading edge node using

CB method, depending on the angular velocity (from 0 to 30)

are equivalent for angular velocities under Ωv = 20, they have different behaviors over this

velocity. To continue, the study checking the validity of the results obtained, it is necessary

to limit the angular velocity panel, considering only Ω≤ 20.

3.5.2 Angular velocities of interest

In this study, the angular velocity is increased until Ωv , and the vibrations of the system are

observed. Figures 3.18(a) and 3.18(b) illustrate the increase of the vibrations of the system as

we increase the angular velocity. From 5 to 15, the maximum of blade tips/casing distance

decreases, but it does not mean that the vibrations do so. As it is illustrated in figures 3.5.2

and 3.5.2, the displacement of the bladed disk increases, but the angular velocity leads the

system to a typical behavior. As we approach the critical velocity Ωc = 15, the vibrations of

the bladed disk are in a special phasing with the deformation of the casing, such that the

distance between the blades and the casing decrease.

The increase of the angular velocity involves an increase of the vibrations of the bladed

disk. The ovale shape of the casing excite the modes which are on pair motor harmonics, and

the dynamical response of the system can be completely describe with pair diameter modes.

3.5 Sensitivity to rotational velocity 47

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

3d

ista

nce

s

time

(a) CM method

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

dis

tan

ces

time

(b) CB method

Figure 3.18 - Evolution of the maximum bladed disk/casing distance depending on the an-

gular velocity; Ω= 5 ( ); Ω= 10 ( ); Ω= 15 ( ); Ω= 20 ( )

0 50 100 150 200 250-10

-7.5

-5

-2.5

0

2.5

dis

pla

cem

ent

(×10−

1

time

(a) Radial displacement

0 50 100 150 200 250-1

-0.5

0

0.5

1

1.5

2d

isp

lace

men

t(×

10−

1)

time

(b) Angular displacement

0 50 100 150 200 250-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0.5

dis

pla

cem

ent

time

(c) Axial displacement

Figure 3.19 - Evolution of the displacements of blade 1 leading edge node depending on the

angular velocity using CM method; Ω = 5 ( ); Ω = 10 ( ); Ω = 15 ( );

Ω= 20 ( )

This phenomenon is highlighted by figures 3.21 for CB method and 3.22 for CM method,

with the influence in terms of modal coefficient of each number of diameter, from 0 to 7. We

can notice that the odd number of diameters have a low influence on the dynamics of the

system compared to the pair number of diameters. However, the influence of pair number

of diameters, particularly 0 and 2, is highly increased when we approach and exceed the

critical velocity for both CB and CM methods. Both CB and CM methods allow to conclude

48 3D model

0 50 100 150 200 250-10

-7.5

-5

-2.5

0

2.5d

isp

lace

men

t(×

10−

1)

time

(a) Radial displacement

0 50 100 150 200 250-1

-0.5

0

0.5

1

1.5

2

dis

pla

cem

ent

(×10−

1)

time

(b) Angular displacement

0 50 100 150 200 250

-1

-0.5

0

0.5

dis

pla

cem

ent

time

(c) Axial displacement

Figure 3.20 - Evolution of the displacements of blade 1 leading edge node depending on the

angular velocity using CB method; Ω = 5 ( ); Ω = 10 ( ); Ω = 15 ( );

Ω= 20 ( )

that the vibrations increase as we approach the theoretical critical velocity Ωc. Because of

its importance for detecting modal interaction between the bladed disk and the casing, it

has been decided to investigate in details the behavior of the two CMS methods around Ωc.

Figures 3.23(a) and 3.23(b) allow us to see some differences between CB and CM methods

at the critical velocity with the presentation of blade 1 trailing edge node/casing distance

and its Fourier transform. More precisely, the graph pictured in figure 3.23(b) presents some

significant differences in terms of amplitude and evolution (the distance with CB method

increases after t = 7 ms while it decreases with CM method). This difference of behavior

between both CMS methods is also underlined in terms of influence of nodal diameters:

figure 3.24 presents differences between the two methods for every nodal diameter. Despite

of the fact that differences can be seen for every nodal diameter, only those relative to nodal

diameter n d = 2 are significant numerically speaking.

