. X?-u and . Xu, vy (sesquilinearity), 3. xu , vy xv , uy (conjugate symmetrydefiniteness). x. , .y is said to be a positive-definite nondegenerate sesquilinear form

R. Bladh, M. P. Castanier, and C. Pierre, Component-Mode-Based Reduced Order Modeling Techniques for Mistuned Bladed Disks???Part II: Application, Journal of Engineering for Gas Turbines and Power, vol.123, issue.1, pp.100-108, 2001.
DOI : 10.1115/1.1338948

. Bladh, Dynamic Response Predictions for a Mistuned Industrial Turbomachinery Rotor Using Reduced-Order Modeling, Journal of Engineering for Gas Turbines and Power, vol.124, issue.2, pp.311-324, 2002.
DOI : 10.1115/1.1447236

R. Bladh, M. P. Castanier, and C. Pierre, Effects of Multistage Coupling and Disk Flexibility on Mistuned Bladed Disk Dynamics, Journal of Engineering for Gas Turbines and Power, vol.125, issue.1, pp.121-130, 2003.
DOI : 10.1115/1.1498267

]. O. Boiteau and O. Nicolas, Algorithme de résolution pour le problème généralisé, Code Aster, EDF Recherche et Développement, 2007.

. Capiez-lernout, Blade Manufacturing Tolerances Definition for a Mistuned Industrial Bladed Disk, Journal of Engineering for Gas Turbines and Power, vol.127, issue.3, pp.621-628, 2005.
DOI : 10.1115/1.1850497

URL : https://hal.archives-ouvertes.fr/hal-00688121

]. M. Castanier and C. Pierre, Using Intentional Mistuning in the Design of Turbomachinery Rotors, AIAA Journal, vol.40, issue.10, pp.2077-2086, 2002.
DOI : 10.2514/2.1542

]. M. Castanier and C. Pierre, Modeling and Analysis of Mistuned Bladed Disk Vibration: Current Status and Emerging Directions, Journal of Propulsion and Power, vol.22, issue.2, pp.384-396, 2006.
DOI : 10.2514/1.16345

. Chatelet, Toward global modelling approaches for dynamic analyses of rotating assemblies of turbomachines, Journal of Sound and Vibration, vol.282, issue.1-2, pp.163-178, 2005.
DOI : 10.1016/j.jsv.2004.02.035

URL : https://hal.archives-ouvertes.fr/hal-00368451

]. J. Chen and C. H. Menq, Periodic Response of Blades Having Three-Dimensional Nonlinear Shroud Constraints, Journal of Engineering for Gas Turbines and Power, vol.123, issue.4, pp.901-909, 2001.
DOI : 10.1115/1.1385828

C. and W. Peng, Dynamic stability of rotating blades with geometric non-linearity, Journal of Sound and Vibration, vol.187, issue.3, pp.421-433, 1995.

B. Choi, J. Lentz, A. J. Rivas-guerra, and M. P. Mignolet, Optimization of Intentional Mistuning Patterns for the Reduction of the Forced Response Effects of Unintentional Mistuning: Formulation and Assessment, Journal of Engineering for Gas Turbines and Power, vol.125, issue.1, pp.131-140, 2003.
DOI : 10.1115/1.1498270

]. R. Christensen and I. F. Santos, Modal Controllability and Observability of Bladed Disks and their Dependency on the Angular Velocity, Journal of Vibration and Control, vol.11, issue.6, pp.801-828, 2005.
DOI : 10.1177/1077546305054596

R. R. Craig and C. J. Chang, On the Use of Attachment Modes in Substructure Coupling for Dynamic Analysis, AIAA Paper 77-405-CP, pp.89-99, 1977.

C. Jr, R. R. Bampton, C. Jr, and M. C. Bampton, Coupling of Substructures for Dynamic Analyses, AIAA Journal, vol.6, issue.7, pp.1313-1319, 1968.

C. R. Jr-1987-]-r and . Craig-jr, A Review of Time-Domain and Frequency Domain Component Mode Synthesis Methods, International Journal of Analytical and Experimental Modal analysis, vol.2, issue.2, pp.59-72, 1987.

]. R. Jr and . Craig, A Review of Time-Domain and Frequency Domain Component Mode Synthesis Methods, Int. J. Anal. and Exp. Modal Analysis, vol.2, issue.2, pp.59-72, 1987.

]. A. Curnier, Computational methods in solid mechanics, 1994.
DOI : 10.1007/978-94-011-1112-6

]. C. Desceliers, Dynamique non linéaire en déplacements finis des structures tridimensionnelles viscoélastiques en rotation, 2001.

