This paper deals with the optimization of vibrating structures as a mean for minimizing unwanted vibration. Presented in this work is a method for automatic determination of a set of preselected design parameters affecting the geometrical layout or shape of the structure. The parameters are selected to minimize the dynamic response to external forcing or base motion. The presented method adjusts the structural parameters by solving an optimization problem in which the constraints are dictated by engineering considerations. Several constraints are defined so that the static deflection, the stress levels and the total weight of the structure are kept within bounds. The dynamic loading acting upon the structure is described in this work by its power spectral density, with this representation the structure can be tailored to specific operating conditions. The uncertain nature of the excitation is overcome by combining all possible spectra into one PSD encompassing all possible loading patterns. An important feature of the presented method is its numerical efficiency. This feature is essential for any reasonably sized problem as such problems are usually described by thousands of degrees of freedom arising from a finite-element idealization of the structure. In this paper, efficient, closed form expressions, for the cost function and its gradients are derived. Those are computed with a partial set of eigenvectors and eigenvalues thus increasing the efficiency further. Several numerical examples are presented where both shape optimization and the selection of discrete components are illustrated.

1.
Kirsh, U., 1993, Structural Optimization: Fundamentals and Applications, Springer Verlag, Berlin.
2.
Haftka, R. T., 1992, Elements of Structural Optimization, Kluver, Dortmund.
3.
Turner
,
M. J.
,
1967
, “
Design of Minimum-Mass Structures With Specified Natural Frequencies
,”
AIAA J.
,
5
, pp.
406
412
.
4.
Taylor
,
J. E.
,
1967
, “
Minimum-Mass Bar for Axial Vibration at Specified Natural Frequency
,”
AIAA J.
,
5
, pp.
1911
1913
.
5.
Kamat
,
M. P.
,
1973
, “
Optimal Beam Frequencies by the FE Displacement Method
,”
Int. J. Solids Struct.
,
9
, pp.
415
429
.
6.
Yamazaki
,
K.
,
Sakamoto
,
J.
, and
Kitano
,
M.
,
1993
, “
Efficient Shape Optimization Technique of a Two-Dimensional Body Based on the Boundary Element Method
,”
Comput. Struct.
,
48
, No. 6, Sep. 17, pp.
1073
1081
.
7.
Berebbia, C. A., 1989, Computer Aided Optimum Design of Structures: Applications, Springer Verlag, Berlin.
8.
Ram, Y. M., and Elhay, S., 1996, “Theory of a Multi-Degree-of-Freedom Dynamic Absorber,” J. Sound Vib., 195, No. 4, Aug 29.
9.
Asami
,
T.
, and
Hosokawa
,
Y.
,
1995
, “
Approximate Expression for Design of Optimal Dynamic Absorbers Attached to Damped Linear Systems
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
,
61
, No. 583, Mar., pp.
915
921
.
10.
Uwes, S., and Pilkey, W. D., 1996, “Optimal Design of Structures Under Impact Loading,” Shock and Vibration, 3, No. 11.
11.
Skelton, R. E., 1989, Dynamic System Control—Linear System Analysis and Synthesis, John Wiley and Sons, New York.
12.
Fletcher, R., 1980, “Practical Methods of Optimization,” Vol. 1, Unconstrained Optimization, and Vol. 2, Constrained Optimization, John Wiley and Sons, New York.
13.
Bucher
,
I.
, and
Braun
,
S. G.
,
1993
, “
Efficient Optimization Procedure for Minimizing the Vibratory Response via Redesign or Modification, Part I—Theory
,”
J. Sound Vib.
,
175
, pp.
433
454
.
14.
Parlett, B. N., 1998, The Symmetric Eigenvalue Problem, SIAM, Philadelphia.
You do not currently have access to this content.