The excitation of the system at an eigenfrequency may be responsible for the numeri-

cal differences between the two methods. Also, the differences between both CMS meth-

ods might be explained by the conclusions of the 2D studies. Indeed, the different studies

have highlighted the high numerical sensitivity of CCM method which is an enriched CM

method. One of the conclusions of chapter 2 was that attachment modes were responsible

for this sensitivity, 3D results tend to show that pseudo-static correction may also be par-

tially responsible for it since no attachment modes are involved in the 3D study.

3.6 Conclusions 49

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.25

0.5

0.75

1

1.25

×10−6

(a) n d = 0

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.25

0.5

0.75

1

1.25

×10−7

(b) n d = 1

replacements

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.5

1

1.5

2

×10−6

(c) n d = 2

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.25

0.5

0.75

1

×10−7

(d) n d = 3

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.75

1.5

2.25

3

3.75

×10−7

(e) n d = 4

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.2

0.4

0.6

0.8

1

×10−7

(f) n d = 5

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.25

0.5

0.75

1

1.25

×10−7

(g) n d = 6

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.2

0.4

0.6

0.8

1

×10−7

(h) n d = 7

Figure 3.21 - Influence of the eight first nodal diameters (0 to 7) in the case of simulations

computed with Craig-Bampton reduced order models. Each graph is plot with

the same scale to emphasize the most important nodal diameters.

3.6 Conclusions

The 3D studies lead to several conclusions. First, the convergence of CB and CM methods

has been demonstrated using the contact code. Then the energy balance has been checked

using CB method, confirming the non dissipative characteristics of the contact algorithm.

The third investigation treated of friction effects on the behavior of the system. It has

been shown that even though the increase of the friction coefficient had no effect using CM

method, its increase implied more noise in the response of the system using CB method.

50 3D model

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.25

0.5

0.75

1

1.25

×10−6

(a) n d = 0

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.5

1

1.5

2

×10−7

(b) n d = 1

replacements

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.5

1

1.5

2

×10−6

(c) n d = 2

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.35

0.7

1.05

1.4

1.75

×10−7

(d) n d = 3

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.6

1.2

1.8

2.4

3

×10−7

(e) n d = 4

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.2

0.4

0.6

0.8

1

×10−7

(f) n d = 5

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.25

0.5

0.75

1

1.25

×10−7

(g) n d = 6

0

0.5

1

1.5

2×10−6

510

1520

Ω 050

100150

200

time 0

0.25

0.5

0.75

1

×10−7

(h) n d = 7

Figure 3.22 - Influence of the eight first nodal diameters (0 to 7) in the case of simulations

computed with Craig-Martinez reduced order models. Each graph is plot with

the same scale to emphasize the most important nodal diameters.

The final study concerned angular velocity effects on both CB and CM methods. A valid

angular velocity panel has been identified for the reduced order models used, and the ef-

fects of an increase of the angular velocity have been observed. CB and CM methods have

equivalent global behavior, with an increase of the vibrations as we exceed the critical ve-

locity. However some differences have been highlighted at this special velocity, showing the

numerical differences between the two methods.

