This paper proposes an inverse structural modification method for eigenstructure assignment (EA), which allows to assign the desired mode shapes only at the parts of interest of the system. The presence of unimposed eigenvector entries leads to a nonconvex problem. Therefore, to boost the convergence to a global optimal solution, a homotopy optimization strategy is implemented based on the convex approximation of the cost function. Such a relaxation is performed through some auxiliary variables and through the McCormick's relaxation of the occurring bilinear terms. The approach handles general assignment tasks, with an arbitrary number of modification parameters and prescribed eigenpairs.

References

References
1.
Ram
,
Y.
, and
Braun
,
S.
,
1991
, “
An Inverse Problem Associated With Modification of Incomplete Dynamic Systems
,”
ASME J. Appl. Mech.
,
58
(
1
), pp.
233
237
.
2.
Bucher
,
I.
, and
Braun
,
S.
,
1993
, “
The Structural Modification Inverse Problem: An Exact Solution
,”
Mech. Syst. Signal Process.
,
7
(
3
), pp.
217
238
.
3.
Sivan
,
D.
, and
Ram
,
Y.
,
1997
, “
Optimal Construction of a Mass-Spring System From Prescribed Modal and Spectral Data
,”
J. Sound Vib.
,
201
(
3
), pp.
323
334
.
4.
Kyprianou
,
A.
,
Mottershead
,
J. E.
, and
Ouyang
,
H.
,
2004
, “
Assignment of Natural Frequencies by an Added Mass and One or More Springs
,”
Mech. Syst. Signal Process.
,
18
(
2
), pp.
263
289
.
5.
Mottershead
,
J.
,
2001
, “
Structural Modification for the Assignment of Zeros Using Measured Receptances
,”
ASME J. Appl. Mech.
,
68
(
5
), pp.
791
798
.
6.
Andry
,
A.
,
Shapiro
,
E.
, and
Chung
,
J.
,
1983
, “
Eigenstructure Assignment for Linear Systems
,”
IEEE Trans. Aerosp. Electron. Syst.
,
19
(
5
), pp.
711
729
.
7.
Richiedei
,
D.
,
Trevisani
,
A.
, and
Zanardo
,
G.
,
2011
, “
A Constrained Convex Approach to Modal Design Optimization of Vibrating Systems
,”
ASME J. Mech. Des.
,
133
(
6
), p.
061011
.
8.
Ouyang
,
H.
,
Richiedei
,
D.
,
Trevisani
,
A.
, and
Zanardo
,
G.
,
2012
, “
Eigenstructure Assignment in Undamped Vibrating Systems: A Convex-Constrained Modification Method Based on Receptances
,”
Mech. Syst. Signal Process.
,
27
, pp.
397
409
.
9.
Ouyang
,
H.
,
Richiedei
,
D.
,
Trevisani
,
A.
, and
Zanardo
,
G.
,
2012
, “
Discrete Mass and Stiffness Modifications for the Inverse Eigenstructure Assignment in Vibrating Systems: Theory and Experimental Validation
,”
Int. J. Mech. Sci.
,
64
(
1
), pp.
211
220
.
10.
Hernandes
,
J.
, and
Suleman
,
A.
,
2014
, “
Structural Synthesis for Prescribed Target Natural Frequencies and Mode Shapes
,”
Shock Vib.
,
2014
, p.
173786
.
11.
Liu
,
Z.
,
Li
,
W.
,
Ouyang
,
H.
, and
Wang
,
D.
,
2015
, “
Eigenstructure Assignment in Vibrating Systems Based on Receptances
,”
Arch. Appl. Mech.
,
85
(
6
), pp.
713
724
.
12.
Allgower
,
E.
, and
Georg
,
K.
,
2003
,
Introduction to Numerical Continuation Methods
,
Society for Industrial and Applied Mathematics
, Philadelphia, PA.
13.
Forster
,
W.
,
1995
, “
Homotopy Methods
,”
Handbook of Global Optimization
(Nonconvex Optimization and Its Applications), Vol.
2
,
R.
Horst
, and
P.
Pardalos
, eds.,
Kluwer Academic Publishers
,
Dordrecht, The Netherlands
, pp.
669
750
.
14.
Dunlavy
,
D. M.
, and
O'Leary
,
D. P.
,
2005
, “
Homotopy Optimization Methods for Global Optimization
,” Sandia National Laboratories,
Report No. SAND2005-7495
.
15.
Vyasarayani
,
C. P.
,
Uchida
,
T.
,
Carvalho
,
A.
, and
McPhee
,
J.
,
2011
, “
Parameter Identification in Dynamic Systems Using the Homotopy Optimization Approach
,”
Multibody Syst. Dyn.
,
26
(
4
), pp.
411
424
.
16.
Al-Khayyal
,
F. A.
, and
Falk
,
J. E.
,
1983
, “
Jointly Constrained Biconvex Programming
,”
Math. Oper. Res.
,
8
(
2
), pp.
273
286
.
17.
McCormick
,
G. P.
,
1976
, “
Computability of Global Solutions to Factorable Nonconvex Programs—Part I: Convex Underestimating Problems
,”
Math Program.
,
10
(
1
), pp.
147
175
.
18.
Andersen
,
E. D.
, and
Andersen
,
K. D.
,
2000
, “
The MOSEK Interior Point Optimizer for Linear Programming: An Implementation of the Homogeneous Algorithm
,”
High Performance Optimization
, Vol.
33
,
Springer
, Dorcrecht, The Netherlands.
19.
Byrd
,
R. H.
,
Hribar
,
M. E.
, and
Nocedal
,
J.
,
1999
, “
An Interior Point Algorithm for Large-Scale Nonlinear Programming
,”
SIAM J. Optim.
,
9
(
4
), pp.
877
900
.
20.
Löfberg
,
J.
,
2004
, “
YALMIP: A Toolbox for Modeling and Optimization in MATLAB
,”
IEEE
International Symposium on Computer Aided Control Systems Design
, Taipei, Taiwan, Sept. 4, pp.
284
289
.
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