In multibody dynamics, the flexibility effects of each body are captured by using a linear combination of elastic mode shapes. If a co-rotational and co-translating frame of reference is used together with eigenvectors of the unconstraint body, which are free-surface modes, some spatial integrals in the floating frame of reference configuration do vanish. The corresponding coordinate system is the so-called Tisserand (or Buckens) reference frame. In the present contribution, a technique is developed for separating an arbitrary elastic mode shape into a pseudo-free-surface mode and rigid body modes. The generated pseudo-free-surface mode has most of the advantageous characteristics of a free-surface mode, and spans together with the rigid body modes the same solution space as it is spanned by the original mode shape. Due to the fact that, in the floating frame of reference configuration, the rigid body motions are already described by special generalized coordinates, only the resulting pseudo-free-surface modes are finally used to capture the flexibility effects of each body. A result of the generated pseudo-free-surface modes is that some of the spatial integrals do vanish and, thus, the equations of motion are significantly simplified. Two examples are presented in order to illustrate and to demonstrate the potential of the proposed method.

References

1.
Shabana
,
A. A.
, 2005,
Dynamics of Multibody Systems
, (3rd edn).,
Cambridge University Press
,
New York
.
2.
Escalona
,
J. L.
,
Valverde
,
J.
,
Mayo
,
J.
, and
Domínguez
,
J.
, 2003, “
Reference Motion in Deformable Bodies under Rigid Body Motion and Vibration. Part I: Theory
,”
J. Sound Vib.
,
264
(
5
), pp.
1045
1056
.
3.
Irschik
,
H.
,
Nader
,
M.
,
Stangl
,
M.
, and
von Garssen
,
H.-G.
, 2009, “
A Floating Frame-of-Reference Formulation for Deformable Rotors using the Properties of Free Elastic Vibration Modes
,”
Proceedings of ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
ASME
, Paper No. DETC2009-86660.
4.
Nader
,
M.
,
Irschik
,
H.
,
Stangl
,
M.
, and
von Garssen
,
H.-G.
, 2010, “
Nonlinear Vibrations of Flexible High-speed Rotors Supported by Visco-elastic Bearings
,”
Proceedings of the 8th IFToMM International Conference on Rotordynamics
.
5.
Tisserand
,
F.
, 1891,
Traite de Mechanique Celeste,
Gauthier-Villars
,
Paris
.
6.
Veubeke
,
B. F.
, 1976, “
The Dynamics of Flexible Bodies
,”
Int. J. Eng. Sci.
14
(
10
), pp.
895
913
.
7.
Canavin
,
J. R.
, and
Likins
,
P. W.
, 1977, “
Floating Reference Frames for Flexible Spacecraft
,”
J. Spacecr. Rockets
,
14
(
12
), pp.
724
732
.
8.
Drab
,
C. B.
,
Haslinger
,
J. R.
,
Pfau
,
R. U.
, and
Offner
,
G.
, 2007, “
Comparison of the Classical Formulation with the Reference Conditions Formulation for Dynamic Flexible Multibody Systems
,”
J. Comput. Nonlinear Dyn.
,
2
(
4
), pp.
337
343
.
9.
Agrawal
,
O.-P.
, and
Shabana
,
A. A.
, 1985, “
Dynamic Analysis of Multibody Systems using Component Modes
,”
Comput. Struct.
,
21
(
6
), pp.
1303
1312
.
10.
Friberg
,
O.
, and
Karhu
,
V.
, 1990, “
Use of Mode Orthogonolization and Modal Damping in Flexible Multibody Dynamics
,”
Finite Elem. Anal. Design
,
7
(
1
), pp.
51
59
.
11.
Yeh
,
H.-F.
, and
Dopker
,
B.
, 1990, “
Deformation Mode Selection and Mode Orthonormalization for Flexible Multibody System Dynamics
,”
Comput. Struct.
,
34
(
4
), pp.
615
627
.
12.
Friberg
,
O.
, 1991, “
A Method for Selecting Deformation Modes in Flexible Multibody Dynamics
,”
Int. J. Numer. Meth. Eng.
,
32
(
8
), pp.
1637
1655
.
13.
Shabana
,
A. A.
, 1996, “
Resonance Conditions and Deformable Body Co-Ordinate Systems
,”
J. Sound. Vib.
,
192
(
1
), pp.
389
398
.
14.
Schwertassek
,
R.
,
Wallrapp
,
O.
, and
Shabana
,
A. A.
, 1999, “
Flexible Multibody Simulation and Choice of Shape Functions
,”
Nonlinear Dyn.
,
20
(
4
), pp.
361
380
.
15.
Nikravesh
,
P.-E.
, and
Lin
,
Y.-S.
, 2005, “
Use of Principal Axes as the Floating Reference Frame for a Moving Deformable Body
,”
J. Multibody Syst. Dyn.
,
13
(
2
), pp.
211
231
.
16.
Heckmann
,
A.
, 2010, “
On the Choice of Boundary Conditions for Mode Shapes in Flexible Multibody Systems
,”
J. Multibody Syst. Dyn.
23
, pp.