3.6 Conclusions 51

0 5 10 15 20 250

0.5

1

1.5

×10−4

frequency

amp

litu

de

(a) Blade 1 trailing edge/casing distance Fouriertransform

0 50 100 150 200 250

0

0.5

1

1.5

2×10−1

time

dis

pla

cem

ent

(b) Blade 1 trailing edge/casing distance

Figure 3.23 - Blade 1 trailing edge/casing distance and its Fourier transform for Ω = 15; CM

method ( ); CB method ( )

52 3D model

0 50 100 150 200 2500

0.25

0.5

0.75

1

1.25

×10−7

(a) Influence n d = 00 50 100 150 200 250

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

×10−8

(b) Influence n d = 1

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

×10−6

(c) Influence n d = 20 50 100 150 200 250

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

×10−8

(d) Influence n d = 3

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

×10−7

(e) Influence n d = 40 50 100 150 200 250

0

0.25

0.5

0.75

1

1.25

×10−8

(f) Influence n d = 5

Figure 3.24 - Influence of the number diameters n d for Ω = 15; on the horizontal axis is the

time in ms, and on the vertical axis is the influence; CM method ( ); CB

method ( )

4.1 Sequential CMS methods 53

4.1.1 Guyan method 554.1.2 CB method 554.1.3 Tests on a 2D model 56

4.2 Applications to a 3D model 58

4.2.1 Convergence 594.2.2 Industrial application 594.2.3 Perspectives 63 4

Approximation of an industrialmistuned bladed disk

The contact algorithm described in the first chapters of this report was used to man-

age contact between tuned bladed disks and a casing. The hypothesis of having a perfectly

tuned bladed disk is quite restrictive but is absolutely necessary for using cyclic symmetry

properties which are required for computing reduced order models. As a consequence, con-

tact was managed between the casing and every blade. However, experiments made with

industrial bladed disks and casing have shown that contact between both structures fre-

quently occurs between the casing and only one blade. The contact between a single blade

and the casing is truly interesting in an industrial prospective because of the consequences

it may have on the behavior of the whole bladed disk. This could be a key for understanding

the initiation of modal interaction between both structures.

The first part of this chapter introduces the theoretical background that will allow us to

manage contact on a single blade: using a double modal reduction, cyclic symmetry and CB

CMS method can be efficiently combined to obtain the reduced order model of mistuned

bladed disk. The second part of the chapter presents the application of this method in the

2D and 3D cases showing the convergence of the models and the results obtained in terms

of modal interaction initiation.

4.1 Sequential CMS methods

Managing contact between a single blade and the casing implies the modification of the

boundary used for the modal reduction of the bladed disk. Because of the numerical prob-

lems encountered with the CM method 1.2, the choice is made to use the CB CMS method

in this chapter. In opposition to what has been done in chapter 1.2, in this chapter the

boundary only contains three nodes on the tip of one blade. The loss of cyclic symmetry in

the consideration of this boundary is directly responsible for the impossibility of computing

reduced order models by using cyclic symmetry properties1.

1The computation of the constraint modes can not be made in the cyclic symmetric space any-more and their computation leads to cumbersome computation times. Technically, the loss of cyclicsymmetry in the boundary conditions implies that the stiffness matrix of the system is not circulantanymore and does not become block diagonal in the cyclic symmetry space.

54 Approximation of an industrial mistuned bladed disk

The point of introducing the double CMS method is to combine the assets of fast com-

putations with the use of cyclic symmetry properties and obtaining a reduced order model

which boundary only takes into account the tip of one blade. The first modal reduction,

using the CB CMS method, is computed with a boundary taking account of every blade, ex-

actly as in chapter 1.2. From this reduced order model composed of boundary DOF and

modal coefficients a second modal reduction is applied. Two methods are compared for

this second modal reduction: in one hand the CB method and in another hand the Guyan

method. The theory is presented using the 2D models in order to simplify the equations and

the visual representations but application to 3D models is exactly identical. Using the

blade tip node

f

f2

θ

u

v

Figure 4.1 - Presentation of the notations on a 2D model

notations introduced in figure 4.1, it is possible to describe the double reduction as follows.

First, thanks to a Craig Bampton method, considering the boundary f containing nd d l f

DOF, and keeping η constraint modes, we can decompose the stiffness matrix as:

K=

Kff Kfη

Kηf Kηη

!

(4.1)

In the second reduction, we divide the boundary f into two parts: the second bound-

ary f 2 containing the DOF located at the top of the first blade (i.e nd d l f 2 DOF), and f 1

containing the DOF at the top of the other blades (nd d l f −nd d l f 2).

