Modal synthesis methods have long been studied because the use of generalized coordinates makes it possible to reduce calculation costs. Our approach uses modes to describe each part of the assembly of several substructures. This method, called “Double Modal Synthesis,” is presented through primal and dual formulations. As modal truncation usually introduces a lack of precision, we will use an $ω2$ development if necessary. These formulations will first be explained using a continuous formulation. A finite element method will then be proposed. Another aim of the paper is to introduce formulations needed to understand the multimodal analysis methods that will be presented in a forthcoming paper.

1.
Hurty
,
W. C.
, 1965, “
Dynamic Analysis of Structural Systems Using Component Modes
,”
AIAA J.
0001-1452,
3
, pp.
678
685
.
2.
Goldman
,
R. L.
, 1968, “
Vibration Analysis by Dynamic Partitioning
,”
AIAA J.
0001-1452,
7
, pp.
1152
1154
.
3.
Hou
,
S. N.
, 1969, “
Review of Modal Synthesis Techniques and a New Approach
,”
Shock Vib. Dig.
0583-1024,
40
, pp.
25
30
.
4.
Jézéquel
,
L.
, and
Setio
,
H. D.
, 1994, “
Component Modal Synthesis Methods Based on Hybrid Models, Part I: Theory of Hybrid Models and Modal Truncation Methods
,”
ASME J. Appl. Mech.
0021-8936,
61
, pp.
100
108
.
5.
Jézéquel
,
L.
, and
Setio
,
H. D.
, 1994, “
Component Modal Synthesis Methods Based on Hybrid Models, Part II: Numerical Tests and Experimental Identification of Hybrid Model
,”
ASME J. Appl. Mech.
0021-8936,
61
, pp.
109
116
.
6.
Qiu
,
J.-B.
,
Ying
,
Z.-G.
, and
Williams
,
F. W.
, 1997, “
Exact Modal Synthesis Techniques Using Residual ConStraint Modes
,”
Int. J. Numer. Methods Eng.
0029-5981,
40
, pp.
2475
2492
.
7.
Shyu
,
W.-H.
,
Ma
,
Z.-D.
, and
Hulbert
,
G. M.
, 1997, “
A New Component Mode Synthesis Method: Quasi-Static Mode Compensation
,”
Finite Elem. Anal. Design
0168-874X,
24
, pp.
271
281
.
8.
Tran
,
D.-M.
, 2001, “
Component Mode Synthesis Methods Using Interface Modes: Application to Structures With Cyclic Symmetry
,”
Comput. Struct.
0045-7949,
79
, pp.
209
222
.
9.
Jézéquel
,
L.
, 1990, “
Procedure to Reduce the Effects of Modal Truncation in Eigensolution Reanalysis
,”
AIAA J.
0001-1452,
28
(
5
), pp.
896
902
.
10.
Jézéquel
,
L.
, 1985, “
Hybrid Method of Modal Synthesis Using Vibration Tests
,”
J. Sound Vib.
0022-460X,
100
(
2
), pp.
191
210
.
11.
Jézéquel
,
L.
, 1980, “
A Method of Damping Synthesis from Substructures Tests
,”
ASME J. Mech. Des.
1050-0472,
102
, p.
1980
.
12.
Craig
,
R. R.
, and
Bampton
,
M. C. C.
, 1968, “
Coupling of Substructures for Dynamic Analysis
,”
AIAA J.
0001-1452,
6
, pp.
1313
1321
.
13.
Weinstein
,
A.
, and
Stenger
,
W.
, 1972,
Intermediate Problems for Eigenvalues: Theory and Ramifications
,