]. D. Ewins, The effects of detuning upon the forced vibrations of bladed disks, Journal of Sound and Vibration, vol.9, issue.1, pp.65-79, 1969.
DOI : 10.1016/0022-460X(69)90264-8

]. D. Ewins, Vibration characteristics of bladed disc assemblies, ARCHIVE: Journal of Mechanical Engineering Science 1959-1982 (vols 1-23), vol.15, issue.3, pp.165-186, 1973.
DOI : 10.1243/JMES_JOUR_1973_015_032_02

]. D. Ewins, Vibration Modes of Mistuned Bladed Disks, Journal of Engineering for Power, vol.98, issue.3, pp.349-355, 1976.
DOI : 10.1115/1.3446180

G. Feiner, ]. D. Feiner, and J. H. Griffin, A Fundamental Model of Mistuning for a Single Family of Modes, Journal of Turbomachinery, vol.124, issue.4, pp.597-605, 2002.
DOI : 10.1115/1.1508384

G. Feiner, ]. D. Feiner, and J. H. Griffin, Mistuning Identification of Bladed Disks Using a Fundamental Mistuning Model???Part I: Theory, Journal of Turbomachinery, vol.126, issue.1, pp.150-158, 2004.
DOI : 10.1115/1.1643913

D. M. Feiner and J. H. Griffin, Mistuning Identification of Bladed Disks Using a Fundamental Mistuning Model???Part II: Application, Journal of Turbomachinery, vol.126, issue.1, pp.159-165, 2004.
DOI : 10.1115/1.1643914

M. Géradin and D. Rixen, Mechanical vibrations. theory and application to structural dynamics, 1993.

. Griffin, ]. J. Hoosac-1984, T. M. Griffin, and . Hoosac, Model Development and Statistical Investigation of Turbine Blade Mistuning, Journal of Vibration Acoustics Stress and Reliability in Design, vol.106, issue.2, pp.204-210, 1984.
DOI : 10.1115/1.3269170

M. Han, M. P. Han, and . Mignolet, A Novel Perturbation-Based Approach for the Accurate Prediction of the Forced Response of Mistuned Bladed Disks, Volume 5: Turbo Expo 2007, 2007.
DOI : 10.1115/GT2007-27352

M. Han, M. P. Han, and . Mignolet, Optimization of Intentional Mistuning Patterns for the Mitigation of the Effects of Random Mistuning, Proceedings of ASME Turbo Expo, 2008.

. Han, Expedient Estimation of the Maximum Amplification Factor in Damped Mistuned Bladed Disks, Volume 5: Turbo Expo 2007, 2007.
DOI : 10.1115/GT2007-27353

]. R. Henry and G. Ferraris, Substructuring and Wave Propagation: An Efficient Technique for Impeller Dynamic Analysis, Journal of Engineering for Gas Turbines and Power, vol.106, issue.1, pp.2-10, 1984.
DOI : 10.1115/1.3239547

K. Huang, ]. B. Huang, and J. H. Kuang, Mode localization in a rotating mistuned turbo disk with Coriolis effect, International Journal of Mechanical Sciences, vol.43, issue.7, pp.1643-1660, 2001.
DOI : 10.1016/S0020-7403(00)00096-5

. Irwanto, ON SUBSTRUCTURING TECHNIQUE: APPLICATION OF LAGRANGE MULTIPLIER METHOD ON CYCLIC STRUCTURES, Journal of Sound and Vibration, vol.247, issue.5, pp.939-945, 2001.
DOI : 10.1006/jsvi.2001.3683

J. Judge, C. Pierre, and O. Mehmed, Experimental Investigation of Mode Localization and Forced Response Amplitude Magnification for a Mistuned Bladed Disk, Journal of Engineering for Gas Turbines and Power, vol.123, issue.4, pp.940-950, 2001.
DOI : 10.1115/1.1377872

J. A. Kenyon, J. H. Griffin, and N. E. Kim, Sensitivity of Tuned Bladed Disk Response to Frequency Veering, Journal of Engineering for Gas Turbines and Power, vol.127, issue.4, pp.835-842, 2005.
DOI : 10.1115/1.1924486

]. P. Ladevèze, M??canique non lin??aire des structures, Revue Fran??aise de G??nie Civil, vol.1, issue.2, 1996.
DOI : 10.1080/12795119.1997.9692126

]. Laîné, Dynamique des rotors, 2005.