K=

Kf2f2 Kf2f1 Kf2η

Kf1f2 Kf1f1 Kf1η

Kηf2 Kηf1 Kηη

(4.2)

The previous notations are implicitly used the same way for mass and damping matri-

ces. The following depends on the method used, and we can chose several type of methods

with different physical meanings to describe synthetically the behavior of the structure.

4.1 Sequential CMS methods 55

4.1.1 Guyan method

This method uses a static approximation of the displacements on f1 thanks to the displace-

ments of f2. Practically, this approximation corresponds to:

Kf2f2 xf2 +Kf2f1 xf1 = 0 (4.3)

So we can build the matrixΨ which represents the DOF of f2 thanks to those of f1:

Ψ=−(Kf2f2 )−1Kf2f1 (4.4)

Following this approximation, a modal calculus enables to approach the behavior of the

DOF of f1 using Ψ .The eigenvalue problem considered here is:

(−ω2M1+K1)(x ) = 0 with: M1 =ΨTMf2f2Ψ and K1 =Ψ

TMf2f2Ψ (4.5)

In [RIX 04], this eigenvalue problem enables to reduce once time again the size of the

problem, and if we keep only ηmodes in this second CMS method, we can write the change

of basis between the physical problem and the double reduced space as follows:

Ψ=ΦGΦCB (4.6)

with ΦCB presented in section 1.2, and

ΦG

=

I 0 0

0 Vν 0

0 0 I

(4.7)

4.1.2 CB method

The method considered here is a classical CB method. The boundary f2 considered herein

contains only the DOF at the top of a unique blade. The constraint modes are so calculated

on f1 and the constraint modes of the first component mode synthesis η. Applying this sec-

ond method enables write the change of variable from the physical problem to the double

reduced problem to:

Ψ=ΦCB2ΦCB (4.8)

with ΦCB presented in section 1.2, and

ΦCB2

=

I 0

−K−1f2f2

Kf2(f1+η) xν

!

(4.9)

with xν the ν first eigenvectors of the problem:

(−ω2M(f1+η)(f1+η)+K(f1+η)(f1+η))x= 0 (4.10)

Two different double CMS methods have been presented to approach the contact sim-

ulation on a kind of mistuning implying a unique blade in the contact phenomena. These

methods have been implemented into the 2D and 3D contact codes, and have been tested

through different aspects. The results obtained are presented in the following.

56 Approximation of an industrial mistuned bladed disk

eplacements

0 0.6 1.2 1.8 2.4 30.0

0.25

0.5

0.75

2

1.25

dis

tan

ces

time ×103

(a) Distance blade-casing depending on thenumber of modes kept in the first Craig Bamp-ton method; η = 0 ( ); η = 22 ( ); η = 44( ); η= 110 ( )

0 0.6 1.2 1.8 2.4 3

40

30

20

10

0

con

tact

effo

rt(.

104

)

time×103

(b) Contact effort depending on the number ofmodes kept in the first Craig Bampton method;η = 0 ( ); η = 22 ( ); η = 44 ( ); η =110 ( )

4.1.3 Tests on a 2D model

It is first necessary to check the convergence of the first CMS method (CB). Figures 4.1.3

and 4.1.3 illustrate this convergence in terms of bladed disk/casing distances and contact

efforts. As soon as more than 22 constraint modes are kept in the modal basis, the conver-

gence is fast. The simulations used an angular velocity of 1; the mechanical parameters of

the models are presented in table 2.1; to initialize the contact, we do not use a load on the

casing anymore, but we apply an initial shape on the bladed disk, on its first bending mode.

