The search for ever lighter weight has become a major goal in the aeronautical industry as it has a direct impact on fuel consumption. It also implies the design of increasingly thin structures made of sophisticated and flexible materials. This may result in nonlinear behaviors due to large structural displacements. Stator vanes can be affected by such phenomena, and as they are a critical part of turbojets, it is crucial to predict these behaviors during the design process in order to eliminate them. This paper presents a reduced order modeling process suited for the study of geometric nonlinearities. The method is derived from a classical component mode synthesis (CMS) with fixed interfaces, in which the reduced nonlinear terms are obtained through a stiffness evaluation procedure (STEP) procedure using an adapted basis composed of linear modes completed by modal derivatives (MD). The whole system is solved using a harmonic balance procedure and a classic iterative nonlinear solver. The application is implemented on a schematic stator vane model composed of nonlinear Euler–Bernoulli beams under von Kàrmàn assumptions.

References

References
1.
Thomas
,
D. L.
,
1979
, “
Dynamics of Rotationally Periodic Structures
,”
Int. J. Numer. Methods Eng.
,
14
(
1
), pp.
81
102
.
2.
Krack
,
M.
,
Salles
,
L.
, and
Thouverez
,
F.
,
2016
, “
Vibration Prediction of Bladed Disks Coupled by Friction Joints
,”
Arch. Comput. Methods Eng.
,
24
(
3
), pp.
589
636
.
3.
Samaranayake
,
S.
,
Bajaj
,
A. K.
, and
Nwokah
,
O. D. I.
,
1995
, “
Amplitude Modulated Dynamics and Bifurcations in the Resonant Response of a Structure With Cyclic Symmetry
,”
Acta Mech.
,
109
(
1–4
), pp.
101
125
.
4.
Vakakis
,
A.
,
1997
, “
Non-Linear Normal Modes (NNMs) and Their Applications in Vibration Theory: An Overview
,”
Mech. Syst. Signal Process.
,
11
(
1
), pp.
3
22
.
5.
Georgiades
,
F.
,
Peeters
,
M.
,
Kerschen
,
G.
,
Golinval
,
J.-C.
, and
Ruzzene
,
M.
,
2009
, “
Modal Analysis of a Nonlinear Periodic Structure With Cyclic Symmetry
,”
AIAA J.
,
47
(
4
), pp.
1014
1025
.
6.
Grolet
,
A.
, and
Thouverez
,
F.
,
2012
, “
Free and Forced Vibration Analysis of a Nonlinear System With Cyclic Symmetry: Application to a Simplified Model
,”
J. Sound Vib.
,
331
(
12
), p.
29112928
.
7.
Kerschen
,
G.
,
Peeters
,
M.
,
Golinval
,
J.
, and
Vakakis
,
A.
,
2009
, “
Nonlinear Normal Modes—Part I: A Useful Framework for the Structural Dynamicist
,”
Mech. Syst. Signal Process.
,
23
(
1
), pp.
170
194
.
8.
Craig
,
R. R.
, and
Bampton
,
M. C. C.
,
1968
, “
Coupling of Substructures for Dynamic Analyses
,”
AIAA J.
,
6
(
7
), pp.
1313
1319
.
9.
Krack
,
M.
,
Panning-von Scheidt
,
L.
, and
Wallaschek
,
J.
,
2013
, “
A Method for Nonlinear Modal Analysis and Synthesis: Application to Harmonically Forced and Self-Excited Mechanical Systems
,”
J. Sound Vib.
,
332
(
25
), pp.
6798
6814
.
10.
Joannin
,
C.
,
Chouvion
,
B.
,
Thouverez
,
F.
,
Ousty
,
J.-P.
, and
Mbaye
,
M.
,
2017
, “
A Nonlinear Component Mode Synthesis Method for the Computation of Steady-State Vibrations in Non-Conservative Systems
,”
Mech. Syst. Signal Process.
,
83
, pp.
75
92
.
11.
Apiwattanalunggarn
,
P.
,
Shaw
,
S. W.
, and
Pierre
,
C.
,
2005
, “
Component Mode Synthesis Using Nonlinear Normal Modes
,”
Nonlinear Dyn.
,
41
(
1–3
), Aug, pp.
17
46
.
12.
Idelsohn
,
S. R.
, and
Cardona
,
A.
,
1985
, “
A Reduction Method for Nonlinear Structural Dynamic Analysis
,”
Comput. Methods Appl. Mech. Eng.
,
49
(
3
), pp.
253
279
.
13.
Slaats
,
P.
,
de Jongh
,
J.
, and
Sauren
,
A.
,
1995
, “
Model Reduction Tools for Nonlinear Structural Dynamics
,”
Comput. Struct.
,
54
(
6
), pp.
1155
1171
.
14.
Muravyov
,
A. A.
, and
Rizzi
,
S. A.
,
2003
, “
Determination of Nonlinear Stiffness With Application to Random Vibration of Geometrically Nonlinear Structures
,”
Comput. Struct.
,
81
(
15
), pp.
1513
1523
.
15.
Nguyen
,
T.
,
2007
, “
Non Linear Dynamic of Coupled Mechanical Systems: Model Reduction and Identification
,” Ph.D. thesis, École Nationale des Ponts et Chaussées hamps-sur-Marne, France.
16.
Lazarus
,
A.
,
Thomas
,
O.
, and
Deu
,
J.-F.
,
2012
, “
Finite Element Reduced Order Models for Nonlinear Vibrations of Piezoelectric Layered Beams With Applications to NEMS
,”
Finite Elem. Anal. Des.
,
49
(
1
), pp.
35
51
.
17.
Zienkiewicz
,
O.
,
Taylor
,
R.
, and
Zhu
,
J.
,
2005
, “
5—Geometrically Non-Linear Problems Finite Deformation
,”
The Finite Element Method Set
,
O.
Zienkiewicz
,
R.
Taylor
, and
J.
Zhu
, eds.,
6th ed.
,
Butterworth-Heinemann
,
Oxford, UK
, pp.
127
157
.
18.
Cameron
,
T. M.
, and
Griffin
,
J.
,
1989
, “
An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems
,”
ASME J. Appl. Mech.
,
56
(1), pp. 149–154.
19.
Thomas
,
O.
,
Sénéchal
,
A.
, and
Deü
,
J.-F.
,
2016
, “
Hardening/Softening Behavior and Reduced Order Modeling of Nonlinear Vibrations of Rotating Cantilever Beams
,”
Nonlinear Dyn.
,
86
(
2
), pp.
1293
1318
.
20.
Grolet
,
A.
, and
Thouverez
,
F.
,
2013
, “
Model Reduction for Nonlinear Vibration Analysis of Structural Systems Using Modal Derivatives and Stiffness Evaluation Procedure
,”
The Fourth Canadian Conference on Nonlinear Solid Mechanics (CanCNSM 2013)
, Montreal, QC, Canada, July 23–26, pp.
1
6
.
You do not currently have access to this content.