M. Touratier, Aeroelastic vibrations and stability in cyclic symmetric domains, International Journal of Rotating Machinery, vol.6, issue.6, pp.445-452, 2000.

]. B. Lalanne, Perturbations Methods in Structural Dynamics and Applications to Cyclic Symmetric Domains, Journal of Engineering for Gas Turbines and Power, vol.127, issue.3, pp.654-662, 2005.
DOI : 10.1115/1.1924430

. Laxalde, Analyse Dynamique des Structures Cycliques Multi-Étages, Actes du XVè Colloque Vibrations, Chocs et Bruit, 2006.

. Laxalde, Dynamics of Multistage Bladed Disks Systems, Journal of Engineering for Gas Turbines and Power, vol.129, issue.4, pp.1058-1064, 2007.
DOI : 10.1115/1.2747641

URL : https://hal.archives-ouvertes.fr/hal-00243153

. Laxalde, Vibration Control for Integrally Bladed Disks Using Friction Ring Dampers, Volume 5: Turbo Expo 2007, 2007.
DOI : 10.1115/GT2007-27087

. Laxalde, Qualitative analysis of forced response of blisks with friction ring dampers, European Journal of Mechanics - A/Solids, vol.26, issue.4, pp.676-687, 2007.
DOI : 10.1016/j.euromechsol.2006.10.002

URL : https://hal.archives-ouvertes.fr/hal-00218249

. Laxalde, Experimental and Numerical Investigations of Friction Rig Damping of Blisks, Proceedings of ASME Turbo Expo, 2008.

M. Legrand, C. Pierre, and P. Cartraud, Full 3D strategies for rotor-stator contact interaction in turbomachinery, Proceedings of the 12th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01008239

. Lim, Vibration Modeling of Bladed Disks Subject to Geometric Mistuning and Design Changes, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, 2004.
DOI : 10.2514/6.2004-1686

S. Lim, C. Pierre, and M. P. Castanier, Predicting Blade Stress Levels Directly From Reduced-Order Vibration Models of Mistuned Bladed Disks, Journal of Turbomachinery, vol.128, issue.1, pp.206-210, 2006.
DOI : 10.1115/1.2098754

M. P. Lin and . Mignolet, EFFECTS OF DAMPING AND DAMPING MISTUNING ON THE FORCED VIBRATION RESPONSE OF BLADED DISKS, Journal of Sound and Vibration, vol.193, issue.2, pp.525-543, 1996.
DOI : 10.1006/jsvi.1996.0300

]. P. Marugabandhu and J. H. Griffin, A Reduced-Order Model for Evaluating the Effect of Rotational Speed on the Natural Frequencies and Mode Shapes of Blades, ASME Journal of Engineering for Gas Turbines and Power, vol.123, pp.772-776, 2003.

]. M. Mignolet and C. Lin, Identification of Structural Parameters in Mistuned Bladed Disks, Journal of Vibration and Acoustics, vol.119, issue.3, pp.428-438, 1997.
DOI : 10.1115/1.2889742

. Mignolet, On the Forced Response of Harmonically and Partially Mistuned Bladed Disks. Part I: Harmonic Mistuning, International Journal of Rotating Machinery, vol.6, issue.1, pp.29-41, 2000.
DOI : 10.1155/S1023621X0000004X

. Mignolet, On the Forced Response of Harmonically and Partially Mistuned Bladed Disks. Part II: Partial Mistuning and Applications, International Journal of Rotating Machinery, vol.6, issue.1, pp.43-56, 2000.
DOI : 10.1155/S1023621X00000051

. Mignolet, Identification of Mistuning Characteristics of Bladed Disks From Free Response Data???Part I, Journal of Engineering for Gas Turbines and Power, vol.123, issue.2, pp.395-403, 2001.
DOI : 10.1115/1.1338949

. Moyroud, A Comparison of Two Finite Element Reduction Techniques for Mistuned Bladed Disks, Journal of Engineering for Gas Turbines and Power, vol.124, issue.4, pp.942-952, 2002.
DOI : 10.1115/1.1415741

URL : https://hal.archives-ouvertes.fr/hal-00508201

. Nikolic, Robust Strategies for Forced Response Reduction of Bladed Discs Based on Large Mistuning Concept, Volume 5: Turbo Expo 2007, 2007.
DOI : 10.1115/GT2007-27183

]. R. Ohayon and C. Soize, Structural Acoustics and Vibration, The Journal of the Acoustical Society of America, vol.109, issue.6, 1998.
DOI : 10.1121/1.1352086

URL : https://hal.archives-ouvertes.fr/hal-00689039

M. Peeters, F. Georgiades, R. Viguié, G. Sérandour, G. Kerschen et al., Development of Numerical Algorithms for Practical Computation of Nonlinear Normal Modes, Proceedings of the International Conference on Advanced Acoustics and Vibration Engineering (ISMA), pp.2373-2386, 2008.