As we can see, there are only a few differences between the curves displayed with 22, 44 and

110 constraint modes. Aware of the convergence of the first CB, we have chosen to keep 110

constraint modes to study the convergence of the second CMS methods. The two methods

presented previously have been tested on the 2D model already presented. The contact al-

gorithm has been adapted to consider and treat the contact on a unique blade. Figure 4.2

presents the evolution of the distances bladed disk-casing using Guyan method in the sec-

ond synthesis. As we can see, a synthesis with zero modes is already good to approach the

dynamical behavior of the system. In fact the constraint modes kept in the first synthesis are

sufficient to describe the dynamic of the bladed disk. The results obtained using a double

Craig Bampton method are shown in figure 4.3. It can easily be seen that we have a con-

vergence of the distance blade-casing following the number of modes kept in the second

synthesis, and clearly a good approximation of the behavior of the system as soon as this

number is over 84. Taking into account the previous results we draw figure 4.4 to compare

the two methods keeping 63 modes in the second synthesis. The two methods enable to

approach the behavior of the system with the same accuracy, considering the calculation

with a unique synthesis (Craig Bampton) keeping 110 constraint modes as a reference. Even

if the two methods can approximate the dynamical behavior of the structure, it is impor-

tant to notice that the double Craig Bampton method considered has 66 DOF, whereas the

Craig Bampton-Guyan method has 176 DOF. It is also necessary,to value the efficiency of the

methods, to consider a fixed number of DOF, and to use the two different methods. Figure

4.1 Sequential CMS methods 57

0 0.6 1.2 1.8 2.4 30

0.25

0.5

0.75

1

1.25

dis

tan

ces

time×103

Figure 4.2 - Distance blade-casing depending on the number of modes kept in Guyan method;

ν= 0 ( ); ν= 21 ( ); ν= 63 ( ); η= 110 ( )

0 0.6 1.2 1.8 2.4 30.0

0.25

0.5

0.75

1

1.25

dis

tan

ces

time

Figure 4.3 - Distance blade-casing depending on the number of modes kept in the double

Craig Bampton method; ν= 0 ( ); ν= 42 ( ); ν= 84 ( ); η= 110 ( )

0 0.6 1.2 1.8 2.4 30

0.25

0.5

0.75

1

1.25

dis

tan

ces

time

Figure 4.4 - Distance blade-casing with 63 modes kept in the second synthesis, depending on

the method chosen; G ν= 63 ( ); CB ν= 63 ( ); η= 110 ( )

58 Approximation of an industrial mistuned bladed disk

0 0.6 1.2 1.8 2.4 30

0.25

0.5

0.75

1

1.25

dis

tan

ces

time

Figure 4.5 - Distance blade-casing observed with the two methods considering the same num-

ber of DOF kept; η = 0 Guyan ν = 63 ( ); η = 110 CB2 ν = 63 ( ); η = 110( )

4.5 presents the distance blade-casing obtained: the reference is a simple Craig Bampton

synthesis with 110 constraint modes, the double CB method has first 110, then 63 constraint

modes kept. Craig Bampton-Guyan method consists in a Craig Bampton synthesis keeping

0 mode and then a Guyan method keeping 63 modes. We can here compare two 66 DOF

methods. The result obtained comparing the distances blade-disk is that, as we have seen

in figure 4.2, the static modes do not enable to approach the dynamical behavior of the sys-

tem, and so the number of Guyan modes kept has no influence on the convergence of the

method. For the same number of DOF considered, the double Craig Bampton method is

more efficient, because of the inclusion of the constraint modes of the structure in the sec-

ond synthesis.

As a conclusion, thanks to the previous results, it is decided to continue and to com-

pute the double CB method in the 3D code to approximate efficiently the dynamical of the

system. For a better apprehension of the methods, figure 4.6 illustrates an example of con-

straint modes calculated in the first and in the second CB, where the boundary varies from

every blade tips to a unique tip. As we can see, in the first CMS the tip of every blade is fixed,

whereas in the second synthesis, only one tip is constraint. To assure that the behavior of

the bladed disk is well represented, it is necessary to check that the double-reduced system

enables to access the n-diameter modes of the structure. A computation of the eigenvec-

tors of the double-reduced system passed in the FE basis leads to a comparison with the

n-diameter modes obtained with the FE model.