141
163
.
17.
Likins
,
P. W.
, 1976, “
Modal Method for Analysis of Free Rotations of Spacecraft
,”
AIAA J.
,
5
(
7
), pp.
1304
1308
.
18.
Shabana
,
A. A.
, 1985, “
Automated Analysis of Constrained Inertia-Variant Flexible Systems
,”
ASME J. Vib., Acoust., Stress, Reliab. Des.
,
107
(
4
), pp.
431
440
.
19.
Shabana
,
A. A.
, 1997, “
Flexible Multibody Dynamics: Review of Past and Recent Developments
,”
J. Multibody Syst. Dyn.
,
1
(
2
), pp.
189
222
.
20.
Schiehlen
,
W.
, ed., 1993,
Advanced Multibody System Dynamics - Simulation and Software Tools
,
Kluwer Academic, Dordrecht
,
The Netherlands
.
21.
Schiehlen
,
W.
, 1986,
Technische Dynamik.
,
Teubner, Stuttgart
,
Germany
.
22.
Gerstmayr
,
J.
, and
Ambr´osio
,
J. A. C.
, 2008, “
Component Mode Synthesis with Constant Mass and Stiffness Matrices Applied to Flexible Multibody Systems
,”
Int. J. Numer. Meth. Eng.
,
73
(
11
), pp.
1518
1546
.
23.
Gurtin
,
M.
, 1972,
The Linear Theory of Elasticity
.,
Springer-Verlag, Berlin
,
Germany
.
24.
Korn
,
A.
, 1910, Ҭ
Uber die Eigenschwingungen eines elastischen K¨orpers bei verschwindenden Druckkomponenten an der Oberfl¨ache
,”
Palermo Rend.
,
30
, pp.
153
184
.
25.
Hurty
,
W. C.
, 1965, “
Dynamic Analysis of Structural Systems using Component Modes
,”
AIAA J.
,
3
(
4
), pp.
678
685
.
26.
Craig
,
R. R.
, and
Bampton
,
M. C. C.
, 1968, “
Coupling of Substructures for Dynamic Analyses
,”
AIAA J.
,
6
(
7
), pp.
1313
1319
.
27.
Craig
,
R. R.
, 1987, “
A Review of Time-Domain and Frequency-Domain Component Mode Synthesis Methods
,”
Int. J. Anal. Exp. Modal Anal.
,
2
(
2
), pp.
59
72
.
28.
Craig
,
R. R.
, and
Kurdila
,
A. J.
, 2006,
Fundamentals of Structural Dynamics
,
John Wiley and Sons Inc.
,
New Jersey
.
29.
Tran
,
D.-M.
, 2001, “
Component Mode Synthesis Methods using Interface Modes. Application to Structures with Cyclic Symmetry
,”
Comput. Struct.
,
79
, pp.
209
222
.
30.
Castanier
,
M. P.
,
Tan
,
Y.-C.
, and
Pierre
,
C.
, 2001, “
Characteristic Constraint Modes for Component Mode Synthesis
,”
AIAA J.
,
39
(
6
), pp.
1182
1187
.
31.
Witteveen
,
W.
, and
Irschik
,
H.
, 2009, “
Efficient Mode-Based Computational Approach for Jointed Structures: Joint Interface Modes
,”
AIAA J.
,
47
(
1
), pp.
252
263
.
32.
Dieker
,
S.
,
Abdoly
,
K.
, and
Rittweger
,
A.
, 2010, “
Flexible Boundary Method in Dynamic Substructure Techniques Including Different Component Damping
,”
AIAA J.
,
48
(
11
), pp.
2631
2638
.
33.
Ma
,
F.
, and
Ng
,
C. H.
, 2004, “
On the Orthogonality of Natural Modes of Vibration
,”
Mech. Res. Commun.
,
31
(
3
), pp.
295
299
.
34.
Courant
,
R.
, and
Hilbert
,
D.
, 1993,
Methoden der mathematischen Physik (4.Auflage
).,
Springer- Verlag, Berlin
,
Germany
.
35.
Witteveen
,
W.
, and
Irschik
,
H.
, 2007, “
Efficient Modal Formulation for Vibration Analysis of Solid Structures with Bolted Joints
,”
In Proceedings of the 25th International Modal Analysis Conference (IMAC-XXV)
2
,
Society of Experimental Mechanics Inc.
36.
Witteveen
,
W.
, and
Sherif
,
K.
, 2011, “
POD Based Computation of Joint Interface Modes
,”
In Proceedings of the 29th International Modal Analysis Conference (IMAC-XXIX)
,
2
,
Society of Experimental Mechanics Inc.
37.
Negrut
,
D.
,
Ottarsson
,
G.
,
Rampalli
,
R.
, and
Sajdak
,
A.
, 2007, “
On an Implementation of the Hilber-Hughes-Taylor Method in the Context of Index 3 Differential-Algebraic Equations of Multibody
,”
J. Comput. Nonlinear Dyn.
,
2
(
1
), pp.
73
85
.
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