]. E. Petrov and D. J. Ewins, Method for Analysis of Nonlinear Multiharmonic Vibrations of Mistuned Bladed Disks With Scatter of Contact Interface Characteristics, Journal of Turbomachinery, vol.127, issue.1, pp.128-136, 2005.
DOI : 10.1115/1.1812781

]. E. Petrov and D. J. Ewins, Effects of Damping and Varying Contact Area at Blade-Disk Joints in Forced Response Analysis of Bladed Disk Assemblies, Journal of Turbomachinery, vol.128, issue.2, pp.403-410, 2006.
DOI : 10.1115/1.2181998

. Petrov, A New Method for Dynamic Analysis of Mistuned Bladed Disks Based on the Exact Relationship Between Tuned and Mistuned Systems, Journal of Engineering for Gas Turbines and Power, vol.124, issue.3, pp.586-597, 2002.
DOI : 10.1115/1.1451753

]. E. Petrov, Explicit Finite Element Models of Friction Dampers in Forced Response Analysis of Bladed Discs, Volume 5: Turbo Expo 2007, 2007.
DOI : 10.1115/GT2007-27980

]. E. Petrov, Method for Sensivity Analysis of Resonance Forced Response of Bladed Discs with Nonlinear Contact Interfaces, Proceedings of ASME Turbo Expo, 2008.

D. Pierre and . Murthy, Aeroelastic modal characteristics of mistuned blade assemblies - Mode localization and loss of eigenstructure, AIAA Journal, vol.30, issue.10, pp.2483-2496, 2002.
DOI : 10.2514/3.11251

A. J. Rivas-guerra and M. P. Mignolet, Local/Global Effects of Mistuning on the Forced Response of Bladed Disks, Journal of Engineering for Gas Turbines and Power, vol.126, issue.1, pp.131-141, 2004.
DOI : 10.1115/1.1581898

. Rivas-guerra, Identification of Mistuning Characteristics of Bladed Disks From Free Response Data??? Part II, Journal of Engineering for Gas Turbines and Power, vol.123, issue.2, pp.404-411, 2001.
DOI : 10.1115/1.1338950

. Rixen, ]. D. Lohman, F. Rixen, and . Lohman, Efficient Computation of Eigenmodes of Quasi-cyclic Structures. International Modal Analysis Conference, 2005.

. Rolls-rolls-plc, The Jet Engine, 2005.

]. S. Rubin, Improved Component-Mode Representation for Structural Dynamic Analysis, AIAA Journal, vol.13, issue.8, pp.995-1006, 1975.
DOI : 10.2514/3.60497

L. Samtech, Rotor module for the analysis of rotating systems, Theoretical Manual, 2001.

. Sever, Experimental and Numerical Investigation of Rotating Bladed Disk Forced Response Using Under-Platform Friction Dampers, Volume 5: Turbo Expo 2007, 2007.
DOI : 10.1115/GT2007-27307

]. A. Sinha, Reduced-Order Model of a Mistuned Multi-Stage Bladed Rotor, Proceedings of ASME Turbo Expo, 2007.

. Slater, Forced response of bladed disk assemblies - A survey, 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, pp.17-24, 1999.
DOI : 10.2514/6.1998-3743

. Song, System Identification of Multistage Turbine Engine Rotors, Volume 5: Turbo Expo 2007, 2007.
DOI : 10.1115/GT2007-28307

]. A. Srinivasan, Flutter and Resonant Vibration Characteristics of Engine Blades, Journal of Engineering for Gas Turbines and Power, vol.119, issue.4, pp.742-775, 1997.
DOI : 10.1115/1.2817053

]. A. Sternchüss and E. Balmès, On the reduction of quasi-cyclic disks with variable rotation speeds, Proceedings of the International Conference on Advanced Acoustics and Vibration Engineering (ISMA), pp.3925-3939, 2006.

. Sternchüss, ]. A. Balmès-2007a, E. Sternchüss, and . Balmès, Réduction de modèles de rotors de turbomachines, Colloque National en Calcul des Structures, pp.641-646, 2007.