4.2 Applications to a 3D model

The method describe and chosen for this study is a double CB method. In the following will

we notice CB1 the first CMS method and CB2 the second.

4.2 Applications to a 3D model 59

(a) example of constraint mode -first CMS

(b) example of constraint mode -second CMS

Figure 4.6 - example of constraint modes obtained in the first and in the second CMS

4.2.1 Convergence

The convergence has been tested on two different points. First, a convergence of the re-

duced models, in terms of eigenfrequencies, and in terms of residual criterion (introduced

in 1.2) is studied. Then, the convergence of the models applying the contact code is tested.

Figures 4.8(a), 4.8(b) and 4.8(c) illustrate the gap in percent between the eigenfrequen-

cies computed on a simple CB1 model with 1120 modes kept and the ones computed on the

double reduced model, depending on the number of fixed interface modes kept in CB2. The

eigenfrequencies computed on double reduced models converge to the ones computed on

the simple CB1 model. As soon as the number of fixed interface modes kept in CB2 exceeds

56, the gap between the two kinds of models in terms of eigenfrequencies is under 1%. Fig-

ures 4.9(a), 4.9(b) and 4.9(c) illustrate the residual between the eigenmodes computed on

a simple CB1 model with 1120 modes kept and the ones computed on the double reduced

model, depending on the number of fixed interface modes kept in CB2. The convergence in

terms of residual criterion is very good. Even without fixed interface modes, it is possible to

approximate 9 modes of the system. The convergence obtained in terms of reduced models

allows us to test the behavior using the contact algorithm. The angular velocity used here is

Ω= 5. Figure 4.10 illustrates the evolution of the blade tip node 1/casing distance increasing

the number of modes in CB2. As we can see, the model with 0, 280 and 560 modes in CB2

give the same results, and as it has been said in the study on the 2D models, the behavior

of the bladed disk can be described using only the constraint modes relative to the blade

considered in the contact.

4.2.2 Industrial application

The contact case considering the contact between a mistuned bladed disk with a blade

longer than the others and the casing has been computed by industrials, but with several

assumptions: first, the whole bladed disk is not modeled, a unique blade is considered

clamped at its foot; secondly, the casing considered is rigid, with a humpback on its sur-

face to involve the contact between the two structures. These computations (noticed CP1)

applied with Ω = 10, using a blade meshed with 18869 DOF imply a cpu time of 24 min per

60 Approximation of an industrial mistuned bladed disk

f=0.21006

(a) 0-diameter mode -CB2

f = 0.21006

(b) 0-diameter mode -FE

f = 0.20981

(c) 1-diameter mode -CB2

f = 0.2098

(d) 1-diameter mode -FE

f = 0.20839

(e) 2-diameter mode -CB2

f = 0.20838

(f) 2-diameter mode -FE

f = 0.20667

(g) 3-diameter mode -CB2

f = 0.20666

(h) 3-diameter mode -FE

f = 0.20643

(i) 4-diameter mode -CB2

f = 0.20643

(j) 4-diameter mode -FE

Figure 4.7 - examples of n-diameter modes with the FE and with the double reduction method

(CB2)

4.2 Applications to a 3D model 61

10000

5000

0

ν= 280ν= 560

ν= 84060

50

40

30

20

10

0

(a) Eigenfrequency 1 to 56

3500

1750

0

ν= 280ν= 560

ν= 84060

50

40

30

20

10

0

(b) Eigenfrequency 57 to 112

300

1500

0

ν= 280ν= 560

ν= 84060

50

40

30

20

10

0

(c) Eigenfrequency 113 to 168

Figure 4.8 - Gap in percent between the eigenfrequencies using a double CB method and the

