Research Papers: Design Automation

Optimal Design of Vibrating Systems Through Partial Eigenstructure Assignment

[+] Author and Article Information
Roberto Belotti

Dipartimento di Tecnica e Gestione
dei Sistemi Industriali,
Università degli Studi di Padova,
Stradella S. Nicola 3,
Vicenza 36100, Italy
e-mail: belotti@gest.unipd.it

Dario Richiedei

Dipartimento di Tecnica e Gestione
dei Sistemi Industriali,
Università degli Studi di Padova,
Stradella S. Nicola 3,
Vicenza 36100, Italy
e-mail: dario.richiedei@unipd.it

Alberto Trevisani

Dipartimento di Tecnica e Gestione
dei Sistemi Industriali,
Università degli Studi di Padova,
Stradella S. Nicola 3,
Vicenza 36100, Italy
e-mail: alberto.trevisani@unipd.it

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received October 16, 2015; final manuscript received April 18, 2016; published online May 16, 2016. Assoc. Editor: Kazuhiro Saitou.

J. Mech. Des 138(7), 071402 (May 16, 2016) (8 pages) Paper No: MD-15-1710; doi: 10.1115/1.4033505 History: Received October 16, 2015; Revised April 18, 2016

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.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


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. [CrossRef]
Bucher, I. , and Braun, S. , 1993, “ The Structural Modification Inverse Problem: An Exact Solution,” Mech. Syst. Signal Process., 7(3), pp. 217–238. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Mottershead, J. , 2001, “ Structural Modification for the Assignment of Zeros Using Measured Receptances,” ASME J. Appl. Mech., 68(5), pp. 791–798. [CrossRef]
Andry, A. , Shapiro, E. , and Chung, J. , 1983, “ Eigenstructure Assignment for Linear Systems,” IEEE Trans. Aerosp. Electron. Syst., 19(5), pp. 711–729. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Hernandes, J. , and Suleman, A. , 2014, “ Structural Synthesis for Prescribed Target Natural Frequencies and Mode Shapes,” Shock Vib., 2014, p. 173786.
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. [CrossRef]
Allgower, E. , and Georg, K. , 2003, Introduction to Numerical Continuation Methods, Society for Industrial and Applied Mathematics, Philadelphia, PA.
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.
Dunlavy, D. M. , and O'Leary, D. P. , 2005, “ Homotopy Optimization Methods for Global Optimization,” Sandia National Laboratories, Report No. SAND2005-7495.
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. [CrossRef]
Al-Khayyal, F. A. , and Falk, J. E. , 1983, “ Jointly Constrained Biconvex Programming,” Math. Oper. Res., 8(2), pp. 273–286. [CrossRef]
McCormick, G. P. , 1976, “ Computability of Global Solutions to Factorable Nonconvex Programs—Part I: Convex Underestimating Problems,” Math Program., 10(1), pp. 147–175. [CrossRef]
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.
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. [CrossRef]
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.


Grahic Jump Location
Fig. 5

Mode shape at 40.135 Hz of the unmodified vibratory feeder

Grahic Jump Location
Fig. 6

Convergence of the method in the second test-case

Grahic Jump Location
Fig. 1

Model of the 5dof test-case

Grahic Jump Location
Fig. 4

Mode shape at 40 Hz of the modified vibratory feeder

Grahic Jump Location
Fig. 3

A 39dof model of the vibratory feeder



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In