. Sternchüss, ]. A. Balmès-2007b, E. Sternchüss, and . Balmès, Reduction of Multistage Rotor Models Through Cyclic Modeshapes, Proceedings of ASME Turbo Expo, 2007.

A. Sternchüss, E. Balmès, P. Jean, and J. Lombard, Model reduction applied to multi-stage assemblies of bladed disks, Proceedings of the International Conference on Advanced Acoustics and Vibration Engineering (ISMA), pp.3611-3625, 2008.

E. Sternchüss, P. Balmès, J. Jean, and . Lombard, Reduction of Multi-Stage Disk Models: Application to an Industrial Rotor, Volume 5: Structures and Dynamics, Parts A and B, pp.98-111, 2009.
DOI : 10.1115/GT2008-50580

]. D. Thomas, Standing waves in rotationally periodic structures, Journal of Sound and Vibration, vol.37, issue.2, pp.288-290, 1974.
DOI : 10.1016/S0022-460X(74)80337-8

]. D. Thomas, Dynamics of rotationally periodic structures, International Journal for Numerical Methods in Engineering, vol.57, issue.1, pp.81-102, 1979.
DOI : 10.1002/nme.1620140107

]. M. Touratier, Floquet waves in a body with slender periodic structure, Wave Motion, vol.8, issue.5, pp.485-495, 1986.
DOI : 10.1016/0165-2125(86)90032-6

]. D. Tran, Component mode synthesis methods using interface modes. Application to structures with cyclic symmetry, Computers & Structures, vol.79, issue.2, pp.209-222, 2001.
DOI : 10.1016/S0045-7949(00)00121-8

]. Tsai, Rotating vibration behavior of the turbine blades with different groups of blades, Journal of Sound and Vibration, vol.271, issue.3-5, pp.547-575, 2004.
DOI : 10.1016/S0022-460X(03)00280-3

. Vidoli, ]. S. Vestroni, F. Vidoli, and . Vestroni, Veering Phenomena in Systems With Gyroscopic Coupling, Journal of Applied Mechanics, vol.72, issue.5, pp.641-647, 2005.
DOI : 10.1115/1.1940666

]. D. Whitehead, Effect of mistuning on the vibration of turbo???machine blades induced by wakes, ARCHIVE: Journal of Mechanical Engineering Science 1959-1982 (vols 1-23), vol.8, issue.1, pp.15-21, 1966.
DOI : 10.1243/JMES_JOUR_1966_008_004_02

]. D. Whitehead, The Maximum Factor by Which Forced Vibration of Blades Can Increase Due to Mistuning, Journal of Engineering for Gas Turbines and Power, vol.120, issue.1, pp.115-119, 1998.
DOI : 10.1115/1.2818061

A. Xie, S. T. Xie, and . Ariaratnam, VIBRATION MODE LOCALIZATION IN DISORDERED CYCLIC STRUCTURES, I: SINGLE SUBSTRUCTURE MODE, Journal of Sound and Vibration, vol.189, issue.5, pp.625-645, 1996.
DOI : 10.1006/jsvi.1996.0040

A. Xie, S. T. Xie, and . Ariaratnam, VIBRATION MODE LOCALIZATION IN DISORDERED CYCLIC STRUCTURES, II: MULTIPLE SUBSTRUCTURE MODES, Journal of Sound and Vibration, vol.189, issue.5, pp.647-660, 1996.
DOI : 10.1006/jsvi.1996.0041

]. Yang and J. H. Griffin, A Reduced Order Approach for the Vibration of Mistuned Bladed Disk Assemblies, Journal of Engineering for Gas Turbines and Power, vol.119, issue.1, pp.161-167, 1997.
DOI : 10.1115/1.2815542

M. Yang and J. H. Griffin, A Reduced-Order Model of Mistuning Using a Subset of Nominal System Modes, Journal of Engineering for Gas Turbines and Power, vol.123, issue.4, pp.893-900, 2001.
DOI : 10.1115/1.1385197

. Zaretsky, Weibull-Based Design Methodology for Rotating Structures in Aircraft Engines, International Journal of Rotating Machinery, vol.9, issue.5, pp.313-325, 2003.
DOI : 10.1155/S1023621X03000290

]. Eigenfrequencies-vshebd, Fourier harmonic coefficients and relative error in frequency, p.68

]. Frequencies-vs3shp, Fourier harmonic coefficients for the rotor and the individual disks, p.82

]. Frequencies-vsfshp, Fourier harmonic coefficients, p.90