eigenfrequencies using a simple CB method with 1120 constraint modes depend-

ing on the number of modes kept in the second CMS method

62 Approximation of an industrial mistuned bladed disk

1.5

0.75

0

ν= 280

ν= 560ν= 840

60

50

40

30

20

(a) Eigenfrequency 1 to 56

3

1.5

0

ν= 280

ν= 560ν= 840ν= 840

60

50

40

30

20

(b) Eigenfrequency 57 to 112

8

4

0

ν= 280ν= 280

ν= 560ν= 840

60

50

40

30

20

(c) Eigenfrequency 113 to 168

Figure 4.9 - Residual between the eigenmodes using a double CB method and the eigenmodes

using a simple CB method with 1120 constraint modes depending on the number

of modes kept in the second CMS method

4.2 Applications to a 3D model 63

10−1

4

3

2

1

0

100 200 300 4000time

dis

tan

ce

Figure 4.10 - Convergence of the bladed disk tip node 1/casing distance depending on the

number of modes kept in the second CB method; ν = 0 ( );ν = 280 ( );

ν= 560 ( )

rotation of the bladed disk.

The double CMS method (noticed CP2) has been tested with the same angular velocity.

the difference is that the contact is initiated by an initial shape of the bladed disk on its first

torsion mode. The number of constraint modes kept in the first CB method is 1120 and the

number of constraint modes kept in the second CB method is 562. These computations CP2

applied with Ω = 10 with reduced order models containing 65 DOF for the bladed disk and

1456 DOF for the casing imply a cpu time of 21 minutes per rotation of the bladed disk.

The use of the double CMS method is interesting on several points:

• on a dynamical point of view, the model used in CP2 is widely richest than the model

in CP1; in fact, the reduced order model contains all the information concerning the

dynamics of the whole bladed disk, whereas the model CP1 contains the behavior of

a unique blade;

• concerning the casing, CP2 allows to take into account a flexible casing, opposite to

the rigid casing in CP1, which does not allows to consider the dynamics of the casing;

• about the time computation issue, CP2 is equivalent to CP1.

4.2.3 Perspectives

The example of application presented in 4.2.1 has to be continued in order to obtain more

information on the contact phenomenon between a kind of mistuned bladed disk with a

blade longer than the other and a casing. In fact, it could be interesting to find conditions

necessary to obtain a recurrent contact on several points of the casing, which would involve

practically a localized wear of the casing.

2considering the results obtained in 4.2.1, we know that we have a low error on the mode consid-ered to initiate the contact

64 Approximation of an industrial mistuned bladed disk

5Conclusion

The interaction between the bladed disk and the casing a phenomena which can be critical

for the structure. Its simulation using reduced order models obtained thanks to CMS meth-

ods highlighted several things on both 2D and 3D applications.

First, the use of CB and CCM methods on a 2D model allowed to compare the CMS

methods on a simple model. This established the convergence of the two methods using

the contact code as we enrich the reduced order model, and the good behavior of the two

CMS methods in terms of energy balance. Then the iterations on the friction coefficient and

the angular velocity highlighted the fast convergence of CB method and the numerical sen-

sitivity of CM method.

Secondly, the convergence of CB and CM method has been checked on an industrial

model; so has been the energy balance. Then, the friction effects have been studied for Cb

and CM method, with little differences between the two CMS methods as we increase µ :

the friction coefficient has no effect on CM reduced order models, whereas it increasing the

noise of the response of the system using CB method. About the angular velocity, it has

been showed that as we exceed the critical speed Ωc=15, the vibrations are increased on the

bladed disk, and we achieve a typical behavior with only the second harmonic responding.

Finally, an application of the contact code using a double CMS method has been made

to simulate a kind of mistuned bladed disk. After the demonstration of the convergence of

the method on both 2D and 3D models, an example of computation and presentation of

the contact zone on the casing has been presented. This final work could serve to present

typical phenomena, in which the casing would be impacted only on a few points precisely

localized.

66 Conclusion

67

68 Conclusion

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